Self-similar analytical model of the plasma expansion in a magnetic field
H. B. Nersisyan, K. A. Sargsyan, D. A. Osipyan, M. V. Sargsyan, H. H. Matevosyan
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov Self-similar analytical model of the plasma expansion in amagnetic field
H B Nersisyan , , K A Sargsyan , D A Osipyan , M V Sargsyan andH H Matevosyan Division of Theoretical Physics, Institute of Radiophysics and Electronics, 0203 Ashtarak,Armenia Centre of Strong Fields Physics, Yerevan State University, Alex Manoogian str. 1, 0025Yerevan, ArmeniaE-mail: [email protected]
Abstract.
The study of hot plasma expansion in a magnetic field is of interest for manyastrophysical applications. In order to observe this process in laboratory, an experiment isproposed in which an ultrashort laser pulse produces a high-temperature plasma by irradiationof a small target. In this paper an analytical model is proposed for an expanding plasma cloudin an external dipole or homogeneous magnetic field. The model is based on the self-similarsolution of a similar problem which deals with sudden expansion of spherical plasma intoa vacuum without ambient magnetic field. The expansion characteristics of the plasma anddeceleration caused by the magnetic field are examined analytically. The results obtained canbe used in treating experimental and simulation data, and many phenomena of astrophysicaland laboratory significance.PACS numbers: 03.50.De, 41.20.Gz, 52.30.-q
Submitted to:
Phys. Scr.
1. Introduction
The problem of sudden expansion of hot plasma into a vacuum in the presence of an externalmagnetic field has been intensively studied in the mid-1960s in connection with the high-altitude nuclear explosions. It has also been discussed in the analysis of many astrophysicaland laboratory applications (see, e.g., [1, 2] and references therein). Such kind of processesarise during the dynamics of solar flares and flow of the solar wind around the Earth’smagnetosphere, in active experiments with plasma clouds in space, and in the course ofinterpreting a number of astrophysical observations [1–5]. Researches on this problem areof considerable interest in connection with the experiments on controlled thermonuclearfusion [6] (a review [1] summarizes research in this area over the past four decades).The expanding plasma pushes magnetic field out which is a consequence of magneticflux conservation. Due to the plasma expansion the surface of the given magnetic flux tubeinside plasma increases. The conservation of the magnetic flux through the surface of the tubeleads to a decrease of the local magnetic field and the formation of the diamagnetic cavity.Thus even if a magnetic field is nonzero initially inside the plasma it tends to zero duringplasma expansion. In addition, the plasma is shielded from the penetration of the external elf-similar analytical model of the plasma expansion in a magnetic field
2. Theoretical model
Usually the motion of the expanding plasma boundary is approximated as the motion withconstant velocity (uniform expansion). In the present study a quantitative analysis of plasmadynamics is developed on the basis of one-dimensional spherical radial model. Within thescope of this analysis the initial stage of plasma acceleration, later stage of deceleration andthe process of stopping at the point of maximum expansion are examined.We consider a collisionless magnetized plasma expanding into vacuum. The relevantequations governing the expansion are those of ideal MHD [15] assuming that thecharacteristic length scales for the plasma flow are much larger than the Debye length andthe Larmor radius of the ions. Thus ∂ρ∂t + ∇ · ( ρ v ) = 0 ,ρ (cid:20) ∂ v ∂t + ( v · ∇ ) v (cid:21) = 14 π [ ∇ × H ] × H − ∇ P,∂ H ∂t = ∇ × [ v × H ] , ∇ · H = 0 , (1) elf-similar analytical model of the plasma expansion in a magnetic field ρ , v , H , and P are the mass density, the velocity, the magnetic field, and the pressurerespectively. These equations must be accompanied by the adiabatic equation of state P ∼ ρ γ ( γ = C p /C V > is the adiabatic index, C p and C V are the heat capacities at constantpressure and constant