Reconstruction of the Circulation in Limited Regions of an Ocean with Open Boundaries: Climatic Circulation in the Tsushima Strait

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1 Oceanology, Vol. 45, No. 6, 05, pp Translated from Okeanologiya, Vol. 45, No. 6, 05, pp Original Russian Text Copyright 05 by Nechaev, Panteleev, Yaremchuk. nglish Translation Copyright 05 by Pleiades Publishing, Inc. MARIN PHYSICS Reconstruction of the Circulation in Limited Regions of an Ocean with Open Boundaries: Climatic Circulation in the Tsushima Strait D. A. Nechaev, G. G. Panteleev, 3, and M. I. Yaremchuk 4 Department of Marine Sciences, University of Southern Mississippi, Mississippi, USA International Arctic Research Center, University of Alaska, Fairbanks, Alaska, USA 3 Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia 4 International Pacific Research Center, University of Honolulu, Honolulu, Hawaii, USA Received December, 04; in final form, March, 05 Abstract Results of numerical modeling to reconstruct the circulation in the region of the Tsushima Strait using the variation method based on a specially developed regional model of the general ocean circulation are presented in this paper. Optimal solutions describing the climatic mean circulation and synoptic circulation in were obtained on the basis of climatic data and current measurements in the strait carried out in The variation algorithm for the data assimilation was implemented on the basis of the modification of the general circulation model developed at the Laboratoire d Oceanographie Dynamique et de Climatologie. Semi-implicit formulation of the finite difference model significantly simplifies the solution of the data assimilation problem. The results of the data assimilation during a 9-month-long period allow us to make conclusions about the applicability and efficiency of the developed variation algorithm for the data assimilation to reproduce the seasonal cycle in limited regions of the World Ocean with open boundaries.. INTRODUCTION The study of the ocean circulation using methods of numerical modeling has a half-century history starting from the pioneering paper by Sarkisyan [0]. During this period, the introduction of effective parameterizations and numerical schemes, as well as enormous progress in computational technology, has significantly improved the quality of reconstructing the oceanic circulation. However, this progress did not reach the level of reconstructing the weather of the ocean as can be observed in meteorology. First of all, this is caused by the significantly smaller deformation radius and the lesser density of the data measurements in the ocean as compared to that in meteorology. The existing models of the ocean circulation [3, 6, 8, 37] were developed as universal instruments for reconstructing the synoptic variability of the ocean and initially were not adapted to analyze the data of oceanic measurements in limited regions of an ocean with open boundaries. From the point of view of the problem of the data assimilation, the disadvantage of traditional oceanic models lies in the necessity for a small time step, which is caused by the explicit resolution of fast physical processes. The small time step predetermines a large number of degrees of freedom in the space of the free model parameters (initial and boundary conditions) and requires enormous computer memory for saving the solution of the direct problem and realization of the minimization process of the discrepancy of the model trajectory from the observed data. In this study, we consider the problem of reconstructing the circulation in a limited region of the ocean on the basis of the general circulation model, which was specially developed for the solution of such problems. As a prototype, we took the Ocean Parallelize OGCM (OPA OGCM) model developed at the Laboratoire d Oceanographie Dynamique et de Climatologie [6]. The numerical scheme of the OPA OGCM was modified for implicit resolution of fast processes related to the barotropic and barocinic adjustments. The developed implicit numerical scheme appeared to be more efficient than the initial explicit scheme for the solution of the data assimilation problems. This allowed us to implement the data assimilation for the region of the Tsushima Strait over a 9-month-long period (Fig. ). The circulation in the Tsushima Strait is determined by the Tsushima Current inflowing through the southern boundary of the region with a mean velocity of 30 cm/s, which transports the Kuroshio water masses into the Sea of Japan. The water exchange across the boundary with the ast China Sea is not sufficiently documented in the literature. Due to the shallow-water southwestern part of the Tsushima Strait, the seasonal fluctuations of the river discharge from the Korean coast significantly influence the coastal water masses and currents. It is likely that, in different seasons, both an inflow and a small outflow of the water through the boundary with the ast China Sea are possible in this region. Some scholars note a periodic inflow of the Sea 76

