Why does the transverse Alfven mode not undergo wave steepening and hence resulting in shock waves?
Initial Question:
I am currently working through the second edition of Introduction to Plasma Physics by <PERSON> & <PERSON>
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. 346
(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. 453