a. Data

Velocity records from more than 2200 historical moorings (over 7000 instruments) were acquired from the Oregon State University Buoy Group, which maintains a database of 1064 deep-water moorings deployed from the 1970s to the 1990s, from the National Oceanographic Data Center (245 moorings), the World Ocean Circulation Experiment (446 moorings), from the Autonomous Temperature Line Acquisition System (ATLAS) database (416 moorings), and from the Woods Hole Oceanographic Institution Buoy Archive (445 moorings; kindly provided by C. Wunsch). These data were initially selected as per Alford (2003a, hereinafter A03) according to the following four criteria:

1. total water depth greater than 3200 m,

2. sampling interval less than 3 h,

3. record length at least six continuous months, and

4. latitude poleward of 2.5°.

A03 required full-depth coverage to resolve flux. Our present focus on energy, which does not, greatly increases the number of usable moorings (690 moorings as compared with A03’s 60).

Coverage (Fig. 1) is adequate in the North Atlantic, and poorer but acceptable in the North Pacific. Elsewhere, substantial geographical voids are present, as to be expected, in the Southern Hemisphere and at low latitude. In addition, potential spatial biases result from the disproportionately large number of records in “interesting places,” for example, near strong currents or topography.

b. WKB scaling

Waves propagating vertically through the ocean’s depth-varying stratification undergo refraction, which affects their vertical wavenumber and amplitude. To account for this effect, velocity measurements are typically “WKB scaled” to enable comparison between different stratification environments (e.g., Leaman and Sanford 1976). Here, each time series is WKB scaled using annual-mean climatological buoyancy frequency, N, computed at each mooring location from the 2001 World Ocean Atlas (Conkright et al. 2002):

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where ũ = u + iυ is the measured rotary velocity, z_i is the measurement depth, and N is the depth-mean stratification at the site.

The effect of WKB scaling is shown for a typical North Atlantic site in Fig. 2. Climatological N ²(z) (Fig. 2a) decreases greatly in depth, but velocity magnitude WKB scales as N^1/2, resulting in amplifications/reductions in ũ_WKB by factors of 0.5–2 (Fig. 2b). Hence, unscaled energy (right, dashed), which scales as N, typically decays by a factor of O(10), but is more nearly constant after scaling (solid). The depth averages of scaled and unscaled energy are equal in a perfectly sampled water column (finite sampling introduces only minor errors dependent on the geometry). Since a goal is the calculation of lateral distribution of depth-averaged energy, this is a desirable property. (An alternate approach, for example, would be to scale values at all sites by the same constant N. Since column-averaged N varies by less than a factor of 2 over the range of sample locations, the choice has little effect on the results.)

c. Spectra

The time series of ũ_WKB are next divided into half-overlapping 30-day blocks, and the rotary frequency spectrum computed for each. The choice of 30 days represents a balance between frequency and temporal resolution. To maximize spectral resolution and precision for our shortened block sizes, the sine multitaper method of Riedel and Sidorenko (1995) is employed. This estimator exhibits superior precision and low bias relative to Welch’s (1967) method for comparable block sizes, without appreciably affecting leakage.

For each block, the spectral estimate Ŝ(ω) is given by the average over all K (=3 for this study) tapers:

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where ω is cyclic frequency and the Kth estimate,

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is computed by applying the time-domain taper

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to the time series. Here M and Δt are the number of measurements and sampling interval. The resulting spectral resolution,

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where T ≡ MΔt = 30 days is the record length, is 0.133 cpd.

Sample time series and spectra from a location at 42°N, 46°W are shown in Fig. 3. Time series of WKB-scaled zonal velocity are presented at left for the entire year (Fig. 3a), together with closeups (Figs. 3b–d) of the periods indicated with gray shading in Fig. 3a. Near-inertial motions are visible in Fig. 3a as “fuzz,” but are resolved as near-inertial motions in subsequent closeups, which depict periods in order of decreasing near-inertial energy.

