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  • View in gallery

    Definition sketch for a two-layered flow (Chen et al. 2014).

  • View in gallery

    Oceanic bathymetric map of the computing area. The black dashed line indicates the 500-m contour. The red asterisks (*) are independent points. The blue stars are the positions of tidal stations (the numbers are tidal stations index), and the black dots are observation positions along the T/P ground tracks.

  • View in gallery

    Initial potential density (the dashed line represents a depth of 500 m).

  • View in gallery

    Prescribed OBCs and the corresponding inverted OBCs with two methods.

  • View in gallery

    The iteration histories of the relative values (log) of the cost function.

  • View in gallery

    Prescribed OBCs and the corresponding inverted OBCs of Fourier coefficients (a) a and (b) b.

  • View in gallery

    Inversion error of Fourier coefficients (a) a and (b) b.

  • View in gallery

    Cotidal charts for the M2 constituent in the SCS. (a) Obtained with the Cressman interpolation. (b) Obtained with the spline interpolation.

  • View in gallery

    The magnitudes of the misfit vector of the results of the two interpolation methods (units: m).

  • View in gallery

    The magnitude of the error vector between simulation and observations at each tidal gauge for two interpolation methods.

  • View in gallery

    Zonal baroclinic velocities at 500-m depth at the end of calculation for (a) Cressman and (b) spline.

  • View in gallery

    The absolute errors between the two interpolation methods in zonal baroclinic velocities at 500-m depth.

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Estimating Smoothly Varying Open Boundary Conditions for a 3D Internal Tidal Model with an Improved Independent Point Scheme

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  • 1 Physical Oceanography Laboratory, CIMST, Ocean University of China, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
  • | 2 Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
  • | 3 Key Laboratory of Marine Resources and Coastal Engineering in Guangdong Province, and School of Marine Sciences, Sun Yat-Sen University, Guangzhou, China
  • | 4 Physical Oceanography Laboratory, CIMST, Ocean University of China, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
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Abstract

An improved independent point (IP) scheme was proposed to estimate the open boundary conditions (OBCs) for a 3D internal tidal model through assimilating the TOPEX/Poseidon (T/P) altimeter data. Under the assumption that the OBCs were spatially and smoothly varying, values at a set of independent points along the open boundary were inverted using the adjoint method and values at other points were interpolated by the spline method. The scheme was calibrated through idealized experiments where the M2 tidal constituent in the northern South China Sea was simulated. The OBCs can be successfully inverted with the improved scheme and were better in spatial smoothness than the results obtained with the Cressman interpolation when embedded in the IP scheme. Simulations in realistic domains showed that the errors between simulations and observations were smaller when the spline interpolation was employed instead of the Cressman interpolation. Three boundary conditions of spline interpolation were used in simulations in realistic domains, and the result of the periodic boundary condition had the smallest error compared with the first and second boundary conditions.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xianqing Lv, xqinglv@ouc.edu.cn; Haibo Chen, chenhb2015@qdio.ac.cn

Abstract

An improved independent point (IP) scheme was proposed to estimate the open boundary conditions (OBCs) for a 3D internal tidal model through assimilating the TOPEX/Poseidon (T/P) altimeter data. Under the assumption that the OBCs were spatially and smoothly varying, values at a set of independent points along the open boundary were inverted using the adjoint method and values at other points were interpolated by the spline method. The scheme was calibrated through idealized experiments where the M2 tidal constituent in the northern South China Sea was simulated. The OBCs can be successfully inverted with the improved scheme and were better in spatial smoothness than the results obtained with the Cressman interpolation when embedded in the IP scheme. Simulations in realistic domains showed that the errors between simulations and observations were smaller when the spline interpolation was employed instead of the Cressman interpolation. Three boundary conditions of spline interpolation were used in simulations in realistic domains, and the result of the periodic boundary condition had the smallest error compared with the first and second boundary conditions.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xianqing Lv, xqinglv@ouc.edu.cn; Haibo Chen, chenhb2015@qdio.ac.cn

