Transient contributions to the forcing of the atmospheric annual cycle

Abstract

The forcing of the global circulation is examined using a primitive equation model and a 38-year reanalysis dataset. One-timestep integrations are initialised with selected sets of initial conditions, and the forcing budget for the mean annual cycle is deduced. This budget consists of sources and sinks of momentum, temperature and humidity which are balanced by dynamical terms. The associated timescale interactions are examined in detail. The time-mean forcing is balanced by time-mean fluxes, annual cycle interactions and transient fluxes. The annual cycle of the forcing is balanced by the interaction of annual cycle anomalies with the time-mean flow and with themselves (this latter cycle-cycle interaction term is found to be important for the moisture supply over West Africa). Transient interactions on other timescales also contribute to the forcing of the annual cycle, but the interaction term between the annual cycle and other timescales is small, as is the storage term associated with seasonal tendencies. This objectively derived empirical forcing is then used to drive the dynamical model. The resulting simple GCM is called DREAM (Dynamical Research Empirical Atmospheric Model). This is the first time this approach has been used with an annual cycle. The systematic errors of DREAM compared to the reanalysis chiefly concern the momentum balance in the southern hemisphere jet. Perpetual season simulations are similar to individual seasons from the annual cycle run, consistent with the small seasonal tendency term in the forcing.

Appendix: forcing budget definitions and calculations

It is convenient to adopt a more flexible notation for the expansion of Eq. (2). The operation of the dynamical model can be expressed as

$$\begin{aligned} (\mathcal{A}+\mathcal{D})({\varvec{\Phi }}) = \mathcal{Q}({\varvec{\Phi }},{\varvec{\Phi }}) + \mathcal{D}({\varvec{\Phi }}) = \mathsf{\Phi }^\dagger \mathsf{Q}{\varvec{\Phi }} + \mathsf{D}{\varvec{\Phi }}, \end{aligned}$$

where $\mathsf{\Phi }^{\dagger }$ is the diagonal matrix whose diagonal is formed from elements of the column vector $\mathbf \Phi$, and $\mathsf{Q}$ and $\mathsf{D}$ are real matrices. If $\mathbf \Phi$ is split into components $\mathbf X$ and $\mathbf Y$, the quadratic term $\mathcal{Q}(\mathbf{X},\mathbf{Y})=\mathsf{X}^{\dagger } \mathsf Q \mathbf Y$ is the column vector with elements $X_i (\sum _j Q_{ij} Y_j )$. Note that $\mathcal{Q}(\mathbf{X},\mathbf{X})=\mathcal{A}(\mathbf{X})$ and $\mathcal{Q}(\mathbf{X},\mathbf{Y}) \ne \mathcal{Q}(\mathbf{Y},\mathbf{X})$.

In this notation the terms in the time mean-annual cycle forcing budget (3) can be written out individually and they are given in Table 1 along with a brief description and a reminder of which component of the forcing $\mathbf{f}$ they contribute to.

Table 1 Terms in the forcing budget

To find all these forcing terms the reanalysis dataset must be sampled in a variety of ways. The dataset ${\varvec{\Phi }}$ consists of 4$\times$ daily data for 38 years, for a total of 55520 data records. The mean annual cycle $(\overline{\varvec{\Phi }} + \widetilde{\varvec{\Phi }})$ is a 365.25-day dataset consisting of 1461 data records. TEND is calculated directly from this dataset as the cyclic centred difference

$$\begin{aligned} \text{ TEND }=\frac{{\varvec{\Phi }}^+ - {\varvec{\Phi }}^-}{12 hrs} \end{aligned}$$

All the other terms are formed from one-timestep integrations of the unforced model.

$$\begin{aligned} \frac{d{\varvec{\Psi }}}{d t} + (\mathcal{A} + \mathcal{D}){\varvec{\Psi }} = 0 \,\,\, \rightarrow \,\,\, (\mathcal{A} + \mathcal{D}){\varvec{\Psi }} = -\frac{d\mathbf{\Psi }}{d t}. \end{aligned}$$

Values of this negative tendency taking various sets of initial conditions from the dataset ${\varvec{\Phi }}_i$ are used to deduce the terms in (3).

Firstly, to find the sum of all the terms in (3) except TEND, the unforced model must be initialised using the entire dataset $\mathbf{\Phi }_i$. The mean and annual cycle from the resulting set of 55520 one-timestep forecasts, together with TEND, will furnish the forcing $\overline{\mathbf{f}}+\widetilde{\mathbf{f}}$. This is all that is needed to find a cyclic forcing for a simple GCM.

To break down the forcing into components we proceed as follows:

To find MM, a single integration is needed, with $\overline{\mathbf{\Phi }}$ as the initial condition. This gives $(\mathcal{A}+\mathcal{D})\overline{\varvec{\Phi }}=\text{ MM }$.

To find the annual cycle contributions MC and CC we use the mean annual cycle $(\overline{\varvec{\Phi }}+\widetilde{\varvec{\Phi }})$ as a set of initial conditions. The negative one-timestep tendencies from these 1461 unforced integrations will deliver MM+MC+CC and since we know MM we can deduce MC+CC.

To separate MC from CC another experiment is required with a set of initial conditions $(\overline{\varvec{\Phi }}+\alpha \widetilde{\mathbf{\Phi }})$. Since $\mathcal{A}$ is quadratic this will deliver $\text{ MM }+2\alpha \text{ MC }+\alpha ^2\text{ CC }$ and algebraic elimination with the previous result will provide MC and CC. The value of $\alpha$ is arbitrary and tests confirm that varying $\alpha$ does not change the result.

To find the transient contributions CT and TT we use the entire dataset $(\overline{\varvec{\Phi }}+\widetilde{\varvec{\Phi }}+{\varvec{\Phi }}')$ as a set of initial conditions as already discussed above. These 55520 unforced integrations will deliver MM + MC + CC + CT + TT and thence CT+TT. To separate CT from TT another set of initial conditions $(\overline{\varvec{\Phi }}+\widetilde{\varvec{\Phi }}+\alpha {\varvec{\Phi }}')$ is used to get $\text{ MM }+\text{ MC }+\text{ CC }+2\alpha \text{ CT } + \alpha ^2\text{ TT }$. CT and TT are again deduced by algebraic elimination. Only the mean and annual cycle components of CT and TT are of interest in our budget for $\overline{\mathbf{f}} + \widetilde{\mathbf{f}}$.

In some of the results the damping and diffusion have been removed, but this is not done by algebraically separating $\mathcal{D}$ from $\mathcal{A}$. Instead the model is simply rerun with damping and diffusion switched off and the the same procedure is followed.