Analysis of the differential-difference schemes of equations for the barotropic ocean

Difference schemes constructed for the problems using a barotropic approximation (Poincaré waves, Kelvin waves, Rossby waves, quasi-geostrophic normal modes, forced free oscillations) allow one to obtain exact solutions for a set of discrete equations. It is shown that the better approximation is achieved on thef-plane, using the grid different from gridsB andC. Translated by Vladimir A. Puchkin.     

Investigation of Transformations of the Local Regions of Pollution in the Northwest Part of the Black Sea Caused by the Motion of a Cyclone

By the method of mathematical simulation, we study the evolution of local discharges of pollutants on the northwest shelf of the Black Sea induced by the motion of a cyclone. We use the nonlinear equations of motion of a homogeneous viscous fluid in the hydrostatic approximation and the equation of turbulent diffusion. A cyclonic formation is represented by a moving axisymmetric area of low pressures. The discharges of pollutants are caused by the action of instantaneous sources located on the sea surface. We perform the analysis of dependences of the paths of motion of polluted regions, the periods of their dissipation, and the depths of penetration of pollutants on the bottom topography and the intensity of diffusion processes. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 6, pp. 17–27, November–December, 2005.     

Generation of seiches by moving baric formations

We consider a plane problem of generation of barotropic seiches in the case of motion of a region of disturbances of atmospheric pressure over a bounded basin. The nonlinear system of equations of long waves with regard for the quadratic dependence of bottom friction is solved by the finite-difference method. The calculations are carried out for three basins of variable depth corresponding to rectangular sections of the Black Sea and Sea of Azov. It is shown that the passage of the baric anomaly over the basin is accompanied by the generation of the lowest seiches. The oscillations of fluid are especially intense in the shallow-water zones of the basins. Seiches exhibit a trend to intensification as the velocity of motion of the atmospheric disturbance increases. The dependence on the width of the baric anomaly is not monotonic. In the shelf zone, the amplitude of oscillations of the level can be several times higher that the pure hydrostatic response of the sea surface. In the analyzed basins, the influence of nonlinearity on the seiches is weak. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 6, pp. 3–15, November–December, 2007.     

Capillary-Gravitational Waves of Finite Amplitude on the Surface of a Homogeneous Liquid

The method of multiple scales is used to deduce equations for three nonlinear approximations of the capillary-gravitational disturbances of the free surface of a layer of a homogeneous liquid of constant depth. In these equations, the space-time variations of the wave profile in the expression for the velocity potential on the liquid surface are taken into account. On this basis, we construct asymptotic expansions up to the quantities of the third order of smallness for the velocity potential and elevations of the liquid surface induced by running periodic waves of finite amplitude. Furthermore, we analyze the dependences of the amplitude-phase characteristics of wave disturbances on the surface tension, depth of the liquid, and the length and steepness of waves of the first harmonic. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 25–34, September–October, 2005.     

Thermohydrodynamics of the ocean generation of seiches by moving baric fronts in bounded basins

We consider a plane problem of barotropic seiches generated by a front of atmospheric pressure moving over a bounded basin. A system of nonlinear equations of long waves is solved by the finite-difference method with regard for the bottom friction and Earth's rotation. The numerical analyses are performed for two basins with distributions of depths typical of the Black Sea. It is shown that the passage of a baric front over the basin leads to the generation of lower seiches. The oscillations of level and the corresponding currents are especially intense in the shallow-water zones of the basins. The seiches become more intense as the velocity of transfer of the atmospheric front increases and the width of the front decreases. Earth's rotation leads to the generation of longshore currents and promotes the process of weakening of residual oscillations of the fluid following the passage of the front. The influence of nonlinearity on seiches is small for the analyzed basins. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 2, pp. 3–18, March–April, 2008.     

On the quasi-geostrophic dynamics of a stratified two-component medium: The ocean

The quasi-geostrophic dynamics of a stratified two-component medium (salty sea water) are described by formulating a closed system of equations containing the temperature and salinity among the sought field variables. The corresponding system consists of the conservation laws of two Lagrange invariants, namely, the quasi-geostrophic potential vorticity and some “thermodynamic” invariant playing the role of a passive tracer. The temperature and salinity fields determined from the values of these invariants are separated into the density and density-compensated parts. In this case, the density part participates immediately in the dynamics, while the density-compensated part is simply transferred by the geostrophic velocity filed with no contribution to the density field. The system thus formulated is used to describe a number of specific features in the dynamics of thermohaline disturbances in zonal geostrophic flows. These features include the sharp-ening of the spatial gradients of the density-compensated distributions in shear currents and breaking of the initial temperature disturbance into two (density and density-compensated) wave packets propagating with different velocities.     

