1. Introduction

The surface layer of the ocean is a major component of the planetary boundary layer. Processes within the surface mixed layer determine the size and direction of momentum, heat, and tracer fluxes between the atmosphere and the ocean interior and so have implications for the planetary climate and its variability (Sullivan and McWilliams 2010). These fluxes also play an important role in linking the oceanic and atmospheric carbon pools through regulation of the sea surface CO₂ fluxes (Mahadevan et al. 2012).

The wind stress on the ocean surface is the direct energy source for many types of fluid motion in the ocean. It is known to generate flows that range from basin-scale (e.g., Marshall and Pillar 2011) to much smaller–scale flows such as Langmuir circulation (Thorpe 2004). A number of pathways to dissipation exist for energy derived from the wind stress in the surface ocean, though their relative importance is poorly understood (Johnston and Rudnick 2009). These include surface wave generation, Langmuir turbulence (Polton et al. 2008), and inertial oscillations (Pollard and Millard 1970). The impact of these processes is largely confined to the surface mixed layer. Below this a highly stratified transition layer inhibits the downward transfer of momentum (Grant and Belcher 2011) and, thus, gives rise to a maximum shear layer widely observed in the global ocean (Johnston and Rudnick 2009). Nonetheless, some of the energy associated with the Langmuir turbulence and near-inertial oscillations is also thought to generate internal waves that then propagate into the ocean interior (e.g., Polton et al. 2008; Dohan and Davis 2011).

Although the time-average shear in the transition layer is generally higher than at other depths, it is also the layer of strongest stratification, and consequently the water column is often at near-critical stability (Richardson number ≈ 1; Johnston and Rudnick 2009). The low Richardson number implies that periods of enhanced shear may induce instabilities, leading to water column mixing and the dissipation of turbulent kinetic energy. D'Asaro (1985) noted that an alignment of the wind and surface current can give rise to such an increase in shear while Rippeth et al. (2009) observed an enhancement in shear associated with wind shear alignment that resulted in a tripling of the observed turbulent dissipation rate in the seasonal thermocline of a temperate shelf sea. Burchard and Rippeth (2009, hereafter BR09) derived a prognostic expression for shear variance in a shallow, stratified tidal sea from a simplified set of the equations of motion. The model demonstrated that the alignment of the wind and shear can lead to a sharp increase in shear variance over a period of hours or less (and so has been termed a shear spike).

Observations from other stratified shelf sea locations, in the northern North Sea (BR09) and the ice-covered Laptev Sea (Lenn et al. 2011), show the coincidence of enhanced dissipation within the thermocline and an increase in shear resulting from the alignment of the shear vector with the surface wind stress. The observations made in the Irish Sea and North Sea had similar levels of wind forcing [O(10⁻¹ N m⁻²)] and stratification [O(10⁻⁴ s⁻²)] to that found in open ocean conditions, suggesting that it may be possible to modify the BR09 model for the open ocean. While much attention has focused on resonant wind-driven mixing at the base of the mixed layer (e.g., D'Asaro 1985; Skyllingstad et al. 1998; Grant and Belcher 2011; Dohan and Davis 2011) these spikes in dissipation occurred without resonant wind forcing, suggesting that they may be a ubiquitous feature of the seasonal thermocline.

In this study, we use data from the Southern Ocean mixed layer and seasonal thermocline to confirm the existence of nonresonant shear spikes in the open ocean. The primary aims of the paper are to modify BR09 for the open ocean surface boundary layer, test the validity of the assumptions underlying the model, and assess its skill in predicting the generation of shear spikes.

The data consist of a slow transect (11 days) of the Drake Passage region made by the RRS James Cook (hereafter Cook) in February 2009 and 242 faster transects (2–5 days) of Drake Passage made by the ARSV Laurence M. Gould (LMG) between 1999 and 2011 (Fig. 1a). An analytical model for the production of shear squared (i.e., S²) in an open ocean environment is derived in section 2. The datasets and methods employed are then described in section 3. The overall structure of the stratification and shear are set out in section 4. The effectiveness of the model is evaluated in section 5 with a summary and discussion following in section 6.

