Abstract
Application of the 4D-var data assimilation technique to the system of the two one-way nested models is discussed. We consider preliminary result of the data assimilation experiments with the nested system involving the Bering Strait high resolution model and a coarser resolution model configured for the Pacific sector of the Arctic Ocean. The observations obtained both in the Chukchi Sea and the Bering Strait regions are assimilated using the "nested" adjoint models. The nested variational data assimilation scheme allows us to obtain a dynamically consistent solution for the both nested models and provides an improved hindcast of the Bering Strait and Arctic Ocean circulations.Motivation for developing a nested 4Dvar data assimilation system
Two-way variational nested data assimilation system is a straightforward approach to resolve common contradictions between the requirements imposed on the design of the system: To maintain a proper resolution of complicated bottom topography and coastline configuration in the key regions of dynamical importance; To keep the dimension of the data assimilation problem reasonably small for numerical efficiency; To account for dramatic difference in data coverage density over the region under investigation, and To reduce the ratio between the number of degrees of freedom of the data assimilation system and the number of the observations without significant simplification of the model physics.The Chukchi Sea and Bering Strait region is chosen as a test region because of the <\p>
1. Importance of accurate estimates of the currents in the Bering Strait for correct prediction of the transport of the Pacific water into the Arctic Ocean; <\p>
2. Availability of continuous velocity observations in the Bering Strait since 1990, SSH, SST and velocity observations in the Chukchi Sea; <\p>
3. Complexity of the local topography and coastline; <\p>
4. Non-regular distribution of the observations in the Chukchi Sea.<\p>
Objectives:
To test different settings of the nested variational data assimilation system and to asses their relative performance; <\p>
To explore the benefits and drawbacks of the increased resolution in the key dynamical region given the configuration of the available observations;<\p>
To assess the quality of the circulation reconstruction in the Bering Strait by the nested data assimilation system.<\p>
4Dvar data assimilation system
The nested data assimilation system is built using the following data assimilation components: Forward Model: the model is a modification of the OGCM designed by Madec et al., 1999. The numerical scheme of the model is implicit for both barotropic and baroclinic modes (Nechaev et al., 2005, Panteleev et al., 2006). The Coriolis terms in the momentum equation are approximated with implicit scheme (Nechaev and Yaremchuk, 2004). Adjoint Model: analytical transposition of the operator of the tangent linear model. Control vector: initial conditions (SSH, T/S, U/V), boundary conditions (T/S , U/V , surface heat/salt fluxes, wind stresses). Time evolution of the functions representing boundary conditions is approximated by piece-wise linear continuous polynomials on 3-day intervals. Coarse resolution (CR) model is configured in the region shown in Figure 1 on a grid with a spatial resolution 10x10km, 12 non-uniform levels with 3m resolution at the surface and 10m near the bottom, and time step 0.1 days. Fine resoluion (FR) model: is configured in the region shown in Figure 1 on a grid with a resolution 5x5 km and the same vertical resolution as CR model. Time step - 0.03 days. Data assimilation time windrow - 1 month.Nested variational data assimilation system
The presented results are obtained with the two setting of "weakly two-way nested" and "strongly two-way nested" variational algorithms. The first guess solution for the nested data assimilation system is specified as an optimal CR solution resulting from application of a conventional 4Dvar procedure for the CR model and a solution of the FR model obtained in a one way nested model run. In case of "weakly two-way nested" approach, the optimization of the CR and FR model control parameters is performed by the repeated solution of the two (CR and FR) 4Dvar problems. The cost functions for CR and FR 4Dvar problems have similar structures. The cost functions contain the terms penalizing the model misfits with observations and the so-called "background" terms penalizing the model misfits with additional (e.g. climatological) data. The two-way information flow between CR and FR data assimilation solutions is conducted by including the CR solution as background data in the FR cost function and vice versa. For the Setting 1 of the "weakly two-way nested" variational algorithm, the "background" data is updated after full convergence of the CR and FR optimization procedures, and optimizations are repeated for the modified cost functions. For the Setting 2, the cost functions are updated after few iterations of the optimization procedures, before the full convergence is achieved. The alternative "strongly two-way nested" variational algorithm is described below.
Figure 1 Local topography and CR/FR model domains of the nested data assimilation system. Grey line - CR model. Blue - dashed Line - FR model.
Data:
Time series of velocity, T/S data at 12 moorings in the Chukchi Sea during the period September 1990 - October 1991 (Figure 2).
T/S data from two hydrological surveys during September - November 1990 (~130 stations) (Figures 1,2).
Realistic (NCEP/NCAR) wind stresses and heat/freshwater surface fluxes.
Figure 2 Locations of the 12 moorings (red circles) and CTD stations (dots) during the fall of 1990.
First guess solutions
Figure 3 Time evolution of the total transport stream functions (in Sv) in the first guess solutions of the CR (left panel) and FR (right panel) models.
Figure 4 Time evolution of the Bering Strait transport in the CR (red) and FR (blue) first guess solutions.
Setting 1 of the nested 4Dvar problem
Figure 5.1 Optimal solution after 2 updates of the cost functions for the Setting 1 of the nested 4Dvar problem. Time evolution of the total transport stream functions (in Sv) of the CR (upper panel) and FR (middle panel) solutions.
Figure 5.2 Time evolution of the Bering Strait transport in the CR (red) and FR (blue) solutions.
Setting 2 of the nested 4Dvar problem
Figure 6.1 Optimal solution after 2 updates of the cost functions for the Setting 2 of the nested 4Dvar problem. Time evolution of the total transport stream functions of the CR (upper panel) and FR (middle panel) solutions.
Figure 6.2 Time evolution of the Bering Strait transport in the CR (red) and FR (blue) solutions.
Strongly two way nested data assimilation system
We propose the "strongly two-way nested" variational algorithm allowing for some reduction of the dimension of the data assimilation problem compared to the "weakly nested" algorithm. Briefly this approach can be summarized as follows:
Conclusions:
1. The Setting 1 of the "weakly two-way nested" 4Dvar algorithm proved to be successful in application to the assimilation of the velocity, SSH and T/S data collected in the Chukchi Sea - Bering Strait region (see Figure 5). This version of the algorithm revealed:
- Fast convergence (two-three updates of the cost functions were sufficient); <\p>
- Significant reduction of the model-data misfits compared to the CR data assimilation solution; <\p>
- Data assimilation into the FR model of the Bering Strait region produced Bering Strait transport ~10% higher than the first guess solution. <\p>
2. The Setting 2 of the "weakly two-way nested" 4Dvar algorithm did not converge, resulting in the growth of the model-data errors in the FR solution, though the CR solution remained to be close to the first guess solution and to the CR solution obtained for the Setting 1. The FR solution is unrealistic containing some artificial features. The FR model Bering Strait transport is reduced with respect to the first guess solution (see Figure 6). <\p>
3. "Strongly two-way nested" 4Dvar approach was realized for DA configuration similar to the setting 1 of the "weakly two-way nested" experiment. Cost function of the "strongly two-way nested" approach was a sum of the CR and FR cost functions of the setting 1. Algorithm proved to be very robust and shown a fast convergence of the optimization procedure. Advantage of this method is a somewhat higher computational efficiency. The method also enforces equal (and consistent with observations) Bering Strait transports in the CR and FR models.<\p>
4. Setting 1 of "weakly two-way nested" data assimilation system provides the CR and FR solutions which are very similar to the solutions of "strongly two-way nested" approach. <\p>
5. Two-way nested 4Dvar is a very promising approach for developing a flexible and efficient data assimilation system in the Arctic Ocean. Additional OSSE experiments are needed<\p>