Horizontal Advection Schemes of a Staggered Grid—An Enstrophy and Energy-Conserving Model

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Abstract
For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjić (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Källberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a second- order as well as a fourth-order (or approximately so) advection scheme for temperature (and moisture) advection is given, preserving the total energy (and moisture) inside the integration region.

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