The rotary spectrum computed with the entire-year record (Fig. 3e) shows the typical broad near-inertial peak (e.g., Fu 1981), together with a narrow peak at the M₂ frequency. Both peaks are well above the level of the internal wave continuum, which shows a slope near −2, as also typical (Fu 1981). Clockwise energy (black) greatly exceeds counterclockwise (gray) at lower frequencies, as expected for linear internal waves and often observed. Typical records exhibit peak frequencies 1.02–1.2 times as great as the local inertial frequency (dashed line); the present record’s peak is near 1.02f.

The sine multitaper spectral estimate from each 30-day period in Figs. 3b–d is plotted in Figs. 3f–h. The spectral resolution Δω ≈ 1/30 cpd, shown by the width of the gray box in Fig. 3f, is obviously coarser than the full-length record (shown by the box in Fig. 3e), as evident in the broadening of the M₂ “line.” However, resolution is still easily sufficient to resolve the near-inertial and tidal peaks (though, e.g., M₂ and S₂ are not separable). Precision (indicated by height of gray boxes) is also reduced somewhat relative to the full-length record, but is still sufficient to resolve all but the weakest peaks. That in Fig. 3h is barely significant.

d. Near-inertial kinetic energy

The kinetic energy of the near-inertial peak (J kg⁻¹) is calculated for each segment by integrating between the half-power points ω_±1/2 (Fig. 3, gray shading) on either side of the near-inertial peak,

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Because our focus is exclusively on the inertial peak, we will henceforth drop the in superscript and refer to the moored kinetic energy KE_moor. Motions with the correct polarization for each hemisphere are isolated by integrating only over negative/positive frequencies in the Northern/Southern Hemispheres, respectively [Figs. 3e–h, black]. Owing to the typical steepness of the falloff on either side of f, the choice of the integration limits (e.g., half-power or quarter-power) does not appreciably affect the calculations.

Both the height and the width of the near-inertial peak vary appreciably (a confirmation of the intermittency of the near-inertial peak). The yearlong record is therefore highly statistically nonstationary in the near-inertial band. The spectra computed with shorter blocks suffer much less in this regard, at the cost of reduced resolution. Note that these observations indicate that both peak height and bandwidth must be taken into account in computing KE_moor.

Here, KE_moor calculated from (6) is indicated at lower left in each panel. The 30-day periods shown (Figs. 3f–h) bracket the value for the yearlong record (Fig. 3e). The associated RMS velocity signals (u_WKB = 3.3, 1.8, and 0.7 cm s⁻¹, respectively) are visually consistent with the panels at left (Figs. 3b–d).

As latitude increases, the near-inertial peak moves closer to the semidiurnal tidal peak. To avoid contamination by tidal motions, the integration is truncated at M₂ − Δω = 1.8025 cpd. At 50°, 55°, and 60°, this is 18%, 11%, and 6% above f, respectively; hence, some underestimation may occur if the peak is wider than these limits. Because the variance in most peaks is contained at frequencies <1.1f, contamination is minimal equatorward of 55°.

Latitudes 25.7°–34.6° {sin⁻¹[½(K₁ ± Δω)]} are, in like manner, potentially affected by diurnal tidal motions. However, visual inspection indicates that in the vast majority of cases the near-inertial peak in spectra from yearlong records dominates the K₁ peak. Hence, we have not attempted to truncate our integrations but acknowledge the possibility of slight overestimation in this latitude range.

As discussed in appendix A, mooring motion leads to errors in KE_moor that are different for each mooring owing to a variety of designs and environment. The most serious potential systematic bias is a possible underestimation when instruments are dragged down by strong currents into depths where WKB refraction reduces observed KE_moor by as much as 40%. In appendix B, confidence limits on KE_moor are discussed. Since the 30-day spectral estimates have but few degrees of freedom, uncertainty is large for individual estimates. However, the large number of instruments in our study allows subsequent averaging and reduction of uncertainty in averages to about a factor of 2.

