1. Introduction
The global mechanical energy budget was posed as an outstanding first-order problem in the theory of the ocean general circulation 14 years ago (Munk and Wunsch 1998), yet today remains largely unsolved. It is typically assumed that the winds working on the surface general circulation provide the dominant power source (Munk and Wunsch 1998; Wunsch and Ferrari 2004). The rate of this forcing has been estimated recently as 0.90 ± 0.05 TW (Scott and Xu 2009) in agreement with other experimental (Xu and Scott 2008; Hughes and Wilson 2008; Wunsch 1998; Scott 1999) and model (von Storch et al. 2007) studies.
The wind forcing is at atmospheric synoptic scales (1000s of km) while the dissipation is at the Kolmogorov scale (millimeters). Many processes are involved in this vast transformation of scale, the key steps of which can briefly be summarized as follows: 1) wind work on the surface geostrophic flow is converted to Ekman pumping that feeds gravitational potential energy below the mixed layer; and 2) this potential energy is then converted to kinetic energy near the deformation scale via baroclinic instability (Killworth and Blundell 2007; Smith 2007) and 3) where a stunted inverse cascade of kinetic energy occurs (Qiu et al. 2008). This is discussed in further detail by Scott and Xu (2009) and Ferrari and Wunsch (2009). We here focus on the dissipation of the geostrophic kinetic energy.
Isolating the mechanism by which this power input to the geostrophic circulation is balanced by dissipation is a much greater challenge, since the main dissipation is presumably largely subsurface and potentially dominated by small-scale and/or intermittent processes. This has important ramifications: this power could potentially provide a key source of energy for the turbulent diapycnal mixing required to drive the meridional oceanic overturning circulation (Kuhlbrodt et al. 2007). Various studies have suggested potential mechanisms, such as ageostrophic instabilities in the ocean interior (Müller et al. 2005) or nonlinear coupling to internal gravity waves (Bühler and McIntyre 2005); meanwhile, the model study of Scott et al. (2011) showed that ~0.42 ± 0.08 TW of this input energy is dissipated via lee wave generation on the ocean bottom. However, to date, none of these proposed mechanisms have been shown individually to be sufficiently strong to dissipate the bulk of this energy, suggesting that a combination of mechanisms may well contribute.
In this study, we focus upon quadratic bottom boundary layer dissipation (BBL dissipation), with the intention of providing an observationally constrained estimate of the energy in subinertial currents (primarily eddies) dissipated globally (the globally integrated BBL dissipation), with assistance from model estimates derived from the Hybrid Coordinate Ocean Model [HYCOM; Chassignet et al. (2007)]. We additionally attempt to establish sensible bounds on the uncertainty of our calculation. This estimate, being a parameterization of bottom drag, inherently does not distinguish between the mechanisms of a turbulent Ekman layer and topographic form drag resulting from lee wave generation.
Idealized quasigeostrophic models typically find that bottom friction is necessary to produce reasonable simulations of realistic ocean-bottom current flow (e.g., Arbic and Flierl 2004, and references therein). Previous studies, however, have assumed that the power dissipated via this mechanism is at most a small proportion of the total power lost by the system; Wunsch and Ferrari (2004), for example, suggested a total global sink due to BBL dissipation of ~0.2 TW, while the review of Müller et al. (2005) assumed it to be negligible. Previous experimental studies, such as those of Sen et al. (2008) and Arbic et al. (2009), have attempted to assess the globally integrated BBL dissipation using broadly similar methods. Sen et al. (2008) calculated a globally integrated BBL dissipation of ~0.22–0.83 TW by averaging a set of 290 moored current meters at depths deeper than 3 km, with potential bias corrections for the real distribution computed using satellite altimetry suggesting an actual value in the lower end of this range. Similarly, using the Naval Research Laboratory Layered Ocean Model (NLOM) and Parallel Ocean Program (POP) global circulation models, with the results compared to 382 current meters, Arbic et al. (2009) estimated a range of ~0.14–0.65 TW.
