ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 31, JANUARY 2014, 188–196

Ocean Model Open Boundary Conditions with Volume, Heat and Salinity Conservation Constraints WANG Qiang1,2 , ZHOU Weidong∗1 , WANG Dongxiao1 , and DONG Danpeng1 1 State

Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou 510301 2 University

of Chinese Academy of Sciences, Beijing 100049

(Received 30 October 2012; revised 30 March 2013; accepted 18 April 2013) ABSTRACT Open boundary conditions (OBCs) for a regional ocean model that can be integrated stably over a long timeframe, as well as satisfy the volume, heat and salinity conservation constraints, were developed. First, the idea that the inward and outward flux information can be treated separately in the OBCs was adopted. Second, in order to maintain the property that the volume, heat and salinity remains conserved in the simulation domain, conservation constraints were added to the OBCs, and an inverse method utilized to solve the constraint equations. Ideal experiments were designed to investigate the conservation property, and the OBCs were found to work efficiently to maintain the volume, heat and salinity conservation. It was found that simulations were comparable to observations when the OBCs were applied to a regional ocean model. Key words: open boundary conditions, modeling, inverse method, ocean circulation Citation: Wang, Q., W. D. Zhou, D. X. Wang, and D. P. Dong, 2014: Ocean model open boundary conditions with volume, heat and salinity conservation constraints. Adv. Atmos. Sci., 31(1), 188–196, doi: 10.1007/s00376-013-2269-y.

1. Introduction In a regional ocean model, it is necessary to use open boundary conditions (OBCs) to solve the model equations. However, OBCs are an ill-posed problem (Bennett and Kloeden, 1978; Oliger and Sundstr¨om, 1978) because there is no achievable boundary condition that acquires a unique solution. Many OBCs have been designed (Chapman, 1985; Palma and Matano, 1998, 2000), and a key success measure is that they should not produce spurious reflection when the disturbance that is generated during the integration in the interior domain propagates outward; otherwise, the interior will be contaminated (Roed and Cooper, 1986). In order to solve this problem, Sommerfeld (1949) applied radiation conditions, in which it was assumed that it is in such a wave-like way that the interior disturbances propagate through the open boundaries. However, a difficulty exists in terms of the choice of the phase speed value, and the nondispersive, shallow water wave speed has generally been chosen (Vastano and Reid, 1967; Wurtele et al., 1967). Orlanski (1976) proposed that the phase speed can be estimated using the interior values, which provides a natural basis for the OBCs to be separated into inward and outward parts and treated individually (Roed and Smedstad, 1984). Raymond and Kuo (1984) applied a ∗

Corresponding author: ZHOU Weidong E-mail: [email protected]

radiation condition in which both normal and tangential phase velocity components were considered in a 2D wave equation. Marchesiello et al. (2001) treated the outward fluxes with a new algorithm to ensure the long-term integration is stable in a regional ocean system. The radiation condition is applied to enable the interior disturbance to propagate outside the domain, while the outside information has to transmit into the interior from the boundary. In order to obtain realistic solutions, it is necessary to correctly integrate the outer information through the boundary. Flather (1976) extended the radiation condition to include prescribed values at the boundary in a 2D equation, combining the wave equation with a 1D continuity equation. They also replaced the phase speed with the speed of external gravity waves in the scheme. Although it is an inexpensive way to calculate OBCs, their physical assumption that surface gravity waves are dominant was rigorous. It has been suggested that the phase speed should be separated into two parts: one from the interior and the other from the outside (Roed and Smedstad, 1984). Gan and Allen (2005) adopted this idea, and their OBCs were set as follows. In the inflow condition, the boundary solution relaxes to the prescribed value, which can be acquired from observational data or other simulation results. In the outflow, the solution combines the disturbance generated in the interior, which is calculated using the radiation equation and the outer prescribed values. Marchesiello et al. (2001) added a nudging term to the radiation equation

© Institute of Atmospheric Physics/Chinese Academy of Sciences, and Science Press and Springer-Verlag Berlin Heidelberg 2014

