Generation Of Lee Waves Research Articles

Eddy saturation and frictional control of the Antarctic Circumpolar Current

AbstractThe Antarctic Circumpolar Current is the strongest current in the ocean and has a pivotal impact on ocean stratification, heat content, and carbon content. The circumpolar volume transport is relatively insensitive to surface wind forcing in models that resolve turbulent ocean eddies, a process termed “eddy saturation.” Here a simple model is presented that explains the physics of eddy saturation with three ingredients: a momentum budget, a relation between the eddy form stress and eddy energy, and an eddy energy budget. The model explains both the insensitivity of circumpolar volume transport to surface wind stress and the increase of eddy energy with wind stress. The model further predicts that circumpolar transport increases with increased bottom friction, a counterintuitive result that is confirmed in eddy‐permitting calculations. These results suggest an unexpected and important impact of eddy energy dissipation, through bottom drag or lee wave generation, on ocean stratification, ocean heat content, and potentially atmospheric CO2.

Spontaneous Wave Generation at Strongly Strained Density Fronts

AbstractA simple analytical model is presented describing the spontaneous generation of inertia–gravity waves at density fronts subjected to strong horizontal strain rates. The model considers fronts of arbitrary horizontal and vertical structure in a semi-infinite domain, with a single boundary at the ocean surface. Waves are generated because of the acceleration of the steady uniform strain flow around the density front, analogous to the generation of lee waves via flow over a topographic ridge. Significant wave generation only occurs for sufficiently strong strain rates α > 0.2f and sharp fronts H/L > 0.5f/N, where f is the Coriolis parameter, N is the stratification, and H and L are the height and width scales of the front, respectively. The frequencies of the generated waves are entirely determined by the strain rate. The lowest-frequency wave predicted to be generated via this mechanism has a Lagrangian frequency ω = 1.93f as measured in a reference frame moving with the background strain flow. The model is intended as a first-order description of wave generation at submesoscale (1 to 10 km wide) fronts where large strain rates are commonplace. The analytical model compares well with fully nonlinear numerical simulations of the submesoscale regime.

Mapping turbulent diffusivity associated with oceanic internal lee waves offshore Costa Rica

Abstract. Breaking internal waves play a primary role in maintaining the meridional overturning circulation. Oceanic lee waves are known to be a significant contributor to diapycnal mixing associated with internal wave dissipation, but direct measurement is difficult with standard oceanographic sampling methods due to the limited spatial extent of standing lee waves. Here, we present an analysis of oceanic internal lee waves observed offshore eastern Costa Rica using seismic imaging and estimate the turbulent diffusivity via a new seismic slope spectrum method that extracts diffusivities directly from seismic images, using tracked reflections only to scale diffusivity values. The result provides estimates of turbulent diffusivities throughout the water column at scales of a few hundred meters laterally and 10 m vertically. Synthetic tests demonstrate the method's ability to resolve turbulent structures and reproduce accurate diffusivities. A turbulence map of our seismic section in the western Caribbean shows elevated turbulent diffusivities near rough seafloor topography as well as in the mid-water column where observed lee wave propagation terminates. Mid-water column hotspots of turbulent diffusivity show levels 5 times higher than surrounding waters and 50 times greater than typical open-ocean diffusivities. This site has steady currents that make it an exceptionally accessible laboratory for the study of lee-wave generation, propagation, and decay.

Evidence for seafloor‐intensified mixing by surface‐generated equatorial waves

AbstractLittle is known about mixing in the abyssal equatorial oceans in spite of its inferred importance for upwelling dense water. Here we present full‐depth microstructure turbulence profiles obtained in the equatorial Pacific that show evidence for intense wind‐generated abyssal mixing. Mixing was intensified over the bottom 700 m where the diffusivity reached 10−3 m2 s−1, of similar intensity to mixing driven by tidal flow over rough topography. However, here the mixing was found over smooth topography. We suggest that the intense mixing could have been driven by surface‐generated equatorial waves through two possible mechanisms: (1) near‐bottom wave trapping as a result of the horizontal component of the Earth's rotation and (2) inertial instability. The generation of lee waves over smooth topography at low latitudes and their subsequent breaking is another viable mechanism for the mixing.

