  The decline of Mars' global magnetic field some 3.8-4.1 billion years ago is
thought to reflect the demise of the dynamo that operated in its liquid core.
The dynamo was probably powered by planetary cooling and so its termination is
intimately tied to the thermochemical evolution and present-day physical state
of the Martian core. Bottom-up growth of a solid inner core, the
crystallization regime for Earth's core, has been found to produce a long-lived
dynamo leading to the suggestion that the Martian core remains entirely liquid
to this day. Motivated by the experimentally-determined increase in the Fe-S
liquidus temperature with decreasing pressure at Martian core conditions, we
investigate whether Mars' core could crystallize from the top down. We focus on
the "iron snow" regime, where newly-formed solid consists of pure Fe and is
therefore heavier than the liquid. We derive global energy and entropy
equations that describe the long-timescale thermal and magnetic history of the
core from a general theory for two-phase, two-component liquid mixtures,
assuming that the snow zone is in phase equilibrium and that all solid falls
out of the layer and remelts at each timestep. Formation of snow zones occurs
for a wide range of interior and thermal properties and depends critically on
the initial sulfur concentration, x0. Release of gravitational energy and
latent heat during growth of the snow zone do not generate sufficient entropy
to restart the dynamo unless the snow zone occupies at least 400 km of the
core. Snow zones can be 1.5-2 Gyrs old, though thermal stratification of the
uppermost core, not included in our model, likely delays onset. Models that
match the available magnetic and geodetic constraints have x0~10% and snow
zones that occupy approximately the top 100 km of the present-day Martian core.
