2. Coupling of Dynamical Core and Parameterization Suite

Consider the general prediction equation for a generic variable ,

where denotes a prognostic variable such as temperature or horizontal wind component. The dynamical core component is denoted and the physical parameterization suite .

A three-time-level notation is employed which is appropriate for the semi-implicit Eulerian spectral transform dynamical core. However, the numerical characteristics of the physical parameterizations are more like those of diffusive processes rather than advective ones. They are therefore approximated with forward or backward differences, rather than centered three-time-level forms.

The *Process Split* coupling is approximated by

where is calculated first from

The *Time Split* coupling is approximated by

The distinction is that in the

As mentioned above, the Eulerian core employs the three-time-level notation in (2.2)-(2.5). Eqns. (2.2)-(2.5) also apply to two-time-level semi-Lagrangian and finite volume cores by dropping centered term dependencies, and replacing -1 by and by .

The parameterization package can be applied to produce an updated field as indicated in (2.3) and (2.5). Thus (2.5) can be written with an operator notation

where only the past state is included in the operator dependency for notational convenience. The implicit predicted state dependency is understood. The

where the first argument of denotes the prognostic variable input to the dynamical core and the second denotes the forcing rate from the parameterization package, e.g. the heating rate in the thermodynamic equation. Again only the past state is included in the operator dependency, with the implicit predicted state dependency left understood. With this notation the

The total parameterization package in CAM 3.0 consists of a sequence of components, indicated by

where denotes (Moist) precipitation processes, denotes clouds and Radiation, denotes the Surface model, and denotes Turbulent mixing. Each of these in turn is subdivided into various components: includes an optional dry adiabatic adjustment (normally applied only in the stratosphere), moist penetrative convection, shallow convection, and large-scale stable condensation; first calculates the cloud parameterization followed by the radiation parameterization; provides the surface fluxes obtained from land, ocean and sea ice models, or calculates them based on specified surface conditions such as sea surface temperatures and sea ice distribution. These surface fluxes provide lower flux boundary conditions for the turbulent mixing which is comprised of the planetary boundary layer parameterization, vertical diffusion, and gravity wave drag.

Defining operators following (2.6) for each of the parameterization components, the couplings in CAM 3.0 are summarized as:

TIME SPLIT

The labels

The *Process Split* form is convenient for spectral transform
models. With *Time Split* approximations extra spectral transforms
are required to convert the updated momentum variables provided by the
parameterizations to vorticity and divergence for the Eulerian
spectral core, or to recalculate the temperature gradient for the
semi-Lagrangian spectral core. The *Time Split* form is
convenient for the finite-volume core which adopts a Lagrangian
vertical coordinate. Since the scheme is explicit and restricted to
small time-steps by its non-advective component, it sub-steps the
dynamics multiple times during a longer parameterization time step.
With *Process Split* approximations the forcing terms must be
interpolated to an evolving Lagrangian vertical coordinate every
sub-step of the dynamical core. Besides the expense involved, it is
not completely obvious how to interpolate the parameterized forcing,
which can have a vertical grid scale component arising from vertical
grid scale clouds, to a different vertical grid. [182]
compares simulations with the Eulerian spectral transform dynamical
core coupled to the CCM3 parameterization suite via *Process
Split* and *Time Split* approximations.