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in order to impose the outer information and guarantee the transition between outward and inward fluxes smoothly. Perkins et al. (1997) argued that accumulating dynamical error can be introduced if the volume conservation across open boundaries is lacking. In order to prevent this problem, Marchesiello et al. (2001) added a volume constraint (i.e., defined a normal correction velocity in the external mode solution), which is the total volume of transport through the open boundaries, and uniformly adjusts the barotropic flow there. It is an efficient way to maintain the volume conservation across the open boundaries, but it ignores the distribution of the correction velocity along the boundaries. Apart from volume conservation, heat and salinity also should be conserved for long-term stable integration and physical acquirement, especially during the spinning-up period (Dong, 2008). In doing so, the flux across both of the lateral boundaries and the surface boundaries should be zeros. To determine the correction velocity, linear equations should be solved, which are also ill-conditioned. In order to calculate the ocean circulation, an inverse method has been proposed to solve the linear equations, which are established based on the conservation constraints (Wunsch, 1978). Under all the constraints, the corrected velocity differs least from the starting conditions, which is the calculated guiding principle. The inverse method is an efficient method to calculate the correction velocity that keeps the volume, heat and salinity conserved, and was thus adopted in our calculation of OBCs. In the present study, we propose to implement proper OBCs in the Tsukuba Ocean Model (TOM) (Takano, 1974; Zhou, 2002a, b). TOM is a primitive-equation, free-surface, z-coordinate ocean model with the barotropic and baroclinic momentum equations resolved separately. It adopts efficient and non-split error time integration schemes, and the volume, heat, salinity, energy and enstrophy can be globally conserved. The model has been used previously in an ocean diagnostic calculation (Zhou et al., 2008). The outline of the paper is as follows. The formulation of the OBCs is presented in section 2. Numerical experiments are reported and discussed in section 3. In section 4, results from applying the OBCs to the South China Sea (SCS) are presented. And finally, a summary of the key findings are given in section 5.

2. Formulation The radiation condition is derived from the wave equation and is written as (φ is the model variable)

∂φ ∂φ +C =0, ∂t ∂n

φn−1,B−1 − φn+1,B−1 , φn+1,B−1 + φn−1,B−1 − 2φn,B−2

where the subscript B denotes boundary grid point, n denotes the present time, and the subscript B − 1 means a grid back toward the interior from the boundary. In order to introduce the exterior forcing, there is a nudging term on the right side of Eq. (1) (Marchesiello et al., 2001) 1 ∂φ ∂φ +C = − (φ − φext ) , ∂t ∂n τ when c < 0 (inflow)

φn+1,B = φn−1,B −

2Δt (φn−1,B − φext ) , τ

2Δt (φn−1,B − φext ) −C(φn,B − φn,B−1 ) , τ where τ is the timescale for nudging and φext represents the prescribed values, which can be provided from a model solution with a larger computational domain, or from measurements. Conservation is an important issue that keeps the integration of the regional ocean stable in the long-term. In order to keep the ocean’s volume, heat and salinity conserved, the constraints are written as

φn+1,B = φn−1,B −

KM IM

∑ ∑ (uik,c + uik,r )Δxi Δzk = 0 ,

(4)

∑ ∑ (uik,c + uik,r )Tik Δxi Δzk + Qnet = 0 ,

(5)

∑ ∑ (uik,c + uik,r )Sik Δxi Δzk + Snet = 0 ,

(6)

k=1 i=1 KM IM

k=1 i=1 KM IM

k=1 i=1

where uc is the normal velocity at the boundaries calculated by the Eq. (3), and ur is the correct velocity that we want to acquire. Δx and Δz are the horizontal and vertical grid spaces along the boundaries, respectively; T and S are the temperature and salinity at the boundaries; Qnet and Snet are the surface heat and salinity net flux; and i and k represent respectively the horizontal and vertical grid locations along the boundaries. Conveniently, we define the matrix A and vector ⎤ ⎡ A1 A = ⎣ A2 ⎦ , A3 where A1i = A2i =

(2)

(3)

when c > 0 (outflow)

