1. Introduction
The study of near-inertial oscillations and internal waves began with the advent of moored, self-recording current meter measurements in the 1960s. These instruments revealed considerable variance near the local inertial frequency (Webster 1968) and motivated a series of efforts to better understand near-inertial variability, its predominant generation mechanisms, and its role in other ocean processes such as diapycnal mixing and energy transport: Pollard and Millard (1970) found that the ocean surface mixed layer responds like a damped harmonic oscillator to impulsive wind forcing if momentum is assumed to diffuse instantaneously throughout the surface mixed layer (i.e., the layer acts like a solid, rather than a liquid, at time scales long relative to turbulence but short relative to the general circulation and mesoscale eddies). D’Asaro (1985) used this “slab” model to estimate the average wind energy flux from several long-term wind recording moorings surrounding North America. Alford (2001), Watanabe and Hibiya (2002), and Alford (2003) expanded the use of the slab model to estimate the global flux using long-term global reanalysis surface winds. Since then, a number of studies have investigated the sensitivity of the global wind work calculation to the properties of the input fields and mixed layer climatology (Jiang et al. 2005; Rimac et al. 2013). Direct connections between observations and simulated currents have remained elusive (Alford et al. 2012).
The Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES) is a long-term international tracer release experiment designed to investigate interior mixing in the Antarctic Circumpolar Current (ACC). The studied area extends from the relatively weak and diffuse region of the ACC found in the eastern South Pacific to the more constricted and energetic currents of the Scotia Sea and the South Atlantic. The ship observations presented here come from the first field expedition of the experiment aboard the R/V Roger Revelle in January 2009 and which returned in late February 2009 (Fig. 1). This cruise was tasked with the deployment of profiling floats, RAFOS floats, and sound sources, the CF3SF5 tracer release, and initial tracer sampling (Ledwell et al. 2011). The cruise covered a wide swath of the eastern South Pacific sector of the ACC. On 7 February 2009, the ship weathered a strong storm. This provided an excellent case study for wind generation of mixed layer near-inertial oscillations (NIOs) and downward-propagating near-inertial waves (NIWs). Three Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats deployed during the cruise and active from February through November 2009 show evidence of downward-propagating near-inertial internal wave packets. The energy contained within observed wave packets, and its relationship to the surface wind stress, is investigated in this study.
Study region in the Southern Ocean west of Drake Passage. (a) Cruise track (dashed) and contours of AVISO absolute dynamic topography averaged from 2000 to 2013. Solid contours approximately correspond to the Subantarctic Front, Polar Front, and Southern Antarctic Circumpolar Current Front at 0, −30, and −100 cm respectively. (b) Closeup of the DIMES tracer release location, with the ship track from 2 to 19 Feb 2009. Shipboard data in this region are treated as a stationary time series in Fig. 5. (c) Gridded wind speed and direction from a QuikSCAT descending pass on 7 Feb 2009, illustrating storm size and shape. (d) Trajectories from three EM-APEX profiling floats (colored by serial number) deployed near 58°S and 108°W.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
Section 2 describes the wind and current data used in the analysis. Section 3 describes the methods used to model the surface response to wind and to interpret float velocity profiles. In section 4, the results from one-dimensional mixed layer models are shown and compared with the results from EM-APEX observations. Sections 5 and 6 discuss these results and their contribution to global wind energy flux estimates.
2. Data
Data for this analysis come from (i) the now defunct QuikSCAT satellite, (ii) National Centers for Environmental Prediction (NCEP) reanalysis wind, (iii) cross-calibrated multiplatform (CCMP) wind, (iv) profiling Lagrangian EM-APEX floats, and (v) ship-based meteorological, (vi) ship-based conductivity–temperature–depth (CTD), and (vii) ship-based acoustic Doppler current profiler (ADCP) measurements.
a. Observed wind and currents
Two vector-measuring wind sensors mounted on the bow and main masts of the ship recorded winds every 30 s. A hull-mounted Teledyne-RDI 150-kHz ADCP was used to measure water velocity between 29 and 200 m. The ADCP is sensitive to sea state (because of the bubbles beneath the transducers) and the presence (or absence) of scatterers in the water column. Its range was limited during the times of peak winds because of the bubbles under the hull and during daylight because of the diel migration of zooplankton (scatterers). The ship stayed within a 48-km square region from 3 to 10 February 2009. The majority of this time was spent within a smaller 10-km radius (Fig. 1b). The location was chosen for its weak tides and geostrophic currents that were confirmed by an initial hydrographic survey of the region. Weak background currents combined with the assumption that NIOs have a horizontal scale of at least 50 km (size of the survey region) allow for the reasonable treatment of the ADCP current record as stationary.
b. Gridded winds
We use NCEP-blended sea winds (Zhang et al. 2006) and CCMP ocean surface winds (Atlas et al. 2011) in three distinct comparisons. These products were interpolated onto the ship position to compare with in situ measurements. A subset of this ship-following time series was sampled between 3 and 10 February 2009 to simulate the observed storm. CCMP winds were interpolated onto the float trajectories to drive a float-following slab model to estimate the ability of these products to reproduce surface oscillations over longer time scales.