1. Introduction

Internal tides are ubiquitous in the oceans, and they are internal waves with tidal frequency, generated in a stratified ocean as barotropic tidal currents interact with topography (Garrett and Kunze 2007). The South China Sea (SCS) is well known for its large-amplitude internal tides and solitary internal waves, especially in the northern SCS, where there are strong tidal currents and complicated topography (Duda et al. 2004). Many numerical models and field observations have suggested that the Luzon Strait (LS), blocked by two submarine ridges, is the major generation region of the internal tides observed in the northern SCS (Liu et al. 2004; Niwa and Hibiya 2004; Jan et al. 2008; Farmer et al. 2009; Ramp et al. 2010; Guo et al. 2012; Alford et al. 2015). In research of internal tides in the SCS, Song et al. (2010) introduced the internal tide viscosity to a regional ocean model, leading to significant improvements, not only in terms of the simulated vertical movement of internal tides, but also in terms of the dissipation of barotropic tides and the modeled vertical temperature structure. To investigate the equilibration of numerical simulation (ENS) of internal tide, Jin et al. (2017) used a 3D isopycnic-coordinate internal tide model to simulate the M2 internal tide around the Hawaiian Ridge; the simulation results indicated that a 50 M2 tidal period (25.88 days) run is capable of ensuring ENS for the M2 internal tide. Based on observations of the currents at moorings, many investigations were made to study internal tides in the SCS. Cao et al. (2017) found that both diurnal and semidiurnal internal tides in the northern SCS contain stronger coherent signals than incoherent ones. Xu et al. (2011) studied the vertical structure of internal tides on the continental shelf of the northwestern SCS and demonstrated that the northwest-propagating semidiurnal internal tides of higher mode with small vertical scale probably do not originate from a distant source like LS but are likely generated near the northwestern SCS.

For numerical studies of internal tides, the open boundary conditions (OBCs) must be prescribed to complete the model description at open boundaries. They are very important and have a critical impact on the modeling results. However, a major difficulty faced by regional ocean models is concerned with the treatment of the OBCs (Lardner et al. 1993). In practical ocean modeling, on the other hand, the external data can be obtained either from available observations near the open boundaries (tidal gauge data or satellite data) or, within a nested approach, from numerical models with a larger domain, such as Schwiderski’s (1980) global tidal model and the TPXO global tidal model (Egbert 1997). In realistic oceanic applications, the parameters should be spatially varying and the dimension of the model parameters can be large (approximately 2000 or much more). Meanwhile, the observations are often made at a limited number of locations. As a result, the parameter estimation problem might be ill-posed (Smedstad and O’Brien 1991), and for identifiability refer to Navon (1998). The independent point (IP) scheme, first proposed by Zhang and Lv (2008), is a good solution to this problem, in which OBCs at some selected IPs were taken as control variables and those at the other points along the open boundary were calculated by interpolating values at IPs. By using this scheme, Zhang and Lv (2008) established a 3D numerical barotropic tidal model and inverted the Fourier coefficients of OBCs, the bottom friction coefficient profiles with the IP scheme. As an extension, Zhang and Lv (2010) explored the inversion of OBCs in detail with a 3D numerical barotropic tidal model. Chen et al. (2014) applied the IP scheme to the estimation of OBCs in a numerical internal tide model. To inverse the periodic OBCs, Zhang and Wang (2014) developed a new method based on the adjoint method in 2D tidal models and used it to simulate the M2 constituent successfully. By using adjoint data assimilation, a 3D cohesive sediment transport model is developed by Wang et al. (2016), and the results of parameter sensitivity analysis indicate that the model is sensitive to the inflow open boundary conditions.