In this paper, we examine the BBL dissipation using data from ocean current meters, to attempt to produce a more tightly constrained result on these estimates. We do so both by vastly increasing the number of current meters considered by comparison to previous studies, and also via a more sophisticated analysis, in which the World Ocean is divided into regions which are considered separately and then combined to give an estimate of the globally integrated BBL dissipation. These regions are defined in different ways in different sections of the paper, but are intended to take account of the expected geographic distribution of BBL dissipation and the locations of available current meter data. The intention of this division into subregions is to reduce the major bias introduced by the greater density of records in some regions over others. Two different methods of determining regions are presented in this article; these methods use the same analysis method, but differ strongly in how they divide the data geographically.
We first discuss in section 2 the data sources used for our analysis. Section 3 then describes how we compute BBL dissipation, and discusses the selection of some key parameters in our analysis. Section 4 then describes the common elements of the data analysis for our two region-determination methods.
We next take the simple approach of dividing the globe up into a regular grid and assessing each grid box individually, as described in section 5. In the case of a single grid box, this updates the analysis of Sen et al. (2008) with a greatly increased number of measurements. Increasing the number of such boxes improves resolution, but at the cost of reduced coverage in regions without current meters.
After this, we consider an approach based upon hierarchical clustering of current meters, followed by assignment of ocean points to the most appropriate such cluster using HYCOM estimates, chosen via a cost function incorporating both physical distance and the expected change in dissipation estimated from HYCOM. Unlike the grid-based method, this guarantees a current meter value is assigned to all locations, except those we consider too far from any valid current meter. This analysis is discussed in section 6.
Finally, our conclusions are presented in section 7.
2. Data
a. GMACMD
Current meter data are derived from the Global Multiarchive Current Meter Database (GMACMD) (Scott et al. 2010, 2011). This is a global collection of physical oceanographic time series derived from the research programs of many countries, which have been collected across several decades. The data are primarily sourced from ocean current meters and ACPDs, and have been standardized to a common format. At time of writing, the dataset consists of approximately 50 000 such time series, covering a wide range of depths and geographic locations.
We also impose some temporal filtering criteria on the data. We do not consider any time series with a duration less than 30 days; these are typically the results of short-duration experiments for a specific purpose, and cannot be expected to represent the medium- to long-term current behavior in any statistically significant way. In practice with the data we have available, the shortest time series remaining after this filter is 76 days in duration. Time series are inherently weighted by duration owing to our analysis method, discussed below; accordingly, short time series with consequent significant uncertainty contribute little to our final result.
We also impose a requirement that each series have a measurement frequency of at least once per day; series with either an irregular time step or a time step shorter than this have been averaged to this frequency, while series with a longer time step have been discarded. Around 6% of the time series used have a time step of one day, with the rest more frequent; hence tidal aliasing due to this should be limited. Of the series analyzed, 50% have durations shorter than 330 days and 90% are shorter than 20 months. The longest series is 1335 days. Figure 1 illustrates the distribution of time series durations, separated by depth range; only one series is longer than 838 days, and consequently the abscissa ends at 850 days.
Histogram illustrating the durations of the time series considered. Each bar shows the results in bins of 10-day duration; the dark gray regions indicate time series in the bottom 10% of the ocean, the midgray bars the additional time series in the bottom 20% of the ocean, and the light gray bars the additional time series in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Histogram illustrating the durations of the time series considered. Each bar shows the results in bins of 10-day duration; the dark gray regions indicate time series in the bottom 10% of the ocean, the midgray bars the additional time series in the bottom 20% of the ocean, and the light gray bars the additional time series in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Histogram illustrating the durations of the time series considered. Each bar shows the results in bins of 10-day duration; the dark gray regions indicate time series in the bottom 10% of the ocean, the midgray bars the additional time series in the bottom 20% of the ocean, and the light gray bars the additional time series in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Wright et al. (2012) showed that 90% of the individual current meter time series autodecorrelated within 10 days or fewer, and estimated spatial decorrelation scales as on the order of a hundred kilometers using the method of Ducet et al. (2000); the analysis regions used in this study are larger than this, suggesting that adjacent regions should be minimally correlated. Consequently, we assume that spatial and temporal covariances in our analysis have at most a very small effect.
b. HYCOM
HYCOM is a hybrid isopycnal-sigma-pressure (generalized) coordinate ocean model. We use here a set of analyses performed by the Naval Research Laboratory at the Stennis Space Center, United States. This model has been chosen for three key reasons. First, it is the highest resolution global reanalysis available; second, it has an extended analyzed period (late 2003–present); and third, in the analysis described by Scott et al. (2010) it was one of the best-performing global eddying models examined when compared to current meter data in terms of reproducing the location and behavior of experimentally determined bottom currents.
