Mesinger, F., 1981: Horizontal advection schemes of a staggered grid- An enstrophy and energy-conserving model. Monthly Weather Review, 109 (3), 467-478.

Abstract: For use in a model on the semi-staggered E ( in the Arakawa notation) grid, a number of conserving schemes for the horizonatl advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjic (1977) is generalized to conserve energy in case of divergent flow. A family of analos 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 approxiamtely fourth)order scheme is developed which maintains the total kinetic energy and, in addition, makes no conrtibution 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 intertia-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) inseide the integration region.