## 1. Introduction

With the rapidly increasing amount of available observations, results of model simulations, and newly gained knowledge of physical processes, the development of data assimilation methods is in great demand. One of the areas of application of data assimilation methods is the specification of open boundary conditions (OBCs) for limited-area models. The use of data assimilation techniques improves the model predictions and avoids the ill-posed, point-wise treatment of OBCs (Bennett 1992, 1994; Oliger and Sundstrom 1978). In many data assimilation techniques of specifying OBCs, the latter are considered as control parameters. In this case, OBCs are chosen in such a way as to simultaneously provide the best fit to the governing equations and to the observations (e.g., Bennett 1992, 1994; Zou et al. 1993; Seiler 1993, etc.). The best fit means the minimization of the norm of the deviation between model results and observations. Thus, the interior solution and the available observations are used to calculate the variables on the open boundary. The main restriction of this approach is the need for significant amounts of computer time and memory.

In Shulman and Lewis (1994, 1995), OBCs are chosen by combining the model dynamics with the data assimilation only on the open boundary and its vicinity. It was shown that many well-known, radiation-type OBCs are special cases of the optimized OBCs obtained. The solution of this local data assimilation scheme was easily derived. Therefore, this approach does not significantly impact the computational time and the required memory. At the same time, the comparison between the Reid and Bodine formulation and the optimized version of this condition showed that optimization provides much better results in modeling tidal constituents.

In section 2, we describe the general formulation of the local data assimilation approach to specification of OBCs and provide the detailed description of the barotropic case. The applications of optimized OBCs are demonstrated for the cases of the idealized channel and the Adriatic Sea. The results of tidal and wind-driven simulations as well as sensitivity studies are presented and discussed in sections 3 and 4.

## 2. Derivation of the optimized open boundary conditions

**X**be the variables that we are to specify on the open boundary (sea surface height, velocity, temperature, salinity, etc.). Suppose we know some reference value of vector

**X**, which we denote as vector

**X**

^{o}. The reference values of boundary variables in

**X**

^{o}can be estimated from available observations, another numerical simulation, and approximations based on the governing physics. We introduce the function

*J*(

**X**,

**X**

^{o}) ≥ 0, which represents the difference between model and reference values of variables on the open boundary. Let

*P*

_{t}be the energy flux through an open boundary, which results from the difference in values of vectors

**X**and

**X**

^{o}. Accordingly, let

*M*

_{t}be the mass flux and

*F*

_{t}be the momentum flux through the open boundary, which results from the difference in values of vectors

**X**and

**X**

^{o}. If we suppose that we know some estimates of

*P*

_{t},

*M*

_{t}, and

*F*

_{t}, we might choose the boundary values (vector

**X**) from the following optimization problem: where

*A, B,*and

*C*are operators for calculating energy, mass, and momentum fluxes. Thus, we choose open boundary values by minimizing the deviation between the reference and model boundary values under the integral constraints representing the energy, mass, and momentum fluxes. Solving (1)–(4) is a very complicated problem, and different approaches can be used. Each of these approaches has to take into account that the estimated

*P*

_{t},

*M*

_{t},

*F*

_{t}, and reference values may contain errors.

*S*is the open boundary of the model domain

*D, u*

_{n}and

*η*are the vertically averaged outward normal velocity and the sea surface elevation on the open boundary,

*H*is the depth, and

*g*is the gravitational constant. Suppose that we have some reference values of sea surface elevation on the open boundary in the form of

*η*

^{o}and some reference values of the outward normal velocity in the form of

*u*

^{o}

_{n}

*P*

_{t}is as previously defined and (6) is the energy flux resulting from the differences in model and reference values of sea surface elevation and velocity. In a real-world situation,

*P*

_{t},

*η*

^{o}, and

*u*

^{o}

_{n}

*γ*is a parameter of regularization. The solution of (7) has the following form: where To determine boundary conditions from (8) and (9), we have to choose a value for the parameter

*γ.*There are many different approaches to choosing

*γ.*Most of them rely on the estimate of the error in the input data. In most cases, we do not know the norms of the errors in the estimates of

*P*

_{t}and the reference values. Moreover, these norms will change from time to time. Therefore, the attractive approaches are the ones that do not require the a priori knowledge of the error norms. In the appendix, we briefly describe an approach of choosing the value of

*γ*that provides the maximum of the entropy integral. This approach was used in this study.

