Force-free collisionless current sheet models with non-uniform temperature and density profiles

We present a class of one-dimensional, strictly neutral, Vlasov-Maxwell equilibrium distribution functions for force-free current sheets, with magnetic fields defined in terms of Jacobian elliptic functions, extending the results of Abraham-Shrauner (Phys. Plasmas 20, 102117, 2013) to allow for non-uniform density and temperature profiles. To achieve this, we use an approach previously applied to the force-free Harris sheet by Kolotkov et al. (Phys. Plasmas 22, 112902, 2015). In one limit of the parameters, we recover the model of Kolotkov et al., while another limit gives a linear force-free field. We discuss conditions on the parameters such that the distribution functions are always positive, and give expressions for the pressure, density, temperature and bulk-flow velocities of the equilibrium, discussing differences from previous models. We also present some illustrative plots of the distribution function in velocity space.

Such current sheets as described above can play a crucial role in, e.g, magnetic reconnection processes, for which it is often necessary to consider kinetic length scales (e.g. Ref. 16 Other studies of collisionless reconnection in force-free current sheets have involved the use of approximate force-free equilibria (e.g. Refs. [19][20][21][22][23][24][25][26] or linear force-free equilibria (e.g.
To find VM equilibrium DFs consistent with force-free current sheets involves solving the VM equations in the opposite order from what is usually done; a magnetic field satisfying Equations (1)-(3) is specified, and the DFs are then given by the solution of an inverse problem (e.g. Refs. [30][31][32][33]. As such, finding exact force-free VM equilibria is generally a non-trivial task, and this is reflected in the relatively small number of known solutions. Linear force-free VM equilibria have been discussed in, e.g., Refs. 18, 27, 31, 34-37. The first solution for a nonlinear force-free field was found by Harrison and Neukirch 38 (see also Ref. 39) for the force-free Harris sheet, and these solutions were later extended by Kolotkov et al. 2 to allow for non-uniform density and temperature profiles (with respect to the spatial coordinate). A number of other equilibrium DFs have also been found for this field. Wilson and Neukirch 40 found DFs with an arbitrary dependence on the particle energy; Stark and Neukirch 41 discussed DFs in the relativistic limit; Allanson et al. 33,42 found DFs in terms of infinite sums over Hermite polynomials, with an arbitrarily low plasma beta (in the previous work on the force-free Harris sheet the plasma beta was constrained to be greater than unity); Dorville et al. 43 discussed 'semi-analytic' DFs for a magnetic field which includes the force-free Harris sheet as a special case.
Abraham-Shrauner 1 discussed VM equilibria for a nonlinear force-free magnetic field given in terms of Jacobian elliptic functions. This work can be thought of as a generalisation of some of the previous work, to account for both linear and nonlinear force-free equilibria in one model, since, in one limit of the elliptic modulus, the magnetic field becomes the force-free Harris sheet field, and in another limit it becomes a linear force-free field. The DFs discussed give rise to spatially uniform temperature and density profiles, in a similar way to some of the models mentioned above. In this paper, we will extend this class of DFs to include those consistent with non-uniform temperature and density profiles, using a similar approach used by Kolotkov et al. 2 for the force-free Harris sheet. As for Abraham-Shrauner's DFs, the new DFs we will discuss include both the linear force-free limit and the force-free Harris sheet limit 2 .
The paper is laid out as follows; in Section II, we outline the background theory of 1D VM equilibria; in Section III, we present an overview of the work by Abraham-Shrauner 1 ; we discuss the extension of this work to include non-uniform temperature and density profiles in Section IV, and the velocity space structure of the new DFs is discussed in Section V; we end with a summary in Section VI.

II. 1D VLASOV-MAXWELL EQUILIBRIA
In line with some of the previous work on 1D VM equilibria (e.g. Refs. 18, 38, and 39), we assume that all quantities depend only on the z-coordinate, and that the magnetic field, , can be written as the curl of a vector potential, A = (A x , A y , 0). We will not repeat all of the details here, but the result of the above assumptions is that the problem reduces to solving Ampère's law in the form to find P zz , which is the zz-component of the pressure tensor, defined by where we assume that the DFs can be chosen in such a way that they are compatible with strict neutrality (the scalar potential φ = 0) 31 . Note that we only consider P zz since this is the component of the pressure tensor which is important for the force-balance of the 1D equilibrium. The DFs, denoted by f s , are assumed to be functions of the particle , and the x-and y-components of the canonical momentum, p = m s v + q s A, since these are known constants of motion for a time-independent system with spatial invariance in the x-and y-directions. Once Ampère's law has been solved for P zz , the DF can be found by solving Eq. (7). This is an example of an inverse problem.

