a. Three vectors

If r denotes the position of any point along the open boundary Γ, the boundary values of any variable η is a function of r, η^(b) = η^(b)(r). Let f₁(r), f₂(r), . . . , f_n(r) be a series of known basis functions. We expand the function η^(b)(r) into

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Thus, the determination of the open boundary condition η^(b)(r) becomes the determination of a set of parameters b₁, b₂, . . . , b_n. The n-dimensional vector,

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is called the boundary parameter vector.

Assume that there are m observations (O₁, O₂, . . . , O_m) located at the interior (Fig. 1). An m-dimensional vector O can be constructed by

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which is called the observation vector. Notice that the observational variable is not necessarily the same dimension as the variable at the open boundary. If B is given, we can solve the dynamic system and obtain the solution S. At the same locations where the observations take place, the solutions are S₁, S₂, . . . , S_m, which form a solution vector

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b. Optimization method for determining B

Ocean model performance can be measured by the rms error

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The vector S depends on B. Change of the boundary vector B (boundary conditions) leads to a change of S (solutions). Inversely, we may determine B by minimizing I;

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Substitution of (5) into (6) leads to a set of n equations implicitly solvable for b₁, b₂, . . . , b_n,

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where

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are components of a n × m Jacobian matrix R = {R_ij}.

From a first-guess boundary vector B*, a solution vector S* is obtained by solving the numerical ocean model. The rms between S* and O might not be minimal. We update the boundary parameter vector components by increments {δb_i | i = 1, 2, . . . , n}, and therefore components of the solution vector become

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Substituting (9) into (7) and neglecting higher-order terms leads to a set of n linear algebraic equations for {δb_i},

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where

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Both O_j and S*_j are known quantities. Therefore, the linear algebraic equations (10) have definite solutions when the Jacobian matrix {R_ij} is determined and

P_il(12)

c. Determination of the Jacobian matrix by a multiperturbation method

As soon as the Jacobian matrix is obtained, we can solve (10) to get boundary parameter vector corrections δbi (i = 1, 2, . . . , n). Therefore, the determination of the Jacobian matrix is the key issue. We propose a simple multiperturbation method on the first-guess boundary parameter vector B* to get the Jacobian matrix. The vector B* at the initial state (t = 0) of any numerical ocean model is set to be a zero vector,

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and the first guess for B at time step k is set to be the same as the determined boundary parameter vector at time step k − 1,

BkBkk(14)

We define a set of n-dimensional unit vectors

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and perturb the first guess at each time step by a small fraction of these unit vectors c⁽ⁱ⁾:

B⁽ⁱ⁾Bε_bc⁽ⁱ⁾in,(16)

where ε_b is a small positive number. Using the n + 1 sets of boundary parameter vectors B*, B⁽¹⁾, B⁽²⁾, . . . , B⁽ⁿ⁾, we obtain n + 1 sets of solution vectors for that time step, S*, S⁽¹⁾, S⁽²⁾, . . . , S⁽ⁿ⁾. The Jacobian matrix (8) denotes the rate of change of the solution vector with the boundary parameter vector; therefore, it can be computed by

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where the m-dimensional vectors e⁽¹⁾, e⁽²⁾, . . . , e⁽ⁿ⁾ are defined by

e⁽ⁱ⁾S⁽ⁱ⁾Sin.(18)

d. Iteration process

After the algebraic equation (10) is solved, the boundary parameter correction vector δB ≡ (δb₁, δb₂, . . . , δb_n) is obtained. We replace B* by B* + δB and repeat the process (Fig. 2) until reaching a certain criterion

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and ε is a small positive number (user input). As soon as the inequality (19) is satisfied, the iteration stops and the final B* becomes the optimal boundary parameter vector for the next time step.

e. Reference model

A model with a given boundary condition (called reference boundary condition) is needed for error estimation. We run the model with the given reference boundary condition and obtain the solution for interior points, O = (O₁, O₂, . . . , O_m), which are taken as “observations.” We also expand the reference boundary values into basis functions (1) to obtain the reference boundary vector B^(ref) = (b^ref₁, b^ref₂, . . . , b^ref_n). Such a model (with known boundary conditions) is called a reference model.

f. Smoothing δB

Both observational (instrument) and computational (numerical) errors perturb the values of B. When the reference model results are used, there is no instrument error in observations. But the computational errors still cause errors in B. For prognostic ocean models, the errors of B feed back to the next step model integration. Error accumulation in each time step may cause computational instability. We use linear regression for smoothing δB to reduce high-frequency modes. The procedure is as follows: 1) For the first 11 time steps, we do not do any smoothing, and after the 11th time step, we use the smoothing technique. 2) For each time step k (k > 11), we maintain smoothed values of each component at the previous 11 steps, δb^(s)_j(k − 11), δb^(s)_j(k − 10), . . . , δb^(s)_j(k − 1). Here the superscript s indicates smoothed value. 3) We fit a linear regression to fit the 12 values: δb^(s)_j(k − 11), δb^(s)_j(k − 10), . . . , δb^(s)_j(k − 1), and δb_j(k). 4) We obtain the value at the time step k from the regression, δb^(s)_j(k). 5) We replace δb_j(k) by δb^(s)_j(k). Such a treatment (Fig. 3) will filter out high-frequency noise in the derived open boundary parameter vector.