Impact of topographic internal lee wave drag on an eddying global ocean model

The impact of topographic internal lee wave drag (wave drag hereafter) on several aspects of the low-frequency circulation in a high-resolution global ocean model forced by winds and air-sea buoyancy fluxes is examined here. The HYbrid Coordinate Ocean Model (HYCOM) is run at two different horizontal resolutions (one nominally 1/12° and the other 1/25°). Wave drag, which parameterizes both topographic blocking and the generation of lee waves arising from geostrophic flow impinging upon rough topography, is inserted into the simulations as they run. The parameterization used here affects the momentum equations and hence the structure of eddy kinetic energy. Lee waves also have implications for diapycnal mixing in the ocean, though the parameterization does not directly modify the density. Total near-bottom energy dissipation due to wave drag and quadratic bottom boundary layer drag is nearly doubled, and the energy dissipation due to quadratic bottom drag is reduced by about a factor of two, in simulations with an inserted wave drag compared to simulations having only quadratic bottom drag. With the insertion of wave drag, the kinetic energy is reduced in the abyss and in a three-dimensional global integral. Deflection by partial topographic blocking is inferred to be one reason why the near-bottom kinetic energy can increase in locations where there is little change in dissipation by quadratic bottom drag. Despite large changes seen in the abyss, the changes that occur near the sea surface are relatively small upon insertion of wave drag into the simulations. Both the sea surface height variance and geostrophic surface kinetic energy are reduced on global average by more than twice the seasonal variability in these diagnostics. Alterations in the intensified jet positions brought about by inserting wave drag are not distinguishable from the temporal variability of jet positions. Various statistical measures suggest that applying wave drag only within a fixed distance from the seafloor is not detrimental to the model performance relative to observations. However, the introduction of a novel diagnostic suggests that one way to improve the wave drag parameterization is to allow the vertical deposition of lee wave momentum flux to be spatially heterogeneous.

Where large deep‐ocean waves break

AbstractUnderwater topography like seamounts causes the breaking of large “internal waves” with associated turbulent mixing strongly affecting the redistribution of sediment. Here ocean turbulence is characterized and quantified in the lowest 100 m of the water column at three nearby sites above the slope of a deep‐ocean seamount. Moored high‐resolution temperature sensors show very different turbulence generation mechanisms over 3 and 5 km horizontal separation distances. At the steepest slope, turbulence was 100 times more energetic than at the shallowest slope where turbulence was still 10 times more energetic than found in the open ocean, away from topography. The turbulence on this extensive slope is caused by slope steepness and nonlinear wave evolution, but not by bottom‐friction, “critical” internal tide reflection or lee wave generation.

Internal waves on the upstream side of a large sill of the Mascarene Ridge: a comprehensive view of their generation mechanisms and evolution

In this paper we aim to clarify the generation of Internal Solitary Waves (ISWs) at work to the east of the Mascarene Plateau (Indian Ocean) using Synthetic Aperture Radar (SAR) imagery and MITgcm fully nonlinear and nonhydrostatic simulations. Realistic representations of stratification and bathymetry are used with asymmetric tidal forcing (including the steady South Equatorial Current which is assumed barotropic in the model) along a 2D transect aligned with the propagation direction of the wave signatures identified in the SAR. The combined flow (i.e. steady and tidal currents) is subcritical with respect to first-mode Internal Waves (IWs), but supercritical with respect to higher wave modes. Different types of nonlinear wave trains with distinct origins (i.e. tidal phase and location) have been identified with the combined aid of model and SAR: (1) large-scale primary mode-1 ISWs evolve from the disintegration of a multimodal baroclinic structure that appears on the upstream side of the sill; (2) mode-2 ISW-like waves that evolve from this same baroclinic structure and are arrested over the sill before being released upstream at the change of flow condition; (3) a large mode-2 lee wave is generated downstream of the sill (i.e. on the west side), which is trapped there during maximum westward tidal flow and released upstream when the tide relaxes; and (4) mode-2 ISW-like waves whose length-scales are O (20km) appear some 50km upstream of the sill, after an Internal Tide (IT) beam scatters into the pycnocline, itself originating from critical topography on the leeward (i.e. westward) side of the sill. The underwater sill being investigated is in the mixed-tidal-lee wave regime, where the internal tide release mechanism, lee wave generation and IT beams can coexist. The large-scale mode-2 ISW-like waves that form far upstream from the sill are long-lived features and can be identified in the SAR due to associated short-scale mode-1 ISWs which propagate with the same phase speed, i.e. in resonance. This coupling is also seen in the model, and here it is argued that the formation of those mode-2 ISW-like waves appears to originate from the IT beam after it reflects from the sea surface and interacts with the pycnocline, a generation mechanism referred in the literature as “local generation of ISWs”. This IW generation process may be easily overlooked and could be at work in many more regions of the world than previously thought.