2 76 NCHAV et al. N AST CHINA SA SA OF Fig.. General scheme of the study area and the region modeled. The triangles indicate the location of acoustic current meters in of Japan waters through the northern boundary of the region selected in this study [,, ]. Thus, the Tsushima Strait is a typical example of a region in which the circulation is determined precisely by the conditions at the open boundaries. This paper is organized as follows. In Section, we describe the data used for reconstructing the circulation in the Tsushima Strait. Next, we consider the formulation of the problem of data assimilation. The experiments on reconstructing the circulation in the Tsushima Strait during a 9-month period are described in Sections 4 5. In the Conclusions, we summarize the main results and inferences of the study. A description of the particularities of the semi-implicit numerical scheme of the model is given in the Appendix.. DATA Data of the measurements of the transports and velocities of the currents. Due to the intense marine fishery, moored velocity measurements in the Tsushima Strait are extremely difficult. Therefore, the first longterm measurements of the velocities and transports through the Tsushima Strait were obtained only recently. The measurements were made from May 99 to February 00 with eleven bottom acoustic Doppler current profilers (ADCP) arranged in two lines as shown in Fig. [4]. According to these data, the monthly mean transport in the strait varied from.6 Sv (in January) to 3.5 Sv (in October). Close values of the transport fluctuations within 98 were obtained on the basis of shipborne ADCP [39]. Due to the absence of other data, the seasonal evolution of the climatic transport was determined on the basis of the daily average data estimates in [34, 4] processed with a 90-day smoothing filter. The data of the daily transport across the southern line of the current meters and the results of their filtration with a 30-day smoothing filter are shown in Fig. a for comparison. Of course, the data of the transport measurements during the period of one two years cannot be considered a good estimate of the seasonal climatic mean transport in the strait. Therefore, we assumed that the values of the transport obtained have an error of ±0.6 Sv. Such an estimate is typical for the majority of the known estimates of the transport by the Tsushima Current [, 3, 39 4]. Temperature and salinity. As the initial data, we used the monthly mean climatic distributions of temperature and salinity over a regular grid with a horizontal step of /6 /6 prepared at the Research Institute of Applied Mechanics of Kyushu University, Kassuga, Japan. This dataset is based on the data collected at the Japanese Oceanographic Data Center and on the data from the Far astern Regional Hydrological Institute (Vladivostok). The vertical resolution of the dataset used was m near the surface and increased to 50 m at a depth of 400 m. These fields were linearly interpolated to the nodes of a regular grid with a mesh of 9 9 km, which was used as the model grid. The obtained evolution of the climatic distribution of the temperature field is shown in Fig. 3. The Tsushima Strait is a well and uniformly studied region in which the circulation is determined to a great degree by the Tsushima Current, which flows through the southern boundary. This allows us to assume that the absolute error of the obtained fields of temperature and salinity are uniform in time and in the horizontal plane but decrease with depth. The estimates of the errors of the temperature and salinity fields were OCANOLOGY Vol. 45 No. 6 05

3 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 763 obtained from an analysis of the spatial and temporal variability of the climatic fields and had characteristic values of C and 0.5 at the surface and 0.5 C and 0. at a depth of 0 m, respectively. In 99 00, there were practically no field measurements of the temperature and salinity. Only a small number of hydrographic stations occupied in the periods of deploying and recovering of the current meters were available. Wind stresses and heat and salt fluxes at the surface. The monthly climatic fields calculated in [] were used as the data of the surface wind stresses and fluxes of heat and salt. These distributions were linearly interpolated over the model grid with a step of 9 9 km. The distributions obtained appeared sufficiently smooth and their horizontal scale was approximately equal to 800 km, which exceeds the characteristic size of the study region. An error of 40% from the spatial variability of the corresponding fields was attributed to these atmospheric data. Such a sufficiently large error was introduced so that the optimized fields of the surface wind stresses and fluxes could adjust to the spatial details resolved in the model used in our study. In order to reconstruct the circulation in 99 00, we additionally used the actual wind stresses recalculated from the wind velocity fields at a height of 0 m obtained as a result of a reanalysis of the NCP/NCAR meteorological data ( The results of the NCP/NCAR analysis are presented with a time interval of six hours. Since the objective of our study was to reconstruct the circulation and its variability with periods from one month to one year, the wind fields were filtered to remove the perturbations with smaller periods. 3. DATA ASSIMILATION PROBLM The data assimilation problem can be formulated as the problem of minimization of the value function J over a set of solutions of the numerical model of the ocean circulation [9, 5, 43]. The value function J is usually a quadratic norm of the discrepancy of the model solution from the data of observations. The statistical interpretation of the least square method [4, 45] allows us to consider the value function as an argument of the Gaussian distribution function, whose minimization makes it possible to obtain the most probable state of the system modeled compatible with the data available. In such an interpretation, the matrices of the weight coefficients for different additives of the value function J are covariation matrices of the errors of the corresponding physical values. During the last years, the variation methods of data assimilation have been sequentially implemented for the simplest [8] and significantly more complex [,, 5 7,,, 3, 38] problems. Transport, Sv ( ) (b) (c) Time, days Fig.. (a) Graphs of the transport via the Tsushima Strait. The thin line shows the daily transport according to [4]; the heavy line is the result of filtration of the daily transport using a 30-day filter; the asterisks indicate the evolution of the transport in the channel used as the climatic values. (b) volution of the optimized climatic transport in the strait (solid line), in the straits between Korea and the Tsushima Is. (dark points), and between the Tsushima Is. and Japan (light points). (c) volution of the optimized transport in the strait in (solid line) and in the straits between Korea and the Tsushima Islands (dark points) and between the Tsushima Is. and Japan (light points). Model. The model suggested is a modification of the OPA OGCM model based on the traditional primitive equations of motion at the hydrostatic and Boussinesq approximations. The equations are numerically approximated by a three-layer leapfrog time scheme over a C-grid (according to the classification by Arakawa []). The model is capable of using orthogonal curvilinear horizontal coordinates and generalized vertical z-coordinates. The model equations and the numerical method for their solution are described in [6]. OCANOLOGY Vol. 45 No. 6 05

