Abstract
A data assimilation (DA) method based on the application of the diffusion stochastic process theory, particularly, of the Fokker-Planck equation, is considered. The method was introduced in the previous works; however, it is substantially modified and extended to the multivariate case in the current study. For the first time, the method is here applied to the assimilation of sea surface height anomalies (SSHA) into the Hybrid Coordinate Ocean Model (HYCOM) over the Atlantic Ocean. The impact of assimilation of SSHA is investigated and compared with the assimilation by an Ensemble Optimal Interpolation method (EnOI). The time series of the analyses produced by both assimilation methods are evaluated against the results from a free model run without assimilation. This study shows that the proposed assimilation technique has some advantages in comparison with EnOI analysis. Particularly, it is shown that it provides slightly smaller error and is computationally efficient. The method may be applied to assimilate other data such as observed sea surface temperature and vertical profiles of temperature and salinity.
Acknowledgment
Authors would like to thank Prof. Jiang Zhu of the Institute for Atmospheric Physics, Chinese Academy of Sciences for his invitation to visit Beijing and for many useful advices concerning the content of the present work. The third author would like to acknowledge PETROBRAS and the Brazilian oil regulatory agency ANP (Agencia Nacional de Petroleo, Gas Naturale Biocombustiveis), within the special participation research project Oceanographic Modeling and Observation Network (REMO), and Ministry of Education of Brazil.
Funding
This research was supported by the Russian Science Foundation, grant 14-11-00434.
References
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Appendix A Numerical solution of the 2D Fokker-Planck (the second Kolmogorov) equation (1.7)
In standard notations, where x = u, y = v, equation (1.7) can be rewritten as
Let (A1) be defined in the domain Ω= {0 ≼ x ≼ l1, 0 ≼ y ≼ l2} on the horizontal plane with the boundary conditions p|Γ = 0, b11|Γ = bjj|Γ = 0. Let also the initial condition be p(0, x, y) =p°(x, y). Let the time-grid be introduced with time-step τ as tn+i = tn+ τ, n = 0, 1, 2,..., and on the domain Ω the grid ωh be introduced with the steps h1, h2 in x and y directions, respectively. Equation (A1) is solved with respect to a finite-difference splitting scheme which consists of three stages in each of time-intervals [tn, tn+1]
In (A2)-(A4) all coefficients are known at the time-moment tn. The standard notations of the difference approximations of the derivatives [23] are used as
In canonic form the system of equations (A2) with fixed index can be written as
This system has a tridiagonal matrix and is solved with the tridiagonal matrix algorithm (TDMA-method) [23]. The condition of stability of the TDMA-method |Ci| ≽ | Ai | +|Bi| holds if
This inequality yields
Let the coefficients of equation (1.7)) satisfy the conditions
Then the similar conditions hold for approximations on each time-interval
Therefore the stability condition (A5) has the following form
The system of linear equations (A3) is solved similarly with the same condition of stability. The system (A4) is solved by the simple iteration method which can be effectively parallelized. As a consequence one may state that the proposed numerical scheme for equation (1.7) has an order of approximation
Also, if initially
Then
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