g. Random noises added on “observations”

When observations are taken from the reference model solution, there is no observational (instrument) error. In order to see the effects of observational error on the determination of open boundary conditions, we add a Gaussian-type random variable to each observation at each time step. The probability distribution function is given by

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where δO is a random variable with a zero mean and a standard deviation of σ.

h. Relative errors

For a given observation vector O and unknown open boundary condition, we use the optimization method to obtain the values at the open boundary, B = (b₁, b₂, . . . , b_n). Then we integrate the same model with the computed open boundary condition and get the solutions at the observational points, S = (S₁, S₂, . . . , S_m). The relative errors,

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measure the validity of the optimization method. The smaller the E^(B) and E^(O), the better the optimization method.

a. Jacobian matrix

For any linear ocean model, the relationship between {S_j} and {δb_i} becomes linear. There are no high order terms in (9). The Jacobian matrix can be easily obtained by letting

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for any time step. The algebraic equation (9) becomes

RB⁽ⁱ⁾e⁽ⁱ⁾in.

b. Example—Csanady’s shelf model

We use a steady-state shelf circulation model proposed by Csanady (1978) as an example to show how to obtain the Jacobian matrix R and to determine the open boundary conditions. Consider a long and straight coastline with coordinates such that the y axis coincides with the coast and positive x points offshore. The water depth is a function only of offshore distance, that is, h = h(x). As pointed out by Csanady (1978), this model is appropriate for simulating mean flow along such a coast driven by mean wind stress.

Under the assumptions of homogeneous water and linear bottom friction (i.e., proportional to depth-averaged velocity), the dynamical equations for depth-averaged velocities (u, v) without wind forcing become

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where η is the surface elevation, r the bottom resistance coefficient, f the Coriolis parameter, and g the gravity. Eliminating u and v from (22)–(24) leads to a single equation for the surface elevation η:

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Wang (1982) applied the Csanady model to a region (Fig. 4) with a combined flat shelf and steep slope, depicted by

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where x₀ = 140 km and x₁ = 180 km are the offshore location of the shelf break and slope edge, respectively. Following Wang (1982), we chose f = 10⁻⁴ s⁻¹, and r = 0.1 cm s⁻¹, and use the boundary condition at the coast (x = 0),

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We consider the pure open ocean forcing case; that is, there is no inflow at the backward boundary (y = 0),

ηy(27)

Equation (25) has the form of the one-dimensional heat-conduction equation, with negative y playing the role of time. We solve (25) numerically by the Gaussian elimination method under the boundary conditions (26)–(27) to obtain the first-guess solution at the interior. The spatial increments are Δx = 2 km, Δy = 10 km. We use the five basis functions f₁(y), f₂(y), . . . , f₅(y) (Fig. 5) for η^(b). The first guess for B is the zero vector. For simplicity and no loss of generality, we choose five equally spaced points on an interior line paralleling to the y axis as observation points. In this case,

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is a unit matrix. The Jacobian matrix then becomes

Re⁽¹⁾e⁽²⁾e⁽³⁾e⁽⁴⁾e⁽⁵⁾(29)

c. Error estimation

For simplicity, we choose Wang’s (1982) solution along with the open boundary condition as a reference model for the error estimation. Five observation points are equally spaced and located along lines paralleling to the y axis. These lines are represented by their x coordinate. Wang’s (1982) solution at the five points is taken as observations, O = (O₁, O₂, O₃, O₄, O₅), and the Wang’s open boundary vector is denoted by B^(ref) = (b^ref₁, b^ref₂, b^ref₃, b^ref₄, b^ref₅). Since the model is steady state, we did not smooth δB.

The five observation points are given as O with unknown open boundary condition. Using the optimization method, we obtain values at the open boundary, B = (b₁, b₂, b₃, b₄, b₅). Then, we integrate the Csanady model (24) with the computed open boundary condition, and get the solutions at the same five points, S = (S₁, S₂, S₃, S₄, S₅). Since the observation points are on the lines paralleling the y axis in this study, E^(B) and E^(O) depend only on x. Surprisingly, both E^(B) and E^(O) are extremely small (Fig. 6). When the observation points are chosen near the coast, the relative errors are on the order of 10⁻⁶–10⁻⁷. At the shelf break (x = x₀, x₀ = 140 km), E^(B) is on the order of 10⁻⁶. Passing the shelf break, E^(B) and E^(O) decrease very fast offshore and are on the order of 10⁻¹⁶ near the open boundary.

a. Example—Princeton Ocean Model (POM)

We apply the optimization method to determine the open boundary conditions of a flat bay centered at 35°N and bounded by three rigid boundaries. This bay expands 1000 km in both the north–south and east–west directions. The northern, southern, and western boundaries are rigid, and the eastern boundary is open. The Cartesian coordinate system is chosen with the origin at the southwest corner. The x axis points toward the east, and the y axis toward the north (Fig. 7a). The circulation in the bay is modeled with the Princeton Ocean Model (POM) developed by Blumberg and Mellor (1987). POM is a primitive equation model with a free surface and a level-2 turbulence closure scheme (Mellor and Yamada 1982). A description of the model code can be found in Mellor (1991). We use the 2D version of POM to illustrate the usefulness of the optimization method for determining the open boundary conditions.

