Partial entropic stabilization of lattice Boltzmann magnetohydrodynamics

The entropic lattice Boltzmann algorithm of Karlin et al. [Phys. Rev. E 90, 031302 (2014)] is partially extended to magnetohydrodynamics, based on the Dellar model of introducing a vector distribution for the magnetic field. This entropic ansatz is now applied only to the scalar particle distribution function so as to permit the many problems entailing magnetic field reversal. A 9-bit lattice is employed for both particle and magnetic distributions for our two-dimensional simulations. The entropic ansatz is benchmarked against our earlier multiple relaxation lattice-Boltzmann model for the Kelvin-Helmholtz instability in a magnetized jet. Other two-dimensional simulations are performed and compared to results determined by more standard direct algorithms: in particular the switch over between the Kelvin-Helmholtz or tearing mode instability of Chen et al. [J. Geophys. Res.: Space Phys. 102, 151 (1997)], and the generalized Orszag-Tang vortex model of Biskamp-Welter [Phys. Fluids B 1, 1964 (1989)]. Very good results are achieved.

Figure 1

The kinetic lattice vectors for 2D LB-MHD are, in our $D2Q9$ model, ${\vec{c}}_i=\left(0,0\right),\left(0,\pm 1\right),\left(\pm 1,0\right),\left(\pm 1,\pm 1\right)$. $w_i$ are appropriate weight factors dependent on the choice of lattice: $w_0=\frac{4}{9}$; for speed 1, $w_i=\frac{1}{9}$; and for speed $\sqrt{2}$, $w_i=\frac{1}{36}$.

Figure 2

(a) The initial velocity (dark) and vorticity (gray) in the unstable magnetized jet simulation as a function of $x$. (b) The corresponding initial vorticity $\omega (x,y)$: gray for $\omega >0$ and dark for $\omega <0$.

Figure 3

Evolution of a Kelvin-Helmholtz jet with very weak axial magnetic field: $B_0=0.005U_0$. The column 2D plots are for (a) the vorticity $\omega$, (b) the entropy parameter ${\gamma }^{*}$, and (c) the current $j$. The jet is unstable forming a von-Karman–like vortex street (time $t=44k$). These vortices start to generate secondary smaller vortex streaks ($t=80k$)—where the entropy factor becomes important. The vortex street then becomes unstable $t=344k$ with vortex-vortex reconnection dominating shortly after the break-up of the vortex street. However by $t=500k$ strong subsidiary vortices are generated because of the 2D MHD turbulence with significant corresponding regions of variations of the entropic parameter away from 2. Note that the color scheme is held constant for all time snapshots. Spatial grid ${1024}^2$. During the time evolution, the grayscale is unchanged in each column: (a), (c) gray for positive values, black for negative, while for (b) gray is maximum, black for $\gamma =0$ and white for $\gamma =2$.

Figure 4

Spectral plot of the Kelvin-Helmholtz simulation at $t=500k$ where the grid size is ${1024}^2$ and the slope of the dashed line is $k^{-1.67}$.

Figure 5

A snapshot of the 2D spatial dependence of (a) magnetic field lines (and velocity fields) from a Chen et al. supersonic Alfvenic simulation, their Fig. 6, compared to (b) our entropic LB-MHD simulation on ${1024}^2$ grid for the same initial profiles. $S=1000.$

Figure 6

A snapshot of the 2D spatial dependence of the (a) the magnetic field line contours (and velocity field) for zero initial shear ($V_0=0.0$), from Chen et al. Figure 4 and (b) from our entropic LB-MHD algorithm on a ${1024}^2$ grid.

Figure 7

Evolution of the magnetic field lines from our partial entropic LB-MHD code for zero initial shear velocity in a uniform magnetic field. Snapshots of the field lines are presented at each 8000 (8k) LB time steps. Grid ${1024}^2$. (a) $t=0$, (b) $t=8000$, (c) $t=16000$, (d) $t=24000$, (e) $t=32000$, (f) $t=40000$, (g) $t=48000$, (h) $t=56000$, (i) $t=64000$, (j) $t=72000$, (k) $t=80000$, (l) $t=88000$.

Figure 8

Snapshot of the current lines from (a) partial entropic LB-MHD code on a grid of ${1024}^2$ at time $=226k$, (b) Biskamp-Welter, Fig. 11(a). Gray for positive values, black for negative values.

Figure 9

Plot of the entropic parameter ${\gamma }^{*}$ after 226k timesteps on a ${1024}^2$ grid. $\gamma =2.0$ corresponds to ordinary LB-MHD. Lattice points with ${\gamma }^{*}\ne 2.0$ correspond to the effects of the partial entropic LB-MHD algorithm. Gray for positive values, black for negative, while for (b) gray is maximum, black for $\gamma =0$ and white for $\gamma =2$.

Figure 10

Plot of energy dissipation over time for the Biskamp-Welter profile.