Condition (8) can be considered as a method of boundary value relaxation toward the reference values. The condition has a coefficient of relaxation *λ*_{t} that changes over time and provides the adaptation of the boundary values to the change in the energy flux through the open boundary. Condition (8) is a modification of the boundary condition introduced in Flather (1976), when *λ*_{t} = 1. This condition has been employed by many researchers (e.g., Oey and Chen 1992; Davies and Lawrence 1994; etc.).

*u*

^{o}

_{n}

*λ*

_{t}= 1, is a modification of the boundary condition introduced by Reid and Bodine (1968). Below, the conditions (8) and (10), when

*λ*

_{t}is calculated from (9), are called optimized versions of the Flather and the Reid and Bodine boundary conditions and denoted correspondingly as OFL and ORB. The standard versions of the Flather and Reid and Bodine conditions are denoted as FL and RB.

## 3. Channel simulations with optimized OBCs

Consider a flat-bottom, frictionless, channel closed at one end (Shulman and Lewis 1994, 1995). The channel is forced at the other end by surface height oscillations at the *M*_{2} tidal frequency with the amplitude of 1 m [function *η*^{o} in (8) and (10)]. The length of the channel is 335 km and the depth is 50 m. There are 23 grid points along the channel (23rd is a wall). In Fig. 1 (top) we reproduced the analytical solution plus the results of the simulations for RB and ORB presented in Shulman and Lewis (1995). One can see that the error in predicting the amplitude is around 60% and the phase offset is around 60° for the RB condition. At the same time, the ORB condition almost entirely eliminated errors in predicted amplitudes while more than halving the errors in predicted phases. We conducted the same experiments with the FL and OFL conditions. The function *u*^{o}_{n}

The results of the simulations with the RB condition can be interpreted as the results of an application of the FL condition with the estimate of the *u*^{o}_{n}*λ*_{t} = 1]. This suggests that the FL condition may be sensitive to errors in data used to construct the reference values of the sea surface elevation and velocity (functions *η*^{o} and *u*^{o}_{n}*u*^{o}_{n}*u*^{o}_{n}

From the analytical solution (Officer 1976), the correct phase shift for *u*^{o}_{n}*η*^{o} is 90°. In the third and fourth experiments, we introduced errors in the phase of *u*^{o}_{n}*u*^{o}_{n}*η*^{o} equals 0° and 45° are shown in Fig. 3. The use of the FL condition produces the maximum phase offset from the analytical solution equaling 60°. However, the maximum phase offset is only 12° when we use the OFL condition. The results of the numerical simulations clearly show that predictions using the FL condition are very sensitive to the errors in reference values of the sea surface elevation and velocity. But, the application of the optimized Flather condition results in much smaller errors.

*W*is the wind stress. We have the following initial and boundary conditions: where

*L*is the length of the channel and

*ρ*is the water density. Suppose that

*W*has the following form: where Therefore, we have the wind linearly increasing with time and then constant after time

*ε*. The analytical solution of problem (11)–(15) has the following form: where We choose the following values for the unknown parameters:

*L*= 335000 m,

*ρ*= 1025 kg m

^{−3},

*H*= 50 m,

*ε*= ½ day = 43200 s,

*T*= 4 days,

*a*= 1.7 × 10

^{−4}. The value of

*a*corresponds to wind stress linearly increasing from zero to 7.35 N m

^{−2}during a half day [if we suppose the drag coefficient is 2.6 × 10

^{−3}(Hellerman and Rosenstein 1983), the maximum wind speed is around 48 m s

^{−1}].