Gridded QuikSCAT satellite data (Fig. 1c) show the storm was a large diffuse cyclone spanning 30° of longitude. This storm organized around a low pressure center just before it passed the ship’s location and its cyclonic structure persisted for 2 days before dissipating into a disorganized wind field. Examination of a 10-yr CCMP wind time series at 58°S and 108°W shows storms with wind speed in excess of 20 m s−1 are common with an average occurrence of once every 10 days. QuikSCAT global images show a near-constant chain of large cyclones across the South Pacific. The strong winds and rotation near the inertial frequency made this storm an excellent candidate for the study of wind generation of mixed layer near-inertial motions.
c. EM-APEX profiling floats
Vertical profiles of horizontal velocity were obtained by EM-APEX floats. These floats measure the horizontal electric current produced by the motion of seawater through Earth’s magnetic field (Sanford et al. 2005). The floats are autonomous and typically collect about 300 vertical profiles before exhausting battery power. The data are relayed back for processing through the Iridium satellite network. In addition to velocity measurements, the floats are equipped with a Sea-Bird Electronics SBE 41 CTD. The floats record horizontal velocity, conductivity, salinity, and pressure every 25 s, corresponding to a depth resolution of 2–3 m (Sanford et al. 2005). These floats were programmed to profile in bursts such that ascending profiles are separated by one-half the local inertial period, in this case about 7 h (Fig. 2). The floats also transmit surface location (from GPS), time, and various performance metrics. Details of post-processing techniques used to convert raw data into water velocities are described in Sanford et al. (2005). Each velocity datum has an associated error estimate that represents the residual signal amplitude after processing. Data with error greater than 0.01 m s−1 were removed.
Diagram of EM-APEX float path through one burst cycle. Heavy lines show half-inertial period spaced ascending profiles. The burst begins with the float descending to 2000 m, then surfacing, descending to 1500 m, and surfacing again. The two ascending profiles are separated by approximately one-half the inertial period. The drift time between bursts is adjustable; a 4-day drift is typical in these data.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
3. Methods
The goal of this work is to clarify the connection between high-frequency winds and subsurface near-inertial wave activity. We therefore require (i) a thorough understanding of the spectral and statistical properties of the wind products, (ii) a model for the upper-ocean forcing process, and (iii) a way to diagnose downward-propagating internal wave energy, both from the rotary spectral properties of velocity profiles and from half inertial differences of pairs of profiles. Each of these is discussed below.
a. Wind time series
1) Smoothing and subsampling the ship wind
High-resolution observed winds (30-s sampling) were smoothed with a 6-h running rectangular filter and subsampled to 6-h resolution for comparison with the gridded products (Fig. 3). The smoothed and subsampled wind was used to assess the impact of the 6-h time resolution inherent in the gridded wind products on spectral levels (Fig. 4a) and air–sea energy flux. These fully resolved and smoothed ship wind records are referred to hereinafter as SW and SSW.
Comparison of global gridded wind products to measurements by the ship’s sensors. (a) 10-m wind from NCEP (interpolated onto the cruise track) plotted against measured wind speed. The eastern component of the wind (u) is shown in blue and the northern component (υ) is shown in red. (b) As in (a), but comparing CCMP wind to the measured wind.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
Spectral analysis of wind records. (a) Power spectrum of wind speed from gridded and measured winds. The local inertial frequency is shown by the solid vertical line. Spectrum from the measured winds smoothed using a 6-h rectangular filter match well with the spectra from gridded winds. The response of the smoothing filter is shown by the dashed black line. (b) The coherence between gridded and smoothed observed winds.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
2) Spectral analysis of wind time series
The rotary power spectrum of wind speed was computed for each of the four wind time series (CCMP, NCEP, SSW, and SW) (Fig. 4a). The full 40-day record from 12 January to 20 February 2009 was separated into fifteen 5-day half-overlapping windows. The coherence between the gridded products and the smoothed subsampled ship wind was computed using the same 5-day window (Fig. 4b).
b. Mixed layer models
The upper-ocean response was modeled using two 1D mixed layer models: the Price–Weller–Pinkel (PWP) and the damped slab models (Pollard and Millard 1970; D’Asaro 1985; Price et al. 1986; Plueddemann and Farrar 2006). The slab model is desirable for its simplicity, but replaces important physics with simple parameterizations. It cannot simulate or accommodate the sudden changes in mixed layer depth seen during strong forcing, and the radiation of near-inertial energy by internal waves is parameterized through a damping coefficient. The first of these limitations is remedied by the PWP model (Price et al. 1986), which includes physically based vertical mixing of momentum and buoyancy but continues to neglect the lateral gradients responsible for wave radiation. PWP lacks the numerical simplicity and possibility of analytic solution. Longer PWP simulations require high-frequency heat and freshwater fluxes. These are unavailable and limit PWP’s use to brief simulations of a single storm event. Comparison between the two models here is a vital step for evaluating the importance of mixed layer depth evolution during a forcing event. We focus on the ocean mixed layer response from 3 to 10 February 2009 (Fig. 5). Wind speed was converted to stress using speed-dependent drag coefficient (Large and Pond 1981). Both models were initialized at rest. The PWP model was initialized with a temperature and salinity profile from the ship’s CTD on 3 February 2009.
