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- Author or Editor: Mark A. Merrifield x
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Abstract
The scattering of coastal-trapped waves (CTWs) by a region of irregular shelf bathymetry is determined from a circulation integral of the depth-integrated momentum equations. For relatively weak stratification the conservation of geostrophic mass flux along isobaths is used to show that bottom pressure of the transmitted waves is equal to bottom pressure p b of the incident waves, when mapped along constant depth contours, plus corrections for the effects of frictional spindown and the rate of change of relative vorticity. These corrections result from changes in the incident wave alongisobath velocity, which can be amplified by the convergence of isobaths between the incident and transmitted regions. For the case of the Labrador shelf, the convergence of isobaths south of the (incident) Hudson Strait region leads to a tenfold increase in the production of relative vorticity and in the correction for pressure for a mode 1 incident wave. This leading order increase in vorticity production violates the assumption of constant geostrophic mass flux and implies that the frictional correction, while small, is invalid. However, the transmitted mode 1 and 2 amplitudes determined are insensitive to these corrections and, in agreement with observations, are of similar magnitude and about 180° out of phase.
Abstract
The scattering of coastal-trapped waves (CTWs) by a region of irregular shelf bathymetry is determined from a circulation integral of the depth-integrated momentum equations. For relatively weak stratification the conservation of geostrophic mass flux along isobaths is used to show that bottom pressure of the transmitted waves is equal to bottom pressure p b of the incident waves, when mapped along constant depth contours, plus corrections for the effects of frictional spindown and the rate of change of relative vorticity. These corrections result from changes in the incident wave alongisobath velocity, which can be amplified by the convergence of isobaths between the incident and transmitted regions. For the case of the Labrador shelf, the convergence of isobaths south of the (incident) Hudson Strait region leads to a tenfold increase in the production of relative vorticity and in the correction for pressure for a mode 1 incident wave. This leading order increase in vorticity production violates the assumption of constant geostrophic mass flux and implies that the frictional correction, while small, is invalid. However, the transmitted mode 1 and 2 amplitudes determined are insensitive to these corrections and, in agreement with observations, are of similar magnitude and about 180° out of phase.
Abstract
A network of island tide gauges is used to estimate interannual geostrophic current anomalies (GCAs) in the western Pacific from 1975 to 1997. The focus of this study is the zonal component of the current averaged between 160°E and 180° and 2° to 7° north and south of the equator in the mean flow regions associated with the North Equatorial Countercurrent (NECC) and the South Equatorial Current (SEC), respectively. The tide gauge GCA estimates agree closely with similarly derived currents from TOPEX/Poseidon sea level anomalies. The GCAs in the western Pacific relate to a basin-scale adjustment associated with the El Niño–Southern Oscillation, characterized here using empirical orthogonal functions of tide gauge and supporting sea surface temperature and heat storage data. The dominant EOF mode describes the mature phase of ENSO events and correlates (0.8) with the GCA south of the equator. The second mode describes transitions to and from ENSO events and correlates (0.9) with the GCA north of the equator. The typical scenario then is for the NECC to intensify about 6 months prior to the peak of an El Niño, to remain near mean conditions during the peak stage of El Niño, and to later weaken about 6 months following the peak. In contrast, the SEC generally weakens throughout an El Niño displaying eastward anomalies. This equatorial asymmetry in the GCAs is consistent with a similar asymmetry in the wind field over the western Pacific. The phase differences between the NECC and SEC are less apparent during La Niña events. The GCA results provide further evidence that transitional phases of ENSO are more active north than south of the equator in the warm pool region.
Abstract
A network of island tide gauges is used to estimate interannual geostrophic current anomalies (GCAs) in the western Pacific from 1975 to 1997. The focus of this study is the zonal component of the current averaged between 160°E and 180° and 2° to 7° north and south of the equator in the mean flow regions associated with the North Equatorial Countercurrent (NECC) and the South Equatorial Current (SEC), respectively. The tide gauge GCA estimates agree closely with similarly derived currents from TOPEX/Poseidon sea level anomalies. The GCAs in the western Pacific relate to a basin-scale adjustment associated with the El Niño–Southern Oscillation, characterized here using empirical orthogonal functions of tide gauge and supporting sea surface temperature and heat storage data. The dominant EOF mode describes the mature phase of ENSO events and correlates (0.8) with the GCA south of the equator. The second mode describes transitions to and from ENSO events and correlates (0.9) with the GCA north of the equator. The typical scenario then is for the NECC to intensify about 6 months prior to the peak of an El Niño, to remain near mean conditions during the peak stage of El Niño, and to later weaken about 6 months following the peak. In contrast, the SEC generally weakens throughout an El Niño displaying eastward anomalies. This equatorial asymmetry in the GCAs is consistent with a similar asymmetry in the wind field over the western Pacific. The phase differences between the NECC and SEC are less apparent during La Niña events. The GCA results provide further evidence that transitional phases of ENSO are more active north than south of the equator in the warm pool region.