4 764 NCHAV et al. 4 December 3 4 December Fig. 3. Climatic distributions of the temperature filed at depths of.5 and during the period from, 99 to, 00. The modification of the model included the following: () Implicit description of the Coriolis acceleration using the algorithm described in [33]. This technology makes it possible to eliminate the energy accumulation at the grid frequency, which is peculiar to the approximation of inertia gravity waves over a C-grid, and to increase the time step of the model. () Implicit description of fast processes including surface gravity waves. (3) Semi-implicit description of slow processes including internal gravity waves, baroclinic adjust- OCANOLOGY Vol. 45 No. 6 05

5 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 765 ment, etc. This approach allows us to avoid the appearance of grid oscillations near the vertical boundaries of the domain and in the regions with sharp changes in the bottom topography even at large integration times. Modification () is based on the approach suggested in [3] and described in detail in [33]. The method of implicit approximation of fast processes (modification ()) is widely spread and used in different OGCM models [,, 7]. In our case, implicit description of fast processes is complicated by the presentation of the Coriolis acceleration. Therefore, we put a short description of the method in Appendix A. Semi-implicit approximation of slow processes is realized by means of iterations using the codes of the linearized and conjugated models. This approach to the stabilization of the model solution can be simply adapted to any model with linearized and conjugated codes. Implicit description is introduced as a correction to the solution of the standard explicit scheme of the model at each time step and can be realized separately for different physical processes described by the initial model. The suggested approach is briefly described in Appendix B. Our experience showed that the implicit approximation leads to a significant decrease in the time splitting of the problem, which is particular to explicit three-layer schemes. In addition to the modifications of the OPA OGCM described above, we developed the codes of the linearized and conjugated models for the version of the modified OPA OGCM with vertical z coordinates. The codes of the linearized and conjugated models were obtained by direct differentiation of the direct model code. At present, the blocks of the conjugated model for calculating the turbulence exchange coefficients (using analytical parameterization of the diffusivity KPP [4] as standard turbulent closures in the OPA OGCM for calculating the vertical diffusion and the approach [36] for the coefficients of the horizontal mixing) are not completed yet. Therefore, for the data assimilation, we used fixed (though varying in time and space) turbulent exchange coefficients. In order to describe the vertical convection, we used the traditional algorithm for convective mixing of the vertical temperature and salinity profiles at each time step. In the case when the solution of the direct problem is conserved at each time step, the linearized and conjugated models are exact analytical sequences of the direct model with fixed coefficients of turbulent exchange. The analytical deduction of the conjugated problem code allows us to determine the value function gradient from the components of the control vector to the computer accuracy. Since the conjugated model is also a sequence of the direct problem, no splitting by time steps was noted in the solution of this problem. The model equations are solved in the domain Ω with four types of boundaries: the free surface Γ : {z = 0}, the ocean bottom Γ : {z = H}, and the open Γ 3 and rigid Γ 4 vertical boundaries. The set of the boundary conditions includes the condition of continuity of the momentum flux at the ocean surface Γ, the Dirichlet condition for the normal velocity component at the open boundary Γ 3, and the zero flow condition at Γ 4 used together with the slip condition for the tangent velocity and the condition of quadratic friction at the bottom. For the vertical velocity, we formulate the kinematic boundary condition at the surface and the zero flow condition at the bottom. For the equations of heat and salt transport, we specify fluxes at the surface, conditions of zero transport at rigid boundaries, and the values of temperature and salinity at the open boundaries. According to the theory of data assimilation [4, 9, 5, 45], the solution of the model equation system is optimized by means of adjusting the free model parameters, which comprise the components of the control vector Y. In our case, the vector of model control Y consists of a set of initial conditions (deviation of the ocean elevation surface field ζ, fields of temperature and salinity C, (T, S), and the horizontal velocity u) and the set of the boundary conditions at the open lateral boundaries (temperature/salinity, normal component of the horizontal velocity u n ) and at the ocean surface (fluxes of heat and salt F µ, distribution of the surface wind stress t). Thus, Y = [(ζ, C µ, u) t = 0, (u n, C µ ) Γ3, (F µ, t ) Γ ], µ =,. While modeling the circulation in the Tsushima Strait, we used a regular spatial grid with a horizontal resolution of 9 km and irregular vertical step. The calculation area had three segments of the open boundary Γ 3 along the latitudes 33.4 N, 35.5 N, and along the 6.5 longitude (Fig. ). The bottom topography field was taken from the TOPO-5 topographic database []. The minimal depth of the calculation area was specified at m, which excluded a part of the shallow-water shelf with an indented coast in the southeastern part of the Korea Peninsula from our consideration. The implicit model allowed us to use a time step of. h. In all the experiments, the time problems were integrated over a 9-month period (from May to February ). The solution of the direct problem was saved at each time step for further use to solve the inverse conjugated problem. The unknown boundary conditions at the open boundary were specified with a time step of days and linearly interpolated for intermediate time moments. The time interpolation of the boundary conditions allowed us to significantly reduce the dimension of the control vector and realize the calculation using a standard personal computer. The processor time needed for the time integration of the implicit scheme appears to be 0 % greater than the time needed for the solution using an explicit scheme. However, using the implicit scheme, we get the following: (a) it is possible to apply practically exact equations of the conjugated problem, which requires saving the solution of the direct problem at each time step; (b) the number of the degrees of free- OCANOLOGY Vol. 45 No. 6 05

