Thoughts on prognostically modeling an eddying double-gyre ensemble mean

We address the question of separating the ocean’s deterministic response to time-dependent forcing from its intrinsic chaotic variability. Because the forcing is neither stationary nor periodic and spatial homogeneity is precluded by both the forcing pattern and boundary conditions, statistical analysis must rely on ensemble averaging. Here, we define this as the arithmetic mean over realizations with equally probable initial conditions. Ideally, one could compute the ensemble mean directly without performing numerous realizations, but this requires knowledge or closure of the second-order statistics — the classical turbulent-closure problem, here recast for a non-equilibrium, geophysical setting. Building on the ideas of nonlinear midlatitude ocean adjustment (Dewar 2003), we examine this problem using idealized quasigeostrophic (QG) double-gyre ensembles subjected to episodic temporal variations in wind forcing. Our objective here is not to develop a subgrid parameterization of unresolved eddies, but rather to construct and test prognostic equations for the ensemble mean itself, using the simplest possible closure assumptions. We find that the performance of ensemble mean closures is highly dependent on the spatiotemporal structure of the forcing. Under slowly varying forcing, approximate closures reproduce the mean evolution reasonably well; under rapidly varying, near-zero-mean forcing, the simplest ensemble-mean closures fail, even at the level of basin-averaged total energy and enstrophy. In both regimes, the ensemble-mean response is not simply the accumulated imprint of the applied forcing, but instead appears as a continuing, non-equilibrated dialogue between the mean and eddy fields.