Abstract
Large semidiurnal vertical displacements (≈100 m) and strong baroclinic currents (≈0.5 m s−1; several times as large as barotropic currents) dominate motions in Mamala Bay, outside the mouth of Pearl Harbor, Hawaii. During September 2002, the authors sought to characterize them with a 2-month McLane moored profiler deployment and a 4-day intensive survey with a towed CTD/ADCP and the Research Vessel (R/V) Revelle hydrographic sonar. Spatial maps and time series of turbulent dissipation rate ϵ, diapycnal diffusivity Kρ , isopycnal displacement η, velocity u, energy E, and energy flux F are presented. Dissipation rate peaks in the lower 150 m during rising isopycnals and high strain and shows a factor-of-50 spring–neap modulation. The largest Kρ values, in the western bay near a submarine ridge, exceed 10−3 m2 s−1. The M 2 phases of η and u increase toward the west, implying a westward phase velocity cp ≈ 1 m s−1 and horizontal wavelength ≈60 km, consistent with theoretical mode-1 values. These phases vary strongly (≈±45°) in time relative to astronomical forcing, implying remotely generated signals. Energy and energy flux peak 1–3 days after spring tide, supporting this interpretation. The group velocity, computed as the ratio F/E, is near ≈1 m s−1, also in agreement with theoretical mode-1 values. Spatial maps of energy flux agree well with results from the Princeton Ocean Model, indicating converging fluxes in the western bay from waves generated to the east and west. The observations indicate a time-varying interference pattern between these waves that is modulated by background stratification between their sources and Mamala Bay.
Abstract
Large semidiurnal vertical displacements (≈100 m) and strong baroclinic currents (≈0.5 m s−1; several times as large as barotropic currents) dominate motions in Mamala Bay, outside the mouth of Pearl Harbor, Hawaii. During September 2002, the authors sought to characterize them with a 2-month McLane moored profiler deployment and a 4-day intensive survey with a towed CTD/ADCP and the Research Vessel (R/V) Revelle hydrographic sonar. Spatial maps and time series of turbulent dissipation rate ϵ, diapycnal diffusivity Kρ , isopycnal displacement η, velocity u, energy E, and energy flux F are presented. Dissipation rate peaks in the lower 150 m during rising isopycnals and high strain and shows a factor-of-50 spring–neap modulation. The largest Kρ values, in the western bay near a submarine ridge, exceed 10−3 m2 s−1. The M 2 phases of η and u increase toward the west, implying a westward phase velocity cp ≈ 1 m s−1 and horizontal wavelength ≈60 km, consistent with theoretical mode-1 values. These phases vary strongly (≈±45°) in time relative to astronomical forcing, implying remotely generated signals. Energy and energy flux peak 1–3 days after spring tide, supporting this interpretation. The group velocity, computed as the ratio F/E, is near ≈1 m s−1, also in agreement with theoretical mode-1 values. Spatial maps of energy flux agree well with results from the Princeton Ocean Model, indicating converging fluxes in the western bay from waves generated to the east and west. The observations indicate a time-varying interference pattern between these waves that is modulated by background stratification between their sources and Mamala Bay.
