Two-layer Ekman model for wind-induced shear flow near the sea surface

A simple two-layered model for steady wind-induced shear flow near the sea surface has been formulated. Basic assumptions of Ekman's theory are retained, including horizontal uniformity, infinite depth and constant (but differing) vertical eddy viscosities in the respective layers. Employing information coming from observational data, the parameters of the model (depth of the surface layer and the two eddy viscosities) are evaluated and optimized. The results thereby obtained favour the presence of a high shear layer, about 1 m thick, immediately below the sea surface. The eddy viscosity in that layer increases approximately linearly with wind speed but remains comparatively small, being generally less than 20 cm² s⁻¹ for wind speeds less than 20 m s⁻¹. In contrast, the eddy viscosity directly below the layer is two orders of magnitude larger and increases more steeply with wind speed.     

Numerical modelling of stratified flow: a spectral approach

A numerical formulation is developed to solve the three-dimensional hydrodynamic equations which describe flow in a stratified sea.Arbitrary continuous physically realistic variations of density and eddy viscosity can be included in the model, which is sufficiently flexible to be applicable to sea areas of any horizontal extent and depth. A continuous current profile from sea surface to sea bed, is computed with the model. A method for expanding computed current profiles in terms of vertical modes is proposed and the contribution of these modes to the current profiles is considered.The time variation of the wind-induced circulation of a stratified lake in response to a suddenly imposed and maintained wind stress is examined. Calculations show that the wind-driven surface current is modulated by the internal seiche motion of the lake.     

Tidal and wind-induced circulation within the Southeastern limit of the Bay of Biscay: Pasaia Bay,Basque Coast

The Basque coastal area, in the southeastern Bay of Biscay, can be characterised as being more influenced by land climate and inputs, than other typically ‘open sea’ areas. The influence of coastal processes, together with the presence of irregular and steep topography, complicate greatly the water circulation patterns. Water movement along the Basque coastal area is not well understood; observations are scarce and long-term current records are lacking. The knowledge available is confined to the surface currents: the surface water circulation is controlled mainly by wind forcing, with tidal and density currents being weak. However, there is a lack of knowledge available on currents within the lower levels of the water column; likewise, on the main time-scales involved in the water circulation. This study quantifies the contribution of the tidal and wind-induced currents, to the overall water circulation; it identifies the main time-scales involved within the tidal and wind-induced flows, investigating difference in such currents, throughout the water column, within Pasaia Bay (Basque coast). Within this context, extensive oceanographic and meteorological data have been obtained, in order to describe the circulation. The present investigation reveals that the circulation, within the surface and the sub-surface waters, is controlled mainly by wind forcing fluctuations, over a wide range of meteorological frequencies: third-diurnal, semidiurnal and diurnal land–sea breezes; synoptic variability; frequencies, near fortnightly periods; and seasonal. At the lower levels of the water column, the main contribution to the water circulation arises from residual currents, followed by wind-induced currents on synoptic time-scales. In contrast, tidal currents contribute minimally to the overall circulation throughout the water column.     

Importance of subgrid-scale parameterization in numerical simulations of lake circulation

Two subgrid-scale modeling techniques––Smagorinsky’s postulation for the horizontal eddy viscosity and the Mellor–Yamada level-2 model for the vertical eddy viscosity––are applied as turbulence closure conditions to numerical simulations of resolved-scale baroclinic lake circulations. The use of the total variation diminishing (TVD) technique in the numerical treatment of the advection terms in the governing equations depresses numerical diffusion to an acceptably low level and makes stable numerical performances possible with small eddy viscosities resulting from the turbulence closure parameterizations. The results show that, with regard to the effect of an external wind stress, the vertical turbulent mixing is mainly restricted to the topmost epilimnion with the order of magnitude for the vertical eddy viscosity of 10⁻³ m² s⁻¹, whilst the horizontal turbulent mixing may reach a somewhat deeper zone with an order of magnitude for the horizontal eddy viscosity of 0.1–1 m² s⁻¹. Their spatial and temporal variations and influences on numerical results are significant. A comparison with prescribed constant eddy viscosities clearly shows the importance of subgrid-scale closures on resolved-scale flows in the lake circulation simulation. A predetermination of the eddy viscosities is inappropriate and should be abandoned. Their values must be determined by suitable subgrid-scale closure techniques.     

