Water exchange in fjords induced by tidally generated internal lee waves

Tidal processes are examined that control the water exchange between two basins of the Trondheimsfjord through a narrow channel with sills. For this purpose, a non-hydrostatic numerical model based on the laterally averaged Reynolds equations in the Boussinesq approximation was developed. The model takes into account the real vertical fluid stratification, variable bottom topography and variable cross-section of the fjord. Numerical experiments were performed to investigate tidally generated internal waves and their influence on the water exchange.The model produces both baroclinic tides and tidally generated lee waves. It was found that, for the Skarnsund strait which connects the Middle Fjord and the Beitstadfjord, the internal tides generated over the Skarnsund sills are very weak. Their amplitudes do not exceed 1 m.The intense short internal waves, which are identified as unsteady lee waves, comprise the basic input of the total internal wave field. These waves are generated by tidal currents at sill breaks, are trapped by topography in the generation area and grow by continuing feedback into large-amplitude waves. As the tidal flow slackens, they move upstream as freely propagating waves.As essentially nonlinear responses, the lee waves cause a nonlinear water transport. The detailed analysis of the residual currents produced by unsteady lee waves (which are propagating in both directions from the Scarnsund sills) has shown, in particular, that the residual currents can reach values as high as 0.27 m s⁻¹.It was also found that such currents exert a considerable effect on the water exchange through the Skarnsund strait between the adjacent basins. This mechanism can play an important role in water renewal and formation of the Beitasdfjord waters.     

Determination of parameters of low-frequency internal waves in the coastal zone of a fringing sea using field measurements and basing on the nonlinear theory

A method is presented to determine the quadratic nonlinearity parameter and amplitude of low-frequency internal gravity waves in the coastal zone of a fringing sea, based upon their propagation rate dependence on local value of pycnocline vertical displacement produced by the waves. To test the method, the internal wave field observations in the coastal zone of the Sea of Japan are used. The testing results show that the internal wave parameters calculated using the proposed method and the experimental data are in a good agreement with those calculated from theoretical formulas.     

两方程湍流模型对内背风波数值模拟的影响

为了深入理解非静力近似下的波-湍相互作用问题,本研究在σ坐标的海洋环境研究和预报模型(MERF)中引入常用的Mellor-Yamada两方程湍混合参数化方案(MY2.5),评估垂向湍混合对小尺度背风波传播过程的影响.瞬时状态场的模拟结果表明,无论是否为非静力近似条件,上述湍参数化方案的引入都会减弱背风波传播的模拟效果.从时间平均场的试验结果来看,垂向湍混合过程会显著减小非静力近似和静力近似之间的差异.此外,能量收支分析的诊断结果表明,MY2.5方案会显著抑制陆坡地形下的背风波传播过程,进而将更多的潮能转化到不可逆的湍混合过程中.     

Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance

Atmospheric cyclones with strong winds significantly impact ocean circulation, regional sea surface temperature, and deep water formation across the global oceans. Thus they are expected to play a key role in a variety of energy transport mechanisms. Even though wind-generated internal gravity waves are thought to contribute significantly to the energy balance of the deep ocean, their excitation mechanisms are only partly understood.The present study investigates the generation of internal gravity waves during a geostrophic adjustment process in a Boussinesq model with axisymmetric geometry. The atmospheric disturbance is set by an idealized pulse of cyclonic wind stress with a Rankine vortex structure. Strength, radius and duration of the forcing are varied. The effect upon wave generation of stratification with variable mixed-layer depth is also examined.Results indicate that internal gravity waves are generated after approximately one inertial period. The outward radial energy flux is dominated by waves having structure close to vertical mode-1 and with frequency close to the inertial frequency. Less energetic higher mode waves are observed to be generated close to the sea floor underneath the storm. The total radiated energy corresponds to approximately 0.02% of the wind input. Deeper mixed-layer conditions as well as weaker stratification reduce this fraction.The low energy transfer rates suggest that other processes that drive vertical motion like surface heat fluxes, turbulent motion, mixed region collapse and storm translation are essential for significant energy extraction by internal gravity waves to occur.     

On intense internal waves in the coastal zone of the Peter the Great Bay (the Sea of Japan)

Spatiotemporal variability of the internal gravity waves in the Sea of Japan shelf zone (in the Peter the Great Bay) is analyzed basing on the results of the experimental data processing. It is shown that, as for autumn water structure, the tidal internal waves moving towards the coast are transformed due to the non-linear effects and may reach significant amplitudes (of about 10 meters). Such waves are usually referred to as strongly non-linear. The main characteristics of such intense waves were estimated. An example of the disintegration of solitary temperature depression into the package of short-period intense internal waves is given.     

