Fast dynamical spin-up of ocean general circulation models using Newton–Krylov methods

Numerical models of the ocean play an important role in efforts to understand past climate variability and predict future climate changes. In many studies, ocean models are driven by forcings that are either time-independent or vary periodically (seasonally) and it is often highly desirable or even essential to obtain equilibrium solutions of the model. Existing methods, based on the simple, expedient idea of integrating the model until the transients have died out, are too expensive to use routinely because the ocean takes several thousand years to equilibrate. Here, we present a novel approach for efficiently computing equilibrium solutions of ocean models. Our general approach is to formulate the problem as a large system of nonlinear algebraic equations to be solved with a class of methods known as matrix-free Newton–Krylov, a combination of Newton-type methods for superlinearly convergent solution of nonlinear equations, and Krylov subspace methods for solving the Newton correction equations. As an initial demonstration of the feasibility of this approach, we apply it to find the equilibrium solutions of a quasi-geostrophic ocean model for both steady forcing and seasonally-varying forcing. We show that the matrix-free Newton–Krylov method converges to the solutions obtained by direct time integration of the model, but at a computational cost that is between 10 and 100 times smaller than direct integration. A key advantage of our approach is that it can be applied to any existing time-stepping code, including ocean general circulation models and biogeochemical models. However, effective preconditioning of the linear equations to be solved during the Newton iteration remains a challenge.