Collaborative research: toward reanalysis of the Arctic Climate System - sea ice and ocean reconstruction with data assimilation

NSF AWARD # 0629311

PI and Co-PIs: A.Proshutinsky, G.Panteleev, D.Nechaev, J.Zhang, R.Lindsay

Collaborators: R.Woodgate

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The *.pdf file can be downloaded here

The webpage for this project has been moved to:

http://www.whoi.edu/science/PO/arcticgroup/projects/andrey_project2/indexAP.html

Objectives: The major project objectives for the first year of research were to:

(1) Test the incremental approach procedure; (2) Reconstruct circulation of the Chukchi Sea for 1990 (October-December) employing ice-ocean data assimilation (3) Estimate ice-ocean heat fluxes and water transport via key sections

Pilot study: Reconstruction of the Chukchi Sea circulation during October-December 1990

Data: (1) Hydrophysicals surveys during September and October 1990 and October 1991 (2) 11 moorings with CTD and current meters (3) Meteorological data

Figure 1. Blue dots - CTD surveys during September and October 1990 and October 1991. Red dots - 11 moorings. Yellow line - model domain for the future study. Green line - model domain for the pilot study.

Incremental data assimilation approach.

The incremental approach was proposed by Courtier et al., (1994) to reduce the computational burden of the 4D-VAR data assimilation. This approach can be considered as an approximate way to minimize the cost function constrained by the state-of-the-art ("complex") model through the series of quadratic cost function minimizations under a simpler linear dynamical constraints ("simple" model, describing evolution of small perturbations to the complex model solution). The complex forward model is used to estimate the cost function and the misfits between the complex model solution and observations. The corrections to the control vector of the complex model are calculated as control of the simple model producing the perturbations, which compensate the complex model-data misfits in the Least Squares sense.

The formal best choice of the linear model for the incremental approach is the exact tangent linear to the non-linear or "complex" model. In this case the incremental approach becomes equivalent to the conventional 4D-Var data assimilation algorithm, though no gain in computational cost is achieved. In practice, the incremental approach utilizes an approximation to the tangent linear model with reduced dimension of the control vector, and/or simplified dynamics, and/or coarse grid resolution

For optimization of the PIOMAS solution with respect to the oceanic observations we propose to implement the incremental approach with the SIOM tangent linear code. A consistent description of the perturbation dynamics in SIOM and convergence of the incremental approach will be ensured by similarly configured spatial grid and bottom topography, by linearization of the SIOM in the vicinity of PIOMAS solution, and by utilization of the time and space varying eddy diffusion coefficients obtained from PIOMAS. The 4D-Var assimilation of the misfits between the PIOMAS solution and data into the tangent linear SIOM will provide us with the corrections, which will be introduced into the PIOMAS fields t o improve the PIOMAS solution. In other words, the function of the 4D-Var assimilation with SIOM tangent linear model will be to redistribute observational information among all model variables and to project this information from data locations onto the boundary of the SIOM domain. In the traditional incremental approach, the complex and simple models are supposed to have common control variables (that is - the common open boundary). We propose the following modification of the traditional procedure:

(1) All simulations with PIOMAS will be performed in the model's "native" region with the boundary at approximately 43 N, while SIOM calculations will be performed in a smaller domain (north from 60øN).

(2) The corrections to the PIOMAS fields will be introduced through the restoring terms in the momentum and thermodynamic equations associated with the control parameters of the SIOM model used in the conventional 4D-var method. For example, we plan to incorporate additional terms into the PIOMAS code within a 5-10 grid cell buffer zone around the SIOM open boundary and in the vicinity of the major rivers. In the prognostic equation for PIOMAS variables the restoring terms will be defined as r(x-xb)c'(xb)/Tinc where c'(xb) is the optimal correction computed by SIOM on the open boundary with coordinates xb , r(x-xb) is the shape function distributing the corrections over the buffer zone. The time scale Tinc for the restoring terms will be the same in all equations and will be defined through several numerical experiments. Our preliminary estimate of Tinc based upon prior estimate of the data density and PIOMAS solution error variance is in the range of 1 week to 1 month

Figure 2. Data flow chart for the data assimilation procedure employing an incremental data assimilation method.

