An applied mathematics perspective on stochastic modelling for climate.

Systematic strategies from applied mathematics for stochastic modelling in climate are reviewed here. One of the topics discussed is the stochastic modelling of mid-latitude low-frequency variability through a few teleconnection patterns, including the central role and physical mechanisms responsible for multiplicative noise. A new low-dimensional stochastic model is developed here, which mimics key features of atmospheric general circulation models, to test the fidelity of stochastic mode reduction procedures. The second topic discussed here is the systematic design of stochastic lattice models to capture irregular and highly intermittent features that are not resolved by a deterministic parametrization. A recent applied mathematics design principle for stochastic column modelling with intermittency is illustrated in an idealized setting for deep tropical convection; the practical effect of this stochastic model in both slowing down convectively coupled waves and increasing their fluctuations is presented here.

Madden-Julian Oscillation analog and intraseasonal variability in a multicloud model above the equator.

The Madden-Julian Oscillation (MJO) is the dominant component of tropical intraseasonal variability, and a theory explaining its structure and successful numerical simulation remains a major challenge. A successful model for the MJO should have a propagation speed of 4-7 m/s predicted by theory; a wavenumber-2 or -3 structure for the planetary-scale, low-frequency envelope with distinct active and inactive phases of deep convection; an intermittent turbulent chaotic multiscale structure within the planetary envelope involving embedded westward- and eastward-propagating deep convection events; and qualitative features of the low-frequency envelope from the observational record regarding, e.g., its zonal flow structure and heating. Here, such an MJO analog is produced by using the recent multicloud model of Khouider and Majda in an appropriate intraseasonal parameter regime for flows above the equator so that rotation is ignored. Key features of the multicloud model are (i) systematic low-level moisture convergence with retained conservation of vertically integrated moist static energy, and (ii) the use of three cumulus cloud types (congestus, stratiform, and deep convective) together with their differing vertical heating structures. Besides all of the above structure in the MJO analog waves, there are accurate predictions of the phase speed from linear theory and transitions from weak, regular MJO analog waves to strong, multiscale MJO analog waves as climatological parameters vary. With all of this structure in a simplified context, these models should be useful for MJO predictability studies in a fashion akin to the Lorenz 96 model for the midlatitude atmosphere.

Coarse-grained stochastic models for tropical convection and climate.

Prototype coarse-grained stochastic parametrizations for the interaction with unresolved features of tropical convection are developed here. These coarse-grained stochastic parametrizations involve systematically derived birth/death processes with low computational overhead that allow for direct interaction of the coarse-grained dynamical variables with the smaller-scale unresolved fluctuations. It is established here for an idealized prototype climate scenario that, in suitable regimes, these coarse-grained stochastic parametrizations can significantly impact the climatology as well as strongly increase the wave fluctuations about an idealized climatology.

Stochastic and mesoscopic models for tropical convection.

A new way to parametrize certain aspects of tropical convection through stochastic and mesoscopic models is developed here. The technical idea is to adapt tools from statistical physics and materials science to model important unresolved features of tropical convection. This new strategy consists of modeling the unresolved effects of convective inhibition in a coarse mesh mesoscopic parametrization through a "heat bath" model involving a stochastic spin flip model with very natural interaction rules for convective inhibition combined with a suitable external potential defined by the coarse mesh values. In turn, the values of the order parameter from this heat bath alter the vertical mass flux in regions of deep convection. Both stochastic and systematic deterministic mesoscopic parametrizations are developed here. The deterministic mesoscopic models derived in this fashion exhibit new phenomena such as multiple radiative equilibria in suitable parameter regimes. The simplest first numerical experiments reported here with the mesoscopic deterministic parametrization qualitatively reproduce several key features of the observational record regarding convectively coupled tropical waves. The systematic stochastic modeling strategy proposed here could also be very useful for capturing other features of tropical convection such as those involving cloud radiation feedbacks.