Abstract

This study developed a hydrodynamic reduced-order model (ROM) to regenerate surface currents in Gorgan Bay, Iran. The developed ROM was based on linking a three-dimensional hydrodynamic model, MIKE3-FM, with a data reduction technique, proper orthogonal decomposition (POD). The MIKE3-FM model was first run to simulate surface currents in the bay under a real wind scenario for two years starting July 1, 2010. Thereafter, time and space steps of 6 hours and 500 m, respectively, were chosen to capture 2,920 snapshots of the simulated surface currents using the MIKE3-FM model on 1,937 grids in the bay. The snapshots were then used as input for the POD model to develop the ROM. By applying the POD on the snapshots, necessary spatial and temporal components of surface currents used to develop the ROM were calculated. Having spatial and temporal terms, two ROMs for regeneration of surface currents U and V in two directions x and y, respectively, were developed. Analysis of ROM results revealed they accurately regenerated surface currents using only the first ten modes (among 2,920 modes). Comparison of MIKE3-FM and ROMs developed by the first ten modes revealed there were only negligible differences between their results when they simulated and regenerated, respectively, U and V, in the bay.

INTRODUCTION

Hydrodynamic models describe the motion of water in coastal environments and provide important information on flow characteristics. The outputs of these models include spatiotemporal variations of water surface elevation, velocity, and temperature. Obtaining acceptable insight on the fluctuation of hydrodynamic objectives is important to adopt suitable decision support tools related to a water body.

There are various commercial and free access numerical models that appropriately simulate hydrodynamic variables in water bodies. These models simulate the spatiotemporal variations of the hydrodynamic variables in one-, two-, and three-dimensions (1-D, 2-D, and 3-D, respectively). However, theoretically, the best choice for the hydrodynamic simulation of a water body is a 3-D model that informs users about variations of flow in all three directions. MIKE3-FM, a 3-D model introduced by the Danish Hydraulic Institute (DHI) in Denmark (DHI 2007), is a commonly used model for hydrodynamic simulation of coastal environments and bays (e.g., Fourniotis & Horsch 2010; Eriksson & Engqvist 2013; Sabatino et al. 2016; Sedigh et al. 2016). Zhang (2006) simulated tidal elevations and tidal current velocities in Singapore's coastal waters using two 3-D models, MIKE3-FM and the Princeton Ocean Model, and showed that the two models were in agreement with field measurements. This study also found that the tidal current velocities simulated by the two models were in agreement with field measurements. Passenko et al. (2008) validated and compared the hydrostatic and non-hydrostatic versions of the MIKE3-FM model in the Gulf of Finland and the Baltic Sea. To assess model robustness, the results were compared with observed sea levels in Helsinki. The results indicated that both versions of MIKE3-FM adequately simulated the sea level fluctuations. The MIKE3-FM model was used to simulate the barotropic response which developed in the Gulf of Patras, in western Greece, due to tidal and wind forcing (Fourniotis & Horsch 2008). Sharbaty (2012a) used the same model to calculate water level fluctuations and velocity components in Gorgan Bay, Iran, and results showed that they were affected by the wind-induced force and the bed topography. Sharbaty (2012b) also investigated wind-induced flow using the MIKE3-FM model in the Caspian Sea and concluded that the surface currents were affected by Volga River flow, prevalent wind patterns, and topography. Small-scale hydrodynamics that control sediment transport at Sao Marcos Bay, along the Ponta da Madeira Port Complex in Brazil, were assessed using the MIKE3-FM model (Samaritano et al. 2013). Li et al. (2014) investigated the tidal dynamic change due to the bridge pier construction in the Jiaozhou Bay using the MIKE3-FM model. Cavalcante et al. (2016), using the MIKE3-FM model and oceanographic surveys, showed that the main factor that influenced the currents in the southern parts of the Persian Gulf was localized peak winds. Pan & Liu (2015) used the same model to investigate the hydrodynamic response in the Yangtze estuary to a typhoon, and concluded that the model successfully simulated typical typhoon impacts. Also, simulated water levels and water surface velocity showed good agreement between the MIKE3-FM model and measured data at Port Assaluyeh in the Persian Gulf (Faghihifard & Badri 2016).