Observed and modeled upper-ocean wind and current time series during the storm period. (a) Wind stress determined from shipboard wind measurements is shown by the solid black line. The gray line shows wind direction with 0 radians = east. Dashed line shows the angular rate of change at the inertial frequency (58°S). (b) Shipboard ADCP currents (east component) shown in color. Background density field from ship CTD shown in gray contours (0.01 kg m−3 σ0 isolines). Three time series of mixed layer depth diagnosed from EM-APEX, ship CTD, and the PWP model are shown by black lines. (c) The bandpassed ADCP measurements, slab model currents, and PWP model currents (black, blue, and red, respectively). Thick lines indicate eastward currents (u) and thinner lines indicate northward currents (υ). (d) Energy flux into near-inertial motions (τ · U) between wind stress and surface currents. (e) Wind work 
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
1) Slab model
Energy radiation from the mixed layer is parameterized using a damping factor r = 0.04|f| (determined empirically by fitting model output to ADCP currents).
2) Price–Weller–Pinkel model
The PWP model (Price et al. 1986) was originally developed to model the influence of heat, salt, and momentum fluxes on the diurnal cycle of upper-ocean stratification at the ocean surface. PWP uses forward time stepping to evolve upper-ocean temperature, salinity, and velocity with depth. We used a 900-s time step and 5-m depth levels to 300 m. For better comparison with the slab model, our implementation of the PWP model also included the linear drag/wave radiation parameter r = 0.04|f|. Though numerically more complex than the slab model, the ability to simulate changes in mixed layer depth because of the wind forcing makes PWP an important component of this study. Buoyancy fluxes were not included to facilitate comparison between the models, and so this approach is limited to short (single wind event) simulations; in the absence of restratifying buoyancy fluxes, mixed layer depth can only increase.
3) Mixed layer depth
Mixed layer depth H is important for accurately modeling the amplitude of the ocean current response to wind variations. When H is constant, |Z| scales as 1/H. For the storm response period, the slab model mixed layer depth was held constant at 80 m, which matches vertical density profiles near the time of the storm. Other slab model studies have used smoothed or climatological mixed layer depths (Alford 2001; Plueddemann and Farrar 2006). These time-varying mixed layers accurately modulate the seasonal cycle of mixed layer current amplitude, but do not change quickly to simulate a storm response. This leads to poor simulation of current speed for any particular storm. Underestimating the mixed layer depth will result in overestimating wind work. Our choice of a constant mixed layer depth simplifies the analysis and is motivated by the presence of slablike inertial oscillations to 80 m in the prestorm period and by poststorm vertical density profiles. This choice also minimizes the root-mean-square difference between model and ADCP currents. As seen in Fig. 5b, stratification and shear are present throughout the upper 80-m highlighting the inherent inaccuracy of the slab (instantaneous vertical mixing) approximation. In the PWP model, mixed layer depth evolves through shear instability parameterizations. This allows PWP to better simulate currents while the mixed layer is deepening.
4) Isolating the inertial-band model response
We chose not to separate the inertial component from the model output using the substitution Z = ZI + ZE into Eq. (1), where 
5) Wind energy flux
6) Error of energy flux estimates
These wind work estimates are highly sensitive to both the methods used to obtain them and the data from which they are obtained, evidenced by the large scatter in results (Table 1). The flux computation τ · Z is quite sensitive to the phasing of the wind and currents. The 6-hourly forcing is often unable to reproduce the exact phasing of observed wind. Mixed layer depth is a further source of error in these estimates, in the slab model the work is proportional to 1/H. The mixed layer depth can be tuned to minimize the error between the observed and modeled currents (as we have done here). This favors a deeper slab model mixed layer as the data are fit to poststorm (deeper mixed layer) currents. The vast majority of the wind work is done at the onset of the storm (Fig. 5). Tuning the mixed layer depth to reproduce the observed energy flux results in a shallower layer. This highlights the importance of the initial stratification in determining the wind work. Peak wind for this storm varies from 21 to 33 m s−1. Here, we use a speed-dependent drag coefficient that ignores sea state (Large and Pond 1981) to derive the stress. It should be noted that the peak stress corresponds with large surface waves. This further compounds the uncertainty of the wind stress used here. Given these sources of error, any confidence limits on the wind work results seem quite arbitrary. A better criterion to validate the model response is to compare with the ship’s ADCP, as we have done.
Wind energy input into internal wave band motions across all model scenarios from 3 to 10 Feb 2009. The asterisk (*) denotes observed energy input computed from shipboard acoustic Doppler current profilers (SADCP) and observed winds.
