  Techniques from numerical bifurcation theory are very useful to study
transitions between steady fluid flow patterns and the instabilities involved.
Here, we provide computational methodology to use parameter continuation in
determining probability density functions of systems of stochastic partial
differential equations near fixed points, under a small noise approximation.
Key innovation is the efficient solution of a generalized Lyapunov equation
using an iterative method involving low-rank approximations. We apply and
illustrate the capabilities of the method using a problem in physical
oceanography, i.e. the occurrence of multiple steady states of the Atlantic
Ocean circulation.