The Impact of Finite-Amplitude Bottom Topography on Internal Wave Generation in the Southern Ocean

Abstract Direct observations in the Southern Ocean report enhanced internal wave activity and turbulence in a kilometer-thick layer above rough bottom topography collocated with the deep-reaching fronts of the Antarctic Circumpolar Current. Linear theory, corrected for finite-amplitude topography based on idealized, two-dimensional numerical simulations, has been recently used to estimate the global distribution of internal wave generation by oceanic currents and eddies. The global estimate shows that the topographic wave generation is a significant sink of energy for geostrophic flows and a source of energy for turbulent mixing in the deep ocean. However, comparison with recent observations from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean shows that the linear theory predictions and idealized two-dimensional simulations grossly overestimate the observed levels of turbulent energy dissipation. This study presents two- and three-dimensional, realistic topography simulations of internal lee-wave generation from a steady flow interacting with topography with parameters typical of Drake Passage. The results demonstrate that internal wave generation at three-dimensional, finite bottom topography is reduced compared to the two-dimensional case. The reduction is primarily associated with finite-amplitude bottom topography effects that suppress vertical motions and thus reduce the amplitude of the internal waves radiated from topography. The implication of these results for the global lee-wave generation is discussed.

Lee wave generation rates in the deep ocean

Using the world's largest data set of in situ ocean current measurements, combined with a high-resolution topography roughness data set, we use a model-assisted hierarchical clustering methodology to estimate the global lee wave generation rate at the ocean floor. Our analysis suggests that internal wave generation contributes 0.75 ± 0.19 TW (±2 standard deviation) to the oceanic energy budget but with a strong dependence on the Brunt-Vaisala (buoyancy) frequency climatology used. This estimate is higher than previous calculations and suggests that internal wave generation may be a much more significant contributor to the global oceanic mechanical energy budget than had previously been assumed. Our results imply that lee wave generation and propagation may be a dominant sink of at least half and potentially the overwhelming majority of ocean surface wind work on the geostrophic circulation.

Impact of parameterized lee wave drag on the energy budget of an eddying global ocean model

The impact of parameterized topographic internal lee wave drag on the input and output terms in the total mechanical energy budget of a hybrid coordinate high-resolution global ocean general circulation model forced by winds and air-sea buoyancy fluxes is examined here. Wave drag, which parameterizes the generation of internal lee waves arising from geostrophic flow impinging upon rough topography, is included in the prognostic model, ensuring that abyssal currents and stratification in the model are affected by the wave drag.An inline mechanical (kinetic plus gravitational potential) energy budget including four dissipative terms (parameterized topographic internal lee wave drag, quadratic bottom boundary layer drag, vertical eddy viscosity, and horizontal eddy viscosity) demonstrates that wave drag dissipates less energy in the model than a diagnostic (offline) estimate would suggest, due to reductions in both the abyssal currents and stratification. The equator experiences the largest reduction in energy dissipation associated with wave drag in inline versus offline estimates. Quadratic bottom drag is the energy sink most affected globally by the presence of wave drag in the model; other energy sinks are substantially affected locally, but not in their global integrals. It is suggested that wave drag cannot be mimicked by artificially increasing the quadratic bottom drag because the energy dissipation rates associated with bottom drag are not spatially correlated with those associated with wave drag where the latter are small. Additionally, in contrast to bottom drag, wave drag is a non-local energy sink.All four aforementioned dissipative terms contribute substantially to the total energy dissipation rate of about one terawatt. The partial time derivative of potential energy (non-zero since the isopycnal depths have a long adjustment time), the surface advective fluxes of potential energy, the rate of change of potential energy due to diffusive mass fluxes, and the conversion between internal energy and potential energy also play a non-negligible role in the total mechanical energy budget. Reasons for the <10% total mechanical energy budget imbalance are discussed.

