Noise-induced drifts in weather and climate

We consider the possibility that unparameterized sub-grid scale forcing in GCMs may not only affect the variability but also the mean of climate variables. In an NWP con text, such a forcing may not only affect the spread but also the mean of forecast ens embles. If the statistics of the forcing are approximated as spatially coherent white noise, the net effect of the noise on the ensemble spread and mean can be determined analytically through the moments of the Fokker-Planck equation. The noise can affect the mean even in a linear model through multiplicative interactions with the forecas t variables, i.e if its statistics are state-dependent That this multiplicative noise effect on the mean can be susbtantial will be demonstrated through examples of Rossb y wave propagation on an ambient flow with a stochastically varying component and/or with stochastically varying wave damping. The standard Fokker-Planck theory is strict ly not applicable if the noise is red rather than white. It will be shown, however, t hat consistent and accurate analytical approximations can be derived for red noise fo rcing which reduce to the Fokker-Planck results in the white noise limit. In addition to providing extensions to the standard theory, these approximations will be useful in testing implementations of stochastic forcing in GCMs and NWP models. In some inst ances they may even make explicit stochastic integrations unnecesary.