Coarse-grained stochastic models for tropical convection and climate

Abstract

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 birth͞death 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. The current practical models for prediction of both weather and climate involve general circulation models (GCMs) where the physical equations for these extremely complex f lows are discretized in space and time and the effects of unresolved processes are parametrized according to various recipes. With the current generation of supercomputers, the smallest possible mesh spacings are Ϸ50 -100 km for shortterm weather simulations and of order 200 -300 km for shortterm climate simulations. There are many important physical processes that are unresolved in such simulations such as the mesoscale sea-ice cover, the cloud cover in subtropical boundary layers, and deep convective clouds in the tropics. An appealing way to represent these unresolved features is through a suitable coarse-grained stochastic model that simultaneously retains crucial physical features of the interaction between the unresolved and resolved scales in a GCM. In recent work in two different contexts, the authors have developed both a systematic stochastic strategy (1) to parametrize key features of deep convection in the tropics involving suitable stochastic spin-f lip models and also a systematic mathematical strategy to coarse-grain such microscopic stochastic models (2) to practical mesoscopic meshes in a computationally efficient manner while retaining crucial physical properties of the interaction. This last work (2) is general with potential applications in material sciences, sea-ice modeling, etc. Crucial new scientific issues involve the fashion in which a stochastic model effects the climate mean state and the strength and nature of f luctuations about the climate mean. The main topic of this article is to discuss development of a family of coarse-grained stochastic models for tropical deep convection by combining the systematic strategies from refs. 1 and 2 and to explore their effect on both the climate mean and f luctuations for an idealized prototype model parametrization in the simplest scenario for tropical climate involving the Walker circulation, the east-west climatological state that arises from local region of enhanced surface heat f lux, mimicking the Indonesian marine continent.

An applied mathematics perspective on stochastic modelling for climate

“…In the simulations from Khouider et al (2003), and those presented below, DxZ80 km. The coarse-grained stochastic CIN model is coupled to this basic parametrization.…”

“…An appealing way to represent these unresolved features is through a suitable coarse-grained stochastic model that simultaneously retains crucial physical features of the interaction between the unresolved and resolved scales in a GCM. In work from 2002 and 2003, two of the authors have developed a new systematic stochastic strategy (Majda & Khouider 2002;Khouider et al 2003) to parametrize key features of deep convection in the tropics involving suitable stochastic spin-flip models and also a systematic mathematical strategy to coarse grain such microscopic stochastic models to practical mesoscopic meshes in a computationally efficient manner while retaining crucial physical properties of the interaction. As regards tropical convection, crucial scientific issues involve the fashion in which a stochastic model effects the climate mean state and the strength and nature of fluctuations about the climate mean.…”

“…Following the general procedure developed and tested in Katsoulakis & Vlachos (2003) and Katsoulakis et al (2003a,b), the simplest local version of the systematic coarse-grained stochastic process is developed in Khouider et al (2003) and summarized here.…”

“…The typical values used here are hZ500 m and HZ16 km, while u 0 Z2 m s K1 and C 0 D Z 0:001, corresponding to a momentum spin up time of approximately 3 days. The explicit values for the other constants used in (3.9) and elsewhere in this section can be found in Majda & Shefter (2001) and Khouider et al (2003).…”

“…The key mathematical development involves systematic strategies to coarse grain such stochastic spin-flip models to achieve computational efficiency while retaining crucial features of the microscale interactions (Katsoulakis & Vlachos 2003;Katsoulakis et al 2003a,b). The use of such stochastic lattice models to parametrize key features of tropical convection is briefly reviewed (Majda & Khouider 2002;Khouider et al 2003). For the coupling of continuum models like a GCM to a stochastic lattice model as well as in many diverse applications, an applied mathematics Platonic ideal model has recently been introduced and analysed by Katsoulakis et al (2004Katsoulakis et al ( , 2005Katsoulakis et al ( , 2006Katsoulakis et al ( , 2007hereafter KMS).…”