Abstract
A 3-month mooring deployment (August–November 2002) was made in 2425-m depth, on the south flank of Kaena Ridge, Hawaii, to examine tidal variations within 200 m of the steeply sloping bottom. Horizontal currents and vertical displacements, inferred from temperature fluctuations, are dominated by the semidiurnal internal tide with amplitudes of ≥ 0.1 m s−1 and ∼100 m, respectively. A series of temperature sensors detected tidally driven overturns with vertical scales of ∼100 m. A Thorpe scale analysis of the overturns yields a time-averaged dissipation near the bottom of 1.2 × 10−8 W kg−1, 10–100 times that at similar depths in the ocean interior 50 km from the ridge. Dissipation events much larger than the overall mean (up to 10−6 W kg−1) occur predominantly during two phases of the semidiurnal tide: 1) at peak downslope flows when the tidal stratification is minimum (N = 5 × 10−4 s−1) and 2) at the flow reversal from downslope to upslope flow when the tidal stratification is ordinarily increasing (N = 10−3 s−1). Dissipation associated with flow reversal mixing is 2 times that of downslope flow mixing. Although the overturn events occur at these tidal phases and they exhibit a general spring–neap modulation, they are not as regular as the tidal currents. Shear instabilities, particularly due to tidal strain enhancements, appear to trigger downslope flow mixing. Convective instabilities are proposed as the cause for flow reversal mixing, owing to the oblique propagation of the internal tide down the slope. The generation of similar tidally driven mixing features on continental slopes has been attributed to oblique wave propagation in previous studies. Because the mechanical energy source for mixing is primarily due to the internal tide rather than the surface tide, the observed intermittency of overturn events is attributed to the broadbanded nature of the internal tide.
Abstract
A 3-month mooring deployment (August–November 2002) was made in 2425-m depth, on the south flank of Kaena Ridge, Hawaii, to examine tidal variations within 200 m of the steeply sloping bottom. Horizontal currents and vertical displacements, inferred from temperature fluctuations, are dominated by the semidiurnal internal tide with amplitudes of ≥ 0.1 m s−1 and ∼100 m, respectively. A series of temperature sensors detected tidally driven overturns with vertical scales of ∼100 m. A Thorpe scale analysis of the overturns yields a time-averaged dissipation near the bottom of 1.2 × 10−8 W kg−1, 10–100 times that at similar depths in the ocean interior 50 km from the ridge. Dissipation events much larger than the overall mean (up to 10−6 W kg−1) occur predominantly during two phases of the semidiurnal tide: 1) at peak downslope flows when the tidal stratification is minimum (N = 5 × 10−4 s−1) and 2) at the flow reversal from downslope to upslope flow when the tidal stratification is ordinarily increasing (N = 10−3 s−1). Dissipation associated with flow reversal mixing is 2 times that of downslope flow mixing. Although the overturn events occur at these tidal phases and they exhibit a general spring–neap modulation, they are not as regular as the tidal currents. Shear instabilities, particularly due to tidal strain enhancements, appear to trigger downslope flow mixing. Convective instabilities are proposed as the cause for flow reversal mixing, owing to the oblique propagation of the internal tide down the slope. The generation of similar tidally driven mixing features on continental slopes has been attributed to oblique wave propagation in previous studies. Because the mechanical energy source for mixing is primarily due to the internal tide rather than the surface tide, the observed intermittency of overturn events is attributed to the broadbanded nature of the internal tide.
Abstract
Microstructure observations over a small seamount on the Kaena Ridge, Hawaii, showed asymmetry in the along- and across-ridge directions. The ∼400-m-high seamount is on the southern edge of the ridge (centered at 21°43′49″N, 158°38′48″W), 42 km northwest of Oahu. A 1-km-resolution numerical simulation shows that the flow within the depth range of the seamount tends to be accelerated around the seamount rather than going up and over it. The flow patterns, however, are more complicated than for an isolated seamount because of the influence of the ∼3000-m-high Kaena Ridge. Comparison with the numerical simulations indicates that the across-ridge asymmetry, in which dissipation on the north-northeastern side of the seamount was higher and more concentrated toward the bed than on the south-southwestern side, is consistent with an M
2 tidal beam generated at the northern edge of the ridge. The along-ridge asymmetry, with higher dissipation on the east-southeastern flank than on the west-northwestern flank, is in qualitative agreement with M
2 shear variance from the model simulation. The average observed dissipation rate over the seamount was
Abstract
