Basic questions about Alfven waves in solar corona I'm a mathematician trying to understand a paper in mathematical physics (Alfven Reflection and Reverberation in the Solar Atmosphere, by Paul Cally), and although I understand most of the pde theory required I'm struggling with the physics behind it. I'm sure many of these questions are basic for people who work in this field, but I don't know of a good reference. I hope someone here can help me.
We are looking at the wave equation in one dimension, $U_{tt} = a(x)^2 U_{xx}$, where here $a(x)$ is a function of $x$. As a first example, if $a(x) = exp(C x)$ for constant $C$, then the solution can be expressed as a linear combination of compositions of Bessel functions of the first and second kind and exponentials in $x$, all times a complex exponential in t. My first question is, we would be looking at this for all real $x$, correct? Or are we looking at $x > 0$? $x>0$ makes more sense to me, because then this is somehow distance from the center of the sun or something, but if so do we need to worry about boundary conditions at $x=0$? Because those don't seem to be mentioned.
Also, it's given that $a(x) = |B_x|/\sqrt{\rho}$, and then rho is described as the density, but what is $|B_x|$? It might be the magnetic permeability, which is set to $1$, but I'm not sure.
One of the issues that this paper is dealing with is that in this model waves can reach infinity in finite time and then reflect and return, which is physically unrealistic, and the Alfven travel time is given as $1/(C a(x))$, but where does this identity come from?
Thanks in advance for any help.
Greg
 A: To adress part of your questions. This reference by An et al. adresses your questions.
The starting point is $a(x) = |B_x|/\sqrt{\rho}$. This is an equation for the wave-propagation perpendicular to the sun. $B_x$ here is the vertical magnetic field.
For the simplified example $a(x) = \exp(C x)$, it is assumed that the magnetic field $B_x$ remains constant (instead of decreasing with increasing distance from the sun). What remains is the density which decreases exponentially $\rho \propto e^{-\lambda x}$ and therefore $a(x) \propto e^{\lambda/2 x} \equiv e^{Cx}$. That means, we are interested in $C>0$ and $x>R_\odot$, i.e., outside of the sun. The actual boundary condition would be the surface of the sun at $x = R_\odot$. Then, I guess the solution is finite on the boundary and the boundary condition only fixes the amplitude, but I could be wrong here.
A: An Alfven waves is just a magnetic perturbation that propagates, i.e., has a finite phase speed (though they need not have a finite group velocity).

My first question is, we would be looking at this for all real $x$, correct?

Yes, this is just the spatial position with respect to the surface of the sun.  The reason being that an Alfven wave incident on the surface would reflect off the boundary (actually, it could reflect well before hitting the surface depending on where it started and it's properties).

$x > 0$ makes more sense to me, because then this is somehow distance from the center of the sun or something, but if so do we need to worry about boundary conditions at $x = 0$?

Yes, you worry about this boundary but you can treat it as an infinitely conducting reflective wall.  There are specific rules about how Maxwell's equations behave across such boundaries.  I am guessing the paper ignores this boundary as they are only concerned with outward propagating waves (i.e., anti-sunward).

Also, it's given that $a\left( x \right) = \lvert B_{x} \rvert / \sqrt{ \rho }$, and then rho is described as the density, but what is $\lvert B_{x} \rvert$?

$\lvert B_{x} \rvert$ is just the magnetic field component along the $x$-direction, i.e., the inhomogeneity direction here which is radial from the sun center.  It's a way of saying the magnetic field only varies along one direction and one can assume in this case that $\lvert B_{x} \rvert$ is equivalent to the magnitude.

It might be the magnetic permeability, which is set to 1, but I'm not sure.

No, the author is just assuming that the permeability is the vacuum permeability, $\mu_{o}$, and then they use a unit system where this factor has a magnitude of unity.

One of the issues that this paper is dealing with is that in this model waves can reach infinity in finite time and then reflect and return, which is physically unrealistic...

Yes, but it's not meant to be actual infinity.  It's an approximation for very large distances relative to the starting location.  Such approximations are used to simplify equations by doing things like a Taylor expansion.

...and the Alfven travel time is given as $1/\left( C \ a\left( x \right) \right)$, but where does this identity come from?

The term $a\left( x \right) = \lvert B_{x} \rvert / \sqrt{ \rho }$ is an expression for the Alfven speed.  The proper version of this uses the magnitude of the magnetic field instead of $B_{x}$ and there would be a factor of $\mu_{o}$ under the radical in the denominator.  If you assume that $a\left( x \right) \propto e^{C \ x}$ then the units of $C$ must be that of inverse length, i.e.,  the inverse of some characteristic scale height of the atmosphere.  Then from dimensional analysis you can see that the units of $1/\left( C \ a\left( x \right) \right)$ must be time.  That is, the time it takes for an Alfven wave to propagate through one scale height of the atmosphere.
Update
I noticed a typo/error in one of the expressions in the paper Cally [2012] (doi:10.1007/s11207-012-0052-3).  In their Equation 2 I think the exponent should be negative, not positive.  That is, the expression should be:
$$
a = a_{1}\left( x \right) = a_{o} e^{-x/2h} \tag{2}
$$
Also note that $a\left( x \right)$ is not, in general, a constant with altitude, $x$.  This is because the magnetic field does not depend upon a half-integer power of $x$ relative to the mass density, i.e., $\lvert B_{x} \rvert / \sqrt{ \rho }$ is not independent of $x$, in general.
The characteristic time scale given by $1/\left( C \ a\left( x \right) \right)$ is just the time necessary for an Alfven wave to propagate through one scale height, $h \propto C^{-1}$.  Since $a\left( x \right)$ is not constant, the propagation time will not be either given that $C$ is a constant.
Note that the derivatives of $a\left( x \right)$ need not be continuous at the boundary, i.e., here the solar surface.  This all depends upon the models used for the magnetic field and mass density but it is often the case that the Alfven speed in Earth's magnetosphere shows a kink near the plasmapause due to the abrupt jump in number density.
