A family of Vlasov-Maxwell equilibrium distribution functions describing a transition from the Harris sheet to the force-free Harris sheet
UUnder consideration for publication in J. Plasma Phys. A family of Vlasov-Maxwell equilibriumdistribution functions describing a transitionfrom the Harris sheet to the force-free Harrissheet
T. Neukirch † , F. Wilson and O. Allanson School of Mathematics and Statistics, University of St Andrews, St Andrews, UK, KY16 9SS Space and Atmospheric Electricity Group, Department of Meteorology, University ofReading, Reading, RG6 6BB, UK.(Received xx; revised xx; accepted xx)
We discuss a family of Vlasov-Maxwell equilibrium distribution functions for currentsheet equilibria that are intermediate cases between the Harris sheet and the force-free (or modified) Harris sheet. These equilibrium distribution functions have potentialapplications to space and astrophysical plasmas. The existence of these distributionfunction had been briefly discussed in by Harrison & Neukirch (2009 a ), but here itis shown that their approach runs into problems in the limit where the guide field goesto zero. The nature of this problem will be discussed and an alternative approach willbe suggested that avoids the problem. This is achieved by considering a slight variationof the magnetic field profile, which allows a smooth transition between the Harris andforce-free Harris sheet cases.
1. Introduction
Current sheets are important for the structure and dynamical behaviour of manyplasma systems. In space and astrophysical plasmas current sheets play a crucial rolein magnetic activity processes by, for example, aiding the release of magnetic energyby magnetic reconnection. Current sheet equilibria are often used as starting points forstudying the dynamic behaviour of plasmas in, e.g., the solar atmosphere, the solar windand planetary magnetospheres.Many astrophysical plasmas can be described as collisionless and in this case therelevant equilibria are solutions of the steady-state Vlasov-Maxwell (VM) equations (e.g.Schindler 2007). Since current sheets are strongly localised in space, they can often bewell approximated by one-dimensional (1D) models (see, e.g. Roth et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. force-free , and satisfy † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] M a y T. Neukirch, F. Wilson and O. Allanson the condition j × B = 0 , i.e. the current density and magnetic field are aligned with eachother.The Harris sheet magnetic field is kept in force-balance by a pressure gradient, but onecan also keep the system in a macroscopic force balance by adding a non-uniform guidefield to the system while the plasma pressure is constant. The resulting configurationis often called the force-free Harris sheet. Equilibrium distribution functions for thisconfiguration have been found, for example, by Harrison & Neukirch (2009 a ); Neukirch et al. (2009); Wilson & Neukirch (2011); Abraham-Shrauner (2013); Kolotkov et al. (2015); Dorville et al. (2015); Allanson et al. (2015, 2016); Wilson et al. (2017); Wilson et al. (2018); Neukirch et al. (2020) (for further references on force-free Vlasov-Maxwellequilibria, see e.g. Moratz & Richter 1966; Sestero 1967; Channell 1976; Correa-Restrepo& Pfirsch 1993; Attico & Pegoraro 1999; Bobrova et al. b ;Vasko et al. a ) (e.g. Wilson et al. et al. et al. et al. et al. et al. et al. et al. et al. a ) also discussed the case of collisionlesscurrent sheets that are intermediate cases between the Harris sheet and the force-free Harris sheet, i.e. cases for which the macroscopic force-balance is provided by acombination of the plasma pressure gradient and the gradient of the magnetic pressurecomponent provided by the non-uniform guide field. These equilibria and their DFs self-consistently describe the transition from the Harris sheet to the force-free Harris sheet (orvice versa), but have so far not been studied in any detail. Hence, in this paper we presentan investigation of these collisionless current sheet equilibria. As this investigation willshow, there are actually some problems with the DFs presented in Harrison & Neukirch(2009 a ), which limit their usefulness in practice. To circumvent these issues, we presenta family of slightly modified magnetic field profiles and corresponding DFs which avoidthese problems, but still describe a transition between the Harris sheet and force-freeHarris sheet as limiting cases.We remark that in this paper we focus on a case in which the plasma temperatureis uniform across the current sheet (i.e. an isothermal case). Distribution functions fornon-isothermal force-free current sheets have been found by, e.g., Kolotkov et al. (2015),Wilson et al. (2017), and Neukirch et al. (2020). In principle the analysis carried out herecould be generalised to these non-isothermal cases.The paper is structured as follows; in Section 2, we briefly discuss the macroscopicequilibria of the Harris sheet, the force-free Harris sheet and the intermediate cases.We then discuss the corresponding VM equilibrium distribution functions as given byHarrison & Neukirch (2009 a ) in Section 3 and illustrate the problem associated with theintermediate cases in the limit when the guide field amplitude tends to zero. In Section4, we present a modified magnetic field model, which allows us to avoid these problemswith the distribution function. We close with our summary and conclusions in Section 5.
