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## Abstract

Coherence maps are a useful tool to study the oceanic response to atmospheric forcing. For a specific frequency band these maps display the coherence between the oceanic current (or pressure) at a single mooring location and the atmospheric forcing field at other locations as a function of separation. This paper calculates such coherence maps from a simple linear quasigeostrophic model forced by a statistically stationary and homogeneous wind field. The calculated coherence maps show values less than one. Such values are not due to the presence of noise but are a consequence of the ocean being forced at many locations. The maps also show characteristic patterns with maxima either at the mooring location or away from it. The locations of the maxima do not indicate the locations of the forcing but instead reflect the scales of the atmospheric forcing spectrum and of the Green’s function of the potential vorticity equation. Coherence maps can be used to estimate the Green’s function in a multiple regression analysis. The presence of noise or nonlinearities in the system can be inferred from the multiple coherence, which is a number. Emphasis is on understanding the information content of coherence maps, not on reproducing observed maps. The results can be generalized to other systems where response and forcing are related by a Green’s function.

## Abstract

Coherence maps are a useful tool to study the oceanic response to atmospheric forcing. For a specific frequency band these maps display the coherence between the oceanic current (or pressure) at a single mooring location and the atmospheric forcing field at other locations as a function of separation. This paper calculates such coherence maps from a simple linear quasigeostrophic model forced by a statistically stationary and homogeneous wind field. The calculated coherence maps show values less than one. Such values are not due to the presence of noise but are a consequence of the ocean being forced at many locations. The maps also show characteristic patterns with maxima either at the mooring location or away from it. The locations of the maxima do not indicate the locations of the forcing but instead reflect the scales of the atmospheric forcing spectrum and of the Green’s function of the potential vorticity equation. Coherence maps can be used to estimate the Green’s function in a multiple regression analysis. The presence of noise or nonlinearities in the system can be inferred from the multiple coherence, which is a number. Emphasis is on understanding the information content of coherence maps, not on reproducing observed maps. The results can be generalized to other systems where response and forcing are related by a Green’s function.

## Abstract

The quasi-geostrophic response of the ocean to stochastic forcing by wind stress and atmospheric pressure is investigated using a linear, continuously stratified, β-plane oceanic model with a flat bottom. We consider a spectral representation of the forcing and response fields, and we estimate the oceanic response using a vertical normal mode expansion. Model spectra of the wind stress, wind stress curl and surface pressure fields are constructed. In the wavenumber-frequency range of quasi-geostrophic eddies, the observations suggest that because of their short correlation time scale, the forcing fields are, to a reasonable approximation, white in frequency space and symmetric in wavenumber space. Forcing by the wind stress has the dominant role. The oceanic response can be off-resonant or resonant. In the off-resonant case, we predict oceanic wavenumber-frequency response spectra. In case of resonance we estimate total energy transfer rates by integrating the oceanic response over depth and wavenumber (in the range 2π/4000 km^{−1}–2π/50 km^{−1}) and we distinguish between the barotropic and the total baroclinic response, the latter being obtained by summing the contribution of all baroclinic modes.

The barotropic response is resonant at practically all eddy frequencies, and the baroclinic response is resonant at frequencies smaller than the maximum frequency of the first baroclinic Rossby wave. In midlatitudes, we find comparable energy input rates into barotropic and baroclinic modes, of the order of 3 × 10^{−4} W m^{−2}. In high latitudes the input is comparable for barotropic Rossby waves and smaller for baroclinic ones. The total energy input rate by resonant forcing is only one order of magnitude smaller than the energy input rate from the mean atmospheric circulation into the general oceanic circulation. It is smaller, but comparable with the rate of energy conversion from the mean oceanic circulation into quasi-geostrophic eddies by barotropic and baroclinic instabilities. At medium and high frequencies, the baroclinic response is off-resonant. The model predicts red frequency spectra that are consistent with temperature observations in the central North Pacific. In particular, the seasonal variability of the observed eddy field is reproduced. A comparison with observations in the western North Atlantic also suggests that local stochastic forcing by the atmosphere is an important generating mechanism for the eddies in regions of low eddy activity.