III. ABRAHAM-SHRAUNER'S MODEL
In this section we discuss some properties of the the model developed by Abraham-Shrauner 1 , in order to give context to the discussion we will present in Section IV. In Abraham-Shrauner's work, a nonlinear force-free current sheet profile is considered, described by the magnetic field where B 0 is a constant, L is the current sheet half-thickness, and sn and cn are Jacobian elliptic functions 44 with the modulus k suppressed (where 0 ≤ k ≤ 1). In the limit k → 0, sn(z/L) → sin(z/L) and cn(z/L) → cos(z/L), and so the magnetic field (8) becomes the linear force-free field B = B 0 (sin(z/L), cos(z/L), 0). In the limit k → 1, sn(z/L) → tanh(z/L) and cn(z/L) → sech(z/L), giving the force-free Harris sheet magnetic field (Eq. (4)). The vector potential, A, used by Abraham-Shrauner 1 is given by where dn is also an elliptic function. This can be seen by using standard integrals 45 and by choosing the integration constants such that, when k → 1, A x → 2B 0 Larctan(e z/L ), A y → − ln(cosh(z/L)) -the vector potential components used in some of the previous work on the force-free Harris sheet (note also that an alternative gauge for A is discussed for the force-free Harris sheet by Allanson et al. 33 ).
The current density is given by and so the force-free parameter α is given by Note that, in the limit k → 0, dn(z/L) → 1, and so α is constant (the linear force-free case), but is otherwise a function of position (the nonlinear force-free case).
It is assumed that the pressure has the form P zz (A x , A y ) = P 1 (A x )+P 2 (A y ); Ampère's law in the form of equations (5) and (6) can then be solved for P zz in terms of the macroscopic parameters, which gives where P t1 and P t2 are constants. This expression can then be used in Eq. (7) to determine the DF, which can be written in terms of the constants of motion as f s (H s , p xs , p ys ) = n 0s e −βsHs √ 2πv th,s 3 a 0s − where a 0s is a dimensionless constant, u xs and u ys are constant parameters with the dimension of velocity, β s = (k B T s ) −1 and v th,s = (β s m s ) −1/2 . In the limit k → 1, this DF takes the form of that discussed in Refs. 38 and 39 for the force-free Harris sheet. In the opposite limit, i.e. k → 0, it takes a general form which is similar to that described in Refs. 18, 31, and 37, but with a shift in p xs and p ys (this corresponds to a regauging of the vector potential).
Note that a number of relations exist between the parameters of the model, to ensure positivity of the DFs, strict neutrality, and consistency between the microscopic and macroscopic descriptions of the equilibrium (see Ref. 1 for further details). Using these relations, the equilibrium density, pressure and temperature can be expressed as where a 0 and n 0 are constant parameters that are introduced when the strict neutrality condition (φ = 0) is imposed. The expressions (15)- (17) are independent of the elliptic modulus k; this can be seen for P zz through the force-balance equation where P T is the total pressure, since B 2 = |B| 2 = B 2 0 for the magnetic field (8), which is independent of k. Since, in this case, P zz = (β e + β i )n/(β e β i ), it follows that the density and temperature will also be independent of k. As can be seen from the expressions (15) and (17), Abraham-Shrauner's model has density and temperature profiles that are constant across the current sheet, in a similar way to the models discussed in Refs. 18, 33, 38-42. In Section IV, we discuss how the method of Kolotkov et al. 2 can be used to extend the model to have spatially non-uniform density and temperature profiles across the current sheet, while still maintaining a constant pressure as is required for a force-free equilibrium (see e.g. Ref. 18).

