Application of the finite analytic numerical method to a flow-dependent variational data assimilation
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Abstract: An anisotropic diffusion filter can be used to model a flow-dependent background error covariance matrix, which can be achieved by solving the advection-diffusion equation. Because of the directionality of the advection term, the discrete method needs to be chosen very carefully. The finite analytic method is an alternative scheme to solve the advection-diffusion equation. As a combination of analytical and numerical methods, it not only has high calculation accuracy but also holds the characteristic of the auto upwind. To demonstrate its ability, the one-dimensional steady and unsteady advection-diffusion equation numerical examples are respectively solved by the finite analytic method. The more widely used upwind difference method is used as a control approach. The result indicates that the finite analytic method has higher accuracy than the upwind difference method. For the two-dimensional case, the finite analytic method still has a better performance. In the three-dimensional variational assimilation experiment, the finite analytic method can effectively improve analysis field accuracy, and its effect is significantly better than the upwind difference and the central difference method. Moreover, it is still a more effective solution method in the strong flow region where the advective-diffusion filter performs most prominently.
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Figure 1. The numerical solution of Eq. (25) was obtained by the upwind difference method (UDM) and the finite analytic method (FAM), and the exact solution.
Figure 2. The numerical solution of Eq. (26) derived from the upwind difference method (UDM) and the finite analytic method (FAM), and the exact solution, where
$ \Delta x = 0.1 $ (a) and$ \Delta x = 0.2 $ (b), respectively.
Figure 4. The numerical solutionsof Eq. (27) obtained by the upwind difference method (UDM) (a) and the finite analytic method (FAM) (b), and their error distribution (c and d), where the spatial step is 0.05 × 0.05.
Figure 5. The numerical solutions of Eq. (27) obtained by the upwind difference method (UDM) (a) and the finite analytic method (FAM) (b), and their error distribution (c and d), where the spatial step is 0.02 × 0.02.
Figure 6. The numerical solutions of Eq. (27) obtained by the upwind difference method (UDM) (a) and the finite analytic method (FAM) (b), and their error distribution (c and d), where the spatial step is 0.01 × 0.01.
Figure 7. The distribution of the true sea surface temperature (SST) field (a), the observations (b), and the flow field (c) used in the assimilation experiment.
Figure 9. The analyzed sea surface temperature (SST) field obtained by the upwind difference method (a), the central difference method (b), and the finite analytic method (c), where the diffusion coefficient is 0.8.
Figure 10. The analyzed sea surface temperature (SST) field obtained by the upwind difference method (a), the central difference method (b), and the finite analytic method (c), where the diffusion coefficient is 0.6.
Figure 11. The analyzed sea surface temperature (SST) field obtained by the upwind difference method (a), the central difference method (b), and the finite analytic method (c), where the diffusion coefficient is 0.2.
Figure 13. The analyzed sea surface temperature (SST) fields with the upwind difference method (a, d, and g), the central difference method (b, e, and h), and the finite analytic method (c, f, and i), where the diffusion coefficient in the left is 0.8, that in the middle is 0.6, and that in the right is 0.2.
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