  Accretion disks whose matter follows eccentric orbits can arise in multiple
astrophysical situations. Unlike circular orbit disks, the vertical gravity in
eccentric disks varies around the orbit. In this paper, we investigate some of
the dynamical effects of this varying gravity on the vertical structure using
$1D$ hydrodynamics simulations of individual gas columns assumed to be mutually
non-interacting. We find that time-dependent gravitational pumping generically
creates shocks near pericenter; the energy dissipated in the shocks is taken
from the orbital energy. Because the kinetic energy per unit mass in vertical
motion near pericenter can be large compared to the net orbital energy, the
shocked gas can be heated to nearly the virial temperature, and some of it
becomes unbound. These shocks affect larger fractions of the disk mass for
larger eccentricity and/or disk aspect ratio. If the orbit can be maintained
despite orbital energy loss, diverse initial structures evolve in only a few
orbits so that they follow a limit-cycle characterized by a low-entropy
midplane and a much higher entropy outer layer. In favorable cases (such as the
tidal disruption of stars by supermassive black holes), these effects could be
a potentially important energy dissipation and mass loss mechanism.