We use the data-assimilative version of HYCOM (HYCOM-DA) as opposed to the free-running version, which performed more poorly in the multimodel analysis of Scott et al. (2010). The model operates on a nominal 
Our HYCOM analyses are produced as daily snapshots, from which we compute 5-day means at each grid point. Scott et al. (2010) examines high-frequency variations in HYCOM-DA, and concludes that while high frequencies do arise in the model, the average of 5 daily snapshots gives close to geostrophic currents, suggesting that geostrophic currents dominate as in the real ocean.
Consequently, high-frequency variations are already averaged out of these results, allowing simplification of the method used to compute BBL dissipation by comparison to that used for our current meter analyses (see section 3 below); Scott et al. (2010) showed that these 5-day means were essentially geostrophic. We use analyses covering the period from 2004–10.
3. BBL dissipation calculation methods
The additional contribution to the result due to higher-frequency interactions was estimated in a previous regional study using a dataset which overlapped substantially with that used here (Wright et al. 2012), and was found to contribute around 15% in shallower regions than those considered here (100 m < depth < 250 m) of the North Atlantic, with a dropoff to much lower levels in deeper waters.
a. The quadratic bottom drag coefficient, cd
A key term in Eqs. (4) and (5) is the quadratic bottom drag coefficient, cd: for both our model and current meter analyses, our result is directly proportional to cd, making the value chosen critical to our results. Accordingly, we would like to use the best possible value for this constant. However, there is little consensus in the literature as to exactly what the value of this should be, with values ranging from as low as 0.5 × 10−3 (Shearman and Lentz 2003; Duncan et al. 2003) to as high as 8.0 × 10−3 (Döös et al. 2004). Several recent studies using current meter data (Sen et al. 2008; Arbic et al. 2009; Wright et al. 2012) have used a value of 2.5 × 10−3, but the justification for this has rested primarily upon models of tidal flows, which may not accurately produce a value of cd suitable for nontidal measurements of the deep ocean.
It should be noted that, in all probability, the quadratic BBL momentum drag cannot be parameterized globally with just a single value of cd, and that some spatial variability should be expected. A full investigation of this variability is, however, beyond the scope of this study. Table 1 illustrates the values of cd estimated by a range of recent studies. The values used in the global POP and HYCOM runs compared in Scott et al. (2010) are included for comparison. Upon examination, a trend is observed in the values presented in the table. All values of cd ≤ 1 × 10−3 are obtained from measurement studies, whereas the higher values, with the exception of the significant outlier study of Döös et al. (2004), are all related to model studies. This is consistent with the significantly reduced bottom currents seen in model studies (Scott et al. 2010).
Literature estimates of the coefficient of quadratic drag, cd. “Type” column indicates whether the study used model estimates (M) or experimental data (X).
Accordingly, we take two different values of cd for this study. For our current meter analyses, we use a value of 1.00 × 10−3; this is toward the upper end of the measurement studies described, and we believe represents a good consensus value. For the HYCOM parts of our study, meanwhile, we use a value of 2.20 × 10−3. This is for two reasons: firstly, this value is a good consensus value for the model studies described, and secondly this is the value for which the HYCOM analyses have been tuned, accordingly representing the optimal value to obtain sensible results from the model.
b. Height range
To obtain an estimate of BBL dissipation, we ideally wish to measure and use the velocity of the flow just above the BBL. In practice, however, we cannot consider just those narrowly clustered about the bottom if we wish to adequately cover the globe with our measurements. Accordingly, we first need to study the current velocities in our data to obtain a measure of how much they vary with height above the ocean bottom, and use the results of these analyses to define the height range we wish to study.