6 766 NCHAV et al. dom for the direct and indirect problems significantly decreases, which leads to the fact that the data assimilation problem becomes better conditioned; (c) due to the sharp decrease in the exchanges with the disk and memory, the requirements for the computer power needed to solve the data assimilation problem significantly decrease. Value function. In order to solve the problem of reconstructing the circulation in ocean regions with open boundaries, we specify the value function J, in which the actual measurements of selected physical parameters are used together with additional controlling additives [4] that provide the corresponding smoothness of the solution obtained: s + W C J = J C + J u, J C = [ W C ( C µ C µ *) µ =, Ω, t ( C µ ) s ]dωdt + [ W B ( B µ ) ] dsdt, J u = 0 k W u, k z = 0 ( u u k *) + W V, n n =, N ud z d s l V n * + W u ( u) dωdt L n H s + W ζ ( ζ) W τ ( t t* ) s [ + + W τ ( t) ] dsdt. z = 0 () Here, u is the vector of the horizontal velocity; ζ is the displacement of the elevation surface; C = (T, S) are the temperature and salinity; t and B µ are the surface wind stresses and heat and salt fluxes; N is the number of segments at the open boundary; V n is the estimate of the transport through segment n of the open boundary L n ; s s s s s and W C, W u, W ζ, W B, W τ, W C, W u, W ζ, W B, W τ are the covariations of the errors of the corresponding data. The asterisks indicate observed fields. The smoothing terms are proportional to the squared Laplace s operator from the corresponding model fields. The introduction of these terms allows us to suppress high frequency noises contained in the data of the observations and, thus, to make the problem regular. The functional J contains two groups of terms J C and J u corresponding to the baroclinic (temperature, salinity, heat and salt fluxes at the surface) and barotropic (surface elevation, velocity, and surface wind stress) variables. The sense of the different terms J C and J u of the value function is clear: minimization of these terms should provide smoothness of the model solution and its proximity to the data of the observations. Ω The spatial distributions of the error covariations of the actual data W C, W u, W ζ, W B, W τ were discussed above in Section. The covariations of the regularizing s s s s s terms of the value function W C, W B, W u, W ζ, W τ were estimated from the analysis of the corresponding s 4 s 4 s scales: W C = L C / C s, W B = L C / B s, W u = L 4 u / V s, s W ζ = L 4 s u / ζ s, W τ = L 4 τ / τ s. The characteristic scales of the velocity and displacement of the ocean surface elevation were chosen as V s = 5 cm/s, ζ s = 0 cm. The estimates for C s, B s, τ s were obtained from the character- istics of the spatial variability of the data. We used spatially uniform scales: L τ = 500 km, L C = 50 km, and L u = 50 km. The mathematical foundations of the value function J minimization under the restrictions in the form of the model equations are described in a number of publications [4, 9, 4, 45]. We used the standard approach: after specifying the first approximation of the control vector Y, we obtained the solution of the model equations. After this, knowing the model solution, we calculated J and the gradient of J on the basis of the model variables, which include distributions of the temperature, salinity, velocity, surface elevation, and fluxes of the momentum, heat, and salt at the ocean surface at each time step of the direct problem. Next, the system of the conjugated equations of the model was integrated back in time, which resulted in the calculation of the gradient of J with respect to the control variables. After this, the gradient and minimization algorithm [] were used to find a more exact new estimate of the control vector for the model. The procedure was repeated until the minimum of the value function was found with the given accuracy. Thus, we determined the optimal control vector Y opt, which gives the model solution corresponding to the minimum of the functional J. Owing to the nonlinearity of the model, the value function J can have several minima in the space of the control vectors. However, the gradient methods of minimization are capable of determining only the closest minimum to the first approximation. Therefore, the choice of the realistic first approximation for the model solution is very important. In order to prepare the best first approximation, which would be the closest to the observed data, we used a two-step procedure. First, we optimized the diagnostic fields of the velocity and surface elevation for the beginning, middle, and end of each month from May to February. This problem was solved by means of minimization of the value function J u within a stationary diagnostic model under fixed distributions of temperature, salinity, and surface wind stresses. The control vector of the diagnostic version included the boundary conditions for the normal velocity at the open boundary, and, in our case, it had a dimension of approximately 500 elements, while the geometric size OCANOLOGY Vol. 45 No. 6 05