(a) Map of Drake Passage with bathymetry in grayscale and contour lines at 1500 m, 3000 m, and 4500 m. The solid line shows the track of the high-resolution James Cook transect, while the crosses mark the Laurence M. Gould crossings. (b) A schematic of the vertical structure used to calculate the bulk shear. The gray-shaded areas represent the layers over which velocity is averaged, H is the mixed layer depth, and h_s is the distance between the centers of mass of the upper and lower layers.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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(a) Map of Drake Passage with bathymetry in grayscale and contour lines at 1500 m, 3000 m, and 4500 m. The solid line shows the track of the high-resolution James Cook transect, while the crosses mark the Laurence M. Gould crossings. (b) A schematic of the vertical structure used to calculate the bulk shear. The gray-shaded areas represent the layers over which velocity is averaged, H is the mixed layer depth, and h_s is the distance between the centers of mass of the upper and lower layers.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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(a) Map of Drake Passage with bathymetry in grayscale and contour lines at 1500 m, 3000 m, and 4500 m. The solid line shows the track of the high-resolution James Cook transect, while the crosses mark the Laurence M. Gould crossings. (b) A schematic of the vertical structure used to calculate the bulk shear. The gray-shaded areas represent the layers over which velocity is averaged, H is the mixed layer depth, and h_s is the distance between the centers of mass of the upper and lower layers.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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2. Shear production model

The central idea of the model is that interaction between shear and local wind forcing is an important source for the production or dissipation of shear variance and that an expression for this can be derived from a simplified set of the equations of motion. The model provides a prognostic equation for changes in shear variance that can be tested by comparison with observations. The validity of these assumptions is tested against the observations in sections 4 and 5.

In the model, the assumption has been made that dynamically important shear can be represented by a two-layer damped-slab model with an upper layer subject to wind forcing. Both of the layers are subject to an interfacial friction that acts to smooth velocity gradients between the layers. This interfacial friction is intended to parameterize relatively laminar vertical momentum fluxes rather than the much larger fluxes that arise when shear instabilities develop, as the model is intended to predict the generation of such instabilities rather than describe their subsequent effects.

BR09 based their continental shelf sea model for shear production on a one-dimensional momentum balance between linear and Coriolis acceleration, the surface wind and bottom tidal stresses, and interfacial friction. For the surface open ocean where vertical velocity gradients due to tides are weak, we can express the vertically-averaged momentum balance, following Pollard and Millard (1970), in the respective layers as

View Expanded

View Expanded

where the surface and lower layers are indicated by the subscripts s and l, u is the horizontal velocity vector, k is the vertical unit vector, t is time, ∂/∂t is the time derivative, f is the Coriolis parameter, ρ is potential density, τ_w is the relative wind stress vector, τ_i is the interfacial frictional stress vector, and H is the mixed layer depth (MLD). Note that surface stress is taken to be the relative wind stress, calculated from the velocity of the wind relative to the surface ocean current. Hereafter, the body forces owing to wind and interfacial friction are denoted as T_w and T_i, respectively, which are the stresses divided by ρ (making the assumption that spatial and temporal variations in ρ are much smaller than its mean value, so it can be taken as a constant).

The damped-slab mixed layer model assumes that diffusion of momentum is rapid within each layer, such that the divergence of the wind and interfacial boundary stresses can be treated as body forces applied equally over the whole layer. The nonlinear advective term between layers in the absence of shear instability is thought to be weak (Polton et al. 2008) and so is neglected. As we are only concerned with the mechanism responsible for the generation of shear spikes, we assume there to be no shear instability, although in consideration of mixing we note the shear instability will actually be the dominant process for the dissipation of kinetic energy (Plueddemann and Farrar 2006). As such, the model understates the destruction of S² after shear spikes where this is brought about by instabilities. Finally, the vertical pressure gradient term is excluded as an implicit assumption that the water column is statically stable has been made. The role of horizontal pressure and velocity gradients is a subject of on-going research.

The momentum balance of (1) can be adapted into shear form. The relationship between velocity and bulk shear is as follows:

View Expanded

where h_s is the distance between the centers of mass of the upper layer (above the mixed layer base) and lower layer (below the mixed layer base), which is assumed to be constant in time. Following the approach of BR09, we can substitute (2) into (1) by subtracting (1b) from (1a) and dividing all terms by h_s, giving

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We then take the dot product of this equation with S to obtain an equation for shear variance:

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where the second term on the left-hand side is zero (as it represents the Coriolis force rotating shear without changing its variance). Rearranging (4) then gives

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or

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where P(S²) denotes the production/destruction of S² (dependent on the relative alignment of the wind and shear vectors) and D(S²) denotes the negative-definite destruction of S² by interfacial friction.