(1)

where n denotes the normal direction to the boundary and c is the phase speed. The phase speed of the disturbance is determined locally by (Gan and Allen, 2005) C=

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A3i =

KM

∑ Δxi Δzk ,

k=1 KM

∑ Tik Δxi Δzk ,

k=1 KM

∑ Sik Δxi Δzk ,

k=1

⎤

⎡ =⎣

1

2⎦ 3

,

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where 1

2

3

= = =

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closed ocean basin under the given wind was integrated for eight years to reach equilibrium (see the kinetic energy in Fig. 3). Under the wind forcing, a jet was located in the middle of the domain. Cyclonic and anticyclonic currents occurred respectively in the northern and southern parts of the domain (Fig. 4a). To resolve the beta effect, the currents had to be asymmetrical. In response to high (north) and low (south) sea surface height (SSH), high (north) and low (south) SST

KM IM

∑ ∑ uik,c Δxi Δzk ,

k=1 i=1 KM IM

∑ ∑ Tik uik,c Δxi Δzk + Qnet ,

k=1 i=1 KM IM

∑ ∑ Sik uik,c Δxi Δzk + Snet ,

k=1 i=1

Then, Eqs. (4)–(6) can be written as Aur = −

.

(7)

Where A is a 3×IM matrix (IM>> 3). Equation (7) is an underdetermined system, and the inverse method has been adopted to solve it (Wunsch, 1978). Different horizontal grid spaces and water depths at the boundaries can affect the correct velocity, and a weight diagonal matrix W has been defined to avoid this issue. The elements are Wik = Δxi Δzk , and Eq. (7) is converted to A ur = − 1

,

(8)

Fig. 1. Model grid, with the gray line representing the location of the open boundary in the experiments.

1

where A = AW − 2 , ur = W − 2 ur . Using singular value decomposition (SVD), the matrix A can be written as A = U LV T (L is the eigenvalue matrix, and Lll = λl đand U and V are the eigenvector of the A AT and AT A respectively, e.g. A AT Ul = λl2 Ul , AT A Vl = λl2 Vl ), and we get the solution ur =

UlT Vl . l=1 λl 3

∑

(9)

where the λl (l = 1, 2, 3) is the eigenvalue of the matrix A . Equation (9) is the particular solution of minimum norm among all the possible solutions. Under the guiding principle that corrected velocity differs least from the starting conditions and satisfies the constraints, Eq. (9) is the required solution. Using the weight matrix, we can obtain the correct velocity from the ur , and then (ur + uc ) is the velocity at the open boundaries.

Fig. 2. Vertical temperature (◦ C) profile used in the numerical experiments averaged from the WOA01 of the North Pacific (a) and zonal wind speed (b).

3. Numerical experiment Ideal experiments were designed to test the conservation property of the OBCs. A rectangular domain with a closed boundary represents the ocean basin in Fig. 1. The vertical profile of temperature was calculated from the World Ocean Atlas 2001 (WOA01) (Boyer et al., 2005) of the North Pacific (Fig. 2a). Salinity throughout the whole domain was set as constant (35 psu). The zonal wind is shown in Fig. 2b and the meridional wind was set as zero. Heat and water fluxes at the sea surface were not considered. The model domain was set from 100◦ to 130◦ E and from 20◦ to 35◦ N, with a horizontal resolution of 0.5◦ × 0.5◦ and an ocean depth of 3750 m. The

Fig. 3. Time evolution of total kinetic energy (J) of the large domain integration.

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Fig. 4. Distribution of the (a) SSH and (b) SST in the pseudoobservation calculation.

