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- Author or Editor: Pijush K. Kundu x

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## Abstract

A linear, two-dimensional, continuously stratified, viscous model has been developed to study the inertial oscillations generated by a propagating wind field. The model, an extension of that of Kundu and Thomson, includes the presence of a coast and superposition due to distributed forcing. These two effects generate a large subsurface oscillation, provided the wind spectrum has energy near the inertial frequency. The presence of the coast causes an additional blue shift of the frequency, and a downward flux from the surface-coast corner. The superposition of responses with random phases does not cancel out but initially increases the rms amplitude as (time)^{½}. The model spectra have a blue shift that increases with depth and can also contain secondary peaks at higher frequencies if the speed of propagation is not too large. For a given propagation speed the blue shift, and hence the downward flux from the surface, is larger in the deep ocean where the gravity wave speeds *c _{n}
* are larger. A calculation in the open ocean with a thermocline shows a decrease of the inertial oscillations with depth, and a clear upward phase and downward energy propagation. Although the model subsurface oscillations are large enough to explain the observations, they are too highly correlated in the vertical and horizontal directions. It is suggested that the variations perpendicular to the direction of propagation, and the β-effect, should be included in the model in order to explain the incoherence of the observed oscillations.

## Abstract

A linear, two-dimensional, continuously stratified, viscous model has been developed to study the inertial oscillations generated by a propagating wind field. The model, an extension of that of Kundu and Thomson, includes the presence of a coast and superposition due to distributed forcing. These two effects generate a large subsurface oscillation, provided the wind spectrum has energy near the inertial frequency. The presence of the coast causes an additional blue shift of the frequency, and a downward flux from the surface-coast corner. The superposition of responses with random phases does not cancel out but initially increases the rms amplitude as (time)^{½}. The model spectra have a blue shift that increases with depth and can also contain secondary peaks at higher frequencies if the speed of propagation is not too large. For a given propagation speed the blue shift, and hence the downward flux from the surface, is larger in the deep ocean where the gravity wave speeds *c _{n}
* are larger. A calculation in the open ocean with a thermocline shows a decrease of the inertial oscillations with depth, and a clear upward phase and downward energy propagation. Although the model subsurface oscillations are large enough to explain the observations, they are too highly correlated in the vertical and horizontal directions. It is suggested that the variations perpendicular to the direction of propagation, and the β-effect, should be included in the model in order to explain the incoherence of the observed oscillations.

## Abstract

Nearly two months of current meter data taken during the summer of 1973 at eleven depths at a station off the coast of Oregon in 100 m of water have been analyzed. The spectra show an 8% increase in the frequency of the inertial peak (ω≈0.064 cph) above the local *f* (=0.059 cph). Because of the close proximity of the tidal frequencies to the local *f*, a sharp bandpass filter centered at ω = 0.064 cph was used to isolate the inertial motions. The results showed that the amplitude of the inertial oscillations decayed slowly with depth, but the decay within about 10 m of the bottom was more rapid. A lagged correlation of the inertial currents clearly showed an upward propagation of phases throughout the water column, at a speed of about 0.1 cm s^{−1} within the depth range 20–60 m, but generally higher both above and below this mid-depth. The inertial currents were found to turn clockwise (looking down) with depth, which corresponds to an upward phase and downward energy propagation, and the vertical phase speeds implied by the rates of turning agreed remarkably well with the lagged correlation calculations. The vertical wavelength was found to be of the order of the water depth. The vertical flux of energy into the bottom boundary layer during the occurrence of inertial bursts was estimated to be of the same order as the rate of turbulence production within the boundary layer, signifying that the inertial bursts can cause appreciable boundary layer stirring. The average hodographs of the horizontal velocity vectors were found to be ellipses of axis ratio 1.03–1.28; a majority of them had their major axes aligned roughly perpendicular to the coast, signifying propagation in that direction. It was found that the simple wind-forced model of Pollard and Millard does qualitatively reproduce many of the observed features of the inertial currents in the surface layer.