Finally, the interfacial friction body force T_i can be parameterized using a quadratic friction law (following BR09) as

View Expanded

where c_i is the drag coefficient. This allows (5) to be restated as

View Expanded

This simple theoretical model predicts that alignment of wind and shear produces the maximum value for the production of S². Note, however, that, if the shear vector is rotating, S² continues to increase after the point of alignment and only reaches its maximum value where (5) is zero, that is, when wind and shear are 90° out of phase.

The model addresses changes in S² at time scales of hours or less, over which time scales the MLD generally varies much less than the other terms and so is held constant in the model. However, the variations in MLD at monthly time scales are comparable to the variations in the other factors and so are considered in the observations of the annual cycle in section 4a. The effectiveness of the model is considered further in section 5.

3. Data and methods

b. Methods

The buoyancy frequency squared (i.e., N²) was calculated as

View Expanded

where g is gravitational acceleration, ρ is the potential density, z is the depth, and ρ₀ is a reference potential density.

The MLD was defined for the Cook data [following the definition of the Dong et al. (2008) Southern Ocean MLD climatology] as the shallowest depth where the potential density was 0.03 kg m⁻³ greater than at the surface. The upper layer, defined for the Cook bulk shear, extended from the ADCP depth bin centered at 31.5-m depth (having excluded the shallowest layer centered at 23.5 m due to occasional missing observations) to the depth closest to the MLD. The mean MLD on the Cook transect was 50 m, so the mixed layer velocity was typically the average of three ADCP depth bins. A gap of 24 m—which accounts for the portion of the water column with the highest gradients—is left between the upper layer and the lower layer. The lower layer is taken to be 32 m thick to give a representative velocity with reduced instrument noise. The size of the gap is based on a conservative estimate for the thickness of the high shear transition layer from Johnston and Rudnick (2009). The MLD for the LMG bulk shear layers were given by the Dong et al. (2008) climatology. The LMG upper layer extended from the first gridded depth at 26 m to the depth closest to the MLD. The LMG lower layer was up to 50 m thick where the data were available, with a 24 m gap between the two layers. The bulk shear is then the difference between the layer-averaged velocities divided by the distance between their centers of mass. A schematic illustrating the vertical structure of the bulk shear calculation is shown in Fig. 1b. A comparison of the bulk shear and the shear taken between successive ADCP depth bins for the Cook transect (not shown) confirms that the bulk shear has less high frequency variability than the interfacial shear but generally captures the variability in the interfacial shear at periods longer than a few hours.

The bulk shear time series are presented without interpolation, with details of any filters used noted in the relevant figure captions. When the bulk shear is used in Fourier spectral analysis, linear interpolation within transects is used to fill in missing data. When the LMG data are used, the Fourier components are computed transect by transect and then averaged to produce the spectra, with each transect providing two degrees of freedom. The separation of the data into transects acts as a natural windowing operation, so no further filters have been applied. The LMG ADCP data are also used to estimate a monthly latitude–depth climatology of shear in Drake Passage in section 4a. For this calculation, the shear for each 15-min-averaged velocity profile is estimated as the vertical centered difference of velocity with depth. The shear profiles for each transect are averaged in 0.1° latitude bins, and the transects are then averaged together by month.

The wind data are used to estimate the 10-m wind components using the boundary layer profile of Large and Pond (1981). The relative wind speed was calculated by subtracting the surface ocean current (using the shallowest ADCP depth bins as a proxy) from the calculated 10-m wind before using the Large and Pond (1981) parameterization to calculate the relative wind stress. This parameterization assumes that the atmosphere is of neutral stability, and the air density was taken as 1.25 kg m⁻³ throughout. Large and Pond (1981) note that there is an error in the calculated stress of less than 10% in converting the wind speed from the observed wind to the 10-m wind and this error decreases with wind speed. If the magnitude of the atmospheric stability parameter (the Monin–Obukhov length) is on the order of the observation height this can introduce a further 20% error in the calculated stress. We assume, however, that these errors are bounded in the evaluation of the model as stronger winds tend to induce neutral stability in the atmospheric boundary layer and, furthermore, these errors do not affect the direction of the wind stress, which is the most critical part of our analysis. For the Cook dataset there were no observations of wind speed greater than the 25 m s⁻¹ upper limit, and only 0.7% of the observations in the LMG dataset exceeded this limit.