However, the Cressman interpolation traditionally used in the IP scheme can result in unsmooth open boundary curves. As an improvement, a new IP scheme was first proposed by Pan et al. (2017), in which the Cressman interpolation was replaced by the spline interpolation, to obtain smooth open boundary curves. Spline interpolation is a form of interpolation whose interpolant uses a special type of piecewise polynomial called spline. The spline interpolation is widely used and is often better than polynomial interpolation because the interpolation error can be made very small (Knott 1999). Based on the spline interpolation, Wijnands et al. (2016) proposed a new mathematical model for tropical cyclone wind speeds, and through comparison with an earlier linear wind model, they showed that the spline model could produce a more accurate wind estimate. Yaghoobi et al. (2017) described a scheme based on a cubic spline interpolation that is applied to approximate the variable-order fractional integrals and is extended to solve a class of nonlinear variable-order fractional equations with time delay. The advantage of the spline interpolation over the Cressman and kriging interpolations was shown in the experiment made by Pan et al. (2017), which showed that the curve obtained with the spline interpolation is much smoother and closer to the given curve than the Cressman and kriging interpolations. Therefore, the proposed method with the spline interpolation generated more accurate results than the traditional statistical method. By using this new IP scheme with a 2D tidal model, Pan et al. (2017) made a series of simulations in the Bohai and Yellow Seas; the results showed that the errors between simulations and observations were smaller when the spline interpolation was used. Guo and Pan (2017) validated this new IP scheme with twin experiments, and the results showed that the prescribed nonlinear distribution of bottom friction coefficients are better inverted with the surface spline interpolation. In this work, the new IP scheme was proposed to estimate the OBCs for a 3D internal tidal model through assimilating the TOPEX/Poseidon (T/P) altimeter data in the northern SCS. And three kinds of boundary conditions were used in simulations in realistic domains to further discuss the influence of a new IP scheme on numerical study.

This paper is organized as follows. Section 2 shows the optimization method. A series of simulations of idealized test cases and realistic domains are conducted in sections 3 and 4, respectively. A discussion and conclusions are presented in section 5 and 6, respectively.

2. Optimization method

a. Model description

A two-layered isopycnic-coordinate numerical model with the adjoint assimilation method described in Chen et al. (2012) is used in this paper (Fig. 1), and only the M2 constituent is considered. Assuming the potential density in each layer is constant, the layer-averaged, nonlinear, time-dependent continuity and momentum equations of each layer subject to the hydrostatic approximations are derived from the primitive 3D governing equations. Here, the subscripts 1 and 2 refer to the upper and lower layers, respectively. Using spherical coordinates in the horizontal direction and isopycnic coordinates in the vertical, we obtain the internal mode equations of the forward model as follows:
  • Upper layer:
    e1a
    e1b
    e1c
  • Lower layer:
    e2a
    e2b
    e2c
Here, is the time; and are the east longitude and north latitude, respectively; and are horizontal velocities in and , respectively, is the time-varying layer mass; and , , where and are the undisturbed layer thickness and interface (surface for ) elevation above the undisturbed level, respectively. The term is the radius of Earth; is the gravitational acceleration; is the Coriolis parameter; , where represents the angular speed of Earth’s rotation; is the horizontal eddy viscosity coefficient; is the Laplacian operator; and
eq1
where and are the interface and bottom friction coefficient, respectively; ; and . In the forward model, , and are the main outputs and are called the state variables in this paper.
Fig. 1.
Fig. 1.

Definition sketch for a two-layered flow (Chen et al. 2014).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

The barotropic currents are defined as
eq2
where .
Integrating the continuity equations by Eq. (1a) + Eq. (2a) and integrating the momentum equations by Eq. (1b) + Eq. (2b) and Eq. (1c) + Eq. (2c), we get the 2D external mode as follows:
e3a
e3b
e3c
Here is the undisturbed water depth, is the vertically averaged density, and . Note that the second terms of and represent the resultant effect of the horizontal density gradients in all layers on the sea surface and are the main causes of the surface manifestation of internal tides. The terms and are the same as those in Chen et al. (2014).
The adjoint method is a powerful tool for parameter estimation. The basic idea of the adjoint method is quite simple: a model is defined by an algorithm and its control parameters, including initial conditions, boundary conditions, and empirical parameters (Chen et al. 2014). Based on the governing Eqs. (1a)(2c) of the forward model, its adjoint model can be constructed as follows. First, a cost function is defined as
e4
where is the layer index, denotes the computing area of both time and space, represents the generalized control parameters, and are the model results, and and are the observations. Here and are the weight matrices and theoretically should be the inverse of the observation error covariance matrix; that is, the cost function is weighted more heavily toward the observations that are more accurate or important. Then a Lagrangian function is defined in this paper as
e5
where , , and are called adjoint variables of the state variables , and , respectively.
According to the typical theory of the Lagrangian multiplier method, we have the following first-order derivates of the Lagrangian function with respect to all the variables and parameters:
e6
e7
e8
Equation (6) returns the governing Eqs. (1a)(2c). The adjoint equations can be derived from Eq. (7). From Eq. (8) we can obtain the gradients of the cost function with respect to control parameters.