## Abstract

Nearly two months of current meter data taken during the summer of 1973 at eleven depths at a station off the coast of Oregon in 100 m of water have been analyzed. The spectra show an 8% increase in the frequency of the inertial peak (ω≈0.064 cph) above the local *f* (=0.059 cph). Because of the close proximity of the tidal frequencies to the local *f*, a sharp bandpass filter centered at ω = 0.064 cph was used to isolate the inertial motions. The results showed that the amplitude of the inertial oscillations decayed slowly with depth, but the decay within about 10 m of the bottom was more rapid. A lagged correlation of the inertial currents clearly showed an upward propagation of phases throughout the water column, at a speed of about 0.1 cm s^{−1} within the depth range 20–60 m, but generally higher both above and below this mid-depth. The inertial currents were found to turn clockwise (looking down) with depth, which corresponds to an upward phase and downward energy propagation, and the vertical phase speeds implied by the rates of turning agreed remarkably well with the lagged correlation calculations. The vertical wavelength was found to be of the order of the water depth. The vertical flux of energy into the bottom boundary layer during the occurrence of inertial bursts was estimated to be of the same order as the rate of turbulence production within the boundary layer, signifying that the inertial bursts can cause appreciable boundary layer stirring. The average hodographs of the horizontal velocity vectors were found to be ellipses of axis ratio 1.03–1.28; a majority of them had their major axes aligned roughly perpendicular to the coast, signifying propagation in that direction. It was found that the simple wind-forced model of Pollard and Millard does qualitatively reproduce many of the observed features of the inertial currents in the surface layer.

## Abstract

The structure of the stratified turbulent upper mixed layer of the ocean has been numerically investigated by using the turbulence closure model of Gibson and Launder, under the action of an impulsive wind stress τ_{0} and zero surface heat flux. The values of buoyancy and Coriolis frequencies assumed are *N* = 0.94 × 10^{−2} s^{−1} and *f* = 10^{−4} s^{−1}, respectively. The solutions indicate that the turbulent diffusion terms, small in general, transfer kinetic energy downward, although its effect on the deepening is negligible. Let *t _{i}
* be the time in inertial periods and

*u*

_{*}be the friction velocity. Then for 0.05 <

*t*< 0.3, the rate of increase of potential energy in the water column varies as ∂(PE)/∂

_{i}*t*∝

*t*

^{½}, rising to a maximum of ∼1.1

*u*

_{*}

^{3}and implying a mixed layer depth

*h*∝

*t*

^{½}as in the Pollard-Rhines-Thompson (PRT) model. For 1 <

*t*< 6, ∂(PE)/∂

_{i}*t*decreases only slightly from a quasi-steady value of ∂(PE)/∂

*t*≈ 0.25

*u*

_{*}

^{3}, implying a deepening rate slightly smaller than the Kraus-Turner

*h*∝

*t*

^{⅓}. The reason for this difference in behavior for the two time ranges is the separation of the flow into a depth-independent inertial oscillation and a quasi-steady shearing flow that carries almost all the turbulent stresses in the water column. The mechanism for deepening is always the lifting of heavier mass by the locally generated turbulence at the base of the mixed layer. For very large times (

*t*> 12), ∂(PE)/∂

_{i}*t*drops sharply, and no deepening was detected with a vertical resolution of 1 m. The assumption necessary to derive the PRT energy equation, namely, that the depth-integrated dissipation nearly balances τ

_{0}·(U

_{0}− Û),where U

_{0}is the surface velocity and Û the depth-averaged velocity, is approximately valid. For

*t*< 0.25, the PRT bulk Richardson number criterion is equivalent to a local critical gradient Richardson number criterion, and is due to the self-similarity of the solutions and the consequent thickening of the “interface.” The self-similarity breaks down for larger times, either because of the Coriolis forces becoming more important or because of the appearance of a sharp interface due to a nonlinear mechanism, whichever is earlier. An imposition of a kinetic energy input at the sea surface, so as to simulate the wind-wave flux, has certain desirable features.

_{i}## Abstract

The structure of the stratified turbulent upper mixed layer of the ocean has been numerically investigated by using the turbulence closure model of Gibson and Launder, under the action of an impulsive wind stress τ_{0} and zero surface heat flux. The values of buoyancy and Coriolis frequencies assumed are *N* = 0.94 × 10^{−2} s^{−1} and *f* = 10^{−4} s^{−1}, respectively. The solutions indicate that the turbulent diffusion terms, small in general, transfer kinetic energy downward, although its effect on the deepening is negligible. Let *t _{i}
* be the time in inertial periods and

*u*

_{*}be the friction velocity. Then for 0.05 <

*t*< 0.3, the rate of increase of potential energy in the water column varies as ∂(PE)/∂

_{i}*t*∝

*t*

^{½}, rising to a maximum of ∼1.1

*u*

_{*}

^{3}and implying a mixed layer depth

*h*∝

*t*

^{½}as in the Pollard-Rhines-Thompson (PRT) model. For 1 <

*t*< 6, ∂(PE)/∂

_{i}*t*decreases only slightly from a quasi-steady value of ∂(PE)/∂

*t*≈ 0.25

*u*

_{*}

^{3}, implying a deepening rate slightly smaller than the Kraus-Turner

*h*∝

*t*

^{⅓}. The reason for this difference in behavior for the two time ranges is the separation of the flow into a depth-independent inertial oscillation and a quasi-steady shearing flow that carries almost all the turbulent stresses in the water column. The mechanism for deepening is always the lifting of heavier mass by the locally generated turbulence at the base of the mixed layer. For very large times (