volume, respectively) and the equation of conservation of entropy,which expresses the fact that the plasma dynamics is adiabatic in the absence of dissipation.Using the thermodynamic relation between entropy, pressure, and internal energy as well asequation (1) the equation for pressure reads [16] ∂P∂t + ( v · ∇ ) P + γP ( ∇ · v ) = 0 . (2)The equation of conservation of energy is derived from the system of equations (1) andis given by [15] ∂ε∂t + ∇ · Q = 0 , (3)where ε and Q are the energy and energy flux densities, respectively, with ε = ρv Pγ − H π , Q = v (cid:18) ρv γPγ − (cid:19) + 14 π [ H × [ v × H ]] . (4)Here the energy ε consists of the kinetic (first term), the internal (second term) and themagnetic field (third term) energies. The last term in the energy flux density Q represents thePoynting vector (let us recall that in ideal MHD the electric field is given by E = − c [ v × H ] ).With the theoretical basis presented so far, we now take up the main topic of this paper.This is to study the dynamics of the plasma boundary expanding in an ambient magnetic field.Consider the magnetic dipole p and a plasma spherical cloud with radius a ( t ) located at theorigin of the coordinate system. The dipole is placed in the position r from the center ofthe plasma cloud ( a ( t ) < r ). The orientation of the dipole is given by the angle θ p betweenthe vectors p and r . We denote the strength of the magnetic field of the dipole by H ( r ) .The energy, which is transferred from plasma to electromagnetic field is the mechanical workperformed by the plasma on the external magnetic pressure H ( r ) / π . Taking into accountthis effect and the energy conservation (3) integrated over the spherical plasma volume Ω p (with r a ( t ) ) the equation of balance of plasma energy is as follows: πγ − Z a ( t )0 P ( r, t ) r dr + 2 π Z a ( t )0 ρ ( r, t ) v ( r, t ) r dr (5) + Z Ω H ( r )8 π d r = W , where Ω is the volume of the spherical shell a r a ( t ) , a = a (0) ( a ( t ) > a ) and W are the initial radius and energy of the plasma. When the plasma cloud is introducedinto a background magnetic field, the plasma expands and excludes the background magneticfield to form a magnetic cavity. The magnetic energy of the dipole in the excluded volumeis represented by the last term in equation (5). Initial plasma velocity is supposed to be v ( r,
0) = v m ( r/a ) at r a and v ( r,
0) = 0 at r > a , where v m is the initial velocity ofthe plasma boundary ( v m = ˙ a (0) ).The obtained energy balance equation can be effectively used if profiles of velocity v ( r, t ) , pressure P ( r, t ) , and mass density ρ ( r, t ) are known functions of the plasma radius a ( t ) . We will take these dependencies from the solution of a similar problem which deals withsudden expansion of spherical plasma into a vacuum without ambient magnetic field [16]. elf-similar analytical model of the plasma expansion in a magnetic field r = a . The self-similar solutions arecharacterized by a velocity distribution linearly dependent on r (see, e.g., [16]). At r a ( t ) v ( r, t ) = r ˙ a ( t ) a ( t ) , (6)where unknown a ( t ) is the radius of sharp plasma boundary while ˙ a ( t ) is the velocity of theboundary. The specification of the mass density profile at r a ( t ) is given by ρ ( r, t ) = Γ (cid:0) + q (cid:1) π / Γ (1 + q ) Ma ( t ) (cid:20) − r a ( t ) (cid:21) q (7)and equation (6) for the velocity, automatically satisfies the continuity equation in (1) for anarbitrary function a ( t ) and for an arbitrary parameter q . Here M = const is the total mass ofplasma cloud and Γ( z ) is the Euler function. Substitution of v ( r, t ) into the entropy equation(2) gives at r a ( t ) the following solution for the pressure P ( r, t ) = p max (cid:20) a a ( t ) (cid:21) γ (cid:20) − r a ( t ) (cid:21) s , (8)where s is an arbitrary parameter and p max is the thermal pressure at the center of the sphericalplasma cloud at t = 0 . In addition the quantities v ( r, t ) , ρ ( r, t ) and P ( r, t ) vanish at r > a ( t ) , v ( r, t ) = ρ ( r, t ) = P ( r, t ) = 0 . Substituting (7) and (8) into the fluid equation of motion (1)and neglecting the term involving