Although the results of hydrodynamic simulations using MIKE3-FM and other models generate a significant amount of useful information, managing the large set of outputs has always been a fundamental challenge for decision-makers. This problem arises especially when the number of computational cells and running times are increased. Therefore, the development of a reduced-order model (ROM) that appropriately manages the large set of simulated hydrodynamic variables with a simple mathematical structure, can be viewed as an important step towards more practical results. A hydrodynamic ROM aims to rapidly capture the essential features of flow. To develop a ROM, basic properties of the objective parameter must already be simulated and then captured by a data reduction approach such as proper orthogonal decomposition (POD). The ROM is then presented in a simple mathematical structure that precisely regenerates the objective parameter. A ROM based on POD may be applied using basis functions, Galerkin projection, or the Galerkin-free method. Rowley et al. (2004) presented low dimensional equations of motion by application of Galerkin projection ROM to the compressible Navier–Stokes equations. They concluded that the resulting equations were much simpler than the full compressible fluids equations. Esfahanian & Ashrafi (2009) applied a Galerkin-free ROM to solve 1-D and 2-D shallow water equations. Noori et al. (2015) regenerated the nitrate concentration in the Karkheh Reservoir using a ROM developed by basis functions. In another study, a basis function ROM was developed for simulation of sea surface temperature over the northwest portion of the Indian Ocean (Noori et al. 2017). While the ROM based on POD approaches has been widely used in the field of computational fluid dynamics (Ravindran 2000; Sirisup et al. 2005; Xiao et al. 2013; Stabile & Rozza 2018), a basis function ROM can be viewed as a useful approach to appropriately manage the large set of simulated hydrodynamic variables (Noori et al. 2017). However, since few research studies have aimed to investigate the capability of the basis function ROM for real-world sites (Noori et al. 2015, 2017), it can be argued that additional studies should be carried out to properly demonstrate the usefulness of the method for regeneration of simulated hydrodynamic variables in extensive water bodies such as Gorgan Bay.

In light of the above, the main objective of this research is to provide a ROM based on the POD method for regeneration of surface currents in Gorgan Bay, Iran. Although the ROM approach has been used in many controlled laboratory flow experiments, it has not yet been applied to a real case study. As such, this research presents a novel approach in the field of hydraulic engineering.

STUDY AREA

Gorgan Bay, located in the southeast corner of the Caspian Sea, covers an area of approximately 400 km2 and has a length of about 70 km (Figure 1). It is isolated from the Caspian Sea by the Minyakale Peninsula. Its maximum depth is 6.5 m, with an average depth of 1.5 m, which increases from west to east towards the southern side of Ashouradeh Peninsula. As shown in Figure 1, this bay is permanently connected to the Caspian Sea at the mouth (with a width of 3.5 km and a depth of 2 to 4 m) and the Strait of Khozeini (with a length of 1.8 km and a depth of 2.5 to 5 m).

Figure 1

Location of Gorgan Bay in Iran.

Figure 1

Location of Gorgan Bay in Iran.

Gorgan Bay has significant ecosystem diversity and was among the first wetlands to be registered in the Ramsar Convention on Wetlands in 1975. In addition, it was selected as one of nine Biosphere Reserves by UNESCO in 1976. Gorgan Bay is not valuable only for its cartilaginous fish, white fish, mullet, etc., but also because it is an important wildlife refuge that hosts a variety of birds throughout the year. In addition, about 40% of Iran's caviar comes from this area, which is internationally recognized as an important ecological area and is considered to be environmentally strategic.

Due to reduced discharge from the Volga River that provides around 80% of inflow to the Caspian Sea, lower rainfall levels, global warming, increased evaporation and other factors, the water level in the Caspian Sea has fallen about 8 cm in recent years. As reported by the Gorgan Bay Restoration Program (funded by the Golestan Regional Water Authority), the consequences of a continued decline in sea level are severe for Gorgan Bay (Pouya Tarh Pars Consultant Engineers 2017).

The aforementioned issues justify research on the hydrodynamic characteristics of Gorgan Bay in order to try and address environmental damage. Several studies have tried to simulate Gorgan Bay hydrodynamics using numerical models, particularly the MIKE Package Software (Sharbaty 2012a; Ranjbar & Zaker 2016, 2018), although the large set of outputs was a significant challenge for decision-makers. This research aims to introduce a ROM to effectively manage the large set of simulated results generated by the MIKE3-FM model with a simple mathematical structure. The MIKE3-FM model was used because the hydrodynamic processes are fundamentally 3-D. Also, 1-D and 2-D models do not fully describe wind-driven mixing (Martin 1985). Wind-induced forces are known as the most important factor affecting hydrodynamic processes in the Gorgan Bay (Ranjbar & Zaker 2018). In addition, since only the field measurements of surface currents were available for the model calibration, it was necessary to use a 3-D model to properly distinguish between bottom and surface currents.

