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Callum J. Shakespeare

Abstract

The generation of internal waves at abyssal hills has been proposed as an important source of bottom-intensified mixing and a sink of geostrophic momentum. Using the theory of Bell, previous authors have calculated either the generation of lee waves by geostrophic flow or the generation of the internal tide by the barotropic tide, but never both together. However, the Bell theory shows that the two are interdependent: that is, the presence of a barotropic tide modifies the generation of lee waves, and the presence of a geostrophic (time mean) flow modifies the generation of the internal tide. Here we extend the theory of Bell to incorporate multiple tidal constituents. Using this extended theory, we recalculate global wave fluxes of energy and momentum using the abyssal-hill spectra, model-derived abyssal ocean stratification and geostrophic flow estimates, and the TPX08 tidal velocities for the eight major constituents. The energy flux into lee waves is suppressed by 13%–19% as a result of the inclusion of tides. The generated wave flux is dominated by the principal lunar semidiurnal tide (M2), and its harmonics and combinations, with the strongest fluxes occurring along midocean ridges. The internal tide generation is strongly asymmetric because of Doppler shifting by the geostrophic abyssal flow, with 55%–63% of the wave energy flux (and stress) directed upstream, against the geostrophic flow. As a consequence, there is a net wave stress associated with generation of the internal tide that reaches magnitudes of 0.01–0.1 N m−2 in the vicinity of midocean ridges.

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Callum J. Shakespeare

Abstract

Curvature can play a significant role in the dynamics of density fronts at small scales and in low-latitude regions of the ocean. Fronts can be displaced from balance by rapid forcing and undergo an adjustment toward a more stable state or be strained and sharpened by surrounding flow in a process known as frontogenesis. This study investigates the role of curvature in adjustment and frontogenesis using the idealized configuration of an axisymmetric eddy and associated circular front. As a result of the curvature, the balanced state of this system is not geostrophic balance, where pressure and Coriolis forces exactly balance, but cyclogeostrophic balance, where pressure and Coriolis forces combine to supply a net inwards centripetal force on fluid parcels. The parameter range for which cyclogeostrophically balanced states exist for a given unbalanced initial condition is determined. This parameter range is smaller for anticyclonic fronts (i.e., fronts curved around a warm core), which have larger angular velocities than comparable straight fronts, implying they are more likely to break down during adjustment. The reverse is true for cyclonic fronts. A model for the sharpening of a curved front in a background strain flow, analogous to the model for a straight front, is developed. Relative to a straight front subject to the same strain rate, vertical velocities are weaker for an anticyclonic front and stronger for a cyclonic front. Anticyclonic fronts collapse to a near discontinuity during frontogenesis far more rapidly than cyclonic fronts for the same strain rate.

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Callum J. Shakespeare

Abstract

A simple analytical model is developed to describe wave generation during frontogenesis forced by a horizontal strain field. In contrast to previous models, neither geostrophic nor hydrostatic balance is assumed. The generated waves are trapped in the strain field and form steady bands of enhanced vertical flow on either side of the surface front on scales from 1 to 100 km. The predictions of the analytical model are confirmed by comparison with fully nonlinear numerical simulations.

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Callum J. Shakespeare and Andrew McC. Hogg

Abstract

Recent numerical modeling studies have suggested significant spontaneous internal wave generation near the ocean surface and energy transfers to and from these waves in the ocean interior. Spontaneous generation is the emission of waves by unbalanced, large Rossby number flows in the absence of direct forcing. Here, the authors’ previous work is extended to investigate where and how these waves exchange energy with the nonwave (mean) flow. A novel double-filtering technique is adopted to separate first the wave and nonwave fields, then the individual upward- and downward-propagating wave fields, and thereby identify the pathways of energy transfer. These energy transfers are dominated by the interaction of the waves with the vertical shear in the mean flow. Spontaneously generated waves are found to be oriented such that the downward-propagating wave is amplified by the mean shear. The internal waves propagate through the entire model depth while dissipating energy and reflect back upward. The now-upward-propagating waves have the opposite sign interaction with the mean shear and decay, losing most of their energy to the nonwave flow in the upper 500 m. Overall, in the simulations described here, approximately 30% of the wave energy is dissipated, and 70% is returned to the mean flow. The apparent preferential orientation of spontaneous generation suggests a potentially unique role for these waves in the ocean energy budget in uniformly drawing net energy from mean flow in the upper-ocean interior and transporting it to depth.

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Leif N. Thomas and Callum J. Shakespeare

Abstract

A simple analytical model is used to elucidate a potential mechanism for steady-state mode water formation at a thermohaline front that involves frontogenesis, submesoscale lateral mixing, and cabbeling. This mechanism is motivated in part by recent observations of an extremely sharp, density-compensated front at the North Wall of the Gulf Stream. Here, the intergyre, along-isopycnal, salinity–temperature difference is compressed into a span of a few kilometers, making the flow susceptible to cabbeling. The sharpness of the front is caused by frontogenetic strain, which is presumably balanced by submesoscale lateral mixing processes. The balance is studied with the simple model, and a scaling is derived for the amount of water mass transformation resulting from the ensuing cabbeling. The transformation scales with the strain rate, equilibrated width of the front, and the square of the isopycnal temperature contrast across the front. At the major ocean fronts where mode waters are found, this isopycnal temperature contrast decreases with increasing density near the isopycnal layers where mode waters reside. This implies that cabbeling should result in a convergent diapycnal mass flux into mode water density classes. The scaling for the transformation suggests that at these fronts the process could generate 0.01–1 Sverdrups (Sv; 1 Sv ≡ 106 m3 s−1) of mode water. These formation rates, while smaller than mode water formation by air–sea fluxes, should be independent of season and thus could fill select isopycnal layers continuously and play an important role in the dynamics of mode waters on interannual time scales.

