Behavior of Compressed Plasmas in Magnetic Fields
Gurudas Ganguli, Chris Crabtree, Alex Fletcher, Bill Amatucci
aarxiv.org manuscript No. (will be inserted by the editor)
Behavior of Compressed Plasmas in Magnetic Fields
Gurudas Ganguli · Chris Crabtree ∗ · Alex Fletcher · Bill Amatucci
Received: date / Accepted: date
Abstract
Plasma in the earth’s magnetosphere is subjected to compressionduring geomagnetically active periods and relaxation in subsequent quiet times.Repeated compression and relaxation is the origin of much of the plasma dy-namics and intermittency in the near-earth environment. An observable man-ifestation of compression is the thinning of the plasma sheet resulting in mag-netic reconnection when the solar wind mass, energy, and momentum floodsinto the magnetosphere culminating in the spectacular auroral display. Thisphenomenon is rich in physics at all scale sizes, which are causally intercon-nected. This poses a formidable challenge in accurately modeling the physics.The large-scale processes are fluid-like and are reasonably well captured in theglobal magnetohydrodynamic (MHD) models, but those in the smaller scalesresponsible for dissipation and relaxation that feed back to the larger scaledynamics are often in the kinetic regime. The self-consistent generation of thesmall-scale processes and their feedback to the global plasma dynamics re-mains to be fully explored. Plasma compression can lead to the generation ofelectromagnetic fields that distort the particle orbits and introduce new fea-tures beyond the purview of the MHD framework, such as ambipolar electricfields, unequal plasma drifts and currents among species, strong spatial andvelocity gradients in gyroscale layers separating plasmas of different character-istics, etc.
These boundary layers are regions of intense activity characterizedby emissions that are measurable. We study the behavior of such compressedplasmas and discuss the relaxation mechanisms to understand their measur-able signatures as well as their feedback to influence the global scale plasmaevolution.
G. Ganguli · C. Crabtree ∗ · A. Fletcher · B. AmatucciPlasma Physics DivisionNaval Research LaboratoryWashington, DC 20375Tel.: +01-202-767-2401Fax: +01-202-767-3553 ∗ E-mail: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] A ug Gurudas Ganguli et al.
The holy grail of much of modern science is the comprehensive knowledge ofthe coupling between the micro, meso, and macro scale processes that char-acterize physical phenomena. This is particularly important in magnetizedplasmas which typically have a very large degree of freedom at all scale sizes.The statistically likely state involves a complex interdependence among allof the scales. In the unbounded space plasma undergoing global compressionduring geomagnetically active periods the multiplicity of spatio-temporal scalesizes is astoundingly large. The statistically likely state has mostly been ad-dressed by global magnetohydrodynamic (MHD) or fluid models, which ignorethe contributions from the small-scale processes that can be locally dominant.This was understandable in the past when the early space probes could hardlyresolve smaller scale features. Also, single point measurements from a movingplatform made in evolving plasma are not ideal for resolving the small-scaledetails of a fast time scale process. Statistical ensembles generated throughmeasurements from repeated satellite visits in a dynamic plasma washes outmany small-scale features that evolve rapidly. Therefore, the need for under-standing the contributions from the small-scale processes was not urgent.However, there are pitfalls in relying on global fluid models alone for anaccurate assessment of satellite measurements that essentially represent thelocal physics. These models ignore the kinetic physics, which often operate atfaster time scales at the local level and are necessary for dissipation, which isimportant for relaxation and feedback to form a steady state that satellitesmeasure. For example, the large-scale MHD models cannot account for theambipolar effects and hence they are inadequate for the physics at ion andelectron gyroscales, which are now being resolved by multi-point measurementsfrom modern space probe clusters, such as NASAs Magnetospheric Multi-ScaleSatellite (MMS) [1] , the Time History of Events and Macroscale Interactionsduring Substorms (THEMIS) mission [2], and the European Space AgencysCluster mission [3].Global scale kinetic simulations that can resolve gyroscales are still notpractical. These simulations suffer perennial issues such as insufficient massratios, insufficient particles per cell, or use implicit algorithms that ignore thesmall scale features. Thus, these simulations are incapable of accurately resolv-ing the gyroscales for capturing ambipolar effects, which as we show in Sec.2, can be critical to the comprehensive understanding of the physics neces-sary for interpreting satellite observations. With technological breakthroughsin the future, resolution of gyroscales in global models will become possible.It is, therefore, necessary to assess the origin of small-scale processes responsi-ble for relaxation and their feedback mechanisms for a deeper understandingand also to motivate future space missions with improved instrumentation tosearch for them in nature. The objective of this article is to highlight the fun-damental role of plasma compression in the inter-connectedness of physicalprocesses at local and global levels in general, and in particular in the earthsimmediate plasma environment through specific examples. ehavior of Compressed Plasmas in Magnetic Fields 3
Although the large-scale models are not yet suitable for addressing thesmaller scale physics, they are necessary for understanding the global mor-phology and global transport of mass, energy, and momentum that creates thecompressed plasma layers when plasmas of different characteristics interface.In the near term, before first principles kinetic global models become prac-tical, the large-scale fluid models should be extended to include small scale(sub-grid) kinetic physics that is discussed in this article so that the effects ofnatural saturation and dissipation of compression can be accounted for on alarger scale. Clearly, therefore, the knowledge of large and small scale processesare like the proverbial two sides of a coin, both equally necessary for a com-prehensive understanding of the salient physics. Since the role of smaller scaleprocesses was not central to most previous studies we focus our analysis hereto their self-consistent origin and their contributions to the overall plasma dy-namics. Arguably, small-scale structures will be increasingly resolved by futuretechnologically-advanced space probes, so there is now a need to accuratelyunderstand their cause and effect.
To understand the physics of compressed layers it is best to consider specificexamples of such layers that arise naturally. Weak compressions, which arecharacterized by scale sizes much larger than an ion gyrodiameter and af-fect both the ions and electrons similarly, are not of interest here. Large-scalemodels can address them. The focus of this article is on stronger compressions,characterized by scale sizes comparable to an ion gyrodiameter or less, whichaffect ions and electrons differently and lead to ambipolar effects that are be-yond the scope of electron-MHD (eMHD) frameworks [4]. To address suchconditions, we construct the equilibrium plasma distribution function withinthe compressed layers and analyze the field and flow structures they sup-port in the metastable equilibrium with self-consistent electric and magneticfields as well as their inherent spatial and velocity gradients. This specifies thebackground plasma condition, which can then be used as the basis to studytheir stability, evolution, and feedback to establish steady state structures.Such small-scale structures, with scale sizes comparable to ion and electrongyroscales, are being resolved with modern space probes [5].We use relevant constants of motion to construct the appropriate distri-bution function subject to Vlasov-Poisson or Vlasov-Maxwell constraints asnecessary. Given the background parameters the solutions provide the self-consistent electrostatic and vector potentials, which then fully specify theequilibrium distribution function, f ( v , Φ ( x ) , A ( x )) where Φ ( x ) and A ( x )are electrostatic and vector potentials. In effect, the potentials are Bernstein-Green-Kruskal (BGK) [6] or Grad-Shafranov [7,8] like solutions. With thedistribution function fully specified, its moments readily provide the staticbackground plasma features and their spatial profiles. As input parameters, i.e. , boundary conditions, we can use the output from global models if they Gurudas Ganguli et al. lobeplasma sheet boundary layerplasma sheetlobe lobeboundary layerplasma sheet density
Fig. 1 (a) Model profile of density at plasma sheet-lob interface. (b) Particle flux datafrom ISEE 1 (March 31, 1979) versus UT for two energy channels (2 keV and 6 keV). Figure(b) reproduced from Figure 1 of Romero et al. [10] can accurately produce them. But since these layers are on the order of iongyroradii and smaller, which the current generation global models cannot ac-curately resolve, we rely on high-resolution in situ observations to obtain theinput parameters. Given the boundary conditions we allow the density and thepotential to freely develop subject to no constraints except quasi-neutrality.This provides the self-consistent distribution function, as was demonstratedfor plasma sheaths by Sestero [9].2.1 Vlasov-Poisson System: Plasma Sheet-Lobe InterfaceConsider the compressed plasma layer that is observed at the interface of theplasma sheet and the lobe in the earths magnetotail region [10] as sketchedin Fig. 1a. The plasma sheet boundary layer is one of the primary regionsof transport in the magnetosphere [11]. This layer separates the hot (thermalenergies > ∼ − ) plasma of the plasma sheet,which is embedded in closed magnetic field lines of the earth, from the cold(thermal energy ∼
10s of eV) and tenuous (density ∼ − ) plasma inopen field lines in the lobe. During geomagnetically active periods, known assubstorms, when the coupling of the solar wind energy and momentum tothe magnetosphere is strong for southward interplanetary magnetic field, thequantity of magnetic flux and the field strength in the tail lobes increases [12, ehavior of Compressed Plasmas in Magnetic Fields 5 To obtain the equilibrium distribution function of such boundary layers weconsider the region (see inset in Fig. 1a) where the magnetic field lines arenearly straight so that the curvature that exists close to the equatorial planecan be neglected. The neglect of the curvature may be justified because its scalesize, L (cid:107) , is much larger than the gradient scale size, L ⊥ , across the magneticfield, i.e. , L ⊥ ∼ ρ i (cid:28) L || , and L (cid:107) = ( ∂ log( B ) /∂s ) − where s is the positionalong the magnetic field line and ρ i is the ion gyroradius. This simplifies theproblem by reducing it to essentially one dimension across the magnetic fieldin the x-direction in which the spatial variation is much stronger than it isalong the magnetic field.To represent the pressure gradient in the x-direction we construct a distri-bution function using the relevant constants of motion, which are the guidingcenter position, X g = x + v y /Ω α , and the Hamiltonian, H α ( x ) = m α v / q α Φ ( x ), Ω α = q α B/ ( m α c ) is the cyclotron frequency where the subscript α represents the species, m α is the mass, q α is the charge and Φ ( x ) is the elec-trostatic potential, so that it is approximately a Maxwellian far away from theboundary layer on either side f α ( X gα , H α ( x )) = N α ( πv tα ) / Q ( X gα ) exp (cid:18) − H α ( x ) T α (cid:19) . (1)The magnetic field is assumed in the z direction and the pressure gradientis normal to the magnetic field in the x direction. (Note that this is not theGSM coordinate system.) The electron and ion thermal velocity is given by v tα , T α = m α v tα / Q α is thedistribution of guiding centers, the shape of which is motivated by the observeddensity structures across the layer and is given by, Q α ( X gα ) = R α X gα < X g α R α + ( S α − R α ) (cid:16) X gα − X g α X g α − X g α (cid:17) X g α < X gα < X g α S α X gα > X g α . (2) N α R α and N α S α are the densities in the asymptotic high (plasma sheet) andlow-pressure (lobe) regions respectively, but in the transition layer the densityand its spatial profile is determined self-consistently. The quantity | S α − R α | is proportional to the pressure difference between the asymptotic regions and | X g α − X g α | represents the distance over which the pressure changes. These Gurudas Ganguli et al. quantities determine the magnitude and the scale-size of the electrostatic po-tential, which in turn determines the characteristics of the emissions that areexcited at the boundary, as elaborated in Section 3. Different values of theparameters X g α and X g α may be chosen to reproduce the observed den-sity profile. Hence, the values of the parameters R α , S α , X g α , and X g α are model inputs determined from observations. These parameters reflect theglobal plasma condition, i.e. , the compression. Hence, they causally connectthe small scale processes to the larger scale dynamics.The density structure within the boundary layer is obtained in terms ofthe electrostatic potential as the zeroth moment of the distribution function,Eq. (1), n α ( x ) ≡ (cid:90) f α ( v , Φ ( x )) d v = N α ( R α + S α )2 exp (cid:18) − eΦ ( x ) T α (cid:19) I α ( x ) (3)where I α ( x ) = 1 ± (cid:18) R α − S α R α + S α (cid:19) (cid:18) ξ α − ξ α (cid:19) × [ ξ α erf( ξ α ) − ξ α erf( ξ α )] + 1 √ π (cid:2) exp( − ξ α ) − exp( − ξ α ) (cid:3) (4)erf is the error function, ξ , α = Ω α ( x − X g , α ) /v tα , and ± refers to the speciescharge. The quasi-neutrality, (cid:80) α q α n α ( x, Φ ( x )) = 0 , then determines Φ ( x ),which in the limit that the Debye length is smaller than the plasma scale length(which is well satisfied here) is equivalent to solving Poissons equation. Theexistence of the transverse electric field reflects the strong spatial variabil-ity and nonlocal interactions that exist across the magnetic field due to thedifference in the electron and ion distributions with their characteristic spa-tial variations. With Φ determined the distribution function is fully specifiedand higher moments can be obtained. This distribution function satisfies theVlasov-Poisson system and is similar to the BGK class of solutions.As in the previous studies [10,15], the temperature variation across thelayer is ignored in the above. However, there is a temperature gradient be-tween the plasma sheet and the lobe that can affect the static backgroundproperties. The effects of the temperature gradient can be accounted for byconsidering two different types of plasma population characterized by theirrespective temperature and density in the asymptotic regions of the plasmasheet and the lobe, assuming isothermal condition exists in both the regionsaway from the boundary layer. While the plasma sheet population, denotedby subscript ps, goes to zero in the lobe, achieved by setting R α,ps = 1 and S α,ps = 0, the lobe population, denoted by subscript l , does just the oppositein the same interval | X g α − X g α | by setting R α,l = 0 and S α,l = 1.To obtain Φ quasi-neutrality must be maintained between all populations,i.e., (cid:88) α q α ( n α,ps ( x, Φ ( x )) + n α,l ( x, Φ ( x )) = 0 (5) ehavior of Compressed Plasmas in Magnetic Fields 7 − . . . . . n / N (a) − . . . . . T xx / T (b) H + e − − x/ρ i . . . . e φ / T e (c) − x/ρ i . . . . . p xx / p (d)
12 (H + )2 (e − ) Fig. 2
Comparison between an equilibrium with a single uniform temperature, labeled 1in the figure, and an equilibrium with a uniform temperature to the left of the layer and adifferent uniform temperature to the right of the layer, labeled 2 in the figure. (a) Densityof two models. (b) Temperatures across the layer for model 2. (c) Electrostatic potential forboth models. (d) Pressures across the layer for both models. For both models the parametersare as follows x g i,e = 0 , x g i,e = 0 . , . R i,e = 1 . , . S i,e = 0 . , . T e /T i = 1 . m i /m e = 1836 . T e,i /T e,i = 0 . , . The assumption that the transition in both density and temperature takesplace in the same interval is for simplicity and can be relaxed. If the intervalsdiffer somewhat, then the details of the spatial variation in the potential pro-file can be affected. However, these are higher level details and may not beobservable due to averaging by the waves that are spontaneously generated bythe highly non-Maxwellian distribution functions that develop as we elaboratein Sec. 3.In addition to the transverse electric field, the interface between the plas-masheet and the lobe is also characterized by ion and electron bi-directionalbeams along the magnetic field [16]. In Sec. 2.2.3 we argue that the origin ofthese beams could be related to the curvature in the magnetic field aroundthe equatorial region, which we ignored here, and not necessarily due to thereconnection process as it is usually assumed.
To understand the effects of a temperature gradient in the boundary layer wefirst consider a case where the density and the temperature gradients are inthe same direction and then in the opposite direction. Fig. 2 is a comparisonof the attributes for an equilibrium with only one temperature as was ana-
Gurudas Ganguli et al. − − . . n/N T ixx /T i T exx /T e − − − . − . .
00 (b) eφ/T e − − x/ρ i −
10 (c) v ye /v ti v yi /v ti Fig. 3
Generation of an electric field by a temperature gradient and no imposed densitygradient. (a) Density and Temperatures across the layer. (b) Electrostatic potential. (c)Flow velocities normalized to the ion thermal velocity defined to the left of the layer. Theparameters are as follows a i,e = 0 , a i,e = 0 . , . R i,e = 1 , S i,e = 1 , T e /T i = 1 . m i /m e = 1836 . T e,i /T e,i = 0 . , . lyzed in Romero et al. [10] and the two temperature model as described inSec. 2.1.1, i.e. , different populations in the lobe and the plasma sheet eachcharacterized by their respective temperature and density. The temperaturegradient of both populations is in the same direction as the density gradient.This example underscores the kinetic origin of the equilibrium electric field.In the two temperature model, the temperature reduces by a factor of 50 go-ing from the high density side to the low density side, thus the total pressuredrop from plasma sheet to lobe is larger. From a fluid perspective (eMHD) onewould expect that the larger pressure gradient must induce a larger electricfield to maintain the pressure gradient, however, as one sees in panel (c) thisis not the case. The electrostatic potential and the magnitude of the electricfield is reduced. This is because the ambipolar effect, which scales as ( ρ i − ρ e )averaged over the distribution, has been reduced by the decrease in the tem-perature, as ambipolar effects vanish with temperature. The x -axis of bothplots is normalized to the constant thermal ion gyroradius calculated to theleft of the layer. However, in the two temperature model the actual thermalion gyroradius decreases by a factor of (cid:112) T l /T ps (cid:39) .
2, where T l is the tem-perature of the lobe plasma and T ps is the temperature of the plasma sheet.This means that the ratio of the ion to electron gyroradius has decreased andthus the kinetic source of the electrostatic potential has reduced.As further illustration of the ambipolar effect we show an extreme case inFig. 3 where we have chosen the asymptotic density to be the same on either ehavior of Compressed Plasmas in Magnetic Fields 9 side of the layer by choosing the distribution of the guiding centers, Q ( X g ),to be a constant but have allowed the Temperature to fall from T e to 0 . T e across the layer. We can see that the temperature gradient creates a changein the difference between the ion and electron gyroradius which generates theambipolar electric field and the density in the layer adjusts to accommodate theambipolar potential even though the guiding center distribution is constant.We note that in this case there is a clear electron flow channel within the layermostly due to E × B drift and sheared flow in both the ions and electrons thatcan be the source of instabilities as discussed in Section 3. This also impliesthat for the temperature gradient driven modes [17,18,19] the effect of theself-consistent electric field must be examined. It is important to understand the origin and nature of the flows and currentsin the compressed plasma layers because they are the sources of free energyfor waves that determine the nonlinear evolution of the layers. The bulk flowcharacteristics change as the layer widths become less than an ion gyrodiame-ter. The flows are associated with the density and temperature gradients andthe ambipolar electric field that develop in the layer as a consequence of thecompression. The resulting E × B drift may not be identical for the electronsand the ions as we elaborate in the following.From the Vlasov equation we can calculate the equilibrium momentumbalance and using the geometry of our equilibrium we can solve for the fluid(or bulk) flow in the y direction as V α = − cE x B + cB q α n α dP αxx dx (6)where the first term is the E × B drift, V E , and the second term is the diamag-netic drift, V ∇ p . While this relationship is completely general for this geometryand applies to fluid and kinetic plasmas, the relative strength of each drift mayvary between fluid and kinetic approaches. This is because individual particleorbits are important in the kinetic approach but not in the fluid approach. It isespecially important in narrow layers when the particle orbits become speciesdependent (Sec. 3) and the ambipolar effects dominate the physics. This leadsto unique static background conditions, which influences the dynamics andhence the observable signatures, as we shall see in Sections 3 and 4.In Fig. 4 we show the fluid flows in panel (a), the electron drift componentsin panel (b), and the ion drift components in panel (c) for the case presentedin Fig. 2 with no temperature gradient. The layer width is larger than theelectron gyroradius but smaller than the ion gyroradius. Note that the fluidvelocity of the electrons is far larger than the ions. In addition, the electron E × B drift and the diamagnetic drift are in the same direction within the layerwhereas for the ions these drifts are in the opposite direction. When the iondrifts combine these components within the layer mostly cancel and the net ionfluid flow becomes negligible compared to the electrons. Thus the Hall current − . − . . . . . . . v e /v ti v i /v ti − . − . . . . . . . . . v E /v te v ∇ p /v te − . − . . . . . . . x/ρ i − v E /v ti v ∇ p /v ti Fig. 4
Comparison of fluid flows and drift velocities. (a) electron and ion flows, (b) electrondrifts, (c) ion drifts. The parameters are as follows x g i,e = 0 , x g i,e = 0 . , . R i,e =1 . , . S i,e = 0 . , . T e /T i = 1 .