Data are analyzed by season to observe temporal patterns. Here austral summer is taken to be January–March, austral fall is April–June, austral winter is July–September, and austral spring is October–December.

4. Characteristics of stratification and shear

The model set out in section 2 assumes the upper ocean has a two-layer structure that surrounds a stratified transition layer where shear is concentrated. In this section, we describe the seasonal variability of Drake Passage upper-ocean shear and stratification to test this assumption and investigate the statistical and rotary properties of bulk shear.

a. Annual cycle

The temperature and salinity profiles from the Cook transect in Fig. 2a provide a snapshot of Drake Passage hydrography. It shows that in the north of Drake Passage the surface layer was not homogeneous but, instead, had a weak thermal stratification (Figs. 2a,b). In the south of Drake Passage there was a thermal inversion with static stability maintained by strong haline stratification (Fig. 2a). However, this region had a uniform mixed layer at the surface above a sharp transition layer (Fig. 2b).

Hydrographic profiles of the James Cook transect. (a) The potential temperature profiles with 0.1 psu salinity contours overlain showing the contrast between the thermally stratified region in the north of Drake Passage and the salinity stratified region in the south. (b) Potential density profiles from a selection of individual CTD casts show that the mixed layers were more sharply defined in the south of the Passage than the north.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Hydrographic profiles of the James Cook transect. (a) The potential temperature profiles with 0.1 psu salinity contours overlain showing the contrast between the thermally stratified region in the north of Drake Passage and the salinity stratified region in the south. (b) Potential density profiles from a selection of individual CTD casts show that the mixed layers were more sharply defined in the south of the Passage than the north.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Hydrographic profiles of the James Cook transect. (a) The potential temperature profiles with 0.1 psu salinity contours overlain showing the contrast between the thermally stratified region in the north of Drake Passage and the salinity stratified region in the south. (b) Potential density profiles from a selection of individual CTD casts show that the mixed layers were more sharply defined in the south of the Passage than the north.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Seasonal variability in the vertical water column structure characterizes the calculated LMG monthly mean N² sections (Fig. 3). Beginning in the austral spring (October–November in Fig. 3), surface layer stratification develops above 150-m depth. This intensifies and shoals throughout the austral summer and peaks in March when the MLD is at ~40 m depth. As fall sets in, this stratification weakens and is substantially eroded by late winter (August in Fig. 3). In addition, there is a clear tendency for stratification to be stronger in the south of the passage throughout the year (Fig. 3). Note that at the southern end of the latitude range the tracks are relatively spaced out (Fig. 1a), so an assumption has been made that the primary gradients are meridional to perform this analysis of the monthly data.

Estimation of log₁₀N² in Drake Passage for the Laurence M. Gould transects based on a 0.1° objective mapping of temperature and salinity: N² is observed for each transect separately and then averaged by month; values of N² less than zero in any profile are excluded. The Dong et al. (2008) mixed layer depth climatology is marked by black dots.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Estimation of log₁₀N² in Drake Passage for the Laurence M. Gould transects based on a 0.1° objective mapping of temperature and salinity: N² is observed for each transect separately and then averaged by month; values of N² less than zero in any profile are excluded. The Dong et al. (2008) mixed layer depth climatology is marked by black dots.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Estimation of log₁₀N² in Drake Passage for the Laurence M. Gould transects based on a 0.1° objective mapping of temperature and salinity: N² is observed for each transect separately and then averaged by month; values of N² less than zero in any profile are excluded. The Dong et al. (2008) mixed layer depth climatology is marked by black dots.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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The effect of the seasonal cycle on shear variance is apparent from Fig. 4. From October into the austral summer (Fig. 4) the peak shear variance increases and the depth of the shear maximum shoals, being shallowest in February–April. After May the shear variance weakens by about an order of magnitude while the depth of the shear maximum also deepens. An analysis of shear variance over the upper 1000 m in Drake Passage by Firing et al. (2011) demonstrated that this is, indeed, the portion of the water column with the highest shear variance in Drake Passage.