e. Mixed-layer flux and kinetic energy

We compare our estimates of KE_moor with estimates of mixed-layer flux and energy computed by running the Pollard and Millard (1970) slab model forced with NCEP–National Center for Atmospheric Research (NCAR) reanalysis winds (Kalnay et al. 1996) at the location and period of each mooring. The model is run using a latitude-dependent damping, r = 0.3f, and climatological monthly-mean mixed-layer depth (Levitus and Boyer 1994). Flux, F_ML, is then computed as described in detail by Alford (2001, 2003b). Over these time scales, the integrated mixed-layer energy,

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is simply related to the flux via

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Because r varies by less than a factor of 2 over the range of latitudes sampled here, KE_ML and F_ML have identical statistics, seasonal cycle, and spatial dependence. In the following, calculations and plots are of F_ML, but we include KE_ML axes using the value of r at 40° for comparison.

The F_ML depends only weakly on the choice of damping r; however, (8) implies that energy depends on the damping coefficient, KE_ML ∼ r⁻¹. Both quantities are potentially underestimated by up to a factor of 2 owing to the finite temporal (Alford 2003b) and spatial resolution (Jiang et al. 2005) of the NCEP winds, or overestimated by approximately the same factor owing to inadequacies in the slab model (Plueddemann and Farrar 2006). To maximize quantitative certainty, the model was tuned by selecting r = 0.3f [about 2 times that used by Alford (2001, 2003b], maximizing agreement between slab-model and observed KE_ML at the 14 moorings from the database with instruments shallow enough to be in the climatological mixed layer for part of the year. A sample calculation at 41°N (Fig. 4) indicates that agreement is good between model and observed kinetic energy, even though the instrument is within the climatological mixed layer for only about 100 days (Fig. 4b). As will be seen in section 3c, KE_ML is also in good agreement with midlatitude values observed by Park et al. (2005), who estimate KE_ML from many Argo floats (these drift while uploading data, during which time inertial loops can be fit to their trajectories). Hence, though a rigorous error bound is not possible, we feel F_ML and KE_ML are accurate to better than a factor of 2 (probably better in northern midlatitudes, which contain most of the tuning data). Importantly, the magnitude of F_ML is only important for our estimates of the replenishment time scale and does not impact our directly observed KE_moor results.

Because waves propagate downward and equatorward from the mixed-layer forcing region, KE_ML and F_ML would be best estimated at the horizontal location determined by ray tracing from each observation depth back to the surface. However, for simplicity we compute the mixed-layer energy and flux at each mooring location. Because rays reach the bottom within 400 km of their source (Garrett 2001) except within 10° of the equator and poles (where we have no data), this is a justifiable simplification provided our focus is gross seasonal energy comparisons rather than event-by-event comparison (and far preferable to the uncertainties introduced by “reverse ray tracing” to determine the best forcing location for each mooring).

a. Case studies

Sample 2-yr time series of near-inertial kinetic energy are shown in Fig. 5. Local winter, defined arbitrarily as November–February and May–August for the Northern/Southern Hemispheres, is shaded gray. In each panel, the WKB-scaled near-inertial energy KE_moor (solid colors) and the energy flux from the wind to the mixed layer, F_ML (dashed; note different units and scale factor), are plotted. To attempt to illustrate representative variability, examples are selected from the Northern (upper) and Southern (lower) Hemispheres, as well as from the Atlantic (left) and Pacific (right) Oceans. Ordinate limits are the same for all four panels.

Considerable variability in KE_moor is seen both in individual records and as a function of location. In addition, the small number of degrees of freedom of these 30-day spectral estimates results in large formal error bars (upper left; see appendix B). However, the following qualitative observations may be made, which will be confirmed statistically (using all moorings) in the following sections:

• Winter enhancement of KE_moor is evident at some depths at each location. It is often seen at great depth (most clearly in Fig. 5a, which shows a factor-of-10 seasonal modulation at all depths).

• The magnitude and phasing of seasonal modulation of mixed-layer energy (dashed) is very similar in each example to that observed in KE_moor. At one site (Fig. 5c), even shorter-period fluctuations in KE_moor and KE_ML appear correlated.

