Relativistic Hydrodynamics and Magnetohydrodynamics
The dynamic of a typical astrophysical fluid governed by the inviscid Euler equations, both in the Newtonian and in the relativistic regime, is characterized by the generation and propagation of non-linear waves, such as shock waves. If we want to understand the dynamics of these waves within the fluid we must first treat an idealized situation, i.e. the evolution of a fluid initially composed of two states with different and constant values of velocity, pressure and density. Such a simplified problem is called a Riemann Problem
From a mathematical point of view a Riemann problem is an initial value problem for a given set of hyperbolic partial differential equations with initial conditions represented by two constant states:
Just like in Newtonian hydrodynamics, also in the relativistic regime the evolution of the system (the solution of the Riemann problem) will produce on of the following wave patterns
2S: One shock wave propagating to the left and one shock wave propagating to the right
SR: One shock wave propagating to the left and one rarefaction wave propagating to the right (or vice-versa)
2R: One rarefaction wave propagating to the left and one rarefaction wave propagating to the right
Click on video to see the evolution of the pressure in a generic Riemann problem generating a SR wave pattern
In special relativistic Hydrodynamics the exact solution of the Riemann problem takes advantage of the Rankine-Hugoniot conditions when treating shock waves, and of the self similar character of the flow when treating rarefaction waves. Moreover, it is possible to know in advance which of the possible three wave pattern will actually take place, by looking at the relative normal velocity vx12 between the two initial states of the Riemann problem.
In fact, vx12 is a monotonic function of the pressure across the contact discontinuity p3 and assumes a different form according to the wave pattern produced (2S, SR, 2R). For each initial data, the comparison of vx12 with the relevant limits predicts the wave pattern that will be produced.
The method proposed has two advantages:
1- If used in a numerical code based on exact Riemann solvers, it performs more efficiently with respects to a method that does not predict the final wave pattern
2- It puts into evidence the occurrence of a genuine special relativistic effect, not present in the Newtonian regime, produced when there are non zero velocities tangential to the two initial states.
A new special relativistic effect: while maintaining the initial conditions (p, r, vx) not altered, a change in the tangential velocities produces a shift from one wave-pattern to another.
Example: consider a well document Riemann Problem: Sod’s problem:
The wave pattern changes from RS to 2S when the tangential velocity in the right state is 0.9, and this is a relativistic effect!
Click on video to see the transition from the SR to the 2S wave pattern when the tangential velocity in the right state is increased. The solution is plotted at the same final time but for different values of the tangential velocity.
Essential bibliography:
1 The analytical solution of the Riemann problem in relativistic hydrodynamics, Marti J. M., Muller E., JFM, 258, 317 (1994)
2 An improved exact Riemann solver for relativistic hydrodynamics, Rezzolla L., Zanotti O., JFM, 449, 395 (2001)
3 New relativistic effects in the dynamics of nonlinear hydrodynamical waves, Rezzolla L., Zanotti O., Phys. Rev. Lett., 89, 114501 (2002)
4 The exact solution of the Riemann problem in relativistic magnetohydrodynamics, Giacomazzo B., Rezzolla L., JFM, 562, 223 (2006)
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