We do this by examining the dependence of the dissipation on the height above the bottom within the BBL. We first conservatively assume that meters in the bottom 10% of the ocean and more than 10 m above the ocean floor are well-correlated with flows in the BBL; this assumption has previously been adopted in several studies, including Sen et al. (2008), Arbic et al. (2009), and Wright et al. (2012), but is also motivated by the availability of measurements to compare to for the following analysis.
Next, we determine all available current meters that are on the same mooring as a current meter in the bottom 10% of the ocean and that operated over the same time period; that is to say, we search for sets of two or more measurements which correspond exactly in latitude, longitude, and time but which differ in height above the seafloor and for which the lowest measurement is in the bottom 10%.
We then normalize all BBL dissipation values to the value of the deepest measurement on the mooring; since we have previously imposed the requirement that at least one measurement be in the bottom 10%, this guarantees that the deepest measurement will be at most this high above the sea floor. This then gives us a measure of how much the speed varies with relative height above the sea floor at the location of each individual current meter mooring.
Finally, we separate the results by seafloor depth and depth as a proportion of this seafloor depth, and repeatedly resample (statistically bootstrap) these values within the box to obtain a distribution of values within the box and a set of uncertainty bounds on this value. Figure 2 illustrates the results of this analysis, divided up by seafloor depth on the abscissa and depth as a proportion of seafloor depth on the ordinate. Boxes have been left blank (white) when insufficient data exist for the bootstrapping process—we choose this value as at least five current meters present in the box. The resulting value of each box is the median of the bootstrapped distribution of ratios of the BBL in that box to the BBL for the lowest meter; that is to say, it represents the center of the distribution of random samplings of the measurements within the box. Observe that values in the bottom 10% itself (lowest row of the figure) are all almost exactly equal to unity, consistent with this.
Estimates of De as a fraction of De in the bottom 10% of the ocean. The horizontal axis shows seafloor depth in m, the vertical axis the depth of the meters analyzed as a proportion of this seafloor depth. Data have been binned into horizontal bins of width 350 m and vertical bins of 10% of ocean depth. Color shows the mean ratio of De to the value of De computed for the bottom 10% of the ocean. Bins containing fewer than five measurements have not been computed.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Estimates of De as a fraction of De in the bottom 10% of the ocean. The horizontal axis shows seafloor depth in m, the vertical axis the depth of the meters analyzed as a proportion of this seafloor depth. Data have been binned into horizontal bins of width 350 m and vertical bins of 10% of ocean depth. Color shows the mean ratio of De to the value of De computed for the bottom 10% of the ocean. Bins containing fewer than five measurements have not been computed.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Estimates of De as a fraction of De in the bottom 10% of the ocean. The horizontal axis shows seafloor depth in m, the vertical axis the depth of the meters analyzed as a proportion of this seafloor depth. Data have been binned into horizontal bins of width 350 m and vertical bins of 10% of ocean depth. Color shows the mean ratio of De to the value of De computed for the bottom 10% of the ocean. Bins containing fewer than five measurements have not been computed.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
From these results, we select three depth ranges to consider. The first such range is the case of measurements in the bottom 10% of the ocean; this is our most conservative estimate, but has the smallest number of measurements. Second, we consider measurements within the bottom 20% of the ocean, including those in the bottom 10%. This is less conservative than the above measurement and consequently increases the number of available measurement locations, but should be broadly in agreement with the result in the bottom 10% as, with the exception of some outliers at depths ~3500 m, values of BBL dissipation in the bottom 20% typically vary by less than a factor of 1.25 × of the value in the bottom 10%; indeed, values in this range are typically slightly lower than those in the bottom 10%, and accordingly may produce an underestimate. Finally, we consider values in the bottom 30% of the ocean; as Fig. 2 shows, some bins have saturated at 3 × the value of the bottom 10%, and accordingly this may represent an overestimate; however, for most sea floor depth ranges this is not the case and accordingly the result should still be reasonable.
