Ratio of non-linear resistivity to the usual Spitzer value as a function of plasma current in the 13-moment extended MHD fluid model.
A systematic treatment of microscopic collisions to extend fluid models of plasmas
At the microscopic level, binary collisions between charged particles conserve two fundamental quantities, namely momentum and energy. This conservation must be present in fluid models in order to describe the macroscopic evolution of plasmas correctly. A systematic treatment of collisional effects is presented to derive fluid models beyond the usual assumption of “thermal equilibrium”. Such extended models will greatly help model and understand cold and moderate temperature plasmas as, for example, at the edge of fusion devices. Understanding and controlling the edge plasma is crucial for maintaining confinement of the ultra-hot core where the fusion reactions occur.
The Science
The collective behaviour of charged particles colliding against one another in plasmas gives rise to very rich and diverse phenomena. Perhaps most importantly, it is only through collisions that the plasma in a fusion reactor reaches a quiescent state known as “thermal” equilibrium that is free from macroscopic instabilities. The computational challenge of kinetic simulations of collective behaviour is almost overwhelming even for the most powerful computers 20 years from now because of the long-range electromagnetic force and the sheer number of particles, up to involved. Instead, one constructs statistical averages, or so-called fluid theories, to simplify the representation of plasmas close to thermal equilibrium. It is much easier to understand the dynamics of fluids than to understand the collective interactions of one hundred billion trillion charged particles. Averaging the almost-infinite detail of the “micro” to obtain a simple “macro” description is no easy task, particularly if one wishes to ensure that fundamental conservation laws, e.g. momentum and energy conservation, are satisfied. We have derived a general solution to the problem of obtaining a macroscopic description of plasmas in the limit of so-called gazing collisions – where the scattering angle from a single collision is small but the large number of accumulated collisions leads to a significant deviation of the particle’s free trajectory – known as Landau’s approximation. From this approach, existing fluid models such as the widely used magnetohydrodynamics (MHD) model can be extended to include the collisional contributions of distribution functions that are not in thermal equilibrium, thereby capturing more microscopic details in the macroscopic picture.
The Impact
The main advantage of fluid models is a reduction in complexity. The rigorous treatment of collisions enables a systematic extension of the commonly used Braginskii equations to include finer details of the microscopic interactions and deviations from thermal distribution functions. Tracking the evolution of higher-order moments of plasma characteristics is essential to reproducing the complex collective phenomena occurring in cold and moderate temperature plasmas such as in the edge of fusion devices and in various astrophysical plasmas. The procedure can equally be applied to the study of the formation of galaxies – where the force between microscopic objects is also proportional to the inverse square of the distance.
Summary
This research shows that the statistical effect of Coulomb collisions in a plasma can be expressed exactly in terms of what are called multi-index Hermite-polynomial moments of the distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The result can be applied to both the classic transport theory of plasmas, that relies on the Chapman-Enskog method, as well as to deriving collisional fluid equations that follow Grad’s moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum-and energy-transfer rate.
@article{pfefferle-exact,
title = {Exact collisional moments for plasma fluid theories},
author = {D Pfefferl\'{e} and E Hirvijoki and M Lingam},
doi = {10.1063/1.4979992},
year = {2017},
date = {2017-04-20},
journal = {Physics of Plasmas},
volume = {24},
number = {4},
pages = {042118},
abstract = {The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum- and energy-transfer rates.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum- and energy-transfer rates.