Mathematische Modelle zur Simulation von Zirkulationsströmungen in einem See

The three-dimensional, finite-difference model of R.L. Street (Stanford University) has been adapted at the simulation of wind-induced currents in the northern basin of the Lake of Lugano. The model makes use of complete Navier-Stokes equations and a dynamical algorithm to calculate the eddy viscosity. The model predictions are compared with data collected in the measure campaign of 1979. A further application of the model is the cold inflow simulation in a stratified basin with inclusion of turbulent vertical exchange processes. A two-dimensional version of the model, which assumes an hydrostatic pressure distribution, is utilized to simulate the thermal sommer development of the Lake of Lugano, during 40 days. The model includes the thermical interactions and the wind influence. The results obtained show the possible application of the three-dimensional model to the currents simulation in lake basins. The two-dimensional model may be utilized to predict the development of a basin subject to a thermal load.      

On the sensitivity of computed surges to open-boundary formulation

A coarse-grid (resolution of order 7 km) model of the west coast of Britain is used to examine the sensitivity of computed storm-surge elevations and currents to a range of open-boundary conditions. The storm-surge period 1 to 26 March 1994 is used for this comparison, as it is a time of significant wind activity. Also current measurements in the North Channel of the Irish Sea together with coastal elevation measurements are available for model validation. Elevations and currents previously computed with a coarse-grid shelf-wide model can also be incorporated into the open-boundary condition to examine the influence of far-field effects. Initial model calculations with no far-field input show the importance of including shelf-wide effects from either the external shelf model, or by using observations from coastal gauges interpolated along the open boundary of the west-coast model. Provided the west-coast models open boundary is taken sufficiently far away from the region of interest, in this case the Irish Sea, then either a radiation condition or an elevation-specified condition is appropriate provided far-field effects are taken into account. If these are not included, then neither boundary condition is successful. For the radiation condition it is necessary to include both elevations and currents from a far-field model in order to reproduce the surge. In the case of an elevation-specified boundary condition far-field effects can be incorporated in hindcast calculations by including observed sea-level changes. In a storm-surge prediction calculation the radiation condition with a far-field model is required. Calculations show that computed elevations are spatially more coherent than currents, with flows through the western Irish Sea showing the greatest sensitivity to open-boundary formulation during storm events.Responsible Editor: Phil Dyke     

Sea breezes drive currents on the inner continental shelf off southwest Western Australia

In southwest Western Australia, strong and persistent sea breezes are common between September and February. We hypothesized that on the inner continental shelf, in the absence of tidal forcing, the depth, magnitude, and lag times of the current speed and direction responses to sea breezes would vary though the water column as a function of the sea breeze intensity. To test this hypothesis, field data were used from four sites were that were in water depths of up to 13 m. Sites were located on the inner continental shelf and were on the open coast and in a semi-enclosed coastal embayment. The dominant spectral peak in currents at all sites indicated that the majority of the spectral energy contained in the currents was due to forcing by sea breezes. Currents were aligned with the local orientation of the shoreline. On a daily basis, the sea breezes resulted in increased current speeds and also changed the current directions through the water column. The correlation between wind–current speeds and directions with depth, and the lag time between the onset of the sea breeze and the response of currents, were dependent on the intensity of the sea breezes. A higher correlation between wind and current speeds occurred during strong sea breezes and was associated with shorter lag times for the response of the bottom currents. The lag times were validated with estimates of the vertical eddy viscosity. Solar heating caused the water column to stratify in summer and the sea breezes overcame this stratification. Sea breezes caused the mixed layer to deepen and the intensity of the stratification was correlated to the strength of the sea breezes. Weak sea breezes of <5 m s⁻¹ were associated with the strongest thermal stratification of the water column, up to 1°C between the surface and bottom layers (6 and 10 m below the surface). In comparison, strong sea breezes of >14 m s⁻¹ caused only slight thermal stratification up to 0.5°C. Apart from these effects on the vertical structure of water column, the sea breezes also influenced transport and mixing in the horizontal dimension. The sea breezes in southwest Western Australia rotated in an anticlockwise direction each day and this rotation was translated into the currents. This current rotation was more prominent in surface currents and in the coastal embayment compared to the open coast.     