the magnetic field H yields a second-order differentialequation governing the motion of the plasma boundary a ( t ) . The problem considered isnot isentropic in general except the case when s = qγ . In the latter case of the isentropicexpansion the equation of state is given by P ρ − γ = const . Throughout in this paper we willassume that q > and s > . The self-similar solution given by equations (6)-(8) has beenconsidered by many authors. We refer to [16] (see also the references therein) for details andfor some applications of this solution.There are several limitations and shortcomings of the self-similar model consideredabove. It should be emphasized that in general the plasma expansion process with and withoutambient magnetic field is not self-similar since there is a characteristic length scale, that is,the initial radius of the plasma a . However, during the later stage (when a ( t ) ≫ a ) theplasma expansion asymptotically approaches to the self-similar regime since the role of theradius a becomes less important. This property is supported by the asymptotic solution ofthe system of partial differential equations (1) and (2) [16, 17] as well as by the numericalsimulations [16] for the free expansion (i.e. at H = 0 ). In addition, it has been shown thatat the beginning stage of the expansion the deviation of the solution (6)-(8) from non-similarone is small and is additionally reduced due to the spatial average in the equation of energybalance (5).Equations (6)-(8) are an exact solution in the case of expansion into a vacuum withoutmagnetic field. However, equation (8) does not satisfy the boundary condition, P ( a ( t ) , t ) = H / π , which is imposed in the case of expansion into an ambient magnetic field. Therefore,in the present case with nonzero magnetic field for the validity of the equations (6)-(8) mostcritical is the domain close to the boundary of the plasma, r ≃ a ( t ) . On the other handif the magnetic pressure is smaller than the plasma average pressure ¯ P , P mag ≪ ¯ P , thedifference between the exact solution in the magnetic field and free expansion model issmall and is localized in a narrow area near the surface of the cloud. These deviations are elf-similar analytical model of the plasma expansion in a magnetic field a ( t ) /a ≫ . Average plasmapressure drops significantly and plays no role in energy balance equation (5) during this stage.In accordance with the above boundary condition local pressure near the plasma edge mustbe equal to the magnetic pressure outside. It causes deviation from the profile equation (7)and accumulation of plasma in this area. This is confirmed independently by the numericalsimulations [18]. In the limiting case when all plasma is localized near the front, one canexpect an increase of the kinetic energy and longer stage of plasma deceleration as comparedwith the self-similar expansion model.Another critical domain for the violation of the self-similar solution (6)-(8) is the finalstage of expansion when the plasma is fully stopped by the magnetic field. As mentionedabove the average plasma pressure is strongly reduced compared to the magnetic pressure andthe equations (6)-(8) clearly become invalid in this case. However, if the critical time interval ∆ t , where ¯ P < P mag , is much smaller than the typical time scale of the plasma flow (up tothe full stop) the contribution of this interval to the overall plasma dynamics is negligible andthe use of the self-similar solution is justified.In the case of dipole magnetic field the volume integral in the last term of equation (5)has been evaluated in [10]. The result reads Z Ω H ( r )8 π d r = p r [ Q ( ηx ( t )) − Q ( η )] , (9)where η = a /r < , x ( t ) = a ( t ) /a (note that a ( t ) /r = ηx ( t ) < ), and Q ( η ) = 1(1 − η ) (cid:2) η (cid:0) − η (cid:1) (cid:0) θ p − (cid:1) (10) +8 η (cid:0) θ p (cid:1)(cid:3) − θ p −
