A general pressure gradient formulation for ocean models. Part I: Scheme design and diagnostic analysis

A Jacobian formulation of the pressure gradient farce for use in models wit h topography-following coordinates is proposed. II can be used in conjuncti on with any vertical coordinate system and is easily implemented. Vertical variations in the pressure gradient are expressed in terms of a vertical in tegral of the Jacobian of density and depth with respect to the vertical co mputational coordinate. Finite difference approximations are made on the de nsity held, consistent with piecewise linear and continuous fields, and acc urate pressure gradients are obtained by vertically integrating the discret e Jacobian from sea surface. Two discrete schemes are derived and examined in detail: the first using st andard centered differencing in the generalized vertical coordinate and the second using a vertical weighting such that the finite differences are cen tered with respect to the Cartesian z coordinate. Both schemes achieve seco nd-order accuracy for any vertical coordinate system and are significantly more accurate than conventional schemes based on estimating the pressure gr adients by finite differencing a previously determined pressure field. The standard Jacobian formulation is constructed to give exact pressure gra dient results, independent of the bottom topography, if the buoyancy field varies bilinearly with horizontal position, x, and the generalized vertical coordinate, s, over each grid cell. Similarly, the weighted Jacobian schem e is designed to achieve exact results, when the buoyancy field varies line arly with z and arbitrarily with x, that is, b(x,z) = b(0)(x) + b(1)(x)z. When horizontal resolution cannot be made fine enough to avoid hydrostatic inconsistency, errors can be substantially reduced by the choice of an appr opriate vertical coordinate. Tests with horizontally uniform, vertically va rying, and with horizontally and vertically varying buoyancy fields show th at the standard Jacobian formulation achieves superior results when the con dition for hydrostatic consistency is satisfied, but when coarse horizontal resolution causes this condition to be strongly violated, the weighted Jac obian may give superior results.