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We demonstrate the large deviation principle in the small noise limit for thethree dimensional stochastic planetary geostrophic equations of large-scaleocean circulation. In this paper, we first prove the well-posedness of weaksolutions to this system by the method of monotonicity. As we know, a recentlydeveloped method, weak convergent method, has been employed in studying thelarge deviations and this method is essentially based on the main result of\cite{ba2} which discloses the variational representation of exponentialintegrals with respect to the Brownian noise. The It\^{o} inequality andBurkholder-Davis-Gundy inequality are the main tools in our proofs, and theweak convergence method introduced by Budhiraja, Dupuis and Ganguly in\cite{ba3} is also used to establish the large deviation principle.

Large deviation principle for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise