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Next: 3.18.2 Implementation of the Up: 3.18 Global Ocean State Previous: 3.18 Global Ocean State   Contents

3.18.1 Overview

This experiment illustrates the optimization capacity of the MITgcm: here, a high level description.

In this tutorial, a very simple case is used to illustrate the optimization 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 adjustment $ Q_\mathrm{netm}$ that attempts to bring the model climatology into consistency with observations (Levitus dataset, Levitus and T.P.Boyer [1994a]). The files for this experiment can be found in the verification directory under tutorial_global_oce_optim.

This adjustment $ 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. [002a] 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:

$\displaystyle J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2$ (3.91)

where $ \overline{T}_i$ and $ \overline{T}_i^{lev}$ are, respectively, the model and observed potential temperature at each grid point $ i$ . The differences are weighted by an a priori uncertainty $ \sigma_i^T$ on observations (as provided by Levitus and T.P.Boyer [1994a]). 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 (Fig. 3.21a). 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 adjustment to insure it is "reasonable", i.e. of order of the uncertainties on the observed surface heat flux:

$\displaystyle J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^Q_i} \right]^2$ (3.92)

where $ \sigma^Q_i$ are the a priori errors on the observed heat flux as estimated by Stammer et al. (2002) from 30% of local root-mean-square variability of the NCEP forcing field (Fig 3.21b).

The total cost function is defined 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 components. The adjoint model then yields 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 then in the sense to reduce $ J$ -- the procedure is repeated until convergence.

Fig. 3.22 shows the results of such an optimization. The model is started from rest and from January-mean temperature and salinity initial conditions taken from the Levitus dataset. The experiment is run a year and the averaged temperature over the whole run (i.e. annual mean) is used in the cost function (3.92) to evaluate the model3.1. Only the top 2 levels are used. The first guess $ Q_\mathrm{netm}$ is chosen to be zero. The weights $ \lambda_1$ and $ \lambda_2$ are set to 1 and 2, respectively. The total cost function converges after 15 iterations, decreasing from 6.1 to 2.7 (the temperature contribution decreases from 6.1 to 1.8 while the heat flux one increases from 0 to 0.42). The right panels of Fig. (3.22) illustrate the evolution of the temperature error at the surface from iteration 0 to iteration 15. Unsurprisingly, the largest errors at iteration 0 (up to 6$ ^\circ $ C, top left panels) are found in the Western boundary currents. After optimization, the departure of the model temperature from observations is reduced to 1$ ^\circ $ C or less almost everywhere except in the Pacific Equatorial Cold Tongue. Comparison of the initial temperature error (top, right) and heat flux adjustment (bottom, left) shows that the system basically increased the heat flux out of the ocean where temperatures were too warm and vice-versa. Obviously, heat flux uncertainties are not the sole responsible for temperature errors and the heat flux adjustment partly compensates the poor representation of narrow currents (Western boundary currents, Equatorial currents) at $ 4\times4^\circ$ resolution. This is allowed by the large a priori error on the heat flux (Fig. 3.21). The Pacific Cold Tongue is a counter example: there, heat fluxes uncertainties are fairly small (about 20 W.m$ ^2$ ), and a large temperature errors remains after optimization.

In the following, section 2 describes in details the implementation of the control variable $ Q_\mathrm{netm}$ , the cost function $ J$ and the I/O required for the communication between the model and the line-search. Instructions to compile the MITgcm and its adjoint and the line-search algorithm are given in section 3. The method used to run the experiment is described in section 4.

Figure 3.21: A priori errors on potential temperature (left, in $ ^\circ $ C) and surface heat flux (right, in W m$ ^{-2}$ ) used to compute the cost terms $ J_1$ and $ J_2$ , respectively.
\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/global_oce_optim/Error.eps}

Figure 3.22: Initial annual mean surface heat flux (top right in W.m$ ^{-2}$ ) and adjustment obtained at iteration 15 (bottom right). Averaged difference between model and observed potential temperatures at the surface (in $ ^\circ $ C) before optimization (iteration 0, top right) and after optimization (iteration 15, bottom right). Contour intervals for heat flux and temperature are 25 W.m$ ^{-2}$ and 1$ ^\circ $ C, respectively. A positive flux is out of the ocean.
\includegraphics[width=\textwidth,height=.3\textheight]{s_examples/global_oce_optim/Tutorial_fig.eps}

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