A New Split-Weight Scheme for Finite-$\beta$ Gyrokinetic Plasmas

The original split-weight scheme for finite-$\beta$ simulations [1], which separates the perturbed particle distribution into an adiabatic part and a non-adiabatic part, is generalized to include spatial inhomogeneities. The new scheme requires an additional separation of the fast particle response associated with quasi-static bending of the magnetic field lines. While the original scheme follows the non-adiabatic response, $\delta h$, in time, where $\delta h = F - (1 + \psi) F_0$, $F$ is the distribution, $F_0$ is the background, $ \psi \equiv \phi + \int (\partial A_\parallel / \partial t) d x_\parallel/c $ and $\phi$ and $A_\parallel$ are the perturbed potentials, the new scheme makes use of $\hat {\bf b} \cdot \nabla (F_0+\delta g)=0, $ where $\hat {\bf b}=\hat {\bf b}_0+\delta {\bf B}/B_0$, and further separates the plasma response as $ F=(1 + \psi) F_0 +\delta g %+\int dx_{||}{\kappa}_n\cdot (\nabla A_{||}\times \hat {\bf b}_0) +\delta h, $ where $ \delta g = \int dx_{||}{\kappa} \cdot (\nabla A_{||}\times \hat {\bf b}_0) $ and $\kappa$ is the zeroth order spatial inhomogeneity. The new $\delta h$ is again followed in time. The results for finite-$\beta$ stabilization of drift waves and ion temperature gradient modes in slab geometry using the new scheme with a $\beta (\equiv c_s^2/v_A^2 )$ as high as $10\%$ and a grid size of the order of the electron skin depth, are in agreement with those discussed in Refs. [2] and [3]. This work is supported by the DoE OASCR Multi-Scale Gyrokinetics (MSG) Project. [1] W. W. Lee, J. Lewandowski, Z. Lin and T. S. Hahm, Phys. Plasmas {\bf 8}, 4435 (2001). [2] J. V. W. Reynders, Ph. D. Thesis, Princeton University (1992). [3] J. C. Cummings, Ph. D. Thesis, Princeton University (1995).