0, and m i /m e = 1836 . is mostly generated by the electron flows and localized over electron scales.This can be understood in the following way. The ions have a large gyroradiuscompared to the scale size of the electric field and the density gradient. Sothe orbit-averaged E × B drift experienced by the ions is a fraction of what isexpected from the zero gyroradius limit. This shows up in a fluid representationas in Eq. 6, by the development of a fluid diamagnetic drift component in theopposite direction to reduce the net ion flow. Note that for layer widths largerthan an ion gyrodiameter the ambipolar electric field will be negligible and thenet current will be due to electron and ion diamagnetic drifts in the oppositedirections.In narrow layers of widths comparable to the ion gyroradius but larger thanan electron gyroradius the kinetic origin of the electric field from compressionof a plasma is shown in Figure 5. In this figure we keep all parameters of theequilibrium the same but vary the width of the layer δx = X g − X g , overwhich the density changes by a factor of 100. As we decrease the layer width themaximum electric field seen in the layer increases (as one would expect fromfluid theory) until the layer width gets below the ion gyroradius and thensaturates asymptotically. The ambipolar electric field becomes strong whenthe density gradient scale size, L n , becomes less than an ion gyrodiameter.Consequently, on average there are insufficient electrons, with much smallergyroradii, to charge neutralize the ions over their large gyro-orbit. As a result,a charge imbalance is generated proportional to ( ρ i − ρ e ) averaged over thedistribution, which leads to the electric field. As δx reduces, this imbalanceincreases because there are fewer electrons that can overlap the larger extent ehavior of Compressed Plasmas in Magnetic Fields 11 − − − δx/ρ i . . . . . . ρ i e E x T e Maximum Electric Field
Fig. 5
Maximum electric field as a function of the layer width normalized to the ion gy-roradius. The ion and electron layer locations are the same. The parameters are as follows R i,e = 1 . , . S i,e = 0 . , . T e /T i = 1, and m i /m e = 1836 . of the ion orbit. When δx falls below an ion gyroradius then there are hardlyany electrons that can do the job and, as a result, the value of the averaged( ρ i − ρ e ) reaches saturation asymptotically. Hence the electric field saturatesand its scale size becomes independent from L n . In contrast, in a fluid model(e.g. eMHD) L/L n = 1 remains valid throughout the layer because the electricfield is directly proportional to the density gradient for constant temperature.The proportionality of the electric field with the pressure gradient breaks downas the ambipolar electric field saturates for gradient scales smaller than an iongyroradius.In Fig. 6 we consider the case in which the temperature and the densitygradients in the transition layer are in the opposite directions. We model thisby two plasma populations in either side of the layer with characteristic densityand temperatures. While the guiding center density ( i.e. , Q ( X g )) of the lowtemperature population in the left of the transition region drops by a factor oftwo across a layer that has a width of δx = 0 . ρ i , the guiding center densityof the high temperature population in the right of the layer rises by a factor of2 in the same interval. The pressures are the same in the asymptotic regionsto the left and the right of the layer. Quasi-neutrality determines the detailsof the spatial variation of the density and temperature of each species in thelayer. Panel (a) shows the electron and ion pressures and the densities. Onecan see that the ion pressure falls across the layer, while the electron pressurerises. This can be understood in the following way. Since the layer width ismuch larger than the electron gyroradius the population on the left and righteffectively mix only within the layer. While the electron temperature increases across the layer, the density falls. However, the density reduction does notfall as much as the guiding center density because it is partly compensatedby the ambipolar electric field. Consequently, the electron pressure inside thelayer rises. Since the ion gyroradius is much larger than the layer width theions effectively mix on a scale larger than the layer width. So the ion tem-perature change is much smaller than the electrons across the layer. However,quasi-neutrality forces the ion density to be identical to the electrons, whichdecreases across the layer from left to right. The combination of these twoeffects lowers the ion pressure in the layer. Panel (b) shows that the net elec-tron fluid flow dominates the net ion flow. The individual drift components areplotted in panels (c) and (d). Both the ion and electron E × B and diamagneticdrifts are in opposite directions. In contrast, Fig. 4 showed that in the absenceof a temperature gradient the electron E × B and diamagnetic drifts were inthe same direction. This was because both the ions and electrons experiencedthe identical pressure gradient within the layer. In this case, from panel (a) inFig. 6, we see that the electron and ion pressure gradients are in the oppositedirections within the layer even though asymptotically the pressure is constanton either side of the layer.While setting the asymptotic pressure to be equal on either side of the layerwas not a sufficient condition to avoid the production of a pressure gradientin the layer, by reducing the asymptotic electron temperature ( i.e. , pressure)on one side it is possible to create a region where the electron pressure isalmost constant across the layer. We illustrate this in Fig. 7. In this case theelectron pressure is almost constant across the layer and consequently theelectrons have only a small diamagnetic drift as can be seen in panel (c) eventhough, asymptoticly, there is a pressure difference. From panel (d) we seethat the ion E × B and the diamagnetic drift cancel each other leading tonegligible net ion flow as seen in panel (b). Thus, the net flow within the layeris primarily due to electron EXB drift. This shows that depending on theboundary condition, as in this case with different pressures in the asymptoticregions, it is possible to generate a layer with no diamagnetic current but anelectron Hall current. This is typically, the situation in the dipolarization frontsas we shall discuss in Section 2.2 (See also Fu et al. [5]). Also, as we will see inSection 3, this condition can lead to waves around the lower hybrid frequencydriven by the gradient in the electron E × B flow that can be misinterpreted tobe the lower hybrid drift instability, which results in a different nonlinear statethat is measurable. Interestingly, the eMHD description of such layers with anegligible pressure gradient would predict a stable condition. This underscoresthe importance of the kinetic details of compressed plasma layers for accuratelyanalyzing satellite data and assessing the salient physics. Satellites measurethe local physics that operates in the layers where the fluid concept does nothold. ehavior of Compressed Plasmas in Magnetic Fields 13 − . − . . . . . . . . . n/N p exx /p e p ixx /p i − . − . . . . . . .
512 (b) n/N T exx /T e T ixx /T i − . − . . . . . . . . . v e /v ti v i /v ti − . − . . . . . . . − . . .
025 (d) v E /v te v ∇ p /v te − . − . . . . . . . x/ρ i −
101 (e) v E /v ti v ∇ p /v ti Fig. 6
Equilibrium where the guiding center density falls by a factor of two from theleft to the right and the temperature in the right asymptotic region is twice as high asthe temperature in the left asymptotic region. (a) Density and Pressures. (b) Density andTemperatures. (c) Electron and ion fluid velocities. (d) Electron drift velocities normalizedto the electron thermal velocity defined to the left of the layer. (e) Ion drift velocitiesnormalized to the ion thermal velocity defined to the left of the layer. The parameters areas follows x g i,e = 0 , x g i,e = 0 . , . R i,e = 1 , S i,e = 0 . , . T e /T i = 1 .
0, and m i /m e = 1836 . T e,i /T e,i = 2 . , . β , is large such as a dipolarization front (DF) [20,21,22]. Thetypical geometry of a DF is sketched in Fig. 8. DFs are observationally char-acterized by a rapid rise in the northward component of the magnetic field,a large earthward flow velocity, a sharp drop in the plasma density, and theonset of broadband wave activity [23]. These changes in plasma parametersare due to a flux tube rapidly propagating past the observing spacecraft. DFsare often observed during bursty bulk flow (BBF) events [24,22], during whichlarge-scale magnetic flux tubes that have been depleted of plasma by some − . − . . . . . . . . . n/N p exx /p e p ixx /p i − . − . . . . . . .
512 (b) n/N T exx /T e T ixx /T i − . − . . . . . . .
501 (c) v e /v ti v i /v ti − . − . . . . . . . . .
02 (d) v E /v te v ∇ p /v te − . − . . . . . . . x/ρ i −
101 (e) v E /v ti v ∇ p /v ti Fig. 7
Equilibrium where the guiding center density falls by a factor of two and the iontemperature in the right asymptotic is twice as high as the temperature in the left asymp-totic region but the electron temperature is only 1.5 times less. (a) Density and Pressures.(b) Density and temperatures. (c) Electron and ion fluid velocities. (d) Electron drift veloc-ities normalized to the electron thermal velocity defined to the left of the layer. (e) Ion driftvelocities normalized to the ion thermal velocity defined to the left of the layer. The param-eters are as follows x g i,e = 0 , x g i,e = 0 . , . R i,e = 1 , S i,e = 0 . , . T e /T i = 1 . m i /m e = 1836 . T e,i /T e,i = 2 . , . event (likely transient reconnection) propagate rapidly towards the Earth toequalize the quantity pV / [25], where p is the plasma thermal pressure and V is the flux tube volume. Flux tubes that have been depleted more thanneighboring flux tubes will have a larger earthward velocity, leading to a com-pression of the plasma at the edge as the faster moving flux tube overtakesthe slower moving flux tube (See Figure 9). This compression maintains theplasma gradients in a narrow layer with widths comparable to an ion gyrora-dius or smaller as the flux tube propagates Earthward. A kinetic equilibriumsolution to the Vlasov-Maxwell system is necessary since the change in themagnetic field by compression in DFs can be sufficiently large especially inhigh β plasmas [26].To address such conditions the model discussed in Sec. 2.1 can be general-ized to include the electromagnetic effects by considering the Vlasov-Maxwell ehavior of Compressed Plasmas in Magnetic Fields 15 𝐸 𝑥∇𝑛∇𝑇𝑧̂𝑥( 𝑣⃗𝐵 dipolarization front
Fig. 8
Equitorial dipolarization front geometry. . . . . . . . X (GSE)0 . . P V / t < t = 0 t > Fig. 9 . Profile of
P V / in typical magnetotail. Some event depletes flux tubes with somemaximum depletion. The earthward speed of the DF is proportional to ∆P V / whichcauses the front to steepen as it propagates. set of equations instead of the Vlasov-Poisson system of Sec. 2.1 as shownbelow: v · ∇ r f α ( r , v ) + q α m α (cid:18) E + v × B c (cid:19) · ∇ v f α ( r , v ) = 0 , ∇ · E = (cid:88) α πq α (cid:90) d v f α ( r , v ) , ∇ × B = 4 πc (cid:88) α q α (cid:90) d v v f α ( r , v ) , (7)In the frame of the DF propagating towards the earth the variation in thenormal direction (with scale size of an ion gyroradius) is orders of magnitudestronger than in the orthogonal directions. Hence for small scale physics itbecomes essentially a one-dimensional model, similar to the plasmasheet lobeinterface discussed in Sec. 2.1. The local magnetic field is in the z directionand varies in the x direction, i.e. B = B ( x ) e z , while a nonuniform electric fieldalso varies in the x direction, i.e. E x ( x ) as sketched in Fig. 7. We introduce avector potential, A , where B = ∇ × A and A = A ( x ) e y . The Hamiltonian is H α ( x ) = p x m α + 12 m α (cid:104) p y − q α c A ( x ) (cid:105) + p z m α + q α Φ ( x ) (8) where p x , p y , and p z are the canonical momenta. The Hamiltonian only de-pends on x and is independent of t , y , and z so H , p y , and p z are constantsof motion, where p y = m α v y + m α Ω α a ( x ). Since the system has only one de-gree of freedom, the dynamics is completely integrable. With a ( x ) = A ( x ) /B and B is the upstream background magnetic field it follows that the guidingcenter position, X gα = p y m α Ω α = a ( x ) + v y Ω α (9)is a constant of motion as well. The construction of the distribution function is similar to that described in Sec.2.1.1, except that we now obtain the moments as a function of a ( x ) and thensolve a ( x ) as a function of x to obtain the spatial profiles of the parameters ofinterest [26]. The moments of the distribution provide the physical attributesof the equilibrium configuration, in particular their spatial variations. Thezeroth moment (density) is n α ( a ) ≡ (cid:104) f α (cid:105) = (cid:90) d v f α ( v , Φ ( a )) = N α ( R α + S α )2 exp (cid:18) − q α Φ ( a ) T α (cid:19) I α ( a )(10)Note the dependence of various quantities on a ( x ) in Eqs. 10, instead of just x as in Sec. 2.1; a ( x ) will be determined from the first moment ( i.e. the currentdensity). The electrostatic potential is found via quasineutrality, n e (cid:39) n i , asbefore: Φ ( a ) = T e T i q e T i − q i T e log (cid:20) N e ( R e + S e ) I e N ( R i + S i ) I i (cid:21) (11)Because ∇ n (cid:54) = 0 and ∇ B (cid:54) = 0, and the electric field, E = −∇ Φ ( a ), are in the x direction, the only nonzero component of the flow is in the y direction. Theflow is u yα ( a ) ≡ (cid:104) v y f α (cid:105) /n α = 1 n α (cid:90) d v v y f α ( v , Φ ( a ))= ± exp (cid:16) − q α φ ( a ) m α v tα (cid:17) N α ( R α − S α ) v tα [erf( ξ α ) − erf( ξ α )]4 n α ( ξ α − ξ α ) (12)and includes the diamagnetic drift, ∇ B drift and E × B drift.The magnetic field produced by the current density inherent in the equi-librium distribution function is found by the Ampere law, dB z dx = − πc j y , (13)where j y = (cid:80) α q α n α u yα is the current density. With B z , the vector potentialis found via dadx = B z B (14) ehavior of Compressed Plasmas in Magnetic Fields 17 − x/ρ i B z B (a) − x/ρ i . . . . . . n N (b)0 5 10 β e m a x i m u m ρ i e | E x | T e (c) − x/ρ i A y B (d) β e = 0 . β e = 0 . β e = 1 . β e = 10 . Fig. 10
Electromagnetic effects on equilibrium. (a) Magnetic field for different values of β e .(b) Density. (c) Maximum electric field seen over the layer as as function of β e . (d) Vectorpotential as a function of position. The legend in panel (d) refers to panels (a),(b), and (c).The parameters are as follows a i,e = 0 , a i,e = 0 . , . R i,e = 1 . , . S i,e = 0 . , . T e /T i = 1 .
0, and m i /m e = 1836 . with appropriate initial conditions. Eqs. 13 and 14 effectively forms the Grad-Shafranov equation and may not have a readily apparent closed-form solutionbut can be integrated numerically. The current density in Amperes law canbe written explicitly as a function of the vector potential a ( x ). Thus we cannumerically solve Eqs. 13 and 14 for the function a ( x ) which then provides amapping to x . All plasma parameters that have been determined as a functionof a ( x ) can now be found as a function of x . An electrostatic approximationis equivalent to specifying a ( x ) explicitly ( e.g. for a uniform magnetic field, a ( x ) = x ).We can continue and consider higher order moments. For the pressuretensor all off diagonal terms vanish and p αxx = p αzz = n α T α . The remainingcomponent, p αyy , which we do not repeat here involves an integral over v y andcan be performed in a manner similar to Eq. 12. Fig. 10 shows the electromagnetic effects on the static background structure.To illustrate the difference we choose the input parameters to be the same asin Fig. 2 but we increase β e . As seen from panels (a) and (c) the electric and Fig. 11
Geometry along the magnetic field line of a DF. In a typical DF the variationof plasma parameters across the magnetic field is stronger than the variation along themagnetic field which reduces the problem to 1D. Since the plasma parameters ( T , | B | ) aredifferent at the two points the electrostatic potentials assumes different values, which leadsto a potential difference ( Φ , − Φ , ) along the magnetic field causing the parallel electricfield. magnetic fields increase with β e . Panel (b) indicates that the density gradientsteepens with increasing β e , which explains the increase in the electric field.Panel (d) shows that as long as β e is less than unity the electromagnetic effectson static structures are minimal. Hence, the use of the simpler electrostaticmodel of Section 2.1.1 to understand the static background features is suf-ficient. However, in dipolarization fronts higher β e is typical. Ganguli et al. [27] and Fletcher et al. [26] have analyzed the MMS data in detail and illus-trated the difference between the electrostatic and electromagnetic models fora specific observation. In the above discussion of the equilibrium structure of a DF we considered thestronger variation normal to the magnetic field and ignored the slower variationalong the field. For a typical DF the transverse electric field is strongest ata particular point; for example marked P in Fig. 11. As we move from thispoint along the magnetic field, to point P , the x and z coordinates rotateby an angle θ as indicated in Fig. 11. Since the local values of the magneticfield, temperature, density, etc. are different at positions P and P along themagnetic field, the electrostatic potential will vary, giving rise to an electricfield along the magnetic field direction proportional to the potential difference ehavior of Compressed Plasmas in Magnetic Fields 19 between the two positions, Φ − Φ . Since Φ (cid:39) Φ ( B ( s )), the parallel electricfield is E (cid:107) ( s ) ≡ − ∂Φ ( B ( s )) /∂s = ( x/L (cid:107) ) E x ( x ). Fig. 3c of [27] shows that E (cid:107) peaks in the electron layer and varies in x for a typical DF. Non-thermal plasmaparticles subjected to E (cid:107) will be accelerated along the magnetic field to forminhomogeneous beams or flows. The generation of the beam along the fieldline by this process provides the physical basis for a non-reconnection originof the observed beams and its causal connection to the global compression.Existence of E (cid:107) indicates that the off-diagonal terms of the pressure tensor, P α = m α (cid:82) ( v − u )( v − u ) f α d v , are non-zero and are necessary to balanceit in equilibrium, i.e. , en ( x ) E (cid:107) = − ( ∇ · P α ( x )) · s = − ( ∂ x p xx b x + ∂ x p xz b z ) , (15)where b x = sin( θ ) and b z = cos( θ ), and to leading order ∂/∂y = ∂/∂z → x direction at a given locationalong the magnetic field. These equilibrium features along the magnetic fieldcan also be important to the dynamics of the compressed plasma layers andaffect the measurable quantities such as spectral character of the emissionsand particle energization. This is discussed in sections 3.3.4 and 4.3.2.3 Vlasov-Maxwell System: Field reversed geometry in the magnetotailWhile the electromagnetic effects of compression are important in DFs, es-pecially when the plasma β is large, electromagnetic effects are essential forthe magnetic field reversal geometry and current sheets. Current sheets areimportant in magnetic fusion experiments and magnetospheric, solar, and as-trophysical dynamics because the reversed magnetic field geometry can leadto magnetic reconnection and thus a large-scale reconfiguration of the system.The formation of the current sheet is the result of a global compression withopposing magnetic fields and the resulting nonlinear reconnection is often fur-ther driven by compression of a large fluid scale current sheet down to kineticscales [28,29,20,30]. We now extend the above equilibrium boundary layermethodology to the case of a current sheet with magnetic field reversal [31] toinvestigate the effects of an inhomogeneous ambipolar electric field resultingfrom global compression that cannot be transformed away. Traditionally thefield reversed case has been addressed by the Harris equilibrium [32] whichis restrictive because it is a specialized distribution designed to produce den-sity and potential gradients such that there is no net electric field by usinga transformation to a uniform velocity frame (described below). As a result,this distribution is inflexible and unable to account for the observed spatiallylocalized structures such as embedded [33,34,35] and bifurcated current sheets[36,37,38,39] that develop during active periods when the plasmasheet thinsdue to large scale compression causing the current sheet to structure. We re-move this inflexibility by constructing a solution to the Vlasov equation that is a generalization of the Harris equilibrium [32] with the inclusion of a non-uniform guiding-center distribution Q α ( x gα ), f α ( x, v ) = N α ( πv tα ) / Q α ( x gα ) exp (cid:18) − E α − U α p y + m α U α T α (cid:19) (16)where the definitions of the various quantities are as before. For Q α → U α → x g, , α (or equivalently a , )and allow the system to develop the density, flows, current, and temperaturestructures self-consistently. Then we can compute the density of each species n α = (cid:90) d v f α ( x, v )= N α exp (cid:18) − q α φT α − U α m α Ω α aT α (cid:19) I α ( a ) (17)where I α ( a ) = 1( πv tα ) / (cid:90) dv y Q α (cid:18) a + v y Ω α (cid:19) exp (cid:18) − ( v y − U α ) v tα (cid:19) (18)As in the Harris equilibrium [32] we choose U e /v te = − U i /v ti ( ρ e /ρ i ) by trans-forming to the frame where this is satisfied, and use quasi-neutrality to solvefor the electrostatic potential. Interestingly, the potential does not depend on U α and has a similar form to the cases considered for the plasma sheet-lobeinterface and for the dipolarization front, eφT e = 11 + T e T i log (cid:18) N i I i ( a ) N e I e ( a ) (cid:19) . (19)In the Harris equilibrium the choice of transformation to a uniformly driftingframe is typically made so that quasi-neutrality may be satisfied without anelectrostatic potential. This choice corresponds to a uniform drift where theinhomogeneity in the E × B drift is balanced by the inhomogeneity in thediamagnetic drift so that this transformation can be done globally. While themathematical simplicity and elegance of the transformation is appealing, itconstrains the system from developing substructures as the current sheet thinsdue to global compression. Introduction of the guiding center distribution, Q α ,relaxes this constraint and allows for nonuniform flows to develop in responseto global compression. Nevertheless the transformation still can be made tosimplify the expressions. ehavior of Compressed Plasmas in Magnetic Fields 21 Next, we calculate the current density using the second moment as, j yα = q α (cid:90) dv y v y f α = q α N α v ta exp (cid:18) − q α φT α − U α m α Ω α aT α (cid:19) J α ( a ) (20)where J α ( a ) = 1( πv tα ) / (cid:90) dv y v y v tα Q α (cid:18) a + v y Ω α (cid:19) exp (cid:18) − ( v y − U α ) v tα (cid:19) . (21)Considering a single ion species and electrons we can write down from Am-pere’s law the equation, ρ i d adx = β i (cid:20) exp (cid:18) − eφT i (cid:19) J i ( a ( x )) − N e v te N i v ti exp (cid:18) eφT e (cid:19) J e ( a ( x )) (cid:21) exp (cid:18) − U i a ( x ) v ti ρ i (cid:19) (22)where β i = 8 πN i T i /B , ρ i = v ti /Ω i , and Ω i = | e | B / ( m i c ). B is a ref-erence magnetic field value, which in the following, takes the value of themagnetic field in the asymptotic limit away from the layer for Q α = 1 in theHarris limit. Unlike the potential, the density and current depend on U α . Wenote that Eq. 22 has the form of an equation of motion, where x is the time-variable and a is the position like variable. With the solution of Eq. 22 (usingEq. 19) the equilibrium is fully specified. In the limit of constant guiding centerdistribution, φ = 0, N i = N e , J i = U i /v ti and J e = U e /v te , and Ampere’slaw becomes d adx = β i L H (cid:20) T e T i (cid:21) exp (cid:18) − a ( x ) L H (cid:19) (23)where L H = ρ i v ti /U i is the single scale size associated with the Harris equilib-rium [32]. Eq.(23) has solutions a ( x ) = L H log(cosh( x/L ))+ L H / β i + β e ).This is the usual Harris sheet vector potential [32]. Because the Harris sheethas only one length scale, L H , it is unable to develop substructures in responseto the compression. Introduction of another scale, L , associated with Q α , inthe generalized Harris equilibrium, Eq. (16), removes this limitation. L is de-pendent on the compression through the parameters, x g , α as discussed insections 2.1 and 2.2. This makes the generalized Harris equilibrium a moreaccurate representation of reality.Using the same linear ramp functions Q α ( x gα ) as used in Secs 2.1 and 2.2we can calculate explicity the functions I α and J α , for the generalized Harris a / i B z / B a i a i a e a e Fig. 12
Phase plane analysis for the case when the current due to the density layer isin the same direction as the Harris current. For this case a i,e = 1 . , . a i,e = 0 . , . R i,e = 0 . , . S i,e = 1 . , . U i /v ti = 0 . T e /T i = 1 .