A unified linear theory of wavelike perturbations under general ocean conditions

A linearized instability analysis model with five unknowns was proposed to describe disturbance motions under general oceanic background conditions, including large-scale current shear, density stratification, frontal zone, and arbitrary topography. A unified linear theory of wavelike perturbations for surface gravity waves, internal gravity waves and inertial gravity waves was derived for the adiabatic case, and the solution was then found using Fourier integrals. In this theory, we discarded the assumptions widely accepted in the literature concerning derivations of wave motions such as the irrotationality assumption for surface gravity waves, the rigid-lid approximation for internal gravity waves, and the long-wave approximation for inertial gravity waves. Analytical solutions based on this theory indicate that the complex dispersion relationships between frequency and wave-number describing the propagation and development of the three types of wavelike perturbation motions include three components: complex dispersion relationships at the sea surface; vertical invariance of the complex frequency; and expressions of the vertical wave-number (phase). Classical results of both surface waves and internal waves were reproduced from the unified theory under idealized conditions. The unified wave theory can be applied in the dynamical explanation of the generation and propagation properties of internal waves that are visible in the satellite SAR images in the southern part of the China Seas. It can also serve as the theoretical basis for both a numerical internal-wave model and analytical estimation of the ocean fluxes transported by wavelike perturbations.     

Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.     

Internal gravity waves in a stably stratified boundary layer

Often, a combination of waves and turbulence is present in the stably stratified atmospheric boundary layer. The presence of waves manifest itself in the vertical profiles of variances of fluctuations and in low-frequency contributions to the power spectra. In this paper we study internal waves by means of a linear stability analysis of the mean profiles in a stably stratified boundary layer and compare the results with observed vertical variance profiles of fluctuating wind and temperature along a 200 m mast. The linear stability analysis shows that the observed mean flow is unstable for disturbances in a certain frequency and wavenumber domain. These disturbances are expected to the detectable in the measurements. It is shown that indeed the calculated unstable frequencies are present in the observed spectra. Furthermore, the shape of the measured vertical variance profiles, which increase with height, is explained well by the calculated vertical structure of the amplitude of unstable Kelvin-Helmholtz waves, confirming the contribution of waves to the variances. Because turbulence and waves have quite distinct transport properties, estimates of diffusion from measurements of variances would strongly overestimate this diffusion. Therefore it is important to distinguish between them.     

Internal tides in the coastal zone of the Sea of Japan in autumn

The results are presented of hydrological studies in the coastal zone of the Sea of Japan in autumn in different years. It is revealed that the typical density stratification of water is formed there in autumn. The amplitudes are estimated of regular (with the periodicity of tidal harmonic M₂) vertical displacements of isotherms in the intermediate layer and maximum values of flow velocity in the surface and bottom layers. It is demonstrated that temperature perturbations are induced at the shelf edge and propagate to the coastal zone with the velocity that is close to the velocity of the first mode of internal gravity waves with the frequency of the tidal harmonic M₂.     

The influence of bottom topography on internal seiches in stratified media

Standing internal waves, so-called seiches, are ubiquitous in reservoirs and lakes. Although the stratification in such basin is often continuous, the modeling of seiches has been confined mostly to two-layer models. Such models are unable to give reliable insights into the vertical structure of the seiches, which might be crucial for the understanding of vertical mixing in natural water basins. To obtain this kind of information a two-dimensional computer model has been developed, which takes both continuous stratification and bottom topography into account. The results of this model are presented. The computed seiche modes reveal that (1) several large-scale modes can exist with similar eigenfrequencies, (2) the modes have a tendency to develop narrow jets and (3) only the lowest modes are strongly influenced by the bottom topography.     

Kinetic equations and stationary energy spectra of weakly nonlinear internal gravity waves

An ensemble of random-phase internal gravity waves is considered in the dynamical framework of the Euler–Boussinesq equations. For flows with zero mean potential vorticity, a kinetic equation for the mean spectral energy density of the waves is obtained under hypothesis of Gaussian statistics with zero correlation length. Stationary scaling solutions of this equation are found for almost vertically propagating waves. The resulting spectra are anisotropic in vertical and horizontal wave numbers. For flows with small but non-zero mean potential vorticity, under the same statistical hypothesis applied to the wave part of the flow, it is shown that the vortex part and the wave part decouple. The vortex part obeys a limiting slow dynamics equation exhibiting vertical collapse and layering which may contaminate the wave-part spectra. Relation of these results to the in situ atmospheric measurements and previous work on oceanic gravity waves is discussed.     