Uncertainties in the results shown in Fig. 2 have been omitted for clarity, but are typically small, ~10% at 95% confidence for the bottom 30% of the ocean.
c. Current meter BBL dissipation estimates
We compute BBL dissipation directly from the velocity time series for each current meter using Eq. (5). For each current meter, we compute the dissipation at each time step, and then average over all available time steps to obtain a single value for each current meter. We have discarded several outlier meters from our analysis, either owing to their lying in an atypically strong current which we do not believe impacts upon the bottom dissipation or because of anomalous results, as follows: a cluster of 6 m near the mouth of the Red Sea due to high-velocity exchange flows (Murray and Johns 1997), a cluster of 17 m near the Ross Sea due to high-velocity downslope gravity currents (Gordon et al. 2009), a cluster of 4 m near Gibraltar due to strong currents flowing between the Atlantic Ocean and Mediterranean Sea, and 10 additional statistical extreme outliers, with time-mean BBL dissipation values lying significantly outside the distribution of current meters.
Figure 3 shows the geographic locations of the time series remaining after applying these selection criteria to our dataset: crosses indicate those meters in the bottom 10% of the ocean, + signs the additional meters in the bottom 20% of the ocean, and circles the additional meters in the bottom 30%. The geographic distribution of these measurements is highly irregular, with a strong bias toward boundaries of the North Atlantic.
Global mean BBL dissipation computed from HYCOM averaged over the period 2004–10. Crosses indicate locations of current meters in the bottom 10% of the ocean, + signs additional current meters in the bottom 20% of the ocean, and circles additional current meters in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Global mean BBL dissipation computed from HYCOM averaged over the period 2004–10. Crosses indicate locations of current meters in the bottom 10% of the ocean, + signs additional current meters in the bottom 20% of the ocean, and circles additional current meters in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Global mean BBL dissipation computed from HYCOM averaged over the period 2004–10. Crosses indicate locations of current meters in the bottom 10% of the ocean, + signs additional current meters in the bottom 20% of the ocean, and circles additional current meters in the bottom 30% of the ocean.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
d. HYCOM BBL dissipation estimates
We use Eq. (4) to compute the BBL dissipation at each point on the HYCOM model grid for all times. Figure 3 shows the average for each point over the time period analyzed. Note the considerable range of values, extending from 10−7–100 W m−2: this illustrates that the great majority of BBL dissipation is generated in a comparatively small proportion of the ocean. In particular, the model data suggest that potentially as much as 105 times as much energy per unit area is dissipated via quadratic BBL dissipation in key areas such as the Gulf Stream, Kuroshio, and a broad band across the Southern Ocean when compared to most of the area of the Pacific Ocean or to the mid-Atlantic. It is thus crucial that our calculation of the global BBL dissipation from current meter data properly accounts for these regions of strong dissipation areas. The globally integrated sum of the time-averaged HYCOM BBL dissipation is 0.42 ± 0.07 TW.
It is important to note that several changes to model parameters were made during the analysis period used. Figure 4 illustrates this by showing the monthly mean globally integrated BBL dissipation computed from the HYCOM estimates. Vertical lines on the figure indicate changes to the model parameters, with the version number indicated adjacent to the line. As can clearly be seen, changes in the model strongly affect the globally integrated BBL dissipation, particularly the change between version 90.3 and 90.6. Closer study of the results, however, shows that the geographic distribution of BBL remains largely unchanged: the effect of the version changes is to amplify or dampen BBL levels uniformly across the globe rather than changing the balance between regions. As such, the model changes should not affect our analysis negatively. To formally assess this, the full analysis as described below was repeated using each individual year of HYCOM data; the uncertainty caused by this is considerably less than other sources (standard deviation ±0.03 TW); these analyses have been omitted for brevity.
Monthly mean globally integrated BBL dissipation computed from HYCOM. Version changes are indicated by vertical lines, with the version number indicated adjacent to the line. Note the sharp changes in integrated BBL dissipation caused by these changes.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Monthly mean globally integrated BBL dissipation computed from HYCOM. Version changes are indicated by vertical lines, with the version number indicated adjacent to the line. Note the sharp changes in integrated BBL dissipation caused by these changes.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Monthly mean globally integrated BBL dissipation computed from HYCOM. Version changes are indicated by vertical lines, with the version number indicated adjacent to the line. Note the sharp changes in integrated BBL dissipation caused by these changes.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
The HYCOM estimates are used for three separate purposes in our analysis: first to define regions in section 6, second to compute the bias due to the geographic locations of our current meters, and third to establish uncertainty bounds. In the first case, we are only concerned with the shape of the distribution, and use the time-averaged results shown in Fig. 3 as the basis for our calculation; hence, this change does not affect this part of the analysis. In the second and third cases, we are interested only in the relative change; uncertainty bounds are computed relative to the mean value, while required bias correction values are computed relative to the surrounding region. The exaggerated variability of the HYCOM estimates will tend to increase the size of our uncertainty bounds, resulting in a more conservative estimate; this effect will however be small, since temporal variability provides only a small contribution to our overall uncertainties.