2. The macroscopic picture: Harris sheet, force-free Harris sheet andintermediate cases
The macroscopic force balance for one-dimensional (1D) collisionless current sheetequilibria, with spatial variation only in the z -direction, is determined by (e.g. Mynick ntermediate VM equilibria et al. et al. ddz (cid:20) B x ( z ) + B y ( z ) µ + P zz ( z ) (cid:21) = 0 . (2.1)For collisionless equilibria, we are usually dealing with a pressure tensor and P zz is theonly component of the pressure tensor which contributes to the force balance equation.In this paper we focus on the family of equilibria defined by B ( z ) = B (cid:18) tanh( z/L ) , B y B cosh( z/L ) , (cid:19) , (2.2) P zz ( z ) = B − B y µ cosh ( z/L ) + P b , (2.3)where L represents the half-thickness of the current sheet, and P b (cid:62) is a constantbackground pressure. For completeness, we mention that the current density is given by j ( z ) = B µ L (cid:18) B y sinh( z/L ) B cosh ( z/L ) , ( z/L ) , (cid:19) . (2.4)The case B y = 0 gives the Harris sheet (Harris 1962), which is a widely used 1D VMequilibrium in, e.g., reconnection studies, Often a constant guide field component is addedto the Harris sheet field, which is not included in our magnetic field model here. When B y = B , we obtain the force-free Harris sheet, for which both the pressure P zz = P b and the magnetic pressure ( B x + B y ) / µ = B / µ are constant. For < B y < B weget intermediate cases between the Harris sheet and the force-free Harris sheet. Figure 1shows the magnetic field, pressure, and current density profiles for the Harris sheet, twointermediate cases, and force-free Harris sheet. Note that in the figure we have set thebackground pressure P b (measured in units of B / (2 µ ) ) to B y /B + 0 . .At the macroscopic level discussed so far, there is no problem with varying B y /B and in particular with letting this ratio go to . This changes, however, when we considerthe microscopic picture.
3. The microscopic picture
1D Vlasov-Maxwell equilibria
We assume a 1D Cartesian setup, in which all quantities depend only on the z -coordinate, and consider magnetic field profiles of the form B = ( B x , B y , , for which B = ∇ × A (for vector potential A = ( A x , A y , ). In this paper we will always imposeconditions on the microscopic parameters of the DFs such that the electric potential φ (and hence the electric field) vanishes (this can always be achieved for the cases wediscuss here, see e.g. Neukirch et al. f s , are functions of the particle energy, H s = m s ( v x + v y + v z ) / ,and the x - and y -components of the canonical momentum, p s = m s v + q s A (for m s themass and q s the charge of species s , respectively), since these are known constants ofmotion for a time-independent system with spatial invariance in the x - and y -directions.Under the assumptions described above, the VM equations reduce to Ampère’s law inthe form d A x d z = − µ ∂P zz ∂A x (3.1) d A y d z = − µ ∂P zz ∂A y , (3.2) T. Neukirch, F. Wilson and O. Allanson (a) (b)(c) (d)
Figure 1.
Magnetic field, pressure, and current density profiles for (a) the Harris sheet, (b) and(c) intermediate cases with B y /B ≈ . and ≈ . , respectively, and (d) the force-free Harrissheet. The background pressure P b (measured in units of B / (2 µ ) ) has been set to B y /B +0 . in each case. where P zz is the only component of the pressure tensor that plays a role in the force-balance of the 1D equilibrium, defined by P zz ( A x , A y ) = (cid:88) s m s (cid:90) v z f s ( H s , p xs , p ys )d v. (3.3)For a specified magnetic field profile, therefore, one needs to determine P zz ( A x , A y ) such that the vector potential associated with the given magnetic field is a solution ofAmpère’s law. Regarding Equation (3.3) as an integral equation for f s and solving it,will give DFs that self-consistently reproduce this macroscopic field profile(e.g. Channell1976; Alpers 1969; Mottez 2003). For some examples of the application of this approach,see Harrison & Neukirch (2009 b , a ); Neukirch et al. (2009); Wilson & Neukirch (2011);Abraham-Shrauner (2013); Kolotkov et al. (2015); Allanson et al. (2015, 2016); Wilson et al. (2017); Wilson et al. (2018).3.2. The distribution functions