0, and m i /m e = 1836 . equilibrium I α ( a ) = 12 ( R α + S α )+ b α ( R α − S α )2 | b α | ( ξ α − ξ α ) (cid:20) √ π (cid:16) e − ξ α − e − ξ α (cid:17) + ξ α Erf( ξ α ) − ξ α Erf( ξ α ) (cid:21) J α ( a ) = u α R α + S α )+ b α ( R α − S α )2 | b α | ( ξ α − ξ α ) (cid:20) u α √ π (cid:16) e − ξ α − e − ξ α (cid:17) −
12 (1 − u α ξ α ) Erf( ξ α ) + 12 (1 − u α ξ α ) Erf( ξ α ) (cid:21) (24)where we have normalized distances by ρ i so that a iα = x giα /ρ i and wehave defined ξ iα = ( − b α u α − a/ρ i + a iα ) /b α where u α = U α /v tα and b α =sign( q α ) ρ α /ρ i is negative for electrons.There are two general cases of the differential equation where the effects ofthe non-uniform flow are important. Both are achieved by choosing a α , a α such that the guiding center distribution changes on a scale comparable to theion gyroradius. This leads to a current due to an ambipolar electric field drift,which corresponds to a global compression on the current sheet, in additionto the current that supports the current-sheet in the Harris equilibrium dueto the drift U α in the distribution functions. There are two cases to consider(1) when this additional current is in the same direction as the Harris currentor (2) when it is in the opposite direction to the Harris current. In this paper,we only review the case when these currents are aligned. For the alternativecase see Crabtree et al. [31].In this case we can examine the possible categories of equilibria by exam-ining the phase-plane analysis of Eq. 22. We do this by solving the differential ehavior of Compressed Plasmas in Magnetic Fields 23 − − − A y ρ i B (a) − − − . . n / N (b) − − − . . e φ / T e (c) − − − x/ρ i . . j e / ( e n v t e ) (d) R α = . R α = 1 Fig. 13
Embedded thin current sheet. (a) Vector potential, a/ρ i , (b) density, (c) potential,and (d) Electron current density across layer. In all panels the blue curve corresponds tothe case with a density gradient achieved by setting R i,e = 0 . , . R i,e = 1 and the rest of the parameters areas follows a i,e = 1 . , . a i,e = 0 . , . S i,e = 1 , U i /v ti = 0 . T e /T i = 1 .
0, and m i /m e = 1836 . equation numerically and plotting da/dx = B z /B vs a/ρ . In Figure 12 weshow the phase-plane figure for the case when the currents are in the samedirection. In this case we find three different kinds of equilibria that are de-termined by the choice of initial conditions for B z /B and a/ρ . The choiceof the initial point, e.g. the value of a at B z = 0, is in general arbitrary. Innature all initial values are possible. The choice of a particular one dependson the global condition, which is beyond the purview of this model but maybe obtained from a global model. However once the initial condition is deter-mined our model can predict the resulting sub-structures of the current sheetcorresponding to the level of the global compression. This level is representedby both the initial point and the choice of parameters a ,i,e and a ,i,e in theguiding center density function Q α . The particular choices of the a ,i,e and a ,i,e are indicated by vertical lines in the figure. The first type of solution (inblack) is a Harris-like equilibrium because the solutions remain in the asymp-totic regime of the guiding center distribution ( i.e. where Q α (cid:39) const. ) sothere is no significant additional current. The second type of solution (in blue)reaches its turning point at B z = 0 within the guiding center distributiongradient and has solutions that are flattened in the phase plane. The thirdtype of solution (in red) completely traverses the gradient region and becomeselongated in the phase plane.In Figure 13 we show the equilibrium attributes corresponding to the blueregion of curves in Figure 12. For reference, we added the Harris solutionin orange. The density gradient scale is comparable to the ion gyroradiusand is self-consistently determined. This generates an ambipolar electrostaticpotential that cannot be transformed away (panel (c)). The small dip in density(as opposed to a peaked density) is necessary to create the electric field in the proper direction (away from the current sheet) to generate a current thatis in addition to the Harris current. Also note that around x = 0, wherethe magnetic field vanishes and hence magnetic confinement of the particlesbecomes weak, the electrostatic potential peaks. Consequently, around thispoint the particles can be electrostatically confined. As a result, the velocityprofile peaks around the null point, which is midway between the turningpoints of the electrostatic potential (Fig. 14). This creates an ideal situation inwhich the velocity gradient driven waves (Sec. 3) can originate in the vicinityof the null region and contribute to anomalous resistivity [40] necessary forthe magnetic reconnection process. Further details are discussed in Crabtree et al. [31]. The case without a density gradient, i.e. the Harris case, is shown inorange in the figure and correspondingly has no electrostatic potential. In panel(d) we show that the current density across the layer consists of a thin centralcurrent sheet, of scale size ∼ L , due to the electron Hall current, embedded in abroader current sheet of scale size ∼ L H due to the bulk drifting component ofthe distribution function (the U α drift). This solution resembles an embeddedthin current sheet which are commonly observed in situ by spacecraft [33,34,35]. In Figure 16 we show the individual drift components. The electrons havea small gyro-orbit compared to the electric field scale size and thus have astandard E × B drift in the ambipolar electric field. The ions have a largerorbit and thus the orbit averaged electric field sampled is smaller, thus thetotal flow of the ions is reduced. This is the source of the additional current.The existence and the magnitude of the electrostatic potential around themagnetic null (Fig. 13c) leads to another interesting question, i.e. , how doesthe electrostatic potential affect the individual particle orbits around the mag-netic null? For the 1D equilibria considered here, the particle orbits are all in-tegrable and the details of how the figure eight orbits [41] are modified by theelectric field are discussed in Crabtree et al. [31]. An open question remainswith the addition of a B x (north-south component in our coordinates) so thatthe magnetic field becomes approximately parabolic. Will the orbits still bechaotic near the null-sheet as they are in the case without an electric field [42]?If so, how does the electrostatic potential affect the extent of the region overwhich they are chaotic? How does the electrostatic potential affect the onsetcondition for chaos if chaotic orbits can still survive? These questions remainto be debated and answered in the future.Current sheet thinning, which is the result of a global compression, is oftenobserved in the magnetotail just prior to the onset of reconnection [28,29,20,30]. With a thin embedded current sheet there are narrow layers of electronflow with large flow shear which can drive many kinds of instabilities, thatwould not exist in a standard Harris equilibrium. These shear-flow driveninstabilities (discussed in Sec. 3) can provide a source of anomalous resistivityfor the onset of magnetic reconnection. Lower-hybrid drift instabilities (LHDI)have been extensively studied in Harris sheets [43,44,45,46] because of theirpotential to provide a source of anomalous resistivity, however, these studieswere done in a Harris equilibrium where the LHDI is confined away from themagnetic null because LHDI favors strong magnetic field and strong density ehavior of Compressed Plasmas in Magnetic Fields 25 − − − − . − . − . .
00 (a) V E /v te V ∇ Pe /v te V ye /v te − − − x/ρ i −
101 (b) V E /v ti V ∇ Pi /v ti V yi /v ti Fig. 14
Embedded thin current sheet. (a) Electron drifts and the total fluid velocity acrossthe layer normalized to the electron thermal velocity. (b) Ion drifts and total fluid velocitynormalized to the ion thermal velocity. The parameters are the same as in Fig. 13 − − − A y ρ i B (a) − − − n / N (b) − − − . . e φ / T e (c) − − − x/ρ i . . j e / ( e n v t e ) (d) R α = . R α = 1 Fig. 15
Bifurcated current sheet. (a) Vector potential, a/ρ i , (b) density, (c) potential, and(d) Electron current density across layer. (c) Electron current density. In all panels the bluecurve corresponds to the case with a density gradient achieved by setting R i,e = 0 . , . R i,e = 1. For both casesthe the solution curve for the vector potential was chosen by selecting A = 0 the rest ofthe parameters are as follows a i,e = 1 . , . a i,e = 0 . , . S i,e = 1 , U i /v ti = 0 . T e /T i = 1 .
0, and m i /m e = 1836 . gradients. With compression we expect current sheets to develop kinetic scalefeatures as shown here, and also observed in the in situ data, such that thesource of the instability can be closer to the magnetic field reversal regionand thus can play a significant role in reconnection. This is a topic for furtherinvestigation.In Figure 15 we show the vector potential in panel (a), the density in panel(b), the electrostatic potential in panel (c) and the electron current density inpanel (d) as a function of the distance across the layer where the magnetic fieldreversal is located at x = 0. The orange curves correspond to the Harris sheet − − − − . − . − . .
00 (a) V E /v te V ∇ Pe /v te V ye /v te − − − x/ρ i −
202 (b) V E /v ti V ∇ Pi /v ti V yi /v ti Fig. 16
Bifurcated current sheet. (a) Electron drifts and the total fluid velocity acrossthe layer normalized to the electron thermal velocity. (b) Ion drifts and total fluid velocitynormalized to the ion thermal velocity. The parameters are the same as in Fig. 15 solution with no ambipolar electric field and the blue curves correspond to thenew generalized Harris solution. This solution corresponds to the class of redcurves in Fig. 12 where we chose a = 0 at the field reversal. Figure 15 showsthat near the guiding center gradient on either side of the field reversal thereis a strong electron Hall current that is stronger than the current of scalesize L H supported by the uniformly drifting component of the distributionfunction ( i.e. the current due to U α ) but in the same direction. In Figure 16we show the electron drifts (in panel a) and ion drifts in panel (b) as well asthe total fluid velocities. We see that the E × B drift of the electrons (panela) is in the same direction as the diamagnetic drift in the layer which leads toa strong net sheared flow of electrons. Whereas with the ions (panel b) theyare in opposite directions. This figure shows that the electrons experience asignificant E × B drift but the ions do not because narrow electric fields existon scales a fraction of the ion gyroradius.The current sheet solution shown in Figures 15 and 16 resemble a bifurcatedcurrent sheet that have been commonly observed by spacecraft in the magne-totail. Such bifurcated current sheets have also been observed in 1D particlein cell simulations [28]. In these simulations the starting point was a Harrisequilibrium and then the layer was compressed by applying time-dependentin-flows at the boundaries (in x in our coordinate system). A steady state wasreached in the simulation after compression that resembled the bifurcated equi-librium shown here in Figure 15d. Thus, there are simulation studies showingthat by further compressing a Harris current sheet one can develop ambipolarelectric fields which drive an electron current and form a bifurcated currentsheet that are consistent with the Vlasov equilibrium solutions discussed here.As in Secs 2.1 and 2.2, we find that even in the field reversed magneticfield geometry as the plasma is compressed an electrostatic potential is self-consistently generated. This introduces plasma flows that are highly sheared. ehavior of Compressed Plasmas in Magnetic Fields 27 As we study in Sec. 3 below, such sheared flows have a natural tendency torelax through emissions that ultimately leads to a new reconfigured steadystate. Further details of the current sheet behavior during active periods andits importance to the magnetic reconnection process is discussed in Crabtree et al. [31].
From Sec. 2 we can conclude that in collisionless environment plasma com-pression generates an ambipolar electric field across the magnetic field whenthe layer width becomes less than an ion gyrodiameter. It also describes somenatural examples of plasma compression but this can also happen in labora-tory devices. The amplitude and gradient of the ambipolar field is proportionalto the intensity of the compression, which creates the pressure gradient thatforms in the layer. It is therefore reasonable to identify the transverse ambipo-lar electric field as a surrogate for the compression for practical purposes. Itis interesting that the electric field is a better surrogate for the compressionthan the pressure gradient because, as we discussed in Sec. 2.1, density andtemperature gradients could combine to reduce the pressure gradient in thelayer but still lead to intense electric fields as the scale size of the layer re-duces with increasing compression. With this identification it becomes possibleto quantitatively address the plasma response to compression by studying thevariety of linear and nonlinear processes that are triggered by the transverseelectric field.At the kinetic level the collective behavior in plasma is sensitive to theindividual particle orbits. The particle orbits are affected by the electric fieldgradient, which develops self-consistently as a result of the compression. Theorbit distortion could be quite substantial and can affect the character of thewaves emitted and their nonlinear evolution as well as saturation properties.Hence, we review the particle orbit modifications due to inhomogeneous trans-verse electric field.3.1 Particle orbit modification due to localized transverse electric fieldIn a uniform magnetic field the charged particle orbit modification to the gyro-motion introduced by a uniform transverse electric field is a uniform E × B drift and this electric field can be transformed away in the moving frame. Sincethe E × B drift is mass and charge independent, both the electron and iondrifts are identical, which implies that there is no net transverse current. Thisis no longer true for an inhomogeneous electric field and has implications forplasma fluctuations. In realistic plasmas, both in nature and the laboratory, thetransverse electric field encountered is inhomogeneous. For example, we foundin Sec. 2 the ambipolar electric field that arises self-consistently due to plasmacompression is highly nonuniform. Therefore, we analyze the modifications toparticle orbits that such electric field inhomogeneity introduces. Consider a uniform magnetic field, B , in the z-direction and an inhomo-geneous electric field, E ( x ), in the x -direction. The energy per mass for acharged particle in this field configuration is K ( x ) = v x / v y / eΦ ( x ) /m ,where Φ ( x ) is the external electrostatic potential, i.e. , E = − dΦ ( x ) /dx .The equations of motion for a charged particle in the x - and y -directions are,˙ v x = Ωv y − ΩV E ( x ) , (25)˙ v y = − Ωv x (26)where V E = − cE ( x ) /B is the E ( x ) × B drift and dots imply time derivative.Integrating Eq. 26 we obtain a constant of motion X g = x + v y /Ω , whichis the guiding center position when the electric field is absent. Expressing v y = Ω ( X g − x )and using it in a Hamiltonian formulation we get, H ( x ) = v x Ω X g − x ) + eΦ ( x ) /m = v x / G ( x ) (27)Minimizing the pseudo potential G ( x ) at x = ξ , dGdx (cid:12)(cid:12)(cid:12)(cid:12) x = ξ = − Ω ( X g − ξ ) + em dΦ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = ξ = 0 (28)we obtain the guiding center position ξ = x + ( v y − V E ( ξ )) /Ω , when an elec-tric field is present. For an inhomogeneous electric field this expression is animplicit function for ξ . These definitions help understand the modification tothe E × B drift due to the inhomogeneity in the electric field.At steady state the time-averaged y -drift can be obtained from Eq. 25, i.e. , (cid:104) ˙ v x (cid:105) = 0 = Ω (cid:104) v y (cid:105) − Ω (cid:104) V E ( x ) (cid:105) . Expanding around the guiding center positionand retaining terms up to O (1 /L ), where L is the scale size of the transverseelectric field, the time averaged y -drift is, (cid:104) v y (cid:105) = (cid:104) V E ( x ) (cid:105) = V E ( ξ ) + (cid:104) ( x − ξ ) (cid:105) V (cid:48)(cid:48) E ( ξ ) / ... (29)The first order term, (cid:104) ( x − ξ ) (cid:105) , is oscillatory and vanishes on time averagingand (cid:104) v y (cid:105) is time independent. Thus, in general v y = u y + (cid:104) v y (cid:105) , where u y is theoscillatory component of the velocity in the y-direction. Using the definitionof the guiding center, x − ξ = − ( v y − V E ( ξ )) /Ω , in Eq. 29 we can express (cid:104) v y (cid:105) as, (cid:104) v y (cid:105) = V E ( ξ ) + V (cid:48)(cid:48) E ( ξ ) (cid:104) u y (cid:105) Ω η ( ξ ) + O ( V (cid:48)(cid:48) ) (30)where η ( ξ ) = 1 + ( dV E ( ξ ) /dξ ) /Ω . The parameter η is a comparison of theinfluences of the electric and magnetic fields on particle orbits. It is also ameasure of the velocity shear strength, and hence of the plasma compression. η − ω s = dV E /dx , and the gyrofrequency, Ω . In the limit ω s → − Ω the particle orbits become ballistic as in a field freeenvironment. In the limit ω s (cid:29) Ω the particles execute trapped orbits in theelectrostatic potential and the electric field dominates. In between the particlesrespond to both electric and magnetic fields. Because of spatial variability ehavior of Compressed Plasmas in Magnetic Fields 29 there may be regions where each of these effects could be pronounced. Thismakes the typical particle orbits much different from the ideal gyro-orbits ina magnetic field, which can affect the collective plasma dynamics. Besidesthe usual E × B drift represented by the first term in the right hand sideof Eq. 30, there is also a mass dependent second order term. While thereis no transverse current in the zeroth order, a second order current arisesdue to electric field curvature, which is proportional to the magnitude of thecompression. This is an important modification to the mean or bulk plasmatransverse flow, which is a fluid property. We shall see in Sec. 3.3 that thisterm is an important contributor to plasma collective effects and hence cannotbe ignored with respect to the order unity term in Eq. 30.There is another important kinetic effect due to the electric field inhomo-geneity that affects the individual particle orbits. To understand this we castthe equation of motion in the guiding center frame [47],˙ v x = η ( ξ ) Ωu y + V (cid:48)(cid:48) E ( ξ )2 Ω (cid:0) (cid:104) u y (cid:105) − u y (cid:1) , (31)˙ u y − Ωv x (32)Taking the time derivative and multiplying by ˙ u y , Eq. 32 becomes ˙ u y ¨ u y = − Ω ˙ u y ˙ v x . Substituting ˙ v x from Eq. 31 yields another constant of motion, w ⊥ = v x + η ( ξ ) u y − V (cid:48)(cid:48) E ( ξ ) Ω (cid:32) u y − (cid:104) u y (cid:105) u y (cid:33) , (33)which reduces to the perpendicular velocity for uniform electric case when L → ∞ . Using this and solving Eqs. 31 and 32 for the particle velocities andorbits we get, v x = w ⊥ sin( (cid:112) η ( ξ ) Ωτ + ϕ ) − V (cid:48)(cid:48) E ( ξ ) w ⊥ η ( ξ ) / Ω sin(2 (cid:112) η ( ξ ) Ωτ + 2 ϕ ) , (34) u y = w ⊥ (cid:112) η ( ξ ) cos( (cid:112) η ( ξ ) Ωτ + ϕ ) − V (cid:48)(cid:48) E ( ξ ) w ⊥ η ( ξ ) Ω cos(2 (cid:112) η ( ξ ) Ωτ + 2 ϕ ) , (35)From Eq. 35 (cid:104) u y (cid:105) = w ⊥ / (2 η ) + O ( V (cid:48)(cid:48) E ) can be calculated so that (cid:104) v y (cid:105) (Eq. 30)becomes, (cid:104) v y (cid:105) = V E ( ξ ) + V (cid:48)(cid:48) E ( ξ ) w ⊥ Ω η ( ξ ) + O ( V (cid:48)(cid:48) E ) (36)Integrating the velocities particle orbits are, x − x = − w ⊥ (cid:112) η ( ξ ) Ω (cid:104) cos( (cid:112) η ( ξ ) Ωτ + ϕ ) − cos( ϕ ) (cid:105) + V (cid:48)(cid:48) E ( ξ ) w ⊥ η ( ξ ) Ω (cid:104) cos(2 (cid:112) η ( ξ ) Ωτ + 2 ϕ ) − cos(2 ϕ ) (cid:105) (37) y − y = w ⊥ η ( ξ ) Ω (cid:104) sin( (cid:112) η ( ξ ) Ωτ + ϕ ) − sin( ϕ ) (cid:105) − V (cid:48)(cid:48) E ( ξ ) w ⊥ η ( ξ ) / Ω (cid:104) sin(2 (cid:112) η ( ξ ) Ωτ + 2 ϕ ) − sin(2 ϕ ) (cid:105) + (cid:104) v y (cid:105) τ (38)A major departure from the uniform electric field case is an effective renor-malization of the gyrofrequency. To leading order in the field gradient Ω → ¯ Ω = √ ηΩ . Hence, even the oscillatory part of the particle orbits is dependenton the electric field gradient and the effective gyrofrequency becomes spatiallydependent even when the magnetic field is uniform.Depending on the magnitude and sign of the electric field gradient, η canbe positive or negative. This has implications for particle orbits. Consider aweak electric field gradient, i.e. , ρ/L < ρ is the particle gyroradius,and η >
