How Does Ion-Neutral Drag Affect Wave Propagation in a Magnetized Plasma?

[leoflindall]

Homework Statement

Consider a uniform static background medium in which the magnetic field is given by B ₀z and the density of the ions and electrons is [tex]\rho[/tex]_0i.

The background medium also contains neutrals with a density of [tex]\rho[/tex]_0n. Consider a wave that propagates in the Z direction which to first order in small quantities causes no perturbation in the density, but does have a perturbation in the magnetic field in the x direction.

Assume that the drag force per unit volume on the ions due to the neutrals is given by

[tex]\alpha[/tex][tex]\rho[/tex]_n[tex]\rho[/tex]_i (V_n - V_i )

Assume that w₁, the angular frequency of the wave is real. Take B_x, V_n, and V_i to vary as exp (iwt - ikz) with k = k_i - ik_i where k_r and k_i are real numbers.

Calculate k_i in the limit that [tex]\alpha[/tex] [tex]\rho[/tex]_nand [tex]\alpha[/tex] [tex]\rho[/tex]_i are both greater than w. (second term should read alpha roe i , but for some reason latex is showing it as another roe...)

In the magnetic induction equation, assume that the conductivity is infinite. Include the frictional term in the x componant of the equation of motion for the charged fluid. You will also have to use the x componant of the equation of motion for the neutral fluid to first order in small quantities that equation gives;


[tex]\rho[/tex]_0n dv_n/dt = [tex]\alpha[/tex][tex]\rho[/tex]_0n[tex]\rho[/tex]_0i (V_i - V_n ),

Homework Equations

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The Attempt at a Solution

I am very stuck with this question, can anybody give me any hints as to the best way to approach this question.

from the infintie conductivity i have derived that the the mangetic field in the z direction divided by the density is constant, but I am pretty sure that is a generla result.

I can't see how to takle this question, so any help would be greatly appreciated!

I'm not looking for an answer, I'm just trying to understand the question and what i need to do to solve it.

Many Thanks

Leo

 

[mark726]

Dear Leo,

Thank you for your post. This is a very interesting problem and I can see why you might be stuck. To start off, it might be helpful to review the basics of magnetohydrodynamics (MHD) and the equations that govern it.

In this problem, we are dealing with a uniform background medium which means that the magnetic field and density are constant throughout the medium. However, we are interested in the perturbations caused by a propagating wave in the x direction. This means that we will have to consider the equations of motion for both the charged fluid (ions and electrons) and the neutral fluid.

The first equation to consider is the continuity equation, which tells us how the density of a fluid changes with time. In this case, we are told that there is no perturbation in the density, so the continuity equation will not be affected by the wave. Next, we can look at the equation of motion for the charged fluid, which will include the drag force per unit volume given by \alpha\rhon\rhoi (Vn - Vi). This equation will also include the Lorentz force due to the perturbation in the magnetic field in the x direction. Finally, we can look at the equation of motion for the neutral fluid, which will also include the drag force per unit volume.

To solve this problem, you will need to use the equations of motion and the continuity equation to find the dispersion relation for the wave, which will relate the angular frequency w1 to the wavevector k. From there, you can use the given expressions for Bx, Vn, and Vi to find the value of ki in the limit given. Remember to use the fact that kr and ki are real numbers.

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your problem!