ASCI/Alliances Center for Astrophysical ASCI/Alliances Center for Astrophysical Thermonuclear FlashesThermonuclear Flashes

FLASH MHD

Timur Linde

FLASH MHD

Timur Linde

This work was supported by the ASCI Flash Center at the University of Chicago under DOE contract B341495.This work was supported by the ASCI Flash Center at the University of Chicago under DOE contract B341495.

0)(t

=ρ⋅∇+∂ρ∂

V

ô g BB I VVV

⋅ ∇+ ρ =⎟⎟⎠

⎞⎜⎜⎝

⎛− ⎟

⎠

⎞⎜⎝

⎛+ + ρ ⋅ ∇+∂ρ∂2

B21

pt

()()()() ()() T B2

1p E

t

E2∇κ ⋅ ∇+ × ∇ η × ⋅ ∇+ ⋅ ⋅ ∇+ ⋅ ρ =⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅ −⎟

⎠

⎞⎜⎝

⎛+ + ⋅ ∇+∂

∂B B ô V V g B V B V

( ) ( )BBVVBB

×∇η×−∇=−⋅∇+∂

∂

t

The equations of magnetohydrodynamics (MHD) describe the behavior of electrically conducting fluids, in particular of astrophysical plasmas, in the presence of magnetic fields.

In appropriate non-dimensional units the equations that the FLASH code uses are:

() B B⋅ ∇ −

()()() B B B B V⋅ ∇ ∇⋅ ε + ⋅ ∇ ⋅ −

( ) ( )BBV ⋅∇∇ε+⋅∇−

B⋅ ∇

Terms controlling in the ideal MHD limit.

• General equation of state and variable transport coefficients.

• Ability to solve ideal and visco-resistive problems with heat conduction.

• Second-/fourth-order, TVD and WENO (in progress) algorithms.

• Uniform programming interface for all algorithms.

• Built-in magnetic monopole self-cleaning mechanism.

Magnetic monopole control in ideal MHD simulationsMagnetic monopole control in ideal MHD simulations

EquationsEquations

Current code capabilitiesCurrent code capabilities

Our current strategy with regard to numerically created magnetic monopoles is to accept their spurious existence. In order to properly implement this strategy we utilize the method of Powell et al. (J. Comput. Phys., v. 154, p. 284, 1999). At the same time we believe that in certain situations artificial monopoles can damage computations. Therefore our implementation of the ideal MHD equations contains diffusion-like terms (similar to those suggested by Marder, J. Comput. Phys., v. 68, p. 48, 1987) that inhibit the growth of magnetic monopoles at the scales at which the monopoles are created. The resulting algorithm is local and very efficient; our analysis shows that it is quite capable of suppressing the generation of magnetic monopoles. We retain the option of using the elliptic projection method (Brackbill and Barnes, J. Comput. Phys., v. 35, p. 426, 1980), however it is rarely needed. error accumulation in the Orszag-Tang

problem. Br

⋅∇

Magnetic field intensity in the Orszag-Tang problem. Solution without monopole control.

Magnetic field intensity in the Orszag-Tang problem. Solution with monopole control.

Monopole contamination without control. Monopole contamination with control.

Example problem: magnetic Rayleigh-Taylor instabilityExample problem: magnetic Rayleigh-Taylor instability

We compute three linearly unstable cases with the density jump of 20 and with non-dimensional gravity of 0.1 (dimensionless units are based on the parameters of the light fluid). The initial condition is perturbed by the k=4 wave mode. In the first simulation the magnetic field strength is set to zero; in the other two the magnetic field with the strength of 0.25 is imposed parallel and perpendicular to the initial interface. As expected in the non-magnetized case the system is nonlinearly unstable. In the parallel magnetized case the system exhibits nonlinear stability while in the perpendicular case it is still nonlinearly unstable although the corresponding instability growth rate is reduced by magnetic tension. A detailed analysis of the magnetized R-T instability growth rates is in progress.

Growth of bubbles and spikes in the RT problem Non-magnetized case at t=1.5 Parallel case at t=1.5 Perpendicular case at t=1.5