0. To leading order in the gradient the equation of motion may besimplified to ¨ v x = − η ( x ) Ω v x + O ( V (cid:48)(cid:48) E ), which shows that the particle orbit iseither oscillatory or divergent depending on the sign of η ( x ). Depending on themagnitude of the gradient, the effective gyroradius, ¯ ρ = v t / ¯ Ω , can be largeror smaller compared to the uniform electric field case for which η = 1. Thiswill be reflected in the averaged equilibrium quantities as larger or smallertemperatures and affect plasma distribution functions, as we shall discuss indetail in Sec. 3.2. While the η → η → ∞ leads to strong magnetization, which effectively is electrostatic confinementof the particles. This property may be especially consequential to the chaoticorbits [48] in the neighborhood of the null sheet in the magnetic field reversedgeometry in the earths magnetotail when there is guiding magnetic field normalto the current sheet. As discussed in Sec. 2.3, an electrostatic potential self-consistently develops around the null sheet that has not been considered inthe studies of the chaotic particle orbits in this region.In the weak gradient limit, the higher-order derivatives of the electric fieldare not important but they become critical for stronger gradients when η < η < v x = | η ( x ) | Ω v x + O ( V (cid:48)(cid:48) E ) indicat-ing that the restoring nature of the force becomes divergent and the particleaccelerates along the electric field. Gavrishchaka [49] studied the strong gra-dient limit. He showed that for strong gradients, multiple guiding centers canarise and the particles do not accelerate indefinitely unless the electric fieldis linear, which is a pathological case. Higher order derivatives prevent indefi-nite linear acceleration, which results in modified orbits that are no longer theideal gyromotion. Effectively, the particle acquires a larger gyroradius arounda new guiding center. As shown in Sec. 2, this can have major implications tothe equilibrium properties when η i becomes small and negative in the narrowlayers with ρ i > L > ρ e .When the scale size of localization reduces much below the gyroradius thegyro-averaged electric field experienced by the particle reduces until a limit isreached below which the electric field becomes negligible [49]. Consequently,the particle E × B motion is drastically reduced if not eliminated. In plasmasthis can lead to an interesting regime when ρ i (cid:29) L > ρ e in which the ions do ehavior of Compressed Plasmas in Magnetic Fields 31 not experience the E × B drift but the electrons do. For short time scale pro-cesses, such that Ω i (cid:28) ω < Ω e , the ions effectively behave as an unmagnetizedfluid while the electrons remain magnetized. This gives rise to a Hall currenteven in a collisonless uniform plasma. In plasmas undergoing compression, orrelaxing from it, the scale size of the electric field varies in time, which affectsthe particle orbits differently at different stages of compression or relaxation.These changes in particle orbits affect the collective dynamics resulting inthe observed spectral characteristic that includes broadband emission as wediscuss in Sec. 3.3.3.2 Analytical distribution functionTo understand the ramifications of the orbit modification discussed in Sec.3.1 on plasma collective effects it is necessary to develop a kinetic formalismto analyze the stability of plasmas including localized DC electric fields. Fordoing so we must obtain a representative zeroth order distribution functionappropriate for the initial equilibrium state characterized by a homogeneousmagnetic field and an inhomogeneous electric field in the transverse direction.In Sec. 2 we found such a distribution function for arbitrary magnitude ofthe compression but it is a solution that uses special functions and does notlend itself transparently to perturbative analysis of the stability properties,which is ideal for a general understanding of the plasma response to localizedelectric fields. In this Section we construct an analytical distribution functionfor weak shear, i.e. for ρ/L < η >
0, using the constants of motion H ( x )and the guiding center position ξ , which will then be perturbed in Secs 3.3to understand the stability of the Vlasov equilibrium state of a compressedplasma. Consider the equilibrium distribution function introduced by Ganguli et al. [47], f ( H ( x ) , ξ ) = N (cid:112) η ( ξ ) g ( ξ ) e − β t H ( x ) e − β t H (cid:107) ( ξ ) , (39) g ( ξ ) = e β t (cid:20) em Φ ( ξ )+ V E (cid:21) , (40)where N = n ( β t / (2 π )) / , β t = 1 /v t , H (cid:107) ( ξ ) = ( v z − V (cid:107) ( ξ )) / v t = (cid:112) T /m isthe thermal velocity, and V (cid:107) ( ξ ) is an inhomogeneous drift along the magneticfield. In constructing the distribution function two requirements are imposed:1) the velocity integrated distribution function should produce a constant den-sity so that a static electric field generated in a quasi-neutral plasma withouta significant density gradient can be studied. However, a density gradient, asprevalent in the compressed layers discussed in Sec. 2 , can be included through n ( ξ ) when necessary, and 2) although any function constructed out of con-stants of the motion is a Vlasov solution, the particular choice must reduce tothe fluid limit when the temperature T → (cid:15) = ρ/L < V (cid:107) ( ξ ) = 0 thedistribution function can be simplified. Using v y = u y + (cid:104) v y (cid:105) in the argument of distribution function Eq. 39 and expanding the argument around the guidingcenter position it can be simplified to, f (cid:39) n (cid:112) η ( x )(2 πv t ) / e − ( v x +( v y − V E ( x )) /η ( x )+ v z ) / (2 v t ) + O ( (cid:15) ) (41)where terms up to O ( V (cid:48) E ) are retained. For a uniform electric field, i.e. , V (cid:48) E = 0, η = 1 and w ⊥ = v x + ( v y − V E ) . Eq. 41 reduces to a Maxwellian distributionwith v y shifted by a constant V E velocity. Since the E × B drift is identicalfor both electrons and ions in collisionless plasma there is no relative drift be-tween the species to feed energy to waves and hence the distribution is stable.This shows that global compression results in a deviation from a Maxwelliandistribution through the velocity gradient, which is a source of free energy forwaves. In a collisionless environment compression triggers a relaxation mech-anism to reach a steady state through the emission of waves and hence bydissipating the velocity gradient. The dependence of the distribution functionon the spatial gradient of the velocity through the parameter η and its asym-metric appearance in the distribution function is noteworthy. It shows thatthe temperature in the y direction is preferentially affected by the localizedelectric field across the magnetic field in the x direction, which introduces anasymmetry and breaks the gyrotropy of the distribution function. This resultsin a difference in the temperature in the x and y directions orthogonal to themagnetic field [27].In the following sections we will analyze how the electric field gradient canexcite broadband waves that can relax the gradients and hence the compres-sion.Transforming to the cylindrical coordinates ( w ⊥ , ϕ , v z ) by using theJacobian, | J | = √ ηw ⊥ V (cid:48)(cid:48) E w ⊥ cos( ϕ )2 η / Ω (42)the velocity integrals can be performed to obtain n ( x ) = n (1 + O ( (cid:15) ))[47].This shows that a large localized static electric field can be maintained in aquasi-neutral plasma across the magnetic field with negligible density gradient,as is observed in the earths auroral region [50].3.3 Stability of the Vlasov equilibriumElectric fields encountered in both laboratory and natural plasmas are nonuni-form, albeit with varying degree of nonuniformity. For example, in Sec. 2 weshowed that the ambipolar electric field that develops self-consistently in acompressed plasma is highly nonuniform. In Sec 3.2 we established that suchelectric fields make the equilibrium distribution function non-Maxwellian andtherefore introduces a source of free energy for waves. In collisionless plas-mas these waves are a natural response to compression since they relax theshear in the electric field so that a steady state can be achieved. Due to the ehavior of Compressed Plasmas in Magnetic Fields 33 strong spatial variability across the magnetic field the plane wave or WKBapproximations will break down. Also, some of the modes due to transverseflows discussed below are essentially nonlocal in nature with no local limit.Hence, the analysis of these waves must be treated as an eigenvalue problem.Their dispersion relation is usually a differential or an integral equation. Inthe following we highlight the key aspects of the derivation of the eigenvaluecondition and refer the readers to [47] for details.Linearizing the analytical equilibrium distribution given in Eq. 39 witha nonuniform density, N ( ξ ), we get f ( x, v , t ) = f ( x, v ) + f ( x, v , t ). Sincethe inhomogeneity is in the x -direction the fluctuating quantities, e.g. , theelectrostatic potential, is periodic in y - and z - directions but localized inthe x -direction, i.e. , φ ( r (cid:48) , t ) = exp[ − i ( ωt (cid:48) − k y y (cid:48) − k z z (cid:48) )] φ ( x (cid:48) ) where φ ( x (cid:48) ) = (cid:82) dk (cid:48) x exp( ik x x (cid:48) ) φ k ( k (cid:48) x ). Then, the perturbed density fluctuation may be ob-tained as n ( x ) = (cid:82) d v f ( x, v ). Using the orbits given in Eqs. 37 38 it canbe shown that, n ( k x ) = − eβ t πm (cid:90) (cid:90) (cid:90) dx d v dk (cid:48) x φ ( k (cid:48) x ) f ( ξ, w ⊥ ) (cid:104) e i ( k (cid:48) x − k x ) x − e i ( k (cid:48) x − k x )¯ ξ F (cid:105) , (43) F = ( ω − k y V g ) (cid:88) l,l,m,m J l (cid:48) ( σ (cid:48) ) J m (cid:48) (ˆ σ (cid:48) ) J l ( σ ) J m (ˆ σ ) ω − ( l (cid:48) − m (cid:48) ) ¯ Ω − k y (cid:104) v y (cid:105) e i { m − m (cid:48) ) − ( l − l (cid:48) ) } ϕ e i { lδ − l (cid:48) δ (cid:48) − m ˆ δ + m (cid:48) ˆ δ (cid:48) } , (44)where J m ( σ ) are Bessel functions, σ (cid:48) = k (cid:48)⊥ w ⊥ /Ω , k (cid:48) ⊥ = k (cid:48) x /η + k y /η , δ (cid:48) =tan − ( k (cid:48) x √ η/k y ), ˆ σ (cid:48) = ˆ k (cid:48)⊥ ˆ w ⊥ / (12 Ω ), ˆ w ⊥ = V (cid:48)(cid:48) E w ⊥ /Ω , ˆ δ (cid:48) = tan − (2 k (cid:48) x √ η/k y ),ˆ k (cid:48) ⊥ = k (cid:48) x /η + k y / η , ¯ Ω = √ ηΩ , and ¯ ξ = x + u y /Ω . V g is the bulk fluid driftin the plasma and is given by V g ( ξ ) = 1 η ( ξ ) Ωβ t f ∂f ∂ξ = V E ( ξ ) − V (cid:48)(cid:48) E ( ξ ) ρ η ( ξ ) − (cid:15) n ρΩη , (45)so that, ω − k y V g ( ξ ) = ω − k y V E ( ξ ) + k y V (cid:48)(cid:48) E ( ξ ) ρ η ( ξ ) − k y (cid:15) n ρΩη ≡ ω + ω − ω ∗ , (46)where ω = ω − k y V E ( ξ ) is the local Doppler shifted frequency, ω = k y V (cid:48)(cid:48) E ( ξ ) ρ / η is a frequency that is introduced due to the second derivative, i.e. the curva-ture, of the electric field, ω ∗ = k y ρ(cid:15) n Ω/η is the diamagnetic drift frequency,and (cid:15) n = ρ/L n where the density gradient scale size L n = [( dn/dx ) /n ] − .A number of noteworthy features arise compared to the uniform electricfield case. Unlike the trivial case when a global Doppler shift is appropriate, inthe nonuniform case a local doppler shift arises and no global transformationcan eliminate this spatially dependent shift. Because of the spatial inhomo-geneity the plane wave assumption in the direction of the inhomogeneity isno longer possible. Higher harmonics of quantized eigenstates are possible,which can broaden the frequency and wave vector bandwidth of the emissions.The transverse electric field becomes an irreducible feature defining the bulk plasma and affects its dielectric properties including the normal modes of thesystem. New time scales, represented by the frequencies ω and ω , are intro-duced. A resonance with the bulk plasma flow arises that can affect the fluid(macro) stability. Landau and cyclotron resonances with individual particlesare affected through orbit modifications altering the kinetic (micro) stabilityof the plasma. Consequently, the transverse electric field can affect both thereal and imaginary parts of the dispersion relation and therefore affect boththe real and imaginary parts of the frequency of oscillations. This can vastlyalter the known waves that characterize a plasma with uniform magnetic fieldand their nonlinear behavior. Under certain conditions the transverse electricfield can suppress some waves while in others waves can be reinforced [51].In addition, an entirely new class of oscillation becomes possible due to aninhomogeneity in the wave energy density introduced by the variable Dopplershift [52].Quasi neutrality, i.e. , (cid:80) α (cid:82) dk x exp( ik x x ) n α ( k x ) = 0, gives the generaldispersion condition for the waves, in the electrostatic approximation, whichis an integral equation and cumbersome to solve. However, for weak gradients, i.e. ρ/L < η ∼
1, and k x (cid:39) k (cid:48) x , some simplifications are possible. Forexample, ˆ σ (cid:48) ∝ ( ρ/L ) (cid:28) J (ˆ σ ) ∼ J (ˆ σ (cid:48) ) ∼ m = m (cid:48) = 0. Furthermore, k x (cid:39) k (cid:48) x implies σ (cid:48) (cid:39) σ and δ (cid:48) (cid:39) δ .In the O ( ρ/L ) term in the denominator of Eq. 44 we may replace w ⊥ , thatappears in (cid:104) v y (cid:105) , by 2 v t . This simplifies F considerably to, F = ( ω + ω − ω ∗ ) (cid:88) l (cid:48) ,l J l (cid:48) ( σ (cid:48) ) J l ( σ ) ω − ω − l (cid:48) Ω e [ i ( l (cid:48) − l ) ϕ + ilδ − il (cid:48) δ (cid:48) ] . (47)It is interesting to note that the electric field curvature related frequency, ω , that appears in the numerator of Eq. 47 originates from the fluid plasmaflow, while the one in the denominator originates from the individual par-ticle orbit due to its kinetic behavior and will be absent in the fluid frame-work. With these simplifications and transforming coordinates from Cartesian,( x, v x , v y , v z ), to cylindrical, ( ξ, w ⊥ , ϕ, v z ), the velocity integrals can be readilyperformed to obtain the density fluctuations, n ( x ) = eβ t πm (cid:90) dk x exp( ik x x ) (cid:90) (cid:90) dξdk (cid:48) x φ ( k (cid:48) x ) exp[ i ( k (cid:48) x − k x ) ξ ] n ( ξ ) × (cid:40) (cid:88) l (cid:32) ω + ω − ω ∗ √ | k (cid:107) | v t (cid:33) Z (cid:32) ω − ω − lΩ √ | k (cid:107) | v t (cid:33) Γ l (¯ b ) (cid:41) (48)where Z ( ζ ) = ( π ) − / (cid:82) ∞−∞ dt exp( − t ) / ( t − ζ ) is the plasma dispersion func-tion, Γ n (¯ b ) = exp( − ¯ b ) I n (¯ b ), ¯ b = ( k ⊥ ρ ) , and I n (¯ b ) is the modified Besselfunction. The weak gradient condition allows the expansion Γ l (¯ b ) = Γ l ( b ) − Γ (cid:48) l ( b ) ρ k x + O (( ρk x ) ), where b = ( k y ρ ) so that the remaining integrals can ehavior of Compressed Plasmas in Magnetic Fields 35 be easily performed to obtain, n ( x ) = − ω p πv t q (cid:34) − (cid:88) n (cid:32) ω + ω − ω ∗ √ | k (cid:107) | v t (cid:33) Z (cid:32) ω − ω − nΩ √ | k (cid:107) | v t (cid:33) dΓ n ( b ) db ρ d dx +1 + (cid:88) n (cid:32) ω + ω − ω ∗ √ | k (cid:107) | v t (cid:33) Z (cid:32) ω − ω − nΩ √ | k (cid:107) | v t (cid:33) Γ n ( b ) (cid:35) φ ( x ) (49)which, in conjunction with the quasi-neutrality condition or the Poisson equa-tion, provides the electrostatic dispersion eigenvalue condition in the form of asecond order differential equation. Using the fluid model for ions the derivationhas also been generalized to the electromagnetic regime [53,54,55]. We first consider the linear plasma response to a weak compression where theelectric field scale size
L > ρ i . As discussed in Sec. 3.1, in this case both ionsand electrons experience identical electric field magnitude since on averagethey sample the electric field throughout their gyro-motion. Hence, to thezeroth order, their E × B drift will be identical. Under this condition thefluctuating density for both the ions and the electrons is given by Eq. 49with respective mass and charge, which leads to the electrostatic dispersionrelation under the quasi neutrality condition, (cid:80) α q α n α = 0 . Ignoring termsof the order of ( m e /m i ) and considering low frequency waves, ω < ω LH = ω pi / (1 + ω pe /Ω e ) / , where ω LH is the lower-hybrid frequency, the n = 0cyclotron harmonic term for the electrons is sufficient. Then the eigenvaluecondition is, (cid:20) ρ i A ( x ) d dx + Q ( x ) (cid:21) φ ( x ) + O ( (cid:15) ) = 0 , (50)where A ( x ) = − (cid:88) n (cid:32) ω + ω − ω ∗ √ | k (cid:107) | v ti (cid:33) Z (cid:32) ω − ω − nΩ i √ | k (cid:107) | v ti (cid:33) dΓ n ( b ) db , (51) Q ( x ) = 1 + (cid:88) n (cid:32) ω + ω − ω ∗ √ | k (cid:107) | v ti (cid:33) Z (cid:32) ω − ω − nΩ i √ | k (cid:107) | v ti (cid:33) Γ n ( b )+ τ (cid:34) (cid:32) ω + ω /τ µ − ω ∗ /τ √ | k (cid:107) | v te (cid:33) Z (cid:32) ω − ω /τ µ √ | k (cid:107) | v te (cid:33)(cid:35) , (52)and τ = T i /T e , µ = m i /m e . There are two branches of oscillations driven bythe electric field in this equilibrium configuration [47]. These branches do notrequire a density gradient so in the following analysis we set ω ∗ = 0. Kelvin-Helmholtz Instability Branch − − b0 . . . . . γ / Ω i k k /k ⊥ = 0 . k k /k ⊥ = 0 . k k /k ⊥ = 0 . Fig. 17
Kinetic solutions (for three values of k (cid:107) /k ⊥ ) showing that the KH modes arestrongly Landau damped. The parameters used are (cid:15) = ρ i /L = . τ = T i /T e = 5, ¯ V E = V E /v ti = 2, µ = m i /m e = 1837, and no density gradient. For low frequencies, such that ω (cid:28) nΩ i , the n = 0 , ± E ( x ) = E sech ( x/L ), L = 10 ρ i , shown in Fig. 17, indicatethat the KH mode is strongly Landau damped.The KH instability is the quintessential shear flow driven instability in-voked in innumerable applications in the fluid phenomenology both in spaceand laboratory plasmas. It is extensively invoked in large-scale fluid models inspace plasmas. If long wavelengths, i.e. , k (cid:107) →
0, or cold plasma, i.e. , T → T i ≥ T e , which is usually the case in the magnetosphere andthe T → k (cid:107) →
0, such that the parallelphase speed of the waves is larger than the ion and electron thermal speedsthe KH modes can be damped by finite Larmor radius (FLR) effects if theperpendicular wavelengths are sufficiently short, which is likely in the thincompressed layers. In this case A ( x ) and Q ( x ) reduces to, A ( x ) = (cid:18) ω + ω ω − ω (cid:19) Γ (cid:48) ( b ) + (cid:18) ω − ω ( ω − ω ) − Ω i (cid:19) Γ (cid:48) ( b ) (53) ehavior of Compressed Plasmas in Magnetic Fields 37 Q ( x ) = 1 − (cid:18) ω + ω ω − ω (cid:19) Γ ( b ) + (cid:18) ω − ω ( ω − ω ) − Ω i (cid:19) Γ ( b ) (54)The Bessel functions diminish the magnitude of the source term for the KHmodes, which is proportional to ω as will become clear in Eq. 55. If theperpendicular wavelength is also sufficiently long such that b = ( k y ρ i ) (cid:28) Γ ( b ) ∼ − b , Γ (cid:48) ( b ) ∼ − Γ ( b ) ∼ b/
2, and Γ (cid:48) ( b ) ∼ /
2. With thesevalues and in the low frequency limit, Ω i > ω > ω , the order unity terms in Q ( x ) cancel out making the second order terms proportional to ( ρ i /L ) as theleading order in the eigenvalue condition, which then yields the classical fluidKH mode equation [56,57], (cid:20) d dx − k y + k y V (cid:48)(cid:48) E ( x ) ω − k y V E ( x ) (cid:21) φ ( x ) = 0 (55)In producing the fluid limit the frequency ω in the numerator of Q ( x ),which originates from the fluid plasma property, combines in equal part withthe one in the denominator, which originates from the kinetic plasma property,to constitute the source term proportional to V (cid:48)(cid:48) E that feeds the KH instability.Another kinetic effect is gyro-averaging. As a result, the fluid flow dueto the E × B drift and its derivatives become smaller as the scale size of thevelocity shear becomes comparable or less than an ion gyroradius. This reducesthe curvature of the flow and hence lowers the KH source term (see Fig. 19).This shows that the kinetic effects are deeply entrenched in the KH mechanism,which can modify the source term substantially. The kinetic effects can bestrong enough to stabilize the instability in a large portion of the parameterspace allowed to it within the fluid framework thereby limiting its applicability.In addition Keskinen et al. [58] and Satyanarayana et al. [59] have shown thata density gradient has a stabilizing effect on the KH modes.It is important to realize that in the fluid limit all the order unity termsexactly cancel each other in Eq. 50, making the otherwise negligible second-order terms responsible for KH instability as leading terms. This is critical tothe recovery of the KH eigenvalue condition in the fluid limit, implying thatthe KH limit is sensitive to the choice of the initial distribution function. Anumber of different initial distribution functions are possible and were triedbut only the particular one described by Eq. 39 yielded the classical KH eigen-value condition in the fluid limit [47,60]. Since many distribution functions arepossible but not all of them lead to the KH modes, the robustness of the KHinstability in warm plasma becomes questionable in comparison to the Inho-mogeneous Energy Density Driven Instability (IEDDI) discussed below, whichdoes not depend on a particular choice and therefore may be more ubiquitous. Inhomogeneous Energy Density Driven Instability Branch
The above discussion on the Kelvin-Helmholtz limit also implies that in thekinetic regime for shorter wavelengths such that the wave phase speed is largerthan or of the order of the ion thermal velocity but smaller than the electronthermal velocity, i.e. v te > ( ω − nΩ i ) /k (cid:107) ≥ v ti , and ω ∼ nΩ i the second Fig. 18
Growth rate vs frequency for IEDDI instability as a function of b = ( k y ρ i ) / incolor. For these calculations k (cid:107) /k ⊥ = 0 . (cid:15) = ρ i /L = . a = 1 . τ = 5, µ = 1837, andno density gradient.ehavior of Compressed Plasmas in Magnetic Fields 39 order terms in A ( x ) and Q ( x ) may be neglected with respect to the orderunity terms. This regime leads to a different branch of oscillations arising dueto the inhomogeneity in the wave energy density introduced by the velocityshear [52]. Unlike the KH instability the IEDDI can be enhanced by a densitygradient [61,62,63]. Fig. 18 shows the typical linear spectrum of the IEDDI.The background electric field profile used is E ( x ) = E sech ( x/L ) with L =3 . ρ i , τ = 5, V E /v ti = .