IV. EXTENSION TO NON-UNIFORM TEMPERATURE/DENSITY CASE
To extend the model of Abraham-Shrauner 1 to have non-uniform temperature and density profiles, we consider a DF of the form (where γ > 0) i.e. a modification of Abraham-Shrauner's DF. This corresponds to assuming that the p xs -dependent population has a different energy dependence than the p ys -dependent population, through the factor γ. We effectively also have two separate constant background populations (through the constants a 0s and b 0s ) whose energy dependences differ. These two populations have been included to allow the limit k → 0 to exist, and to ensure this we assume that the constants a 0s and b 0s scale with the elliptic modulus k in the following way; for constantsā 0s andb 0s . Note that we have defined the constants in this way so that we have a model that works for all k values between 0 and 1, but for finite small k (or large u xs /v th,s ), the k-dependent parts of a 0s and b 0s can become very large, which may lead to, e.g., a large maximum density, which may not be physically appropriate. If we were only interested in a particular finite small value of k, we could redefine the constants to avoid such issues.
By calculating the number density (n s = f s d 3 v) of the modified DF (19), and imposing the condition φ = 0 (n i (A x , A y ) = n e (A x , A y )), we obtain the neutrality relations (A1)-(A8) in Appendix A. We can then express n s = n as and the pressure can be calculated from the DF through Eq. (7) as  (19) can then be written as Sufficient conditions for the positivity of the DF (24) across the whole phase space can be derived by assuming that the functions are both positive, and are given bȳ whereā 0 andb 0 are defined in Appendix A. Note that these conditions are well defined in the limit k → 0. Since 0 ≤ k ≤ 1, γ > 0 and the exponential term in Eq. (27) has a minimum value of unity, we see thatā 0 ≥ 0.
The new DF (24) describes an equilibrium with non-uniform density and temperature profiles; we can show this by writing them as functions of z using Equations (9), (10), (A13)-(A15) and the definitions ofā 0 andb 0 , which gives where the uniform value of the pressure is given by which is independent of the modulus k (for the same reasons as discussed in Section III), and is similar to the expression found by Kolotkov et al. 2 for the force-free Harris sheet.
Note, however, that this time the density depends on k, due to the introduction of the γ factors in the DF (the pressure can no longer be written as P zz = (β e + β i )n/(β e β i ) as it can in the uniform temperature model). It can be seen that, for γ = 1, we recover the constant density/temperature case of Abraham-Shrauner 1 .
Provided the DF (24) is positive over the whole phase space, then the density, pressure and temperature will also be positive everywhere. Note, however, that the opposite is not true, i.e. a positive density and pressure do not imply a positive DF. We ensure that the DF is positive by choosing parameters in such a way that the conditions (27) and (28) are satisfied (for both ions and electrons). Figure 1 shows profiles of the density and temperature for different values of γ, with k = 0 (the linear force-free case). Figure   opposite behaviour is seen when γ < 1, i.e. a depletion/enhancement of the quantities.
Similar features are seen by Kolotkov et al. 2 (which we obtain in the limit k → 1), but note that the density and temperature are not periodic in this case, and so, for a particular γ value, there is either an enhancement or depletion of the density/temperature (not both).
Through these expressions, we see the role played by the parameters u xs and u ys , which can also be written in terms of the ratio of the species gyroradius, r g,s , to the current sheet half-width, L, by using Eq. (A16) (similarly to Neukirch et al. 39 ) as The current density can be calculated from the bulk flow velocity as and has components j x = n 0 eγ(u xi − u xe )sn(z/L)dn(z/L) (40) j y = n 0 e(u yi − u ye )cn(z/L)dn(z/L) Using Equations (A11) and (A17), we can show that these expressions are equivalent to those obtained macroscopically from Ampère's law (Eq. (11)).
In the models in e.g. Refs. 1, 38, and 39, the spatial structure of the current density is determined solely by the structure of the bulk flow velocity since the density is constant, in contrast to the classic Harris sheet model 46 , where the bulk flow velocity is constant, and it is the spatial dependence of the density that determines the structure of the current density.
In this extended model (and also that of Kolotkov et al. 2 ), however, both the bulk-flow velocity and density are spatially dependent, and so the spatial structure of the current density is determined from the product of the two quantities.

A. Limiting values of k
In the limit k → 1, the number density, temperature, and pressure (Equations (29)- (31)) go to the form discussed by Kolotkov et al. 2 for the force-free Harris sheet, and the DF (24) becomes the Kolotkov DF (note that our notation is slightly different).
In the limit k → 0, the field becomes linear force-free, and we get a DF of the form which is a modified form of the DF obtained in the k → 0 limit of the DF (14). The density and temperature have the form given by Equations (29) and (30) respectively, where sn(z/L) = sin(z/L).