Monte Carlo methods were used to better estimate the effects these uncertainties introduce into the integrated wind work. Each wind stress time series was randomized with the Gaussian noise of an amplitude set by its root-mean-square difference with the fully resolved wind. Integrated wind work was computed for each realization. The values from ±2 standard deviations from the ensemble mean are shown in Table 1 as the 95% confidence limits (CL).
7) Float-following slab model
Over the 9-month deployment of the three EM-APEX floats, the upper-ocean stratification changed gradually (Fig. 6). CCMP winds were sampled along the EM-APEX drift trajectories to produce float-following wind time series. These winds were used to force the slab model. Coriolis frequency variations because of the moving model domain were included (though they do not have a large effect). Mixed layer depth interpolated from monthly, Argo-based climatology (de Boyer Montégut et al. 2004) gave similar values to those derived from the EM-APEX float CTD measurements. Float profiles and climatology both show winter mixed layer depths in excess of 400 m, but density-compensating temperature and salinity variance indicate that these deep “mixed” layers are not actively mixing. This suggests these depths do not represent the vertical extent of wind-forced inertial oscillations and should not be used with the slab model. Average turbulent vertical velocities derived from wind stress time series (D’Asaro 2001) indicate that the time scale of vertical momentum transport in deep mixed layers is often greater than half of the inertial period. It is unlikely that wind-driven, turbulent, vertical momentum transport could drive the near-instant current response assumed in the slab and PWP models over such a large vertical range. In consideration of this, we constructed a mixed layer depth time series based on weak stratification seen in the float record. The average monthly mixed layer depth was chosen as the depth where monthly averaged buoyancy frequency begins to increase (Fig. 6).
Monthly averaged (top) buoyancy frequency squared (N2) and (bottom) shear squared (|dU/dz|2) plotted on a repeating logarithmic scale from 10−7 to 10−4. Monthly averaged mixed layer depth using 0.05 and 0.1 kg m−3 criteria shown as open circles and triangles, respectively. The mixed layer time series used with the float-following slab model is shown by the dashed line.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
c. Identifying near-inertial internal waves
1) Half inertial differencing
The result of this operation applied to a velocity profile showing a likely near-inertial wave is shown in Fig. 7. This method works well as long as inertial signals dominate, but it is vulnerable to the aliasing of higher frequencies. For example, simulations indicate M2 variance, if present, is aliased into the half inertial difference. However, a frequency spectrum from the ship ADCP time series (not shown) indicates that near-inertial variance is much greater than M2 in this data. As a test of the half inertial method, we have also made use of additional profiles collected by the EM-APEX to construct a high-passed view of currents (not shown) by removing the burst average velocity profile. The extent to which half inertial differences and high-passed profiles, which include all measured internal wave variance, agree is evidence that the near-inertial band dominates high-frequency motions in this sector of the Southern Ocean. This agreement strengthens the argument that downward-propagating beams (Fig. 8) are indeed near-inertial wave packets.
(a) Ascending profiles of horizontal velocity. Eastward currents (u) are shown in blue, northward currents (υ) are shown in red. (b) Eastward and northward inertial-band currents after differencing. (c) Inertial-band current speed (black) and 250-m average speed (red).
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
Kinetic energy of inertial-band motions observed by EM-APEX. (top) Column-integrated (0 to 1500 m) kinetic energy for each float. (bottom) Kinetic energy from half inertially differenced velocity profiles from February to November 2009 from each EM-APEX float. Gray contours show potential density anomaly as seen by each float at 0.1 kg m−3 spacing.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
We identify downward-propagating, near-inertial waves in the EM-APEX velocity profiles based on the following criteria: mirror imaging in half inertial pairs indicating a dominance of inertial-band energy (Sanford 1975), clockwise rotation of horizontal velocity with depth (Leaman and Sanford 1975), and downward progression of kinetic energy maxima over time (Rossby and Sanford 1976) (Fig. 8). The relationship of these downward-propagating wave packets to the wind-forcing and mixed layer inertial oscillations is discussed in the results section.
2) Single-wave packet spectral analysis
A wave packet observed by three floats from 20 February to 31 March 2009 was selected for a more focused analysis of vertical structure and downward energy flux. The 2000-m record was separated into seven half-overlapping 500-m segments. For each data segment, a linear fit was removed from the record and the Hamming tapering window (a modified cosine window) was applied to the data (Harris 1978; Welch 1967). Directional and rotary shear spectra were calculated in each segment (Leaman and Sanford 1975). Spectra were then time averaged in each depth range (Fig. 9).
