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Initial Question:

I am currently working through the second edition of Introduction to Plasma Physics by Donald A. Gurnett & A. Bhattacharjee

In section 8.3.8 on page 304, they state that:

.. on a qualitative level, both the slow and fast magnetosonic modes have the proper nonlinear characteristics to lead to wave steepening. That this is the case can be seen from a simple inspection of Eqs. (6.5.5), (6.5.14), (6.5.17), and (6.5.19), which shows that the phase velocities of both the slow and fast magnetosonic modes increase as the pressure and/or the magnetic field strength increase (i.e., the larger amplitude part of the disturbance tends to overtake the smaller amplitude part of the disturbance). Thus, slow and fast shocks arise from steepening of the slow and fast magnetosonic modes...

where the relevant equations are $$ \text{Speed of sound: } V_s^2 = \frac{\gamma P_0}{\rho_{m0}} = \frac{\gamma k_B T_0}{m} \tag{6.5.5} $$ $$ \text{Alfven speed: } V_A = \frac{B_0}{\sqrt{\mu_0 \rho_{m0}}} \tag{6.5.14} $$ $$ v_p^2 = \frac{1}{2}(V_A^2 + V_B^2) - \frac{1}{2}\left[\left(V_A^2 - V_s^2\right)^2 + 4V_A^2 V_s^2 \sin^2\theta^2\right]^{1/2} \tag{6.5.17} $$ $$ v_p^2 = \frac{1}{2}(V_A^2 + V_B^2) + \frac{1}{2}\left[\left(V_A^2 - V_s^2\right)^2 + 4V_A^2 V_s^2 \sin^2\theta^2\right]^{1/2} \tag{6.5.19} $$

Here, equations (6.5.17) and (6.5.19) corresponds to the phase velocity of the slow and fast magnetosonic modes when considering small amplitude waves propagating in an homogeneous ideal MHD fluid (see section 6.5 on page 206). This is the reason why (6.5.5) and (6.5.14) have a subscript $0$ for some of the terms since they are considering: $$ P = P_0 + P_1 \\ B = B_0 + B_1 \\ \rho_m = \rho_{m0} + \rho_{m1} $$

where the subscript 0 represents the equilibrium state and the subscript 1 represents the perturbation.

At first glance:

Their above statements seem to make sense, since $V_A$ contains $B_0$ and $V_s$ contains $P_0$, so indeed, if $B_0$ or $P_0$ increases, the phase velocity $v_p$ increases.

However, reading further:

However, they then state, for the transverse Alfven mode (this is again obtained from section 6.5):

...On the other hand, the transverse Alfvén mode, which is linked to the intermediate shock, does not tend to steepen, since the phase velocity, which is determined by the zero-order background magnetic field, is independent of the magnetic field perturbation..

Again, if we look at the phase velocity of the transverse Alfven mode derived in section 6.5 using the small amplitude waves, the phase velocity is: $$ v_p^2 = V_A^2 \cos^2\theta \tag{6.5.18} $$

Indeed, since $V_A$ is a function of $B_0$, the phase velocity does not constitute the perturbation $B_1$.

Going back to the first glance

But take a look at equations (6.5.17) and (6.5.19) describing the slow and fast magnetosonic modes, they also do NOT contain the perturbation terms since both $V_A$ and $V_s$ are functions of the zero order terms $B_0$ and $P_0$ respectively. Furthermore, if we read their first statement again:

...(i.e., the larger amplitude part of the disturbance tends to overtake the smaller amplitude part of the disturbance)...

Suddenly this doesn't make sense, because the phase velocities of the slow and fast magnetosonic modes do not contain the perturbation term.

So my question is:

How did Gurnett & Bhattacharjee deduce from the phase velocities of the slow and fast magnetosonic modes, $(6.5.17)$ and $(6.5.19)$ respectively, result in wave steepening from the larger amplitude part (the zero order $B_0$ and $P_0$) overtaking the smaller amplitude part (the perturbation $B_1$ and $P_1$) if the perturbation term of the pressure and magnetic field, $B_1$ and $P_1$ are not present in $(6.5.17)$ and $(6.5.19)$?

Apologies for the long question, hopefully it is clear enough.

Edit 1:

From trying to understand honeste_vivere's answer in the comments, I would try to paraphrase it in my own words.

Transverse Alfven Mode

Consider the eigenvectors of the transverse Alfven mode:

Eigenvectors of transverse Alfven mode (taken from page 210 of Gurnett & Bhattacharjee)

Here, the perturbations of the fluid velocity are given by $\tilde{\mathbf{U}}$ and $\tilde{\mathbf{B}}$. It is clear that the perturbations are orthogonal to the direction of wave propagation of the transverse Alfven mode (given by $\mathbf{k}$).

Wave steepening in a sound wave occurs because the larger amplitude part (the rarefaction) overtakes the smaller amplitude part (the compression) in a wave. Therefore, it is clear that wave steepening is a phenomenon only occurring in longitudinal waves.