Estimation of log₁₀S² in Drake Passage. Shear has been calculated here as centered differences around ADCP depth bins rather than as a bulk shear. The Dong et al. (2008) mixed layer depth climatology has been marked by white dots to illustrate that high shear usually occurs below the base of the mixed layer. The data show that monthly mean shear tends to be higher and more spatially concentrated in the south of the Passage and in summer.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Estimation of log₁₀S² in Drake Passage. Shear has been calculated here as centered differences around ADCP depth bins rather than as a bulk shear. The Dong et al. (2008) mixed layer depth climatology has been marked by white dots to illustrate that high shear usually occurs below the base of the mixed layer. The data show that monthly mean shear tends to be higher and more spatially concentrated in the south of the Passage and in summer.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Estimation of log₁₀S² in Drake Passage. Shear has been calculated here as centered differences around ADCP depth bins rather than as a bulk shear. The Dong et al. (2008) mixed layer depth climatology has been marked by white dots to illustrate that high shear usually occurs below the base of the mixed layer. The data show that monthly mean shear tends to be higher and more spatially concentrated in the south of the Passage and in summer.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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These stratification and shear data suggest that the model assumption of a two-layer structure is more appropriate for the south of the Passage than the north and for the austral summer, fall, and spring than winter.

b. Bulk shear statistics and rotary properties

Insight into the range of applicability of the model can be gained by considering the statistics of bulk shear observed throughout the year from the LMG, which show the extent to which it is subject to spiking, and its rotary properties that illustrate the extent to which the near-inertial processes of interest dominate.

The probability distribution functions (PDFs) of bulk shear variance (S², Fig. 5) show that it has a non-Gaussian distribution with a distinctly leptokurtic profile (i.e., with a higher peak and fatter tails) in all seasons. This property is confirmed by taking the kurtosis of the distribution (Table 1), where kurtosis values of 12 or greater are found in all seasons (versus a value of three for a Gaussian distribution), which shows that enhancement of shear is an intermittent process with bursts of high shear interspersed among longer intervals of much lower shear, particularly in summer, fall, and spring. The (normalized) standard deviations show that the degree of variation is relatively uniform throughout the year (Table 1).

Probability distribution functions of bulk shear variance in Drake Passage for the Laurence M. Gould data where (a) shows the PDFs for standard deviations less than 3 and (b) extends this for the tail of the distribution with a rescaled y axis. A Gaussian distribution has been included for comparison. These PDFs have been calculated for each season individually with respect to the standard deviation of that season. The value at each point indicates the proportion of observations within 0.25 standard deviations of that normalized standard deviation.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Probability distribution functions of bulk shear variance in Drake Passage for the Laurence M. Gould data where (a) shows the PDFs for standard deviations less than 3 and (b) extends this for the tail of the distribution with a rescaled y axis. A Gaussian distribution has been included for comparison. These PDFs have been calculated for each season individually with respect to the standard deviation of that season. The value at each point indicates the proportion of observations within 0.25 standard deviations of that normalized standard deviation.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Probability distribution functions of bulk shear variance in Drake Passage for the Laurence M. Gould data where (a) shows the PDFs for standard deviations less than 3 and (b) extends this for the tail of the distribution with a rescaled y axis. A Gaussian distribution has been included for comparison. These PDFs have been calculated for each season individually with respect to the standard deviation of that season. The value at each point indicates the proportion of observations within 0.25 standard deviations of that normalized standard deviation.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Table 1.

Statistics of bulk shear variance for the Laurence M. Gould data where the layer depths have been calculated using the Dong et al. (2008) MLD climatology. The standard deviations have been normalized by the respective mean values.

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The highest mean values of bulk S² are found in austral summer, fall, and spring (Table 1). The mean values in winter are somewhat lower, despite this being the season of highest wind stress variance in Drake Passage (Thompson et al. 2007). Taken in conjunction with the monthly profiles of S² calculated between successive ADCP depth bins in Fig. 4, this could mean the kinetic energy input by the wind, particularly in the north of the Passage, follows a markedly different pathway to dissipation than elsewhere. We return to this point in the discussion. Therefore, the two-layer assumption of the model may not be applicable in the regime of the northern Drake Passage.

The inertial frequency in Drake Passage ranges from 0.070 cycles per hour (cph) (14.25-h period) in the north to 0.075 cph (13.25 h) in the south. The rotary spectra of the Cook bulk shear peaks at an anticyclonic rotation of 0.072 cph (Fig. 6a), very close to the mean inertial frequency of 0.073 cph. There is much less shear variance rotating cyclonically. The 95% confidence intervals are, however, relatively large owing to the few degrees of freedom in this individual transect, with only the largest peaks significantly different from zero.