The area depicted in Fig. 7a is called domain A, where the boundary conditions are known at the three rigid boundaries (northern, southern, and western), and unknown at the eastern boundary. The eastern boundary of domain A is connected to a mirror image of domain A (about x = 1000 km) forming a closed rectangular domain (Fig. 7b), called domain B. The POM model was integrated from the following initial conditions,

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under no surface heat or salinity fluxes and zonal surface pseudo–wind stress varying with latitude (Fig. 8):

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The time step was chosen as 2 min. The horizontal resolution was 50 km. Bottom stress is parameterized by the quadratic drag relation. Horizontal kinematic viscosity is set to be 500 m² s⁻¹.

b. Boundary vector

c. Reference model

We choose the POM model solution for domain A computed with the reference open boundary condition determined by the two time series b₁(t), b₂(t) (Fig. 9) as a reference model. We ran the reference model for 90 days with known boundary conditions (three rigid and one reference open boundary condition). Figure 11 shows the horizontal velocity of the reference model run for two different times (30 day and 90 day). We will use the reference model results to verify the optimization method.

d. Three cases

A random number generator (Fortran function, Ranf) was used to produce random disturbances for each observational point independently with mean value of zero and standard deviation of 0.01 m s⁻¹. In order to test the performance of the optimization method, we ran three cases: 1) without smoothing on δB and without random noise added to the observations, 2) with smoothing on δB and without random noise added to the observations, and 3) with both smoothing on δB and random noise added to the observations.

Eighteen randomly picked points are treated as observational points (Fig. 12). The reference model solution at the 18 points is the observations, O = (O₁, O₂, . . . , O₁₈). Using the optimization method, we obtain the temporally varying b₁(t) and b₂(t) for the three cases. For case 1 (Fig. 13a), both b₁ and b₂ fit the reference values (Fig. 9c) very well until the 35th day. After that day, b₁ and b₂ change rapidly with time and finally blow up at the 45th day. For case 2 (Fig. 13b), both b₁ and b₂ fit the reference values (Fig. 9c) very well. For case 3 (Fig. 13c), both b₁ and b₂ are very close to the reference values (Fig. 9c) with small perturbations. Figure 13 tells us that smoothing on δB is a key issue for this method.

We integrate the POM for domain A with the computed open boundary conditions from b₁ and b₂ for the three cases. The 30th day’s horizontal velocity fields (Fig. 14) for the three cases all agree quite well with the reference model results (Fig. 11a), and the 90th day’s horizontal velocity fields (Fig. 15) for cases 2 and 3 agree quite well with the reference model results (Fig. 11b).

e. Error estimation

Similar to the linear case, we use 18 interior observations O, the reference boundary vector B^ref = (b^ref₁, b^ref₂), the open boundary vector B = (b₁, b₂), and the interior solution S = (S₁, S₂, . . . , S₁₈) for error estimation.

The boundary errors E^(B) for the three cases are shown in Fig. 16. When there is no smoothing on δB (case 1), E^(B) keeps very small values (10⁻¹²–10⁻¹³) for the first few days. After the eighth day, E^(B) increases exponentially with time and reaches the order of 1 at the 45th day. This indicates that we cannot use the optimization method without smoothing on δB. When there is smoothing on δB and no noise added to the observations (case 2), E^(B) has larger values (∼10⁻²) than case 1 at the beginning, then rapidly decreases with time during the first 10 days and gradually decreases with time afterward. After 10 days of integration, the magnitude of E^(B) is on the order of 10⁻⁴–10⁻⁵. When there is smoothing on δB and noise added to the observations (case 3), E^(B) fluctuates around 10^−2.5 (≃3.162 × 10⁻³) with the maximum value near 10^−1.5 (≃0.03) and the minimum value around 10⁻⁴. Figure 16 indicates that smoothing on δB is very important for this method.

The interior errors E^(O) for the three cases are shown in Fig. 17. When there is no smoothing on δB (case 1), E^(O) keeps very small values (10⁻¹³) for the first few days and then increases exponentially with time and reaches the order of 1 at the 45th day. When there is smoothing on δB and no noise added to the observations (case 2), E^(O) has larger values (∼10⁻²) than case 1 at the beginning then rapidly decreases with time during first 10 days and gradually decreases with time afterward. After 10 days of integration, the magnitude of E^(O) is on the order of 10⁻⁴–10⁻⁵. When there is smoothing on δB and noise added to the observations (case 3), E^(O) fluctuates around 10^−2.2 (≃6.3 × 10⁻³) with the maximum value near 10⁻² and the minimum value around 10^−2.4 (≃3.9 × 10⁻³). Both Figs. 16 and 17 indicate that smoothing on δB is very important for determining open boundary conditions.