Harrison & Neukirch (2009 a ) used Channell’s method (Channell 1976) to find the ntermediate VM equilibria f s ( H s , p xs , p ys ) = n s (cid:0) √ πv th,s (cid:1) e − β s H s (cid:2) e β s u ys p ys + a s cos( β s u xs p xs ) + b s (cid:3) . (3.4)where n s is a typical particle density for species s , β s = ( k b T s ) − is the usual inversetemperature parameter and v th,s = k b T s /m s = ( m s β s ) − is the square of the thermalvelocity of species s . As discussed in detail in, for example, Neukirch et al. (2009), theadditional parameters a s , b s , u xs and u ys have to satisfy further constraints to (i) havea positive DF ( b s > a s (cid:62) ), (ii) guarantee that the electric potential vanishes, and(iii) ensure that the magnetic vector potential associated with the given macroscopicmagnetic field is a solution of Ampère’s law.The DF (3.4) is the sum of the Harris sheet DF (Harris 1962) and an additional part,depending on H s and p x,s f s ( H s , p xs , p ys ) = f s,Harris ( H s , p ys )+ n s (cid:0) √ πv th,s (cid:1) e − β s H s ( a s cos( β s u xs p xs ) + b s ) , (3.5)where the Harris sheet DF is given by f s,Harris ( H s , p ys ) = n s (cid:0) √ πv th,s (cid:1) e − β s ( H s − u ys p ys ) . (3.6)Harrison & Neukirch (2009 a ) pointed out that by varying a s the DF (3.4) can in principledescribe all the intermediate cases between the force-free and Harris cases and it looksas if in the limit a s → one should recover the Harris sheet DF.However, if one looks more carefully one finds that the parameter a s has to satisfy therelation (similary to e.g. Neukirch et al. a s = B y B exp (cid:32) u xs v th,s (cid:33) exp (cid:32) u ys v th,s (cid:33) , (3.7)where u xs = 4 B y β s q s L . (3.8)This particular form for a s results from the consistency relations that the distributionfunctions have to satisfy for the electric potential calculated from the quasi-neutralitycondition to vanish identically (this is a pre-requisite for being able to apply the methodby Channell (1976)). Hence, in the limit B y → a s does not go to zero, but to ∞ , whichis unacceptable for the distribution function. For finite B y , a s is finite but it increasesrapidly as a function of B y .This leads to further unwanted properties of the DF, which we illustrate in Figures 2and 3. Figure 2 shows the DF as a function of v x /v th,s , for z/L = 0 . and v y = v z = 0 ,and how it changes as B y /B decreases from . to . . The maximum values of theDFs in Figure 2 are normalised to unity (i.e. we have divided the DFs by their maximumvalue for a given ratio B y /B ). As one can clearly see in Figure 2 the DF develops moreand more maxima and minima in the v x -direction, due to the dominance of the cosineterm in the distribution function caused by the increase in a s . It must be suspected thatthis filamentation in velocity space might lead to instabilities. We also point out thatthe parameter b s has to increase as well so that b s > a s to keep the DF positive (in theplots we have used b s = 1 . a s ). Figure 3 shows on a logarithmic scale how the maximumvalue of the DF (for the fixed values of z , v y and v z ) increases dramatically as B y /B decreases. T. Neukirch, F. Wilson and O. Allanson (a) (b)(c) (d)
Figure 2.
Variation of the DF with v x /v th,s for four different values of B y /B : panel (a) B y /B = 1 . , panel (b) B y /B = 0 . , panel (c) B y /B = 0 . , panel (d) B y /B = 0 . .Here, each DF has been normalised by its maximum value and we have chosen z/L = 0 . and v y = v z = 0 . Figure 3.
Variation of the DF maximum with B y /B . ntermediate VM equilibria B y /B → does not existand that hence there is no smooth transition to the Harris sheet DF, but that the familyof DFs will not be very useful even for finite, but small values of the ratio B y /B . Thequestion that arises is: can one find a family of VM equilibrium DFs which provides asmooth transition from the force-free Harris sheet to the Harris sheet?
