Two-layer Ocean Research Articles

Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow

The results of a theoretical study on Benjamin–Ono (BO) soliton evolution are presented in a simple model of a two-layer ocean with a shear flow and viscosity. The upper layer is assumed to move with a constant speed relative to the lower layer with a tangential discontinuity in the flow profile. It is shown that in the long-wave approximation, such a model can be appropriate. If the flow is supercritical, i.e., its speed (U) exceeds the speed of long linear waves (c1), then BO solitons experience “explosive-type” enhancement due to viscosity, such that their amplitudes increase to infinity in a finite time. In the subcritical regime, when U<c1, BO solitons experience very slow decay due to viscosity. Soliton amplitude decays with time as A∼t−1/2 or A∼t−1/3, depending on whether both layers are weakly viscous (the former case) or only the lower layer is viscous (the latter case). Estimates of "explosion time" are presented for real oceanic parameters.

Two-Layer Ocean Circulation Model with Variational Control of Turbulent Viscosity Coefficient

Two-Layer Ocean Circulation Model with Variational Control of Turbulent Viscosity Coefficient

Two-Layer Ocean Circulation Model with Variational Control of Turbulent Viscosity Coefficient

The development of a variational method for solving the problem of quasi-geostrophic dynamics in a two-layer periodic channel is considered. The development of the method is as follows. First, the formulation of the variational problem is generalized: the turbulent exchange coefficient of a quasi-geostrophic potential vorticity (QGPV) is included in the control vector. Secondly, the solution area more accurately describes the size of the Antarctic Circumpolar Current (ACC). Using the selection of linear meridional transport and the expansion of the solution in a Fourier series, the problem is reduced to a nonlinear system of ordinary differential equations (ODEs) in time. The doubly connected domain leads to the fact that the solution of the ODE must satisfy an additional stationary relation that determines the transport of the ACC. The variational algorithm is reduced to solving a system of forward and adjoint equations minimizing the mean squared error of the stationary relation. The QGPV turbulent exchange coefficient is determined in the process of solving the optimal problem. The numerical runs are carried out for a periodic channel simulating the water area of the ACC in the Southern Ocean. The characteristics of stationary current regimes are studied for different values of the model parameters. Typical is a sinusoidal circulation in both layers with a linear transfer with the wind, depending on the bottom topography. In some cases, under the sinusoidal, in the lower layer, a cellular circulation is formed, and sometimes an undercurrent occurs. In this case, the solution of the optimal problem is characterized by a low value of the turbulent viscosity coefficient and a low transport in the lower layer.

Variational Method for Solving the Quasi-Geostrophic Circulation Problem in a Two-Layer Ocean

Variational Method for Solving the Quasi-Geostrophic Circulation Problem in a Two-Layer Ocean

Scattering of flexural-gravity waves due to a crack in a floating ice sheet in a two-layer fluid in the context of blocking dynamics

The influence of wave blocking on the scattering of a flexural-gravity wave by a linear crack in a thin ice cover resting over a two-layer ocean having a rigid flat seafloor is investigated. The wave dispersion curve reveals the existence of multiple propagating wave modes within the blocking frequencies, either in the surface or interface mode. The Sommerfeld radiation condition depends on multiple propagating wave modes within the blocking frequency and contributes to wave energy propagation. The solution process involves the appropriate transition of wave modes within the blocking frequencies, which is obtained with the help of the dispersion curve. The reflection and transmission coefficients are generalized in the case of multiple propagating wave modes, and the associated energy balance relation is derived using Green's integral theorem. The scattering matrix is generated to describe all the possible transmitted and incident wave modes. The role of lateral compressive force and the density ratio on the scattering process, ice deflection, and interface elevation are shown graphically. This study reveals the occurrence of removable and jump discontinuities in the reflection and transmission coefficients at the saddle point as well as blocking frequencies and at the frequency for which incident wave mode changes. Irregular plate deflection and interface elevation patterns are found due to the superposition of multiple propagating wave modes within the primary and secondary blocking frequencies.