The vertical structure of tidal currents

Abstract

The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = ε and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition ε>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth.

The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity ε or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is shown to depend upon accurate specification of both eddy viscosity and the bed stress condition.     

Floating marine debris surface drift: Convergence and accumulation toward the South Pacific subtropical gyre

Whatever its origin is, a floating particle at the sea surface is advected by ocean currents. Surface currents could be derived from in situ observations or combined with satellite data. For a better resolution in time and space, we use satellite-derived sea-surface height and wind stress fields with a 1/3° grid from 1993 to 2001 to determine the surface circulation of the South Pacific Ocean. Surface currents are then used to compute the Lagrangian trajectories of floating debris. Results show an accumulation of the debris in the eastern-centre region of the South Pacific subtropical gyre ([120°W; 80°W]–[20°S; 40°S]), resulting from a three-step process: in the first two years, mostly forced by Ekman drift, the debris drift towards the tropical convergence zone (30°S). Then they are advected eastward mostly forced by geostrophic currents. They finally reach the eastern-centre region of the South Pacific subtropical gyre from where they could not escape.     

LES of turbulent surface shear stress and pressure-gradient-driven flow on shallow continental shelves

Turbulent shear flows on shallow continental shelves (here shallow means that the interaction with the solid, no-slip bottom is important) are of great importance because tide- and wind-driven flows on the shelf are drivers of the transfer of momentum, heat, and mass (gas) across the air–sea interface. These turbulent flows play an important role because vertical mixing and current are vectors for the transport of sediment and bioactive material on continental shelves. Understanding the dynamics of this class of flows presents complications because of the presence of a free surface and also because the flow can be driven by a pressure gradient (a tidal current), a stress at the free surface (a wind-driven current), or a combination of both. In addition, the flow can be modified by the presence of a wave field that can induce Langmuir circulation (Langmuir, Science 87:119–123, 1938). Large eddy simulation is used to quantify the effects of pressure gradient and wind shear on the distinctive structures of the turbulent flow. From these computations, an understanding of the physics governing the turbulence of pressure-driven and wind-driven flows, how they can interact in a normal or a tangential direction, and the effect of wave forcing on these flows is obtained.     

Modeling the impact of wind and waves on suspended particulate matter fluxes in the East Frisian Wadden Sea (southern North Sea)

Suspended particulate matter (SPM) fluxes and dynamics are investigated in the East Frisian Wadden Sea using a coupled modeling system based on a hydrodynamical model [the General Estuarine Transport Model (GETM)], a third-generation wave model [Simulating Waves Nearshore (SWAN)], and a SPM module attached to GETM. Sedimentological observations document that, over longer time periods, finer sediment fractions disappear from the Wadden Sea Region. In order to understand this phenomenon, a series of numerical scenarios were formulated to discriminate possible influences such as tidal currents, wind-enhanced currents, and wind-generated surface waves. Starting with a simple tidal forcing, the considered scenarios are designed to increase the realism step by step to include moderate and strong winds and waves and, finally, to encompass the full effects of one of the strongest storm surges affecting the region in the last hundred years (Storm Britta in November 2006). The results presented here indicate that moderate weather conditions with wind speeds up to 7.5 m/s and small waves lead to a net import of SPM into the East Frisian Wadden Sea. Waves play only a negligible role during these conditions. However, for stronger wind conditions with speeds above 13 m/s, wind-generated surface waves have a significant impact on SPM dynamics. Under storm conditions, the numerical results demonstrate that sediments are eroded in front of the barrier islands by enhanced wave action and are transported into the back-barrier basins by the currents. Furthermore, sediment erosion due to waves is significantly enhanced on the tidal flats. Finally, fine sediments are flushed out of the tidal basins due to the combined effect of strong erosion by wind-generated waves and a longer residence time in the water column because of their smaller settling velocities compared to coarser sediments.