*t*> 12), ∂(PE)/∂

_{i}*t*drops sharply, and no deepening was detected with a vertical resolution of 1 m. The assumption necessary to derive the PRT energy equation, namely, that the depth-integrated dissipation nearly balances τ

_{0}·(U

_{0}− Û),where U

_{0}is the surface velocity and Û the depth-averaged velocity, is approximately valid. For

*t*< 0.25, the PRT bulk Richardson number criterion is equivalent to a local critical gradient Richardson number criterion, and is due to the self-similarity of the solutions and the consequent thickening of the “interface.” The self-similarity breaks down for larger times, either because of the Coriolis forces becoming more important or because of the appearance of a sharp interface due to a nonlinear mechanism, whichever is earlier. An imposition of a kinetic energy input at the sea surface, so as to simulate the wind-wave flux, has certain desirable features.

_{i}## Abstract

It has been shown that the phase angle of the complex correlation coefficient is a good measure of the average relative angular displacement (veering) between a pair of two-dimensional vector series. The correlation coefficient between the low-frequency (ω<0.6 cpd) components of the current vectors at 20 m and 5 m heights from the ocean bottom at a station near the Oregon coast reveals an Ekman veering of 6°.

## Abstract

It has been shown that the phase angle of the complex correlation coefficient is a good measure of the average relative angular displacement (veering) between a pair of two-dimensional vector series. The correlation coefficient between the low-frequency (ω<0.6 cpd) components of the current vectors at 20 m and 5 m heights from the ocean bottom at a station near the Oregon coast reveals an Ekman veering of 6°.

## Abstract

The excitation of coastal inertial oscillations by a rapidly varying wind is investigated. It is shown that the mean-square response to a completely random forcing is ϕ¯^{2} ∝ ∫ ϕ_{δ}
^{2}dt, where ϕ_{δ} is the response to impulsive forcing and the integral is over the record length. The rms response therefore initially increases with time as *t*
^{½}, and reaches stationarity in the decay scale for ϕ_{δ}. As in the random-walk problem, the *t*
^{½} increase is a result of the superposition of uncorrelated steps. Continuous random forcing preferentially increases subsurface amplitudes, since the energy flux from the coast-surface corner causes a surface decay and a subsurface growth of ϕ_{δ}.

With assumed parameters, a step-input wind forcing of 1 dyn cm^{−-2} generates inertial oscillations of 4 cm s^{−1} in the surface layer and 0.7–1.5 cm s^{−1} below. With a random wind in the range (−0.5, 0.5) dyn cm^{−2}, the surface values increase to 8–11 cm s^{−1} and the subsurface values to 3–7 cm s^{−1}. With an observed wind-forcing the surface and subsurface amplitudes are 10–17 cm s^{−1} and 5–9 cm s^{−1}, respectively. Compared to the step-input wind, the oscillations due to a randomly varying wind are less coherent in the vertical and more intermittent in time.

## Abstract

The excitation of coastal inertial oscillations by a rapidly varying wind is investigated. It is shown that the mean-square response to a completely random forcing is ϕ¯^{2} ∝ ∫ ϕ_{δ}
^{2}dt, where ϕ_{δ} is the response to impulsive forcing and the integral is over the record length. The rms response therefore initially increases with time as *t*
^{½}, and reaches stationarity in the decay scale for ϕ_{δ}. As in the random-walk problem, the *t*
^{½} increase is a result of the superposition of uncorrelated steps. Continuous random forcing preferentially increases subsurface amplitudes, since the energy flux from the coast-surface corner causes a surface decay and a subsurface growth of ϕ_{δ}.

With assumed parameters, a step-input wind forcing of 1 dyn cm^{−-2} generates inertial oscillations of 4 cm s^{−1} in the surface layer and 0.7–1.5 cm s^{−1} below. With a random wind in the range (−0.5, 0.5) dyn cm^{−2}, the surface values increase to 8–11 cm s^{−1} and the subsurface values to 3–7 cm s^{−1}. With an observed wind-forcing the surface and subsurface amplitudes are 10–17 cm s^{−1} and 5–9 cm s^{−1}, respectively. Compared to the step-input wind, the oscillations due to a randomly varying wind are less coherent in the vertical and more intermittent in time.

