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3.18.1 Overview

Mean surface heat flux as a control variable : $ Q_\mathrm{netm}$

This experiment illustrates the optimization (or data-assimilation) capacity of the MITgcm. Using an ocean configuration with realistic geography and bathymetry on a $ 4\times4^circ$ spherical polar grid, we estimate a time-independent surface heat flux correction $ Q_\mathrm{netm}$ that brings the model climatology into consistency with observations (Levitus climatology). The files for this experiment can be found in the verification directory under tutorial_global_oce_optim.

This correction $ Q_\mathrm{netm}$ (a 2D field only function of longitude and latitude) is the control variable of an optimization problem. It is inferred by an iterative procedure using an `adjoint technique' and a least-squares method (see, for example, Stammer et al. (2002) and Ferreira et al. (2005)).

The ocean model is run forward in time and the quality of the solution is determined by a cost function, $ J_1$, a measure of the departure of the model climatology from observations:

$\displaystyle J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2$ (3.108)

where $ \overline{T}_i$ is the averaged model temperature and $ \overline{T}_i^{lev}$ the annual mean observed temperature at each grid point $ i$. The differences are weighted by an a priori uncertainty $ \sigma_i^T$ on observations (Levitus and Boyer 1994). The error $ \sigma_i^T$ is only a function of depth and varies from 0.5 at the surface to 0.05 K at the bottom of the ocean, mainly reflecting the decreasing temperature variance with depth. A value of $ J_1$ of order 1 means that the model is, on average, within observational uncertainties.

The cost function also places constraints on the correction to insure it is "reasonnable", i.e. of order of the uncertainties on the observed surface heat flux:

$\displaystyle J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^x_i} \right]^2$ (3.109)

where $ \sigma^x_i$ are the a priori errors (2d field from ECCO ..... Fig ?).

The total cost function is obtained as $ J=\lambda_1 J_1+ \lambda_2 J_2$ where $ \lambda_1$ and $ \lambda_2$ are weights controlling the relative contribution of the two mcomponents. The adjoint model then provides the sensitivities $ \partial J/\partial Q_\mathrm{netm}$ of $ J$ relative to the 2D fields $ Q_\mathrm{netm}$. Using a line-searching algorithm (Gilbert and Lemaréchal 1989), $ Q_\mathrm{netm}$ is adjusted in the sense to reduce $ J$ -- the procedure is repeated until convergence.

In the following example, the configuration is identical to the "Global ocean circulation" tutorial where more details can be found. In each iteration, the model is started from rest with temperature and salinity initial conditions taken from Levitus dataset and run for a year. The first guess $ Q_\mathrm{netm}$ is chosen to be zero.

The experiment employs two executables: one for the MITgcm and its adjoints and one for the line-search algorithmi (offline optimization). The implementation of the control variable $ Q_\mathrm{netm}$, the cost function $ J$ and the I/O required for the commmunication betwwen the model and the line-search are described in details in section 2. The compilation of the two executables is given in section 3. A method to run the experiment is described in section 4.

Gilbert, J. C., and C. Lemaréchal, 1989: Some numerical experiments with variable-storage quasi-Newton algorithms. Math. Programm., 45, 407-435.

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