Similar to the forward model, the adjoint model also consists of the internal and external modes. Actually, the equations derived from Eq. (7) are considered as the internal mode and the external mode can be derived from the internal mode in a similar way as the external mode of the forward model is derived. The details of both the internal and external modes of the adjoint model can be found in Chen et al. (2012).

b. Gradients of the cost function with respect to the OBCs

In this model the Flather conditions are employed by the external mode and the relaxation conditions are employed by the internal mode. Assume that at a certain open boundary grid point the height of water level of the M2 tidal force at the nth time step is subject to
e9
where denotes the frequency of the M2 constituent; is the time step length; and are the Fourier coefficients, which are taken as the adjustable parameters of the model in this study. The gradients of the cost function with respect to and are
e10
respectively, where are the same as those in Chen et al. (2012); the gradients of the cost function with respect to the OBCs can thus be calculated. Then the OBCs are optimized with the IP scheme described in the following section.

c. Optimization with the IP scheme

In this study, the performances of two interpolation methods—the traditional Cressman method and the spline method—are compared. The Cressman method is an improvement to the inverse distance weighting interpolation method, which has been widely used in many fields (Goodin et al. 1979; Franke 1982; Willmott et al. 1985; Biau et al. 1999; Largueche 2006). In these two methods, let be a series of independent OBCs, N is the number of the independent OBCs, and is the result of the interpolation of . Then the relation can be written as
e11

In the Cressman method, the weighting coefficient , where , in which R is the influence radius and is the distance between the and points (Cressman 1959). The spline method allows a smoother and more realistic interpolated curve, and the detailed derivation of the weighting coefficient in the spline method can be found in Pan et al. (2017), so it will not be repeated here. The gradients of the cost function with respect to the independent OBCs can be derived from Eq. (8), which yields , and note that the term is computed using Eq. (10).

During assimilation, the cost function is minimized with the steepest decent (SD) method; for a detailed algorithm for implementing the SD method, refer to the work of Chen (2012).

3. Simulations of idealized test cases

a. Numerical simulation design

A series of simulations of idealized test cases are implemented to evaluate the feasibility of this 3D internal tidal model and the improved IP scheme. In the present study, the model is tested in a regional ocean in the SCS: 17°30′–24°10′N, 116°–124°E (Fig. 2). The space resolution in this model is 5′ × 5′, and there are 97 × 81 horizontal grids total in this area. The vertical profile of the initial potential density is shown in Fig. 3, which is computed using the profile at 19°N, 121°E from the World Ocean Atlas 2005 (WOA05). It can be seen from Fig. 3 that the density is almost unchanged as the depth gets deeper than 500 m. Therefore, the undisturbed interface of this two-layered model is placed at the depth of 500 m. The angular frequency of the M2 tide is 1.405 078 902 5 × 10−4 s−1, and the whole time step is 496.863s (1/90 of the period of the M2 constituent). The horizontal eddy viscosity coefficient is chosen to be Ah = 1000 m2 s−1. The coefficients of the bottom and interface friction are taken as κ = 0.002 and Aυ = 0.03 m2 s−1, respectively.

Fig. 2.
Fig. 2.

Oceanic bathymetric map of the computing area. The black dashed line indicates the 500-m contour. The red asterisks (*) are independent points. The blue stars are the positions of tidal stations (the numbers are tidal stations index), and the black dots are observation positions along the T/P ground tracks.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

Fig. 3.
Fig. 3.

