Wednesday, May 30, 2012

Commemorating Hannes Alfven's Achievements (1)

As most researchers, workers in plasma physics are aware, this year marks the 70th anniversary of a ground breaking research letter ('Existence of electromagnetic -hydrodynamic waves') in Nature, that was to change the face of physics forever. Though only a half page long, it was the content that mattered, describing a new type of low frequency oscillation for a magnetized plasma.

As with many novel insights, the content was long disregarded (or openly dismissed) mainly because the new waves couldn't be demonstrated experimentally at the time. However, over decades and as experimental - lab techniques grew more refined (including experiments conducted in the space environment via satellites and on Space Shuttles) the novel waves we call "Alfven" soon began to occupy an important role in space and plasma physics.

It helps at this stage to examine a bit of the nature of Alfven waves, conceived by Hannes Alfven, and which were finally experimentally demonstrated in lab plasmas by 1960.

Alfven waves, by the way, 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

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) ^½

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) = B^2/ u_o

where B is the magnetic induction and u_o is the magnetic permeability for free space, u_o = 4 π x 10^-7 H/m)

Examining the origin of these waves always starts with setting out the basic equations for what we call “ideal MHD”, e.g. one of the equations is: @B / @t = Curl (v X B). We then 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 we substitute these back into the original ideal MHD equations  Then, after using a LOT more math (which I will spare readers) we end up with the general Alfven wave quation:
 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, and c_s is the ion sound speed.  One will then take the preceding equation and resolve it into x, y components, viz.

x-component of wave:

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

y-component of wave:

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]^½

v(A) = B_o/ [u_o rho_o]^½

since B_o is in the x –direction

A more refined and useful form is obtained by incorporating the x and z-components and solving the resulting simultaneous equations, then doing some simplification to get:

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(Θ) 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”


Visualize 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 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

Why the maddening complexity with these waves and being able to quantify them? Mainly because Alfven used both the electromagnetic theory of James Clerk Maxwell in combination with hydrodynamics (or fluid dynamics) which hitherto had been established as separate disciplines of physics. But by marrying them via the sort of steps outlined above (to do with Alfven waves) Alfven succeeded in opening up an entirely novel area called magnetohydrodynamics.

Oddly, and ironically, at the time they were conceived these waves had no practical basis, or applications. But by the 1950s, when thermonuclear research took off, they emerged as it became possible to generate high temperature plasmas artificially on Earth.  The rest, as they say, is history.

Next: Some further contributions of Hannes Alfven.

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