Spectral barotropic model for the prediction of synoptic currents in the open part of the ocean

We consider a quasigeostrophic spectral model used for the prediction of synoptic currents in the barotropic ocean. The spectral method is based on the expansion of the current function in a double series in cosines. An algorithm of numerical evaluation of the nonlinear term in the equation of potential eddy is described in detail. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 4, pp. 62–69, July–August, 2006.     

Investigation of the sensitivity of the Mellor–Yamada parametrization to the choice of finite-difference analogs in a numerical three-dimensional model of the operative prediction of currents in the black sea

Within the framework of the Mellor–Yamada approach, we realize a numerical scheme for the calculation of the coefficients of turbulent viscosity and diffusion in the z -system of coordinates for the three-dimensional model of operative prediction of currents in the Black Sea. Some discrete analogs of the equations for turbulent kinetic energy and turbulence macroscale are studied. Their high sensitivity to the choice of finite-difference approximations is demonstrated. On the basis of the comparison of the results of prognostic experiments with the data of observations, we choose the best approximation of the term used to describe the generation pf turbulence energy.     

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。     

The effect of large-scale bottom irregularities on the dynamics of Rossby waves in the ocean

This paper discusses, in terms of the geometrical optics approximation, how large-scale bottom irregularities influence the propagation of Rossby waves in the ocean. To describe the major peculiarities of the phenomenon, a two-layer model is applied, with the depth of the upper layer being considerably smaller than that of the lower layer. However, even with the bottom topography being allowed for, the wave motion is described by two Rossby wave modes, namely, a barotropic mode and a baroclinic mode. It is demonstrated that barotropic mode transformation caused by large irregularities of the sea-floor may lead to wave interaction, resulting in their anomalous distribution. Translated by Vladimir A. Puchkin.     

Current structure and volume transport of the Soya Warm Current in summer

ADCP, CTD and XBT observations were conducted to investigate the current structure and temperature, salinity and density distributions in the Soya Warm Current (SWC) in August, 1998 and July, 2000. The ADCP observations clearly revealed the SWC along the Hokkaido coast, with a width of 30–35 km and an axis of maximum speed of 1.0 to 1.3 ms⁻¹, located at 20–25 km from the coast. The current speed gradually increased from the coast to a maximum and steeply decreased in the offshore direction. The SWC consisted of both barotropic and baroclinic components, and the existence of the baroclinic component was confirmed by both the density front near the current axis and vertical shear of the alongshore current. The baroclinic component strengthened the barotropic component in the upper layer near the axis of the SWC. The volume transport of the SWC was 1.2–1.3 SV in August, 1998 and about 1.5 SV and July, 2000, respectively. Of the total transport, 13 to 15% was taken up by the baroclinic component. A weak southeastward current was found off the SWC. It had barotropic characteristics, and is surmised to be a part of the East Sakhalin Current.     

Focusing of internal waves generated by the nonuniform motion of a region of atmospheric pressures

In the linear statement of the problem, by using the long-wave approximation, we study the possibility of generation of significant disturbances in a continuously stratified fluid by the nonuniform motion of a region of atmospheric pressures. We consider the case where the disturbances of the medium propagating from a part of the trajectory of motion of the generator simultaneously arrive at a selected point of the space (focus). The amplitude of disturbances in the focus is evaluated for an arbitrary motion of the region of pressures, which may lead to the analyzed effect. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 22–29, September–October, 2006.     

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.     

Numerical experiment on the computation of the equatorial circulation on the basis of a conservative model

A finite-difference system of motion equations for a baroclinic ocean with an irregular bottom is developed on the basis of the box method and the method of indefinite coefficients. The numerical model is shown to possess the property of conserving momentum, mass, and energy. In the case of barotropic motion, the finite-difference scheme retains potential enstrophy. A numerical experiment on the adaptation of hydrophysical fields in the Equatorial Atlantic in summer was carried out within the framework of the model developed. We managed to obtain a series of features of the equatorial circulation when the coefficients of turbulent exchange and diffusion were small enough.Translated by Mikhail M. Trufanov.     

