Electromagnetohydrodynamics

Q: What are the dispersion curves for EMHD equations?

The dispersion curves for EMHD equations show the dependence of nondimensional phase velocity of the wave on the nondimensional wave vector k. They represent the wave speed on the conductivity coefficient for a fixed wave vector or on the wave vector at constant conductivity. The solutions have simple interpretations in limiting cases. For large k or small n, there are two EM and two sound waves moving in opposite directions. For small k or large n, there are two fast and two slow MHD waves. In the middle region, the behavior becomes more complicated, and physical interpretation requires solving the EMHD equations directly. Transformations of fast MHD waves into EM waves and sound waves into slow MHD waves are suggested in certain cases. The solutions in the intermediate region are presented as simple dots in Figs. 5 and 6.

Question

1. What is magnetohydrodynamics (MHD) and its limitations?

2. What equations describe EM wave formation in magnetized plasma?

3. What is the equation for H in the large conductivity limit?

4. What are the two solutions for linear waves in an almost ideal plasma with nm=0?

Accepted Answer

Magnetohydrodynamics (MHD) is a field that describes the behavior of magnetized media, introduced by H.Alfven in 1942. It considers the interaction between magnetic fields and conducting fluids, but does not account for displacement electrical currents. MHD is suitable for slow processes, as it assumes that the fluid moves slowly enough for the magnetic field to remain frozen in the fluid. In practice, most processes with conducting magnetized fluids on Earth satisfy these conditions. However, in laboratory experiments involving rapid plasma clods colliding with magnetic walls, displacement currents and electromagnetic wave generation become significant. In astrophysical objects with strongly magnetized objects moving at relativistic speeds, such as X-ray pulsars, AGN, and GRBs, it is necessary to consider displacement currents to describe transformations between MHD and electromagnetic waves. To study these processes, equations derived from hydrodynamic and Maxwell equations, known as electromagnetohydrodynamic (EMHD) equations, are used. These equations consider a plasma with zero viscosity and heat conductivity, but finite electric conductivity. The EMHD equations help describe the formation of MHD shock waves and electromagnetic waves in magnetized plasmas. The propagation of weak linear waves in a magnetized plasma is described by a dispersion equation, which includes MHD waves, sound waves, and electromagnetic waves in limiting cases, as well as new types of behavior in general situations. Overall, MHD provides a framework for understanding magnetized fluid behavior, but its limitations necessitate the use of EMHD equations for more complex scenarios.

Accepted Answer

The equations describing EM wave formation in magnetized plasma involve the displacement current, Maxwell's equations, and Ohm's law. By neglecting the Hall effect and thermodiffusion, the equations can be written as EQUATION. By expressing E from Ohm's law and deriving j, we obtain EQUATION. Excluding j, we get H t = rot[u x H] - rot(n m rotH) + c 4p t rot E s. The equation for the magnetic field is EQUATION. Additional equations for velocity u, matter density r, and pressure P are needed, which in the absence of viscosity and thermal conduction have the form EQUATION EQUATION. The equations for fluid motion in the fields H and E are EQUATION EQUATION. These equations describe the behavior of EM waves in magnetized plasma.

Accepted Answer

In the large conductivity limit (n m 1), the equation for the ideal MHD [2] is obtained. This equation describes the situation where MHD processes and electromagnetic waves are important at the same time. Accurate results on the properties of the electromagnetic wave, formed in the collision of the gas flow with the magnetic wall, can be obtained using equation (4) in the 1D case (y = z = 0).

Accepted Answer

In an almost ideal plasma with nm=0, there are two solutions for linear waves. The first solution corresponds to the limit of slow MHD wave in the case of its propagation across the magnetic field (static perturbation), while the second one corresponds to fast MHD wave in the same case. The frequency of the fast MHD wave differs from the usual MHD due to effects of the electric field. In vacuum case with uAc, this wave transforms into an electromagnetic wave, while in the opposite case, one can obtain a usual value for MHD, where the displacement current is neglected. The relativistic values allowed for the magnetosonic speed should still be non-relativistic, csc, as in traditional MHD equations. In the presence of low dissipation, the static perturbation and fast MHD wave are damping. With a small addition to the second solution of (24) in the form EQUATION, a solution is found by inserting (25) into (23). In limiting cases of very dense (csc-c) and very rarefied (uAc) medium, the wave damping does not occur even with finite nm. For weak damping of the static perturbation, one can formally obtain the increment EQUATION, which is not applicable in the field-free case.

Accepted Answer

The four types of waves in magnetized plasma are sound waves (SW), conversion of waves, magnetized ideal MHD plasma, and plasma with strong damping. These waves correspond to different modes based on parameters a and s. The transition region from magnetized ideal MHD plasma to plasma with EM waves requires more detailed analysis. The dispersion relation curves for these waves are defined by the functions x r (k) and x i (k), which are obtained by solving the dimensionless dispersion equation. The limits of ideal MHD and plasma with strong damping are clearly seen in the plots for high and low values of z, respectively.

Accepted Answer

In astrophysical objects, relativistic effects may be important and we may expect a, s 1. The topology of wave modes takes place for a < a b 2.82s for low s 1, and this ratio changes with the increase of s to a b 2.75s at s = 0.1, where the value of Alfven velocity approaches relativistic value. For relativistic values of a and s, the qualitative presentation, similar to Figs. 1,2, takes place for Alfven velocities below the threshold value a thr, depending on s (see Table I below). The connections between EM and SMHD waves occur at z = n/k ~ 1 with a limiting case of z = 1 for a, s approaching unity, while the connections between FMHD and sound waves occur at z < 1. The region without waves takes place if a > a b 2.82s at a 1 for low s values, i.e., in highly magnetized plasma. At small z = n/k, sound and EM waves are present, while in the relativistic regime with a 1, the system deviates from regimes 1 and 2. The connection of electromagnetic waves with fast MHD wave occurs at 1 < z < 2 with the limiting value of z = 1 at a, s 1. The connection between sound waves and static perturbations happens at z ~ 1. The vicinity of wave transitions region (Case 2, imaginary parts) for parameters s = 0.03, a = 0.15 shows the region near the transition point of EM waves with fine grid to resolve large gradients in the real part of the solution. The behavior of different wave modes in electromagnetic waves in astrophysical objects is complex and depends on the values of a, s, and z.

Accepted Answer

The dispersion curves for EMHD equations show the dependence of nondimensional phase velocity of the wave on the nondimensional wave vector k. They represent the wave speed on the conductivity coefficient for a fixed wave vector or on the wave vector at constant conductivity. The solutions have simple interpretations in limiting cases. For large k or small n, there are two EM and two sound waves moving in opposite directions. For small k or large n, there are two fast and two slow MHD waves. In the middle region, the behavior becomes more complicated, and physical interpretation requires solving the EMHD equations directly. Transformations of fast MHD waves into EM waves and sound waves into slow MHD waves are suggested in certain cases. The solutions in the intermediate region are presented as simple dots in Figs. 5 and 6.

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Most frequently asked questions

1. What is magnetohydrodynamics (MHD) and its limitations?

2. What equations describe EM wave formation in magnetized plasma?

3. What is the equation for H in the large conductivity limit?

4. What are the two solutions for linear waves in an almost ideal plasma with nm=0?

5. What are the four types of waves in magnetized plasma?

6. What is the behavior of different wave modes in electromagnetic waves (EM) in astrophysical objects like pulsars, X-ray sources, and AGN?

7. What are the dispersion curves for EMHD equations?

Citations

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