METHODOLOGY

The methodology used in this research is based on linking a hydrodynamic simulation model (MIKE3-FM) and a data reduction technique (POD). The numerical model simulates spatiotemporal variations of hydrodynamic variables (surface currents) in Gorgan Bay. The POD uses the simulated dataset of the spatial distribution of surface currents taken at a fixed time (snapshot), and then identifies the dominant modes of surface velocities U and V in two directions, x and y, respectively, in the bay. Thereafter, the ROM is developed based on the dominant modes calculated by the POD method. The accuracy of the developed ROM results is evaluated through the comparison of the regenerated U and V data by ROM and the simulated data by the MIKE3-FM model. Figure 2 shows the step by step methodology used in this research.

Figure 2

Step by step methodology used in the study.

Figure 2

Step by step methodology used in the study.

MIKE3-FM model

Model description

The software package developed by DHI (MIKE3-FM) that was selected for 3-D simulation of Gorgan Bay has been shown to perform well in hydrodynamic and water quality modeling of different types of water bodies (Zhang 2006; Fourniotis & Horsch 2008; Abily et al. 2013; Vojinovic et al. 2013; Payandeh et al. 2015; Zeinoddini et al. 2015; Vo & Gourbesville 2016). The model is based on the Reynolds-averaged Navier–Stokes equations considering Boussinesq approximation and hydrostatic pressure for incompressible fluids. To conserve continuity in the equation-solving process, time discretization of equations is usually performed using a semi-implicit method so that terms of horizontal and vertical planes are discretized by explicit and implicit methods, respectively. The MIKE3-FM model applies a finite volume method to discretize the continuity, momentum, and transfer equations. Discretization of equations in this model is carried out through two flexible triangular and rectangular grids. This model uses structured and unstructured grids for hydrodynamic simulations along the vertical and horizontal planes, respectively. Note that water density in the MIKE3-FM model is considered to be a function of both temperature and salinity.

Model inputs

Input data required for the hydrodynamic simulation of the bay by the MIKE3-FM model include precipitation, evaporation, river inflows, wind speed and direction, mean water-level fluctuations, atmospheric pressure, air humidity, and bathymetric information. The topography of the bay has a gentle slope from the banks to the midsection. The slopes in the northern and eastern parts of the bay are greater than those in the southern and western area. Wind speed and direction data in a two-year period, starting from July 1, 2010, were taken from the weather station at Bandar Turkmen. The prevailing wind directions in the area are from the west and northwest. Other meteorological data such as precipitation, evaporation, temperature, relative humidity, and river inflows were provided by the National Meteorological Organization of Iran. Also, hydrological data such as river inflows were taken from the Iranian Water Resources Management Department. Historical data of water levels in the main inlet of the bay (item no. 1 in Figure 1) was considered as the open boundary condition. The average annual temperature, relative humidity, evaporation, and precipitation were 17.7 °C, 74%, 1,016 mm, and 471.7 mm, respectively. The simulation was conducted under conditions of real wind between July 1, 2010 and July 1, 2012.

Model set-up

This research used the MIKE3-FM model that was calibrated and verified for simulation of hydrodynamic variables in Gorgan Bay by Ranjbar & Zaker (2016, 2018). The results obtained over the two-year study period were used to develop the ROM. A brief overview of the calibration and verification steps is discussed here. As with most papers, reports, and books that introduce the MIKE3-FM model (McClimans et al. 2000; DHI 2007; Bolaños et al. 2014; Payandeh et al. 2015), including all the details of the model is avoided here and only the model configuration is explained.

The governing equations for local continuity and horizontal momentum in x and y directions are given as Equations (1)–(3), respectively (DHI 2007): 
formula
(1)
 
formula
(2)
 
formula
(3)
where t is the time; x, y, and z are the Cartesian coordinates; η is the surface elevation; d is the still water depth (i.e., the average water depth at any moment, excluding local variation caused by wind action and waves); h = η + d is the local water depth; u, v, and w are the velocity components in x, y, and z directions; f = 2Ωsin(ϕ) is the Coriolis parameter; Ω is the angular rate of revolution; ϕ is the geographic latitude; g is the gravitational acceleration; ρ is the density of water; sxx, sxy, syx, and syy are the components of the radiation stress tensor; vt is the vertical turbulent (or eddy) viscosity; pa is the atmospheric pressure; and  is the reference density of water. S is the discharge magnitude due to point sources and (us, vs) is the velocity by which the water is discharged into the ambient water (DHI 2007).