Microstructure observations over a small seamount on the Kaena Ridge, Hawaii, showed asymmetry in the along- and across-ridge directions. The ∼400-m-high seamount is on the southern edge of the ridge (centered at 21°43′49″N, 158°38′48″W), 42 km northwest of Oahu. A 1-km-resolution numerical simulation shows that the flow within the depth range of the seamount tends to be accelerated around the seamount rather than going up and over it. The flow patterns, however, are more complicated than for an isolated seamount because of the influence of the ∼3000-m-high Kaena Ridge. Comparison with the numerical simulations indicates that the across-ridge asymmetry, in which dissipation on the north-northeastern side of the seamount was higher and more concentrated toward the bed than on the south-southwestern side, is consistent with an M
2 tidal beam generated at the northern edge of the ridge. The along-ridge asymmetry, with higher dissipation on the east-southeastern flank than on the west-northwestern flank, is in qualitative agreement with M
2 shear variance from the model simulation. The average observed dissipation rate over the seamount was
Abstract
Fjords along the western Antarctic Peninsula are episodically exposed to strong winds flowing down marine-terminating glaciers and out over the ocean. These wind events could potentially be an important mechanism for the ventilation of fjord waters. A strong wind event was observed in Andvord Bay in December 2015, and was associated with significant increases in upper-ocean salinity. We examine the dynamical impacts of such wind events during the ice-free summer season using a numerical model. Passive tracers are used to identify water mass pathways and quantify exchange with the outer ocean. Upwelling and outflow in the model fjord generate an average salinity increase of 0.3 in the upper ocean during the event, similar to observations from Andvord Bay. Down-fjord wind events are a highly efficient mechanism for flushing out the upper fjord waters, but have little net impact on deep waters in the inner fjord. As such, summer episodic wind events likely have a large effect on fjord phytoplankton dynamics and export of glacially modified upper waters, but are an unlikely mechanism for the replenishment of deep basin waters and oceanic heat transport toward inner-fjord glaciers.
Abstract
Fjords along the western Antarctic Peninsula are episodically exposed to strong winds flowing down marine-terminating glaciers and out over the ocean. These wind events could potentially be an important mechanism for the ventilation of fjord waters. A strong wind event was observed in Andvord Bay in December 2015, and was associated with significant increases in upper-ocean salinity. We examine the dynamical impacts of such wind events during the ice-free summer season using a numerical model. Passive tracers are used to identify water mass pathways and quantify exchange with the outer ocean. Upwelling and outflow in the model fjord generate an average salinity increase of 0.3 in the upper ocean during the event, similar to observations from Andvord Bay. Down-fjord wind events are a highly efficient mechanism for flushing out the upper fjord waters, but have little net impact on deep waters in the inner fjord. As such, summer episodic wind events likely have a large effect on fjord phytoplankton dynamics and export of glacially modified upper waters, but are an unlikely mechanism for the replenishment of deep basin waters and oceanic heat transport toward inner-fjord glaciers.
Abstract
Over the past century, tide gauges in Hawaii have recorded interdecadal sea level variations that are coherent along the island chain. The generation of this signal and its relationship to other interdecadal variability are investigated, with a focus on the last decade. Hawaii sea level is correlated with sea surface height (SSH) over a significant portion of the North Pacific Ocean, and with the Pacific–North America (PNA) index, which represents teleconnections between tropical and midlatitude atmospheric variations. Similar variations extend well below the thermocline in World Ocean Atlas temperature. Comparison with NCEP reanalysis wind and pressure shows that high (low) sea level phases around Hawaii are associated with an increase (decrease) in the strength of the Aleutian low. The associated wind stress curl pattern is dynamically consistent with observed sea level anomalies, suggesting that sea level at Hawaii represents large-scale changes that are directly wind-forced in concert with the PNA. Atmospheric modulation, as opposed to Rossby wave propagation, may explain the linkage of Hawaii sea level to North American sea level and ENSO events. A wind-forced, baroclinic Rossby wave model replicates some aspects of the interdecadal SSH variations and their spatial structure but fails to predict them in detail near Hawaii. The accuracy of wind products in this region and over this time period may be a limiting factor. Variations in mixed layer temperature due to surface heat flux anomalies may also contribute to the interdecadal sea level signal at Hawaii.