12 ln 1 + η − η . Substituting equations (6)-(8) into (5) and integrating over r yields first-order differentialequation for a ( t ) , which already satisfies initial condition ˙ a (0) = v m , ˙ x ( τ ) + βx γ − + α [ Q ( ηx ( τ )) − Q ( η )] = 1 . (11)Here two dimensionless quantities are introduced α = p W r , β = π / p max a ( γ − W Γ(1 + s )Γ( + s ) < (12)which determine the magnetic and the thermal energies, respectively, in terms of the totalinitial energy W . The latter is easily obtained from equation (5) and reads W = 3 M v m q ) + π / p max a γ − s )Γ( + s ) . (13)From equations (12) and (13) it is seen that β < . New dimensionless variables areintroduced as follows: x ( τ ) = a ( t ) /a , τ = t/t , t = a /u m , where u m = [2(5 +2 q ) W / M ] / is the velocity of plasma expansion, achieved asymptotically at t → ∞ in thecase of expansion into a vacuum without magnetic field (i.e. at α = 0 ).The total energy of the plasma cloud at time t is obtained from equation (11) W ( t ) = W − p r [ Q ( ηx ( t )) − Q ( η )] . (14) elf-similar analytical model of the plasma expansion in a magnetic field Q ( η ) monotonically increases with the argument and the plasma cloudenergy decreases with time.Consider also the case of uniform magnetic field when H = const . In this case thevolume integral in equation (9) is replaced by ( H / a ( t ) − a ) and the differential equation(11) for the plasma boundary reads ˙ x ( τ ) + βx γ − + σ [ x ( τ ) −
1] = 1 , (15)where σ = W mag /W , W mag = (4 πa / P mag is the initial magnetic energy in the plasmavolume, and P mag = H / π is the magnetic field pressure.Equations (11) and (15) coincide with the equation of the one-dimensional motion ofthe point-like particle in the potential U ( x ) which is determined by second and third terms ofequations (11) and (15). The distance x s of the plasma cloud motion up to the full stop (thestopping length) at the turning point is determined by U ( x s ) = 1 . In particular, it is easier toobtain the stopping length in the case of homogeneous magnetic field and at vanishing thermalpressure ( β = 0 ). Then from equation (15) one obtains the equation of motion ˙ x = 1 + σ − σx . (16)It is seen that in this case the stopping length is given by x s = (1 + 1 /σ ) / . The solution ofequation (16) can be represented in the form t = t √ σ + 1 (cid:20) x ( t ) F (cid:18) x ( t ) x s (cid:19) − F (cid:18) x s (cid:19)(cid:21) , (17)where F ( z ) = F (cid:0) , ; ; z (cid:1) and the latter is the hypergeometric function. Substituting inequation (17) x ( t ) = x s we obtain the corresponding stopping time as a function of themagnetic field and the plasma kinetic energy t s = t √ σ + 1 " C (cid:18) σ + 1 σ (cid:19) / − F (cid:18) σσ + 1 (cid:19) . (18)Here C = F (1) = √ π Γ (cid:0) (cid:1) / Γ (cid:0) (cid:1) ≃ . is a constant. At vanishing ( σ ≪ )and very strong ( σ ≫ ) magnetic fields the stopping time becomes t s ≃ CR m /v m = C ( a /v m ) σ − / , t s ≃ R m / v m a = (2 a / v m ) σ − , respectively, where the radius R m = (6 W /H ) / is obtained by equating the initial kinetic energy W of an initiallyspherical plasma cloud to the energy of the magnetic field that it pushes out in expanding tothe radius R m . It is worth mentioning that in the case of weak magnetic field, σ ≪ , and atvanishing thermal pressure ( β = 0 ) the stopping time does not depend on the initial plasmaradius, t s ∼ ( M/v m P mag ) / .We now turn to the general equations determined by (11) and (15). At the initial stage ofplasma expansion ( t ≪ t ) from these equations we obtain x ( t ) ≃ v m ta + 34 h (cid:18) tt (cid:19) , (19)where h = β ( γ − − κ , κ = α ηQ ′ ( η ) and κ = σ for the dipole and homogeneous magneticfields, respectively. Here the prime indicates the derivative with respect to the argument. Thusat the initial stage the plasma cloud may get accelerated or decelerated depending on the signof the quantity h (in other words on the relation between thermal and magnetic pressures).For instance, in the homogeneous magnetic field the acceleration occurs when p max > p c ,where p c = 43 √ π Γ (cid:0) + s (cid:1) Γ (1 + s ) P mag (20) elf-similar analytical model of the plasma expansion in a magnetic field -2 -1 -2 -1 x ( t ) t (cid:1) = 0.1 (cid:0) = 100 (cid:2) = 2 · (cid:3) = 5/3; (cid:4) = 0.5; (cid:5) = 10 -2 d x ( t ) / d t t (cid:6) = 0.1 (cid:7) = 100 (cid:8) = 2 · (cid:9) = 5/3; (cid:10) = 0.5; (cid:11) = 10 -2 Figure 1.
