Ergodic Dynamics of the Coupled Quasigeostrophic Flow-energy Balance System

The authors consider a mathematical model for the coupled atmosphereocean system, namely, the coupled quasigeostrophic flow-energy balance model. This model consists of the large scale quasigeostrophic oceanic flow model and the transport equation for oceanic temperature, coupled with an atmospheric energy balance model. After reformulating this coupled model as a random dynamical system (cocycle property), it is shown that the coupled quasigeostrophic-energy balance fluid system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic, namely, for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long. 1. Mathematical model We consider large scale geophysical flows modeled by the quasigeostrophic flow equation in the horizontal xy−plane, in terms of vorticity q(x, y, t), and the transport equation for the oceanic temperature T (x, y, t), coupled with the atmospheric energy balance equation proposed by North and Cahalan [16] for the air temperature Θ(x, y, t), on the domain D = {(x, y) : 0 ≤ x ≤ l, 0 ≤ y ≤ l}: Θt =∆Θ − (a+Θ) + Sa(x, y)− b(y)(So(x, y) + Θ− T (x, y)) + ẇ, ∂q ∂t =ν∆q − rq + Pr Ra ∂yT − J(ψ, q + βy), ∂T ∂t =∆T − J(T, ψ), (1) where ψ(x, y, t) is the stream function, β ≥ 0 is the meridional gradient of the Coriolis parameter, ν > 0 is the viscous dissipation constant, and r > 0 is the Ekman dissipation constant. Furthermore, q(x, y, t) = ∆ψ(x, y, t) is the vorticity, a > 0 is a constant parameterizing the effect of the earth’s longwave radiative cooling, b(y) is the latitudinal fraction of the earth covered by the ocean basin, Pr is the Prandtl number, and Ra is the Rayleigh number. Note that Sa(x, y) and So(x, y) are empirical functions representing the effects (on atmosphere and ocean, respectively) of the shortwave solar radiation. Moreover, J(g, h) = gxhy − gyhx is the Jacobian operator and ∆ = ∂xx + ∂yy is the Laplacian operator. All these equations are in non-dimensionalized forms. The fluctuating noise ẇ(x, y, t) is usually of a shorter time scale than the response time scale of Date: May 20, 2004.