  The creation of high energy density ($\gtrsim10^6$ joules per cm$^3$)
over-critical plasmas in a large volume has essential applications in the study
of warm dense matter, being present in the hot cores of stars and planets. It
was recently shown that femtosecond Bessel beams enable creating over-critical
plasmas inside sapphire with sub-wavelength radius and several tens of
micrometers in length. Here, the dependence of field structure and absorption
mechanism on the plasma density transverse profile are investigated by
performing self-consistent Particle-In-Cell (PIC) simulations. Two { limiting}
cases are considered: one is a homogeneous step-like profile, that can sustain
plasmon formation, the second is an inhomogeneous Gaussian profile, where
resonance absorption occurs. Comparing experimental absorption measures to
analytical predictions allows determining the plasma parameters used in PIC
simulations. The PIC simulation results are in good agreement with experimental
diagnostics of total absorption, near-field fluence distribution, and far-field
radiation pattern. We show that in each case an ambipolar field forms at the
plasma surface due to the expansion of the hot electrons and that electron
sound waves propagate into the over-critical region.