• In the Southern Hemisphere examples (bottom), seasonal modulation of both KE_moor and KE_ML appears weaker than in the Northern Hemisphere.

• When WKB-scaled, no obvious depth trend is discernible in KE_moor in these examples (subsequent statistics will reveal a depth decay during winter).

The seasonal modulation of deep KE_moor at these sites, and its similarity in amplitude and phase to wind-forced mixed-layer energy, are suggestive that the inertial motions are associated with wind forcing. Because of the variability evident in these few examples, and large error bars in single spectral estimates, the remainder of the paper will focus on composites of all moorings. Because we wish to eliminate near-surface influences, we will henceforth consider only instruments deeper than 300 m.

b. Depth dependence

We first examine the depth dependence of WKB-scaled moored energy (Fig. 6). We first focus on the Northern Hemisphere, since coverage is so much better and to simplify the presentation. All values are plotted versus depth (Fig. 6a) and versus height above bottom (Fig. 6b) as dots, with darker grays indicating winter months. Oceanographers’ predilection to place instruments at standard depths is clear. The high degree of variability, which is greater at shallow depths, documents the well-known intermittency of near-inertial motions.

Data are next binned versus month and averaged in 500-m depth intervals (colored lines). The number of values in each month’s average, indicated in the legend (bottom left), varies somewhat from month to month since most records do not span exactly a year. The mean number in each depth bin is used to determine 95% confidence limits as described in appendix B (upper left, gray), which are less than a factor of 2 above 4000 m; below this, errors grow owing to the paucity of deep measurements.

The magnitude of bin-averaged KE_moor is ≈(1–10) × 10⁻⁴ J kg⁻¹ (1.4–4.4 cm s⁻¹); individual values can be an order of magnitude higher or lower. In comparison with the unscaled quantity (Fig. 2), WKB-scaled KE_moor decreases much less in depth during all months. This lack of strong depth dependence, also observed by Fu (1981), greatly simplifies subsequent analysis, since energy at a given location and time can be efficiently characterized by the depth mean. However, in winter months (blue), a clear decay by a factor of ≈4–5 is significant at the 95% level (left, gray) from 500 to 3500 m, with more irregular and marginally significant structure below 4000 m. In summer, WKB-scaled energy is nearly constant, to within measurement error.

These conclusions are sharpened somewhat when plotted versus height above bottom instead of depth (Fig. 6b), perhaps owing to the more uniform sampling and resultant reduction of the “standard depth” biases in (Fig. 6a), in addition to hypsometry effects. Here, summer KE_moor is constant or decreases by less than a factor of 2; a clear depth decay by about a factor of 4 from 5000 to 500 m above bottom is seen in winter. As a result of this decay, seasonal modulation is nearly undetectable within 500 m of the bottom. This reduction is a possible signature of topographic generation of inertial waves, which would not be expected to show a seasonal cycle. Above this, the observed winter decay with depth is consistent with an energy source at the surface (though meridional propagation of near-inertial waves complicates 1D interpretations). The observed seasonal cycle of not only the magnitude of KE_moor but also its degree of depth decay are strongly suggestive that the surface energy source is the wind, whose forcing is strongest in the winter.

c. Seasonal cycle

To examine seasonal variability more closely, depth-mean KE_moor, F_ML, and their ratio (which yields a replenishment time scale) are averaged within 10° latitude bands and plotted by month to form a canonical seasonal cycle (Fig. 7). In contrast to the depth-month view just presented, which averaged all northern latitudes together, the seasonal cycle is here examined as a function of latitude. (We continue to focus for the moment on the Northern Hemisphere.) Each quantity is plotted as a thick line whose thickness indicates the 95% confidence limits.

Marked seasonal cycles well in excess of error bounds are seen in the latitude bands centered on 30°, 40°, and 50° (orange, yellow, green). The amplitude of the former two is about 4–5, with the latter closer to 2–3. The highest latitude range (blue) shows a “two-plateau” cycle, marginally resolved, where energy in June–December exceeds February–April energy by a factor of about 2. Energy in the 15°–25° band (which is the most data poor, and strongly weighted toward latitudes >20; see Fig. 1) is constant to within measurement errors.