Abstract
Over the past century, tide gauges in Hawaii have recorded interdecadal sea level variations that are coherent along the island chain. The generation of this signal and its relationship to other interdecadal variability are investigated, with a focus on the last decade. Hawaii sea level is correlated with sea surface height (SSH) over a significant portion of the North Pacific Ocean, and with the Pacific–North America (PNA) index, which represents teleconnections between tropical and midlatitude atmospheric variations. Similar variations extend well below the thermocline in World Ocean Atlas temperature. Comparison with NCEP reanalysis wind and pressure shows that high (low) sea level phases around Hawaii are associated with an increase (decrease) in the strength of the Aleutian low. The associated wind stress curl pattern is dynamically consistent with observed sea level anomalies, suggesting that sea level at Hawaii represents large-scale changes that are directly wind-forced in concert with the PNA. Atmospheric modulation, as opposed to Rossby wave propagation, may explain the linkage of Hawaii sea level to North American sea level and ENSO events. A wind-forced, baroclinic Rossby wave model replicates some aspects of the interdecadal SSH variations and their spatial structure but fails to predict them in detail near Hawaii. The accuracy of wind products in this region and over this time period may be a limiting factor. Variations in mixed layer temperature due to surface heat flux anomalies may also contribute to the interdecadal sea level signal at Hawaii.
Abstract
Motivated by observations of enhanced near-inertial currents at the island chain of Palau, the modification of wind-generated near-inertial oscillations (NIOs) by the presence of an island is examined using the analytic solutions of Longuet-Higgins and a linear, inviscid, 1.5-layer reduced-gravity model. The analytic solution for oscillations at the inertial frequency f provides insights into flow adjustment near the island but excludes wave dynamics. To account for wave motion, the numerical model initially is forced by a large-scale wind field rotating at f, where the forcing is increased then decreased to zero. Numerical simulations are carried out over a range of island radii and the ocean response detailed. Near the island, wind energy in the frequency band near f can excite subinertial island-trapped waves and superinertial Poincaré waves. In the small-island limit, both the Poincaré waves and the island-trapped waves are very near f, and their sum resembles the Longuet-Higgins analytic solution but with increased amplitude near the island. The flow field can be viewed as primarily a far-field NIO locally deflected by the island plus an island-trapped contribution, leading to enhanced near-inertial currents near the island, on the scale of the island radius. As the island size is increased, the island-trapped wave frequency deviates further from f and its amplitude depends strongly on the frequency bandwidth and wavenumber structure of the wind forcing. In the large-island limit, the island-trapped wave resembles a Kelvin wave, and the sum of incident and reflected Poincaré waves suppresses the near-inertial current amplitude near the island.
Significance Statement
Strong, impulsive winds over the ocean excite currents that rotate in the opposite direction to Earth’s rotation. This work examines how these wind-generated currents, known as near-inertial oscillations (NIOs), are modified by the presence of an island. Around small islands, the primary response is locally enhanced near-inertial currents. Alternatively, around large islands, near-inertial currents are weaker. Understanding how these currents behave should provide insight into the physical processes that drive current variability near islands and spur local mixing.
Abstract
Motivated by observations of enhanced near-inertial currents at the island chain of Palau, the modification of wind-generated near-inertial oscillations (NIOs) by the presence of an island is examined using the analytic solutions of Longuet-Higgins and a linear, inviscid, 1.5-layer reduced-gravity model. The analytic solution for oscillations at the inertial frequency f provides insights into flow adjustment near the island but excludes wave dynamics. To account for wave motion, the numerical model initially is forced by a large-scale wind field rotating at f, where the forcing is increased then decreased to zero. Numerical simulations are carried out over a range of island radii and the ocean response detailed. Near the island, wind energy in the frequency band near f can excite subinertial island-trapped waves and superinertial Poincaré waves. In the small-island limit, both the Poincaré waves and the island-trapped waves are very near f, and their sum resembles the Longuet-Higgins analytic solution but with increased amplitude near the island. The flow field can be viewed as primarily a far-field NIO locally deflected by the island plus an island-trapped contribution, leading to enhanced near-inertial currents near the island, on the scale of the island radius. As the island size is increased, the island-trapped wave frequency deviates further from f and its amplitude depends strongly on the frequency bandwidth and wavenumber structure of the wind forcing. In the large-island limit, the island-trapped wave resembles a Kelvin wave, and the sum of incident and reflected Poincaré waves suppresses the near-inertial current amplitude near the island.
Significance Statement
Strong, impulsive winds over the ocean excite currents that rotate in the opposite direction to Earth’s rotation. This work examines how these wind-generated currents, known as near-inertial oscillations (NIOs), are modified by the presence of an island. Around small islands, the primary response is locally enhanced near-inertial currents. Alternatively, around large islands, near-inertial currents are weaker. Understanding how these currents behave should provide insight into the physical processes that drive current variability near islands and spur local mixing.