The dynamics of the plasma cloud expanding in a dipole magnetic field. Shown arethe scaled radius a ( t ) /a (left panel) and the velocity ˙ a ( t ) /u m (right panel) of the plasmaboundary vs time (in units of t ) at γ = 5 / , β = 0 . , η = 10 − , θ p = 0 and for α = 0 . (solid lines), α = 100 (dashed lines) and α = 2 × (dotted lines). (i.e. at h > ) and continues until x ( t ) reaches some value x c > given by x c =( p max /p c ) / γ . The time interval t < t c of the acceleration is determined from theequation of motion (15). The critical radius x c and time t c correspond to the beginning ofplasma deceleration. Further plasma motion at t > t c is an expansion with slowing-downvelocity. It ends up at the turning point which corresponds to the maximum of expansion, U ( x s ) = 1 . However, in the opposite case of the low thermal pressure with p max < p c theplasma systematically get decelerated in the whole time interval of its dynamics.A characteristic stopping time of plasma motion up to the full stop at the turning point isgiven by the integral of the equations (11) and (15) τ s = Z x s dy p − U ( y ) ≃ s x s − U ′ ( x s ) . (21)Calculating time t s needed for plasma to reach this point, one can simplify the integrandtaking into account that the main contribution comes from the vicinity of upper limit ofintegration. This approximation is expressed by the second part of equation (21). Inthe case of weak and homogeneous magnetic field this yields universal expressions, t s ∼ ( M/u m P mag ) / and a s ∼ u m t s . It is worth mentioning that in the case of weak magneticfield the stopping time and length do not depend on the initial plasma radius but depend onthe thermal pressure (or temperature) (cf. these relations with those obtained above). At verystrong magnetic fields, a s ≃ a + (1 / v m t s and t s ∼ M v m /a P mag , and the stoppingcharacteristics of the plasma essentially depend on the initial radius but are now independenton the thermal pressure. The similar estimates can be found for the dipole magnetic field.However, we note that the latter case significantly differs from the homogeneous field situationconsidered above. Since in the vicinity of the dipole the magnetic field is arbitrary large thestopping length cannot naturally exceed r for any thermal energy of the plasma ( x ( t ) < /η in (11)). For a weak magnetic field this simply yields a s ≃ r and t s ≃ r /u m .As an example in figure 1 we show the results of model calculations for the normalizedradius a ( t ) /a (left panel) and the velocity ˙ a ( t ) /u m (right panel) as a function of time (inunits of t ) at γ = 5 / , β = 0 . , η = 10 − , θ p = 0 and for different values of the parameter α . In this figure the dimensionless strengths α of the dipole magnetic field are chosen suchthat the coefficient h in equation (19) is positive, h > , for solid and dashed lines and elf-similar analytical model of the plasma expansion in a magnetic field -3 -2 -1 -3 -2 -1 x s ( a ) a (cid:12) = 0.1 (cid:13) = 0.5 (cid:14) = 0.9 (cid:15) = 5/3; (cid:16) = 10 -2 t s ( a ) a (cid:17) = 0.1 (cid:18) = 0.5 (cid:19) = 0.9 (cid:20) = 5/3; (cid:21) = 10 -2 Figure 2.