Rotary spectral analysis of the bulk shear vector in the (a) Cook and (b) Laurence M. Gould transects averaged by season (s⁻²); 95% χ² confidence intervals are shown. Anticyclonic spectra are plotted as solid lines; cyclonic spectra are plotted as dashed lines. (c) The spectral coherence between S² and |τ| (red), d_tS² and P(S²) (blue), d_tS² and d_tτ (green) are shown with 95% confidence intervals assuming two degrees of freedom for each of 161 transects of sufficient length included here. Gray shading in (a)–(c) indicates the inertial frequency range of Drake Passage.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Rotary spectral analysis of the bulk shear vector in the (a) Cook and (b) Laurence M. Gould transects averaged by season (s⁻²); 95% χ² confidence intervals are shown. Anticyclonic spectra are plotted as solid lines; cyclonic spectra are plotted as dashed lines. (c) The spectral coherence between S² and |τ| (red), d_tS² and P(S²) (blue), d_tS² and d_tτ (green) are shown with 95% confidence intervals assuming two degrees of freedom for each of 161 transects of sufficient length included here. Gray shading in (a)–(c) indicates the inertial frequency range of Drake Passage.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Rotary spectral analysis of the bulk shear vector in the (a) Cook and (b) Laurence M. Gould transects averaged by season (s⁻²); 95% χ² confidence intervals are shown. Anticyclonic spectra are plotted as solid lines; cyclonic spectra are plotted as dashed lines. (c) The spectral coherence between S² and |τ| (red), d_tS² and P(S²) (blue), d_tS² and d_tτ (green) are shown with 95% confidence intervals assuming two degrees of freedom for each of 161 transects of sufficient length included here. Gray shading in (a)–(c) indicates the inertial frequency range of Drake Passage.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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The magnitude of the LMG bulk shear variance differs between seasons (Fig. 6b) in line with the means shown in Table 1. The spectra in spring and fall have a similar shape with a peak close to the inertial frequency and the variance falls off more slowly at lower frequencies. In summer and winter the peak shear variance occurs at 0.045 cph (22 h), with the remaining variance spread broadly around this (Fig. 6b). On account of the much higher degrees of freedom in this calculation, the shear variance is shown to be significantly different from zero in all seasons at frequencies less than approximately 0.15 cph (6.7 h).

The higher mean, kurtosis, and shear variance at near inertial frequencies of bulk shear in summer, fall, and spring suggest that the model is more applicable in these periods than in winter.

5. Model evaluation

Here, we employ a dual approach to evaluating the proposed open-ocean shear spike generation model with the Drake Passage observations. First, we perform a detailed comparison of d_tS² (with the subscript denoting differentiation) and predicted shear squared production P(S²) for the Cook transect as set out in (5). Second, we expand our model evaluation over all seasons using the LMG data to provide a statistical test of model skill. The dissipation of S² by interfacial friction is shown in the appendix to be insignificant with respect to the production of S² and is omitted from the comparison.

In the Cook transect the model can be evaluated in terms of four distinct regimes found over the transect. The first regime applies to the period from the start of the transect until the start of 10 February (i.e., between 56.8° and 58.1°S) where the weakest S² (Fig. 7b), wind stress (Fig. 7a), and stratification (Fig. 2b) during the transect were found. In this interval there were spikes that coincided with wind shear alignment such as midway through 8 February and early on 9 February (Fig. 7c), though the amplitude of the actual changes was much larger than that predicted by the model. These spikes occurred during periods when the shear vector in Fig. 7d rotated at a near-inertial frequency (i.e., approximately parallel to the black lines in Fig. 7d, which simulate an anticyclonic rotation at the local inertial frequency for comparison purposes). Overall, however, it seems that the model's applicability in this regime of weak stratification and wind stress is low.

