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Boris-HLLD: a new MHD Riemann solver for reducing Alfven speed
- 1. Boris-HLLD: a new MHD Riemann solver for reducing Alfven speed Tomoaki Matsumoto（Hosei Univ） Takahiro Miyoshi (Hiroshima Univ) Shinsuke Takasao (Nagoya Univ/NAOJ) and James Stone (IAS) Application: MHD simulation of circumbinary disk ASTRONUM 2019 7/4, 1:55-2:20 PM
- 2. New MHD Riemann solver: Boris-HLLD 2 Matsumoto, Miyoshi, Takasao, ApJ, 874, 37 (2019) (arXiv:1902.02810) TM Takahiro Miyoshi Shinsuke Takasao
- 3. Troublesome MHD simulations 3 Very low density Very fast Alfven speed Very small timestep Larger timestep Reducing Alfven speed Easy to install
- 4. Back ground • Boris correction (Boris 1970) reduces Alfven speed, bounded by c: vA < c – c can set at a low value artificially. • It is widely used in both astrophysics and space science. – Star formation: Allen+ 03 – Accretion disks: Miller & Stone 00, Parkin 14 – Solar physics: Rempel 17 – Magnetosphere: Lyon+ 04 – Space weather: Gombosi+02, Toth+ 12, c.f., Talks of Sokolov and Toth • However, conventional Riemann solvers have been used, e.g., Lax- Friedrichs, HLL. • A high-resolution Riemann solver (e.g., HLLD) is desired. • We developed a new MHD Riemann solver with Boris correction, based on HLLD solver. 4
- 5. Ideal MHD equations with Boris correction 5 Simplified version of Boris (1970) correction. A form of the semi-relativistic MHD equations. additional inertia from displacement current. Reduced Alfven speed
- 6. Features of Boris-HLLD Riemann solver 6 HLL(E) Original HLLD Boris-HLLD (this work) # of jump conditions 2 5 5 # of states in Riemann fan 1 4 4 Resolve CD? No Yes Yes Preserve positivity? Yes Yes ? t x UL UR 0 SM SR SL SR * UL * UR * UR **UL ** SL * SL < 0 < SM *
- 8. Linear Alfven waves 8 Speed of light Propagationspeed Wave speed is bounded by c Theoretical prediction Wave speed approaches classical VA
- 9. Shock tube problem, Brio & Wo test 9 Boris-HLLD reduces wave speed. Boris-HLLD captures CD sharply. Boris-HLLD shows NO overshoot at shock wave. Density Velocity (vx)
- 10. Stability tests with linear waves 10 Magnetic field 1. Magneto-sonic wave (fast/slow wave) 2. Alfven wave 3. sound wave (fast/slow wave) Parameters to be changed: gas velocity, Alfven speed, sound speed
- 11. Stability tests; magneto-sonic waves 11 Stable Unstable Unstable Stable Unstable Unstable Boris-HLLD is stable when Gasvelocity Gasvelocity Alfven speed Sound speed
- 12. Stability tests; Alfven waves 12 Stable Stable
- 13. Stability tests; sound waves 13 Stable Stable
- 14. Orszag-Tang vortex problem 14 Classical MHD Boris c=2 (>vmax)Boris c=10 (>vmax) Boris c=1 (<vmax) Almost same Crash Gas velocity exceeds c
- 15. Circmbinary disks of protobinary systems: MHD effects 15 Matsumoto & Stone in preparation James Stone (IAS) TM
- 16. How to make “initial condition” 16 t t = –10 rev t = 0 revHD MHD “Initial condition” Uniform, vertical B t = 10 rev Making “steady state” Variable speed of light W/o Boris correction, only 1.5 rotation periods can be followed. x-y plane x-z plane
- 17. 17 x-y plane x-z plane logdensitylogdensity C.f., Machida+ 04, 05, Kuruwita+17 Fast outflows are driven by circumstellar disks. Slow outflow is driven by circumbiary disk. MRI in circumbinary disk
- 18. Boris correction is effective in outflows 18 Iso-surfaces beta = 10^{-10} Very low beta!
- 19. Summary • We develop a new MHD scheme (Boris-HLLD), based on – Boris correction (reducing Alfven speed) and – HLLD scheme (high-resolution shock-capturing scheme). – Alfven speed is bounded by the reduced speed of light c. – The scheme is stable when speed of light is several times larger than gas speed: c > (a few) u • Boris-HLLD is applied to circumbinary disk simulations. – It enables us to follow a long-term evolution (>10 rotation period in MHD phase) – A model with very low beta region is solved with time-steps that are not extremely small. 19
Editor's Notes
- I am Tomoaki Matsumoto from Hosei university. Today, I would like to talk about Boris-HLLD, a new MHD Riemann solver for reducing Alfven speed. In this talk, at first, I will show you the new MHD scheme and also several test problems. Next, I will show an application of the new scheme to the problem of a circumbinary disk in context of star formation.
- The first topic is a new MHD Riemann solver. In this work, I collaborate with Miyoshi-san and Takasao-san. Miyoshi-san is a developer of the original HLLD Riemann solver. The HLLD Riemann solver is now widely used in the MHD community.