7 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 767 of the region was boxes. This way, we calculated optimized distributions of the diagnostic fields of the velocity and surface elevation. The evident mathematical meaning of this step is to find the optimal boundary conditions for the normal velocity component, which give a physically correct solution within the domain and minimize the value function J u at fixed distributions of the temperature and salinity. As the second step, we performed the calculation of the direct model between May and February (5400 time steps); we selected the diagnostic fields calculated for May being a result of the first step as the initial conditions. At each time step, the boundary conditions for the normal component of the velocity were determined using the method of linear interpolation between diagnostic solutions. In order to determine the boundary conditions for the temperature and salinity, we used linear interpolation between the climatic fields available at those segments of the open boundary in which the normal component of the velocity was directed inwards (inflow), while the Orlanski radiation condition was specified for the outflow segments. The boundary conditions at the ocean surface were linearly interpolated from the known climatic fields. The evolution of the temperature (Fig. 4a) and salinity (Fig. 4b) fields at a depth of 45 m is shown in Fig. 4. The lack of coordination between the partly optimized boundary conditions for the velocity and the unoptimized boundary conditions for the temperature and salinity along the open boundaries lead to the appearance of computational boundary layers in the temperature field at a depth of (Fig. 4a, February ) and gradual divergence of the model solution from the data of the observations (Fig. 3). For example, the mean temperature at the end of the model period (Fig. 4a, ) differs from the mean climatic temperature in February by 3 ë, which approximately corresponds to 70% of the temperature variability in the region of the Tsushima Strait. Starting from the first approximation obtained (Figs. 4a, 4b), we minimized the value function J, which characterizes the discrepancy of the model solution from all the data available. The piece-wise linear interpolation of the boundary conditions allowed us to reduce the dimension of the control vector to elements. It is clear that various components of the control vector (velocity, temperature, salinity, etc.) differently influence the solution. For example, for the minimization of the value function J, we used a method of sequential increasing of the control vector dimension. Actually, the minimization of the value function was carried out using the following scenario: at the first step, we used the initial and boundary conditions for the velocity and the initial distribution of the surface elevation as the control vector, while all the rest of the parameters of the control vector were fixed. At the next stage, the initial and boundary conditions for the temperature and salinity were added to the control vector. Only at the third stage were all the possible components of the control vector optimized. Numerous experiments demonstrated that such a minimization scheme makes it possible to minimize the value function most rapidly and requires smaller computational resources. Usually, approximately iterations were needed to minimize the value function, which allowed us to reduce the norm of the gradient by a factor of One minimization procedure required two days for the calculation with an Athlon 00+ computer. 4. RSULTS OF TH RCONSTRUCTION OF TH OPTIMIZD CLIMATIC FILDS OF TH TMPRATUR, SALINITY, AND VLOCITY IN TH TSUSHIMA STRAIT The data of the temperature, salinity, wind stress, and fluxes of heat and salt at the ocean surface, as well as the values of the mean transport in the strait, were used to reconstruct the climatic mean circulation. The measurements of the velocity were not used directly, i.e., the value of the weight factor W u, k in () was specified as zero. The optimized fields of the temperature and velocity obtained as a result of minimizing the value function J are shown in Figs. 5a and 5b. ven a visual comparison of the optimized (Fig. 5a) and climatic (Fig. 3) fields of temperature shows that the optimization of the boundary and initial conditions performed made it possible to significantly improve the reconstruction of the circulation in the Tsushima Strait as compared to the solution of the first approximation (Fig. 4a). The error of the deviation of the optimized temperature field from the climatic distribution is within 40% and only slightly depends on time. We recall that, at the initial moment, the temperature field in the solution of the first approximation (Fig. 4a) completely coincides with the climatic temperature distribution, but, already after the second month of the integration, the discrepancy of the model solution is equal to 30 50% and, by the 0th month of the model integration, it increases up to 60 70%. The evolution of the mean climatic fields of the temperature and salinity of the first approximation and the fields of the optimized solution at depths of and is shown in Fig. 6 for comparison. One can see that the differences between the average values of the climatic and optimized fields of temperature and salinity are not significant, whereas the differences between the climate and the solution of the first approximation reach ë and 0.. It is likely that the difference that strong indicates that the boundary conditions used in the solution of the first approximation are not consistent and illustrates well that the optimized solution is preferable. The velocity fields of the optimized (Fig. 5b) and not optimized (Fig. 4b) solutions only slightly differ from each other in the internal points of the study region. In our opinion, this is explained by the barotro- OCANOLOGY Vol. 45 No. 6 05

8 768 NCHAV et al. (a) December December Fig. 4. volution of the (a) temperature ( C) and (b) velocity fields at depths of.5 and obtained as the fields of the first approximation. The velocity scale is shown. pic nature of the circulation in the region and the close integral transports via the strait. The main differences in the velocity fields shown in Figs. 5a and 5b are observed along the open boundaries of the region, i.e., at the places where the variation algorithm used directly causes a variation in the normal component of the velocity, which is a control parameter. It is interesting to emphasize the following particular features of the optimized solution. (a) Two clearly manifested branches of the currents one off the Korean coast and the other off the Japanese coast were formed in the northern part of OCANOLOGY Vol. 45 No. 6 05