It should be noted that the Caspian Sea is a closed water body and does not experience any tide (Beni et al. 2013). However, due to the effect of wind (DeMarchis et al. 2014) and topography on coastal area hydrodynamics, the impacts of these parameters was considered in the MIKE3-FM model that was calibrated and verified by Ranjbar & Zaker (2016, 2018). Also, the model had both open and closed boundaries. Historical data of water levels in the main inlet of the bay (item no. 1 in Figure 1), and also the bay's bottom and coastlines, were considered as the open and closed boundaries, respectively. Additional details on the calibration and verification of the model are given by Ranjbar & Zaker (2016, 2018).

To solve the partial differential equations governing the MIKE3-FM model, the solution domain should be a discrete set of computational interconnected cells. For this reason, the bathymetry file of Gorgan Bay in the MIKE ZERO MESH GENERATOR environment was created using a DEM file under the coordinate system of WGS-1984-UTM-ZONE-39N. Also, to satisfy the stability condition in solving the equations, the Courant–Friedrichs–Lewy (CFL) condition (Table 1) was considered equal to 0.8. Gorgan Bay has low salinity compared to other coastal and marine water bodies, and therefore water density has very little effect on bay hydrodynamics. Thus, in this study, the density effect on the flow was considered barotropic.

Table 1

MIKE3-FM equations used in the model configuration

Horizontal eddy viscosities (Smagorinsky) 
 
 The horizontal eddy viscosities 
 Constant 
 A characteristic length 
  The deformation rate 
Vertical eddy viscosities (k-ɛ turbulent) 
 
 The vertical eddy viscosities 
 The turbulent kinetic energy per unit mass (TKE) 
 The dissipation of turbulent kinetic energy 
 An empirical constant 
Courant–Friedrichs–Lewy (CFL) 
 
 Courant–Friedrichs–Lewy number in hydrodynamic flows 
 The total water depth 
 Time step interval 
 Characteristic length scale in the x- and y-direction, respectively 
 Velocity components in the x- and y-direction, respectively 
 The gravitational acceleration 
Quadratic roughness coefficient 
 
 The bottom stress 
 The water density 
 The von Karman constant 
 Distance above the seabed 
 The bed roughness length scale 
 The flow velocity above the bottom 
Horizontal eddy viscosities (Smagorinsky) 
 
 The horizontal eddy viscosities 
 Constant 
 A characteristic length 
  The deformation rate 
Vertical eddy viscosities (k-ɛ turbulent) 
 
 The vertical eddy viscosities 
 The turbulent kinetic energy per unit mass (TKE) 
 The dissipation of turbulent kinetic energy 
 An empirical constant 
Courant–Friedrichs–Lewy (CFL) 
 
 Courant–Friedrichs–Lewy number in hydrodynamic flows 
 The total water depth 
 Time step interval 
 Characteristic length scale in the x- and y-direction, respectively 
 Velocity components in the x- and y-direction, respectively 
 The gravitational acceleration 
Quadratic roughness coefficient 
 
 The bottom stress 
 The water density 
 The von Karman constant 
 Distance above the seabed 
 The bed roughness length scale 
 The flow velocity above the bottom 

Due to the effect of mesh dimensions on the model accuracy, different mesh sizes were used in the deep layers, and model sensitivity was evaluated in terms of the selected mesh sizes. The results of the sensitivity analysis (SA) showed that the best bathymetry of the bay consisted of 12,882 elements, and 6,824 nodes, in five deep layers with a flexible triangular grid type. The results of SA for horizontal and vertical eddy viscosities using Smagorinsky and k-ε turbulent models (Table 1), respectively, revealed a slight effect of both parameters on the flow in Gorgan Bay. SA results for the bed friction term using both roughness height and quadratic roughness coefficient methods (Table 1) showed that it strongly influenced flow variations in the bay so that the currents were reduced as bed roughness increased.

In the next step, according to the SA results, the model was calibrated for the different values of bed roughness. Calibration results indicated that the best values for bed roughness in the northern, middle, western, and southern parts of the bay were equal to 7.5, 4, 2, and 1 mm, respectively.

