The role of nonhydrostatic dynamics in controlling development of a surface ocean front

Numerical studies of surface ocean fronts forced by inhomogeneous buoyancy loss show nonhydrostatic convective plumes coexisting with baroclinic eddies. The character of the vertical overturning depends sensitively on the treatment of the vertical momentum equation in the model. It is less well known how the frontal evolution over scales of O(10 km) is affected by these dynamics. Here, we compare highly resolved numerical experiments using nonhydrostatic and hydrostatic models and the convective-adjustment parametrization. The impact of nonhydrostatic processes on average cross-frontal transfer is weak compared to the effect of the O(1 km) scale baroclinic motions. For water-mass distribution and formation rate nonhydrostatic dynamics have similar influence to the baroclinic eddies although adequate resolution of the gradients in forcing fluxes is more important. The overall implication is that including nonhydrostatic surface frontal dynamics in ocean general circulation models will have only a minor effect on scales of O(1 km) and greater.     

A comparison of two formulations of barotropic–baroclinic splitting for layered models of ocean circulation

In numerical models of ocean circulation, it is widespread practice to split the fast and slow motions into barotropic and baroclinic subsystems, respectively. In the case of the baroclinic equations, the dependent variables can either be (1) slowly-varying baroclinic quantities, obtained from splitting the original flow variables into barotropic and baroclinic components, or (2) the original unsplit variables, which can vary on both the fast and slow time scales. In the second case, the variables in each layer are adjusted after each (long) baroclinic time step to ensure compatibility with the results produced from the barotropic equations. The second approach can be applied to the layer thickness equation to ensure exact conservation of mass within each layer. In the case of the momentum equations, the second approach amounts to replacing unresolved fast portions of Coriolis and pressure forcing with time averages of well-resolved forcing from the barotropic system. In this study, both approaches for the momentum equations are evaluated, in several test problems, by comparing to analytical solutions or to solutions computed with an unsplit code that uses short time steps. The two methods give very similar results in some simple problems for which analytical solutions are known. However, in some eddying double-gyre simulations, the formulation with unsplit variables requires a significant reduction in the baroclinic time step in order to avoid numerical difficulties that include grid noise and inaccurate representation of the flow field. In contrast, the formulation with split variables does not display such difficulties, and in those same examples it can be used with zero explicit horizontal viscosity. All of these computations employ a two-level time-stepping method that was previously developed by the author.     

Critical Latitude in Tidal Dynamics Using the Kara Sea as an Example

It is well known that, within the linear nonviscous equations of tidal dynamics, the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components unlimitedly increase when approaching the critical latitude. It is also known that the linear equations of tidal dynamics with a constant and specified vertical eddy viscosity indicate the occurrence of significant tidal velocity shears in the near-bottom layer, which are responsible for increasing the baroclinic tidal energy dissipation, the turbulent kinetic energy, and the thickness of the bottom boundary layer. The first circumstance—the growth of the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components—is due to the elimination in the original equations of small terms, which are small everywhere except for the critical latitude zone. The second circumstance—the occurrence of significant tidal velocity shears—is due to the fact that internal tidal waves, which induce the dissipation of the baroclinic tidal energy and the diapycnal diffusion, are either not taken into account or described inadequately. It is suggested that diapycnal diffusion can lead to the degeneration (complete or partial) of tidal velocity shears, with all the ensuing consequences. The aforesaid is confirmed by simulation results obtained using the QUODDY-4 high-resolution three-dimensional finite-element hydrostatic model along the 66.25° E section, which passes in the Kara Sea across the critical latitude.     

Modeling sea dynamics and turbulent zones on high spatial resolution nested grids

The problem of simulating sea dynamics in areas comprising near-shore zones and zones of high turbulence is considered. A mathematical model and the numerical algorithm of its solving are formulated. The model is based on the equations for nonhydrostatic dynamics and includes (k-ε) and (k-ω) parameterization of turbulent processes. The equations of the model are written in a σ-coordinate system. The numerical algorithm for solving the problem is based on the use of implicit schemes owing to the splitting with respect to the physical processes and space coordinates. The model calculations were performed for four nested sea basins with different spatial resolution: the Baltic Sea (3.7-km space resolution), the Gulf of Finland (1.85-km resolution), the Tallinn-Helsinki area (560-m resolution), and Tallinn Bay (93-m resolution). The results of the experiment show that the model well simulates the processes of enhanced turbulent activity in the near-shore zones that affect the local features of the sea characteristics.     

Algorithm of the <Emphasis Type="Italic">k</Emphasis>–ω Turbulence Equations Solution for the Ocean General Circulation Model

The algorithm for splitting k–ω turbulence equations is used to parameterize viscosity and diffusion coefficients in the ocean general circulation model. The k–ω equations are split into stages describing the transport-diffusion and generation-dissipation of the turbulent kinetic energy and frequency function ω. At the generation-dissipation stage, the equations are solved analytically. Calculations of circulation in the North Atlantic–Arctic Ocean for 1948–2009 have been carried out. The experiments demonstrate an adequate reproduction of hydrophysical characteristics and high efficiency of the algorithm. It is shown that considering the climatic annual mean buoyancy frequency in the turbulence equations at the generation-dissipation stage is an important factor in improving the accuracy of simulated fields.     