9 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 769 (b) December December Fig. 4. (Contd.) the region. Between these two branches, the water masses flow either to the east or have an insignificant negative meridional component. Such branching of the Tsushima Current was repeatedly noted in selected papers [, 9, 46]. The results of the studies [, 34] indicate that such branching usually occurs more to the south (by 0.5 ) than was obtained as a result of the optimization. This difference can be explained by the fact that the climatic fields of the temperature and salinity are very smooth over the set of various realizations, OCANOLOGY Vol. 45 No. 6 05

10 770 NCHAV et al. while the intensity and width of both branches of the Tsushima Current strongly change in time [34]. We note that, in the solution of the first approximation (Fig. 4b), the flow through the northern boundary is not characterized by clearly manifested branches and is practically uniformly distributed between Korea and Japan. (b) A return current was formed at the southern boundary near the coast of Japan. The presence of such a current was repeatedly noted [30]. On the basis of the 9-month measurements of the currents in 99 00, the authors of [34] found that a return current near the coasts of Japan was periodically noted in the summer period and practically always appeared in the late autumn and in the winter. According to our results (Fig. 4b), in February, the return current manifests itself more clearly, while in June August, it is either absent or localized in the regions with depths shallower than 00 m. We note that the conclusions about the periodicity (instability) of the return current in the summer [34] were made on the basis of the current measurements at a point with the coordinates 33.4 N, 9, i.e., at a place where the depth exceeds 00 m. Therefore, we have grounds to suppose that the return current at the southern boundary of the Tsushima Strait is quasi-permanent, but its intensity and width are subjected to strong seasonal changes. (c) At the western boundary, the distinguishing features of the optimized solution are poorly manifested. However, we note an increase in the velocities at this boundary as compared to the solution of the first approximation. A graph of the changes in the optimized transport via the strait is shown in Fig. b. With an accuracy to the possible errors, the time changes in the transport obtained satisfy our initial suppositions about the values and errors of the climatic transport (Fig. a). We note that, in the optimized solution, the maximal transport in the strait is observed at the end of July beginning of September (days 0 50 in Fig. ), which agrees with the results in [39]. The curves of the transport between Korea and Tsushima Island and between Tsushima Island and Japan are also shown in Fig. b. We note that the values of the transports in both straits are very close. 5. RSULTS OF TH RSONSTRUCTION OF TH LARG-SCAL CIRCULATION IN TH TSUSHIMA STRAIT IN The long-term data of the measurements of the vertical profiles of the currents at points (Fig. ) in the Tsushima Strait [34] make it possible to reconstruct the evolution of the currents in more detail. Formally, the model we used could be applied to reconstruct the currents at time scales up to the tidal currents. However, in this case, the function approximating the boundary conditions should resolve such time scales. This problem would have a large number of degrees of freedom; therefore, it would require a significant increase in the amount of the data and computer resources. Thus, we limited ourselves to the problem of reconstructing the evolution of the currents during the period from May, 99, to February, 00, with a variability time scale equal to 4 weeks. In order to solve the problem of reconstructing the actual circulation, it is highly desirable to have an overall dataset of velocities (or surface elevations), temperatures, salinities, and meteorological parameters. Unfortunately, the velocities at acoustic current meter sites and the NCP/NCAR data of analysis were practically the only available information on this region during the period from May, 99 to February, 00. The values of the temperature and salinity were known only at 74 hydrographic stations obtained during the periods of deployment (May 7, 99) and recovery (February 8, 00) of the current meters. The reconstruction of the actual circulation during the period from May, 99 to February, 00 was carried out according to the scheme described above. The climatic circulation (Fig. 5) was assumed as the solution of the first approximation, which was later optimized with account for the actual data in The differences in the structure of the value function () were minimal: (a) The term of the functional Σ n =, N W ( dzdl ) V, n u V n *, L n H which provides the proximity of the solution to the estimates of the integral transport in the strait, was not taken into account ( W V, n = 0). Instead, we used the functional Σ k, (u u ) k * W u n responsible for drawing the solution to the data of the velocity observations, which were averaged by a smoothing filter with a period of 30 days. (b) Instead of the climatic data on the surface wind stress, we used the daily average wind field at a height of 0 m (NCP/NCAR), which was converted into tangential wind stresses. (c) In addition to the climatic data on the temperature and salinity, we used the few available measurements of C µ *, µ =, at the beginning and at the end of the study period. This corresponds to the inclusion of an additional functional Σ k W (C µ ) Cµ, rk, C µ *, k into the value function (). It is natural to consider that these data are significantly more exact than the climatic distributions. Therefore, we supposed that their accuracy is 3 times better than the accuracy of the climatic distributions of the temperature and salinity, i.e., 3 W Cµ, rk, W C. The fields of the temperature and currents reconstructed using this method are presented in Figs. 7a OCANOLOGY Vol. 45 No. 6 05