Estimate of energy loss from internal solitary waves breaking on slopes

Abstract. Internal solitary waves (ISWs) emerge in the ocean and seas in various forms and break on the shelf zones in a variety of ways. This results in intensive mixing that affects processes such as biological productivity and sediment transport. As ISWs of depression propagate in a two-layer ocean, from the deep part onto a shelf, two mechanisms are significant: (1) the breaking of internal waves over bottom topography when fluid velocities exceed the wave phase speed that causes overturning of the rear face of the wave, and (2) the changing of polarity at the turning point where the depths of the upper and lower layers are equal. We assume that the parameters that describe the process of the interaction of ISWs in a two-layer fluid with an idealized shelf-slope topography are (1) the nondimensional wave amplitude, normalized on the upper-layer thickness; (2) the ratio of the height of the bottom layer on the shelf to the incident wave amplitude; and (3) the angle of the bottom inclination. Based on a proposed three-dimensional classification diagram, four types of wave propagation over the slopes are distinguished: the ISW propagates over the slope without changing polarity and wave breaking, the ISW changes polarity over the slope without wave breaking, the ISW breaks over the slope without changing polarity, and the ISW both breaks and changes polarity over the slope. The energy loss during ISW transformation over slopes with various angles was estimated using the results of 85 numerical experiments. “Hot spots” of high levels of energy loss were highlighted for an idealized bottom configuration that mimics the continental shelf in the Lufeng region in the South China Sea.

Stability analysis of the boundary value problem modelling a two-layer ocean

<p style='text-indent:20px;'>We study boundedness of solutions to a linear boundary value problem (BVP) modelling a two-layer ocean with a uniform eddy viscosity in the lower layer and variable eddy viscosity in the upper layer. We analyse bounds of solutions to the given problem on the examples of different eddy viscosity profiles in the case of their parameter dependence.</p>

Surface gravity wave interaction with a partial porous breakwater in a two-layer ocean having bottom undulations

Linear surface gravity wave scattering by a partial porous breakwater is studied in a two-layer ocean having an interface with bottom undulations. The partial breakwaters, namely surface-piercing breakwater (SPB) and bottom-standing breakwater (BSB), are considered. The continuity of fluid pressure and mass flux is applied at interface boundaries, while mass conserving conditions are employed to account for slope discontinuity in seabed geometry. Porous breakwaters' reflective and dissipative characteristics are studied under wave-wave interaction. The study reveals that when a mound bottom is present, more incident waves in interface mode are transmitted through BSB, while more waves in surface mode are dissipated. More incident waves in surface mode are transmitted through SPB, while more waves in interface mode are dissipated. In contrast, more transmission occurs in surface mode through both BSB and SPB for a slope-shelf bottom. However, the breakwater dissipates a substantial amount of incident-wave energy in interface mode. Further, a considerable reduction in wave transmission occurs due to higher reflection by the interface-piercing porous breakwaters, and the breakwaters experience high wave loads. Relatively, porous breakwater dissipates less wave energy in shallow depth. The study reveals that the dead water effect is weakened by the porous breakwater.

Hydrodynamic Forces on a Submerged Circular Cylinder in Two-layer Fluid with an Ice-cover

We consider problems based on linear water wave theory concerning the interaction of wave with horizontal circular cylinder submerged in two-layer ocean consisting of a upper layer of finite depth bounded above by an ice-cover and below by an infinite layer of fluid of greater density, the ice-cover being modelled as an elastic plate of very small thickness. Using the method of multipoles, we formulate the problems of hydrodynamic forces on a submerged cylinder in either the upper or the lower layer. The vertical and horizontal forces on the circular cylinder are obtained and depicted graphically against the wave number for various values of flexural rigidity of ice-cover to show the effect of the presence of ice-cover on these quantities. Also when the flexural rigidity and surface density of the ice-cover are taken to be zero, the ice-cover tends to a free-surface. Then all the forces are the same as in the case of two-layer fluid with free surface.

Perturbation Analysis for the Surface Deflection Angle of Ekman-Type Flows with Variable Eddy Viscosity

In the setting of steady wind-driven oceanic currents in a non-equatorial two-layer ocean with constant eddy viscosity in the lower layer and a general continuous eddy viscosity profile in the upper layer, we derive a formula for the change of the surface deflection angle due to a perturbation (in the upper layer) of a given eddy viscosity profile for which the solution to the problem is known. This extends and generalizes previous results for perturbations of a constant eddy viscosity. The perturbation analysis is performed using a recent reformulation of the linear second-order governing equations for Ekman-type flows as a nonlinear first-order Riccati-type equation.

Wave interaction with an array of porous walls in a two-layer ocean of varying bottom topography

