Expert: Philip A. Stahl - 2/10/2007

Question
Can you explain to me how Alfven waves arise? Please use any necessary math to show properties of these waves.

Answer
Hello,


Alfven waves are the most important waves propagating in the solar atmosphere, as well as the Earth’s magnetosphere (underpinning the coupling between it and the ionosphere). They are important in that they efficiently carry energy and momentum along the magnetic field.

One way to get a handle (of sorts) on Alfven waves is to look at the analogy with mechanical waves – say propagating along a string put under tension. Consider the reference frame or coordinate system:

^ y
!
!----------------------------------------->.x


Say x marks the direction of propagation in the above coordinate system, and y is the direction of transverse (wave) displacement. Then the vertical force component is:

F_y =   - T(@y/ @x)

where T is the tension and the bracketed quantity is the partial of y with respect to x. Thus, just as the restoring force for a mechanical wave is the string tension T, the restoring force for an Alfven wave is the magnetic tension. This magnetic version of “tension” accelerates the plasma and is opposed by the inertia of the ions (mainly from proton masses m(p))

Now, the wave speed on a string is related to u (mass per unit length), and T such that:

v =  (T/ u) ^1/2

and as you can see, increasing the string tension increases the wave speed in an analogous way to what magnetic tension does for the Alfven wave. The magnetic tension analog can be expressed (as we shall see) as:

T(M) =  B^2/ u_o

where B is the magnetic induction and u_o is the magnetic permeability for free space (e.g. u_o = 4 pi x 10^-7 H/m)

In what follows we assume a uniform plasma in equilibrium, which will then be subjected to velocity disturbance or perturbation that affects all other key quantities. The treatment is kept as simple as possible (considering the complexity of the subject matter!) , and we don’t veer out of the linear domain. Nevertheless it should be stated at the outset that some details are omitted, or left as work for yourself with hints provided. In this way you will better understand and appreciate the genesis of Alfven waves. In terms of symbols, all have retained their earlier meanings (from previous questions) and this includes the vector operators, DIV, grad, Curl etc.

Examining the origin of these waves always starts with setting out the basic equations for what we call “ideal MHD”:

@ rho/ @ t =  - DIV (rho v)

@ (rho v)/ @ t  = - DIV (rho vv) – grad p + 1/ u_o [(Curl B) X B]

@B / @t = Curl (v X B)

@p/ @t = - v. grad p – gamma p DIV v

where the partial derivative symbols (@) are as in previous answers, v is the fluid velocity, p the pressure, B the magnetic induction, and gamma is the polytropic index, e.g. 5/3 for many cases.

Now,  introduce small perturbed quantitites (e.g. imagine introducing a small perturbation into the plasma velocity such that v_o -> v_1, which will also subject the mass density, fluid pressure and magnetic field to perturbation), such that:

rho = rho_o + rho_1

v = v_1

B = B_o + B_1

p = p_o + p_1

and substitute these back into the original ideal MHD equations to obtain:

@ rho_1/ @t = - rho_o DIV v_1

rho_o (@v_1/ @t) = - grad p_1 + 1/ u_o [(Curl B_1) X B_o]

@B_1/ @t = Curl (v_1 X B_o)

@ p_1/ @t = - gamma p_o DIV v_1

Now, divide through the 2nd equation above by ‘rho’, the mass density:

@v_1/ @t  = -(c_s^2)  grad p_1/ rho -  1/ u_o rho [B_o X Curl B_1]


where ‘c_s’ is the sound speed. (Note that you should be familiar with a vector identity also used to obtain the preceding!)

Now, using this result and the last two equations of the perturbed set, we apply Fourier transforms such for @/ @t and  @/ @k to obtain:

w^2 v_1 – c_s^2 (kv_x)k* + B_o/ u_o rho_o [k X k* X (v_1 X B_o)] = 0

where w denotes the plasma frequency, k is the wave number vector (k* the vector orientation) and the other quantities are as before.

The x-component is:

w^2 v_x – c_s^2 k^2 v_x +  B_z k^2/ u_o rho_o [v_zB_x – v_xB_z] = 0




The y-component is decoupled from the others (x, z) and can be written:

w^2 v_y -  B_x^2 k^2 v_y/ u_o rho_o = 0

or simply:

w^2 =  [B_x^2/ u_o rho_o] k^2

where the quantity in brackets is the Alfven velocity or alternatively written”

v(A) =  [w/ k] =  B_x / [u_o rho_o]^1/2

or  

v(A) =  B_o/ [u_o rho_o]^1/2

since B_o is in the x –direction

For completeness, you should be able to show the z-component equation is:

w^2 v_z -  B_x k^2/ u_o rho_o [v_z B_x – v_x B_z] = 0

Let me back up here and refer you again to the basic wave equation one can obtain by getting the 2nd derivative of:


@v_1/ @t  = -(c_s^2)  grad p_1/ rho -  1/ u_o rho [B_o X Curl B_1]
One can then find the solution in terms of plane waves by assuming:

v_1 = v_1*[exp ik.x – iwt]

for which taking the second derivative, of v_1 with respect to t yields the original equation in w we found earlier. All of this you should be able to work out, but most of it (after taking derivatives) reduces to brute force algebra!

For completeness, I need to note what happens when you solve the preceding (simultaneous) equations in x, and z. You then obtain (again, work this out for yourself!):

w^4 + w^2[-c_s^2 k^2 -  B_x^2 k^2/ u_o rho_o] + c_s^2 k^4 [B_x^2 / u_o rho_o] = 0

or:

w^4 – w^2(c_s^2  + v(A)^2)k^2 + c_s^2 v(A)^2 cos^2 (theta) k^4 = 0

and finally,

w^2 = ½[(c_s^2 + v(A)^2k^2 +/-  [(c_s^2 + v(A)^2 k^4 – 4 c_s^2 v(A)^2 cos^2(theta) k^4]^1/2

Now, if one plots the preceding using for the vertical axis (c_s^2 + v(A)^2) and for the horizontal B_o (e.g. x) one will get what is called “Friedrich’s diagram”


!
!
-----------------
!
!

Imagine superimposed on the above axes, the following graphs:

1)   a small “dumb bell” or figure-8 shaped graph centered at the origin. This will be for what we call “slow mode” waves

2)   a single larger lobe that envelopes the smaller right lobe of the dumb bell. This will be for Alfven waves proper.

3)   A circle shaped graph surrounding both 1, 2 above. This will be for what we call the “fast MHD” mode.  

The critical thing to note here is that the fast mode is the only MHD wave able to carry energy perpendicular to the magnetic field. This has important ramifications for solar flares, as well as magnetospheric effects (such as the aurora). Meanwhile, the phase velocity (w/k) of the slow mode wave perpendicular to the magnetic field is always zero. In the limit where the sound speed c_s^2 < < v(A)^2, and the Alfven speed v(A)^2 << c_s^2, the slow wave disappears. (Which you can easily validate and confirm for the equation in w^2)

Other properties, points to note:

-   the velocity perturbation v_1 is orthogonal to B_o
-   the wave is incompressible since DIV v_1 = ik.v_1 = 0
-   the magnetic field perturbation (B_1) is aligned with the velocity perturbation. Since both are perpendicular to k and B_o
-   the current density perturbation (J_1) exists as a current perturbation perpendicular to k and B_o  e.g.

J_1 = k X Bo

- when c_s^2 << v(A)^2 the fast mode wave becomes a compressional Alfven wave. This has a group velocity equal to its phase velocity w/k

Hope this info helps!