11 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 77 ( ) 4 December December Fig. 5. volution of the optimized (a) temperature ( C) and (b) velocity fields at depths of.5 and reconstructed with account for the mean climatic data on the hydrology, meteorology, and integral transport in the strait. The velocity scale is shown. and 7b. As a result of the optimization performed, the root-mean-square discrepancy between the measured and modeled velocity values decreased from 70 80% (for the solution of the first approximation) to 30 40%, which is well seen from the comparison of Figs. 5b and 7b. The assimilation of the actual data influences the fields of the currents and temperatures in the strait. The most significant changes occurred in the northern part OCANOLOGY Vol. 45 No. 6 05

12 77 NCHAV et al. (b) December December Fig. 5. (Contd.) of the region, where the assimilation distinguished two inflows of water masses through the northern boundary (in June and in November December 99). Both inflows are clearly manifested in the data of the velocity measurements shown in Figs. 5b and 6b. Such inflows of water masses from the Sea of Japan are one of the well-known features of the local circulation []. It is noteworthy that only the inflow in June 99 clearly manifests itself in the optimized temperature field (Fig. 7a). It is likely that this is the consequence of the scarcity of the hydrological measurements in the initial period and of the lack of similar information in the other periods. In December 99 to February 00, the assimilation of the actual velocity data distinguished an OCANOLOGY Vol. 45 No. 6 05

13 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 773 Temperature, ë 4 m m Time, days Salinity, Fig. 6. volution of the temperature (on the left) and salinity (on the right) values averaged over the depths calculated from the climatic data (points), from the fields of the solution of the first approximation (asterisks), and from the fields of the optimized solution (circles) intensification of the return current at the southern boundary as compared to the optimized climatic solution. This fully agrees with the results in [34]. At the western boundary, we have to note a tendency to a decrease in the inflow into the Tsushima Strait from the ast China Sea and even the appearance of a periodic outflow through this boundary. The evolution of the optimized transport in the strait in is shown in Fig c. The fluctuations similar to the fluctuations of the filtered transport in (Fig. a) obtained on the basis of the same velocity data, which have already been assimilated, are clearly seen on the graph. However, the absolute values of the transport optimized in are % lower than the climatic optimized transport, which was obtained using only the information about the transport. We associate this decrease in the total transport with the practical absence of actual data of the temperature and salinity measurements. During minimization of the value function without such data, there is a necessity to adjust the smoothed climatic data to the velocity field observed in We suppose that taking into account such smoothed data leads to a decrease in the absolute values of the transport. We note that, in May (35th day) and in the winter period ( th days), the optimized transport in did not practically differ from the optimized climatic transport. This can be related to the assimilation of the hydrological information at the initial moment and to the winter mixing and barotropization of the circulation in the Tsushima Strait, i.e., to the processes that decrease the influence of the horizontal inhomogeneities of the temperature and salinity on the velocity field. During the analysis of the optimized solution for 99 00, one has to take into account that this solution is optimal for the data assimilation within the assumptions about the spatiotemporal smoothness of the solution. The use of additional data of observations can lead to changes in the optimized solution. We can suppose that these changes can be significant, since the data we used were far from ideal. 6. ANALYSIS OF TH RSULTS AND CONCLUSIONS One of the objectives of this study is to reconstruct the climatic mean velocity field in the region of the Tsushima Strait and to obtain a realistic circulation in on the basis of the entire climatic information and the data on the currents and wind velocity in Owing to the nonlinearity, the solution of this problem depends on the choice of the solution of the first approximation. Therefore, it is desirable that the initial and boundary conditions taken as the first approximation would give an evolutionary solution sufficiently close to the data of the observations available. In order to determine such boundary conditions, we used the solutions of stationary problems of optimizing the normal velocity component at the open boundaries. The final determination of the initial and boundary conditions for the first approximation was made on the basis of optimized fields and climatic distributions of the temperature and salinity as piece-wise functions linear in time. The solution of the model based on such boundary conditions reconstructs only the main features of the velocity field in the Tsushima Strait and, owing to the inconsistency between the different boundary and initial conditions, the solution gradually deviates from the data of the observations used in the calculations. The solution of the problem of minimizing the value function J over the set of the solutions of the model equations allowed us to significantly improve the coincidence between the model solution OCANOLOGY Vol. 45 No. 6 05

14 774 NCHAV et al. ( ). December 4 December N Fig. 7. volution of the (a) temperature ( C) and (b) velocity fields at depths of.5 and reconstructed with account for the meteorological data, the data of the velocity measurements, and the small amount of temperature and salinity data obtained in The velocity scale is shown. and the data available, in particular, to perform a practically ideal reconstruction of the evolution of the mean temperature and salinity values (Fig. 6). The obtained optimized solution reconstructs a number of important features of the circulation at the southern and northern boundaries of the region, which agree well with the OCANOLOGY Vol. 45 No. 6 05