1, and k (cid:107) = 0 . k ⊥ . The spectrum remains relativelyunaffected for an electric field with a top hat profile (Fig. 19), although thegrowth rates reduce as the field profile becomes smoother. This is because theIEDDI does not depend on the local value of a specific derivative of the electricfield like the KH mode.To understand the general characteristics of the two (KH and IEDDI)branches of oscillations we have considered a generic electric field profile, E ( x ) = E A sinh ( x/a ) + 1 (56)where A = 1 / sinh ( x /a ), x = L/ (cid:15) = ρ i /L . At x = x the value of E ( x )reduces to E /
2. For a = x / sinh − (1), A = 1 and E ( x ) = E sech ( x/a ).For a → L and a . In the natural environment, especially undercompression, the static electric fields are likely to be generated with multiscaleprofiles. This also becomes apparent from our equilibrium studies in Sec. 2. InEq. 56 while L determines the overall extent of the localization of the electricfield, a determines its local gradient. For A →
1, the scale lengths a and L become comparable. The first column of Fig. 19 shows the transition ofthe electric field profile in Eq. 56 from a top hat to a smooth sech ( x ) as afunction of increasing a . The second and the third columns of Fig. 19 show thesecond derivative and the gyro-averaged second derivative of the electric field.For a → a decreases.This has a stabilizing effect on the KH mode (see eq. (56) below). On the otherhand, electric field profiles with smaller a favors the IEDDI mechanism as itprimarily depends on the localized nature of the electric field rather than thelocal value of any specific derivative (see Eq. (58)) below. The gyro-averagingeffect becomes more prominent as the external compression increases and thescale sizes shrink compared to the ion gyroradius. (For the KH instability inneutral fluids there is no gyro-averaging, since the particles are not charged,and this stabilizing effect does not exist in a neutral medium.)As discussed in Sec 3.3, the general eigenvalue condition for the IEDDI isan integral equation. For weaker shear it may be approximated to a secondorder differential equation. The numerical solution for the truncated IEDDIeigenvalue condition is easier in the a → a . The potential, Q ( x ) /A ( x ), of the second order differential equa-tion, Eq. (50), becomes stiff and there are a number of roots in close vicinity − − . . . . . . a =0.10 E/E − − − − ρ i E d Edx − − − − D ρ i E d Edx E − − . . . . . . a =0.50 a − − − . − . − . . . . . − − − . − . . . . − − . . . . . . a =1.00 L − − − . − . . . . − − − . − . − . − . . . . − − x/ρ i . . . . . . a =1.89 − − x/ρ i − . − . . . − − x/ρ i − . − . − . − . . . Fig. 19
The left column shows profiles of the electric field for different values of a with (cid:15) = 0 .
3. The middle column shows the profile of the second derivative normalized to the ionthermal gyroradius. The column on the right indicates the gyro-averaged second derivative. of each other. This poses considerable difficulty in tracking the IEDDI rootsby solving the differential equation. Potential barriers develop that obstructthe energy flux away from the negative energy density region created by thelocalized electric field that is necessary for the IEDDI (as elucidated in Eq.61). This may partly be because of the truncation of the integral equation tosecond order. Ganguli et al. [47] had to use a small density gradient in orderto circumvent this difficulty to obtain the roots. ehavior of Compressed Plasmas in Magnetic Fields 41
Thus, unlike the KH modes, the solution to the eigenvalue problem, Eq. 50,with the potential
Q/A given by Eqs. 51 and 52 for the IEDDI is not trivial.As x → ∞ , Eq. 50 has two asymptotic solutions: one that is exponentiallygrowing and the other exponentially decaying. The decaying one is the physi-cal solution, but the growing one can easily contaminate numerical solutions.Furthermore, Q/A has poles scattered around the complex plane that can alsomake finding precise eigenvalues difficult.The effects of the exponentially growing solution can be minimized signif-icantly by using the Riccati transform. This technique was recently appliedto tearing instabilities and an explanation of how and why the method workswas provided [64]. In this method the potential is transformed using u = φ (cid:48) /φ ,where the prime denotes an x derivative. This gives the transformed equation dudx = − QA − u (57)which has asymptotic solutions u ( x → ∞ ) = ± i (cid:114) Q ∞ A ∞ (58)where the + / − refers to growing/decaying solutions. Therefore, the decayingsolution may be chosen at x → ∞ and integrated backwards, using Eq. 57,towards x = 0. For modes with even parity in φ ( x ), i.e. φ (cid:48) (0) = 0, u ( x ) shouldbe zero at x = 0. A complex root finder ( e.g. Muller’s method or Newton’smethod) finds the appropriate eigenvalue, ω , that leads to u (0) = 0. A closeguess for an appropriate ω is still necessary for the root finder to convergereliably.The spiky nature of Q/A can introduce further difficulty, but as long asthe poles do not lie exactly on the real axis, a standard numerical integratorthat controls accuracy will be sufficient. In the case that the poles are on thereal axis ( e.g. both the real and imaginary parts of φ are zero simultaneously),a numerical integrator based on Pad´e approximations is useful [65]. Thesenumerical techniques allow robust solutions to be found without the need toadd any density gradient (as was needed in 1988 [47]).Both the KH and IEDDI branches and their applications have been exten-sively studied in the literature and are not repeated here. Instead, below wereview the physical mechanisms that are responsible for the two branches ofoscillations. Physical Origin of the Kelvin-Helmholtz Instability
Although both the branches mentioned above are sustained by the velocitygradient, they rely on different mechanisms for drawing the free energy fromit. This is best understood by analyzing the energy balance conditions. Forthe KH modes the energy quadrature can be derived as [60], ∂∂t (cid:90) dx (cid:20) | E | π + n m i | cE | B + n m i | x | V E V (cid:48)(cid:48) E ( x ) (cid:21) = 0 , (59) where E = − ik y φ , x = v x / ( ω r − k y V E ( x )), v x = − cE y /B , and E and v are the fluctuating electric field and velocity The first two terms of Eq.59 are due to the fluctuating wave electric field. The first term represents theelectrostatic wave energy density in vacuum, the second term is the wave-induced kinetic energy of the ions. The energy balance condition in Eq. 59indicates that reduction in the equilibrium flow energy, i.e. , ( (cid:104) V E ( x + x ) (cid:105) − V E ( x )) = | x | V E ( x ) V (cid:48)(cid:48) E ( x )+ O ((1 /L ) ), at a given position x , which occurs dueto time averaging by the waves, is available as the free energy necessary for thegrowth of the KH instability. The time averaging removes the first derivativeand therefore the free energy is proportional to the second derivative of thedc electric field. Consequently, to leading order the KH instability is explicitlydependent on the second derivative, i.e. , the curvature, of the electric field.This condition may be a limiting factor to the viability of the KH instabilitycompared to its sister instability, the IEDDI, which does not depend on anyparticular velocity derivative as we discuss next. Physical Origin of the IEDDI
When both the electrons and the ions are cold fluids it leads to the clas-sical KH description as shown above. The ions play the crucial role while theelectrons simply provide a charge neutralizing background. But for k (cid:107) (cid:54) = 0, T e (cid:54) = 0 and for waves with ω ∼ nΩ i the electron response can be adiabatic, i.e. , v te > ( ω − nΩ i ) /k (cid:107) ≥ v ti . In this limit ignoring the ( ρ i /L ) terms in A ( x ) and Q ( x ) we obtain the eigenvalue condition for the IEDDI branch. Tounderstand the physics of this branch of oscillations we may assume the ionresponse to be fluid so that b (cid:28) (cid:34) d d ¯ x − ¯ k y + (cid:18) ω Ω i (cid:19) − (cid:35) φ = 0 (60)where ¯ x = x/ρ s , ρ s = c s /Ω i , c s = T e /m i , and ¯ k y = k y ρ s . Following theprocedure outlined in Ganguli [60] we obtain the condition, S + 2 Ω i (cid:90) ∞−∞ d ¯ x γ ( ω r − k y V E ( x )) | φ | = 0 , (61)where S = ( φ ∗ φ (cid:48) − φφ (cid:48)∗ ) / i is the flux and is a positive real number, γ is thegrowth rate for the IEDDI, φ ∗ is the complex conjugate of φ , and the primesindicate spatial derivatives. In order for Eq. 61 to be valid the second termmust be negative which implies that the product γ ( ω r − k y V E ) < i.e. , γ >
0, is that the Dopplershifted frequency ( ω r − k y V E ) be negative in some region of space.To understand the physical consequences of ( ω r − k y V E ) < D ( ω ) = 1 + τ − Γ ( b ) − (cid:88) n> ω ω − n Ω i Γ n ( b ) = 0 (62) ehavior of Compressed Plasmas in Magnetic Fields 43 I IIII L/2-L/2 E B x Fig. 20
Geometry of Inhomogeneous Energy Density Driven Instability (IEDDI).
The wave energy density is given by, U ∝ ω ∂D∂ω = ω (cid:32)(cid:88) n> ωn Ω i Γ n ( b )( ω − n Ω i ) (cid:33) ≡ ω Ξ ( ω ) (63)Clearly, the ion cyclotron waves are positive energy density waves. However,introduction of a uniform electric field in the x direction initiates an E × B driftin the y -direction and consequently there is a Doppler shift in the dynamicalfrequency, i.e. , ω → ω . The energy density in the presence of the Dopplershift is, U I ∝ ωω Ξ ( ω ), which can be negative provided ωω < L . It isclear that because of the localized nature of the E × B drift in region-I, theenergy density in region-I can become negative provided the Doppler shiftedfrequency ω <
0, while it remains positive in region-II. A nonlocal wavepacket can couple these two regions so that a flow of energy from region-I intoregion-II will enable the wave to grow. In region-I it is a negative energy wavewhile it is positive energy wave in region-II. The situation is complementaryto the two-stream instability. In that case there are two waves one of positiveenergy density and the other of negative energy density at every location andtheir coupling in velocity space leads to the instability. In the IEDDI casethere is only one wave but two regions, one in which the wave energy densityis negative and positive in the other. The coupling of these two regions in theconfiguration space by a wave packet leads to the instability [52].
This simple idea may be quantified further using the wave-kinetic frame-work. The growth of the wave in region-I implies a loss of energy from thatregion. By conservation of energy, this must be the result of convection of en-ergy into region-II in the absence of local sources or sinks. The rate of growthof the total energy deficit in region-I is proportional to the growth rate of thewave, the wave energy density U I in region-I, and the volume of region-I givenby the extent in the x-direction of region-I times a unit area A ⊥ in the planeperpendicular to x . The rate of energy convection through A ⊥ is V g U II , where V g is the group velocity in the x -direction and U II is the wave energy densityin region-II, which is positive since the electric field is absent in this region.We can then write the power balance condition as γU I LA ⊥ = − V g U II A ⊥ , (64)which implies that the growth rate of the IEDDI is γ ∝ − U II /U I . Conse-quently, if U I is negative then the growth rate is positive showing that thegrowth of the wave can be sustained by convection of energy into region-IIfrom region-I. On the other hand, if U I is positive then the convection of en-ergy out of region-I would lead to a negative growth rate and, therefore, todamping of the waves. This shows that if the wave energy density is sufficientlyinhomogeneous to change its sign over a small distance then it can supportwave growth. This is in contrast to the KH mechanism in which there is anexchange of energy between the medium and the wave via local plasma flowgradient (Eq. 59). In the IEDDI mechanism such an exchange is not necessary.Instead, as described in Eq. 61, the IEDDI is dependent on energy transportfrom one region to another such that the sign of energy density changes.In addition to the driving mechanism described above, dissipative mech-anisms are also present in a realistic system. If the energy gained from thedc electric field is larger than the energy dissipated the wave can exhibit anet growth. It is important to note that this phenomenon is not restricted toa resonant group of particles in velocity space. The only requirement is that( ω r − k y V E ) < Magnetron Analogy of the IEDDI.
A nonlinear description of the waveparticle interaction responsible for IEDDIwas given by Palmadesso et al. [67]. It was shown that the fluctuating waveelectric field E leads to an average secular (ponderomotive) force F y ∼ O ( γE y ) in the y -direction (see Fig. 21). This leads to a F y × B drift in the x -direction, which in the small gyroradius limit is u x ∝ − γ ( ω − k y V E ) − E y ,leading to a shift in the particle position in the x -direction given by δx = (cid:82) u x dτ ∼ E y . As there is dc electric field E ( x ) in the x direction there is apotential energy gain given by E ( x ) δx if ( ω r − k y V E ) <
0. Since the particle ehavior of Compressed Plasmas in Magnetic Fields 45 𝐸 " 𝑥𝐵 " 𝐹⃗ ’ 𝑧̂ 𝑥*𝑦* 𝐹 ⃗ ’ ×𝐵 Fig. 21
Geometry of the ponderomotive force and nonlinear particle drift. motion is perpendicular to F y there can be no net increase in the particleenergy. Thus, the energy gained by the particles by falling in the potential ofthe dc electric field in the x -direction is lost to the waves in the y -direction.Consequently, E y grows and F y is further enhanced, which closes a positivefeedback loop as shown in Fig. (22). This leads to the instability in a waysimilar to a magnetron.The second order ion drift in the direction of the electric field constitutes apolarization current that reduces the magnitude of the external electric field.Such polarization current was observed in the Particle-in-Cell (PIC) simulationof the IEDDI by Nishikawa [68]. As compression increases the self-consistent electric field becomes more intenseand narrower in scale size. In the intermediate compression regime the scalesize is narrower than an ion gyroradius but larger than an electron gyroradius, i.e. , ρ i > L > ρ e . As discussed in Sec. 3.1, the ions in this regime do notexperience the electric field over their entire gyro-orbit. Consequently, theions experience a lower gyro-averaged electric field than the electrons. Forsufficiently localized electric field the ions experience vanishingly small electricfield. In this regime for intermediate frequencies and short wavelengths, i.e. , Ω i < ω < Ω e and k y ρ i > > k y ρ e , the ions behave as an unmagnetized !" grows $ % &' = ) *!" + % ,' = $ % &'× . / %5 /5 Fig. 22
Positive feedback loop for IEDDI instability. plasma species but the electrons are magnetized. The cyclotron harmonics forthe ions can be integrated to rigorously show their unmagnetized character[61]. Since the wave frequency is much smaller than the electron cyclotronfrequency it will suffice to consider only the n = 0 cyclotron harmonic termfor the electrons. Also, for simplicity, we assume that the velocity shear that theelectrons experience is small enough so that we may use η = 1 for the electrons.The ions do not experience a Doppler shift so the phase speed of the wavescan remain larger than the thermal velocity, which allows the assumption offluid ions in which the density perturbation is given by [61], n i ( x ) = 14 πq i ω pi ω (cid:18) k y + k (cid:107) − d dx (cid:19) φ ( x ) (65)However the electrons experience a spatially varying Doppler shift. The phasespeed of the waves can become comparable to the electron thermal velocity atsome locations. So for generality we use the kinetic response for the electron, ehavior of Compressed Plasmas in Magnetic Fields 47 which leads to their density perturbation n e ( x ) = − ω pe πv te q e (cid:34) − (cid:32) ω + ω e − ω ∗ √ | k (cid:107) | v te (cid:33) Z (cid:32) ω − ω e √ | k (cid:107) | v te (cid:33) dΓ n ( b e ) db ρ e d dx +1 + (cid:32) ω + ω e − ω ∗ √ | k (cid:107) | v te (cid:33) Z (cid:32) ω − ω e √ | k (cid:107) | v te (cid:33) Γ ( b e ) (cid:35) φ ( x ) . (66)Combining Eqs. 65 and 66 with the Poisson equation we get the general eigen-value condition of the EIH instability in the kinetic limit that includes theelectron diamagnetic drift.0 = d φdx + (cid:16) − ω pi ω (cid:17) ( k y + k (cid:107) ) − ω pe v te (cid:104) (cid:16) ω + ω e − ω ∗ √ | k (cid:107) | v te (cid:17) Z (cid:16) ω − ω e √ | k (cid:107) | v te (cid:17) Γ ( b e ) (cid:105) − ω pi ω + ω pe Ω e (cid:16) ω + ω e − ω ∗ √ | k (cid:107) | v te (cid:17) Z (cid:16) ω − ω e √ | k (cid:107) | v te (cid:17) dΓ n ( b e ) db φ ( x )(67)In the long wavelength ( k (cid:107) → k y →
0) limit Eq. 67 reduces to, d φdx − ( k y + k (cid:107) ) φ + (cid:32) ω pe Ω e + ω pe (cid:33) ω ( ω − ω LH ) (cid:34) k y ( V (cid:48)(cid:48) E − Ω/L n ) ω − k (cid:107) Ω e ω (cid:35) φ ( x ) = 0(68)Eq. 68 includes the modified two-stream instability [69], which was not inFletcher et al. [26] since k (cid:107) = 0 was assumed. The modified two-stream insta-bility dispersion relation can be recovered if the electric field curvature and thedensity gradient are neglected in Eq. 68. Including the density gradient Eq. 68represents the lower-hybrid drift instability [70]. The lower-hybrid drift modesdepend upon the density gradient and hence their growth relaxes the densitygradient. If the density gradient is ignored but V (cid:48)(cid:48) E (cid:54) = 0 then Eq. 68 reducesto the eigenvalue condition for the electron-ion hybrid (EIH) instability [61]where the free energy is obtained from the sheared electron flow through fasttime averaging by the perturbations [61] similar to the KH modes discussedearlier. The growth of the EIH waves relaxes the velocity shear.From Eq. 68 it is clear that the intermediate frequency waves depend ona double resonance ω (cid:39) ω LH (cid:39) k y V E ( x ). The spatial variation of k y V E ( x )is particularly important because at some point in x the argument of the Z function in Eq. 67 can become of the order of unity so that Landau dampingcannot be ignored unless k (cid:107) is sufficiently small. Hence, the limit k (cid:107) → k (cid:107) (cid:54) = 0. It is included in the last term in -5 0 5-0.4-0.200.20.40.60.81 -5 0 5-0.4-0.200.20.40.60.81 Fig. 23
Eigenfunctions for L n /L = 1 and L n /L = ∞ and α = 1 with k y L chosen tomaximize the growth rate. Eq. 68 but its contribution is minimal because for k (cid:107) → ω pe > Ω e , so ω LH (cid:39) (cid:112) ( Ω i Ω e ) and the firstfactor in the third term of Eq. 68 is about one. In the k (cid:107) → (cid:40) d d ¯ x − ¯ k + (cid:18) ¯ ω ¯ ω − (cid:19) ¯ k ( α s ¯ V (cid:48)(cid:48) E (¯ x ) − LL n )¯ ω − ¯ kα s ¯ V E (¯ x ) (cid:41) φ (¯ x ) = 0 , (69)where ¯ x = x/L , ¯ ω = ω/ω LH , ¯ k = k y L , ¯ V E = V E /V , V = cE /B , α s = V /LΩ e is the shear parameter.Fig. 23 shows two solutions to Eq. 69 ( i.e. the real and imaginary partsof the eigenfunctions). Fig. 24 is a plot of the linear growth rate and thereal frequency obtained from solving the eigenvalue condition given in Eq69. The eigenfunctions and eigenvalues were found via a shooting methodin which the large ¯ x solution goes to zero at infinity. The density profile is n ( x ) = n tanh( x/L n ) and the electron flow profile by E ( x ) = E sech ( x/L )are chosen to match the self-consistent low β [27] DF discussed in Sec. 2.2 andits parameters are based on the MMS observations. As the shear parameteris increased, implying higher compression, the growth rate increases. The realfrequency is around the lower-hybrid frequency while Doppler shifting broad-ens the frequency spectrum. The bandwidth increases with shear parameter.In the two cases shown, the growth peaks for k y L ∼
1. The wavelengthis much longer than ρ e since L (cid:29) ρ e . As L n /L is reduced, the wavelengthsbecome shorter and in the limit of uniform electric field ( L → ∞ ) it is wellknown that k y ρ e ∼ x are stillcontinuously dependent on k y . The parallel wave vector, k (cid:107) , is assumed to bezero. In Sec. 4.2 the nonlinear evolution of this equilibrium condition and its ehavior of Compressed Plasmas in Magnetic Fields 49 Fig. 24
Linear growth rate as a function of real frequency, colored by associated k y L value.On the left is the fluid case, where the electric field balances the density gradient. On theright is the limit of the kinetic case. Reproduced from Figure 14 of Fletcher et al. [26]. -0.6 -0.5 -0.4 -0.3 -0.2-2-101234 -1 Fig. 25
The ratio of the two driving terms in Equation 69 as a function of x (left) and asa function of layer width (right). Reproduced from Figure 15 of Fletcher et al. [26]. observable signatures are studied by PIC simulation and show that the spectralbandwidth becomes even broader nonlinearly as lower frequency waves arenaturally triggered with increasing L .Since Eq. 69 contains both density and electric field gradients, an interest-ing question is which one of these is responsible for the waves?To answer this question Fig. 25 compares the relative strength of the LHDand the EIH terms in Eq. 69. The left plot shows the ratio of these EIH toLHD instability source terms for the low beta MMS parameters [27,26], whichcan be reproduced by our electrostatic equilibrium model discussed in Sec.2.1 with R i = R e = 1, S i = S e = 0 . x g e = − . ρ i , x g e = − . ρ i , x g i = − . ρ i , x g i = 0 . ρ i , n = 0 .