e. Ekman layer arrest
A potentially important consideration is the possibility that there may be arrest of the Ekman layer at the ocean floor in regions where the ocean floor has a sharp high-gradient, due to the slope and flow rate (Garrett et al. 1993). When this occurs, current velocities may rapidly drop to near-zero values, and no BBL dissipation will take place. To assess whether this is taking place in our dataset, we have considered the variability of current flows within the bottom 100 m of the ocean: as the Ekman layer arrest process is theorised to take place very close to the sea floor, measurements of current flows over this range when the Ekman layer is arrested should be easily observable in this height range.
Figure 5 presents the results of this analysis, with height above the sea floor on the ordinate and the ratio of kinetic energy at this height to that in the bottom 50% of the whole ocean on the abcissa. The solid line is the median of the distribution, the light gray bands the 68% confidence interval of the bootstrap distribution, and the dark gray bands the 95% confidence interval. We observe kinetic energy increasing with proximity to the sea floor, with values at 10 m above the sea floor approximately 50% as high as in the ocean above, in qualitative agreement with the observations of Thompson (1977), Uehara and Miyake (2000), and Oey and Lee (2002). Further study of this effect will be an important direction for future study using this dataset.
Ratio of kinetic energy in the bottom boundary layer to kinetic energy in the ocean above, as a function of height above the sea floor.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Ratio of kinetic energy in the bottom boundary layer to kinetic energy in the ocean above, as a function of height above the sea floor.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Ratio of kinetic energy in the bottom boundary layer to kinetic energy in the ocean above, as a function of height above the sea floor.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
We hence conclude that we find no evidence of Ekman layer arrest in our data. Currents in the bottom 30 m are stronger, not weaker, than currents above this level, consistent with the results of Sen et al. (2008) and in sharp contrast to Fig. 10 of Garrett et al. (1993).
For comparison, we also studied individual kinetic energy–height profiles from the 12 locations where we possessed at least five current meters on the same vertical mooring within the bottom 300 m of ocean locations deeper than 2000 m, at least one of which was in the bottom 30 m. These results were broadly in agreement with those presented in Fig. 5, but were felt to be too few in number to be statistically significant.
4. Analysis
As discussed in section 1, we would like to estimate the globally integrated BBL dissipation by dividing the globe into regions, computing each region separately, and then integrating over the globe, using two different region selection techniques. The methods used to generate regions are discussed in detail in sections 5 and 6. Once the regions are generated, all are analyzed in the same way.
a. Current meter estimates
For each region, we determine the mean current meter–derived BBL dissipation value and scale it by area appropriately. The area-weighted sum of these regional estimates provides an estimate of the globally integrated BBL dissipation.
b. Sampling uncertainty
To provide an estimate of uncertainty due to the locations of our current meters (the “sampling uncertainty”), we have repeatedly resampled the HYCOM estimates for each region using sets of Mn points in each region, where Mn is the number of physical current meters present in the same region. We have then used these resamplings to compute the expected distribution of possible BBL dissipation values in each region if the current meters were randomly distributed in the region. By summing globally and considering these distributions, we have then generated global sampling uncertainty bounds.
Initially, our HYCOM 5-day mean files are averaged to produce 30-day means of BBL dissipation over the duration of available estimates on the original grid. This duration corresponds to the thirty-day minimum period we have imposed upon the current meter data. From the 7 yr of HYCOM estimates we use, this gives us 85 individual estimates of the 30-day-mean BBL dissipation at each gridbox.
We then take our defined regions and select 10 000 points from each, randomly distributed across space and time. These values are scaled by the area of each region and summed, giving us a distribution of 10 000 estimates of the BBL dissipation for each region, which are summed to give 10 000 individual estimates of the global BBL dissipation for the whole globe.
