Reflection at a conducting surface Boundary Conditions: Not sure if I am understanding this correctly or not

Question

In Griffiths Introduction to Electrodynamics (4th edition), when discussing the boundary conditions for a dielectric to (imperfect) conductor interface for a monochromatic plane wave, Griffith claims that,

For Ohmic conductors (J=σE) there can be no free surface current since this would require an infinite electric field at the boundary",

and, he adds (in an excerpt on pg 425) that

In Section 9.4.2 I argued that there can be no surface currents in an
ohmic conductor (with finite conductivity). But there are volume currents, extending in (roughly) to the skin depth. As the conductivity increases, they are squeezed into a thinner and thinner layer, and in the limit of a perfect conductor they become true surface currents.

Now, he doesn't ever explicitly state whether the current densities that he is referring to are induced by consequence of the EM wave penetrating into the conductor (where it decays) or if he is speaking in general. However, the fact that he says for an imperfect conductor there are volume currents extending into the conductor a distance (roughly) equal to the skin depth, makes me think that he is referring to currents which should result from the EM wave in the conductor (not a "preexisting current" which is already flowing in the conductor prior to incidence if that makes sense). Can anyone confirm or deny if I am correct or not? I know this will probably seem like a dumb question to some of you, but, I wanted to be as sure as possible that I wasn't misunderstanding anything. Thank you for your time!

To elaborate a bit more...consider a cylindrical (imperfect so E does not have to =0 on interior) conducting wire, where both circular ends are at a potential difference of V (uniform electric field in the wire). The current density should also be uniform if wire is of homogenous conductivity by ohms law and so the volume current does not only extend as far as the skin depth like Griffiths claims in the quote from my original post but exists uniformly distributed in the wire. Therefore, if Griffiths is not specifically referring to currents induced by EM wave penetration into the conductor, then I can't see how this is not paradoxical.

Answer 1

At the end of the page where it says

For Ohmic conductors ($\mathbf{J}=σ\mathbf{E}$) there can be no free surface current since this would require an infinite electric field at the boundary"

it also calculates the reflection ($E_R$) and transmission ($E_T$) of an incident EM wave ($E_I$) with polarisation parallel to the conductor (so that it would induce a surface current). The result is that: $$ E_{R} = \left (\frac{1-\tilde\beta}{1+\tilde \beta} \right ) E_I \quad \text{and} \quad E_T = \left (\frac{2}{1+\tilde \beta} \right ),$$ where $\tilde \beta$ depends on the conductivity $\sigma$.

For a perfect conductor, he adds, where $\sigma \rightarrow \infty$ and hence $\tilde \beta \rightarrow \infty$, you get: $$ E_R = -E_I\quad \text{and} \quad E_T = 0, $$ that is the wave is perfectly reflected, there is no electric field inside any volume of the conductor, but notice that the net electric field at the surface goes to $0$, because $E_R$ and $E_I$ have opposite signs. This makes sense and connects to the idea that electric field lines, in equilibrium, are always perpendicular to a conducting surface.

So, for a perfect conductor no electric field penetrates and there is no surface current. All good.

For an imperfect conductor, there is an internal field (that decays with the skin depths $s \propto 1/\sqrt{\sigma}$ and a surface current. But since the surface current is not the only current, because of the charges moving in regions where $E_T \neq 0$, you don't have a pure surface current and physics is saved again.

makes me think that he is referring to currents which should result from the EM wave in the conductor (not a "preexisting current" which is already flowing in the conductor prior to incidence if that makes sense

I think it's incorrect to say that an "EM wave" is in the conductor, because that implies it's a waveguide. What you mean is probably AC current and no external wave. This also leads to the skin effect (because of time dependent EM fields inside the conductor) and in that case you can indeed not reach a surface current, because the resistance of an Ohmic conductor goes as $\rho \ell/A$, where $A$ is the cross-sectional area which would tend to $0$ for pure surface current.