Maximum entropy production states in quasi‐geostrophic dynamical models

Abstract

AbstractMathematical expressions for the time‐mean production rate of statistical, hydrodynamical entropy are introduced in the context of the two‐level quasi‐geostrophic equations and Gibbsian thermodynamics. These form the basis of an investigation into the hypothesis put forward by Paltridge that the climatic‐mean fields are those which maximize this quantity. the entropy production rate is treated as a functional of the surface wind field (obtained by linear extrapolation) and the temperature field as defined in the two‐level equations of quasi‐geostrophic dynamics. Using the calculus of variations, two coupled second‐order differential equations are obtained for these fields and they are solved analytically and numerically for betaplane and spherical‐polar geometries respectively. A global torque constraint is applied using the method of Lagrange multipliers.The zonal‐mean surface wind equation derived is found to be identical to Green's surface wind equation obtained from his vorticity transfer theory, apart from the redefinition of some empirical constants. an important difference regarding the method of satisfaction of the torque constraint results in realistic surface winds in spherical geometry in contrast to those found by White (1977) using an extension of Green's original work.The extremal flow field of an equivalent barotropic model of a wind‐driven ocean basin circulation is also investigated and is found to resemble a superposition of Fofonoff's non‐linear inertial solution and a wind stress forced circulation.