Roberts and D. Marshall (1998) (RM98) proposed a new biharmonic operator for use in z-level models. Ideally, this operator adiabatically dissipates structures at the grid scale. This property is useful for both eddy permitting models, where enstrophy cascades to the small scales and so must be dissipated, and coarse models, where problems with advection schemes can pollute the solution with spurious noise. This operator is motivated for numerical reasons, not from any fundamental arguments. Stated quite simply, the goal is to enhance the advective nature of the simulation without sacrificing adiabaticity. Without some way to suppress the grid noise, an advectively dominant solution can become a sea of noise. If the manner to suppress the grid noise does not take into account the needs of adiabaticity, such as the traditional horizontal biharmonic operator or dissipative advection schemes, then the solution may also be of low value due to the loss of water mass integrity. These competing needs are potentially addressed by this new operator.
RM98 termed their operator a ``biharmonic GM'' operator since it represents a straightforward generalization of the usual ``Laplacian GM'' operator. Yet, as shown by RM98 and in the following, their biharmonic operator does not generally dissipate APE, whereas the usual GM operator does. Therefore, the RM98 operator is perhaps better considered one of the many possible adiabatic biharmonic operators. Indeed, an alternative adiabatic operator, which always dissipates APE, is mentioned in the following. It turns out, however, that the RM98 operator is more readily discretized using the triad approach already used for the Redi diffusion (Section 34.1) and GM skew-diffusion (Section 34.1.6) processes.