samples/texts/2974107/page_2.md

Aligned interpolation and application to drift kinetic semi-Lagrangian simulations with oblique magnetic field in cylindrical geometry

G. Latu, M. Mehrenberger, M. Ottaviani, E. Sonnendrücker

December 24, 2014

Abstract

We introduce field aligned interpolation for Semi-Lagrangian schemes, adapting a method developed by Hariri-Ottaviani [7] to the semi-Lagrangian context. This approach is validated on the constant oblique advection equation and on a 4D drift kinetic model with oblique magnetic field in cylindrical geometry. The strength of this method is that one can reduce the number of points in the longitudinal direction. More precisely, we observe that we gain a factor $\frac{|n|}{|n+m\boldsymbol{i}|}$ (where $\boldsymbol{i}$ is the inverse safety factor), with respect to the classical approach, for the typical function $\sin(m\theta + n\varphi)$.

1 Introduction

In gyrokinetic simulations, it is observed that solution structures follow the field lines of the (strong) magnetic field and numerical methods have to be adapted to benefit from this fact. Different strategies exist for dealing with field alignement in gyrokinetic codes (see [9], [5] for example). We explore here an idea developed recently in [7] and adapt it in the context of a semi-Lagrangian code.

Our example of validation will be the following 4D drift-kinetic equation in cylindrical geometry, with oblique magnetic field. We look for $f = f(t, r, \theta, z, v)$ satisfying

$${\mathrm{\partial }}_{t}f+[\varphi ,f]+v{\mathrm{\nabla }}_{\parallel }f-{\mathrm{\nabla }}_{\parallel }\varphi {\mathrm{\partial }}_{v}f=0,\backslash partial\_t\; f\; +\; [\backslash phi,\; f]\; +\; v\; \backslash nabla\_\{\backslash parallel\}\; f\; -\; \backslash nabla\_\{\backslash parallel\}\; \backslash phi\; \backslash partial\_v\; f\; =\; 0,$$

with

$$[\varphi ,f]=-\frac{{\mathrm{\partial }}_{\theta }\varphi }{r{B}_0}{\mathrm{\partial }}_{r}f+\frac{{\mathrm{\partial }}_r\varphi }{r{B}_0}{\mathrm{\partial }}_{\theta }f,{\mathrm{\nabla }}_{\parallel }=\mathbf{b}\cdot \mathrm{\nabla },[\backslash phi,\; f]\; =\; -\backslash frac\{\backslash partial\_\{\backslash theta\}\backslash phi\}\{rB\_0\}\backslash partial\_r\; f\; +\; \backslash frac\{\backslash partial\_r\backslash phi\}\{rB\_0\}\backslash partial\_\{\backslash theta\}f,\; \backslash quad\; \backslash nabla\_\{\backslash parallel\}\; =\; \backslash mathbf\{b\}\; \backslash cdot\; \backslash nabla,$$

so that

$${\mathrm{\partial }}_{t}f-\frac{{\mathrm{\partial }}_{\theta }\varphi }{r{B}_0}{\mathrm{\partial }}_{r}f+(\frac{{\mathrm{\partial }}_r\varphi }{r{B}_0}+v\frac{b_{\theta }}{r}){\mathrm{\partial }}_{\theta }f+v{b}_z{\mathrm{\partial }}_{z}f-(b_{\theta }\frac{{\mathrm{\partial }}_{\theta }\varphi }{r}+b_z{\mathrm{\partial }}_z\varphi ){\mathrm{\partial }}_{v}f=0,(1)\backslash partial\_t\; f\; -\; \backslash frac\{\backslash partial\_\backslash theta\; \backslash phi\}\{r\; B\_0\}\; \backslash partial\_r\; f\; +\; \backslash left(\; \backslash frac\{\backslash partial\_r\; \backslash phi\}\{r\; B\_0\}\; +\; v\; \backslash frac\{b\_\backslash theta\}\{r\}\; \backslash right)\; \backslash partial\_\backslash theta\; f\; +\; v\; b\_z\; \backslash partial\_z\; f\; -\; \backslash left(\; b\_\backslash theta\; \backslash frac\{\backslash partial\_\backslash theta\; \backslash phi\}\{r\}\; +\; b\_z\; \backslash partial\_z\; \backslash phi\; \backslash right)\; \backslash partial\_v\; f\; =\; 0,\; \backslash quad\; (1)$$

for $(r, \theta, z, v) \in [r_{\min}, r_{\max}] \times [0, 2\pi] \times [0, 2\pi R] \times [-v_{\max}, v_{\max}]$. The self-consistent potential $\phi = \phi(r, \theta, z)$ solves the quasi-neutral equation without zonal flow

$$-({\mathrm{\partial }}_r^2\varphi +(\frac{1}{r}+\frac{{\mathrm{\partial }}_r{n}_0}{n_0}){\mathrm{\partial }}_r\varphi +\frac{1}{r^2}{\mathrm{\partial }}_{\theta }^2\varphi )+\frac{1}{T_e}\varphi =\frac{1}{n_0}(\int f-f_{eq}dv)\mathrm{.}-\backslash left(\backslash partial\_r^2\; \backslash phi\; +\; \backslash left(\backslash frac\{1\}\{r\}\; +\; \backslash frac\{\backslash partial\_r\; n\_0\}\{n\_0\}\backslash right)\; \backslash partial\_r\; \backslash phi\; +\; \backslash frac\{1\}\{r^2\}\; \backslash partial\_\backslash theta^2\; \backslash phi\backslash right)\; +\; \backslash frac\{1\}\{T\_e\}\; \backslash phi\; =\; \backslash frac\{1\}\{n\_0\}\; \backslash left(\backslash int\; f\; -\; f\_\{eq\}\; dv\backslash right).$$

Here the oblique magnetic field B whose norm is B (which can depend on r) writes

$$\mathbf{B}=B\mathbf{b},\mathbf{b}=b_z\widehat{\mathbf{z}}+b_{\theta }\widehat{\theta },b_{\theta }=\frac{c}{\sqrt{1+c^2}},b_z=\frac{1}{\sqrt{1+c^2}},c=\frac{ir}{R},\backslash mathbf\{B\}\; =\; B\backslash mathbf\{b\},\; \backslash quad\; \backslash mathbf\{b\}\; =\; b\_z\backslash hat\{\backslash mathbf\{z\}\}\; +\; b\_\backslash theta\backslash hat\{\backslash theta\},\; \backslash quad\; b\_\backslash theta\; =\; \backslash frac\{c\}\{\backslash sqrt\{1+c^2\}\},\; \backslash quad\; b\_z\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+c^2\}\},\; \backslash quad\; c\; =\; \backslash frac\{ir\}\{R\},$$