  In this work we generalize the notion of convective inhibition to apply in
cases where there is an infinite reservoir of condensible species (i.e., an
ocean). We propose a new model for the internal structure and thermal evolution
of super-Earths with hydrogen envelopes. We derive the criterion for convective
inhibition in a generalizes phase mixture from first principles thermodynamics.
We then investigate the global ocean case using a water-hydrogen system, for
which we have data, as an example. We then extend our arguments to apply to a
system of hydrogen and silicate vapor. We then employ a simple atmospheric
model to apply our findings to super-Earths and to make predictions about their
internal structures and thermal evolution. For hydrogen envelope masses roughly
in the range between 0.1-10% Earth's mass, convective contact between the
envelope and core may shut down because of the compositional gradient that
arises from silicate partial vaporization. For envelope hydrogen masses that
cause the associated basal pressure to exceed the critical pressure of pure
silicate (of order a couple kilobars), the base of that envelope and top of the
core lie on the critical line of the two-phase hydrogen-silicate phase diagram.
The corresponding temperature is much higher than convective models would
suggest. The core then cools inefficiently, with intrinsic heat fluxes
potentially comparable to the Earth's internal heat flux today. This low heat
flux may allow the core to remain in a high entropy supercritical state for
billions of years, but the details of this depend on the nature of the
two-component phase diagram at high pressure, something that is currently
unknown. A supercritical core thermodynamically permits the dissolution of
large quantities of hydrogen into the core.