## Abstract

The circulation forced by an inflow of water through an eastern ocean boundary is investigated using two linear, viscid, and continuously stratified models. One of the models has a flat bottom, and solutions are obtained analytically; the other has a continental shelf, and solutions are found numerically. Without vertical mixing all the inflow continues across the ocean. With vertical mixing, however, part of it bends poleward to generate a coastal circulation. The presence of a shelf displaces the coastal currents offshore, but otherwise changes their structure and magnitude very little. Solutions suggest that the southward bending of the throughflow from the Pacific into the Indian Ocean may contribute to the Leeuwin Current off western Australia, but that it is not the dominant mechanism for driving the circulation there.

## Abstract

The circulation forced by an inflow of water through an eastern ocean boundary is investigated using two linear, viscid, and continuously stratified models. One of the models has a flat bottom, and solutions are obtained analytically; the other has a continental shelf, and solutions are found numerically. Without vertical mixing all the inflow continues across the ocean. With vertical mixing, however, part of it bends poleward to generate a coastal circulation. The presence of a shelf displaces the coastal currents offshore, but otherwise changes their structure and magnitude very little. Solutions suggest that the southward bending of the throughflow from the Pacific into the Indian Ocean may contribute to the Leeuwin Current off western Australia, but that it is not the dominant mechanism for driving the circulation there.

## Abstract

A solution for a concentrated line front translating at speed *U* is given. It is shown that the frequency is near-inertial if *U*≫*c*
_{1}, where *c*
_{1} is the long internal wave speed of the first baroclinic mode. Each more has a charactristic frequency ω_{
n
} associated with it. The spectra contain a near-inertial primary peak, composed of the higher modes, whose blue shift increases with depth. They also contain secondary peaks at higher internal wave frequencies if *U* is only slightly larger than *c*
_{1}. The flow field is intermittent, and involves a continuous interchange of energy between the surface layer and the stratified interior. The dominant period of this intermittency is the beating period of the first mode with a purely inertial oscillation. Short periods of apparent subinertial motion are also generated. Several features of the solution are in agreement with observations.

## Abstract

A solution for a concentrated line front translating at speed *U* is given. It is shown that the frequency is near-inertial if *U*≫*c*
_{1}, where *c*
_{1} is the long internal wave speed of the first baroclinic mode. Each more has a charactristic frequency ω_{
n
} associated with it. The spectra contain a near-inertial primary peak, composed of the higher modes, whose blue shift increases with depth. They also contain secondary peaks at higher internal wave frequencies if *U* is only slightly larger than *c*
_{1}. The flow field is intermittent, and involves a continuous interchange of energy between the surface layer and the stratified interior. The dominant period of this intermittency is the beating period of the first mode with a purely inertial oscillation. Short periods of apparent subinertial motion are also generated. Several features of the solution are in agreement with observations.

## Abstract

Velocity measurements from the continental shelf off Oregon taken during the Coastal Upwelling Experiment CUE-2 in the summer of 1973 are utilized to investigate momentum, vorticity and mass balance relationships for subinertial frequency (ω < 0.6 cpd) current fluctuations. Measurements from stations in water of depths of 54, 100 and 200 m are utilized. By a comparison of the magnitude of terms involving horizontal velocities in the linear momentum and in the nonlinear, depth-integrated momentum equations, support is found for the linear geostrophic balance of the alongshore velocity in the onshore-offshore momentum equation and, in the depth range 100 m ≤ *H* ≤ 200 m, for a linear ageostrophic balance in the alongshore momentum equation. Evidence is also found to support the validity of a linear depth-integrated vorticity balance, again for depths 100 m ≤ *H* ≤ 200 m. In this balance, which is similar to that in the theory for continental shelf waves, the interaction of the onshore velocity with the onshore-offshore bottom slope of the continental shelf forms the primary vortex stretching mechanism. The mass balance equation from idealized two-dimensional coastal upwelling models, wherein the depth integral of the interior, inviscid onshore velocity *U* equals the offshore Ekman layer transport −τ/ρ_{0}
*F*, where τ is the alongshore component of the wind stress, is investigated by comparing the time-dependent behavior of *U* and τ/ρ_{0}
*f*. It is found that the correlation of *U* and τ/rho;_{0}
*f* is of the proper sign to support this relation and that, in general, the magnitudes of these two terms are similar, but that the correlation is not especially high, presumably due to three-dimensional effects.