The generation of non-linear internal waves by a barotropic tide in the area of a solitary bottom elevation

A mathematical model is suggested for calculating current, density, and pressure fields in the area of a solitary bottom rise (seamount). The model is based on a set of non-linear differential equations governing the motion of an inviscid continuously stratified fluid. The algorithm for solving the equations is based on the splitting technique. The model has been used to compute non-linear baroclinic waves generated by a barotropic tide in the seamount area. Translated by Vladimir A. Puchkin.     

Algorithm for non-hydrostatic dynamics in the Regional Oceanic Modeling System

A non-hydrostatic algorithm for the Regional Oceanic Modeling System (ROMS) is proposed. It is based on a decomposition technique for hydrostatic and non-hydrostatic pressure. The algorithm has a pressure-correction scheme with split-explicit time-stepping for baroclinic and barotropic vertical modes with a free surface. The algorithm implementation requires solving a Poisson equation for a non-hydrostatic pressure that has a non-symmetric matrix in discrete form. The efficiency of a different class of solvers and preconditioners were tested. The algorithm is successfully implemented with several examples where non-hydrostatic effects are important. These include standing external gravity waves; strongly nonlinear internal wave generation and transformation; stratified shear instability and its associated mixing; and nonlinear internal tidal generation over a ridge. The corresponding changes in the pre-processing and post-processing infrastructure in the existing hydrostatic ROMS code were performed to implement parallel elliptic solvers and a new set of dynamical equations.     

Nonhydrostatic tidal dynamics in the area of a seamount

The nonhydrostatic boundary problem for an arbitrary three-dimensional domain with a seamount is considered. The problem is integrated into curvilinear boundary-fitted coordinates on a nonuniform grid. In order to identify nonhydrostatic effects the grid is condensed on the slopes of the seamount preserving a coarse resolution in the rest of the domain, where the problem is solved in the hydrostatic approximation. Calculation results for the nonhydrostatic tidal dynamics and hydrology of the Strait of Messina in the area of a seamount are presented.     

三维斜压台风模式I.数值方法

一种多重移动套网格斜压台风模式已被应用于海洋环境数值预报。模式采用坐标系的原始方程组作为控制方程。现在用于国家海洋环境预报中心的模式垂直方向为非等距４层,水平方向为ＡｒａｋａｗａＢ型格式,所采用的差分格式满足动量和能量守恒原理。模式控制方程组分离成平流过程和适应过程二组方程,并根据大气运动不同过程的特性,分别采用不同的时间步长和不同的积分方法。预报和后报结果显示该数值方法不仅可以缩短机时,而且可以得到稳定的预报结果。     

Similarity parameters and a centrifugal convective instability of horizontally inhomogeneous circulations of the Hadley type

The stability of the zonal axisymmetric quasi-geostrophic hydrostatic solution to the equations of atmospheric dynamics that is determined by the horizontal temperature gradient is studied. Time-dependent regions of unstable solutions specified by the Rayleigh number describe ordinary convective (baroclinic) processes and the long-term weak growth of disturbances under the action of the centrifugal forces arising from the Earth’s rotation. Comparison with a centrifugal hydrodynamic instability is made. The spatiotemporal structure of the corresponding geophysical fields is described.     

三维斜压台风模式Ⅰ.数值方法

一种多重移动套网格斜压台风模式已被应用于海洋环境数值预报.模式采用σ坐标系的原始方程组作为控制方程.现在用于国家海洋环境预报中心的模式垂直方向为非等距4层,水平方向为Arakawa B型格式,所采用的差分格式满足动量和能量守恒原理.模式控制方程组分离成平流过程和适应过程二组方程,并根据大气运动不同过程的特性,分别采用不同的时间步长和不同的积分方法.预报和后报结果显示该数值方法不仅可以缩短机时,而且可以得到稳定的预报结果.     

Research on hydrodynamics model test for deepsea open-framed remotely operated vehicle

This paper presents the features of newly designed hydrodynamics test for the scaled model of 4500 m deepsea open-framed remotely operated vehicle(ROV),which is being researched and developed by Shanghai Jiao Tong University.Accurate hydrodynamics coefficients measurement and spatial modeling of ROV are significant for the maneuverability and control algorithm.The scaled model of ROV was constructed by 1:1.6.Hydrodynamics coefficients were measured through VPMM and LAHPMM towing test.And dynamics model was derived as a set of equations, describing nonlinear and coupled 5-DOF spatial motions.Rotation control motion was simulated to verify spatial model proposed.Research and application of hydrodynamics coefficients are expected to enable ROV to overcome uncertainty and disturbances of deepsea environment,and accomplish some more challengeable and practical missions.     