The Impact of Small-Scale Topography on the Dynamical Balance of the Ocean

AbstractThe impact of small-scale topography on the ocean’s dynamical balance is investigated by quantifying the rates at which internal wave drag extracts (angular) momentum and vorticity from the general circulation. The calculation exploits the recent advent of two near-global descriptions of topographic roughness on horizontal scales on the order of 1–10 km, which play a central role in the generation of internal lee waves by geostrophic flows impinging on topography and have been hitherto unresolved by bathymetric datasets and ocean general circulation models alike. It is found that, while internal wave drag is a minor contributor to the ocean’s dynamical balance over much of the globe, it is a significant player in the dynamics of extensive areas of the ocean, most notably the Antarctic Circumpolar Current and several regions of enhanced small-scale topographic variance in the equatorial and Southern Hemisphere oceans. There, the contribution of internal wave drag to the ocean’s (angular) momentum and vorticity balances is generally on the order of ten to a few tens of percent of the dominant source and sink terms in each dynamical budget, which are respectively associated with wind forcing and form drag by topography with horizontal scales from 500 to 1000 km. It is thus suggested that the representation of internal wave drag in general circulation models may lead to significant changes in the deep ocean circulation of those regions. A theoretical scaling is derived that captures the basic dependence of internal wave drag on topographic roughness and near-bottom flow speed for most oceanographically relevant regimes.

The Cyclic Behavior of Mountain Gravity Waves Generated by Flow over Topography

The cyclic behavior of lee wave systems, generated by stratified flow over mountains is investigated by the Advanced Regional Prediction System (ARPS) model. The results show that, surface friction has a direct impact upon the number and timing of mountain gravity waves cycle generation. Cyclic generation of mountain lee waves and down-slope winds was found to be extremely sensitive to the magnitude of the surface drag coefficient, where mountain waves amplitude and intensity varies with the magnitude of the drag coefficient, and the interaction of mountain waves and boundary layer process determinates the wave characteristics. For the typical drag Cd = 10–3, surface friction promotes the formation of the stationary mountain lee waves and hydraulic jump, especially, promotes boundary layer separation, the generation of low-level turbulent zones and rotor circulation or reversal flow within boundary layer. When drag coefficient becomes Cd = 10–4, lee waves remain steady states and the first evolution cycle maintains much longer than that of Cd = 10–3. In the case of the highest drag coefficient Cd = 10–2, surface friction suppresses wave breaking and the onset of hydraulic jump, and reduces greatly the amplitude and intensity of lee waves and down slope wind.

Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography

The rate of generation of internal gravity waves in the lee of small length scale topography by geostrophic flow in the World Ocean was estimated using linear theory with corrections for finite amplitude topography. Several global data sets were combined for the calculation including an ocean circulation model for the near-bottom geostrophic flow statistics, over 500 abyssal current meter records, historical climatological data for the buoyancy frequency, and two independent estimates of the small scale topographic statistical properties. The first topography estimate was based on an empirically-derived relationship between paleo-spreading rates and abyssal hill roughness, with corrections for sedimentation. The second estimate was based on small-scale (<100 km) roughness of satellite altimetry-derived gravity field, using upward continuation relationships to derive estimates of abyssal hill roughness at the seafloor at scales less than approximately 20 km. The lee wave generation rate was found to be between 0.34 to 0.49 TW. The Southern Hemisphere produced 92% of the lee wave energy, with the Southern Ocean dominating. Strength of the bottom flow was the most important factor in producing the global pattern of generation rate, except in the Indian Ocean where extremely rough topography produced strong lee wave generation despite only moderate bottom flows. The results imply about one half of the mechanical power input to the ocean general circulation from the extra-equatorial wind stress of the World Ocean results from abyssal lee wave generation. Topographic length scales between 176 m and 2.5 km (horizontal wavelengths between 1 and 16 km) accounted for 90% of the globally integrated generation.