Abstract
Most of the M 2 internal tide energy generated at the Hawaiian Ridge radiates away in modes 1 and 2, but direct observation of these propagating waves is complicated by the complexity of the bathymetry at the generation region and by the presence of interference patterns. Observations from satellite altimetry, a tomographic array, and the R/P FLIP taken during the Farfield Program of the Hawaiian Ocean Mixing Experiment (HOME) are found to be in good agreement with the output of a high-resolution primitive equation model, simulating the generation and propagation of internal tides. The model shows that different modes are generated with different amplitudes along complex topography. Multiple sources produce internal tides that sum constructively and destructively as they propagate. The major generation sites can be identified using a simplified 2D idealized knife-edge ridge model. Four line sources located on the Hawaiian Ridge reproduce the interference pattern of sea surface height and energy flux density fields from the numerical model for modes 1 and 2. Waves from multiple sources and their interference pattern have to be taken into account to correctly interpret in situ observations and satellite altimetry.
Abstract
Most of the M 2 internal tide energy generated at the Hawaiian Ridge radiates away in modes 1 and 2, but direct observation of these propagating waves is complicated by the complexity of the bathymetry at the generation region and by the presence of interference patterns. Observations from satellite altimetry, a tomographic array, and the R/P FLIP taken during the Farfield Program of the Hawaiian Ocean Mixing Experiment (HOME) are found to be in good agreement with the output of a high-resolution primitive equation model, simulating the generation and propagation of internal tides. The model shows that different modes are generated with different amplitudes along complex topography. Multiple sources produce internal tides that sum constructively and destructively as they propagate. The major generation sites can be identified using a simplified 2D idealized knife-edge ridge model. Four line sources located on the Hawaiian Ridge reproduce the interference pattern of sea surface height and energy flux density fields from the numerical model for modes 1 and 2. Waves from multiple sources and their interference pattern have to be taken into account to correctly interpret in situ observations and satellite altimetry.
Abstract
Full-depth velocity and density profiles taken along the 3000-m isobath characterize the semidiurnal internal tide and bottom-intensified turbulence along the Hawaiian Ridge. Observations reveal baroclinic energy fluxes of 21 ± 5 kW m−1 radiating from French Frigate Shoals, 17 ± 2.5 kW m−1 from Kauai Channel west of Oahu, and 13 ± 3.5 kW m−1 from west of Nihoa Island. Weaker fluxes of 1–4 ± 2 kW m−1 radiate from the region near Necker Island and east of Nihoa Island. Observed off-ridge energy fluxes generally agree to within a factor of 2 with those produced by a tidally forced numerical model. Average turbulent diapycnal diffusivity K is (0.5–1) × 10−4 m2 s–1 above 2000 m, increasing exponentially to 20 × 10−4 m2 s–1 near the bottom. Microstructure values agree well with those inferred from a finescale internal wave-based parameterization. A linear relationship between the vertically integrated energy flux and vertically integrated turbulent dissipation rate implies that dissipative length scales for the radiating internal tide exceed 1000 km.
Abstract
Full-depth velocity and density profiles taken along the 3000-m isobath characterize the semidiurnal internal tide and bottom-intensified turbulence along the Hawaiian Ridge. Observations reveal baroclinic energy fluxes of 21 ± 5 kW m−1 radiating from French Frigate Shoals, 17 ± 2.5 kW m−1 from Kauai Channel west of Oahu, and 13 ± 3.5 kW m−1 from west of Nihoa Island. Weaker fluxes of 1–4 ± 2 kW m−1 radiate from the region near Necker Island and east of Nihoa Island. Observed off-ridge energy fluxes generally agree to within a factor of 2 with those produced by a tidally forced numerical model. Average turbulent diapycnal diffusivity K is (0.5–1) × 10−4 m2 s–1 above 2000 m, increasing exponentially to 20 × 10−4 m2 s–1 near the bottom. Microstructure values agree well with those inferred from a finescale internal wave-based parameterization. A linear relationship between the vertically integrated energy flux and vertically integrated turbulent dissipation rate implies that dissipative length scales for the radiating internal tide exceed 1000 km.