355 cm − , T e = 654 .
62 eV, T i /T e = . B = 12 .
55 nT. It shows that even for weak compression, as in thecase considered, the EIH term is three times as large as the LHD term. Inthe stronger compression high beta case (Fig. 26), the EIH term is more thanan order or magnitude larger. The right plot shows the maximum of the ratioof EIH/LHD terms as the compression is increased. This plot was made byusing the same parameters as the low β case and compressing and expandingthe layer via choice of x g α and x g α . Clearly, the EIH instability dominatesover the LHD instability as long as the scale size of the density gradient iscomparable to ion gyroradius or less.Magnetic field gradients result in a stronger EIH instability [71], but aweaker LHD instability [72]. In Sec. 2.1 we showed that a gradient in thetemperature can also develop, which can make the pressure gradient in thelayer (and hence the diamagnetic drift) weaker, but not significantly affectthe ambipolar electric field. This also favors the EIH instability over the LHDinstability. Thus, the EIH mechanism will dominate wave generation and hencethe nonlinear evolution in a compressed plasma system in the intermediatefrequency range.In general, the self-consistent generation of an ambipolar electric field isunavoidable in warm plasmas with a density gradient scale size comparable toor less than the ion gyroradius. This raises an interesting question: How ubiq-uitous in nature is the classical LHD instability? To examine this we generalizethe Fig. 25 results to include electromagnetic effects in the equilibrium condi-tion and compare the relative strengths of the two drivers of the electrostaticinstability in Eq. 66: 1) α s ¯ V (cid:48)(cid:48) E (¯ x ), which is the shear-driven EIH instability,and 2) − L/L n , which is the density gradient-driven LHD instability. By usingthe electromagnetic equilibrium model of section 2.2 we can investigate themagnitude of these two driving terms. In general we find that for T i /T e > T e /T i <
1, whichis typical in laboratory plasmas, LHD tends to dominate. Fig. 26 shows thesame ratio of terms for different values of β e . As β e increases, the EIH termalso becomes more dominant because the ambipolar electric field intensifieswith β as shown in Figure 10. For typical conditions in the magnetotail (high β e and T i /T e ), the EIH term is greater than the LHD term. The dominanceof the EIH over LHD wave becomes further evident in the nonlinear analysisin Sec. 4. As compression increases further so that ρ i (cid:29) L ≥ ρ e then even higher fre-quency modes with ω ≤ Ω e are possible. For these modes the ions do not playany important role other than charge neutralizing background and they maybe ignored. The dispersion relations will become similar to the KH and IEDDIdiscussed in Sec. 3.3.1 but for the electron species. By symmetry for ω < Ω e the electron KH modes can be recovered and for ω ∼ nΩ e the electron IEDDIcan be recovered. ehavior of Compressed Plasmas in Magnetic Fields 51 -1 . . Fig. 26
EIH to LHD instability growth term ratio vs temperature ratio and β e for the equi-librium model where the width of the transition layer is equal to the ion thermal gryoradius.Red indicates that the EIH drive term dominates.2 Gurudas Ganguli et al. V (cid:107) ( x )In sections 3.3.1 through 3.3.3 we discussed the waves that are driven bythe shear in transverse flows. However, as discussed in Sec. 2.2.3, large-scalemagnetic field curvature can lead to a potential difference along the mag-netic field. This originates because the global compression is strongest at aparticular point and decreases away from it and hence the transverse elec-trostatic potential generated by compression also decreases proportionatelyaway from this point along the magnetic field. The potential difference alongthe field line results in a magnetic field aligned electric field as sketched inFig. 11. Non-thermal particles can be accelerated by the parallel electric fieldto form a beam along the magnetic field direction, with a transverse spatialgradient, i.e. , dV (cid:107) /dx . The gradient in the parallel flow is also a source forfree energy. This has been established both theoretically [73,74,75,76,77] andthrough laboratory experiments [73,78,79,80,81]. Like its transverse counter-part the spatial gradient in the parallel flow can also support a hierarchy ofoscillations. Below we summarize the physical origin of these waves.Consider a uniform magnetic field in the z direction with a transverse gra-dient in the flow along the magnetic field ( dV (cid:107) /dx ). The background plasmacondition is sketched in Fig. 27. Unlike the transverse flow shear, the parallelflow shear does not affect the particle gyro-motion, which simplifies the anal-ysis considerably. For simplicity consider a locally linear flow, i.e. , V (cid:107) ,α ( x ) = V (cid:107) ,α + ( dV (cid:107) ,α /dx ) x where V (cid:107) ,α and dV (cid:107) ,α /dx are constants and α representsthe species and let dV (cid:107) ,e /dx = dV (cid:107) ,i /dx ≡ dV (cid:107) /dx . Transforming to the ionframe ( i.e. , V (cid:107) ,i = 0) so that V (cid:107) ,e ≡ V (cid:107) represents the relative electron-ionparallel drift. Although a nonlocal eigenvalue condition is desirable, a locallimit exists for the parallel flow shear driven modes.First consider the general dispersion relation for waves with ω (cid:28) Ω e sothat only the n = 0 cyclotron harmonic of the electrons is sufficient. For thiscondition the dispersion relation is [77],1 + (cid:88) n Γ n ( b ) F ni + τ (1 + F e ) = 0 , (70) F ni = (cid:32) ω √ | k (cid:107) | v ti (cid:33) Z (cid:32) ω − nΩ i √ | k (cid:107) | v ti (cid:33) − k y k (cid:107) Ω i dV (cid:107) dx (cid:34) (cid:32) ω − nΩ i √ | k (cid:107) | v ti (cid:33) Z (cid:32) ω − nΩ i √ | k (cid:107) | v ti (cid:33)(cid:35) (71) F e = (cid:32) ω − k (cid:107) V (cid:107) √ | k (cid:107) | v te (cid:33) Z (cid:32) ω − k (cid:107) V (cid:107) √ | k (cid:107) | v te (cid:33) + k y k (cid:107) µΩ i dV (cid:107) dx (cid:34) (cid:32) ω − k (cid:107) V (cid:107) √ | k (cid:107) | v te (cid:33) Z (cid:32) ω − k (cid:107) V (cid:107) √ | k (cid:107) | v te (cid:33)(cid:35) (72) ehavior of Compressed Plasmas in Magnetic Fields 53 𝑣⃗ 𝑥 𝑧̂ 𝑥&𝑦& Fig. 27
Geometry for parallel shear flow.
In the absence of shear (i.e., dV (cid:107) /dx ≡ V (cid:48)(cid:107) = 0), the dispersion relation reducesto the case of a homogeneous flow as discussed by Drummond and Rosenbluth[66] and applied to space plasmas by Kindel and Kennel [82]. Low Frequency Limit: Sub-Cyclotron Frequency Waves.
We first dis-cusss low (sub-cyclotron) frequency ion-acoustic waves for which only the n = 0 cyclotron harmonic term for the ions is sufficient. For long wavelength, i.e. b (cid:28) Γ ( b ) ∼
1, and Eq. 70 simplifies to, σ + τ ˆ σ + σ ξ Z ( ξ ) + τ ˆ σ ξ e Z ( ξ e ) = 0 (73)where ξ = ω/ ( √ | k (cid:107) | v ti ), ξ e = ( ω − k (cid:107) V (cid:107) ) / ( √ | k (cid:107) | v te ), σ = (1 − k y V (cid:48)(cid:107) /k (cid:107) Ω i ),ˆ σ = 1 + k y V (cid:48)(cid:107) / ( k (cid:107) Ω i µ ). Assuming the ions to be fluid ( ξ (cid:29)
1) and electronsto be Boltzmann ( ξ e (cid:28)
1) and equating the real part of Eq. 73 to zero we get, ω = k z c s σ/ ˆ σ ∼ k z c s σ (74)where ˆ σ ∼ µ (cid:29)
1. In the absence of shear ( dV (cid:107) /dx = 0, i.e. σ = 1) the classical ion acoustic limit is recovered. If σ < ω r = 0 in the drifting ion frame. The D’Angelo instabilityhas been the subject of numerous space and laboratory applications [83,84,85]. The σ > et al. [75]. In thisregime Eq. 73 indicates that it is possible to obtain a shear modified ion-acoustic (SMIA) wave with interesting properties. Eq. 74 indicates that shearcan increase the parallel phase speed ( ω r /k (cid:107) ) of the ion acoustic mode by thefactor σ . For a large enough σ the phase speed can be sufficiently increasedso that ion Landau damping is reduced or eliminated. Consequently, a muchlower threshold for the ion acoustic mode can be realized even for T i > T e .The growth rate expression for the SMIA instability is given by Gavrishchaka et al. , [75], γ | k (cid:107) | v ti = (cid:114) π σ τ (cid:20) τ / µ / (cid:18) V (cid:107) σc s − (cid:19) − σ exp( − σ / τ ) (cid:21) (75)The classical ion-acoustic wave growth rate is recovered for σ = 1. From Eq.75 it is clear that σ can rapidly lower the ion Landau damping as seen from theexponential dependence of the second term in the bracket. The critical drift isobtained from Eq. 75 by setting the growth rate to zero and minimizing overthe propagation angle ( k (cid:107) /k y ) as is plotted in Fig. 28 with dV (cid:107) /dx = 0 . Ω i . Itis found that even a small shear can reduce the critical drift for the ion acousticinstability by orders of magnitude and put it below that of the classical ioncyclotron wave [66] for a wide range of τ = T i /T e but the shear modifiedion acoustics waves propagate more obliquely than their classical counterpart.This is a major departure from the conclusion of Kindel and Kennel [82]; thatamong the waves driven by a field aligned current in the earths ionosphere thecurrent driven ion cyclotron instability has the lowest threshold. Kindel andKennel’s conclusion had extensively guided the interpretation of in-situ datafor a long time until Gavrishchaka et al. reexamined the data [86] with shearmodified instabilities in mind. Low Frequency Limit: Ion Cyclotron Frequency Waves.
To study the ion cyclotron frequency regime we return to Eq. 70 but relaxthe constraints of low frequency and long wavelength used to study the shearmodified ion acoustic waves. We first examine how a gradient in the parallelplasma flow affects the threshold condition for ion cyclotron waves by analyzingthe expression for critical relative drift for the ion cyclotron waves in small andlarge shear limits. For the marginal stability condition ( γ = 0) the imaginarypart of the dispersion relation, Eq. 70, is set equal to zero, i.e. , (cid:88) n Γ n (cid:34)(cid:32) ξ − k y V (cid:48)(cid:107) k (cid:107) Ω i ξ n (cid:33) Im Z ( ξ n ) (cid:35) + τ (cid:32) k y V (cid:48)(cid:107) k (cid:107) Ω i µ (cid:33) ξ e Im Z ( ξ e ) = 0 (76)where ξ n = ( ω − nΩ i ) / ( √ | k (cid:107) | v ti ). ehavior of Compressed Plasmas in Magnetic Fields 55 − τ = T i T e V d e v t i CDEIACDEIC | V d / Ω i | = . Fig. 28
Critical Drift vs temperature ratio. Blue curve is for the classical current drivenelectrostatic ion acoustic mode (CDEIA). Orange curve is for the shear modified ion acoustic-instability.
Dividing Eq. 76 throughout by ξ and considering the electrons to be adi-abatic, i.e. , ξ e (cid:28)
1, we get, (cid:88) n Γ n (cid:32) − k y V (cid:48)(cid:107) k (cid:107) Ω i (cid:18) − nΩ i ω r (cid:19)(cid:33) exp − (cid:32) ω r − nΩ i √ | k (cid:107) | v ti (cid:33) + τ / µ / (cid:32) k y V (cid:48)(cid:107) k (cid:107) Ω i µ (cid:33) (cid:18) ω r − k (cid:107) V (cid:107) c ω r (cid:19) = 0 (77)Under ordinary conditions ( k y /k (cid:107) )( dV (cid:107) /dx ) /Ω i (cid:28) µ , which implies that theshear in the electron flow is not as critical as it is in the ion flow and can beignored. Since only a specific resonant cyclotron harmonic term dominates, Eq.77 can be simplified by considering only that resonant term in the summationto obtain an expression for the critical relative drift, V (cid:107) c = ω r k (cid:107) (cid:34) Γ n ( b ) µ / τ / (cid:40) − k y V (cid:48)(cid:107) k (cid:107) Ω i (cid:18) − nΩ i ω r (cid:19)(cid:41) exp (cid:32) − ( ω r − nΩ i ) k (cid:107) v ti (cid:33)(cid:35) (78) For no shear, V (cid:48)(cid:107) = 0, the critical drift reduces to, V (cid:107) c = ω r k (cid:107) (cid:34) Γ n ( b ) µ / τ / exp (cid:32) − ( ω r − nΩ i ) k (cid:107) v ti (cid:33)(cid:35) (79)This is the critical drift for the homogeneous current driven ion cyclotroninstability (CDICI) [66]. Since the relative sign between the two terms withinthe bracket is positive and each term is positive definite, the critical drift isalways greater than the wave phase speed and increases for higher harmonicssince ω r ∼ nΩ i .From Eq. 78 it may appear that for small but non-negligible and positivevalues of ( k y V (cid:48)(cid:107) /k (cid:107) Ω i )(1 − nΩ i /ω r ) there can be a substantial reduction inthe critical drift for the current driven ion cyclotron instability because ofreduction in the ion cyclotron damping. However, this is not possible and canbe understood by rewriting Eq. 78 as, V (cid:107) c V (cid:107) c = 1 − (cid:32) − ( ω r /k (cid:107) ) V (cid:107) c (cid:33) (cid:32) k y V (cid:48)(cid:107) k (cid:107) Ω i (cid:33) (cid:18) − nΩ i ω r (cid:19) (80)where the second term represents the correction to the critical drift for thecurrent driven ion cyclotron instability due to shear. A necessary condition forthe CDICI is that V (cid:107) > ω r /k (cid:107) . For a given magnitude of | dV (cid:107) /dx | /Ω i (cid:28)
1, itis clear from Eq. 80 that the shear correction is small unless the ratio k y /k (cid:107) can be made large. However, as k y increases, the real frequency of the waveapproaches harmonics of the ion cyclotron frequency and consequently (1 − nΩ i /ω r ) becomes small which makes the shear correction small. Alternately,when k z decreases the wave phase speed increases and the condition V (cid:107) >ω r /k (cid:107) is violated. Thus, for realistic (small to moderate) values of the shearmagnitude, the reduction in the threshold current for the current driven ioncyclotron instability by a gradient in the ion parallel flow is minimal at best.This is unlike the current driven ion acoustic mode case as discussed in theprevious section.Although shear is ineffective in reducing the threshold current for the ioncyclotron instability, it allows for a novel method to extract free energy fromthe spatial gradient of the ion flow, which does not involve a resonance ofparallel phase speed with the relative drift speed. To illustrate this we returnto Eq. 78 and consider the limit ( k y V (cid:48)(cid:107) /k (cid:107) Ω i )(1 − nΩ i /ω r ) (cid:29)
1, in which Eq.78 reduces to, V (cid:107) c = ω r k z (cid:34) − Γ n ( b ) µ / τ / (cid:40) k y V (cid:48)(cid:107) k (cid:107) Ω i (cid:18) − nΩ i ω r (cid:19)(cid:41) exp (cid:32) − ( ω r − nΩ i ) k (cid:107) v ti (cid:33)(cid:35) , (81)For ω r > nΩ i each term of Eq. 81 is still positive but the relative sign betweenthem is now negative, which allows for V (cid:107) c = 0. In this regime the ion flow gra-dient can support ion cyclotron waves. This can be understood by examining ehavior of Compressed Plasmas in Magnetic Fields 57 the relevant terms in the growth rate [77], γΩ i ∝ τ / µ / (cid:18) V (cid:107) ( ω r /k z ) − (cid:19) − (cid:88) n Γ n (cid:40) − k y V (cid:48)(cid:107) k (cid:107) Ω i (cid:18) − nΩ i ω r (cid:19)(cid:41) exp (cid:32) − ( ω r − nΩ i ) k (cid:107) v ti (cid:33) , (82)The first term in the bracket represents a balance between growth due tothe relative field-aligned drift and electron Landau damping while the secondterm represents cyclotron damping. Provided the drift speed exceeds the wavephase speed and the magnitude of the first term is large enough to overcomethe cyclotron damping a net growth for the ion cyclotron waves can be realized.This is the classical case where inverse electron Landau damping leads to wavegrowth [66]. For the homogeneous case ( i.e. , dV (cid:107) /dx = 0), the second term ispositive definite and always leads to damping. However, if ( k y V (cid:48)(cid:107) /k z Ω i )(1 − nΩ i /ω r ) > V (cid:107) = 0. This possibility forwave growth is facilitated by velocity shear via inverse cyclotron damping andfavors short perpendicular and long parallel wavelengths, which makes theterm proportional to shear large even when the magnitude of shear is small.A necessary condition for ion cyclotron instability due to inverse cyclotrondamping is, (cid:18) − nΩ i ω r (cid:19) (cid:18) k y k (cid:107) dV (cid:107) /dxΩ i (cid:19) = (cid:18) − nΩ i ω r (cid:19) (cid:18) V py V pz dV (cid:107) /dxΩ i (cid:19) > V py and V pz are ion cyclotron wave phase speeds in the y and z direc-tions.Another noteworthy property introduced by the ion flow gradient is in thegeneration of higher harmonics. From Eq. 79 we see that in the homogeneouscase the n th harmonic requires a much larger drift than the first harmonic.However, for ω r ∼ nΩ i the critical shear necessary to excite the n th harmonicof the gradient driven ion cyclotron mode, can be expressed as,( dV (cid:107) /dx ) c Ω i ∼ τ / µ / (cid:18) k (cid:107) k y (cid:19) (cid:18) τ − Γ ( b ) Γ n ( b ) (cid:19) , (84)For short wavelengths, i.e. , b (cid:29) Γ n ∼ / √ πb and hence, to leading or-der, the critical shear is independent of the harmonic number. Consequently,a number of higher harmonics can be simultaneously generated by the shearmagnitude necessary for exciting the fundamental harmonic. This is quanti-tatively shown in Fig. 29 (also in Gavrishachaka [76]), which indicates about20 ion cyclotron harmonics can be generated for typical ionospheric plasmaparameters. This figure also shows that when the Doppler broadening due toa transverse dc electric field is taken into account the discrete spectra aroundindividual cyclotron harmonics overlap to form a continuous broadband spec-trum such as those found in satellite observations. This remarkable ability ofvelocity shear to excite multiples of ion cyclotron harmonics simultaneouslyvia inverse cyclotron damping is similar to the ion cyclotron maser mechanism ω/ Ω i . . . γ / Ω i (a)0 . . . . . . . . . k y ρ i . . . γ / Ω i (b)0 5 10 15 20 ω/ Ω i + k y V E / Ω i . . . γ / Ω i (c) Fig. 29
First 20 ion cyclotron harmonics. (a) Growth rate vs frequency, (b) Growth rate vs k y ρ i , (c) growth rate vs doppler shifted frequency with V E = . v ti . Here V de = 0 (current-free case), V (cid:48) d = 2 Ω H + , µ = 1837, and Ω e /ω pe = 8 . [87] that results in broadband spectral signature. However, important differ-ences with the ion cyclotron maser instability exist. The ion cyclotron maserinstability is an electromagnetic non-resonant instability while we discuss theelectrostatic limit of a resonant instability. Also, in this mechanism the back-ground magnetic field is uniform unlike the ion cyclotron maser mechanism. High Frequency Limit. ehavior of Compressed Plasmas in Magnetic Fields 59
As discussed in the previous section multiple harmonics of the ion cyclotronfrequencies can be generated by the shear in parallel flows. In the presence of aparallel sheared flow and a transverse electric field the waves generated at thecyclotron harmonics can overlap due to Doppler shift, which can result in abroadband spectrum. Romero et al. [88] discussed the intermediate and higherfrequency modes due to parallel flow shear in which ions can be assumed asan unmagnetized species but the electrons remain magnetized for waves in thefrequency range Ω i < ω < Ω e and wavelengths in the range k y ρ i > > k y ρ e .For even shorter time scales with frequencies ω > Ω e both ions and elec-trons behave as unmagnetized species. Mikhailovoskii [89] has shown that flowshear in this regime can drive modes around the plasma frequency.Thus, the combination of low, intermediate, and high frequency emissionsthat are generated by parallel velocity shear can also lead to a broadbandspectral signature similar to that due to transverse velocity shear. Summarizing the survey of shear driven waves in sections 3.3.1 3.3.4 it canbe concluded that the linear response of a magnetized plasma to compres-sion is to generate shear driven waves with frequencies and wave-vectors thatscale as the compression. In Sec. 2 we showed that plasma compression self-consistently generates ambipolar electric fields that lead to sheared flows bothalong and across the magnetic field. This establishes the causal connectionof the shear-driven waves with plasma compression. Cumulatively, the gradi-ent in the parallel and perpendicular flows constitute a rich source for wavesin a broad frequency and wave vector band. In a collisionless environmenttheir emission is necessary to relax the stress that builds up in the layer dueto compression. Fig. 30 schematically shows the impressive breadth of thefrequency range involved with these waves starting from much below the ioncyclotron frequency and stretching to above the electron cyclotron and plasmafrequencies that can be generated by a magnetized plasma system undergoingcompression.In the dynamic phase the relaxing gradient can successively excite thenext lower frequency wave in the hierarchy when the gradient scale size issufficiently relaxed to turn off the higher frequency wave, or vice-versa witha steepening gradient [90]. Both relaxation and compression are longer timescale processes compared to the shear driven wave time scales. This can resultin emissions in a very broad frequency band in a quasi-static background thatis usually observed in the in-situ data. As a proof of principle a recent labo-ratory experiment has demonstrated this phenomenon in a limited frequencyrange that was possible within the constraints of a laboratory device [91] aswe elaborate in Sec. 5. Frequency overlap due to Doppler shift and nonlinearprocesses, such as scattering, vortex merging, etc. , can smooth out the spec-trum and contribute to seamless frequency broadening as typically observedby satellites. This naturally raises a question of how these waves affect the
Inhomogeneous Flow
Kelvin-Helmholtz InstabilityInhomogeneous Energy Density Driven InstabilityElectron-Ion Hybrid Instability
Perpendicular Flow
D’Angelo Instability,Shear-Modified Ion Acoustic InstabilityShear-Modified Ion Cyclotron InstabilityShear-Modified Lower Hybrid Instability
Parallel Flow
Electrostatic Oscillations
Kelvin-Helmholtz InstabilityInhomogeneous Energy Density Driven InstabilityElectron-Ion Hybrid Instability (whistler)
Perpendicular Flow
D’Angelo Instability,Shear-Modified Ion Acoustic Instability
UnexploredUnexplored
Parallel Flow
Electromagnetic Oscillations ! > $ % & < ( % !~ $ % &~( % $ * < ! < $ % & > ( % Compression ! < $ * &~& +* Unmagnetized plasma limit: Instability at plasma frequency
Fig. 30
Hierarchy of compression driven waves as a function of the magnitude of thecompression shown in the first column as shear scale size and the associated wave frequencies.Green corresponds to those cases which have been theoretically predicted and experimentallyvalidated in the laboratory. Yellow corresponds to the cases which have been theoreticallypredicted but yet to be validated in a laboratory experiment. White indicates the cases thatare expected to be there by symmetry arguments but yet to be rigorously analyzed. plasma-saturated state that a satellite observes. This is the topic of discussionin the following section.