海洋内波的控制方程组

从海水质量守恒律、盐量守恒律、动量守恒律、熵变化律和海水状态律出发，给出了低频变密度海洋运动的原始控制方程和它的各类实用形式。在推导的过程中着重明确控制方程各种近似形式的数学物理意义，这使我们在使用这些方程时更严格、更准确。     

Hadley and Rossby regimes in a simple model of convection of a rotating fluid

This paper analyzes the properties of solutions to the equations describing the motion of a stratified fluid in the class of velocity and temperature fields linear in coordinates. For an ideal fluid, these equations, on the one hand, are exact for the corresponding hydrodynamic problem and, on the other hand, are identical to the equations of motion for a heavy top. In a conservative case, the equations of motion of a top share common solutions with the equations of geophysical fluid dynamics and reproduce motions similar to those existing in the theory of the large-scale atmospheric circulation. This study considers the effects of viscosity and heat conduction in the fluid, which are, in a sense, similar to the effect of friction in the case of a top. The influence of deflections of the vectors of gravity and external rotation from their standard directions for a plane-parallel atmosphere is also considered. The regimes of motions that are described by the starting equations and approximations commonly used to model the atmospheric general circulation (the quasi-geostrophic approximation) are analyzed. It is shown that these equations correctly describe the Hadley and Rossby circulation regimes and transitions between them that are observed in numerical and laboratory experiments. Particular attention is given to the consistency between different regimes of the exact equations and their quasi-geostrophic approximations, which is manifested for small Rossby numbers and is generally absent for large Rossby numbers. The asymptotic behaviors of the curves of transition between the Hadley and Rossby regimes under the conditions of breaking the external symmetry of flows are obtained. These asymptotics explain the corresponding transition boundaries for the regimes observed in the known experiments in annuluses.     

The asymptotic theory of the generation of long-wave structures in critical layers of weakly dissipative jet flows

The mechanisms of the large-scale vortex structures formation in zonal jet flows (atmospheric blockings, cyclonic, and anticyclonic vortices) is investigated. Nonlinear perturbations formed during the onset of barotropic instability of a long-wave mode in weakly-dissipative and weakly supercritical jet flows with a symmetric velocity profile are considered in the β-plane approximation. This analysis is performed within the framework of the asymptotic theory based on the concept of a nonlinear critical layer. The equations describing the interaction of a wave with vorticity perturbations in a critical layer are derived. The regimes of a quasi-stationary and nonstationary nonlinear critical layer are considered separately. Combined equations of evolution covering the principle regimes of instability development are proposed. The existence of autowave-type structures characterized by a balance between the energy receipt to the wave and its dissipation are obtained within the framework of a numerical simulation. The dependence of the parameters of generated autowave structures on the shape of the zonal jet profile and the flow supercriticality level is studied.     

Balance of Volume Transports between Horizontal Circulation and Meridional Overturn in the North Pacific Subarctic Region

Horizontal and meridional volume transports on timescales from intra-seasonal to interannual in the North Pacific subarctic region were investigated using a reanalysis dataset for 1993–2001 that was constructed from an assimilation of the TOPEX altimeter and in situ data into an eddy-permitting North Pacific ocean general circulation model. The barotropic flow is excited along east of the Emperor Seamounts by the western intensification dynamics. The volume transport of this flow compensates for that across the interior region east of the Seamounts below the summit depth of the Seamounts. The Oyashio, which is also considered as a compensation flow for the transport in the whole interior region, includes baroclinic as well as barotropic components. Baroclinic transports in the whole interior region exceed those in the western boundary region in the upper (200–1000 m) and lower (2000–5000 m) layers, and the total transport is northward (southward) in the upper (lower) layer. These excesses of the baroclinic transport are balanced by a vertical transport of the meridional overturn. The meridional overturn has a complementary relation to the basin-scale baroclinic circulation in the North Pacific subactic region. This revised version was published online in July 2006 with corrections to the Cover Date.     

The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model

The purpose of this study is to find a combination of optimal numerical algorithms for time-stepping and mode-splitting suitable for a high-resolution, free-surface, terrain-following coordinate oceanic model. Due to mathematical feedback between the baroclinic momentum and tracer equations and, similarly, between the barotropic momentum and continuity equations, it is advantageous to treat both modes so that, after a time step for the momentum equation, the computed velocities participate immediately in the computation of tracers and continuity, and vice versa, rather than advancing all equations for one time step simultaneously. This leads to a new family of time-stepping algorithms that combine forward–backward feedback with the best known synchronous algorithms, allowing an increased time step due to the enhanced internal stability without sacrificing its accuracy. Based on these algorithms we design a split-explicit hydrodynamic kernel for a realistic oceanic model, which addresses multiple numerical issues associated with mode splitting. This kernel utilizes consistent temporal averaging of the barotropic mode via a specially designed filter function to guarantee both exact conservation and constancy preservation properties for tracers and yields more accurate (up to second-order), resolved barotropic processes, while preventing aliasing of unresolved barotropic signals into the slow baroclinic motions. It has a more accurate mode-splitting due to redefined barotropic pressure-gradient terms to account for the local variations in density field, while maintaining the computational efficiency of a split model. It is naturally compatible with a variety of centered and upstream-biased high-order advection algorithms, and helps to mitigate computational cost of expensive physical parameterization of mixing processes and submodels.     

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.