@article{lingam-nonlinear,
title = {Nonlinear resistivity for magnetohydrodynamical models},
author = {M Lingam and E Hirvijoki and D Pfefferl\'{e} and L Comisso and A Bhattacharjee},
doi = {10.1063/1.4980838},
year = {2017},
date = {2017-04-20},
journal = {Physics of Plasmas},
volume = {24},
number = {4},
pages = {042120},
abstract = {A new formulation of the plasma resistivity that stems from the collisional momentum-transfer rate between electrons and ions is presented. The resistivity computed herein is shown to depend not only on the temperature and density but also on all other polynomial velocity-space moments of the distribution function, such as the pressure tensor and heat flux vector. The full expression for the collisional momentum-transfer rate is determined and is used to formulate the nonlinear anisotropic resistivity. The new formalism recovers the Spitzer resistivity, as well as the concept of thermal force if the heat flux is assumed to be proportional to a temperature gradient. Furthermore, if the pressure tensor is related to viscous stress, the latter enters the expression for the resistivity. The relative importance of the nonlinear term(s) with respect to the well-established electron inertia and Hall terms is also examined. The subtle implications of the nonlinear resistivity, and its dependence on the fluid variables, are discussed in the context of magnetized plasma environments and phenomena such as magnetic reconnection.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A new formulation of the plasma resistivity that stems from the collisional momentum-transfer rate between electrons and ions is presented. The resistivity computed herein is shown to depend not only on the temperature and density but also on all other polynomial velocity-space moments of the distribution function, such as the pressure tensor and heat flux vector. The full expression for the collisional momentum-transfer rate is determined and is used to formulate the nonlinear anisotropic resistivity. The new formalism recovers the Spitzer resistivity, as well as the concept of thermal force if the heat flux is assumed to be proportional to a temperature gradient. Furthermore, if the pressure tensor is related to viscous stress, the latter enters the expression for the resistivity. The relative importance of the nonlinear term(s) with respect to the well-established electron inertia and Hall terms is also examined. The subtle implications of the nonlinear resistivity, and its dependence on the fluid variables, are discussed in the context of magnetized plasma environments and phenomena such as magnetic reconnection.
@article{hirvijoki-2016,
title = {Fluid moments of the nonlinear Landau collision operator},
author = {E Hirvijoki and M Lingam and D Pfefferl\'{e}},
url = {http://dx.doi.org/10.1063/1.4960669},
doi = {10.1063/1.4960669},
year = {2016},
date = {2016-08-09},
journal = {Physics of Plasmas},
volume = {23},
number = {8},
pages = {080701},
abstract = {An important problem in plasma physics is the lack of an accurate and complete description of Coulomb collisions in associated fluid models. To shed light on the problem, this Letter introduces an integral identity involving the multivariate Hermite tensor polynomials and presents a method for computing exact expressions for the fluid moments of the nonlinear Landau collision operator. The proposed methodology provides a systematic and rigorous means of extending the validity of fluid models that have an underlying inverse-square force particle dynamics to arbitrary collisionality and flow.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
An important problem in plasma physics is the lack of an accurate and complete description of Coulomb collisions in associated fluid models. To shed light on the problem, this Letter introduces an integral identity involving the multivariate Hermite tensor polynomials and presents a method for computing exact expressions for the fluid moments of the nonlinear Landau collision operator. The proposed methodology provides a systematic and rigorous means of extending the validity of fluid models that have an underlying inverse-square force particle dynamics to arbitrary collisionality and flow.
involved. Instead, one constructs statistical averages, or so-called fluid theories, to simplify the representation of plasmas close to thermal equilibrium. It is much easier to understand the dynamics of fluids than to understand the collective interactions of one hundred billion trillion charged particles. Averaging the almost-infinite detail of the “micro” to obtain a simple “macro” description is no easy task, particularly if one wishes to ensure that fundamental conservation laws, e.g. momentum and energy conservation, are satisfied. We have derived a general solution to the problem of obtaining a macroscopic description of plasmas in the limit of so-called gazing collisions – where the scattering angle from a single collision is small but the large number of accumulated collisions leads to a significant deviation of the particle’s free trajectory – known as Landau’s approximation. From this approach, existing fluid models such as the widely used magnetohydrodynamics (MHD) model can be extended to include the collisional contributions of distribution functions that are not in thermal equilibrium, thereby capturing more microscopic details in the macroscopic picture.