The reflection and diffraction of internal waves from the junction of a slit and a half-space, with application to submarine canyons

We consider the three-dimensional reflection and diffraction properties of internal waves in a continuously stratified rotating fluid which are incident on the junction of a vertical slit and a half-space. This geometry is a model for submarine canyons on continental slopes in the ocean, where various physical phenomena embodying reflection and diffraction effects have been observed. Three types of incident wave are considered: (1) Kelvin waves in the slit (canyon); (2) Kelvin waves on the slope; and (3) plane internal waves incident from the half-space (ocean). These are scattered into Kelvin and Poincaré waves in the slit, a Kelvin wave on the slope and Poincaré waves in the half-space. Most of the discussion is centered around case (1). Various properties of the wave field are calculated for ranges of the parameters c/cot θ, γα and ƒ/ω where cot θ is the topographic slope, c is the internal wave ray slope, α is the canyon half-width, γ is the down-slope wave-number, ƒ is the Coriolis parameter and ω is the wave frequency. Analytical results are obtained for small γα and some approximate results for larger values of γα. The results show that significant wave trapping may occur in oceanic situations, and that submarine canyons may act as source regions for internal Kelvin waves on the continental slope.     

The impact of different vertical diffusion schemes in a three-dimensional oil spill model in the Bohai Sea

Vertical transport is critical to the movement of oil spills in seawater. Breaking waves play an important role by developing a well-defined mixing layer in the upper part of the water column. A three-dimensional （3-D） Lagrangian random walk oil spill model was used here to study the influence of sea surface waves on the vertical turbulence movement of oil particles. Three vertical diffusion schemes were utilized in the model to compare their impact on oil dispersion and transportation. The first scheme calculated the vertical eddy viscosity semi-empirically. In the second scheme, the vertical diffusion coefficient was obtained directly from an Eulerian hydrodynamic model （Princeton Ocean Model, POM2k） while considering wave- caused turbulence. The third scheme was formulated by solving the Langevin equation. The trajectories, percentages of oil particles intruding into water, and the vertical distribution structures of oil particles were analyzed for a series of numerical experiments with different wind magnitudes. The results showed that the different vertical diffusion schemes could generate different horizontal trajectories and spatial distributions of oil spills on the sea surface. The vertical diffusion schemes caused different water-intruding and resurfacing oil particle behaviors, leading to different horizontal transport of oil particles at the surface and subsurface of the ocean. The vertical diffusion schemes were also applied to a realistic oil spill simulation, and these results were compared to satellite observations. All three schemes yielded acceptable results, and those of the third scheme most closely simulated the observed data.     

不稳定边界层下地形重力内波

水槽实验及线性理论研究表明,当低层大气处于近中性或不稳定时,如果地形引起的动力扰动足够强,地形扰动可在上部稳定层结中激发出重力内波,波动反过来影响低层流场,引起动量输送。低层大气处于近中性或不稳定时,地形波同样对大气运动可产生波阻,这应引起模式工作者的重视。最后讨论了大气粘性对中性或不稳定层结下地形波的影响。     

Boundary layer structure over an inhomogeneous surface: Simulation with a non-hydrostatic mesoscale model

A three-dimensional, non-hydrostatic mesoscale model is used to study boundary-layer structure over an area characterized by the city of Copenhagen, the Øresund strait, and adjacent coastal farmland. Simulations are compared with data obtained on June 5, 1984 during the Øresund experiment.Under moderately strong wind conditions, a stable internal boundary layer (IBL) developed over the Øresund strait during the day. Near-surface winds decelerate over water due to diminished vertical momentum transfer.The turbulent kinetic energy field closely reflects the surface roughness distribution due to the imposed relatively strong wind forcing. TKE budgets over water, farmland and a city area are discussed by inspection of vertical profiles of the individual terms. The buoyancy term is used to indicate IBL heights because it changes sign at the boundary between different stability regimes. Measured and simulated dissipation rates show a decrease in the transition zone as the air travels over water and an abrupt increase when the IBL over a downwind city area is intersected. The top of the stable IBL is characterized by a minimum in the vertical TKE profile.     

