Formulating the Equations of Ocean Models

the purpose of this chapter is to formulate the equations of ocean models and to outline solution methods. Global ocean climate models, including those representing mesoscale eddies, are traditionally based on the hydrostatic primitive equations. We nonetheless discuss extensions to the more fundamental non-hydrostatic equations, which are used in certain fine resolution process studies, such as for convection and mixing, and increasingly for coastal and regional modeling. the target audience for this chapter includes students and researchers interested in fundamental physical and numerical aspects of ocean models. We thus aim to present a reasonably concise yet thorough accounting of the rationalization required to pose the problem of ocean modeling. We take a first principles perspective to allow readers with little background in ocean fluid mechanics to follow the full development. this goal necessitates starting from the basics as we develop the model equations and methods. For this purpose, much material was culled from various research papers and textbooks, such as Gill [1982], Pedlosky [1987], Lion et al. [1992], Marshall et al. [1997], Haidvogel and Beckmann [1999], Griffies et al. [2000a], Griffies [2004, 2005], Vallis [2006], Higdon [2006], and Müller [2006]. Our presentation focuses on developing the fluid mechanics of the ocean and weaves into this discussion elements appropriate for the formulation of ocean models. We begin with a discussion of ocean fluid kinematics in Section 2 where we introduce mass conservation as well as the notions of dia-surface transport. Section 3 then focuses on tracer budgets, which follow quite naturally from mass budgets, only with the introduction of possible nontrivial fluxes of tracer which occur in the absence of mass fluxes. Section 4 introduces a dynamical description that arises from the use of Newton’s Second Law of Motion applied to continuous fluid parcels. Section 5 presents the equation of state for density and discusses the material evolution of density. Section 6 derives some energetic properties of the equations of motion, with energetics providing a guiding principle for developing certain numerical solution methods. Section 7 introduces notions of non-equilibrium thermodynamics, a subject which forms the basis for establishing budgets of heat within the ocean. Section 8 linearizes the dynamical equations to deduce various linear modes of motion fundamental to ocean dynamics. these motions also have direct relevance to the development of methods used to solve the ocean equations. they furthermore motivate certain approximations or filters used to simplify the supported dynamical modes, with certain approximations described in Section 9. Section 10 presents an overview of vertical coordinates. The Formulating the Equations of Ocean Models