We now examine how the linear fluctuations induced by the compressionevolve, the dominant nonlinear processes that relax the gradients to establisha steady state, and the measurable signatures of the compression driven waves.For this we need numerical simulations. However, due to the huge disparityin space and time scales it is difficult to simulate the entire chain of physicsin a single simulation. Hence, we focus on limited frequency and wavelengthdomains in order to understand the development of the spectral signature andthe steady state features in the nonlinear stage along with other nonlinearcharacteristics. ehavior of Compressed Plasmas in Magnetic Fields 61 et al. [68,93] used this equilibrium distribution function tosuccessfully simulate the IEDDI and demonstrated that it was another branchof oscillation in magnetized plasmas with transverse electric field distinct fromthe KH instability. The simulation also showed the development of a polariza-tion current along the electric field direction that reduced the magnitude ofthe external electric field as the waves grew and a bursty spectrum of waves,which were consistent with the nonlinear IEDDI (ion magnetron) model ofPalmadesso et al. [67]. More important to this article, as shown in Fig. 31(reproduced from Nishikawa et al. [68]), the growth of the instability relaxedthe flow gradient. This establishes that the strong transverse electric field gra-dients that develop as a response to plasma compression (Sec. 2) can relaxthrough the emission of the shear driven modes discussed in section 3.The IEDDI was later validated in laboratory experiments in NRL [94] andelsewhere [95] as discussed in Sec. 5. These laboratory experiments consistentlyshowed that the IEDDI fluctuations have azimuthal mode number m = 1.Interestingly, Hojo et al [96] showed that there can be no m = 1 KH modein a cylindrical geometry. The KH wave growth peaks for higher m numbersin a cylindrical geometry [97,98,99] while the IEDDI growth maximizes for m = 1 [100]. This is an experimental confirmation that the IEDDI is distinctfrom the KH instability and that they form separate branches of oscillations inmagnetized plasma with transverse sheared flow. Subsequently, Pritchet [101]also tested the Ganguli et al. [47] equilibrium model and concluded that itled to more reliable results although he could not resolve the IEDDI in hissimulation accurately. Fig. 31
The average ion flow velocity v y ( x ) at Ω i t = 0, 160, and 240 reproduced fromFigure 5 of Nishikawa et al. [68]. i.e. , ρ i > L > ρ e . Also in this region wave power around the lower hybrid frequencyrange has been observed. The generation of both electrostatic and electromag-netic waves around the lower hybrid frequency by velocity gradient has beenextensively studied. Simulations [40] indicate that these waves produce anoma-lous viscosity and relax the velocity gradients to reach a steady state. In thefollowing sections we study the nonlinear evolution of these waves leadingto formation of the steady state and the observable signatures by numericalsimulation. In understanding the behavior in the compressed plasma layer formed atthe plasma sheet-lobe interface (Sec. 2.1) Romero et al. [102] used the Gan-guli et al. [47] equilibrium (Eq. 41) for the electrons and an unmagnetizedMaxwellian distribution for the ions in a 2D electrostatic PIC model to sim-ulate the spontaneous generation of the intermediate frequency EIH wavesdiscussed in Sec. 3.3.2. The localized electric field used in the simulation wasin the intermediate scale length defined by ρ i > L > ρ e and was self-consistentwith the density gradient. The simulation was motivated by the ISEE satelliteobservation in the plasma sheet-lobe interface as shown in Fig. 1. Spontaneous ehavior of Compressed Plasmas in Magnetic Fields 63 Fig. 32
Spatial profiles of the electron cross-field flow at different times indicating therelaxation of the velocity gradient. Reproduced Fig. 16 of Romero et al. [40]. growth of the lower hybrid waves was seen in the boundary layer. The wavesnonlinearly formed vortices. The scale size of the vortices was comparable tothe velocity gradient scale size. Fig. 32, (reproduced from Romero and Ganguli[40]), shows that the growth of the EIH waves relaxed the velocity gradientsimilar to that observed in the IEDDI simulation of Nishikawa et al. [93]. In-terestingly, the density gradient was not relaxed by the EIH instability. Thedifference in the two simulation was that in the Nishikawa et al. [93] simulationof Ion cyclotron IEDDI the electric field was localized over a distance largerthan ρ i while in the Romero and Ganguli [40] simulation it was localized overa smaller distance. The inference that can be drawn from the two simulationsis that if the initial compression is large such that L < ρ i , then the growthof the lower hybrid waves could relax the velocity gradient so that L > ρ i atsteady state. While this saturates the lower hybrid waves, the flow shear willbe in the right magnitude to trigger the lower frequency IEDDI. When IEDDIrelaxes the gradient even further so that L (cid:29) ρ i then the KH modes could betriggered and so on. This nonlinear cascade to appropriate frequencies as thebackground gradient scale changes is how the shear driven modes can lead toa broadband signature of the emissions that are observed in the compressedplasmas [14]. In addition, the Nishikawa et al. [93] simulation showed the co-alescence of smaller vortices into larger ones implying that the wavelengthsbecome larger with time due to nonlinear vortex merging. Thus, these lowerhybrid waves have large wavelengths, roughly of the order of the shear scalelength rather than an ρ e as expected due to the LHDI, as discussed in Section3.3.2. The spatio-temporal scales associated with the cascading frequenciesare so large that it is difficult to simulate the entire bandwidth in a singlesimulation.The initial Romero et al. simulation [102] was followed up with more de-tailed studies of the nonlinear signatures of these waves, effects of magneticfield inhomogeneity on these waves, as well as their contribution to viscosity Fig. 33
Plasma density, n , (left) and electrostatic potential, φ , (right) at t (cid:39) /ω LH .Waves in the y direction and vortices are both visible. Reproduced from Figure 16 of Fletcher et al. [26]. and resistivity, which provide the steady state and feedback to the larger scaledynamics [40,71]. More recently, the Romero et al. [40] simulation model was applied to theDF plasmas [26] and generalized to the electromagnetic regime [103]. Theplasma parameters used in the simulation [26] were ω pe /Ω e = 3 . β e =0 . m e /m i = 1 / cE /B = 0 . v te . The simulation time is175 /ω LH , the spatial domain is 21 ρ i by 21 ρ i (1200 by 1200 cells), boundariesare periodic in all directions, and 537 million particles were used.Fig. 33 (from Fletcher et al. [26]) shows a snapshot of the plasma densityand electrostatic potential from the simulation at t (cid:39) /ω LH . These imagesshow only a part of the simulation domain in order to make features morevisible. Kinking is seen in the density. Vortices are formed on the lower density(right) side of the layer as well; these are visible in the potential (for example,one vortex is located at ( x/ρ i , y/ρ i ) (cid:39) (1,-1.5)). Wave activity in the y directionwith k y L ∼ x position; the layer iscentered near x/ρ i = 0. It is similar to what a satellite would measure if itwere flying through the simulated layer or a DF would propagate past theobserving satellite. There are broadband waves spread around and above the ehavior of Compressed Plasmas in Magnetic Fields 65 Fig. 34
Wavelet spectrum of the electric field as a function of position near t (cid:39) /ω LH .The density gradient is steepest near x/ρ e = 0. Reproduced from Figure 17 of Fletcher et al. [26]. lower hybrid frequency. The lower frequency power ω/ω LH (cid:39) . y direction and accompanying electric fieldin the x direction is significantly relaxed, indicating the dominance of shear-driven instability (EIH) over the density gradient-driven instability (LHD).Fig. 35 shows these two separate source terms responsible for the EIH and theLHD instabilities respectively (as in the numerator of the last term Eq. 69) andthe field energy as a function of simulation time. Instability growth and waveemission occurs before t = 20 /ω LH . The dotted black line is the theoreticallinear growth predicted by Eq. 69. During the growth phase, the EIH sourceterm (and thus the velocity shear) is clearly falling, suggesting that the shearis the source of free energy for the waves. The simulation reaches a saturatedstate at t (cid:39) /ω LH .4.3 Ion cyclotron waves in parallel sheared flowsIn Secs 4.1 and 4.2, we studied the nonlinear evolution of sheared transverseflows. We found that spontaneous generation of shear-driven waves relaxes the -4 Fig. 35
The driving terms for the EIH instability and LHD instability (left) and the fieldenergy fraction (right) in the simulation as a function of time. Reproduced from Figure 18of Fletcher et al. [26]. velocity gradient that leads to saturation. The frequency and wavelengths ofthese waves scale as the shear magnitude. Nonlinear vortex merging results inlonger wavelengths. Relaxation of stronger shear leads to weaker shear whichcan then drive lower frequency modes. This cascade leads to the broadbandspectrum of emissions that are often observed. Now we examine the nonlinearbehavior of parallel flow shear driven waves.The nonlinear evolution of the parallel flow shear driven modes discussedin Section 3.3.4 was investigated with PIC simulations by Gavrishchaka et al. [76]. The simulations included full ion dynamics but used a gyrocenter ap-proximation for the electrons. To clearly resolve short wavelength modes 900particles per cell were used with grid size ∆ = λ D = 0 . ρ i and mass ratio µ = 1837. A drifting Maxwellian (H+, e-) plasma is initially loaded, with equalion and electron temperatures. The magnetic field is slightly tilted such that k (cid:107) /k y = 0 .
01. A parallel drift velocity V (cid:107) ( x ) is assigned to ions to obtain aninhomogeneous velocity profile. The magnitude of the flow is initially specifiedand not reinforced during the simulation. To characterize the role of spatialgradients in the flow, the relative drift between the ions and the electrons, i.e. , field aligned current, is kept at a minimum. Its value does not exceed3 v ti locally while on average it is negligible. Periodic boundary conditions areused in both x and y directions. The magnitude of shear | dV (cid:107) /dx | max = 2 Ω i is used for the simulation. For this case the simulation box size was specifiedby L x = 64 λ D and L y = 64 λ D .The saturated spectral signature in the simulation without and with auniform transverse electric field shown in Fig. 36. On the left is the wavespectrum without a transverse electric field. In this simulation several ioncyclotron harmonics are excited with discrete harmonic structure. While onthe right a uniform transverse dc electric field is included with V E = 0 . v i . ehavior of Compressed Plasmas in Magnetic Fields 67 Fig. 36
Nonlinear spectral signature from a PIC simulation. (left) Frequency spectrumwithout a transverse DC electric field. (right) Frequency spectrum including a transverseDC electric field. Figures reproduced from Fig. 3 of Ganguli et al. [77].
The washing out of the harmonic structure and broadening of the spectrumdue to overlap of the discrete spectra around ω = 0 and multiple cyclotronharmonics becomes evident. Larger Doppler broadening either by large V E orlarge bandwidth, ∆k y , or a combination of both, could lead to an even broaderspectrum.The meso-scale effect of the parallel flow shear driven instability (normal-ized by their initial values) is given in Fig. 37 (reproduced from Gavrishchaka et al. , [76]). To highlight the role of shorter wavelength ion cyclotron waves thelonger wavelengths are removed by using a (64 × λ d size simulation box inthis case. Fig. 37 illustrates that the effect of the ion cyclotron wave generationis relaxation of the flow gradient due to wave-induced viscosity. This is similarto the effect of the transverse shear driven waves but not as strong. This maybe because the orbit modifications due to a localized transverse electric fieldis absent in this case. Thus the primary conclusion is that the compressiongenerated velocity shear either in parallel or transverse flow leads to broad-band emissions accompanied by relaxation of the velocity gradient that leadsto a steady state and determines the observed features that are measured bysatellites.In the above we discussed only the part of the simulation that showed theformation of the broadband spectral signature and relaxation of the velocityshear due to these waves, which is central to this article. However, the simula-tion also explained a number of interesting auroral observations that are not Fig. 37
Electrostatic wave potential obtained from PIC simulations after Ω i t = 40 (a), 60(b), 100 (c), and the corresponding ion velocity parallel to the magnetic field (d) shown bysolid, dashed, and dot-dashed lines, respectively. Reproduced from Fig. 3 of Gavrishchaka et al. [76]. elaborated here. For a detailed account of these we refer to Gavrishchaka etal. [75,76] and Ganguli et al. [77]. In Secs. 2-4, we outlined the theoretical foundation for understanding com-pressed plasma behavior and showed evidence of its characteristics in un-controlled natural plasmas from in situ data gathered from satellites. Thechallenge with in situ data is in characterization of a specific phenomenon inconstantly evolving plasmas subject to uncertain external forces. As a result,typically there are many competing theories of space plasma phenomena thatare difficult to distinguish unambiguously. Because of this difficulty, scaledlaboratory experiments have become a valuable tool in understanding spaceplasma processes. Not every aspect of space plasmas can be faithfully scaled inthe laboratory. Large MHD scale phenomena are especially challenging. Butothers, such as cause and effects of waves and various coherent processes in themeso and micro scales, which are difficult to resolve by in situ measurements inspace, are amenable to laboratory scaling. In the modern era, satellite clusterswith multi-point measurements have been used to overcome some of the dif-ficulties with resolving the space-time ambiguity in measurements made froma single moving platform. While they help, they are expensive and there arestill limitations of measurements made from a moving platform. An area wherelaboratory experiments can contribute substantially is in the understanding ofthe effects of highly localized regions of strong spatial variability, such as thestrong gradients over ion or electron gyroscales associated with compressedplasmas discussed in this article. These phenomena can be scaled reasonablywell in the laboratory. The Space Chamber at the US Naval Research Labora- ehavior of Compressed Plasmas in Magnetic Fields 69
Fig. 38
NRL Space Physics Simulation chamber. Main chamber section (1.8 m by 5 m) ison the right. Source chamber (0.55 m by 2 m) is on the left. tory (NRL) is especially designed for understanding space plasma phenomena,such as the behavior of compressed plasmas.The NRL Space Physics Simulation Chamber (SPSC), shown in Fig. 38,consists of two sections that can be operated separately or in conjunction.The main chamber section is 1.8 m in diameter and 5 m long, while the sourcechamber section provides an additional 0.55 m-diameter, 2-m long experimen-tal volume. The steady-state magnetic field strength in the main and sourcechamber sections can be controlled up to 220 G and 750 G respectively, gener-ated by 12 independently controlled water-cooled magnets capable of shapingthe axial magnetic field. Each section has a separate plasma source. The mainchamber has a 1-m x 1-m hot filament plasma source capable of generatingplasmas with a range of density n ∼ − cm − , electron temperature T e ∼ . − T i ∼ .
05 eV. The source chamber has ahelicon source capable of generating 30-cm diameter plasmas with the follow-ing parameters: n ∼ − cm − , T e ∼ − T i ∼ . − ) 10 − ∼ − electron temp. (eV) ∼ . ∼ . − ∼ . . . ∼ . ∼ .
04 up to 750 G (SC)& 250 G (MC)plasma freq. (Hz) 10 − × − ion gyrofrequency (Hz) ∼
30 (O + ) ∼
60 (H + ) ∼ − (Ar + )electron gyrofrequency (Hz) ∼ ∼ − ω pe /Ω e ∼ . − ω/ν en > (cid:29) ∼ − β − − − − − − − Table 1
Comparison of plasma parameters in the ionosphere, the Radiation Belts (RB),and the NRL SPSC.
We discuss a few experiments performed in the NRL Space Chamber andelsewhere that were designed to understand the effects of strong velocity andpressure gradients typical of compressed plasmas. As discussed in Sec. 2, thelocalized electric field can be considered a surrogate for the global compression.Thus, by studying the plasma response to localized electric fields we can gleanthe physical processes that characterize a compressed plasma layer.5.1 Low Frequency Limit: Transverse Velocity GradientIn the 1970s, the NASA S3-3 satellite observed emissions around the ion cy-clotron frequency in uniform density plasma at auroral altitudes where spa-tially localized DC electric fields were large [50]. Kelly and Carlson [104] re-ported intense shear in plasma flow velocity at the edge of an auroral arcassociated with short wavelengths fluctuations, the origin of which was a mys-tery. They noted that, A velocity shear mechanism operating at wavelengthsshort in comparison with the shear scale length, such as those observed here,would be of significant geophysical importance. Kintner [105] described thedifficulty for exciting the current-driven ion cyclotron waves [82] in the lowerionosphere where the magnitude of the field-aligned current is usually belowthe threshold and yet bulk heating of ions suspected due to ion cyclotron wavesis detected.In addition to space observations, there were laboratory experiments, al-though unconnected with the space observations, reporting ion cyclotron wavescorrelated to localized transverse dc electric fields [106,107]. The generationmechanism of these ion cyclotron waves was not clear.These observations led to theoretical analysis at NRL, described in Sec.3.3.1, which suggested that the Doppler shift by a localized transverse electric ehavior of Compressed Plasmas in Magnetic Fields 71
Fig. 39
Segmented disk and biased multi-ringed electrode in the NRL SPSC. field could make the energy density of the ion cyclotron waves negative in theelectric field region while it is positive outside. A flow of energy between theregions with opposite signs of wave energy density can lead to an instability [52,47]. Because the necessary condition for instability is that the energy must flowfrom one region to another with opposite sign of energy density, the instabilityis essentially nonlocal. It was a promising mechanism for understanding anumber of mysterious observations in the auroral region including low altitudeion heating [108], which was a front burner issue of the time. So, its validationand detailed characterization in the laboratory became an important topic.Using a segmented disc electrode, shown in Fig. 39, in the West VirginiaUniversity Q-machine Amatucci et al. [109] showed that sub-threshold field-aligned current could support the ion cyclotron instability if a radially localizedstatic electric field produced by biasing the segments is introduced (see Fig.40). This explained the observation of ion cyclotron waves for sub thresholdcurrents in the auroral region noted by Kintner [105]. It was not possible toeliminate the axial current totally in the experiment because the inner seg-ment of the electrode was biased and drew electrons. Subsequently, Amatucci et al. [110] demonstrated that by increasing the magnitude of the transverseelectric field and virtually eliminating the axial current with biased ring elec-trodes (Fig. 39), the electrostatic ion cyclotron waves could be sustained bya sheared transverse flow alone. These experiments were later followed up by
Fig. 40
Threshold value of current density as a function of transverse, localized, dc electric(TLE) field strength. Current densities are normalized to the zero-TLE-strength value. Errorbars represent one standard deviation. Reproduced from Figure 2 of Amatucci et al. [109].