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Callum J. Shakespeare and Andrew McC. Hogg

Abstract

The action of the barotropic tide over seafloor topography is the major source of internal waves at the bottom of the ocean. This internal tide has long been recognized to play an important role in ocean mixing. Here it is shown that the internal tide is also associated with a net (domain integrated) momentum flux. The net flux occurs as a result of the Doppler shifting of the internal tide at the point of generation by near-bottom mean flows. Linear theory is presented that predicts the amplitude of the wave momentum flux. The net flux scales with the bottom flow speed and the topographic wavenumber to the fourth power and is directed opposite to the bottom flow. For realistic topography, the predicted peak momentum flux occurs at scales of order 10 km and smaller, with magnitudes of order 10−3–10−2 N m−2. The theory is verified by comparison with a suite of idealized internal wave-resolving simulations. The simulations show that, for the topography considered, the wave momentum flux radiates away from the bottom and enhances mean and eddying flow when the tidal waves dissipate in the upper ocean. Our results suggest that internal tides may play an important role in forcing the upper ocean.

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Callum J. Shakespeare and Andrew McC. Hogg

Abstract

Recent theories, models, and observations have suggested the presence of significant spontaneous internal wave generation at density fronts near the ocean surface. Spontaneous generation is the emission of waves by unbalanced, large Rossby number flows in the absence of direct forcing. Here, spontaneous generation is investigated in a zonally reentrant channel model using parameter values typical of the Southern Ocean. The model is carefully equilibrated to obtain a steady-state wave field for which a closed energy budget is formulated. There are two main results: First, waves are spontaneously generated at sharp fronts in the top 50 m of the model. The magnitude of the energy flux to the wave field at these fronts is comparable to that from other mechanisms of wave generation. Second, the surface-generated wave field is amplified in the model interior through interaction with horizontal density gradients within the main zonal current. The magnitude of the mean-to-wave conversion in the model interior is comparable to recent observational estimates and is the dominant source of wave energy in the model, exceeding the initial spontaneous generation. This second result suggests that internal amplification of the wave field may contribute to the ocean’s internal wave energy budget at a rate commensurate with known generation mechanisms.

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Callum J. Shakespeare and John R. Taylor

Abstract

A 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.

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Callum J. Shakespeare and Andrew McC. Hogg

Abstract

An analytical model of the full-depth ocean stratification and meridional overturning circulation for an idealized Atlantic basin with a circumpolar channel is presented. The model explicitly describes the ocean response to both Southern Ocean winds and the global pattern and strength of prescribed surface buoyancy fluxes. The construction of three layers, defined by the two isopycnals of overturning extrema, allows the description of circulation and stratification in both the upper and abyssal ocean. The system is fully solved in the adiabatic limit to yield scales for the surface layer thickness, buoyancies of each layer, and overturning magnitudes. The analytical model also allows scaling of the Antarctic Circumpolar Current (ACC) transport. The veracity of the three-layer framework and derived scales is confirmed by applying the analytical model to an idealized geometry, eddy-permitting ocean general circulation model.

Consistent with previous results, the abyssal overturning is found to scale inversely with wind stress, whereas the North Atlantic overturning and surface-layer thickness scale linearly with wind stress. In terms of the prescribed surface buoyancy fluxes, increased negative fluxes (buoyancy removal) in the North Atlantic increase the North Atlantic overturning and surface-layer thickness, whereas increased positive fluxes in the middle and low latitudes lead to a decrease in both parameters. Increased negative surface buoyancy fluxes to the south of Drake Passage increase the abyssal overturning and reduce the abyssal buoyancy. The ACC transport scales to first order with the sum of the Ekman transport and the abyssal overturning and thus increases with both wind stress and southern surface buoyancy flux magnitude.

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Callum J. Shakespeare and Leif N. Thomas

Abstract

Submesoscale-resolving numerical simulations are used to investigate a mechanism for sustained mode water formation via cabbeling at thermohaline fronts subject to a confluent strain flow. The simulations serve to further elucidate the mechanism and refine the predictions of the analytical model of Thomas and Shakespeare. Unlike other proposed mechanisms involving air–sea fluxes, the cabbeling mechanism, in addition to driving significant mode water formation, uniquely determines the thermohaline properties of the mode water given knowledge of the source water masses on either side of the front. The process of mode water formation in the simulations is as follows: Confluent flow associated with idealized mesoscale eddies forces water horizontally toward the front. The frontogenetic circulation draws this water near adiabatically from the full depth of the thermohaline front up to the surface 25 m, where resolved submesoscale instabilities drive intense mixing across the thermohaline front, creating the mode water. The mode water is denser than the surrounding stratified fluid and sinks to fill its neutral buoyancy layer at depth. This layer gradually expands up to the surface, and eddies composed entirely of this mode water detach from the front and accumulate in the diffluent regions of the domain. The process continues until the source water masses are exhausted. The temperature–salinity (TS) relation of the resulting mode water is biased to the properties of the source water that has the larger isopycnal TS anomaly. This mechanism has the potential to drive O(1) Sv (1 Sv ≡ 106 m3 s−1) mode water formation and may be important in determining the properties of mode water in the global oceans.

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