Initial potential density (the dashed line represents a depth of 500 m).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

As shown in Fig. 2, most parts of the four boundaries of the computing area are open boundaries. It has been shown by numerous studies (Jan et al. 2008; Miao et al. 2011) that the M2 barotropic tide generated in the LS with the widest tidal beam propagates eastward into the Pacific Ocean and thus the eastern boundary can be treated as an important resource of the tidal force in the computing area here. In this section, for simplicity and no loss of generality, we focus only on the eastern OBCs. The total number of open boundary points of interest is 77. Not all of the points along the T/P ground tracks are locate in the center of computational grids, and it may have more than one observation in the same one grid. Therefore, we choose the mean value of all the observations in the same one grid as the center value of this grid, and the positions serve as observation sites in idealized simulations (Fig. 2). The forward model is run with a certain prescribed distribution of OBCs. Assuming the forward model is perfect, the model-generated results of the surface tidal currents (i.e., tidal currents in the upper layer) at these observation points are taken as the pseudo-observations. Then, an initial value (taken as zero here) of the OBCs is assigned to run the forward model. The difference between the simulated values and the pseudo-observations plays the role of the external force of the adjoint model. The optimized OBCs can be obtained through the backward integration of the adjoint equations. With the abovementioned procedure repeated, the OBCs will be optimized continuously and the difference between simulated values and pseudo-observations will be diminished. Meanwhile, the difference between the prescribed and inverted OBCs will also be decreased.

b. Numerical results

1) Comparison between different independent point strategies

In this section, for simplicity, we only focus on the inversion of the Fourier coefficient a. The Fourier coefficient a is constructed by the trigonometric function , where j is the meridional indexes of the open boundary point and is the total number of the open boundary points of interest. Along the eastern boundary, b (=0) is treated as the known and prescribed spatial distributions of a is given as Fig. 4. Because the given spatial distribution is a periodic curve, we choose the periodic boundary condition of spline interpolation in this section. During the iterative minimization of the cost function, the OBCs are optimized with the SD method. The iterative minimization terminates once a convergence criterion is met. The Cressman and spline interpolation methods are used in this section (five independent points are given for both methods).

Fig. 4.
Fig. 4.

Prescribed OBCs and the corresponding inverted OBCs with two methods.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

As shown in Fig. 5, both of the cost functions drop dramatically compared with their own initial values, which indicates that the data assimilation and optimization of OBCs are also efficient when the IP scheme is used. The inverted distributions are compared with the prescribed ones in Fig. 4. As we can see, both inverted distributions are consistent with the prescribed one. However, the difference between the open boundary curve inverted with the spline interpolation and the prescribed curve is much smoother. The mean absolute errors (MAEs) of OBCs inverted with the Cressman and spline interpolation methods are 3.36 and 0.44 cm, respectively, showing that the errors in experiments adopting the spline interpolation are much smaller, demonstrating the feasibility of the model and the IP scheme. Considering the influence of the spline method on the results in the interior computing area, the MAEs between the simulation results with pseudo-observations are given in Table 1. All the numerical errors of tidal currents U and V of the spline method are less than about a half of those of the Cressman method. The replacement of the Cressman interpolation method with the spline method results in a decrease of errors, which is confirmed in Fig. 6 in section 4b.

Fig. 5.
Fig. 5.

The iteration histories of the relative values (log) of the cost function.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

Table 1.

Results of experiments in section 3b(1). Inversion errors are MAEs for U and V.

Table 1.
Fig. 6.
Fig. 6.

Prescribed OBCs and the corresponding inverted OBCs of Fourier coefficients (a) a and (b) b.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

2) Simulations with artificial errors

In this section, both Fourier coefficients a and b along the open boundary shown in Fig. 6 are inverted. The twin experiments are designed as follows. Considering that the practical in situ observations definitely contain noises, we add artificial errors into the pseudo-observations. In this section, four experiments (E1-C, E1-S, E2-C, E2-S, in which C and S represent the Cressman and spline interpolations, respectively) are carried out, in which two IP schemes are used and the maximum percentage errors are either 5% (E1-C, E1-S) or 10% (E2-C, E2-S). The inverted distributions are compared with the prescribed ones in Fig. 6. Furthermore, in order to have a clearer comparison, we also show the inversion error of all four experiments on every open boundary point in Fig. 7.