Model of the Baltic Sea’s dynamics with nonhydrostatic effects on a triangular grid

A numerical model of the sea’s thermohydrodynamics with a finite-difference approximation of the equations of the nonhydrostatic dynamics on a grid with a triangular form of the horizontal section of its element is formulated. The slope of the lower side of the grid’s bottom cell is determined by the given linear profile of the bottom relief. Within the shallow-water approach, the dispersion relations of the B and C grids and the developed discrete model are compared; the results of tests for different approximations of the bottom relief for a rectangular basin are given and analyzed. The developed model of the thermohaline dynamics is used for studying the influence of the nonhydrostatic effects on the circulation of the Baltic Sea and a part of Vistula Bay. The comparison of the simulation results obtained according to the version with the full equation of the vertical momentum and to that using the hydrostatic approach shows the influence of the effects of the nonhydrostatic dynamics on the structure of the simulated fields even with small horizontal resolution (the step of the grid is 3.5 km). This is manifested in the strengthening of the field of the vorticity and the increasing of the sea level gradients and the velocities of the horizontal currents, whose growth reaches 1.5 cm/s.     

Implementation of baroclinic terms in a three-dimensional hydrodynamic model and its application to Anzali Port, Iran

Baroclinic terms have been implemented in a three-dimensional fully hydrodynamic model developed by Badiei et al. [2008. A three-dimensional non-hydrostatic boundary fitted model for free surface flows. International Journal for Numerical Methods in Fluids, 56(6), 607-627] modifying its momentum equations to account for density gradients and utilizing the scalar (salinity, temperature, etc.) conservation equation (SCE) and a state equation for the calculation of density. In the solution of advection-diffusion terms of the governing Navier-Stokes equations (NSE) and SCE, a symmetric splitting method was applied to ensure the long-term stability of simulations. Correction terms proposed by Ruddic et al. (1995) were applied to SCE to ensure the conservation of the scalar quantity. In the presence of baroclinic terms, the zero gradient pressure in the vertical direction in the vicinity of surface and bottom boundaries assumed by Badiei et al. [2008. A three-dimensional non-hydrostatic boundary fitted model for free surface flows. International Journal for Numerical Methods in Fluids, 56(6), 607-627] created spurious currents. This problem was solved by assuming a hydrostatic pressure variation at those boundaries. The ability of extended model was validated by comparing its results with an experimental test case. The simulation of hydrodynamic and salt intrusion at Anzali Port located at the southern coasts of Caspian Sea in Iran was carried out by the model with both barotropic and baroclinic modes. The simulated results with baroclinic mode show a better agreement with measured data as compared to the results of barotropic mode that clearly demonstrate the significance of baroclinic terms in the simulation of cyclic intrusion of salt wedge into the Port Basin.     

Simulation of the internal tide in the Strait of Bab el Mandeb (the Red Sea)

The baroclinic tide in the Strait of Bab el Mandeb is simulated by solving a three-dimensional boundary-value problem in boundary-fitted coordinates for the equations of dynamics, density constituents, and turbulence characteristics. Results related to simulations of the barotropic tide in the Red Sea and to the internal tide in the Strait of Bab el Mandeb are presented.     

A reduced-dynamics variational approach for the assimilation of altimeter data into eddy-resolving ocean models

A new method of assimilating sea surface height (SSH) data into ocean models is introduced and tested. Many features observable by satellite altimetry are approximated by the first baroclinic mode over much of the ocean, especially in the lower (but non-equatorial) and mid latitude regions. Based on this dynamical trait, a reduced-dynamics adjoint technique is developed and implemented with a three-dimensional model using vertical normal mode decomposition. To reduce the complexity of the variational data assimilation problem, the adjoint equations are based on a one-active-layer reduced-gravity model, which approximates the first baroclinic mode, as opposed to the full three-dimensional model equations. The reduced dimensionality of the adjoint model leads to lower computational cost than a traditional variational data assimilation algorithm. The technique is applicable to regions of the ocean where the SSH variability is dominated by the first baroclinic mode. The adjustment of the first baroclinic mode model fields dynamically transfers the SSH information to the deep ocean layers. The technique is developed in a modular fashion that can be readily implemented with many three-dimensional ocean models. For this study, the method is tested with the Navy Coastal Ocean Model (NCOM) configured to simulate the Gulf of Mexico.     

Nonlinear inertial oscillations of a multilayer eddy: An analytical solution

Nonlinear axisymmetric oscillations of a warm baroclinic eddy are considered within the framework of an reduced-gravity model of the dynamics of a multilayer ocean. A class of exact analytical solutions describing pure inertial oscillations of an eddy formation is found. The thicknesses of layers in the eddy vary according to a quadratic law, and the horizontal projections of the velocity in the layers depend linearly on the radial coordinate. Owing to a complicated structure of the eddy, weak limitations on the vertical distribution of density, and an explicit form of the solution, the latter can be treated as a generalization of the exact analytical solutions of this form that were previously obtained for homogeneous and baroclinic eddies in the ocean.     

On Instability of Geostrophic Current with Linear Vertical Shear at Length Scales of Interleaving

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability.     

Multilevel hydrothermodynamic model of a deep basin

A hydrothermodynamic model based on the traditional system of differential equations is discussed. The model uses conservative finite-difference schemes based on the methods of identifying the barotropic and baroclinic velocity components and on the complete inversion of the dynamic operator. Test computations for the Black Sea basin have been conducted.Translated by Vladimir A. Puchkin.