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are displayed respectively (Fig. 4b). The results of this rectangular ocean were used as pseudo observations (hereafter referred to simply as “the observations”), and the western part of the domain was separated as a regional ocean with an open east boundary—denoted by the gray line in Fig. 1. Using the obs in the sixth year as initial conditions, two versions of the regional ocean with the same sea surface forcing but different OBCs were integrated for two years, and both reached equilibrium. Two experiments were designed: one was integrated with OBCs calculated using Eq. (3) without correction (exp1); and the other was integrated with OBCs calculated using Eq. (3) with correction (exp2). The simulation results are shown in Fig. 5. The left panel is the distribution of SSH, and the right panel is the SST. The simulated SSH distribution in exp1 (Fig. 5b1) and exp2 (Fig. 5c1) were very similar, both reproducing the main SSH structure well, and both resembling each other well. However, SST in exp1 was much lower than that in the observations. Figure 6 shows the vertical profile of zonal velocity along a section through one grid-space back toward the interior from the east boundary. In order to investigate the difference between the two experiments and the observations, the results of each experiment minus observations are presented

Fig. 5. Comparison among the observations, exp1 and exp2. Left panel is SSH; right panel is SST; a1 and a2 are the results from observations; b1 and b2 are the results from exp1; c1 and c2 are the results from exp2.

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Fig. 6. Vertical profile of zonal velocity along the section one grid-space back toward the interior from the east boundary: (a) exp0; (b) exp1; (c) exp2; (d) exp1 minus exp0; (e) exp2 minus exp0; (f) exp2 minus exp1.

in Figs. 6d and e. It can be seen that both two experiments underestimated the upper wind-driving circulation, but with exp2 producing better results. The difference between exp1 and exp2 (Fig. 6f) indicates that the correction method can improve the simulation of the upper wind-driving circulation. The vertically integrated temperature flux was also investigated (Fig. 7). Associated with the underestimated upper wind-driving circulation, the temperature flux of exp1 deviated from the observations, especially at the large velocity locations. Under the constraints, the temperature flux of exp2 was closer to the observations. Differences of the temperature flux between the experiments and the observations showed that the deviation of exp1 behaved like the upper wind-driving circulation pattern, and the deviation of exp2 was largely reduced. In order to investigate volume and temperature conservation throughout the entire simulation domain, the averaged SSH and volume-weighted average temperature of the domain were calculated for the observations, exp1 and exp2 (Fig. 8). Differences in averaged SSH were not obvious, but some improvement was also found. It can be seen that the averaged temperature of the observations was in equilibrium. Although the result from exp2 was a little smaller than from the observations, it did reach equilibrium. The averaged temperature decreased without reaching equilibrium at the end of integration, which led to much lower SST due to heat loss through the open boundary. The differences of temperature profile between the obser-

vations and simulations are shown in Fig. 9. It can be seen that the differences mainly occurred in the upper 500 m, with improved results in exp2. However, the simulation results of exp2 at depths of around 500 m were not improved, and even become worse. Therefore, it may be the case that the OBCs are more useful for upper-ocean simulation, being less sensitive in the deeper ocean. However, overall, the simulation can still be improved. In summary, the experimental results confirmed that the conservation constraints on the OBCs are necessary and that our proposed OBCs work effectively.

4. Application The OBCs were applied in a regional ocean model including the South China Sea and part of the western Pacific Ocean. The model has a horizontal resolution of 0.25◦ and 25 levels in the vertical direction. ETOPO2 (Marks and Smith, 2006) was used to prescribe the topography and climatological wind-stress was calculated using the ERA40 dataset (Uppala et al., 2005). The model was initialized by the climatological hydrographic data (WOA01) in January with a resolution of 0.25◦ . The relaxation scheme was used to replace the surface heat and water flux, relaxing toward the WOA01, with a relaxation timescale of 30 days. The Simple Ocean Data Assimilation (SODA) reanalysis product (Carton and Giese, 2008) was used to provide the lateral forcing. The model was integrated from a state of rest for ten

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Fig. 8. Time evolution of the volume-weighted average SSH (m) (upper panel) and temperature (◦ C) (bottom panel) in observations, exp1 and exp2.