Oblique wave scattering by a breakwater consisting of an array of thin porous walls in a two-layer ocean with varying bottom topography is investigated by using linear wave theory. Further, wave trapping is studied by considering an impermeable seawall. The entire bottom profile is assumed to be a shelf-type comprising a varying bottom of finite length and two uniform bottoms of semi-infinite lengths. Porous walls extending from bottom to free-surface are assumed at a lower water-depth level (lee side). Impingement of waves, which approach from a deeper depth level (seaside), on walls is considered. The Fourier method or method of eigenfunction expansion is applied for the uniform bottom, whereas an approximation technique, called a mild-slope equation, is used for the varying bottom. Solutions from these two methods are matched at interfaces under physical conditions. An explicit solution is derived in the form of a system of algebraic equations. The effect of several bottom configurations on wave interaction with porous walls is analyzed. Reflection and transmission coefficients and wave force on porous walls and seawall are analyzed for the parameters related to waves, bottom, and porous medium. The study reveals that porous walls are very effective in reducing wave transmission for bottom profiles dominated by waves. Wave forces on porous walls are less in the two-layer ocean than in a homogeneous ocean. In wave trapping, seawall attains higher wave-force for incoming surface waves than for incoming interfacial waves. The ratio of water-depth levels and wave incident angles are also analyzed for the mitigation of wave forces. The force on seawall increases with the increased depth ratio in surface wave incidence, while the force decreases with the increased depth ratio in interface wave incidence. The findings may be useful for coastal engineers to understand wave scattering by multiple porous structures in the stratified ocean.

Долгопериодная изменчивость океанической циркуляции при различной интенсивности ветрового воздействия

Within the framework of a numerical model of a two-layer ocean with the depth of layers corresponding to average oceanic conditions, the evolution of large-scale circulation under the action of an external vorticity flow of different intensity with constant dissipation parameters is studied. The characteristic time scales of long-period oscillations of current energy at different values of wind impact, jet flow parameters, and the time-average level of total energy in the mode of long-period oscillations at different wind impact intensity are analyzed. The stability of the long-period oscillation regime under various initial conditions of the problem is discussed.

The deflection angle between a wind-forced surface current and the overlying wind in an ocean with vertically varying eddy viscosity

The angle between the wind stress that overlies the ocean and the resulting current at the ocean surface is calculated for a two-layer ocean with uniform eddy viscosity in the lower layer and for several assumed eddy viscosity profiles in the upper layer. The calculation of the deflection angle is greatly simplified by transforming the linear, second-order, vertical structure equation to its associated nonlinear, first-order, Riccati equation. The transformation to a Riccati equation can be used as an alternate numerical scheme, but its main advantage is that it yields analytic expressions for several eddy viscosity profiles.

Steady flow in a frictionless homogenous ocean

A mathematical model is developed to study the free (frictionless) steady horizontal flow which can occur in a homogeneous ocean of constant depth. The flow satisfies the dynamic constraint that the vertical component of absolute vorticity is constant along a streamline. The conclusion is reached that in an enclosed ocean a free steady circulation cannot have any slow broad eastward currents. The eastward currents must occur as narrow streams of high velocity and high relative vorticity. Intensified currents are present along the eastern and western coasts. The theory which is developed for the homogeneous ocean of constant depth can be applied to the two-layer ocean if the horizontal divergence of flow is negligible. If the horizontal divergence is not negligible, then the intensification of poleward currents is more pronounced and that of equatorward currents less pronounced as compared with the homogeneous ocean.

Gravity wave interaction with a porous breakwater in a two-layer ocean of varying depth

Oblique gravity wave interaction with a thick porous breakwater is analyzed in a two-layer ocean having bottom undulation. The plane-wave solution in water of constant depth is matched with the solution of mild-slope equation in the region of varying depth. The breakwater is assumed to be extended from free-surface to seabed. Two different physical aspects such as (i) wave scattering and (ii) wave trapping by porous breakwater near an impermeable wall are investigated. Reflection and transmission coefficients due to interaction of surface and interfacial waves with the breakwater are analyzed for parameters associated with waves, breakwater and seabed profiles. In this study, it is found that porous breakwater reduces transmission considerably in surface and interfacial waves for bottoms over which more wave transmission occurs at lee side. Wave reflection is found to be zero remarkably at some incident angles, depth ratios and bottom topographies. Distance between wall and breakwater is optimized to attain less force on wall The present study will enhance the understanding of scattering and trapping of waves from a porous breakwater under wave–wave interaction in a two-layer ocean of continental shelf. The findings are likely to be useful for creation of tranquility zones and protection of coasts.

ДОЛГОПЕРИОДНАЯ ИЗМЕНЧИВОСТЬ СИСТЕМЫ КРУПНОМАСШТАБНОЙ ЦИРКУЛЯЦИИ ОКЕАНА И МЕЗОМАСШТАБНЫХ ВИХРЕЙ КАК ЯВЛЕНИЕ САМООРГАНИЗАЦИИ

Within the framework of the numerical model of a two-layer ocean with the depth of layers corresponding to the average ocean conditions, the evolution of large-scale circulation under the action of an external stationary vorticity flow under constant dissipation parameters is studied, the time scales of long-period oscillations of the energy of flows are analyzed. The temporal variability of the enstrophy spectrum of the system of flows is considered, the connection of oscillations of the total energy of large-scale circulation with oscillations of the position of the center of mass of the enstrophy spectrum of the system is revealed.

