Getting a Handle on Alfven Waves (1)

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. 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 we 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) = B² / m _o


where B is the magnetic induction and m_o  is the magnetic permeability for free space ( 4π 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”:

1) ¶ r / ¶ t  = - DIV ( r v)

2) ¶ (r v)/ ¶ t  = - DIV (r vv) – grad p + 1/ m _o [(Curl B) X B]

3) ¶ B / ¶t = Curl (v X B)

4) ¶p/ ¶ t = - v. grad p –  g p DIV v

where the partial derivative symbols (@) are as before, v is the fluid velocity, p the pressure, B the magnetic induction, and ‘gamma’ = - d ln p/ d ln V where V denotes volume

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

r = r _o  +   r _l

v = v  _l

B = B _o + B _l

p = p _o + p _l

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


5) ¶ r _l / ¶ t = -  r _o  DIV v _l

6) r _o  (¶v _l/ ¶ t) = - grad p _l + 1/ m _o  [(Curl B _l) X B _o ]

7) ¶ B  _l / ¶t  = Curl (v  _l  X B _o )

8)  ¶ p  _l  / ¶t = - g p _o DIV v _l


Now, divide through the 2nd equation above by  the mass density r:

¶ v _l / ¶t  = -(c _s  ²) grad  p  _l  / r  - 1/ m _o  r  [B _o X Curl B ₁]


where ‘c _s’ is the sound speed. (Note that the reader 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:

w²  v _l –  c _s  ²  (kv _x )k* +  B _o / m _o  r _o  [k X k* X ( v _l  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. In the next instalment we’ll obtain the x and y components of the velocity but readers can try in the meantime to do it themselves!