Numerical modeling of hydrodynamics and sediment transport—an integrated approach

Point measurement-based estimation of bedload transport in the coastal zone is very difficult. The only way to assess the magnitude and direction of bedload transport in larger areas, particularly those characterized by complex bottom topography and hydrodynamics, is to use a holistic approach. This requires modeling of waves, currents, and the critical bed shear stress and bedload transport magnitude, with a due consideration to the realistic bathymetry and distribution of surface sediment types. Such a holistic approach is presented in this paper which describes modeling of bedload transport in the Gulf of Gdańsk. Extreme storm conditions defined based on 138-year NOAA data were assumed. The SWAN model (Booij et al. 1999) was used to define wind–wave fields, whereas wave-induced currents were calculated using the Ko?odko and Gic-Grusza (2015) model, and the magnitude of bedload transport was estimated using the modified Meyer-Peter and Müller (1948) formula. The calculations were performed using a GIS model. The results obtained are innovative. The approach presented appears to be a valuable source of information on bedload transport in the coastal zone.     

A three-layered spectral model with application to wind-induced motion in the presence of stratification and a bottom slope

The main features of a new three-dimensional spectral model for the motion of a stratified sea are described. The model is applied to determine wind-induced elevations and currents over an idealised shelf-edge slope during the summer season of thermal stratification. A wind stress field spanning the slope is applied in the form of a rectangular pulse through time, the direction of the stress being parallel to the shelf edge. It is shown that strong vertical motion occurs near the top of the slope in the wind region; inertial currents are excited. Outside the wind region, long-period waves trapped over the slope and propagating along it may be identified.     

Experiments with a numerical model of coastal currents and tidal mixing fronts

A three-dimensional numerical model is used to simulate the development of disturbances on shelf-sea coastal currents and fronts. The model, which has a free surface, uses a finite difference grid ☐ scheme based on sigma coordinates. It has a semi-implicit scheme for the barotropic flow and a hydrid advection scheme to retain sharp fronts. The results demonstrate that (i) eddy formation follows changes at the inflow of a coastal current, (ii) a simple radiation boundary condition at the outflow produces nearly identical results for different outflow boundary positions, (iii) eddy growth, with matching behaviour of surface and bottom fronts, follows a small displacement on a tidal mixing front and (iv) effects of friction and mixing can significantly alter the behaviour of the front and the relative strength of the cyclonic and anticyclonic eddies formed.     

Seasonal variation of the <Emphasis Type="Italic">M</Emphasis><Subscript>2</Subscript> tide

The seasonal cycle of the main lunar tidal constituent M ₂ is studied globally by an analysis of a high-resolution ocean circulation and tide model (STORMTIDE) simulation, of 19 years of satellite altimeter data, and of multiyear tide-gauge records. The barotropic seasonal tidal variability is dominant in coastal and polar regions with relative changes of the tidal amplitude of 5–10 %. A comparison with the observations shows that the ocean circulation and tide model captures the seasonal pattern of the M ₂ tide reasonably well. There are two main processes leading to the seasonal variability in the barotropic tide: First, seasonal changes in stratification on the continental shelf affect the vertical profile of eddy viscosity and, in turn, the vertical current profile. Second, the frictional effect between sea-ice and the surface ocean layer leads to seasonally varying tidal transport. We estimate from the model simulation that the M ₂ tidal energy dissipation at the sea surface varies seasonally in the Arctic (ocean regions north of 60°N) between 2 and 34 GW, whereas in the Southern Ocean, it varies between 0.5 and 2 GW. The M ₂ internal tide is mainly affected by stratification, and the induced modified phase speed of the internal waves leads to amplitude differences in the surface tide signal of 0.005–0.0150 m. The seasonal signals of the M ₂ surface tide are large compared to the accuracy demands of satellite altimetry and gravity observations and emphasize the importance to consider seasonal tidal variability in the correction processes of satellite data.     

