TEM wave propagation and no skin depth effect

From the EM wave equation solution in conductors, we find that EM waves cannot propagate in a conductor since they are attenuated heavily and in a distance of the skin depth attenuated by 'e'. So how can they traverse in a transmission line. At first I was satisfied since I interpreted that the EM wave does not travel along the line instead it tries to radiate out from the line which would create an E field along the line that will create the current in the transmission line and it will then give us the skin depth on around the surface of the conductor.
But I was reading Microwave Engineering by David Pozar and in that he takes a case of TEM wave propagation in the line. And takes the E and H fields along the line 0! (z direction) This is in the Transmission line chapter when he derives the Telegrapher's equations by Maxwell's Equations. So I don't understand how come there is no effect of skin depth or how could such a wave propagate in the transmission line conductor?

For a lossless TEM line, E × H (cross product) indicates the direction of propagation, along the length of the line (use the right hand rule, with Ex and Hy, your thumb points along the z axis). In the losslesss case, skin depth is zero. This can not happen in real life, but we come very close with superconductors. Ez and Hz must be zero for TEM propagation in the z direction.

When there is resistance, the current penetrates into the conductor. Due to the resistance, there is now voltage along the length of the conductor (by Ohm's Law). This means that there is an Ez component. Thus a lossy line is no longer a true TEM line. The Ez also means there will be dispersion for both propagation velocity and characteristic impedance. For very small resistance (typically the case), the Ez and the dispersion are very small and we say we have a quasi-TEM line.

A TEM line must have the same dielectric everywhere, like coax or stripline. Microstrip has air above and substrate below. Even in the lossless case, this causes both Ez and Hz to be non-zero. Thus microstrip is non-TEM and dispersive. If we don't go too high in frequency, however, dispersion is very small and we can still call it quasi-TEM.

The skin depth equation is calculated based on a plane wave normally (at right angles) incident on an infinite conducting plane. The plane wave then tries to propagate down into the conducting plane but both the wave number (beta) and characteristic impedance are complex with real and imaginary parts equal. This means that the plane wave penetrates a bit then bounces back and most of the incoming plane wave becomes a reflected outgoing plane wave.

The situation in microstrip is completely different. There is no normally incident plane wave, we have a microstrip quasi-TEM mode flowing horizontally along the length of the line. Even so, it turns out, the current density inside the conductor is essentially the same as if there were a normally incident plane wave. For good conductors, it does not matter whether the exitation is a microstrip mode flowing horizontally along a line or a plane wave coming from above. The skin depth current is the same.

It is interesting that you ask this question. This bothered me a lot a few years ago and I know of no one else who was also troubled by it. It would be interesting to solve the problem exactly for not-so-good conductors (any Ph. D. candidates out there?) and see the difference in current distribution between plane wave excitation and microstrip excitation.

Thank you for the detailed reply, although I am still trying to grasp it fully.