4. Alternative intermediate cases
In this section, we will consider an alternative magnetic field profile to that in Eq.(2.2), of the form B ( z ) = B (cid:18) tanh( z/L ) , λ cosh( λz/L ) , (cid:19) , (4.1)where we have defined the abbreviation λ = B y B . (4.2)A similar, albeit not totally identical magnetic field profile has previously been used byHuang et al. (2017) to study instabilities using this type of collisionless current sheet. Wewill show that this magnetic field profile can be used to consistently describe a transitionfrom the Harris sheet to the force-free Harris sheet.For completeness we here also state the current density and zz -component if thepressure tensor associated with this field, which are given by j ( z ) = B µ L (cid:18) λ sinh( λz/L )cosh ( λz/L ) , ( z/L ) , (cid:19) , (4.3) P zz ( z ) = B µ (cid:20) ( z/L ) + λ (cid:18) − ( λz/L ) (cid:19)(cid:21) + P b , (4.4)respectively, where P b (cid:62) is a constant background pressure. We remark that we havewritten the non-background part of the pressure in such a way that it is always positive,regardless of the value of the positive constant P b .For λ = 0 , the magnetic field (4.1) becomes the Harris sheet field, and for λ = 1 itbecomes the force-free Harris sheet field. Setting P b = P b + B / µ in Eq. (2.3) for λ = 1 ( B y = B ) will make the two pressure functions equal in that case. The range < λ < can be thought of as describing intermediate fields between the Harris and force-freeHarris sheets, although the guide field and pressure profile deviates from the previousintermediate cases. Figure 4 shows magnetic field, pressure and current density profilesfor the Harris sheet ( λ = 0 . ), two intermediate cases with λ = 0 . and λ = 0 . andthe force-free Harris sheet ( λ = 1 . ). The background pressure has been chosen in thesame way as for Figure 1. The only obvious difference to the plots shown in Figure 1 arethe slight dips in the pressure profile (local minima) at the edges of the current sheet.On comparison with Figure 1, we also see that decreasing λ results in a widening of the B y profile, due to the λ factor inside the [cosh( λz/L )] − in the y -component of Eq. (4.1).The amplitude of B y decreases in the same way as in the other case as λ decreases, andeventually heads to zero as λ → .It is straightforward to show that this macroscopic magnetic field profile is consistentwith the DF in equation (3.4). One could suspect that this leads to the same problemwith the limit B y → as before, but when checking the constraints on the parameters T. Neukirch, F. Wilson and O. Allanson (a) (b)(c) (d)
Figure 4.
Magnetic field, pressure, and current density for the alternative intermediate caseswith (a) λ = 0 . , (b) λ = 0 . , (c) λ = 0 . and (d) λ = 1 . . The only noticeable changecompared to the profiles shown in Figure 1 are the slight dips (minima) in the pressure profileat the edge of the current sheet. The changes in the B y and j x profiles are not immediatelyobvious without direct comparison. of the DF one finds that while one still has a s = λ exp (cid:32) u xs v th,s (cid:33) exp (cid:32) u ys v th,s (cid:33) , (4.5)as before, the condition for u xs has changed to u xs = 4 λ B y β s q s L = 4 B β s q s L , (4.6)which no longer varies with B y . Therefore, in this case we indeed find that λ → implies a s → , as desired.It is, however, prudent to also have a look at the DFs and their maximum value as λ → . We show plots of the variation of the DF with v x (for z/L = 0 . and v y = v z = 0 )in Figure 5, for decreasing values of λ . For this case we can actually take the limit λ → without any problem (see panel (d)). As in Figure 2, we have normalised each DF toits maximum value. The variation of this maximum value with decreasing λ is shownin Figure 6. For this case the maximum of the DF actually decreases as λ and hence B y decreases. With a relatively simple modification of the magnetic field profile we havemanaged to eliminate the singular limit. ntermediate VM equilibria (a) (b)(c) (d) Figure 5.
Variation of the DF with v x /v th,s for different values of λ = B y /B . Here, the DFhas been normalised by its maximum value and we have chosen z/L = 0 . and v y = v z = 0 . Asone can see there is very little change as λ decreases.
5. Summary and Conclusions
In this paper, we have discussed collisionless current sheet equilibria that are inter-mediate cases between the Harris sheet (current density perpendicular to magnetic fielddirection) and the force-free Harris sheet (current density exactly parallel to the magneticfield direction). Such a family of Vlasov-Maxwell equilibrium DF had already been brieflymentioned in Harrison & Neukirch (2009 a ). However, as the more detailed investigationpresented in this paper shows, this family of DFs is of limited usefulness due to the factthat first of all the limit of the guide field amplitude B y → is singular in the sensethat the maximum of the DF tends to ∞ and that with decreasing B y the velocityspace structure of the DF in the v x direction becomes more and more filamentary. Weproposed an alternative family of intermediate collisionless current sheet equilibria witha magnetic guide field that has a slightly modified spatial structure. Formally, the DFsassociated with this magnetic field remain the same, but the constraints imposed on theDF parameters by the self-consistency condition now allow the maximum value of theDFs to remain not only finite, but at a reasonable level as B y → .We consider it important both from a theoretical and from a modelling/observationalpoint-of-view that reasonable self-consistent equilibria of collisionless current sheets areavailable not only for the two limiting cases of force-free Harris sheet and normal Harris0 T. Neukirch, F. Wilson and O. Allanson
Figure 6.
Variation of the maximum of the DF with B y /B for the alternative magnetic fieldprofile. As one can see the maximum decreases with decreasing λ and it does not diverge in thelimit λ → . In contrast to Figure 3 here a linear scale can be used for the plot. sheet. While some observations can be explained by, for example, force-free current sheetmodels (e.g. Panov et al. et al. b , a ; Neukirch et al. REFERENCESAbraham-Shrauner, B.
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