## Abstract

Velocity measurements from the continental shelf off Oregon taken during the Coastal Upwelling Experiment CUE-2 in the summer of 1973 are utilized to investigate momentum, vorticity and mass balance relationships for subinertial frequency (ω < 0.6 cpd) current fluctuations. Measurements from stations in water of depths of 54, 100 and 200 m are utilized. By a comparison of the magnitude of terms involving horizontal velocities in the linear momentum and in the nonlinear, depth-integrated momentum equations, support is found for the linear geostrophic balance of the alongshore velocity in the onshore-offshore momentum equation and, in the depth range 100 m ≤ *H* ≤ 200 m, for a linear ageostrophic balance in the alongshore momentum equation. Evidence is also found to support the validity of a linear depth-integrated vorticity balance, again for depths 100 m ≤ *H* ≤ 200 m. In this balance, which is similar to that in the theory for continental shelf waves, the interaction of the onshore velocity with the onshore-offshore bottom slope of the continental shelf forms the primary vortex stretching mechanism. The mass balance equation from idealized two-dimensional coastal upwelling models, wherein the depth integral of the interior, inviscid onshore velocity *U* equals the offshore Ekman layer transport −τ/ρ_{0}
*F*, where τ is the alongshore component of the wind stress, is investigated by comparing the time-dependent behavior of *U* and τ/ρ_{0}
*f*. It is found that the correlation of *U* and τ/rho;_{0}
*f* is of the proper sign to support this relation and that, in general, the magnitudes of these two terms are similar, but that the correlation is not especially high, presumably due to three-dimensional effects.

## Abstract

An analysis is presented of the low-frequency fluctuations [ω<0.6 cycle per day (cpd)] of the currents near the Oregon coast, based on the 1972 and 1973 measurements from moored current meters in CUE-1 and CUE-2. Let *u* and *v* denote the eastward (approximately onshore) and northward (approximately alongshore) components of the currents. The mean alongshore velocity *v* has the structure of a baroclinic coastal jet, whose maximum speed occurs near the surface at a distance of about 15–20 km from the shore, whereas the fluctuating part of *v* has the structure of a roughly barotropic coastal jet whose maximum occurs very near (<4 km) the shore. The standard deviation of *v* is approximately depth-independent whereas that of *u* decreases with depth. As one approaches the coast, the standard deviation of *u* decreases whereas that of *v* rises steeply, consistent with the behavior expected of coastally trapped wave motion. A scatter plot of the velocity fluctuations in a hodograph plane indicates that the fluctuations roughly follow the direction of the local isobaths. The Reynolds stresses in an east-north coordinate system therefore change sign because of the change of direction of the isobaths in the region. The *v* fluctuations seem to be mutually better correlated than the *u* fluctuations throughout the region, suggesting that the *u* components may be affected by “turbulence.” By finding the time lag corresponding to maximum correlation between stations separated alongshore, the velocity fluctuations have been found to propagate northward approximately nondispersively at a mean velocity of about 500 km day^{−1} during 1972 and 120 km day^{−1} during 1973. A method for performing the empirical orthogonal decomposition for two-dimensional vector time series has been formulated and applied to the velocity field over the continental shelf.

## Abstract

An analysis is presented of the low-frequency fluctuations [ω<0.6 cycle per day (cpd)] of the currents near the Oregon coast, based on the 1972 and 1973 measurements from moored current meters in CUE-1 and CUE-2. Let *u* and *v* denote the eastward (approximately onshore) and northward (approximately alongshore) components of the currents. The mean alongshore velocity *v* has the structure of a baroclinic coastal jet, whose maximum speed occurs near the surface at a distance of about 15–20 km from the shore, whereas the fluctuating part of *v* has the structure of a roughly barotropic coastal jet whose maximum occurs very near (<4 km) the shore. The standard deviation of *v* is approximately depth-independent whereas that of *u* decreases with depth. As one approaches the coast, the standard deviation of *u* decreases whereas that of *v* rises steeply, consistent with the behavior expected of coastally trapped wave motion. A scatter plot of the velocity fluctuations in a hodograph plane indicates that the fluctuations roughly follow the direction of the local isobaths. The Reynolds stresses in an east-north coordinate system therefore change sign because of the change of direction of the isobaths in the region. The *v* fluctuations seem to be mutually better correlated than the *u* fluctuations throughout the region, suggesting that the *u* components may be affected by “turbulence.” By finding the time lag corresponding to maximum correlation between stations separated alongshore, the velocity fluctuations have been found to propagate northward approximately nondispersively at a mean velocity of about 500 km day^{−1} during 1972 and 120 km day^{−1} during 1973. A method for performing the empirical orthogonal decomposition for two-dimensional vector time series has been formulated and applied to the velocity field over the continental shelf.