- This cartoon shows the motivation for this work. I think that all of you working on MHD simulations experienced such a situation. If you have a very low density in your computational domain, you have a very high Alfven speed there. Then you have a very short timestep, which brings about time-cosuming computation. If you use Boris-HLLD Riemann solver, it reduces Alfven speed. and you can take a larger timestep. This is a take home message.
- This is background It is known that, so called Boris-correction reduces Alfven speed, so that it is bounded by speed of light c. The speed of light can be set at a low value artificially. It is widely used in both the astrophysics and space science by these people. In this conference, some speakers mentioned the Boris correction. However, conventional Riemann solvers have been used, e.g., Lax-Friedrichs, HLL. A high-resolution Riemann solver (e.g., HLLD) is desired. We now developed a new MHD Riemann solver with Boris correction, based on HLLD solver.
- These are the ideal MHD equations with Boris correction. There are several formulations of the Boris correction, We adopt the simplified version. This a form of semi-relativistic MHD equations. We add only one term to momentum equation, shown in yellow. This is additional inertia because of the magnetic fields. It comes from the displacement current. Because of additional inertia, the gas motion of the Alfven wave is reduced by a factor gamma_A, which is expressed as follows. The gamma factor goes down when VA goes up. So Alfven speed is reduced. The VA is small, gamma_A reaches unity, the MHD equations corresponds to the classical MHD equations.
- The derivation of Riemann solver is boring. I just show you a comparison between the new scheme and the existing scheme. The left column is for the HLL scheme, the middle column for the original HLLD scheme, and the right column is for Boris-HLLD scheme. The Boris-HLLD solver has almost features of HLLD solver. They have 5 jump conditions and 4 states in the Riemann fan. They also resolve contact discontinuities. HLL and HLLD solvers ensure positivity in their solution. Unfortunately, we were not able to prove a positivity of Boris-HLLD solver.
- We move to test problem.
- First, we estimated the propagation speed of linear Alfven waves. The vertical axis is a wave speed normalized by the classical Alfven speed. The horizontal axis is a speed of light normalized by the classical Alfven speed. The dots are observed speeds. and the solid line is a theoretical prediction. When the speed of light is low, propagation speed decreases. bounded by c. When the speed of light is high, propagation speed approaches the classical Alfven speed.
- This is shock tube problem of Brio & Wo. Left and right panels show density and velocity distributions, respectively. Green and orange lines are solutions obtained by Boris-HLLD scheme with c=2 and 3, respectively. For comparison, we show the solutions obtained by the original HLLD and HLL scheme. Here is fast rarefaction wave, compound wave, contact discontinuity, show shock, and fast rarefaction wave. Basically the right side has a high Alfven speed, and Boris correction affect solution more than left side. As we increase c, the solution approaches the classical one. When we decrease c, the travel distance of the fat rarefaction wave decreases here. The inset compares solutions between Boris-HLLD and Boris-HLL schemes. Boris-HLLD captures CD sharply. Boris-HLLD shows NO overshoot at shock wave.
- Now we move to stability tests with linear waves. We consider three types of waves, Magneto-sonic wave perpendicular to the magnetic field. Alfven wave parallel to the magnetic field. Sound wave parallel to the magnetic field. We change parameters of bulk gas velocity, Alfven speed, sound speed.
- This is a result for magneto-sonic waves. This axis is gas velocity and these axes are Alfven speed and sound speed. Each pixel corresponds to amplification factor of each wave. Red region is unstable. Blue region is stable. This graph indicate that Boris-HLLD is stable when gas velocity is less than half of speed of light.
- Next is Alfven wave. Alfven wave is always stable.
- The sound wave is always stable too.
- This is Orszag-Tang vortex problem of the classical HLLD and Boris-HLLD with different speed of light. When the speed of light is larger than maximum gas velocity, the solution is almost same as that of classical MHD. In the case of c=1, the calculation crashed because the maximum gas velocity exceed c.
- We move to the circumbinary disk of protobinary star system. I collaborating with James Stone for this work. I will show some preliminary results.
- This side shows how to make the initial condition. In the case of a disk around a single star, people start at the hydrodynamical equilibrium solution. In the case of the circumbinary, we don’t know the the equilibrium solution. So we take this strategy. In the period of t-10 to 0 rev, we calculate hydrodynamical model without magnetic field. Gas is injected at boundary toward binary stars. We obtain roughly steady state solution, we add a vertical uniform magnetic field. Se use variable speed of light as follows. C is basically 50, but 4 times larger than the maximum gas speed. For comparison, isothermal sound speed is 0.1, and typical velocity of stars is around unity.
- This is a movie. The left panels show density in the x-y plane. The right panel show in the x-z plane. Lower panels are magnification of top panels. As shown here, after the magnetic field sets in, outflows are ejected. Fast outflows are driven by circumstellar disks. Slow outflow is driven by circumbiary disk. MRI is excited in the circumbinary disk.
- This shows where Boris correction is effective. The isosurfaces show the region where Alfven speed reaches the speed of light, and Boris correction is effective therein. This indicates that Boris correction affects only along the outflow region. There we have very low beta, about 10^-10. Even such extremely low beta, we are able to calculate it with moderate timestep because of Boris-correction.