For model calibration and verification, currents in Gorgan Bay were measured over 5 days (6th to 12th November, 2013) in different areas using a Drifter constructed by the Iranian Institute of Oceanography and Atmospheric Science. The differences among the field measurements of the Lagrangian trajectories of water mass by Drifters, with the corresponding model outputs, were considered for model calibration and verification (Ranjbar & Zaker 2018). The Lagrangian particle-tracking method has been widely applied for calibration of numerical models (Liu et al. 2004; Muller et al. 2009; Liu & Weisberg 2011; Haase et al. 2012). The average relative error (%) between the modeled and measured trajectories for the MIKE3-FM model was 14.7%, as reported by Ranjbar & Zaker (2018). The Drifter had a cylindrical shape with a height of 1 m and a cross diameter of 15 cm. The trajectories of the Drifters were recorded by an internal global positioning system device with a time interval of 1 min. More details on Drifter and its precision are given in Ranjbar & Zaker (2018).

ROM

POD description

POD is a data reduction technique that has been broadly used in the field of CFD (Barone et al. 2009; Troshin et al. 2016). This method aims to find a set of basis functions (or modes) that is representative of the members of the ensemble on the sequence of (Noori et al. 2015): 
formula
(4)
where, denotes the set of N simulations of the objective parameter as snapshots and N is the number of snapshots.
The basis functions are admixtures of the snapshots and are presented as Equation (5): 
formula
(5)
In Equation (5) the coefficients should be determined so that the modes (Equation (5)) are parallel to the ensemble (Ly & Tran 2001). In this regard, Equation (6) should be maximized subject to Equation (7) (Esfahanian & Ashrafi 2009): 
formula
(6)
 
formula
(7)
where, is the L2 inner product and is the L2-norm.
It has been demonstrated that the above optimization problem is transformed to an eigenvalue problem (Ashrafi 2012). Therefore, the basis functions are the eigen-functions of the integral equation as follows (Ly & Tran 2001): 
formula
(8)
Putting Equation (5) into Equation (8) gives the eigenvalue problem as follows: 
formula
(9)
where 
formula
(10)
where, L is a non-negative Hermitian matrix (or correlation matrix) and are eigenvectors corresponding to eigenvalues .
Therefore, the problem involves solving for the eigenvectors of a symmetrical matrix with dimension equal to the size of the snapshots (N). Straight calculation demonstrates that function (Equation (11)) is maximized when the coefficients of Equation (5) are the elements of the eigenvector corresponding to the largest eigenvalue of L (Ly & Tran 2002). 
formula
(11)
The obtained basis functions represent the main physics of the system (i.e., they equivalently capture a significant percentage of energy). Using the R basis modes that correspond to the largest R eigenvalues, the regenerated objective parameter can be presented as (Ashrafi 2012): 
formula
(12)
Therefore, a regenerated objective parameter can be presented by the POD method as a combination of a time-dependent term, , and a space-dependent term, (Noori et al. 2015). The time-dependent term can be calculated as in Equation (13) (Ashrafi 2012; Noori et al. 2015): 
formula
(13)

More details on the POD method are provided in Berkooz et al. (1993), Liang et al. (2002), Kerschen et al. (2005), Ashrafi (2012), and Noori et al. (2012, 2013).

ROM development

In order to develop a hydrodynamic basis function ROM that appropriately regenerates the surface currents in Gorgan Bay, U and V were simulated by the MIKE3-FM model, and then extracted on square grids. Indeed, the simulated U and V are used as input data into the POD model. To do this, the grid size and time interval for extraction of MIKE3-FM results should be selected in such a way that they properly show the spatiotemporal variation of U and V in the bay, while they are as large as possible to avoid high computational costs for the development of ROM. It is clear that the minimum values for the time interval and grid size should be higher than the time step selected for running the MIKE3-FM (60 sec) and cell size in this model (47,700 m2), respectively. Also, it is noteworthy that the tidal effects on hydrodynamic processes in the bay are negligible, and only density gradients and wind are the main driving forces in this water body (Kitazawa & Yang 2012; Ranjbar & Zaker 2018). As such, the variation of hydrodynamic variables in Gorgan Bay is very small. However, a primary and rough screening of the MIKE3-FM results revealed that U and V variations were negligible in square grid sizes, and time intervals less than 500 m and 6 hours, respectively. Therefore, in this study a square grid size of 500 × 500 m was selected, which divided the bay surface into 1,937 grids as shown in Figure 3. Also, a time interval of 6 hours was chosen as the time interval that resulted in the extraction of 2,920 snapshots for each of U and V simulated by MIKE3-FM in the bay during the two-year simulation period. By considering the discretization, the spatial and temporal resolutions of the ROM were 500 m and 6 hours, respectively. This means that the desirable resolution should be defined for the ROM model in this step, if one aims to regenerate U and V in the bay with finer/coarser grids and shorter/longer time intervals. However, the selection of very fine grids and very short time intervals will result in increasing the computational costs for POD and ROM application, while not significantly improving the results.