The same analysis is applied to F_ML in Fig. 7b (note wider ordinate scale). As indicated previously, monthly mean F_ML and KE_ML differ only by a multiplicative factor (8), allowing KE_ML values to be indicated at right using the value of r at 40°. The conversion is correct at 40° (yellow), 10% too high at 50°, and a factor of 2 too low at 20%, but these differences are nearly imperceptible in Fig. 7b. For comparison, KE_ML measured by fitting inertial loops to the drift tracks of Argo floats and averaged from 30° to 60°N in the Atlantic (circles) and Pacific (pluses) are overplotted. Since the floats measure mixed-layer depth, they do not rely on a climatology. Though the latitude range sampled by these floats is not the same as here, their magnitude and seasonal cycle agrees well with our 35°–45° curve (which appears to contain most of their data). This agreement gives confidence in the F_ML and KE_ML estimates.

As seen from the case studies, the magnitude of the seasonal cycle of wind forcing in each band is generally similar to that in KE_moor. The strongest modulation is seen spanning 30° and 40°, as for KE_moor, with similar magnitudes. At lower latitudes, F_ML shows weak/undetectable seasonal modulation, like KE_moor. An exception is at the highest latitudes (55°–65°, blue), where F_ML shows a much stronger cycle than KE_moor. [However, its phasing (minimum in April) is the same as for KE_moor.]

The ratio of depth-integrated KE_moor to the energy flux F_ML gives a replenishment time scale,

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which can be interpreted as the time it would take to substantially reduce KE_moor (a factor of e⁻¹) if forcing stopped. The mean depth of the ocean is D, taken as 4000 m. Here, we have taken advantage of the near constancy with depth of KE_moor (Fig. 6) to estimate its depth integral in (9), which would not otherwise be possible from these vertically sparse data.

The value of τ ≈ 10 days at latitudes 35°–45° (Fig. 7c, yellow), increasing to 50–60 days in the Tropics (red). Only at high latitude does τ exceed 100 days during a portion of the year, associated with the F_ML minimum when climatological mixed layers in the North Atlantic are at their deepest (and possibly poorest known owing to the spatial patchiness of deep convection). At all other latitudes, τ shows a weak or nonexistent seasonal cycle, suggesting a temporally constant relationship between KE_moor and F_ML.

For comparison, dissipation time scales computed in like manner for the internal wave continuum bracket these estimates (7 and 70 days for “Abyssal Recipes” and “pelagic” mixing rates of K_ρ = 10⁻⁴ and 10⁻⁵ m² s⁻¹, respectively; Munk 1981). Though not definitive evidence for wind forcing, the observed seasonal cycle and calculated time scales are consistent with it; for example, observation of a strong seasonal cycle and τ ≥ 1 yr would be inconsistent, since energy reserves would be too large to expect a seasonal cycle.

d. Probability density

We next present the statistics of KE_moor and F_ML and at last consider the Southern Hemisphere. Probability density functions (Fig. 8) indicate that both quantities are qualitatively lognormal. The distributions of F_ML and KE_ML (right panels; axes on top and bottom, respectively) are broader (more intermittent) than KE_moor, as expected owing to their strong dependence on wind speed (D’Asaro 1985; Alford 2001). Winter enhancement of KE_moor is once again obvious in the Northern Hemisphere (Fig. 8a). By contrast, austral winter enhancement in the Southern Hemisphere (Fig. 8b) can be barely discerned. The relative scarcity of Southern Hemisphere data is also apparent from the insets, which indicate the number of observations in each month.

