  We classify ``arithmetic convection equations'' on modular curves, and
describe their space of solutions. Certain of these solutions involve the
Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while
others involve the Serre-Tate expansions of the same modular forms; in this
sense, our arithmetic convection equations can be seen as "unifying" the two
types of expansions. The theory can be generalized to one of
  ``arithmetic heat equations'' on modular curves, but we prove that modular
curves do not carry ``arithmetic wave equations.'' Finally, we prove an
instability result for families of arithmetic heat equations converging to an
arithmetic convection equation.