15 RCONSTRUCTION OF TH CIRCULATION IN LIMITD RGIONS 775 (b) N 35.0 December December Fig. 7. (Contd.) independent observations and studies of other authors. Such improvements allow us to consider the solution obtained (Fig. 5) as a realistic estimate of the climatic mean circulation in the Tsushima Strait obtained on the basis of the data available. Meanwhile, we have to note that the estimates of the climatic transport used in this study are based on limited data of measurements of the velocity and transport in the strait; therefore, they can be erroneous to a significant degree. Consequently, the use of other data as cli- OCANOLOGY Vol. 45 No. 6 05

16 776 NCHAV et al. matic estimates of the integral transport can change the fields presented in Fig. 5. At the same time, due to the significant barotropic component of the currents in the Tsushima Strait, the changes in the transport would first influence the absolute values of the velocities and, to a lesser extent, they would influence the general pattern of the currents. The second but very important problem whose solution interested us was the development of the most effective strategy to solve the problem of reconstructing the circulation in a limited region of an ocean with long open boundaries over a long time interval. The problem of determining a realistic solution of the first approximation described above is only one of such technical problems. Another problem is determining the set of the most important controlling parameters of the problem. Numerous experiments demonstrated that minimization of the value function requires lesser calculation resources if the control vector is gradually increasing. Such a procedure of obtaining the optimal solution required approximately external iterations and allowed us to decrease the norm of the functional gradient by a factor of In this study, we made an attempt to reconstruct the circulation in the Tsushima Strait in on the basis of the data of the velocity measurements available and a small number of actual measurements of the temperature and salinity. Our results indicate that data assimilation leads to a more detailed reconstruction of the velocity fields. The lack of hydrological data does not allow us to make a final conclusion about the quality of the reconstructed hydrological fields. An analysis of the optimized temperature fields (Fig. 7a) indicates that the assimilation of a small amount of hydrological data in May significantly changes the temperature fields and makes them much more realistic. Therefore, the significant difference in the temperature fields in June shown in Figs. 5a and 7a and the similar temperature fields in other months possibly indicates that the majority of the temperature fields shown in Fig. 7a are not realistic enough. Despite the fact that the temperature fields in Fig. 7a are close to the climatic fields, the results presented here show that the numerical model of the ocean and the variation mechanism of the data assimilation developed here can be successfully used for the problems of reconstruction and analysis of circulation during sufficiently long time intervals as well as for the problem of reconstruction and forecast of the ocean weather in regions with long open boundaries. The quality of the reconstructed fields would naturally depend on the amount and quality of the data available, which should resolve the processes studied. Thus, in addition to the usual hydrological and velocity data, it is useful to use data on the sea surface temperature and surface elevation as well as other kinds of satellite information. Our experience indicates that the model is well controlled by the parameters of the control vector and is sufficiently effective from the computational point of view. A disadvantage of the model is the lack of parameterization of the diffusivity. In the future, we plan to resolve this problem. ACKNOWLDGMNTS This study was supported by the Office of Naval Research, grant no. NR ; the National Science Foundation, grant no. OC 0-0; the Frontier Research System for Global Change, JAMTC, Japan; the International Arctic Research Center, Fairbanks, USA; and the International Pacific Research Center (IPRC/38), USA. The authors thank M.N. Koshlyakov, A. Ostrovskii, H. Perkins, G. Jacobs, P. Pistek, and W. Teague for the data and fruitful discussion of the results obtained. APPNDIX A: IMPLICIT DSCRIPTION OF FAST PROCSSS The finite difference equations of the model needed to consider an implicit description of the fast processes of the model solution are written in Cartesian coordinates in the following form: t δ t u m f σˆ mv m = gδ x ζ t + R u, m =,, 4, (A) δ t v m f σˆ t + m u m = gδ y ζ t + R v, m =,, 4, (A) u = ( u, v ) = -- u 4 ( m, v m ), w z m =, δ t ζ = w z = 0 + R ζ, δ z w = t δ x u δ y v, (A3) (A4) (A5) = H + uδ x H + v δ y H = 0. (A6) quations (A) and (A) approximate the equations of the evolution of the horizontal components of the momentum, where the vector R u includes the advection of momentum, the gradient of the baroclinic part of the pressure, and the turbulent diffusion of the momentum. quations (A4) and (A6) specify the kinematic boundary conditions at the surface and bottom of the ocean, R ζ is the advection of the surface elevation, and q. (A5) approximates the equation of continuity in the Boussinesq approximation. According to the generally accepted notations, ζ i, j n ζ i, j n u i + --, j, k u i --, j, k δ t ζ = ( )/δt, δ x u = ( )/δx; and the indices i, j, k, n are common for all the variables and are omitted in the equations. The implicit numerical scheme is provided by time averaging of the terms in the equations, for example, ζ t = αζ + βζ n + αζ n, where α and β are not negative, and α + β =. The parameter of the implicit scheme α can be different for different terms of the equation; however, for the sake of OCANOLOGY Vol. 45 No. 6 05