Looking Again At Alfven Waves - In More Detail

 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 in a single direction x  for a string put under tension T.

     Consider the reference frame or coordinate system for a transverse wave on a string:

Where we have in terms of the string tension T:

 T _u  = T sin q

As usual per these approaches, assume q is very small, in which limit, sin q  » tan q

Then:

T sin q = T tan q =  T  (¶  u/ ¶x)

 Taking  the force difference:  

[T _u ] _{x + dx} -   [T _u ] _x  =  ¶  / ¶x (T  (¶  u/ ¶x) dx

    To ensure no net horizontal forces due to tension,  we limit the situation to small slopes. Also, neglect the possibility of a vertical force per unit length, so:

 r dx (¶ ² u/ ¶ t²) = ¶  / ¶x (T  (¶  u/ ¶x) dx

Which simplifies to:

 r  (¶ ² u/ ¶ t²) = T  (¶  ² u/ ¶x² )

Now, transform to the standard wave equation in 1-dimension:

 ¶² u/ ¶x²  -  1/ c² (¶ ² u/ ¶ t²) = 0

 From this we can solve for the velocity  c:

c = (T/ r ) ^½

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 T _u 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.

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 v is the fluid velocity, p the pressure, B the magnetic induction, and g = - d ln p/ d ln V where V denotes volume.   Next we  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 _s²) grad p  _l  / r  - 1/ m _o  r  [B _o x Curl B₁]

where ‘c _s’ is the sound speed. Using this result and the last two equations of the perturbed set, we apply Fourier transforms 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 wave vector orientation) and the other quantities are as before.

It is of interest here to obtain the x, y-components of velocity associated with the wave, by use of Fourier transforms.  The procedure is straightforward and I show it in what follows.

The x-component is:

w²  v _x   –  c _s  ²    k ²    v _x  + B  _z k ²  /  m _o  r _o  [v _z   B  _x  –  v _x  B  _z] = 0

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

 w²  v _y   -  B  _x ²   k ²  v _y / m _o   r _o = 0

Or simply:

w²   = [B  _x ²   / m _o  r _o] k ²

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

v  _A  = [ w/ k] =  B _x  / [m _o  r _o]  ^{1/ 2}

Or :

v  _A   = B_o  / [m _o  r _o]  ^{1/ 2}

(Since B_o    is in the x –direction)

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

w²  v _z  -  B _x  k ²  / m _o  r _o  [v _z  B _x  -  v _x  B  _z] = 0

I  refer  again to the basic wave equation one can obtain by getting the second derivative of:

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

One can then find the solution in terms of plane waves by assuming: 

v  ₁ = v  ₁ [exp ikx – iwt]

For which taking the second derivative, of  v  ₁ with respect to t yields the original equation in w  found earlier. 

For completeness, I note what happens when you solve the preceding (simultaneous) equations in x, and z.

w ⁴ +  w² [- c _s  ²  k  ²  - B _x ²  k  ²  / m _o  r _o ] + 

c _s  ²  k  ⁴    [B _x ²  / m _o  r _o ] = 0

Or:

w ⁴ -   w²  (c _s ²   + v  _A ² )k ²  +  c _s  ² v  _A ²  cos  ² (Θ) k ⁴  = 0

And finally,

w²   = ½[( c _s ² v  _A ²  k ²  +  

 [(c _s ² v _A ²  k ⁴  –  4 c _s ² v _A ²    cos ² (Θ) k⁴ ]^1/2 

Careful variation of Θ  for the above equations leads to special conditions which can also identify the “fast” and “slow” megnetosonic waves

The conditions and associated results are expressed in summary form:

For  Θ   =  0,   w²   =  {  v _A ² k ²  

                                          { c _s ²   k ²  

For  Θ   =  90,   w²   = {(c _s ²   + v  _A ² )k ²  

                                       =  {0

It is convenient to distinguish these waves  of differing velocity using the diagram shown below:

Note in particular the orientation of the wave number vector (k) as well as the lobes for the sound speed  c _s  and the Alfven velocity  v _A - both in relation to (c _s  ²  + v  _A ² ) and the  equilibrium magnetic field intensity,  B _o.  Note (c _s  ²   + v  _A ²)  is called the “fast mode” magneotsonic wave velocity, while  c _s  denotes the “slow mode” velocity.  If one plots the preceding using (c _s  ²   + v  _A ² ) for the vertical axis and B _o (e.g. x)  for the horizontal,  one will also get what is called “Friedrich’s diagram”  shown below:

   The Friedrich diagram  consists of :

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

2)A double lobe enveloped by a larger circle and itself enveloping a smaller double lobe. 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 aspect 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 one can easily validate and confirm for the equation in w² )

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  B _o

- 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

For fast mode waves with k ⊥  B _o   we can present the diagrams below showing the relations between the distinct field directions and applicable velocity as well as wave number vector in relation to B-field perturbations.

Suggested Problems:

1) For the mechanical wave diagram for transverse wave on a string  sketch the x, y-components of the string tension T. Show how the components and differences between them lead to the result:

[T _u ] _{x + dx} -   [T _u ] _x  =  ¶  / ¶x (T  (¶  u/ ¶x) dx

2) (a) A student sketches the  diagram below to show the different modes of plasma waves  which he argues exist in the Sun.

Redraw the diagram to improve its clarity and then explain the reasons for any changes made.  Would the parameters shown really make sense for such waves propagating in the Sun? Why or why not?  Give a quantitative explanation if possible.

(b)Compute the Alfven velocity in a layer of the Sun for which:

  r _o =   0. 2 g/ cm  ³  and B = 0.1 T.