The model run reproduced reasonably well the analytical solution for the closed channel. To test the optimized OBCs, we cut our computational domain on one side, removed five grid cells, and considered a sixth point as the open boundary. On the open boundary, the reference values of the sea surface elevation *η*^{o} and velocity *u*^{o}_{n}

## 4. Modeling tidal and wind-driven circulation in the northern part of the Adriatic Sea

The model used in this study is the *σ*-coordinate, explicit version of the Princeton ocean model (Blumberg and Mellor 1987). This model is a three-dimensional, free-surface, primitive equations model. It includes the Mellor–Yamada turbulence closure submodel, and Smagorinsky diffusivity scheme for horizontal diffusion. The model uses a mode-splitting technique: the separation of vertically integrated equations (external, barotropic mode) from the vertical structure equations (internal, baroclinic mode). In this study, the simulations were conducted using only the barotropic mode of the model. For additional information on the model, the reader is referred to Blumberg and Mellor (1987).

Two orthogonal curvilinear grids were used that cover the entire Mediterranean Sea. The first grid had a relatively fine mesh with 441 × 141 grid points in the horizontal and a grid size ranging from 8 to 12 km in the Adriatic Sea. The second grid had a relatively coarse mesh (221 × 71 grid points) with a grid size ranging from 32 to 61 km in the Adriatic Sea area. The comparison of tidal observations in the region of the northern Adriatic Sea with the results of the simulation using the fine grid model are shown in Table 1. These results provide a good approximation of the observations at six tidal stations: Porto Piave Vecchia, 45.29°N, 12.34°E; Rovinj, 45.05°N, 13.38°E; Porto Corsini, 44.30°N, 12.17°E; Pesaro, 43.55°N, 12.55°E; Ancona, 43.37°N, 13.30°E; and Pula, 44.52°N, 13.50°E. To quantify the errors, we use a weighted average percent error for these six northern Adriatic Sea stations. The average error is 18.6% for amplitudes and 1.7% for phase. The predictions of *M*_{2} tides in the northern Adriatic using the coarser grid model are considerably worse (see Table 2). The average error in overestimating the tidal amplitude is 99%.

To test the OBCs in coupling coarse- and fine-resolution models, a limited-area model (LAM) of the northern part of the Adriatic Sea was produced, based on a portion of the fine grid of the Mediterranean Sea model (Fig. 6). The output from the coarse-resolution model run was interpolated to the open boundary of the LAM to create reference values of sea surface elevation *η*^{o} and velocity *u*^{o}_{n}*M*_{2} tidal amplitudes and 4.4% for phases. The results of simulations with ORB showed much better agreement with the observations (see Fig. 7), with an average error of 12% for amplitudes and 0.9% for phases.

The worst results in predicting tidal amplitudes occur when using the FL condition (Fig. 8). The average error is 106%; this is close to the error of the coarse grid run, the results of which were used to construct the reference values for *η*^{o} and *u*^{o}_{n}*η*^{o} and *u*^{o}_{n}*λ*_{t} in OFL and ORB provides the adaptation of the boundary condition to the energy flux generated by the LAM. One can see that the results of simulations with the optimized OBCs showed much better agreement with the observations than their standard versions.

The next set of experiments was conducted with the wind forcing also included. The mean (Hellerman and Rosenstein 1983) wind stress for February was increased 10 times to get a stronger model response to the wind forcing. To establish the “truth” for later comparisons, the fine grid model of the Mediterranean Sea was forced for 25 days with this wind and *M*_{2} tides. The values of the sea surface elevation and velocity on the open boundary of the LAM were stored as reference values of sea surface elevation *η*^{o} and velocity *u*^{o}_{n}*η*^{o} and *u*^{o}_{n}*η*^{o} reduced by 20% and *u*^{o}_{n}*u*^{o}_{n}*u*^{o}_{n}*M*_{2} tides, equaling 87°). The results are shown in Table 4. The averaged errors of the major axis predictions are 24.6% for FL and 14.8% for OFL. The predictions of phases are reasonably good for both boundary conditions.