Figure 3

Selected square grid size of 500 × 500 m that divided the Gorgan Bay surface into 1,937 grids.

Figure 3

Selected square grid size of 500 × 500 m that divided the Gorgan Bay surface into 1,937 grids.

Having extracted U and V velocities, establishment of the Hermitian matrix L with dimensions equal to snapshots (2,920 × 2,920), was carried out by applying Equation (10). Eigenvalues and eigenvectors for the matrix were then calculated. The format of MIKE3-FM outputs was unusable for the POD method and conversion to a usable format was done by programming in the MATLAB environment. Also, an appropriate code was developed in the MATLAB environment to calculate the eigenvalues and eigenvectors for proper application of the POD method. Generally, all coding to link the MIKE3-FM and ROM models based on POD were performed in the MATLAB environment.

After calculation of eigenvalues and eigenvectors, the modes that include both spatial and temporal terms of the objective parameter were developed as Equations (5) and (13), respectively. According to Equation (5), the spatial term of a basis function (Φ) is a linear combination of eigenvectors and snapshots and, according to Equation (13), the temporal term of a basis function (a) is an inner product of two vectors Φ to snapshots. It is worth noting that a1, a2, … , and an correspond to the first, second, … , and n eigenvalues, respectively. Therefore, an and Φn represent an equal percentage of temporal and spatial variations, respectively, for each variables U and V. Finally, the root mean square error (RMSE) between the results of ROMs developed by different numbers of the first modes, and the corresponding simulated results of MIKE3-FM, were calculated to assess the robustness of the developed ROM.

RESULTS AND DISCUSSION

MIKE3-FM results

The calibrated and verified MIKE3-FM model by Ranjbar & Zaker (2016, 2018) was used to simulate the hydrodynamics of Gorgan Bay under conditions of real wind from July 1, 2010 to July 1, 2012. Figure 4(a) and 4(b) illustrate the mean seasonal simulated U and V velocities, and circulation pattern, in the surface layer of the bay during the summer and autumn, respectively. In these figures, the background colors show the magnitude of U and V, and the circulation pattern is depicted by arrows. According to Figure 4(a), the water circulation pattern in the surface layer of the bay is counter-clockwise during the summer. The main reason for the emergence of such circulation is the direction of prevailing winds from the west and southwest. Current direction near the coast is almost in the direction of the prevailing wind (west to east), and the magnitude of the current depends on the wind speed. Current speed is higher along the shallower north and south coasts of the bay than in the deeper central part. In addition, in the area where the bay is connected to the Caspian Sea, there is some turbulence affected by water transmission between the Caspian Sea and the bay. Figure 4(b) indicates that a clockwise water circulation pattern is dominant for most of the autumn months. According to this figure, relatively tranquil conditions are found in the middle of the bay and the velocity is higher along the coast.

Figure 4

Simulated U and V velocities during the (a) summer and (b) autumn seasons.

Figure 4

Simulated U and V velocities during the (a) summer and (b) autumn seasons.

Considering the above-mentioned conditions, there are stable and tranquil conditions in the middle of the bay and turbulence at the edges. In the Strait of Ashouradeh, where the bay connects to the Caspian Sea, turbulence is higher than in other areas.

Hydrodynamic ROM

Having 2,920 snapshots for each U and V velocity on 1,937 grids in Gorgan Bay, the establishment of a Hermitian matrix L in dimensions 2,920 × 2,920 was carried out by applying Equation (10), and eigenvalues and eigenvectors for the matrix were calculated. Figure 5 shows the first 100 calculated eigenvalues (among 2,920) and conserved energy of the system (U and V variations in Gorgan Bay) for the first ten eigenvalues (among 2,920), respectively.

Figure 5

Results of (a) the first 100 calculated eigenvalues (among 2,920) for U and V, and (b) conserved energy of the system (U and V variations in Gorgan Bay) for the first ten eigenvalues.