Fig. 7.
Fig. 7.

Inversion error of Fourier coefficients (a) a and (b) b.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

From Table 2, we can see that all the cost functions drop dramatically compared with their own initial values (=1), which indicates that the data assimilation and optimization of OBCs are also efficient when the pseudo-observation contains noise. Through Fig. 5 we can draw the same conclusion as in section 3a, that the errors are much smaller when the IP scheme with spline interpolation is adopted. From Fig. 6 and Table 2, we can see that, when the noise is increased, the inversion errors also increased in both IP schemes, but the errors for the spline method are obviously less than those for the Cressman method. This demonstrates that the spline interpolation is better than the Cressman interpolation when embedded into the IP scheme even in the case where noise is added to the pseudo-observation.

Table 2.

Results of experiments in section 3b(2). The cost function is the final value (×10−2). Inversion errors are MAEs for Fourier coefficients a and b.

Table 2.

4. Simulations in realistic domains

a. Numerical simulation design

In simulations in a realistic domain, the M2 tide in the SCS is simulated by assimilating T/P altimetry data, and there are 370 observation points in total. The experimental area and model variables are the same as those of the idealized experiment (section 3a; Fig. 2). What is different from the idealized experiment is that all four boundaries are taken into consideration in this section, and we select 19 IPs along the open boundaries in total (see Fig. 2). To determine the OBCs, two different interpolation schemes (Cressman and spline) and the SD method are used. To further test the feasibility of the IP scheme, observation data at 10 tidal gauges are chosen for comparison (Fig. 2). In the real ocean, the exact boundary condition of the open boundaries is usually unknown. Therefore, three boundary conditions of spline interpolation are used in simulations in realistic domains: the first boundary condition specifies the slope at two end points of the spline, the second open boundary condition indicates that the second derivative at two end points of the spline is known (in this section are given 0), and the periodic boundary condition is same as that in Pan et al. (2017).

b. Numerical results

We can see from Table 3 that the magnitude of error vector of all kinds of conditions is smaller than the result of the Cressman method. And the periodic condition yielded the smallest errors among all the methods. Therefore, we take periodic condition in this experiment. The errors between simulated water levels and observations are presented in Table 3. For all interpolation schemes, MAEs in amplitude and phase between the simulation and T/P data are smaller than 2.7 cm and 6.2°, respectively, implying a successful simulation.

Table 3.

The MAE in simulations in realistic domains. The p, 1, and 2 denote periodic, and the first and the second boundary conditions of the spline method, respectively. And the n means the errors with a new expanded computing area.

Table 3.

Before we examine the simulation of the internal tides, it is necessary to analyze the simulated surface tides. The cotidal charts for the M2 constituent in the SCS are shown in Fig. 8. Both of the cotidal charts, corresponding to the Cressman and spline methods, respectively, show the following characteristics: sinuous oscillations of the cophase lines are very obvious around the Luzon Strait, which are believed to be caused by internal tides; and sinuous oscillations can also be seen in the distribution of tidal amplitudes, which has been confirmed by many observations (Jan et al. 2008; Cao et al. 2017; Miao et al. 2011). To show the difference between the two methods more clearly, the magnitude of the model misfit vector for the two interpolation methods are given in Fig. 9. Obvious differences near the open boundaries, especially in areas A and B, are found.

Fig. 8.
Fig. 8.

Cotidal charts for the M2 constituent in the SCS. (a) Obtained with the Cressman interpolation. (b) Obtained with the spline interpolation.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

Fig. 9.
Fig. 9.