Observations of the wind and shear along with model output in the James Cook transect: (a) magnitude of the wind stress, (b) bulk S², (c) a comparison of d_tS² and P(S²), and (d) the direction of the wind (green stars) and shear (blue circles) vectors where rotation parallel to the black lines indicates anticyclonic rotation at the local inertial frequency. The direction is calculated relative to due north with rotation positive in a clockwise sense. A 3-h moving average filter has been applied to the d_tS² time series in (c). All time series are with respect to Universal Time and local time in the region is 4–5 h behind this.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Observations of the wind and shear along with model output in the James Cook transect: (a) magnitude of the wind stress, (b) bulk S², (c) a comparison of d_tS² and P(S²), and (d) the direction of the wind (green stars) and shear (blue circles) vectors where rotation parallel to the black lines indicates anticyclonic rotation at the local inertial frequency. The direction is calculated relative to due north with rotation positive in a clockwise sense. A 3-h moving average filter has been applied to the d_tS² time series in (c). All time series are with respect to Universal Time and local time in the region is 4–5 h behind this.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Observations of the wind and shear along with model output in the James Cook transect: (a) magnitude of the wind stress, (b) bulk S², (c) a comparison of d_tS² and P(S²), and (d) the direction of the wind (green stars) and shear (blue circles) vectors where rotation parallel to the black lines indicates anticyclonic rotation at the local inertial frequency. The direction is calculated relative to due north with rotation positive in a clockwise sense. A 3-h moving average filter has been applied to the d_tS² time series in (c). All time series are with respect to Universal Time and local time in the region is 4–5 h behind this.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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The second regime covers the period from the start of 10 February to midway through 11 February (i.e., between 58.1° and 59.1°S). This part of the transect had stronger wind forcing and stratification though the latter was still relatively weak (Fig. 2b). There was again agreement between the model and d_tS² on the sign of changes in shear variance at near-inertial frequencies over 10 February, though the amplitude of the actual changes remained somewhat larger. Near midday on 11th February, however, a spike in shear variance of similar amplitude was clearly seen in the data and model prediction. This spike was marked by a near-inertial rotation of the shear vector (Fig. 7d) while, in contrast, the wind had almost no rotational component. The model performed better at identifying when near-inertial shear spikes would occur in this regime as the stratification increased.

The third regime was the period of wind stress close to zero (Fig. 7a) from 11 February until later on 12 February (i.e., between 59.1° and 60.6°S). The model obviously predicted low generation of S², though this interval was notable for three distinct shear spikes (Fig. 7c), which are unexplained by the model. The wind vector underwent a near reversal in direction in this interval.

The fourth regime spans the interval from 12 February to the end of the transect (i.e., south of 59.1°) when there was once again a strong wind stress. The mixed layer depth varied over this region (Fig. 2b) but was characterized by a sharp transition layer throughout. This interval included a double peak in shear production midway through 13 and early on 14 February (Fig. 7c), which were coincident with wind shear alignment, (Fig. 7d), while the clearest agreement of all between the observations and model occurred on 15 February, followed by destruction of shear variance which proceeded more rapidly in the observations than predicted by the model. Over this interval of a few hours the near-inertial anticyclonic rotation of the shear vector was less apparent. This was followed by a period in which S² fell while wind and shear were opposed. This period of strong wind stress and stratification is the most applicable for the model.

A challenge to this analysis is determining whether the near-inertial shear arose from inertial oscillations or near-inertial internal waves. This can be better understood by performing a wavelet analysis on the upper and lower layers of one of the velocity components, in this case zonal velocity. Figure 8 shows the amplitude of oscillations at different rotational periods (without specifying the sense of the rotation). In the north of the passage (7–12 February in Figs. 8a,b) near-inertial rotation of similar amplitude is apparent in both upper and lower layers (where the local inertial period is indicated by the black line). The structure was quite different in the south of the passage, however, where the rotations were much stronger in the upper layer than in the lower layer. This implies that the near-inertial bulk shear arose in the south of the passage simply by the strong inhibition of vertical momentum fluxes through the transition layer, while the bulk shear arose in the north owing to weak inhibition of vertical momentum fluxes and possibly a more subtle mechanism involving a change in phase of the rotation with depth (Dohan and Davis 2011).

A wavelet analysis of the zonal velocity in the (a) upper and (b) lower layers used to calculate bulk shear during the Cook transect. The colorscale (cm s⁻¹) values correspond to the amplitude of oscillations at each rotational period. The analysis highlights the strong vertical diffusion of velocity variance at the local inertial period (indicated by the horizontal black line) in the north (early in time series) compared to the south of the passage (later in the time series).

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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A wavelet analysis of the zonal velocity in the (a) upper and (b) lower layers used to calculate bulk shear during the Cook transect. The colorscale (cm s⁻¹) values correspond to the amplitude of oscillations at each rotational period. The analysis highlights the strong vertical diffusion of velocity variance at the local inertial period (indicated by the horizontal black line) in the north (early in time series) compared to the south of the passage (later in the time series).

