Generalized parallel heat transport equations in collisional to weakly collisional plasmas

Abstract

A new set of two-fluid heat transport equations that is valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates, a set of moment equations describing plasma energy transport along the field lines of a space- and time-dependent magnetic field is derived. No restrictions on the anisotropy of the ion distribution function or collisionality are imposed. In the highly collisional limit, these equations reduce to the classical heat conduction equation (e.g., Spitzer and Härm or Braginskii), while in the weakly collisional limit, they describe a saturated heat flux (flux limited). Numerical examples comparing these equations with conventional heat transport equations show that in the limit where the ratio of the mean free path λ to the scale length of the temperature gradient LT approaches zero, there is no significant difference between the solutions of the new and conventional heat transport equations. As λ/LT→1, the conventional heat conduction equation contains a significantly larger error than (λ/LT)2. The error is found to be O(λ/L)2, where L is the smallest of the scale lengths of the gradient in the magnetic field, or the macroscopic plasma parameters (e.g., velocity scale length, temperature scale length, and density scale length). The accuracy of the flux-limited model depends significantly on the value of the flux limit parameter which, in general, is not known. The new set of equations shows that the flux-limited parameter is a function of the magnetic field and plasma parameter profiles.

Kinetic Particle Simulation Study of Parallel Heat Transport in Scrape-off Layer Plasmas over a Wide Range of Collisionalities

“…As explained for the previous case, since the ion heat flux follows the expression q || = 3q ⊥ [28], the parallel heat is transported away three times faster than the perpendicular heat with a corresponding drop in the parallel temperature. However, since the trapped electrons pass through the Langevin heat bath many times, they tend to remain isotropic.…”

“…Figure 4 (d) shows that over the weakly collisional regime 10 −2 < λ mfp /L || < 1, the ions shift quickly from being isotropic to the parallel temperature being equal to 1/6 of the perpendicular temperature. When the source ions are cold, the anisotropy in the collisionless limit is T || /T ⊥ = 1/3 because the parallel heat flux is three times higher than the perpendicular heat flux [28]. However, when the ion source is the same temperature as the electron source and the adiabatic index is γ A = 3, then the ion sound speed doubles.…”

“…There has even been an attempt to explain the plasma behaviour in the weakly collisional regime [28]. However, even this approach must assume a Maxwellian distribution when providing fourth-order closure terms.…”

“…Therefore, the adiabatic index is necessary to determine the ion sound speed from the local temperature. Finally, fluid codes do not currently separate the temperatures parallel and perpendicular to the magnetic field, which typically differ by a factor of three [28]. Therefore, the temperature anisotropy must be known to adjust the calculated heat flux.…”

An extended kinetic model for the thermal force on impurity particles in weakly collisional plasmas

“…References are examples of those works, presenting several heat flux limiter formulations based on the ion mean free path, steepness of the ion temperature gradient, or the free‐streaming energy flux of ions, for instance. Among those studies, we focus on the heat flux formula derived from the generalized moment equations which would enable us to include the plasma collisionality information into q .…”