Both the winter enhancement seen in the Northern Hemisphere, and its relative lack in the Southern Hemisphere, are replicated in the distributions of F_ML and KE_ML. In the former case, F_ML is clearly greater, and more intermittent, during winter. In the Southern Hemisphere, only a very weak seasonal cycle in KE_moor and F_ML is observable. (As seen next, this is partly due to the strong weighting of the Southern Hemisphere data toward tropical latitudes, which show weak seasonal cycles in both hemispheres.) At the sampled locations, the seasonal cycle of midlatitude storms, which force mixed-layer motions, is much weaker in the Southern Hemisphere. These patterns are replicated in the deep observed inertial kinetic energy.

e. Spatial structure

The spatial dependence of depth-mean KE_moor is averaged over May–August (northern summer; Fig. 9a) and November–February (northern winter; Fig. 9b). The four-month averages, each of which is plotted as a colored dot, show a great deal of variability. This again reflects the intermittency; a single event occurring in one record can strongly influence the mean. Nonetheless, some patterns are apparent. First, elevation of KE_moor in each region during local winter is evident as seen in previous plots. Second, many of the most energetic northern winter values (Fig. 9b) occur in the western parts of ocean basins, where the strongest wind forcing is found (grayscale; Alford 2003b). This trend is marginally seen in the North Pacific, but is fairly well resolved in the North Atlantic where coverage is densest. Of course, mesoscale activity is strongest in the west as well, as would be expected for a generation by geostrophically adjusting mesoscale flows. However, the western intensification is absent during summer (Fig. 9a), again implicating the wind.

The latitude dependence of KE_moor and its seasonal cycle is examined (Fig. 10a) by plotting energy versus latitude (dots; grayscale indicates month) and computing the zonal mean in 5° bins (colors). Supporting results in previous sections, a seasonal cycle is clearly evident for northern latitudes 25°–45°, with the greatest modulation (factor of 5) occurring near 35°–40°, consistent with Fig. 7a.

A seasonal cycle of a factor of 2–3 is barely resolvable at the 95% level in southern midlatitudes between 35° and 50°, but with smaller magnitude than in the Northern Hemisphere. As can be seen from Fig. 9 and the error bars in Fig. 10a, Southern Hemisphere data are weighted toward the Tropics. Hence, the last section’s probability density analysis (which did not separate versus latitude) showed a nearly undetectable seasonal modulation owing to the more numerous low-latitude records.

Broad winter maxima of KE_moor ≈ 5 × 10⁻⁴ J kg⁻¹ are seen near 30°–40° in both hemispheres (coincident with the midlatitude storm track), decreasing by a factor of 3–4 toward higher latitudes. As discussed, some underestimation is possible for latitudes >55° in order to avoid the M2 peak, but the falloff occurs well before this. At lower latitudes, coverage is poor, but both hemispheres show a decrease near 20° relative to the midlatitude value, before rising again approaching the equator. In summer, the latitudinal distribution is much flatter.

High KE_moor values are observed near 28°N, which elevate the mean in this latitude range by a factor of 2–3. As discussed, the possibility of contamination by diurnal tidal motions cannot be ruled out. It could also be the signature of PSI, as argued by van Haren (2005) from investigation of seven moorings along 20°W (those near 28°N are a subset of the moorings from this database). Whether the enhancement is due to PSI or contamination by diurnal motions, it appears to be localized enough so as not to greatly affect the latitudinal distribution of KE_moor.

As indicated in Fig. 7b, mixed-layer kinetic energy F_ML shows a seasonal cycle of comparable magnitude to KE_moor. It shows a narrower midlatitude peak than does KE_moor, reflecting patterns in storm forcing and mixed-layer depth. The falloff of F_ML in the Tropics is an indication that the primary forcing is associated with the winter midlatitude storm tracks rather than with tropical cyclones (Alford 2001).

The lack of this falloff at low latitudes in KE_moor is possible evidence for propagation of near-inertial waves from higher latitudes where forcing is stronger. This observation is quantified as a strong latitudinal trend in Northern Hemisphere τ (Fig. 7c). Under the storm track, τ ≈ 10 days as in Fig. 7c, but increases to 70–100 days at lower latitudes. On the other hand, the increase of τ south of 29°N would also be consistent with near-inertial motions generated via PSI at 29°N that subsequently propagate equatorward.