Figure 5

Results of (a) the first 100 calculated eigenvalues (among 2,920) for U and V, and (b) conserved energy of the system (U and V variations in Gorgan Bay) for the first ten eigenvalues.

It is clear from Figure 5(a) that the first few eigenvalues have considerable value compared to the others. Figure 5(b) indicates that the first eigenvalues of U and V conserve about 65% and 59% of the system energy, and values reach approximately 98% and 97% for the first ten eigenvalues (U and V variation in the bay), respectively.

The spatial terms (or space-dependent terms) of modes (Φ) were calculated as a linear combination of eigenvectors and snapshots. As the spatial term of the first basis function (Φ1) corresponds to the first eigenvalues, it represents about 65% and 59% of the spatial variation of U and V in Gorgan Bay, respectively. Φ2 corresponds to the second eigenvalue and conserves about 10% and 9% of the system energy for U and V, respectively. Equation (13) was applied to calculate the temporal variations of modes. According to this equation, the temporal term of a mode (a) is the inner product of two vectors Φ to snapshots and a1, a2, … , and an correspond to the first, second, … , and n eigenvalues, respectively. Therefore, a1, like Φ1, represents about 65% and 59% of temporal variation of U and V, respectively. Figure 6 shows the results of the 1st to 4th (Figure 6(a) and 6(d) for U and V, respectively), 5th to 8th (Figure 6(b) and 6(e) for U and V, respectively), and 100th, 1,000th, 2,000th, and 2,920th (Figure 6(c) and 6(f) for U and V, respectively) temporal terms of the modes, respectively. The first eight temporal terms of basis functions have considerably greater values than the others, and it is clear from Figure 6 that for both U and V, the higher orders of a have less variation so that variation for a200, a1,000, a2,000, and a2,920 approaches zero.

Figure 6

Results of the temporal terms of the ROM developed for simulation of U and V: (a) and (d) for the 1st to 4th temporal terms of U and V, respectively; (b) and (e) for the 5th to 8th temporal terms of U and V, respectively; and (c) and (f) for the 100th, 1,000th, 2,000th, and 2,920th temporal terms of U and V, respectively.

Figure 6

Results of the temporal terms of the ROM developed for simulation of U and V: (a) and (d) for the 1st to 4th temporal terms of U and V, respectively; (b) and (e) for the 5th to 8th temporal terms of U and V, respectively; and (c) and (f) for the 100th, 1,000th, 2,000th, and 2,920th temporal terms of U and V, respectively.

The number of modes considered to construct the ROM is usually determined by the amount of energy conserved through the eigenvalues. Although there is no specific formula to select the number of modes, a threshold of approximately 98% of energy has been suggested by different researchers (Kunisch & Volkwein 1999; Esfahanian & Ashrafi 2009; Ashrafi 2012). However, in this study the first ten modes that conserve approximately 98% and 97% for U and V were selected to develop the ROM. As shown in Figure 2, the selected number of first modes could be increased if the defined criteria for the modeling process are not satisfied (as described at the end of this section).

Having Φ and a, and separately applying Equation (12) for U and V, resulted in two hydrodynamic ROMs that appropriately regenerate U and V in Gorgan Bay, respectively. Since the first few modes (among 2,920) mainly contribute to the physical condition of the system, the hydrodynamic ROMs were developed using 1, 2, and 10 modes. After developing the ROMs, the models were run to regenerate the hydrodynamic variables (U and V) in Gorgan Bay. Figure 7 shows regenerated U using ROMs developed by 1, 2 and 10 modes, and the MIKE3-FM model in the 10th day (snapshot 10) of the simulation period. Also, regenerated V using ROMs developed by 1, 2 and 10 modes, and MIKE3-FM model in the 49th day (snapshot 49) of the simulation period, is illustrated in Figure 8. To have a good insight into the accuracy of the developed ROM, the difference between MIKE3-FM and ROMs are also illustrated in these figures (Figures 7 and 8). Note that the differences (errors) between the developed ROMs and MIKE3-FM in different parts of the bay result from the number of modes used for construction of ROMs. In this regard, greater numbers of the first modes (that result in the conservation of higher percentages of the system energy) should be selected for the development of ROM, if one aims to reduce the error between ROM and MIKE3-FM. This is reflected in Figures 7 and 8, where considerable errors can be seen in different parts of Gorgan Bay (between the ROM developed by the first mode and the MIKE3-FM model). The errors were reduced by the selection of the first two modes for construction of ROM as shown in the figures. By increasing the number of modes to the first ten, the differences between ROM and the MIKE3-FM model was very small so that it could be neglected for practical applications. Therefore, according to Figures 7 and 8, it is clear that the agreement between MIKE3-FM and ROM developed by the first ten modes is excellent. However, because the first ten modes conserve approximately 98% and 97% of system energy for U and V, respectively, the consistency is not perfect, and there are very small differences between them.