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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A wavelet analysis of the zonal velocity in the (a) upper and (b) lower layers used to calculate bulk shear during the Cook transect. The colorscale (cm s⁻¹) values correspond to the amplitude of oscillations at each rotational period. The analysis highlights the strong vertical diffusion of velocity variance at the local inertial period (indicated by the horizontal black line) in the north (early in time series) compared to the south of the passage (later in the time series).

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0104.1

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Given the intermittent nature of shear spikes, it is instructive to consider a statistical measure of the model skill using the larger LMG dataset (Fig. 6c). We estimated and compared the magnitude-squared coherence (hereafter coherence) between 1) shear-squared and relative surface stress, 2) the time derivatives of the aforementioned, and 3) d_tS² and predicted P(S²). All of the estimated coherences differed significantly from zero at 95% confidence. Coherence was highest between the shear-squared and relative surface stress at low frequencies, suggesting that longer periods of forcing will generate more shear, consistent with previous findings (Grant and Belcher 2011). However, the coherence drops markedly at superinertial frequencies. Notably, the coherence between d_tS² and predicted P(S²) exceeded the other estimates with 95% confidence over most of the superinertial frequencies resolved. The peak coherence between d_tS² and P(S²) occurs at a higher frequency than the inertial period, reflecting the fact that the key time scale here is the rate of relative rotation of the wind and shear, which is likely to be spread over a range of frequencies owing to the rapid advection of atmospheric features rather than focused on the local inertial frequency.

6. Discussion

A simple two-layer theoretical model for the generation of shear spikes in open ocean conditions has been derived and tested against observations from Drake Passage. The model predicts that shear is generated by the alignment of wind and shear vectors and is destroyed by their opposition, with the destruction of shear by the vertical diffusion of momentum owing to interfacial friction playing a negligible role. In Drake Passage the model's highest skill at predicting shear generation was for near-inertial frequencies when the water column had a well-defined surface mixed layer above a strongly stratified transition layer. The model was not readily applicable in regions where the surface layer was lightly stratified throughout.

The observed rate of change of S² decreased sharply on a number of occasions, in Fig. 7c, in a manner not replicated by the model (e.g., late 14 February and early 15 February). This could reflect processes not represented explicitly in the simple model, such as shear instability which gives rise to turbulent momentum fluxes that operate much more efficiently than those parameterized by the interfacial friction term intended to replicate relatively laminar momentum transfer.

In demonstrating the regimes of applicability of the model, climatologies of stratification and shear variance were presented. The climatologies showed that the south of Drake Passage has higher levels of shear and stratification than the north of the passage throughout the year. It also illustrated that in the north of the passage, shear variance was lower in winter than in summer, despite this being the season of highest wind stress magnitude. These differences may indicate a number of distinct pathways to kinetic energy dissipation in the passage that depend on the location and season of energy input. In the south the presence of sharp stratification may mean more kinetic energy is dissipated in the transition layer if a Kelvin–Helmholtz instability develops or is transferred to the internal wave field if no instability develops. In the north the very weak stratification in winter may mean, instead, that more near-inertial kinetic energy is projected onto the barotropic or low vertical wavenumber baroclinic modes, though this would decrease as the stratification becomes stronger in summer. A separate study would be required to confirm this.

A notable feature of Fig. 7d is the tendency for the shear vector rotation to be coherent in phase over long intervals. From late on 13 February to the first half of 16 February, for example, the shear vector had a clear near-inertial rotation that was coherent in phase over an interval in which the ship covered a distance of about 180 km. Although the existence of inertial rotations over such a range is a simple consequence of the length scale of atmospheric motions, the significant phase coherence of this rotation along the track may imply that the shear spiking events analyzed here are of substantial horizontal extent.

The main motivation for investigating shear spikes is as a possible trigger for the development of shear instabilities and thus diapcynal mixing. Although the spatial and temporal resolution of these datasets is too coarse to estimate this directly, the intermittency of shear production predicted by the model is consistent with estimates of dissipation of turbulent kinetic energy made with microstructure instruments (e.g., Gregg et al. 1993). Furthermore, these spikes in shear occurred without a wind stress rotating at the local inertial frequency and so may be more widely prevalent than the resonant case. The shear spiking model does reduce to a continuous production of S² in the special case of a resonant wind. The shear spiking mechanism could have important local effects in mixing the water column, as it could repeatedly provide nutrient fluxes from the interior into the light-rich surface layer over a summer stratified period in the open ocean (Parslow et al. 2001) sustaining a deep chlorophyll maximum.