The applicability of the tangent linear code is limited to corrections with sufficiently small amplitude. If application of the incremental approach results in high amplitude corrections and the inconsistency of the tangent linear model slows down the convergence of the PIOMAS solution to the oceanic observations, we plan to test an alternative approach for the nesting of the SIOM and PIOMAS data assimilation systems and will initialize the full non-linear SIOM model using the results of POIMAS simulations and will look for the solution of the SIOM model which minimizes the model data misfits. The difference between the PIOMAS results and the solution of the SIOM 4D-var data assimilation problem will then be used to set up the nudging terms in PIOMAS model as is described above.

The optimization of the PIOMAS solution will be performed sequentially by assimilation of the oceanic data over several one-year time intervals. Note that sequential optimization of the PIOMAS solution on discrete time intervals still produces continuous in time solution over the entire period of integration, because the optimization of PIOMAS solution does not involve any re-initialization of the PIOMAS solution at the beginning of the new data assimilation interval

4D-var assimilation.

The conventional variational data assimilation method realized in SIOM is similar to a traditional Least Squares problem (Le Dimet and Talagrand, 1986; Thacker and Long, 1988). The optimal solution of the model is found through constrained minimization of a quadratic cost function, where the cost function measures squared weighted distances between the model solution and data and can incorporate other constraints such as the smoothness of the solution. Under a statistical interpretation the optimal solution is the most probable model state for the given data and prior error statistics where the cost function weights are the inverse covariances of the corresponding data errors (Thacker, 1989; Wunsch, 1996).

To determine the optimal solution of the forward model, the cost function is minimized on the space of control vectors of the model. To improve the controllability of the data assimilation algorithm we will implement the weak constraint version of the 4D-var method, which accounts for the model errors. The control vector of the model includes "free" model parameters, such as the grid values of the initial conditions, open boundary conditions and surface fluxes. Additionally the control vector will include correction terms (model errors) in the regions and during the time intervals where, as we expect, the model may have significant dynamical errors (e.g. due to known flaws in the turbulent closure scheme). For example, such corrections might be necessary during the late fall/early winter period, i.e. during the events of strong winter cooling and vertical mixing. The absolute values of the restoring terms will be physically reasonable and will be controlled by the incorporation the quadratic norm of these terms into the cost function. That should minimize the violation of model dynamical equations in SIOM caused by correction terms. The application of the conventional 4D-var data assimilation approach involves (i) running the forward model starting with some prior estimate (the so-called first guess) of the model control to estimate the cost function and the model-data misfits; (ii) running the adjoint model backward in time to compute the gradient of the cost function with respect to control vector, and (iii) application of a descent algorithm to find updated values of the control vector components. The procedure is repeated for the updated model control vector until some convergence criterion is satisfied. Typically the minimization of the cost function requires hundreds of forward and adjoint model runs.

Figure 3. Data flow chart for the data assimilation procedure based on conventional 4Dvar for the ocean data and optimal nudging data assimilation for the ice data.

First iteration. Assimilation of the 9-days low-passed velocity, CTD data and ocean surface heat/salt fluxes from PIOMAS.

SIOM solution. 1-st iteration.

Circulation in the Chukchi Sea for the period October-1990-January 1991 can be seen here:

Figure 6. Mean model-data error for the 4Dvar data assimilation with NCEP/NCAR data (blue) and PIOMAS ocean surface output (red).

After several numerical experiments we came to the following conclusions:

1) Incremental data assimilation approach based on incorporation of the nudging terms into the right part of the momentum equation of PIOMAS does not provide sufficient convergence for the minimization procedure. That can be explained by the influence of the non-local forcing.

2) Separate assimilation of the ocean data into the SIOM and ice data into PIOMAS (Figure 3) is a straightforward procedure that gives good results. Experiments on reconstruction of the circulation in the Chukchi Sea show that utilization of the ice-ocean surface momentum fluxes taken from PIOMAS produce more realistic results than utilization of the NCEP-NCAR surface momentum fluxes and/or wind at 10m (Figure 6). This is probably due to fact that 70% of the ice motion can be explained by wind effect (Andrey, I pomnu bila stateika gde odin iz aftorov bil Roger Colony I on otcenil chto wind opredeliaet 70% dvigenija l'da. Makshtas mne toge govoril pohozgie otcenki.. Ne pomnish' ssillku?)