  Subglacial lakes are isolated, cold-temperature and high-pressure water
environments hidden under ice sheets, which might host extreme microorganisms.
Here, we use two-dimensional direct numerical simulations in order to
investigate the characteristic temperature fluctuations and velocities in
freshwater subglacial lakes as functions of the ice overburden pressure, $p_i$,
the water depth, $h$, and the geothermal flux, $F$. Geothermal heating is the
unique forcing mechanism as we consider a flat ice-water interface. Subglacial
lakes are fully convective when $p_i$ is larger than the critical pressure
$p_*\approx 2848$ dbar, but self organize into a lower convective bulk and an
upper stably-stratified layer when $p_i < p_*$, because of the existence at low
pressure of a density maximum at temperature $T_d$ greater than the freezing
temperature $T_f$. For both high and low $p_i$, we demonstrate that the Nusselt
number $Nu$ and Reynolds number $Re$ satisfy classical scaling laws provided
that an effective Rayleigh number $Ra_{eff}$ is considered. We show that the
convective and stably-stratified layers at low pressure are dynamically
decoupled at leading order because plume penetration is weak and induces
limited entrainment of the stable fluid. From the empirical power law equation
for $Nu$ with $Ra_{eff}$, we derive two sets of closed-form expressions for the
variables of interest, including the unknown bottom temperature, in terms of
the problem parameters $p_i$, $h$ and $F$. The two predictions correspond to
two limiting regimes obtained when the effective thermal expansion coefficient
is either approximately constant or linearly proportional to the temperature
difference driving the convection.
