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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 8

### A coarse grained stochastic multi-type particle interacting model for tropical convection: Nearest neighbour interactions

Pages: 1379 – 1407

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n8.a1

#### Author

#### Abstract

Particle interacting systems on a lattice are widely used to model complex physical processes that occur on much smaller scales than the observed phenomenon one wishes to model. However, their full applicability is hindered by the curse of dimensionality so that in most practical applications a mean field equation is derived and used. Unfortunately, the mean field limit does not retain the inherent variability of the microscopic model. Recently, a systematic methodology was developed and used to derive stochastic coarse-grained birth-death processes which are intermediate between the microscopic model and the mean field limit, for the case of the one-type particle-Ising system. Here we consider a stochastic multicloud model for organized tropical convection introduced recently to improve the variability in climate models. Each lattice is either clear sky or occupied by one of three cloud types. In earlier work, local interactions between lattice sites were ignored in order to simplify the coarse graining procedure that leads to a multi-dimensional birth-death process; Changes in probability transitions depend only on changes in the large-scale atmospheric variables. Here the coarse-graining methodology is extended to the case of multi-type particle systems with nearest neighbour interactions and the multi-dimensional birth-death process is derived for this general case. The derivation is carried under the assumption of uniform redistribution of particles within each coarse grained cell given the coarse grained values. Numerical tests show that despite the coarse graining the birth-death process preserves the variability of the microscopic model. Moreover, while the local interactions do not increase considerably the overall variability of the system, they induce a significant shift in the climatology and at the same time boost its intermittency from the build up of coherent patches of cloud clusters that induce long time excursions from the equilibrium state.

#### Keywords

particle interacting systems, coarse graining, stochastic parametrization, tropical convection, conditional expectation

#### 2010 Mathematics Subject Classification

65C05, 65C35, 70-xx, 76Xxx