垂直切变和地形影响下惯性重力波的发展

采用分层浅水波方程和ＷＫＢ方法，分别讨论了在基流垂直切变和地形的影响下，二维惯性重力波的稳定性及其发展，并根据理论结果试图解释１９９２年６月２１日发生在京津冀的一次飑线演变与惯性重力波之间的关系，这可能为飑线预报提供一种思路     

On the effects of vertical eddy viscosity on rossby and eady waves in the ocean

The effects of vertical eddy viscosity on simple mesoscale waves in the ocean are studied. The decay of Rossby waves is investigated by one-dimensional depth-dependent linear stability problems which are derived for the interior non-viscous or viscous quasigeostrophic flow using parameterizations of the top and bottom boundary layers corresponding to Ekman suction, no-stress and bottom-stress boundary conditions.The non-slip condition at the bottom yielding an O(E_v^1/2)-Ekman layer causes very short damping times for the 0th Rossby mode. This suggests that this boundary condition is not suitable for mesoscale wave studies, because a Rossby wave fit for the MODE eddy can be done satisfactorily without any damping. Reasonable results for damping times of Rossby waves are obtained by prescribing the bottom stress, resulting from the constant-stress layer at the bottom, and the free-slip condition at the surface. The growth rates of Eady waves are reexamined using this bottom-stress condition.Vertical viscosity in the interior of the ocean, e.g. internal wave induced viscosity, may have a significant influence on the dynamics of the mesoscale motions, comparable to that of the boundary layers in some cases. The results are compatible with the sparse observations available.     

对流凝结加热和垂直风切变对行星波垂直传播的影响

本文从理论上分析研究了对流凝结加热和垂直风切变对大气行星波垂直传播的影响。结果表明,对流凝结加热有抑制行星波垂直传播的作用;基本气流的西风(正)垂直切变对行星波的垂直传播有利,而东风(负)切变却对行星波的垂直传播有阻碍作用。 进-步分析还发现,垂直风切变对行星波垂直传播的影响以在中高纬度地区更为显著;对流凝结加热对行星波垂直传播的影响在热带地区更值得重视。并且,垂直风切变和对流凝结加热的影响可能是使得大气中板地波导更显著的重要原因之一。      

Tidal circulation and buoyancy effects in the St. Lawrence Estuary

Abstract

The action of tides on density‐driven circulation, internal gravity waves, and mixing was investigated in the St. Lawrence Estuary between Rimouski and Québec City. Time‐varying fields of water level, currents and density were computed under typical summer conditions using a three‐dimensional hydrostatic coastal ocean model that incorporates a second order turbulence closure submodel. These results are compared with current meter records and other observations. The model and the observations reveal buoyancy effects produced by tidal forcing. The semi‐diurnal tide raises the isopycnals over the sills at the head of the Laurentian Trough and English Bank, producing internal tides radiating seaward. Relatively dense intermediate waters rise from below 75‐m depth to the near surface over the sills, setting up gravity currents on the inner slopes. Internal hydraulic controls develop over the outer sills; during flood, surface flow separation occurs at the entrances of the Saguenay Fjord and the upper estuary west of Ilet Rouge Bank. Early during ebb flow (restratification), the surface layer deepens to encompass the tops of the sills. As the ebb current intensifies, the model predicts the formation of seaward internal jumps over the outer sills, which were confirmed from acoustic reflection observations. As the internal Froude number increases further, flow separation migrates up to sill height. As a result of these transitions, internal bores emanate from the head region one to two hours before low water. We find that the mixing of oceanic and surface waters near the sills is driven by the vertical shear produced during ebb in the channel south of Ilet Rouge, the shear produced in the bottom gravity flood currents, and, to a lesser extent, the processes over the sills.     

On the three-dimensional internal waves excited in the flow of a linearly stratified Boussinesq fluid

Three-dimensional flow of a linearly stratified Boussinesq fluid is studied numerically. The flow is assumed to be confined in a rectangular channel and internal waves are excited by bottom topography. Near resonance of the first vertical internal wave mode, it was found that the reflection of the internal wave at the sidewall is ‘abnormal’ in the sense that the reflection angle is larger than the incident angle and a third wave perpendicular to the sidewall is generated. The waves become straight crested (two-dimensional) as this third wave becomes longer. The whole mechanism is similar to the ‘Mach reflection’ observed in the general stratified fluid in which the usual solitary waves are generated. In the case of the linearly stratified Boussinesq fluid, the abnormal reflection occurs even though the wave near the sidewall has a sinusoidal profile and not a sech² profile. This suggests that the abnormal reflections similar to Mach reflection always occur when the wave amplitude is large enough, irrespective of the wave profile.