The magnitudes of the misfit vector of the results of the two interpolation methods (units: m).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

The magnitude of misfit vector between simulation and tidal gauge data for the two interpolation methods are given in Fig. 10. From Fig. 10, we can clearly see that there are significant differences between the results of the two methods at tidal gauges 2 and 10, which are located in areas A and B, respectively, demonstrating that the new IP scheme can improve the numerical results to some extent. And the most important information we found from this experiment is that the change in interpolation method from the Cressman method to the spline method results in a decrease in errors at almost all tidal gauges, which also can be seen in Table 3. The magnitude of error vector for the spline method is only 4.5 cm, smaller than that for the Cressman method. To conclude, by using the spline scheme, the model accuracy can also be improved in simulations in a realistic domain.

Fig. 10.
Fig. 10.

The magnitude of the error vector between simulation and observations at each tidal gauge for two interpolation methods.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

To obtain a better result, we expanded the original computing area to 15°30′–26°N, 116°10′–127°E (Fig. 11). The model parameters are the same as in section 3a. The periodic condition is used in this test. The MAEs in amplitude and phase between the simulation and T/P data are listed in Table 3. As we can see, the errors here are larger than those computed in the original computing area. Figure 11 shows the zonal baroclinic velocities at 500-m depth at the final time step of simulation for the two interpolation methods. The results of both methods show that the M2 internal tides are generated in the Luzon Strait and propagate away in three direction:, eastward into the Pacific Ocean with the widest tidal beam, westward toward Dongsha Island with the most energy, and southwestward into the deep basin of the SCS with the least energy. However, there are also some differences between the results of the two methods. Figure 12 showed the simulation difference between the two methods. We can see that the misfits mainly appear near the west open boundary, indicating that the new IP scheme can, to some extent, make a difference to the propagation of the M2 constituents in the westward direction.

Fig. 11.
Fig. 11.

Zonal baroclinic velocities at 500-m depth at the end of calculation for (a) Cressman and (b) spline.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

Fig. 12.
Fig. 12.

The absolute errors between the two interpolation methods in zonal baroclinic velocities at 500-m depth.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0155.1

5. Discussion

The simulation results of idealized test cases have demonstrated that the IP scheme involving the spline method is feasible in this 3D tidal model, and it can also improve the model results to some extent. Compared with traditional interpolation method, the new method performs more effectively. In terms of the decline of the cost function, the spline method used fewer steps to get much smaller values than the Cressman method. From the figures of the inversion error (Figs. 6, 7), we can see that the errors of the traditional method at the middle and two ends of the whole open boundary are larger, which may be caused by the fact that the prescribed distribution has a strong nonlinear variation here and this cannot be exactly acquired by the traditional method. In contrast, the overall inversion errors of the spline interpolation are smaller, whereby we can get a much smoother inversion curve, which is more consistent with the physical principles than that obtained with the traditional method. In the real ocean, the nonlinear distributed OBCs may be suitable for many ocean models because nonlinear dynamical processes are very common in the ocean. Therefore, the spline interpolation method is expected to be better than the traditional one in practical applications. Furthermore, the new method not only improves the inversion results of the OBCs, but also improves the results in the interior computing area even in the case where noise is added to the pseudo-observation. The inversion errors of the spline method are only half of those of the traditional method, meaning that by using the IP scheme with spline interpolation, the model precision and inversion results can be improved. Therefore, it makes sense to apply this method to the simulation in a realistic domain.