## 5. Conclusions

The local data assimilation approach for specifying open boundary conditions for limited-area models is proposed. This approach provides the methodology for an optimized determination of variables on the open boundary based on available reference information about boundary values and dynamics of the model near the open boundary. The optimization problem is constrained by the physics of flux of energy through the open boundary. The reference boundary values can be derived from observations, another coarser-resolution model run, and/or from another specification of open boundary conditions. For the barotropic models, the proposed optimized open boundary conditions have a sound physical interpretation as optimized versions of well-known radiation-type open boundary conditions (Reid and Bodine 1968; Flather 1976). The results of tidal and wind-driven simulations for the idealized channel and the northern part of the Adriatic Sea show that the application of the optimized open boundary conditions reduces significantly the error of model predictions compared to the use of nonoptimized counterparts. Therefore, the proposed local data assimilation schemes of specifying open boundary conditions can provide an improvement in the accuracy in simulating model interior flow. Coupling of a coarser-resolution model and a finer-resolution limited-area model and sensitivity studies show that radiation-type open boundary conditions transmit the level of errors in the reference values into the interior domain. This can result in either overestimating or underestimating the amplitudes of sea surface elevation and velocity. But the optimized versions of these conditions correct the energy input from the reference values into the limited-area domain, and thus result in a reduction in errors. Overall, the results of simulations show that optimized open boundary conditions can be used to force barotropic models or the barotropic mode of three-dimensional models (such as the Princeton Ocean Model). Our future research will be focused on extending the proposed local data assimilation techniques to the baroclinic case.

## Acknowledgments

We would like to acknowledge the essential support of this work by Dr. J. K. Lewis. We are also thankful to Dr. A. F. Blumberg for stimulating comments. The comments of the anonymous reviewers were very helpful in improving the manuscript. This work was supported by a research grant from the Office of Naval Research to the Center for Ocean and Atmospheric Modeling, University of Southern Mississippi.

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## APPENDIX

### Regularization of Optimized Open Boundary Conditions

*γ.*We introduce the following notation: and we will discuss the value of the nondimensional parameter

*μ.*Suppose that the sea surface elevation

*η*

_{ex}is a solution of the optimization problem when

*P*

_{t}and

*u*

_{n}are the exact values for the energy flux and velocity. We do not know the function

*η*

_{ex}, but we have the function

*η*(

*μ*) from (8) and (9). Some norm of the product

*μ∂η*/

*∂μ*[corresponding to the first member of the Taylor series of the difference between

*η*(

*μ*) and

*η*

_{ex}] can be used to estimate the difference between

*η*(

*μ*) and

*η*

_{ex}and to estimate the optimal value of

*μ*and

*γ.*Let us introduce the following norm: According to (A1), we have Let us introduce the normalized distribution function: which is, according to (A2), equal to We choose the value for

*μ*according to the maximum entropy method: In this case, by maximizing entropy over all values of

*μ,*we are picking one that makes the fewest unnecessary assumptions (most cautious hypothesis). The solution for (A3) is

Observed and model-predicted amplitudes (cm) and phases (degrees, relative to UTC) for the *M*_{2} tide in the northern Adriatic Sea based on a simulation using the finer grid model of the Mediterranean Sea.

Observed and model-predicted amplitudes (cm) and phases (degrees, relative to UTC) for the *M*_{2} tide in the northern Adriatic Sea based on a simulation using the coarser grid model of the Mediterranean Sea.

The results of LAM simulations when amplitude of *η** ^{o}* is reduced by 20% and

*u*

^{o}

_{n}

*M*

_{2}current ellipse parameters: major axis and orientation of major axis (

*θ*).

The results of LAM simulations when the reference velocity *u*^{o}_{n}*M*_{2} current ellipse parameters: major axis and orientation of major axis (*θ*).