Retrieval of Internal Solitary Wave Amplitude in Shallow Water by Tandem Spaceborne SAR

The accurate estimation of the upper layer thickness in a two-layer ocean is a crucial step in the retrieval of internal solitary wave (ISW) amplitude from synthetic aperture radar (SAR) data. In this paper, we present a method to derive the upper layer thickness and the consequent ISW amplitude by combining two consecutive SAR images with the extended Korteweg-de Vries (eKdV) equation. An ISW case observed twice by the Chinese C-band SAR GaoFen-3 (GF-3) and the German X-band SAR TerraSAR-X (TS-X) with a temporal interval of approximately 11 min in shallow water to the southeast of Hainan Island in the northwestern South China Sea was used to demonstrate the applicability of the method. Using the in situ measurements of temperature and salinity near the observed ISW, the proposed method yielded an ISW amplitude of −4.52 m, in close proximity to −5.66 ± 1.24 m derived by applying the classic Korteweg–de Vries (KdV) equation based on the continuously stratified theory. Moreover, the climatological dataset of the World Ocean Atlas 2013 (WOA13) was also used with the proposed method in the Hainan case, and the results showed that the method can still provide a reasonable estimate of ISW amplitude in shallow water even when in situ oceanic stratification measurements are absent. The application of our method to derive the ISW amplitude from consecutive SAR images seems highly promising with the increasing emergence of tandem satellites in orbits.

Some New Aspects of the Joint Effect of Rotation and Topography on Internal Solitary Waves

AbstractUsing a recently developed asymptotic theory of internal solitary wave propagation over a sloping bottom in a rotating ocean, some new qualitative and quantitative features of this process are analyzed for internal waves in a two-layer ocean. The interplay between different singularities—terminal damping due to radiation and disappearing quadratic nonlinearity, and reaching an “internal beach” (e.g., zero lower-layer depth)—is discussed. Examples of the adiabatic evolution of a single solitary wave over a uniformly sloping bottom under realistic conditions are considered in more detail and compared with numerical solutions of the variable-coefficient, rotation-modified Korteweg–de Vries (rKdV) equation.

АНАЛИЗ ВРЕМЕННЫХ МАСШТАБОВ ДОЛГОПЕРИОДНОЙ ИЗМЕНЧИВОСТИ ВЕТРОВОЙ ОКЕАНИЧЕСКОЙ ЦИРКУЛЯЦИИ В ДВУХСЛОЙНОМ ОКЕАНЕ. ЧАСТЬ 2. СПЕКТРАЛЬНЫЙ АНАЛИЗ

АНАЛИЗ ВРЕМЕННЫХ МАСШТАБОВ ДОЛГОПЕРИОДНОЙ ИЗМЕНЧИВОСТИ ВЕТРОВОЙ ОКЕАНИЧЕСКОЙ ЦИРКУЛЯЦИИ В ДВУХСЛОЙНОМ ОКЕАНЕ. ЧАСТЬ 2. СПЕКТРАЛЬНЫЙ АНАЛИЗ

A Theoretical Model of Long Rossby Waves in the Southern Ocean and Their Interaction with Bottom Topography

An analytical model of long Rossby waves is developed for a continuously-stratified, planetary geostrophic ocean in the presence of arbitrary bottom topography under the assumption that the potential vorticity is a linear function of buoyancy. The remaining dynamics are controlled by equations for material conservation of buoyancy along the sea surface and the sea floor. The mean, steady-state surface circulation follows characteristics that are intermediate to f and f / H contours, where f is the Coriolis parameter and H is the ocean depth; for realistic stratification and weak bottom currents, these characteristics are mostly zonal with weak deflections over the major topographic features. Equations are derived for linear long Rossby waves about this mean state. These are qualitatively similar to the long Rossby wave equations for a two-layer ocean, linearised about a state of rest, except that the surface characteristics in the wave equation, which dominate the propagation, follow precisely the same path as the mean surface flow. In addition to this topographic steering, it is shown that a weighted integral of the Rossby propagation term vanishes over any area enclosed by an f / H contour, which has been shown in the two-layer model to lead to Rossby waves “jumping” across the f / H contour. Finally, a nonlinear Rossby wave equation is derived as a specialisation of the result previously obtained by Rick Salmon. This consists of intrinsic westward propagation at the classical long Rossby speed, modified to account for the finite ocean depth, and a Doppler shift by the depth-mean flow. The latter dominates within the Antarctic Circumpolar Current, consistent with observed eastward propagation of sea surface height anomalies.