Calculation of Pressure in Ocean Simulations

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Abstract
Many state-of-the-art numerical ocean models calculate pressure using the hydrostatic balance, or an equation derived from it. The proper form of this deceptively simple-looking equation, ∂p/∂z = −gρ(S, T, p) (where notation is standard), is nonlinear in the pressure p. In contrast, most numerical models solve the linear equation ∂p/∂z = −gρ(S, T, z). This modification essentially replaces the total pressure, which includes a time-dependent signal, with an approximate time-independent pressure associated with the depth of a model grid point. In this paper, the authors argue that the inclusion of the total pressure when solving the hydrostatic equation can generate a depth-dependent baroclinic pressure gradient equivalent to a geostrophic velocity of several centimeters per second. Further, this effective velocity can increase with depth and is largest in dynamically important areas like western boundary currents. These points suggest that the full feedback of pressure on density should be included in numerical models. Examples of the effect using oceanic data and output from a typical primitive equation model run are discussed. Finally, algorithms for both rigid-lid and free surface models that explicitly include full pressure are derived, and some related numerical issues are discussed.

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