## Abstract

The quasi-geostrophic response of the ocean to stochastic forcing by wind stress and atmospheric pressure is investigated using a linear, continuously stratified, β-plane oceanic model with a flat bottom. We consider a spectral representation of the forcing and response fields, and we estimate the oceanic response using a vertical normal mode expansion. Model spectra of the wind stress, wind stress curl and surface pressure fields are constructed. In the wavenumber-frequency range of quasi-geostrophic eddies, the observations suggest that because of their short correlation time scale, the forcing fields are, to a reasonable approximation, white in frequency space and symmetric in wavenumber space. Forcing by the wind stress has the dominant role. The oceanic response can be off-resonant or resonant. In the off-resonant case, we predict oceanic wavenumber-frequency response spectra. In case of resonance we estimate total energy transfer rates by integrating the oceanic response over depth and wavenumber (in the range 2π/4000 km^{−1}–2π/50 km^{−1}) and we distinguish between the barotropic and the total baroclinic response, the latter being obtained by summing the contribution of all baroclinic modes.

The barotropic response is resonant at practically all eddy frequencies, and the baroclinic response is resonant at frequencies smaller than the maximum frequency of the first baroclinic Rossby wave. In midlatitudes, we find comparable energy input rates into barotropic and baroclinic modes, of the order of 3 × 10^{−4} W m^{−2}. In high latitudes the input is comparable for barotropic Rossby waves and smaller for baroclinic ones. The total energy input rate by resonant forcing is only one order of magnitude smaller than the energy input rate from the mean atmospheric circulation into the general oceanic circulation. It is smaller, but comparable with the rate of energy conversion from the mean oceanic circulation into quasi-geostrophic eddies by barotropic and baroclinic instabilities. At medium and high frequencies, the baroclinic response is off-resonant. The model predicts red frequency spectra that are consistent with temperature observations in the central North Pacific. In particular, the seasonal variability of the observed eddy field is reproduced. A comparison with observations in the western North Atlantic also suggests that local stochastic forcing by the atmosphere is an important generating mechanism for the eddies in regions of low eddy activity.

## Abstract

The effect of equatorial undercurrent (EUC) shear on equatorial upper-ocean mixing is studied using a large eddy simulation (LES) model. This study consists of five numerical experiments of convection with various initial shear profiles: 1) full background shear (EUC shear), 2) same as 1 but with a surface cooling rate reduced by a factor of 10, 3) no shear, 4) stable part the background shear only (velocity constant above 30 m where Ri < 1/4 in experiment 1), and 5) unstable part of the background shear only (velocity constant below 30 m). It is found that flow evolution crucially depends on the background shear. Removal of all or part of the shear profile dramatically degrades the realism of the results. Convection in the mixed layer triggers shear instability, which in turn radiates gravity waves downward into the upper thermocline. Local shear instability can be triggered by downward-propagating internal waves in a marginally stable environment. This local shear instability is the cause of mixing well below the mixed layer. When complete EUC shear is present, internal waves with wavelengths of 200–300 m are generated below the boundary layer, in agreement with observations and linear instability analysis. The total shear profile determines the characteristics of the waves. When the stable shear, or the portion of the shear with Ri > 1/4, is eliminated, the internal waves have smaller wavelengths (about 80 m). When the unstable shear, or the portion of the shear with Ri ≤ 1/4, is eliminated, the intensity of internal waves below the boundary layer is much reduced, but the wavelengths are much larger than the case of convection without shear. In the absence of large-scale forcing to maintain the surface shear, the bulk of the kinetic energy from the mean shear is released in just a few hours after the onset of convection and shear instability. Turbulent kinetic energy budgets with and without shear show some similarities during the early stage of convection but show dramatic differences when the turbulence is fully developed. Namely, the turbulent transport and pressure transport terms are important in the case of convection without shear but are negligible in the case of convection with EUC shear, even though the surface forcing is the same. Local shear instability in a marginally stable mean flow environment is shown to play an important role in transporting heat and momentum into the stratified region below the mixed layer. Turbulence and waves generated by the mean shear instability are shown to be more effective than convective plumes in triggering local instability in the marginally stable region below the mixed layer.

