Abstract
We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov–Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans’ theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris sheet (Allansonet al.,Phys. Plasmas, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma${\it\beta}$, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for${\it\beta}_{pl}=0.05$.
Highlights
An important question in the study of plasmas is to understand the fundamental physics involved in magnetic reconnection
We claim – provided gs is differentiable and convergent – that for values of the magnetisation parameter δs less than some critical value δc, according to 0 < δs < δc ∞, gs is positive for any positive pressure function
Summary The primary result of this paper is the rigorous generalisation of a solution method that exactly solves the ‘inverse problem’ in 1-D collisionless equilibria, for a certain class of equilibria
Summary
An important question in the study of plasmas is to understand the fundamental physics involved in magnetic reconnection. Using the DF found in Harrison & Neukirch (2009a), the first fully kinetic simulations of collisionless reconnection with an initial condition that is an exact Vlasov solution for a nonlinear force-free field was conducted by Wilson et al (2016) Motivated by these and other considerations, this paper presents results on the theory and application of a method that allows the calculation of collisionless kinetic plasma equilibria. J=0 the Weierstrass transform can immediately be inverted to obtain the formal expansion φ(y) = ηjHj(y/2) For this method to be useful in our problem, the pressure function must have a Maclaurin expansion that is convergent over all (Ax, Ay) space. These requirements imply by the matching of (2.8) and (2.11) that
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