Efficient Methods to Simulate Planar Free Surface in the 3D 4th-Order Staggered-Grid Finite-Difference Schemes

We numerically tested accuracy of two formulations of Levander's (1988) stress-imaging technique for simulating a planar free surface in the 4^th-order staggered-grid finite-difference schemes. We have found that both formulations (one with normal stress-tensor components at the surface, the other with shear stress-tensor components at the surface) require at least 10 grid spacings per minimum wavelength ( _min÷h = 10) if Rayleigh waves are to be propagated without significant grid dispersion in the range of epicentral distances up to 15 _dom ^S.Because interior 4^th-order staggered-grid schemes usually do not require more than 6 grid spacings per minimum wavelength, in the considered range of epicentral distances, it was desirable to find alternative techniques to simulate a planar free surface, which would not require denser spatial sampling than _min÷h = 6. Therefore, we have developed and tested new techniques: 1. Combination of the stress imaging (with the shear stress-tensor components at the surface) with Rodrigues' (1993) vertically refined grid near the free surface. 2. Application of the adjusted finite-difference approximations to the z-derivatives at the grid points at and below the surface that uses no virtual values above the surface and no stress imaging. The normal stress-tensor components are at the surface in one formulation, while the shear stress-tensor components are at the surface in the other formulation.The three developed formulations give for the spatial sampling _min÷h = 6 results very close to those obtained by the discrete-wavenumber method. Because, however, the technique with the vertically refined grid near the free surface requires 3 times smaller time step (due to the refined grid), the technique with adjusted finite-difference approximations is the most accurate and efficient technique from the examined formulations in the homogeneous halfspace.     

Comparison of a simple planetary boundary layer model with measurements of a turbulent boundary layer under pack ice

A simple analytical theory which describes the motion in a turbulent planetary boundary layer near a rough sea bed by using a two-layer eddy viscosity model is presented. The vertical structure of the current in the boundary layer is presented, and comparisons are made with data fromMcPhee and Smith (1976, Journal of Physical Oceanography,6, 696–711) obtained from measurements of the turbulent boundary layer under drifting Arctic ice.     

Tidally generated island wakes and surface water cooling over Izu Ridge

The Kuroshio current is well known for generating cold wakes behind islands over Izu Ridge in Northwestern Pacific. Observational data from the geostationary Himawari-8 satellite for 2015–2017 revealed the occurrence of cold waters during the period when the Kuroshio current flows away from the islands. With a focus on tidal currents, this study presents an investigation of dynamical processes responsible for the formation of areas with low sea surface temperature (SST) through the adoption of a high-resolution numerical ocean model for an event that happened in July 2017. Areas with cold water emerged only when tidal currents are included in the numerical model. The model results indicate the cold surface waters are formed in the vicinity of the islands because of upwelling and vertical mixing. Qualitative features of the cold water formation for each island are found to depend on its size, topography, and ambient currents. Near Kozu Island, the tidal excursion is large enough to cause eddy shedding. These shed eddies are stirred by tidal currents to extend the surface cooling effect to wider areas. Near Hachijo Island, a persistent wake is formed by the ambient northward current. Inclusion of tidal currents destabilizes the wake, and consequently leads to the formation of a low SST area, although no clear eddy shedding is detected. The flow patterns around the islands are classified using an additional non-dimensional parameter, defined as the ratio between tidal excursion and island diameter.

     

Relationship between thermally forced surface wind and sea surface temperature gradient

An important part of the influence of the oceans on the atmosphere is through direct radiation, sensible heat flux and release of latent heat of evaporation, whereby all of these processes are directly related to the surface temperature of the oceans. A main effect of the atmosphere on the oceans is through momentum exchange at the air-ocean interface, and this process is directly related to the surface wind stress. The sea surface temperature (SST) and the surface wind stress are the two important components in the air-ocean system. If SST is given, a thermally forced boundary layer atmospheric circulation can be simulated. On the other hand, if the surface wind stress is given, the wind-driven ocean waves and ocean currents can be computed.The relationship between SST and surface wind is a coupling of the atmosphere and the oceans. It changes a one-way effect (ocean mechanically driven by atmosphere, or atmosphere thermally forced by oceans) into two-way air-sea interactions. Through this coupling the SST distribution, being an output from an ocean model, leads to the thermally forced surface winds, which feeds back into the ocean model as an additional forcing.Based on Kuo's planetary boundary layer model a linear algebraic equation is established to link the SST gradient with the thermally forced surface wind. The surface wind blows across the isotherms from cold to warm region with some deflection angle to the right (left) in the Northern (Southern) Hemisphere. Results from this study show that the atmospheric stratification reduces both the speed and the deflection angle of the thermally forced wind, however, the Coriolis' effect increases the wind speed in stable atmosphere (Ri>10^–4) and increases the deflection angle.