Tejero et al. [111] to confirm the electromagnetic IEDDI [53]. These waves,besides validating the theory, were shown to be efficient in ion heating [110] aswas expected [108]. The experiment also showed that the heating profile wasdistinct from the typical Joule heating [112] as shown in Fig. 41. The scalesize of the electric field, L , was greater than the ion gyroradius, ρ i , for theseexperiments.Characterization of the IEDDI in the laboratory was a significant contribu-tion because it clarified the role of localized electric fields in wave generationthereby validating the theory for the origin of these waves and led to numerousapplications to understand satellite observations [113,114,115,116]. In addi-tion, it successfully addressed a major issue in space plasmas, i.e. , ion heatingin the lower ionosphere necessary to initiate the out flow of the heavy grav-itationally bound oxygen ions observed deep inside the magnetosphere [117].These experiments became anchors for a comprehensive ionospheric heatingmodel [90] and inspired sounding rocket experiments to look for corroboratingsignatures in the ionosphere [118,113,119,120]. Subsequently, a comprehensivestatistical survey of satellite data confirmed the importance of static transverseelectric fields to wave generation in the ionosphere [121]. More importantly,these early laboratory experiments started a trend in simulating space plasma ehavior of Compressed Plasmas in Magnetic Fields 73 Fig. 41 (a) Perpendicular ion temperature T i /T i , (b) mode amplitude, (c) Doppler-shiftedmode frequency, and (d) transverse electric field strength plotted as a function of the normal-ized ionneutral collision frequency. A transition from a wave-heating regime ( ν in /Ω i < ν in /Ω i > .
7) is observed as the ionneutral collision frequency isincreased. Reproduced from Figure 3 of Amatucci et al. [112]. phenomena in the controlled environment of the laboratory for detailed char-acterization that helped in the interpretation of in situ data and develop adeeper understanding of the salient physics.5.2 Low Frequency Limit: Parallel Velocity GradientAnother intriguing issue in the ionosphere was the observations of low fre-quency ion acoustic-like waves [122] in the nearly isothermal ionosphere wherethe ion acoustic waves are expected to be ion Landau damped. The originof these low frequency waves became a much-debated issue. As discussed inSec. 3.3.4 Gavrishchaka et al. [75] showed that a spatial gradient in the mag-netic field aligned flow could drastically lower the threshold of the ion acousticwaves by moving the phase speed of the waves away from Landau resonance.In addition, Gavrishchaka et al. [76] also showed that higher frequency wavescan be triggered by spatial gradients in the parallel flow with multi-harmonicion cyclotron emissions. The magnitude of the gradient required for gener-ating either of these waves was very modest. These results could potentially explain a number of auroral observations [86,77] including the NASA FASTsatellite observation of multi-ion harmonic spectrum and spiky parallel electricfield structures [123]. Thus, validation of the Gavrishchaka et al. theory in thelaboratory became an important issue.In a series of Q-machine experiments with inhomogeneous magnetic fieldaligned flow at the University of Iowa [78,79] and West Virginia University[80,81] the existence of both the shear modified low frequency and the ioncyclotron frequency range fluctuations were confirmed and their signatures andproperties were studied. The experiments highlighted the critical role of thespatial gradient in the flow parallel to the magnetic field. A similar situationcan also arise in compressed plasmas in DFs as well as the plasma sheet-lobe interface, as discussed in Sec. 2.2.3. The laboratory validation of thetheory and the characterization of the instability increased the confidence inits application to other regions of space plasmas [124,125].Other low frequency waves due to parallel inhomogeneous flows with adensity gradient were investigated in laboratory experiments by Kaneko et al. [126,127]. They also theoretically analyzed the case and showed that driftwaves can be both destabilized and stabilized by velocity shear in the parallelion flow depending on the plasma conditions and shear strength in the parallelflow. Similar conclusions regarding the drift wave behavior in plasma withperpendicular flow shear was discussed by Gavrishachaka et al. , [49].5.3 Intermediate Frequency Limit: Transverse Velocity GradientAs described in Sec. 2.1, during geomagnetically active periods, global com-pression of the magnetosphere by the solar wind stretches the Earths mag-netotail and a pressure gradient builds up between the low-pressure lobe andthe high-pressure plasmasheet. The boundary between these regions exhibitsa complex structure, which includes thin layers of energetic electrons confinedto the outermost region of the plasmasheet [128,129]. Localized static elec-tric fields in the north-south direction are observed during crossings into theplasma sheet from the lobes [130,131] but their cause and effect was not known.Also, enhanced electrostatic and electromagnetic wave activity is detected atthe boundary layer [14,129,132,133].To understand the plasma sheet-lobe equilibrium properties, a kinetic de-scription of the boundary layer was developed by Romero et al. [10], as de-scribed in Sec. 2.1.1. It showed that with increasing activity level, as theboundary layer scale size approaches an ion gyrodiameter, an ambipolar elec-tric field develops across the magnetic field, which intensifies with the globalcompression. As shown in Sec. 3, for small enough L , ions effectively be-have as an unmagnetized species for intermediate scales ( Ω i < ω < Ω e and k ⊥ ρ i > > k ⊥ ρ e ) and an instability appears around the lower hybrid fre-quency. The wavelength of this instability scales as k ⊥ L ∼ L (cid:29) ρ e [61],which distinguishes it from the lower-hybrid-drift instability with k ⊥ ρ e ∼ ehavior of Compressed Plasmas in Magnetic Fields 75 electric fieldsmall diameter plasma sourcesmall source plasma potential bias voltage V bias Fig. 42 (top)Schematic of creating localized electric fields in laboratory experimentsadapted from Amatucci et al. [109]. On the right is a large plasma source. In front (tothe left in the figure) of the large plasma source is a blocking disk that prevents plasmafrom the large source to stream down the center of the chamber. On the left is a smallersource that can fill in plasma at the center. By biasing the end plates an electric field can becreated between the two plasmas. (bottom) Measured density vs radial position in the NRLSpace Physics Simulation Chamber for different filament current settings on the plasmasource illustrating the experimental control over the plasma density. (bottom) Reproducedfrom Fig. 3 of Amatucci et al. [134] scaling. Hence, laboratory validation and characterization of the EIH wavesdiscussed in Section 3.3.2 became an important topic.While the basic physics of the EIH instability was verified in Japan byMatsubrara and Tanikawa [135] using a segmented end plate to create thelocalized radial electric field and then in India by Santhosh Kumar et al. [136], their experimental geometry did not correspond to the reality of thelobe-plasma sheet system. The challenge was to produce the conditions of a
Fig. 43
Stack plot of the FFT Amplitude vs Frequency as the electric bias is increased(up in the figure) showing that the EIH wave power increases as the applied electric field isincreased. Reproduced from Fig. 8a of Amatucci et al. [134] stretched magnetotail in the lab where the dense plasma sheet is surroundedby tenuous lobe plasma as shown in Fig. 1a of Section 2.1. Amatucci et al. [134] introduced an innovative way to achieve this by using interpenetratingplasmas produced by independent sources with controllable plasma potentialsand densities sketched in Fig. 42. This set up was more representative of therealistic plasma sheet-lobe configuration with a boundary layer of scale sizeon the order or less than an ion gyroradius. The experiment demonstratedspontaneous generation of lower hybrid waves as shown in Fig. 43.Subsequently, DuBois et al. [137,91] used the Amatucci method in theAuburn University Auburn Linear Experiment for Instability Studies (ALEXIS)device and varied the magnetic field to scale the ion gyroradius from largerto smaller than the electric field scale size thereby effectively simulating thevariation of stress that characterizes the relaxation phase of a stressed mag-netotail. This showed the generation of a broadband emission starting fromthe lower hybrid frequency to less than ion cyclotron frequency differing by 5orders of magnitudes in a single experiment as shown in Fig. 44.The DuBois et al. experiment was a proof of principle of the theory [15]which had posited that a compressed boundary layer can relax through theemission of a hierarchy of electric field-driven waves starting from above theelectron gyrofrequency to much below the ion gyrofrequency and could be theprimary source for the observed broadband electrostatic noise. Tejero et al. [111] and Enloe et al. [138] have subsequently shown that the plasma com-pression can also produce electromagnetic emissions but the wave power isprimarily concentrated in the electrostatic regime [139], consistent with the in situ observations [133]. These laboratory experiments have elucidated thesubtler aspects of the magnetotail dynamics, which would be difficult to dis-cern from in situ measurements alone. They also inspired new experimental ehavior of Compressed Plasmas in Magnetic Fields 77
Fig. 44
The log of ω/Ω i is plotted as a function of the ratio ρ i /L which was varied exper-imentally by controlling the magnitude of the magnetic field in ALEXIS. Reproduced fromFig. 5 of DuBois et al. [91] research in the laboratory to understand the physics of the dipolarizationfronts. Besides academic interests, the practical goal of developing a deeper under-standing of space plasma processes is to improve the accuracy of space weatherforecasting. The challenge in a physics-based forecasting model is in account-ing for the physics at multiple scales in a global model. As discussed in thisarticle, spatiotemporal processes in the space plasma environment are multi-scale. It is not feasible to model the wide range of scales from first princi-ples, because of computational limitations and lack of detailed initial and orboundary conditions. Hence, success of simulations, forecasting, and interpre-tation of multi-scale spatiotemporal dynamics critically depends on a realisticformulation including the coupling of physical models describing processes onmicro- and macro scales. Small-scale kinetic processes could significantly influ-ence larger-scale dynamics. However, introduction of small-scale kinetic effectsas anomalous coefficients into larger-scale fluid simulations without runningsmall-scale simulations involves empirical adjustments of coupling parameters taking into account simulation stability and other considerations. Some at-tempts in magnetosphere-ionosphere coupling has been made based on thisconcept [140,141]. Similarly, one can use coarse-grain analogue models withjust a few main elements [142,143] whose characteristics are also inferred fromdeeper multi-scale physical models. Still such physics-based models may notbe accurate enough for certain practical applications. Recent developments inartificial intelligence (AI) and machine learning (ML) offers a new vista fordeeper understanding and forecasting in the space plasma environment.Alternatively, applied modeling of a wide range of complex systems in-cluding space weather forecasting are based on data-driven statistical andML approaches [144,145,146,147,148]. Such empirical approaches could offerpractical solutions with good accuracy given enough training data coveringkey regimes of the considered systems are available. However, performanceof standard ML approaches could quickly deteriorate with severe data lim-itations, high dimensionality and non-stationarity [149,150]. Domain-expertknowledge including physical models based on deeper understanding of theconsidered complex system, such as the kinetic processes discussed in this arti-cle, could play a key role in applications with severe incompleteness of trainingdata because of natural dimensionality reduction and usage of domain-specificconstraints [149,150,151]. Typical practical example of the domain knowledgeincorporation into ML solution is selection of model inputs and drivers usingphysics-based considerations [144,145,146,150]. This procedure of augmentingpurely data driven models with physics based models is a step towards gainingphysical insight into the system.The most successful modern ML frameworks such as deep learning (DL)based on deep neural networks (DNNs) and boosting-based ensemble learn-ing offer even more opportunities for efficient synergetic combination withdomain-expert knowledge [152,153,154,155,156,157,149,150]. First, similar tonatural sciences, both techniques actively use advantages of hierarchical dataand knowledge representations that are capable of crucial reduction of de-pendency on the training data size. This is achieved by layer-by-layer learningwith automated hierarchical feature discovery and dimensionality reduction inDNNs and the intrinsically hierarchical nature of boosting algorithms whereit builds a global-scale model at the first iteration and focuses on more de-tailed modeling of sub-populations, sub-scales and sub-regimes in subsequentiterations [152,153,154,155,156,158,149,150]. For example, in Section 2 weshowed that global compression leads to ambipolar effects on ion and electrongyroscales that generate spatially localized transverse electric fields. In Section3 we showed the linear plasma response to such electric fields, which are muchsmaller scale features. In Section 4 we showed the nonlinear evolution of theseelectric fields and ultimately their saturation to generate macroscopic measur-able features of the larger scale dynamics that satellites measure. These micro-macro coupling processes could be iteratively incorporated into global modelsto produce a much more comprehensive model of the space plasma dynam-ics than currently possible. Such hierarchical physics-based knowledge couldsignificantly improve accuracy in space weather forecasting capability. The ehavior of Compressed Plasmas in Magnetic Fields 79 described nature of these algorithms creates different channels for efficient in-tegration of many pieces of domain-expert knowledge including physics-basedmodels, scaling and constraints. For example, collection of simplified physicalmodels with a few adjustable empirical parameters, e.g. anomalous coefficientscapturing small-scale effects, could be used as base models in boosting algo-rithms to create ensemble of interpretable models with boosted accuracy andstability compared to a single model [149,150,151]. Alternatively, simplifiedphysical models capturing multi-scale effects in an approximate manner canbe used to generate large amounts of synthetic data for all possible regimes.Later actual data can be augmented by this synthetic data to allow a DLframework to discover robust representations that can be used to train orfine-tune DNNs or other ML models [150]. Synergetic combination of ML al-gorithms and physics-based models, such as those discussed in this article andglobal MHD models, could be especially useful for representation and detec-tion of rare events and regimes [159,160]. Further advancements in discoveryof stable and accurate hybrid solutions in complex systems modeling can beachieved by leveraging methods from computational topology which showedpromising results in a wide range of applications [161,162,163,160]. Until sucha time when global models can capture detailed physics at all scale sizes, suchhybrid modeling may be necessary for accurate space weather forecasting.
In this review article we have analyzed the behavior of compressed plasmasin a magnetic field, which is a configuration often encountered both in nat-ural and laboratory plasmas. Compression creates stress, or gradients, in thebackground plasma parameters. When the scale size of the gradient across themagnetic field becomes comparable to an ion gyrodiameter a self-consistentstatic electric field is generated due to ambipolar kinetic effects. This electricfield is highly inhomogeneous. Hence, the localized Doppler shift due to the E × B flow cannot be transformed away, which affects the dieletric propertiesof the plasma including the normal modes. In addition, it affects the individualparticle orbits as well as shears the mean flow velocities both transverse andalong the magnetic field. Velocity shear is a source of free energy for plasmafluctuations. Consequently, a compressed plasma system achieves a higher en-ergy state compared to its relaxed counterpart. The electric field gradient,and by causality the velocity shear, that develops scales with the magnitudeof compression. Thermodynamic properties compel the plasma to seek a lowerenergy state. In response, in a collisionless medium spontaneous generationof emissions follow that dissipate the velocity shear and returns the plasmato a relaxed lower energy state. This makes compressed plasmas to be activeregions with characteristic emissions. In the space environment, these regionsare relatively easy to detect and measure due to large plasma fluctuations.The spectral signature of the emissions is typically found to be broadband infrequency with power mostly concentrated in the electrostatic regime. Hence, they have often been referred to as the broadband electrostatic noise (BEN) inthe literature. But they are also accompanied by some electromagnetic compo-nent [133]. As we discussed in Secs. 3 and 4, the velocity shear has the uniqueability to produce such broadband signatures in which the power is mostlyin the electrostatic regime but with some electromagnetic power as well. Theintensity and bandwidth of the emissions, which scale as the velocity shear, isa diagnostic of the level of compression imposed on the plasma. This is evidentfrom in situ measurements in space plasmas where broadband emissions area hallmark of compressed plasmas found in boundary layers.Although we used the framework provided here to analyze natural plasmaprocesses, it is general and applicable to laboratory experiments as well asto active experiments in space. For example, compressed plasma layers canbe generated locally in the ionosphere by the ionization of exhausts or ef-fluents discharged from rockets [164] or by active chemical release experi-ments [165,166]. In the NASA sponsored Nickel Carbonyl Release Experiment(NICARE) [167] the introduction of electron capturing agents, such as CF3Br,SF6, Ni(CO)4, etc. , in the ionosphere created an electron depleted region inthe ionosphere surrounded by natural oxygen-electron plasma. This generateda boundary layer of positive ions, negative ions, and electron plasma withstrong spatial gradients in their densities. Experimental data indicated a largeenhancement of noise level concurrent with the formation of the negative ionplasma. This resulted in a situation similar to the natural boundary layers,discussed in Sec. 2, in which the negative ion population inside the electron-depleted region diminished to zero outside, while the electron population didthe opposite in a narrow boundary layer [165]. Quasi-neutrality between theelectron, negative ions, and positive oxygen ions led to a strong self-consistentelectrostatic potential in the boundary layer that separated the negative ionplasma from the ambient oxygen-electron plasma. Hybrid simulations showedthe formation of the boundary layer with a localized radial electric field inthe intermediate ( ρ i > L > ρ e ) scale size and spontaneous generation of sheardriven EIH waves that relaxed the boundary layer [168,169].Laboratory experiments of plasma expansion due to laser ablation, in whichthe laser front acts as a piston to compresses the plasma, shows interestingsimilarity with the physics of the dipolarization fronts we discussed in Sec.2. Dipolarization fronts, characterized by a pressure gradient over a narrowplasma layer comparable to an ion gyroradius, are created in the aftermath ofmagnetic reconnection when a stretched magnetic field snaps back towards adipolar configuration. In a laser ablated plasma expansion across an externalmagnetic field similar density gradient structures with scale size comparable toan ion gyroradius accompanied with a cross-magnetic field flow are observed[170]. Due to the piston-like action of the laser front both ions and electronsmove with nearly the same speed across the magnetic field and hence thecross field current is negligible but there is a gradient in the intermediate scalesize in the plasma flows that are generated. Furthermore, as in the dipolar-ization front, waves around the lower hybrid frequency are seen, which werethought to be the lower hybrid drift waves [70] because of their association ehavior of Compressed Plasmas in Magnetic Fields 81 with the density gradient just as in the dipolarization front case. However, thewavelength of the lower hybrid waves was found to be much longer than theelectron gyroradius and comparable to the scale size of the cross-field flow.As we discussed in section 3, the long wavelength signature is not consistentwith the lower hybrid drift waves but similar to that expected from the EIHwaves, which depend on the gradient in the flow and not on a cross-field cur-rent. Long wavelengths are generated by nonlinear vortex merging (see Sec.4). Peyser et al. [171] analyzed a number of experimental cases and comparedthe data with theoretical models. They concluded that the waves were likelyto be the EIH waves; for similar reasons argued for the origin of the emissionsin a dipolarization front in Secs 3 and 4. However, due to the inability in theexperiment to measure the details of the parameters, unambiguous character-ization of the origin of the waves in laser ablated plasma jets was not possible.More recent laser ablation experiments have shown the generation of wavesaround the lower hybrid frequency [172] and their origin is still an open issue.While the plasma response to velocity shears in both perpendicular andparallel flows has been studied separately their combined effect has not beenanalyzed. In nature it is likely that that the velocity shear is in an arbitrarydirection due to magnetic field geometry. In Sec. 2.2.3 we showed in a simplecase how this may be possible. But in that case the scale size of the magneticfield variation was orders of magnitude larger than the electric field variation,which allowed us to cleanly separate the two scale sizes and study them indi-vidually. Effectively, this reduced the problem to one dimension. This may notalways be possible in other instances in nature or in laboratory. In general,the linear response will involve two or three dimensional eigenvalue conditions,which are more difficult to solve. There have been some attempts to addressthe combined effect of parallel and transverse velocity shear [ e.g. , Kaneko et al. [173]] but this topic remains an interesting area of research and deservesfurther attention. In addition, manifestation of the velocity shear effect in amulti-species plasma, which is likely to prevail in some regions in space, is an-other interesting future research topic since shear effect is mass dependent andhence affects different species differently, which introduces relative differencesin properties between species [174].A common feature in the nonlinear evolution of a compressed plasma sys-tem is that spontaneous generation of shear driven waves relaxes the velocitygradient generated by the compression so that a balance can be achieved. Thisbalance, or the steady state, defines the electromagnetic plasma environment.In addition, the shear driven waves contribute to viscosity and resistivity asfeedback to the global physics and modify the meso scale plasma features.Thus, the union of the small and large scale physics is the reality that a satel-lite measures, which underscores the importance of understanding both thesmall and large scale processes and the coupling between them as we have at-tempted to show through natural examples in the earths neighborhood plasmaenvironment. Acknowledgements
This work was partially supported by the Naval Research Laboratorybase program and NASA grant NNH17AE70I. Special thanks to Valeriy Gavrishchaka forreading the manuscript and for valuable discussions.
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