(left) Vertically binned (500-m bins) power spectra of vertical shear normalized by buoyancy frequency (bin averaged N2). The GM76 reference shear spectrum at twice the base energy level is shown by the solid black line. The color scale (the same across all panels) denotes increasing depth. (middle) Counterclockwise (positive kz) rotary power spectrum of vertical shear. (right) Clockwise (negative kz) rotary power spectrum of vertical shear.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
Ratio of counterclockwise to clockwise variance from integrated rotary power spectra. Values above one indicate the majority of the variance is counterclockwise (indicating downward-propagating waves). The inset panel shows the histogram of the log10 of the ratio; the solid black line separates predominantly counterclockwise variance from predominantly clockwise variance.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
4. Results
a. Wind products
Comparison of the two reanalysis products NCEP and CCMP with the filtered, decimated ship wind time series (SSW) shows good correlation (Fig. 3), with CCMP matching the observations more closely than NCEP except at the point of strongest wind. Root-mean-square scatter of NCEP and CCMP around the observations is 10.2 and 6.1 m s−1, respectively, in rough agreement with validation statistics (Atlas et al. 2011).
All three low-resolution (i.e., 6 hourly) wind time series are reduced in amplitude (relative to SW) at frequencies above 1 cycle day−1 (Fig. 4). In the case of the degraded ship wind time series (SSW), this can be partially explained by the spectral response of the 6-h boxcar (running mean) filter applied before decimation to 6-h sampling. Such a filter has a sinc2(pf) behavior (for filter width p and frequency f), resulting in a reduction factor in spectral level at the inertial frequency of 0.53 (intersection of the dashed and solid lines in Fig. 4). Interestingly, the difference in spectral level between the 6-h and fully resolved spectra is roughly a factor of 4, the filter response makes up slightly less than half the attenuation of variance at f. The similarity of the NCEP and CCMP spectral levels to the SSW suggests that their levels are also influenced by a similar filter (though likely one more complex than a simple rectangular window). Gridded winds are coherent with observed winds to just over 1 cycle day−1, but are not significantly coherent at the inertial frequency (Fig. 4). The drop in coherence corresponds to the frequency at which the spectral levels of NCEP and CCMP become significantly different from the spectrum of SW.
b. Model results
1) Ship observation period
During the period of the storm and upper-ocean response observed by the R/V Revelle, both the slab and PWP models are able to largely reproduce the wind-forced evolution of the near-inertial currents (Fig. 5), with PWP additionally simulating deepening of the mixed layer. A difficulty for the models is reproducing the initial mixed layer response to the strongest wind. Initial currents quickly deepen the mixed layer leading to a steep initial decline in current amplitude. PWP performs better in this respect but underestimates the full strength of the initial current response. In the wake of the storm, the current direction and amplitude remain in fairly close agreement until the end of the observation period.
The simulated mixed layer currents from all model scenarios were compared with ship-based ADCP observations. The result of these simulations is the integral 
Wind work estimates from the various slab model runs vary considerably with the wind-forcing time series used (Table 1). Much of this variability can be attributed to a single point in the wind-forcing time series—the maximum wind stress at the onset of the storm. At 1800 UTC 7 February 2009 wind speeds were 29.5, 23.7, 33.2, and 20.5 m s−1 for the CCMP, NCEP, SW, and SSW, respectively, corresponding to wind stresses of 2.53, 1.4, 3.56, and 0.95 Pa. The 6-h resolution does not allow faithful reproduction of the wind variations over the course of the storm, which remained near the maximum speed for less than 2 h. Depending on the timing of the samples, this can result in an over- or underestimation of the rate of acceleration and rotation of the winds that impact the overall momentum input. On average, the wind’s near-inertial energy level reduction because of 6-h filtering and/or sampling (described above and in Fig. 4) should be the dominant effect, but for a single storm there is considerable uncertainty in the result. Monte Carlo testing shows a range of 0.25 to 0.33 kJ m−2 simply from moving the starting point of the subsampling used to generate SSW. It is possible that different data assimilation and interpolation approaches used in NCEP and CCMP also have an influence, but the large single-storm uncertainty outweighs any systematic differences between these datasets. However, SSW slab model wind work (Table 1) is approximately ⅕ that from the SW because of a reduction in peak wind stress. This is somewhat less than the ⅓ ratio between the globally averaged wind work from 6-h NCEP forcing and 1-h forcing found by Rimac et al. (2013). However, the ratio of between highly SW and SSW forced PWP simulations is 0.4. If this wind work reduction comes equally from a reduction in wind stress and mixed layer currents, it would imply a 
2) Lagrangian/float-following slab model
Longer-duration slab model simulations of mixed layer motions (using CCMP winds interpolated to each float’s trajectory) confirm that mixed layer wind energy inputs occur around the time of each subsurface near-inertial wave packet observed by the floats. Additionally, the times are coincident with enhanced tracer depth shear seen by Ledwell et al. (2011). The float-following slab output was compared to float-observed mixed layer currents. A complicating factor is the treatment of mixed layer depth in the Lagrangian model; this can be (i) held constant, (ii) follow climatology (e.g., de Boyer Montégut et al. 2004), or (iii) be derived from float observations (which is done here). One of the greatest challenges to an observationally based approach is that the definition of the mixed layer typically used for large-scale climatology is not appropriate for wind-forced near-inertial current models. Here, we construct a time series of mixed layer depth using the payer of weakest monthly averaged stratification (dashed line, Fig. 6). Additionally, two established mixed layer criteria were applied to the data (Fig. 6), but both criteria fail to capture the weak stratification determining mixed layer current response to the wind. As a result, they dramatically underpredict mixed layer inertial oscillation amplitudes during austral winter.