A 2D-numerical modeling of the generation and propagation of internal solitary waves in the Luzon Strait

The wide presence of internal solitary waves (ISWs) in the northern South China Sea (SCS) has been confirmed by both Synthetic Aperture Radar (SAR) images and in situ observations. These ISWs are believed being generated over the varying topography in the Luzon Strait. They typically propagate westwards into the SCS with a diurnal or semidiurnal period. Their generation sites are, however, not yet solidly identified. To obtain a clear picture of the ISWs, we designed numerical experiments to analyze the generation and propagation of the ISWs in the Luzon Strait using a 2-dimensional non-hydrostatic model. The model current is forced by barotropic or baroclinic currents imposed at open boundaries. The experiments show that the tidal current serves as a kind of triggering force for the ISWs over the submarine ridges in the strait. Under the forcing of tidal currents, depressions are formed near the ridges. The ISWs then split from the depressions through a process different from lee-wave generation mechanism. The appearance of the ISWs is influenced by the strength and period of the forcing current: the ISWs are more likely to be generated by a stronger tidal current. That is why the ISWs in the Luzon Strait are frequently observed during spring tide. Compared with diurnal tidal current, the ISWs generated by semidiurnal tidal current with the same amplitude is much more energetic. It is partly because that the wave beams in diurnal frequency have a larger angle with the vertical direction, thus are more likely to be reflected by the topography slope. The impact of the Kuroshio to the ISWs is also analyzed by adding a vertical uniform or shear current at boundaries. A vertically uniform current may generate ISWs directly. On the other hand, a vertically shear current, which is more realistic to represent the Kuroshio branch, seems to have little influence on the generation process and radiating direction of the ISWs in the Luzon Strait.

Effect of water depth and the bottom boundary layer upon internal wave generation over abrupt topography

The role of water depth and bottom boundary layer turbulence upon lee-wave generation in sill regions is examined. Their effect upon vertical mixing is also considered. Calculations are performed using a non-hydrostatic model in cross-section form with a specified tidal forcing. Initial calculations in deeper water and a sill height such that the sill top is well removed from the surrounding bed region showed that downstream lee-wave generation and associated mixing increased as bottom friction coefficient k increased. This was associated with an increase in current shear across the sill. However, for a given k, increasing vertical eddy viscosity A v reduced vertical shear in the across sill velocity, leading to a reduction in lee-wave amplitude and associated mixing. Subsequent calculations using shallower water showed that for a given k and A v, lee-wave generation was reduced due to the shallower water depth and changes in the bottom boundary layer. However, in this case (unlike in the deepwater case), there is an appreciable bottom current. This gives rise to bottom mixing which in shallow water extends to mid-depth and enhances the mid-water mixing that is found on the lee side of the sill. Final calculations with deeper water but small sill height showed that lee waves could propagate over the sill, thereby reducing their contribution to mixing. In this case, bottom mixing was the major source of mixing which was mainly confined to the near bed region, with little mid-water mixing.

Well-balanced compressible cut-cell simulation of atmospheric flow

Cut-cell meshes present an attractive alternative to terrain-following coordinates for the representation of topography within atmospheric flow simulations, particularly in regions of steep topographic gradients. In this paper, we present an explicit two-dimensional method for the numerical solution on such meshes of atmospheric flow equations including gravitational sources. This method is fully conservative and allows for time steps determined by the regular grid spacing, avoiding potential stability issues due to arbitrarily small boundary cells. We believe that the scheme is unique in that it is developed within a dimensionally split framework, in which each coordinate direction in the flow is solved independently at each time step. Other notable features of the scheme are: (i) its conceptual and practical simplicity, (ii) its flexibility with regard to the one-dimensional flux approximation scheme employed, and (iii) the well-balancing of the gravitational sources allowing for stable simulation of near-hydrostatic flows. The presented method is applied to a selection of test problems including buoyant bubble rise interacting with geometry and lee-wave generation due to topography.