Through the simulations of the idealized test cases, the new method has been demonstrated to be able to improve numerical results. In a realistic domain, we can also find that the results of the spline method are better than those of the Cressman method as expected. Besides, the periodic boundary condition is more suitable to this model than others. The errors between the model result and T/P data for the new IP scheme are smaller than those for the traditional one, which is similar to the idealized simulations and means that the spline method can improve the precision of this model. The same conclusion can be drawn through comparison with some tidal gauges’ data in the computing area, further indicating the feasibility of the new IP scheme. From Fig. 10 we can see that errors at all 10 gauges are small, meaning that by using this internal tidal model, the computing area can be simulated precisely, no matter which method is chosen. To some extent, it demonstrates that this internal tidal model itself has an excellent capability of simulating internal tides. On the other hand, it is not difficult to see that the difference between the simulation errors of two methods also varies from gauge to gauge. In gauges 2, 5, 8, 9, and 10, the results obtained by the spline method are obviously better than those by the traditional one, in gauges 3, 4, and 6, the difference between the two methods is small and the result of the spline method is slightly better, and in gauges 1 and 7, the result of the spline method is slightly worse. On the whole, the simulation results of the spline method are still better. The different OBCs inverted by the two methods yield different simulation result in the interior area. From Figs. 9 and 12, we can see that the difference between simulation results of the two methods has an uneven spatial variation. The difference is remarkably large near the northern and western open boundaries. Figure 9 shows the results of the surface tidal amplitude, and it can be found that the two methods yield more obviously different results in areas A and B near the northern open boundary, where gauges 2 and 10 are located. Considering the fact that the simulation results of the spline method are better than the traditional method at the two gauges (Fig. 10), it is possible and also rational to extend this conclusion to the whole area near the northern open boundary. The superiority of the spline method over the traditional method here may be partly attributed to the complicated land–sea distribution pattern near the northern open boundary, whereby the nonlinear characteristic of ocean dynamical processes are dominant here, which is consistent with the conclusion drawn from the idealized test cases. On the other hand, from Figs. 9 and 12 we can also see that near the western open boundary is another area where the difference in the simulation results of the two methods is large, and gauge 1 is located in this area. It should be noted that at gauge 1 the result of the spline method is slightly worse than that of the traditional method, as shown in Fig. 10. From this, it is rational to some extent to infer that the results of the spline method may be also worse than those of the traditional method in the whole area near the western open boundary, but the difference is believed to be quite limited. This situation may be partly attributed to the fact that the ocean area near the west boundary is very flat and open, whereby the ocean dynamical processes tend to result in an overall linear pattern here. Nevertheless, from the results at all 10 available gauges in Fig. 10, the spline method still holds its overwhelming superiority over the traditional method in internal tidal simulation precision.

6. Conclusions

A two-layered isopycnic-coordinate numerical model with the adjoint assimilation method described in Chen et al. (2012) is used in this paper, and only the M2 constituent is considered in this paper. In this model, the ocean interior is regarded as a stack of isopycnic layers, each with a constant density; the Flather conditions are employed by the external mode; and the relaxation conditions are employed by the internal mode. The isopycnic coordinate can track the depth variance of the thermocline and is very suitable for the simulation of internal tides. To obtain a smoother and more realistic interpolated curve, we adopt the spline method instead of the Cressman interpolation in this work: values at selected IPs are taken as control variables and those at other points are obtained by the spline interpolation instead of the traditionally used Cressman interpolation. The improved scheme was calibrated with a series of idealized experiments. The results showed that OBCs can be successfully inverted by assimilating pseudo-observations and those obtained with the spline interpolation were more accurate and smoother, even in the case where artificial noise was added to pseudo-observations. In simulations in realistic domains, the observations were successfully assimilated through inverting the OBCs with both interpolation methods. But the spline method showed better performance than the Cressman method. And the periodic boundary condition of the spline interpolation showed the best performance in this paper than the other two boundary conditions.

In conclusion, the spline inversion strategy is feasible and better than the Cressman method, and can improve numerical results to some extent. We can obtain much smoother open boundary curves, which are inverted with the spline method, and therefore are more consistent with reality. It is worthwhile to apply the improved scheme to the inversion of other control variables of numerical models.

Acknowledgments

The authors appreciate the suggestions to an earlier edition of the manuscript given by the reviewers and editor. We thank Haidong Pan for the spline numerical program he kindly provided. Partial support for this research was provided by the national key research and development plan through Grant 2016YFC1402304, the key research and development plan of Shandong Province through Grant 2016ZDJS09A02, the National Natural Science Foundation of China through Grants 41606006 and 41371496, the Natural Science Foundation of Zhejiang Province through Grant LY15D060001, and the Ministry of Education’s 111 Project through Grant B07036. The TOPEX/POSEIDON data were provided by PODAAC of the JPL.

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