The half inertial differencing operation amplifies noise in the float measurements, resulting in a noise floor of 0.05 m s−1. The empirical probability distributions of the model results and observations look quite similar (Fig. 11, inset). The Kolmogorov–Smirnov test, applied to these data above 0.05 m s−1, indicates the modeled NIOs, and float observations are drawn from the same underlying distribution at the 95% confidence level. Monthly means of both modeled and observed near-inertial amplitudes (Fig. 11) show a similar seasonal cycle, with the exception of excess observed energy in April 2009. The source of the excess energy is unknown but may be trapped NIWs. In April, the floats were traveling near the northern (anticyclonic) flank of the Polar Front where trapping of subinertial NIWs is likely because of the strong vorticity (Kunze 1985).
Results from the slab model in a float-following reference frame. The current speed from the float-following model is shown in light red (results from three model runs are superimposed), and observed mixed layer currents are shown by gray circles. Colored red and black lines show the monthly averaged mixed layer current speed from the model and observations, respectively. The 95% confidence limits are indicated by vertical error bars. The inset shows the empirical probability density function of the EM-APEX observations (black) and model currents (red). A Kolmogorov–Smirnov test applied to these data does not reject the null hypothesis (α = 0.05) that the data are drawn from the same continuous distribution.
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
c. Float observations of downward-propagating waves
1) Near-inertial wave packet case study, March 2009
We estimate that the storm on 7 February 2009 input 3.51 kJ m−2 of kinetic energy into inertial motions in the ocean surface layer, and the three EM-APEX observed column-integrated kinetic energy over the following month at a level similar to this input (Fig. 8). The EM-APEX data during March 2009 are unique in that multiple floats sampled the same wave packet as they transited the outer edge of an anticyclonic eddy and remained within about 10 km of one another. The burst sampling executed by the floats yields at best four evenly timed current observations at a given depth. These data are insufficient to exactly determine the wave’s intrinsic frequency, but plane wave fits indicate the frequency of the observed waves is not more than 2% greater than the local inertial frequency. Downward energy flux within the packet, diagnosed by combining the energy density with a vertical group velocity of 40 m day−1 estimated from the downward slope of the packet over time (Fig. 8; March 2009), is above 1 mW m−2 at the depth of the largest current amplitude, decreasing below 900-m depth (Fig. 12). The CCW (downward energy propagation) rotary wavenumber spectrum of vertical shear from this period (Figs. 9, 10) is markedly peaked around 5 × 10−3 cpm (200-m vertical wavelength) in contrast to the CW spectrum (upward energy) that more closely matches the Garret–Munk (GM76) (Garrett and Munk 1975; Cairns and Williams 1976) model of the typical background internal wave field.
(left) Vertical energy flux, computed as observed near-inertial internal wave energy density times vertical group velocity, during March 2009. Error bars show the 95% confidence limits. (right) Average near-inertial internal wave energy density over all 2009 data (black line) with 95% confidence limits (gray shading).
Citation: Journal of Physical Oceanography 45, 2; 10.1175/JPO-D-14-0076.1
Vertical ray-tracing solutions (Kunze 1985), assuming a horizontal scale of 20–30 km, show the wave as having been generated within 40 km of the observed location between 5 to 20 February 2009 (i.e., likely by one of the two storms passing through the area on 7 and 19 February).
2) Half inertial period differencing and kinetic energy
Over the 9 months sampled (February to November 2009), a series of downward-propagating near-inertial wave packets were observed by each float. The apparent downward trajectory of the amplitude peaks, combined with the dominance of counterclockwise rotation with depth, indicates downward-propagating energy in these wave packets.
EM-APEX floats over the 9 months following the cruise show depth-integrated (0 to 1500 m) kinetic energy ranging from 1 to 7 kJ m−2 with a long-term mean of 1.6 kJ m−2 (Fig. 8). The time-mean kinetic energy profile from all half inertial differences shows energy decreasing from 6 to 1.5 J m−3 over the measured depth range (Fig. 12). When combined with the 40 m day−1 group velocity observed from the March 2009 wave packet, this gives an energy flux of 3 mW m−2 at the mixed layer base decreasing to 0.7 mW m−2 at 1500-m depth. However, our ability to quantify the vertical group velocity is poor, making this estimate highly uncertain.
3) Shear and strain spectra of the near-inertial band
A rotary spectral decomposition was computed for each velocity profile. Increased variance near the 200-m wavelength is prominent in profiles that also exhibit elevated inertial-band currents in the time domain. The ratio of integrated rotary variance (Fig. 10) shows the predominance of energy in positive (CCW) wavenumbers. Shear spectra from profiles containing near-inertial wave signals are elevated at low wavenumber, significantly departing from the spectral levels predicted in the Garrett–Munk model of the typical background IW field. Depth windowing of velocity data from March 2009 shows the depth evolution of the rotary polarization nine. Near-surface spectra are highly polarized with a CCW/CW ratio near 10. This ratio approaches one with increasing depth.