Large-amplitude internal waves observed in the Kruzenshtern Strait of the Kuril Island Chain and possible water transport and mixing

Vertical sections of temperature and current velocity across the sill in the Kruzenshtern Strait were observed four times in one day in September 1997. The observations captured nonlinear internal waves near the sea surface, which are likely to be tidally generated. Two of these waves had maximum amplitudes near the surface and each contained a trapped core, suggesting that a significant volumetric flux was induced. At mid depths, isotherms moved vertically more than 50 m, which is consistent with the generation of internal lee waves. From water-mass movements, the currents are estimated to flow from the North Pacific to the Sea of Okhotsk in the temperature-minimum layer at a mean speed of 0.2– 0.3 m s - 1 . The mean flow was weaker and the oscillatory component was stronger at mid depths, with the amplitude of the latter around 0.4 m s - 1 , leading to the above lee wave generation. Active temporal variations in potential temperature, particularly those associated with small-scale patches, suggest the occurrence of vigorous water-mass transformation. Vertical and horizontal diffusivities are estimated to be O ( 10 - 2 m 2 s - 1 ) and O ( 10 2 – 10 3 m 2 s - 1 ) , respectively. Because the tidal amplitude was close to the annual minimum at the time of observation, larger-amplitude internal waves and more vigorous water-mass transformation could take place at other times of the year.

Non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions

A three-dimensional non-linear, non-hydrostatic model in cross-sectional form is used to determine the factors influencing the relative importance of the linear, non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions due to tidal forcing. The importance of the free surface elevation term is also considered. Idealised topography representing the sill at the entrance to Loch Etive, the site of a recent measurement programme, is used. Calculations show that the non-linear terms in the energy flux become increasingly important as the sill Froude Number (Fs) increases and the sill aspect ratio is increased. The vertical profile of the stratification, in particular its value close to the sill crest where internal waves are generated, has a significant influence on unsteady lee wave and mixed tidal–lee wave generation and the non-linear contribution to the energy flux. Calculations show that as Fs increases, the energy flux due to the non-linear and non-hydrostatic terms increases more rapidly than the linear term. The importance of the non-linear terms in the energy flux also increases as the sill aspect ratio is increased. Increasing the buoyancy frequency reduces the contribution of the non-hydrostatic and non-linear terms to the total energy flux. Also, as the buoyancy frequency is increased, this reduces unsteady lee wave and mixed tidal–lee wave generation. In essence, these calculations show that the energy flux due to the non-hydrostatic and non-linear terms is appreciable in sill regions.

Influence of bottom frictional effects in sill regions upon lee wave generation and implications for internal mixing

A cross-sectional nonhydrostatic model using idealized sill topography is used to examine the influence of bottom friction upon unsteady lee wave generation and flow in the region of a sill. The implications of changes in shear and lee wave intensity in terms of local mixing are also considered. Motion is induced by a barotropic tidal flow which produces a hydraulic transition, associated with which are convective overturning cells, wave breaking, and unsteady lee waves that give rise to mixing on the lee side of the sill. Calculations show that, as bottom friction is increased, current profiles on the shallow sill crest develop a highly sheared bottom boundary layer. This enhanced current shear changes the downwelling of isotherms downstream of the sill with an associated increase in the hydraulic transition, wave breaking, and convective mixing in the upper part of the water column. Both short and longer time calculations with wide and narrow sills for a number of sill depths and buoyancy frequencies confirm that increasing bottom friction modifies the flow and unsteady lee wave distribution on the downstream side of a sill. Associated with this increase in bottom friction coefficient, there is increased mixing in the upper part of the water column with an associated decrease in the vertical temperature gradient. However, this increase in mixing and decrease in temperature gradient in the upper part of the water column is very different from the conventional change in near-bed temperature gradient produced by increased bottom mixing that occurs in shallow sea regions as the bottom drag coefficient is increased.

Lee waves from a sphere in a stratified flow

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocityUof a stratified fluid of buoyancy frequencyNpast a sphere of radiusaare presented, for either weak or strong stratification, corresponding to either large or small internal Froude numberF=U/(Na), respectively. ForF⋙1, the fluid separates into two regions radially: an inner region of scaleawith three-dimensional irrotational flow unaffected by the stratification, and an outer region of scaleU/Nwith small-amplitude lee waves generated by theO(1) vertical motion in the inner region. ForF⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distanceU/Nabove the bottom of the sphere to the upper dividing streamsurface situated at a distanceU/Nbelow the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by theO(F) vertical motion in the transition layers.The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: forF⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; forF⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. ForF⋙1, the drag coefficient decreases as (lnF+7/4-γ)/(4F4), with γ the Euler constant; forF⋘1, it increases as$(32\surd2)/(15\upi)F^{3/2}$. The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.