  We consider the momentum formulation of the two-dimensional surface
quasi-geostrophic equations forced by random noise, of both additive and linear
multiplicative types. For any prescribed deterministic function under some
conditions, we construct solutions to each system whose energy is the fixed
function. Consequently, we prove non-uniqueness of almost sure Markov
selections of suitable class of weak solutions associated to the momentum
surface quasi-geostrophic equations in both cases of noise.