Strain spectra (not shown), computed from ascending CTD profiles, do not appear elevated in the corresponding wavenumber range. This is consistent with wave motion near the inertial frequency that is expected to be nearly horizontal.
5. Discussion
a. Gridded wind products
The 6-hourly wind products cannot force a realistic one-dimensional model response because of their lack of high-frequency variance. The 10-m wind spectrum is “red,” and near-inertial variance makes up a small fraction of total wind variance. Wind events of importance to NIO/NIW generation occur at and above the inertial frequency. All of the variance driving near-inertial motion at relatively high latitudes occurs very near the Nyquist frequency of 6-h products. Small-amplitude high-frequency winds constantly drive low-amplitude inertial responses in the mixed layer. Integrated wind work from fully resolved observations shows a gradual increase over time that is the result of high-frequency flux. In contrast, the wind work time series generated using 6-h gridded winds is dominated by large steps at higher strength wind events. Our comparison of ship wind to gridded products indicates that the spectral properties of gridded winds might be corrected by accounting for a 6-h filter (Fig. 3a). Higher temporal resolution of wind is necessary to accurately model wind energy flux.
b. Mixed layer models
1) Comparison of wind work to previous estimates
Previous studies highlight the predominance of near-inertial-band energy in the ocean surface (Elipot and Lumpkin 2008). The key results in the series of papers by Alford (2001) and Watanabe and Hibiya (2002) are both the total integrated wind flux estimate and its global distribution. These global estimates require wind and mixed layer depth from reanalysis and climatology, respectively. This study shows the energy flux τ · Z to be directly proportional to mixed layer depth. Climatological mixed layer depths significantly misrepresent the ocean response to impulsive wind forcing when the upper-ocean buoyancy gradient is small. This could lead to significant error in globally integrated wind energy flux estimates; we hope to address this further in future work. We also find that smoothing the highly resolved wind record with a 6-h filter attenuates wind variance in the inertial band by a factor of 4, in rough agreement with a recent study on attenuation of wind variance by decimating in space and time (Rimac et al. 2013). The resulting energy flux from our wind time series vary considerably, indicating that although two wind stress time series can resemble one another statistically, they do not contribute the same energy into the ocean surface. Two factors are integral in influencing the energy flux, wind stress, and actively mixing depth at the onset of the storm. We cannot directly compare our results with previous energy flux estimates because we have simulated one event; however, these results show estimates for one event are influenced by small variations in wind and mixed layer depth.
Models forced with fully resolved wind were able to capture the initial storm response and, in a bulk sense, accurately predict the momentum flux into the upper ocean using only a wind time series. Models forced with low-resolution time series did not perform as well. Model results could be made to fit the observations very well by fine tuning the linear damping/radiation and mixed layer depth terms at the expense of generality. Despite this, no one set of mixed layer depth and damping parameters could reconcile results using low- and high-frequency forcing. Ocean inertial currents are driven by sharp changes in the magnitude and direction of wind stress at and above the inertial frequency. The 6-hourly wind time series either miss these brief impulse events completely and/or overestimate the temporal extent of peak winds. The result is a total wind work estimate that may be much higher (CCMP here) or lower (NCEP here) than the real ocean. During this time period, the CCMP product overpredicted wind work; however, at other times the results were the opposite with the NCEP product overpredicting wind work. In both cases, the results come down to a small number of data points during peak wind periods.
2) Lagrangian mixed layer model
The slab model, forced with CCMP winds following the 9-month float trajectories (Fig. 11), estimates an average energy flux into near-inertial currents of 3 mW m−2. While monthly averages of modeled currents and float observations agree well, these data are not ideal for the computation of the wind energy flux into inertial oscillations. We have seen in our highly resolved case that the 6-h winds can both overestimate and underestimate the real fluxes. The key to matching the modeled currents with the data became the identification of the mixed layer depth. Our mixed layer time series is motivated by near-surface variability in the float CTD data, but it is difficult to construct an objective algorithm that finds a similar trend. The reliance of these results on mixed layer depth highlights both the current lack of understanding of the vertical distribution of near-surface momentum transport and the sensitivity of the slab model to its parameterization.
c. Wave observations
The highest-amplitude near-inertial waves we observe here have a vertical wavelength near 200 m. Internal wave observations frequently occur in the narrow depth range of 500 to 1000 m near the base of geostrophic shear. This hints that these waves may be trapped by the horizontal eddy structure (modulating planetary vorticity; Kunze 1985) and vertical (critical layer) shear. A topic for further study will be to model the three-dimensional near-inertial wave ray paths through a flow field resembling the observed background conditions. Observed wave packets appear to lag periods of energetic wind forcing making one-to-one correspondence between the subsurface waves and wind-driven mixed layer motions difficult to verify. These data highlight the complex three-dimensional structure of the wave field. Propagating waves appear at middepth throughout the EM-APEX record. Wave energy may preferentially collect in some regions (near fronts or within eddies) making observations more likely.