Finally, the 95% and 68% confidence bounds of the distribution are computed, and provided as estimates of the uncertainty of the global BBL dissipation estimate for this set of regions. These estimates are computed as a fraction of the HYCOM globally integrated BBL dissipation for that set of regions and converted to a fraction of the current meter derived globally integrated BBL dissipation.
c. Bias correction factor
We next attempt to determine the bias in our result due to the nonrandom locations of the real current meters. Because of the many considerations, both technical and scientific, involved in the deployment of ocean current meters, we may expect a significant proportion of current meters to be sited in locations of particular scientific interest rather than distributed randomly (Sen et al. 2008; Holloway et al. 2011). The meter location symbols on Fig. 3 illustrate the effect of this on a global scale, demonstrating the significantly greater density of records in the North Atlantic as compared to most other regions, but it is possible that even on a more local scale, current meters may be sited in places of local scientific interest such as the locations of known small-scale currents, and accordingly may exhibit a systematic bias from the expectation value of the surrounding region. Accordingly, we would like to determine as far as possible the effect of this on our results.
To make an estimate of this effect, we determine the (all-time mean) HYCOM values at the nearest grid points to our current meter records, and then analyze these values in the same way as the true current meter values. By comparing the results computed in this way to the sum of the equivalent grid boxes in the HYCOM estimates, we hence obtain an estimate of the systematic bias, and hence the required bias correction factor, between the locations with current-meter sampling as compared to the surrounding region. This allows us to determine the effects of geographic placement on our overall estimates of BBL dissipation.
5. Grid analysis
a. Method
For our first set of analyses, we divide the globe into a regular latitude–longitude grid of regions, on a range of gridscales. This provides an estimate based on a geographical distribution of regions determined without reference to either the current meter locations or the HYCOM estimates. In the simplest case, the number of such regions is one; this hence updates the calculation of Sen et al. (2008) with a greatly increased number of measurements. Regular grids from 2.0°–120°, in 0.5° steps (grid sizes), were assessed; at higher resolutions (smaller gridsizes), a more accurate estimate of the dissipation in regions with coverage is possible in principle, but with the caveat that fewer of the regions thus produced will contain current meters. This thus reduces the proportion of the globe, and hence of the global BBL dissipation, covered by that estimate.
Figure 6 illustrates the process used to obtain our estimates for the grid analysis, at a grid size of 30°. Regions have been defined on a 30° grid, indexed to 0°N, 0°E. The top panel shows the application of these a priori regions to the current meter data; current meters are marked with crosses. Here, the color of each region indicates the mean dissipation of the current meters in that region, while regions with no current meter records are left blank. The estimated value of De from current meters is the sum of the product of each filled gridbox with its nonland area. The middle panel shows the median value of the HYCOM samplings in each region which possesses a current meter; the sampling uncertainty bounds are derived from the full distribution of values calculated here (not shown). Finally, the bottom panel shows the mean of the HYCOM model values closest to the physical current meters; the ratio of this to the middle panel produces our estimate of the location bias correction factor.
Illustration of the grid-based method of computing globally integrated BBL dissipation. Regions have been defined on a 30° grid, indexed to0°N, 0°E. (top) Regions filled with the median current meter value in that region, with regions containing no current meters left blank are shown. (middle) The same regions filled via random-sampling from HYCOM are shown. (bottom) The regions filled by taking the time-mean HYCOM value from the model grid points corresponding to the actual current meter locations are shown. The color scale in each case is marked in log10 mW m−2.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Illustration of the grid-based method of computing globally integrated BBL dissipation. Regions have been defined on a 30° grid, indexed to0°N, 0°E. (top) Regions filled with the median current meter value in that region, with regions containing no current meters left blank are shown. (middle) The same regions filled via random-sampling from HYCOM are shown. (bottom) The regions filled by taking the time-mean HYCOM value from the model grid points corresponding to the actual current meter locations are shown. The color scale in each case is marked in log10 mW m−2.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Illustration of the grid-based method of computing globally integrated BBL dissipation. Regions have been defined on a 30° grid, indexed to0°N, 0°E. (top) Regions filled with the median current meter value in that region, with regions containing no current meters left blank are shown. (middle) The same regions filled via random-sampling from HYCOM are shown. (bottom) The regions filled by taking the time-mean HYCOM value from the model grid points corresponding to the actual current meter locations are shown. The color scale in each case is marked in log10 mW m−2.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
b. Results
Figure 7 shows the results of our grid analysis, plotted against the gridsize in degrees. The top panel shows results for the bottom 10% of the ocean, the middle panel the bottom 20%, and the bottom panel the bottom 30%. Note that, in each case, the values of the abscissa decrease from left to right–since as we reduce the gridsize we generate more individual boxes, this corresponds to an increasing number of regions considered and hence a higher-resolution analysis. Each of these panels is broadly similar, but with some important differences between them.