V. VELOCITY SPACE STRUCTURE OF DF
In this section, we present some illustrative plots of the DF (24)  In the discussion of the plots below, we will refer to cases where the p xs population is 'hotter'/'colder' than the p ys one. This refers to the p xs population having an energy dependence resulting in a 'narrower'/'wider' Maxwellian factor in the DF than the p ys one.
We note, however, that because the DFs are not purely Maxwellian, the temperature cannot be properly defined in terms of the width of the DF, but the widths of the first and second parts of the DF gives us a qualitative measure of the temperature difference between the different populations. This notion of temperature should not be confused with the definition of the temperature given in Eq. (30).
In Figure 3, we plot the electron DF (24) in the v x -direction (for v y = v z = 0) with γ = 1 (i.e. the Abraham-Shrauner DF). We have chosen a set of parameters for which, at z = 0, the DF has a double maximum in v x (these are the same parameters as in Figure 2). We note, however, that it is also possible to choose parameters for which the DF has only a single maximum in v x over the whole phase space, if required (by increasing the density of the background populations appropriately). In Figure 3, and all subsequent figures in this paper, we normalise the DF to have a maximum value of unity. Our main aim in this section is to investigate the effect of changing γ on the velocity space structure of the DF. This is why we have chosen parameters that give a double maximum for γ = 1, since the effect of changing γ is illustrated more clearly in such cases. Figure 4 shows in v x , is therefore 'swamped' by the wider second part for decreasing γ, and we see the behaviour in Figure 4. Possible behaviour of the DF in the v y -direction can be explored heuristically by noting that, for given values v x , v z and z, the DF has the general form

VI. SUMMARY
In this paper, we have presented a class of 1D strictly neutral Vlasov-Maxwell equilibrium DFs for both linear and nonlinear force-free current sheets, with magnetic fields defined in terms of Jacobian elliptic functions, which are an extension of the DFs discussed by Abraham-Shrauner 1 to account for non-uniformities in the temperature and density, whilst still maintaining a constant pressure (with respect to the spatial coordinate), as is required for force-balance of the force-free equilibrium. To achieve this, we have used the method of Kolotkov et al. 2 , which involves modifying the DF of the original case to include temperature differences between the different particle populations in the model, and then ensuring that strict neutrality is satisfied, and that there is consistency between the microscopic and macroscopic parameters of the equilibrium.
The new DF can be regarded as consisting of four particle populations: one depending on p xs , one on p ys , and two background populations. We have derived sufficient conditions on the parameters such that the positivity of the DFs is ensured, and have given explicit expressions for the density, temperature and pressure across the current sheet. Additionally, we have derived the components of the bulk-flow velocity from the DF, to show that the spatial structure of the current density is determined by the product of the spatial structure of the density and bulk-flow velocity, in contrast to the models of, e.g., Abraham-Shrauner 1 and Neukirch et al. 39 , where the current density structure is determined solely by the structure of the bulk-flow velocity, and also in contrast to the Harris sheet case 46 , where it is determined solely by the density structure.
We have investigated limiting cases of the elliptic modulus, k. For k → 1 the magnetic field becomes that of the force-free Harris sheet, and in this limit we recover a DF similar to that found by Kolotkov et al. 2 for this magnetic field. In the limit k → 0, the magnetic field becomes linear force-free, and in Abraham-Shrauner's case the DF takes a form which is similar to one discussed in Refs. 18, 31, and 37, but which is shifted in p xs and p ys . In our extended model, the k → 0 limit simply gives an extension of this shifted DF to include non-uniformity in both the temperature and density.
We have also illustrated graphically the effect of changing the temperature difference between the particle populations in the DF. In the v x -direction, we found that making the p xs part 'colder' than the p ys part can result in rather pronounced double maxima of the DF (due to a cosine term in v x ), but when the p xs part is 'hotter' these maxima are less significant, or the DF becomes single peaked. In the v y -direction, the DF contains two drifting Maxwellians (with the same energy dependence), and two non-drifting Maxwellians (with different energy dependences), and so there is the possibility of double maxima in the DF depending on the relative values of the coefficients of the separate parts.
Double maxima in the DF may lead to velocity space instabilities (e.g. Ref. 47). Due to the increased complexity of the model, however, we have not attempted a systematic study of the velocity space structure, i.e. we have not derived conditions on the parameters such that the DF can be multi-peaked for some z, such has been done by Neukirch et al. 39 and Abraham-Shrauner 1 . This is left for a future investigation. We note, however, that it will be possible to choose the density of the background populations large enough such that there are only single maxima of the DF over the whole phase space.
By calculating two expressions for the pressure P zz , in terms of the macroscopic and microscopic parameters of the equilibrium respectively, and comparing these expressions, we obtain the relations