From the diagram above, it is clear that the perturbations are transverse with respect to the wave propagation, this implies that there is no compression and rarefaction and hence no wave steepening.

Hence this must mean that the phase velocity of the transverse Alfven mode solely relies on the initial magnetic field $B_0$ since the perturbations cannot cause wave steepening. Therefore shockwaves CANNOT occur in the Alfven mode.

Slow and Fast Magnetosonic Modes

Let us now consider the slow and fast magnetosonic modes, the eigenvectors are given by:

Eigenvectors of slow & fast magnetosonic waves (taken from page 212 of Gurnett & Bhattacharjee)

For the fast magnetosonic wave, because $\theta$ is arbitrary, there is a non - zero projection of the fluid and magnetic field perturbations on the wave propagation vector, that is, $\mathbf{k} \cdot \tilde{\mathbf{B}} \neq 0$ and $\mathbf{k} \cdot \tilde{\mathbf{U}} \neq 0$. This allows for the existence of BOTH longitudinal and transverse components of the magnetic field and fluid velocity perturbations. The longitudinal component cause compressions and rarefactions and eventually wave steepening if the rarefactions overtake the compressions.

Similarly, for the slow magnetosonic wave, even though there is no magnetic perturbation, there is still a fluid velocity perturbation $\tilde{\mathbf{U}}$ which causes the same compressions and rarefactions that result in wave steepening.

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    $\begingroup$ The only issue here is that the coordinate basis shown for you Alfven wave has a finite projection of $\mathbf{k}$ onto $\delta \mathbf{E}$, which would be a longitudinal electric field oscillation. This is still fine as the important oscillations, $\delta \mathbf{B}$ and $\delta \mathbf{U}$, are orthogonal to $\mathbf{k}$. Thus, the term $\mathbf{k} \cdot \delta \mathbf{U} = 0$, i.e., incompressible flow. That is, be careful to note that the transverse part is only in relation to $\delta \mathbf{B}$ and $\delta \mathbf{U}$ (I know the book is a little lax on this). $\endgroup$ Jul 13 at 12:56
  • $\begingroup$ @honeste_vivere thank you for your patience! $\endgroup$
    – D. Soul
    Jul 15 at 12:57
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Why does the transverse Alfven mode not undergo wave steepening and hence resulting in shock waves?

Because it is a transverse oscillation. Nonlinear steepening happens in longitudinal oscillations. As long as the Alfven mode remains transverse, it cannot steepen.

Look at your Equation 6.5.18. If you apply the variational principle to the right-hand side, it will go to zero. In the linear, transverse limit, the Alfven mode is just a transverse oscillation, i.e., $\mathbf{k} \cdot \delta \mathbf{E} = 0$ and $\mathbf{k} \cdot \delta \mathbf{B} = 0$ (technically the latter is always true owing to there being no magnetic monopoles).

Note that $\mathbf{k} \cdot \delta \mathbf{V} = 0$ as well, where $\mathbf{V}$ are velocity oscillations associated with the wave. So in the traditional fluid dynamic sense of steepening, the steepening term satisfies $\mathbf{V} \cdot \nabla \mathbf{V} = 0$.

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  • $\begingroup$ Thank you! I can understand your answer but i was wondering if you could explain from the perspective of the book? As i have described above, they mentioned that (6.5.18) does not contain the perturbation term (only controlled by zero order) and hence does not steepen. But (6.5.17) and (6.5.19) also do not contain the perturbation term? $\endgroup$
    – D. Soul
    Jul 12 at 3:47
  • $\begingroup$ Also, when you mentioned applying the variational principle on the RHS, do you mean: $$ V_A^2 \cdot \delta \bigg(\cos^2\theta\bigg) $$ $\endgroup$
    – D. Soul
    Jul 12 at 3:50
  • $\begingroup$ You are correct that this can seem misleading. Note that the sound speed defined in Chapter 6 is done under the linear perturbation approach in an asymptotic limit. The sound speed is defined as $C_{s}^{2} = \tfrac{ \partial P }{ \partial \rho }$, so it is shows that as pressure increases the sound speed increases. The transverse Alfven mode does not have perturbations along the direction of propagation. However, in the context of the fast and slow modes, the Alfven speed used depends on the local magnetic field magnitude... $\endgroup$ Jul 12 at 13:16
  • $\begingroup$ So higher pressure and/or higher magnetic field strength cause larger phase speeds. In principle, the simple transverse Alfven mode does not alter the local Alfven speed, which is why it won't steepen. $\endgroup$ Jul 12 at 13:21
  • $\begingroup$ Thank you! Yes, the explanation was kind of opaque and is really confusing for a someone completely new to the subject, especially in an introductory textbook. Anyway, I tried to paraphrase your answer in my "Edit 1" at the bottom of my original post above. Would you mind helping to check if we are on the same page? $\endgroup$
    – D. Soul
    Jul 13 at 8:04

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