Fig. 7. Vertically integrated temperature flux along the section one grid-space back toward the interior from the east boundary: (a) temperature flux of each experience; (b) difference of temperature flux between experiences (multiplied by 10).

years with the ERA40 wind forcing data, and reached equilibrium according to the kinetic energy. Two experiments were also designed: one was integrated using the OBCs without constraints (called C1) and the other was with constraints (called C2). Comparisons between the simulated SSH and the T/P satellite altimeter observations, as well as simulated temperature (at 125 m) and WOA01, are given in Fig. 10. It can be seen that both C1 and C2 simulated the distribution pattern of SSH well compared to the satellite observations (Fig. 10, left). The negative SSH center to the west of the Philippines and the SCS west boundary currents were reproduced well in both experiments. The path of the Kuroshio Current in the Luzon Strait and the locations of the bifurcation of the north equatorial current were similar in the simulation and the remote sensing results. However, there were significant

Fig. 9. Difference of the temperature profile between the observations and simulations.

differences in the magnitudes of SSH simulated by the two experiments. Firstly, SSH from C1 was higher overall in the SCS and lower in the east of the Luzon Strait compared to the remote sensing results, while the results of C2 were much closer to the observations. Secondly, the difference of SSH between the two sides of the Luzon Strait is an important factor deciding the intrusion of the Kuroshio Current because

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Fig. 10. SSH (left) and temperature at 125 m (right) in Jan: (a1) derived from climatologic monthly mean T/P altimetry data; (a2) derived from woa01; (b1, b2) from model results using OBCs without constraints; (c1, c2) from model results using OBCs with constraints.

the geostrophic balance controls transport in the Luzon Strait above 1500 m (Song, 2006), and C2 produced more reasonable results than C1 in this regard. Lastly, the temperature from C1 was higher than that from WOA01, which indicated the simulated ocean temperature was heated without the conservation constraints. Meanwhile, the result from C2 was much closer to that of WOA01. In summary, the comparisons confirmed the validity of the model results and the OBCs. In order to test the influence of a different domain size on this method, as well as the variability of the model solutions affected by the OBCs, another experiment was performed. Keeping all the forcing functions the same as above, the model domain was relocated to 18◦ –25◦ N and 110◦ –124◦ E, and the resolution was changed to 0.125◦ . The model was integrated from a state of rest. Under January forcing, the model was integrated for 10 years, and then driven by seasonal forcing for another 10 years. The results from the final

year were then exported for analysis. The temperature at 125 m in each season is compared in Fig. 11. It can be seen that the new OBCs effectively adjusted the temperature field with the different simulated domain size. The seasonal variability was also simulated.

5. Discussion and summary In the work reported in the present paper, OBCs for a regional ocean model that can be integrated stably over a long timeframe, as well as satisfy the constraints of volume, heat and salinity conservation, were developed. An important issue is the computational time it takes to calculate the OBCs. Firstly, it is not necessary to renew the correct velocity at each integration step; we carried out some experiments with different steps (1, 10 and 20) and the influence could be omitted. And secondly, the time that each calculation costs is negligi-

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Fig. 11. Temperature (◦ C) at 125 m in each season. The upper panel shows results derived from woa01; the middle panel shows model results using OBCs without constraints; and the bottom panel from model results using OBCs with constraints: (a) winter; (b) spring; (c) summer; (d) fall.

ble. Simple experiments were designed to test the conservation property. When the conservation constraints were added to the OBCs, the average temperature was closer to the observations and was able to reach equilibrium. Whereas, if the constraints were removed, the simulation could not reach equilibrium and the temperature keept decreasing due to heat loss though the east open boundary. The OBCs were also applied to a regional ocean simulation. Comparisons between the model results and observations further confirmed the validity of the OBCs. Under the background of global warming, the regional ocean and even the whole ocean may be not at a state of equilibrium. The solution of a regional ocean model with such OBS is more like an approach to the climatological state, while it is useful to correct the numerical bias. If the change of the regional ocean state has been acquired (observations or global coupled model results), it can be added to the right of Eqs. (4)–(6) as constraints, allowing the simulation to become more realistic. Acknowledgements. This study is funded by the Strategic Priority Research Program of the Chinese Academy of Sciences with Grant No. XDA11010304, and is also supported by National Natural Science Foundation of China (No. 41376026 and 40776009).

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