The normalized stopping length a s ( α ) /a (left panel) and the stopping time t s ( α ) /t (right panel) of the plasma cloud expanding in a dipole magnetic field vs thenormalized dipole magnetic field α at γ = 5 / , η = 10 − , θ p = 0 and for β = 0 . (solid lines), β = 0 . (dashed lines) and β = 0 . (dotted lines). negative, h < , for dotted lines. From the right panel of figure 1 it is seen that at h > (solid and dashed lines) there is a short initial period of acceleration, t . t , when theplasma boundary is accelerated according to equation (19). During this period (which is onlyweakly sensitive to the magnetic field strength) the dimensionless radius a ( t ) /a increases upto − , and at t . t < t c almost all initial total energy W is transferred into kinetic energyof free radial expansion at constant velocity ∼ u m . As expected (see above) the time t c isreduced with increasing the strength of the magnetic field and the free expansion period isshorter for larger α . The further increasing the strength of the magnetic field (figure 1, dottedline) results in a plasma dynamics with systematically slowing-down velocity.For the same set of the parameters γ , η and θ p in figure 2 it is shown the normalizedstopping length (left panel) and the stopping time (right panel) of the plasma cloud as afunction of the dimensionless strength α of the dipole magnetic field for some values of thenormalized plasma thermal pressure β . It is seen that the stopping length and time decreasewith the strength of the magnetic field and practically are not sensitive to the variation of theplasma thermal pressure.Note that at otherwise unchanged parameters the strength of the dipole magnetic field ismaximal at the orientation θ p = 0 and monotonically decreases with θ p . For instance, thestrength H (0) of the dipole magnetic field at the center of the plasma cloud is reduced by afactor of by varying the dipole orientation from θ p = 0 to θ p = π/ . Therefore the effectof the magnetic field shown in figures 1 and 2 is weakened at the orientation θ p = π/ of thedipole. In particular, this results in a larger stopping lengths and times than those shown infigure 2.In this paper we have assumed that the self-similar solutions (6)-(8) and hence theshape of the plasma remain isotropic during plasma expansion. Such assumption is notevident since the distribution of the magnetic pressure on the plasma surface is anisotropicin general. Thus one can expect strong deformation of the initially spherical plasma surface.Consider briefly this effect assuming constant magnetic field H . Taking into account theeffect of the induced magnetic field the total magnetic pressure on the plasma surface is P m ( θ ) = (9 / P mag sin θ , where θ is the angle between the radius vector r and H [8, 10].The magnetic pressure vanishes at θ = 0 , π and is maximal at θ = π/ . Therefore,one can expect that the spherical shape of the plasma is deformed into the ellipsoidal one elf-similar analytical model of the plasma expansion in a magnetic field P mag by the angular dependent one P m ( θ ) . The expansionradius a ( t, θ ) as well as the stopping length a s ( θ ) and time t s ( θ ) should be also anisotropicin this case.
3. Conclusions
An analytical self-similar solution of the radial expansion of a spherical plasma cloud in thepresence of a dipole or homogeneous magnetic field has been obtained. The analysis of theplasma expansion into ambient magnetic field shows that there are processes of acceleration,retardation and stopping at the point of maximum expansion that are very distinct andseparated in space and time. The scaling laws obtained are, in general, the functions of twodimensionless parameters, α (or σ for constant magnetic field) and β , which can be varied bymeans of the choice of the external magnetic field, the thermal pressure and the initial energyof the plasma. It allows to test the different regimes of the plasma dynamics in a wide rangeof external conditions.We expect our theoretical findings to be useful in experimental investigations as wellas in numerical simulations of the plasma expansion into an ambient magnetic field (eitheruniform or nonuniform). One of the improvements of our present model will be the derivationof the dynamical equation for the plasma surface deformation. In this case it is evident thatthe problem is not isotropic with respect to the center of the plasma cloud ( r = 0 ) and afull three-dimensional analysis is required. A study of this and other aspects will be reportedelsewhere. Acknowledgments
This work was made possible in part by a research grant (Project PS-2183) from the ArmenianNational Science and Education Fund (ANSEF) based in New York, USA. Also this work hasbeen supported by the State Committee of Science of Armenia (Project No. 11-1c317).
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