Nonlocal closures for plasma fluid simulations

Abstract

The application of fluid models in studies of transport and macroscopic stability of magnetized, nearly collisionless plasmas requires closure relations that are inherently nonlocal. Such closures address the fact that particles are capable of carrying information over macroscopic parallel scale lengths. In this work, generalized closures that embody Landau, collisional and particle-trapping physics are derived and discussed. A gyro/bounce-averaged drift kinetic equation is solved via an expansion in eigenfunctions of the pitch-angle scattering operator and the resulting system of algebraic equations is solved by integrating along characteristics. The desired closure moments take the form of integral equations involving perturbations in the flow and temperature along magnetic field lines. Implementation of the closures in massively parallel plasma fluid simulation codes is also discussed. This implementation includes the use of a semi-implicit time advance of the fluid equations to stabilize the dominant closure terms which are introduced explicitly. Application of the nonlocal, parallel heat flow closure, q ʈ , in studies of temperature flattening across helical magnetic islands in toroidal geometry reveal a scaling of temperature versus critical island width for flattening of Tϳw d Ϫ1.5 . This result predicts more robust flattening at small island widths when compared to the diffusive scaling, Tϳw d Ϫ1.7 , which assumes a Braginskii-type parallel heat conductivity. Preliminary application of q ʈ to tokamak disruption simulations shows qualitative agreement of wall heat loads with experimental observations, smooth distribution in toroidal angle, and striation in the poloidal direction along the wall.

Computational modeling of fully ionized magnetized plasmas using the fluid approximation

“…Effects associated with charged particles rapidly streaming along magnetic-field lines are computed through solutions of a drift kinetic equation using the method of characteristics ͑see Sec. V A͒ and a basis function expansion in velocity space 54 ͑see Sec. V B͒.…”

Local and Nonlocal Parallel Heat Transport in General Magnetic Fields

“…Reference [2] discusses a finite-element numerical implementation of nonlocal heat transport with applications including temperature flattening across magnetic islands and tokamak disruptions. The use of high-order discretizations has been shown to mitigate numerical pollution of the perpendicular dynamics in finite-difference [3,4] and finite-element methods [4,5].…”

“…However, more general closures, like those in Refs. [2,14], are straightforward to implement by first computing the Green's function numerically. It is also interesting to point out that, for non-integer 1 < α < 2, the Green's function in Eq.…”

“…[2,14]. In these references, the flux is calculated by integrating the transport kernel along the field lines.…”

Dynamics of the major disruption of a DIII-D plasma

“…Future work will include more accurate modeling of the heat flux, including temperature-dependent Braginskii coefficients, use of Landau-fluid closures, 21,22 and a kinetic calculation of heat flux. 23,24 Much of the success of the modeling this discharge is due to the peaked, L-mode pressure profile of the equilibrium studied. Because the pressure gradient was peaked inside the plasma, the resultant mode was largely internal and resulted in little movement of the plasma boundary.…”