Figure 7

Regenerated U using ROM developed by 1, 2, and 10 modes and the MIKE3-FM model in the 10th day (snapshot 10) of the simulation period.

Figure 7

Regenerated U using ROM developed by 1, 2, and 10 modes and the MIKE3-FM model in the 10th day (snapshot 10) of the simulation period.

Figure 8

Regenerated V using ROM developed by 1, 2, and 10 modes and the MIKE3-FM model in the 49th day (snapshot 49) of the simulation period.

Figure 8

Regenerated V using ROM developed by 1, 2, and 10 modes and the MIKE3-FM model in the 49th day (snapshot 49) of the simulation period.

Finally, RMSE between the regenerated U and V by the ROMs and corresponding simulated results by MIKE3-FM was calculated to check the developed ROMs' robustness. Figure 9(a) and 9(b) show the RMSE calculated for a number of developed ROMs in the selective snapshots of the simulation period for U and V, respectively. These figures indicate that the ROMs developed for U and V using ten modes had an order of magnitude of error less than 0.012. This means that by using ten modes the error is less than 0.012. Thus, by selecting the first ten modes, it is possible to regenerate U and V using the developed ROM with an error of order of magnitude less than 0.012. Also, it can be seen from Figure 9(a) and 9(b) that the decreasing trend of RMSE is considerable when one increases the number of modes for construction of the ROMs until the first ten modes. Thereafter, the change in RMSE is not considerable, such that increasing the number of first modes to more than ten does not significantly change ROM accuracy. Therefore, it can be concluded that the first ten modes are the main contributors to the physical condition of the system, such that the ROMs developed by these modes appropriately regenerate U and V in the surface layer of Gorgan Bay.

Figure 9

RMSE calculated for a number of developed ROMs in the selective days of the simulation period for (a) U and (b) V.

Figure 9

RMSE calculated for a number of developed ROMs in the selective days of the simulation period for (a) U and (b) V.

CONCLUSIONS

This study presented a hydrodynamic ROM for regeneration of surface currents in Gorgan Bay, Iran, between July 1, 2010 and July 1, 2012. A link between the MIKE3-FM and POD models was established to calculate the necessary information for developing the ROM (i.e., the spatiotemporal components of the surface currents). Having the spatial and temporal terms, two ROMs for regeneration of surface currents in two directions, x and y, were developed.

The MIKE3-FM results showed that water circulation patterns were counter-clockwise and clockwise in the Gorgan Bay in the summer and autumn seasons, respectively. The simulated results by the MIKE3-FM model also revealed that there were stable and turbulent conditions in the middle and edge areas of the bay. In the Strait of Ashouradeh, where the bay is connected to the Caspian Sea, turbulence was higher than in other areas.

POD results determined that just the first few eigenvalues (out of 2,920 eigenvalues) had considerable value when compared with others. Specifically, it indicated that the first eigenvalues of U and V conserved about 65% and 59% of system energy, respectively. This value reached approximately 98% and 97% for the first ten eigenvalues of U and V, respectively. This suggests that just the first ten eigenvalues were the main contributors to the physical condition of the system (U and V variations in Gorgan Bay). Calculation of the first ten temporal terms of the basis functions for U and V specified that they had considerably greater values than others. Also, it was shown that for both U and V, the higher order temporal terms had less variations and were nearly zero.

Analyzing the results of ROMs revealed that they accurately regenerated the surface currents by application of only the ten first modes (out of 2,920 modes). Comparison of MIKE3-FM and the ROMs developed by the first ten modes also revealed that there were only negligible differences between their results when they simulated and regenerated, respectively, U and V in the bay. RMSE calculated between MIKE3-FM and the ROMs developed for U and V by just the first ten modes resulted in an error of order of magnitude less than 0.012. Thus, by selection of the first ten modes, it was possible to regenerate U and V using the developed ROMs with an RMSE less than 0.012.

CONFLICTS OF INTEREST

The authors declare no conflict of interest.

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