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exercises.md

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HARMPI exercises by Sasha Tchekhovskoy

Please also see the tutorial that explains basic code use.

Hydro problems

To run problems with HARMPI and to analyze the results, please follow this tutorial.

1D hydro problems

  • Bondi accretion

    Set WHICHPROBLEM to BONDI_PROBLEM_1D in decs.h. Note: a good total resolution is 256x1x1

    • Plot the profiles of density at a few times in a simulation. Determine the position of the sonic surface. Hint: look at where v1p variable changes sign. Ordinarily, v1p tells dr/dt of the outgoing (positive, p) fast wave (in this problem there is no magnetic field, so fast waves are sound waves). At large radii the flow barely falls inward, so it will be > 0 but at small radii the flow falls inward supersonically, at the speed of light, so it will be < 0.

Magnetized problems

1D magnetized problems

  • Monopole problem

    Set WHICHPROBLEM to MONOPOLE_PROBLEM_1D in decs.h. Note: a good total resolution is 768x1x1

    • Measure the ratio OmegaF/OmegaH. How does it compare with the standard value, 0.5?
    • What is the initial magnetization (near the black hole) of the plasma? Hint: look at the quantity $\sigma_0 = b^2/4\pi\rho c^2 =$ bsq/rho . Here $b^2/4\pi =$ bsq is the square of the fluid frame magnetic field and $\rho =$ rho is the fluid frame mass density.
    • Make a plot of Lorentz factor vs. radius, $\Gamma(r)$. Hint: $Gamma =$ alpha * uu[0]. What value does the Lorentz factor saturate at? How does it compare to the near-black hole magnetization, $\sigma_0$, of the magnetosphere determined above?

2D magnetized problems

  • BZ-Michel monopole problem

    Set WHICHPROBLEM to BZ_MONOPOLE_2D in decs.h. Note: a good total resolution is 256x256x1

    • Plot the location of the surface in $R$--$z$ plane at which the radial contravariant component of velocity, $u^r$, vanishes. Hint: plot the contour of uu[1] == 0.
    • Plot the dependence of the ratio OmegaF/OmegaH at the horizon as a function of the angle, $\theta$.
    • Compare the prediction of power by the standard Blandford & Znajek (1977) power formula to the simulation results.

  • 2D monopole problem

    Set WHICHPROBLEM to MONOPOLE_PROBLEM_2D in decs.h. Note: a good total resolution is 1152x256x1. It will take some time for the problem to complete depending on the number of cores used.

    • Compare $\Gamma(r)$ to that in the above 1D monopole problem. In which case is the Lorentz factor higher? Why?
    • Verify that the value of the Lorentz factor at the fast surface is $\Gamma =$ (bsq/rho)**0.5. What is the value of $\Gamma$ at the fast surface?

  • Torus problem

    Set WHICHPROBLEM to TORUS_PROBLEM in decs.h. Note: a good total resolution is 256x256x1. It will take some time for the problem to complete depending on the number of cores used.

    • Check by how many cells the MRI wavelength is resolved in the initial conditions. Good resolution is $\gtrsim15$ cells per wavelength, but you could sometimes also get away with $\gtrsim 5{-}10$. Hint: a call to Qmri(dir=2) returns the number of cells per wavelength of the fastest growing MRI mode in the $\theta$-direction.
    • Make a movie of the simulation: logarithm of density shown with color contours overlaid with magnetic field lines. Feel free to ask for guidance on how to make the movie.