# 3. Coupling of Dynamical Core and Parameterization Suite¶

The CAM5.0 cleanly separates the parameterization suite from the dynamical core, and makes it easier to replace or modify each in isolation. The dynamical core can be coupled to the parameterization suite in a purely time split manner or in a purely process split one, as described below.

Consider the general prediction equation for a generic variable ,

(1)¶

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

(2)¶

where is calculated first from

(3)¶

The *Time Split* coupling is approximated by

(4)¶

(5)¶

The distinction is that in the *Process Split* approximation the
calculations of and are both based on the same past
state, , while in the *Time Split* approximations
and are calculated sequentially, each based on the
state produced by the other.

As mentioned above, the Eulerian core employs the three-time-level notation in (2)-(5). Eqns. (2)-(5) also apply to two-time-level finite volume, semi-Lagrangian and spectral element (HOMME) 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 (3) and (5). Thus (5) can be written with an operator notation

(6)¶

where only the past state is included in the operator dependency for
notational convenience. The implicit predicted state dependency is
understood. The *Process Split* equation (2) can also be written in
operator notation as

(7)¶

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 *Time Split* system
(4) and (5) can be written

(8)¶

The total parameterization package in CAM5.0 consists of a sequence of components, indicated by

(9)¶

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 (6) for each of the parameterization components, the couplings in CAM5.0 are summarized as:

TIME SPLIT

(10)¶

PROCESS SPLIT

(11)¶

The labels *Time Split* and *Process Split* refer to the coupling of the
dynamical core with the complete parameterization suite. The components
within the parameterization suite are coupled via time splitting in both
forms.

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.
[Wil02] compares simulations with the Eulerian spectral
transform dynamical core coupled to the CCM3 parameterization suite via
*Process Split* and *Time Split* approximations.