## Abstract

The effect of equatorial undercurrent (EUC) shear on equatorial upper-ocean mixing is studied using a large eddy simulation (LES) model. This study consists of five numerical experiments of convection with various initial shear profiles: 1) full background shear (EUC shear), 2) same as 1 but with a surface cooling rate reduced by a factor of 10, 3) no shear, 4) stable part the background shear only (velocity constant above 30 m where Ri < 1/4 in experiment 1), and 5) unstable part of the background shear only (velocity constant below 30 m). It is found that flow evolution crucially depends on the background shear. Removal of all or part of the shear profile dramatically degrades the realism of the results. Convection in the mixed layer triggers shear instability, which in turn radiates gravity waves downward into the upper thermocline. Local shear instability can be triggered by downward-propagating internal waves in a marginally stable environment. This local shear instability is the cause of mixing well below the mixed layer. When complete EUC shear is present, internal waves with wavelengths of 200–300 m are generated below the boundary layer, in agreement with observations and linear instability analysis. The total shear profile determines the characteristics of the waves. When the stable shear, or the portion of the shear with Ri > 1/4, is eliminated, the internal waves have smaller wavelengths (about 80 m). When the unstable shear, or the portion of the shear with Ri ≤ 1/4, is eliminated, the intensity of internal waves below the boundary layer is much reduced, but the wavelengths are much larger than the case of convection without shear. In the absence of large-scale forcing to maintain the surface shear, the bulk of the kinetic energy from the mean shear is released in just a few hours after the onset of convection and shear instability. Turbulent kinetic energy budgets with and without shear show some similarities during the early stage of convection but show dramatic differences when the turbulence is fully developed. Namely, the turbulent transport and pressure transport terms are important in the case of convection without shear but are negligible in the case of convection with EUC shear, even though the surface forcing is the same. Local shear instability in a marginally stable mean flow environment is shown to play an important role in transporting heat and momentum into the stratified region below the mixed layer. Turbulence and waves generated by the mean shear instability are shown to be more effective than convective plumes in triggering local instability in the marginally stable region below the mixed layer.

## Abstract

This study investigates the sensitivity of the dynamics of the surface equatorial ocean to the parameterization of vertical mixing. A new high-resolution, numerical model of a zonally independent equatorial channel helps to explore this question and includes three parameterizations, all of which increase mixing for decreasing Richardson numbers. It compares the smooth increase of eddy coefficients traditionally used in general circulation models, the dramatic increase of the eddy coefficients for small Richardson numbers recently observed in the equatorial Pacific, and the combination of a mixing mechanism based on the diagnostic adjustment of the water column to noncritical Richardson numbers and of a bulk mixed layer model.

The meridional and vertical velocity fields in the surface layer are very sensitive to the strength of mixing implied by the different parameterizations. For the smooth Richardson number dependence of the eddy coefficients, equatorial upwelling due to easterly winds reaches the surface. The dramatically increasing eddy coefficients for small Richardson numbers yield reduced equatorial upwelling rates in the surface layer. The diagnostic adjustment of the Richardson number shows in the surface layer close to the equator reversed meridional shear and downwelling in response to easterly winds!

A simple model for the low-latitude wind current in the presence of horizontal density gradients reproduces this reversal of the meridional and vertical flows. If the equatorial Ekman number is large, there is a latitude range where within the upper layer the vertically averaged flow and density are dominated by rotation, while the vertical shear of horizontal velocities is strongly influenced by vertical friction. In this region vertical shears point downstream of the wind stress and of the pressure forces due to gradients in density. For an easterly wind the pressure gradient forces surface waters toward the equator and can reverse the vertical shear of meridional velocity and the equatorial vertical velocity. The critical value of the vertical eddy coefficient for this reversal to occur is of the order of 5 × 10^{−3} m^{2} s^{−1}. This value is of the same order as measured in the surface equatorial Pacific and used in general circulation models. The physics of this reversal are so basic it is likely they are active in the ocean and three-dimensional circulation models.

## Abstract

This study investigates the sensitivity of the dynamics of the surface equatorial ocean to the parameterization of vertical mixing. A new high-resolution, numerical model of a zonally independent equatorial channel helps to explore this question and includes three parameterizations, all of which increase mixing for decreasing Richardson numbers. It compares the smooth increase of eddy coefficients traditionally used in general circulation models, the dramatic increase of the eddy coefficients for small Richardson numbers recently observed in the equatorial Pacific, and the combination of a mixing mechanism based on the diagnostic adjustment of the water column to noncritical Richardson numbers and of a bulk mixed layer model.

The meridional and vertical velocity fields in the surface layer are very sensitive to the strength of mixing implied by the different parameterizations. For the smooth Richardson number dependence of the eddy coefficients, equatorial upwelling due to easterly winds reaches the surface. The dramatically increasing eddy coefficients for small Richardson numbers yield reduced equatorial upwelling rates in the surface layer. The diagnostic adjustment of the Richardson number shows in the surface layer close to the equator reversed meridional shear and downwelling in response to easterly winds!