The decay in downward energy in the March 2009 wave packet (Fig. 12) could be interpreted either as being because of turbulent dissipation or simply as a lower boundary of a vertically localized wave packet. If the former, a rate of 8 × 10−10 W kg−1 is inferred, which is considerably larger than the 1 × 10−10 W kg−1 found by the high-resolution profiler measurements reported in Ledwell et al. (2011). Thus, a vertical limit of a transient event appears more likely. It is also possible that vertical group velocity decreases with depth. However, this is not clearly seen in the depth versus time maps (Fig. 8).
6. Conclusions
This case study of near-inertial wind forcing and downward energy propagation in the Southern Ocean illustrates several features of the upper-ocean forcing of internal waves that have long been suspected but have been difficult to observe in entirety. These include the following:
the quantitative connection between episodic near-inertial wind work and the increase of kinetic energy in the mixed layer (Fig. 5 and Table 1);
the sensitivity of wind work calculations to the details of the wind time series, particularly the spectral level at the inertial frequency (Fig. 4), to the maximum wind stress in an event, and to near-surface stratification;
the radiation of the bulk of mixed layer inertial energy in the form of downward-propagating near-inertial internal waves;
the ubiquity of downward-propagating near-inertial wave energy in the upper 1500 m of the southeast Pacific sector of the Southern Ocean (Fig. 8); and
the decay of the downward-propagating energy well above the deep-ocean bottom (Fig. 12).
The range in total wind work inferred from model runs forced by different wind products highlights the sensitivity of these models to both wind stress peaks and temporal resolution. The slab model forced with smoothed winds produced just 21% of the wind work found by the same model forced with fully resolved winds (Table 1). Gridded wind products provide an excellent statistical picture of the scale and duration of large storms, but may not accurately reproduce the shape, location, and speed for any given storm, particularly in regions with little in situ data such as the Southern Ocean. During the period ship winds were available, January and February 2009, the CCMP product appears to do a better job of matching the observations and of creating a realistic model response, but this may simply be fortuitous.
From float velocity data, we observe downward-propagating features in kinetic energy. Analysis of these energetic periods shows counterclockwise rotation with depth, consistent with downward-propagating near-inertial waves. A large fraction of the variance in these packets lies in the vertical wavenumber range 0.001 to 0.01 cpm (100 to 1000 m). The largest-amplitude horizontal velocities are associated with wavenumbers near 0.005 cpm (200 m), usually occurring between 500- and 1200-m depth. This depth range coincides with an increased buoyancy frequency at the base of the winter mixed layer (Fig. 6). Velocity profiles also indicate that the majority of the geostrophic shear in the ACC eddies and fronts decays with a scale of approximately 1000 m, suggesting the possibility that the observed waves are slowed and amplified as the background shear or vorticity decreases with depth Kunze (1985).
Over the full 9-month EM-APEX observation period, both slab model wind work and downward near-inertial energy flux are estimated at approximately 3 mW m−2. Given the above caveats about gridded wind products, the wind work may still be an overestimate in spite of the fact that surface current amplitudes are well predicted by the slab model. Additionally, the downward flux estimate relies on the assumption of constant vertical group velocity and can only be regarded as a ballpark figure. Nevertheless, the agreement suggests that most of the near-inertial wind energy input is able to radiate from the mixed layer as propagating near-inertial internal waves.
Acknowledgments
We gratefully acknowledge the use of several freely available global datasets. NCEP reanalysis 2 data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado. CCMP wind was provided by NASA/JPL PODAAC. The QuikSCAT data used in this study are distributed from the NASA/JPL PODAAC and Remote Sensing Systems. Sea surface altimeter products were produced by Ssalto/Duacs and distributed by AVISO with support from CNES. Thanks to Dr. Eric Kunze providing essential scientific and editorial advice. Useful comments were provided by Brian Chinn and Andy Pickering. Many thanks to the captain and crew of the R/V Roger Revelle. This research was funded through NSF Grants OCE0623177 and OCE1129564.
APPENDIX
Integrating Factor Solution to Pollard and Millard’s Mixed Layer Current Model
We refer to Eq. (A6) as the slab mixed layer model; it is a linear, inhomogeneous, first-order, ordinary differential equation of the type 
The slab model has a particular analytic solution whenever 
This solution technically allows for a time-varying mixed layer depth. While Eq. (A13) can still be evaluated with H = H(t), this does not explicitly conserve momentum in the mixed layer. For example, if a mixed layer deepened by a factor of 2 under increased wind stress, this would not dampen the surface currents and would effectively double the momentum in the model domain. Slowly varying H(t), as is used here in the float-following model, has little effect on the momentum budget over short time scales.
The amplitude of near-inertial currents is proportional to 1/H, an important factor in interpreting the results from this model.
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