Heating of directly transmitted ions at low Mach number perpendicular shocks: New insights from a statistical physics formulation

Abstract

[1] The heating of directly transmitted ions at low Mach number, quasi-perpendicular collisionless shocks is rapid, greater than adiabatic, and exhibits a distinct T ? > T k anisotropy. In this paper we present a theoretical study of the evolution of the ion velocity distribution across a stationary one-dimensional perpendicular model shock profile. A Lagrangian/Hamiltonian formulation of the ion equations of motion is introduced. We argue that the classical statistical physics solution of Liouville's equation in terms of the energy (Hamiltonian) is not applicable to the case of a laminar perpendicular shock. Assuming a Maxwellian incident ion velocity distribution, it is possible to obtain the analytical form for the distribution through the shock in terms of functions of upstream parameters that are independent of the incident temperature. Unlike the classical Hamiltonian solution, we show that contours of equal phase space probability do not correspond to contours of equal energy. It is this property of the velocity distribution that makes anisotropic heating possible. We recover the observed results that the distribution is stretched across the magnetic field direction as it passes through the shock and that it rotates as a whole around the field in the downstream region. We are able to show that in the low-temperature limit, the shape of the distribution remains Gaussian but that this is not the case for higher temperatures. For this Gaussian approximation, lower and upper bounds for the variance of the downstream velocity and therefore the heating are obtained. An efficient method for the numerical computation of the distribution through the shock is proposed and evaluated for typical shock parameter combinations. The downstream behaviour of the distribution is also elucidated.

A statistical physics perspective on the evolution of ion distributions across low Mach number quasi-perpendicular collisionless shocks

“…Like , A is an invariant of the flow but, unlike , it depends only on x (position normal to the shock) and the upstream velocity and not on the other position coordinates (such that f meets a Maxwellian boundary condition far upstream). In this distribution contours of equal phase space probability do not correspond to contours of equal energy and EW03 argue that it is this property of the distribution that makes heating and temperature anisotropy possible. Upstream, the phase shells corresponding to A 2 = constant are circles.…”

“…This enables estimates of the downstream to upstream temperature ratio of between 2 and 4 (consistent with, e.g., Lee and Wu [2000]). EW03 show that through the shock and downstream of the shock, the phase area (Δ x ) enclosed by a curve of constant A 2 is inversely proportional to the modal speed of the ions in the shock normal direction, so that the temperature of the distribution increases through the shock in a manner consistent with the results of Balikhin and Wilkinson [1996]: where = , V u is the bulk plasma velocity upstream (directed antiparallel to the shock normal), B u is the magnitude of the upstream magnetic field, and is the modal value of the canonical momentum corresponding to motion toward the shock ( p 1 = ; see section 2).…”

“…In a subsequent study, Ellacott and Wilkinson [2006] (hereafter referred to as EW06) examine several thermodynamic issues and demonstrate that it is only by taking into account the difference between the mean and modal velocities of the distribution that a change in internal energy is obtained across the shock. Ellacott and Wilkinson [2007] (hereafter referred to as EW07) then use the approach developed in EW03 to delve into quasi‐perpendicular shocks and show that, here too, the required solution to Liouville's equation is not the Hamiltonian and that phase shells of constant probability are not shells of constant energy, which leads to the temperature anisotropy downstream. The temperature increase across the shock is then related to the difference in the radii, R 1 and R 2 , of spheres of constant energy corresponding to the smallest and largest values, respectively, of on the phase shell.…”

Ion thermalization and wave excitation downstream of Earth's bow shock: A theory for Cluster observations of He2+ acceleration

“…These reflected protons have a different velocity from the downstream bulk flow which is dominated by the transmitted protons; they gyrate around the magnetic field in the frame of the transmitted protons with a large temperature anisotropy T ⊥ / T z , where ⊥ and z refer to the directions perpendicular and parallel to the average magnetic field. The directly transmitted protons are also heated more strongly in the direction perpendicular to the magnetic field during their transversal of the shock potential [ Ellacott and Wilkinson , 2003; Liu et al , 2005]. The resulting anisotropic distribution is unstable to the excitation of ion cyclotron waves or mirror mode waves, depending on the value of the plasma β , which is the ratio of the thermal pressure to the magnetic pressure.…”

Multipoint observations of plasma phenomena made in space by Cluster

“…At the present time, there are at least two possible ways to interpret the source of high temperature (∼ few keV) plasmas observed in the magnetosheath. One attributes the hot magnetosheath plasma to heating of the solar wind by instabilities at the shock (Auer et al 1971;Wu and Yoon 1990;Gedalin 1997;Ellacott and Wilkinson 2003). There are many possible instabilities, but none has been specifically identified.…”