A simple model for the low-latitude wind current in the presence of horizontal density gradients reproduces this reversal of the meridional and vertical flows. If the equatorial Ekman number is large, there is a latitude range where within the upper layer the vertically averaged flow and density are dominated by rotation, while the vertical shear of horizontal velocities is strongly influenced by vertical friction. In this region vertical shears point downstream of the wind stress and of the pressure forces due to gradients in density. For an easterly wind the pressure gradient forces surface waters toward the equator and can reverse the vertical shear of meridional velocity and the equatorial vertical velocity. The critical value of the vertical eddy coefficient for this reversal to occur is of the order of 5 × 10^{−3} m^{2} s^{−1}. This value is of the same order as measured in the surface equatorial Pacific and used in general circulation models. The physics of this reversal are so basic it is likely they are active in the ocean and three-dimensional circulation models.

## Abstract

We describe the meridional and seasonal structures of daily mean mixed-layer depth and its diurnal amplitude and their relation to atmospheric fluxes by compositing mixed-layer depth estimates derived from density observations. The diurnal mean mixed-layer depth shows a ridge at the equator, troughs, which vary seasonally in intensity, at 10° to 15°N and 5° to 10°S, and a trough appearing just north of the equator in the second half of the year. This is in contrast to the ridge-trough structure of the top of the main thermocline, which reflects the dynamic topography associated with the equatorial current system. The diurnal amplitude is significantly different from zero for most latitudes year-round, indicating that the diurnal cycle of mixed-layer depth is a widespread phenomenon. For sufficiently strong heating, both the mixed-layer depth and its diurnal amplitude are significantly correlated with Monin-Obukhov length scales based on the mean net heat flux, mean wind stress, and mean shortwave radiation. This suggests a possible parameterization of the mixed-layer depth and diurnal amplitude in terms of the mean atmospheric fluxes for meridional scales of a few degrees and seasonal time scales.

## Abstract

We describe the meridional and seasonal structures of daily mean mixed-layer depth and its diurnal amplitude and their relation to atmospheric fluxes by compositing mixed-layer depth estimates derived from density observations. The diurnal mean mixed-layer depth shows a ridge at the equator, troughs, which vary seasonally in intensity, at 10° to 15°N and 5° to 10°S, and a trough appearing just north of the equator in the second half of the year. This is in contrast to the ridge-trough structure of the top of the main thermocline, which reflects the dynamic topography associated with the equatorial current system. The diurnal amplitude is significantly different from zero for most latitudes year-round, indicating that the diurnal cycle of mixed-layer depth is a widespread phenomenon. For sufficiently strong heating, both the mixed-layer depth and its diurnal amplitude are significantly correlated with Monin-Obukhov length scales based on the mean net heat flux, mean wind stress, and mean shortwave radiation. This suggests a possible parameterization of the mixed-layer depth and diurnal amplitude in terms of the mean atmospheric fluxes for meridional scales of a few degrees and seasonal time scales.

## Abstract

The scattering of oceanic internal gravity waves off random bottom topography is analyzed under the assumptions that (i) the height of the topography is smaller than the vertical wavelength and (ii) the slope of the topography is smaller than the wave slope. For each frequency, scattering redistributes the incoming energy flux in horizontal wavenumber space. The scattered wave field approaches an equilibrium state where the energy flux is equipartitioned in horizontal wavenumber space. For incoming red spectra, this implies a transfer from low to high wavenumbers. For typical internal wave and bottom spectra, about 6.8% of the incoming energy flux is redistributed. While this might be less than the flux redistribution caused by reflection off a critical slope, the scattering process transfers the energy flux to higher wavenumbers than the reflection process. Scattering might thus be equally or more efficient than reflection in causing high shears and mixing near the bottom.

## Abstract

The scattering of oceanic internal gravity waves off random bottom topography is analyzed under the assumptions that (i) the height of the topography is smaller than the vertical wavelength and (ii) the slope of the topography is smaller than the wave slope. For each frequency, scattering redistributes the incoming energy flux in horizontal wavenumber space. The scattered wave field approaches an equilibrium state where the energy flux is equipartitioned in horizontal wavenumber space. For incoming red spectra, this implies a transfer from low to high wavenumbers. For typical internal wave and bottom spectra, about 6.8% of the incoming energy flux is redistributed. While this might be less than the flux redistribution caused by reflection off a critical slope, the scattering process transfers the energy flux to higher wavenumbers than the reflection process. Scattering might thus be equally or more efficient than reflection in causing high shears and mixing near the bottom.