Results for the grid-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) bottom 30% are shown.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Results for the grid-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) bottom 30% are shown.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Results for the grid-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) bottom 30% are shown.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
1) Coverage
We consider first the blue line. This shows, on the right-hand axis, the proportion of global De we expect to capture with the analysis. This is calculated by first computing which gridboxes contain at least one real current meter, then summing up the HYCOM-derived De in each of these boxes and comparing it to the globally summed HYCOM value for all locations (0.424 TW).
As we see, the coverage decreases as we decrease the grid size. This is as we would expect: with very large grid boxes, we should include at least one current meter in every region, whereas at smaller grid sizes, not every box will contain a measurement location. The drop in coverage is not monotonic with decreasing gridsize due to the way the grid is defined: since the same origin point (at 0°N, 0°E) is used for each analysis, a slight change in grid size may shift a box edge sufficiently far in latitude or longitude that it no longer contains a measurement where the “equivalent” box at the previous scale did.
Results are strikingly similar for all three depth analyses, to a much greater degree than might be expected: while there is a very small increase in coverage at some grid sizes as we increase the allowed depth range for our current meters, it is very minor. This shows that, while increasing the number of current meters analyzed, the increased acceptable depth range does not significantly change the proportion of grid boxes which contain a measurement—that is to say, the “new” current meters in the 20% and 30% analyses are in very similar locations to the ones in the bottom 10%, and consequently provide an improvement in the sampling accuracy of our result rather than in coverage.
2) Bias
Next, we examine the red-dashed line. This is our required bias correction factor, as defined in section 4c, with values shown on the far-left axis. Values greater than one indicate that our results are positively biased, values less than one negatively.
Consider first the left side of each panel, where values are consistently less than one. This implies that, compared to other values in the same box, the HYCOM points corresponding to our physical current meters have smaller values than typical for the gridbox they are in. Hence, at these grid sizes, we expect our globally integrated De to be underestimated by this analysis.
In the middle of each panel (~80°–30° grid size), required bias correction values range between 0.6 × and 1.2×—this suggests that, in the model, the current meter locations are reasonably, although not perfectly, representative of the rest of the grid box surrounding them.
At the far right of each panel, required bias correction factors tend toward one—as individual gridboxes become smaller, the probability of a given location being significantly different to those in the gridbox surrounding it becomes significantly smaller, and our estimate becomes less and less biased, at the expense of sharply reduced coverage.
As with coverage, bias is only weakly responsive to changes in grid size. The key difference is that the required bias correction factor becomes smaller (i.e., tends toward 1) with increasing accepted depth. There are also fewer spikes in the curves with increased accepted depth: the 10% panel is the most volatile, the 20%, with the exception of one particularly large spike at around 65° gridsize, more stable but still with some significant spikes, and the 30% is, relatively, the most stable. This is expected, due to the increased number of sampled points within each given gridbox.
3) Globally integrated BBL dissipation and sampling uncertainty
Finally, we consider the solid black line, and the light and dark gray bands surrounding it: these form the core results of our analysis. The line shows the globally integrated BBL dissipation computed at this grid size, using the method outlined in section 4a above, while the bands show the sampling uncertainty we estimate for this value, as described in section 4b. Dark gray indicates the 95% confidence interval, light gray the 68%. These values are shown on the inner left-hand axis.
The most obvious general features of these results are their declining absolute value and shrinking sampling uncertainty, as illustrated by the narrow gray bands, with decreasing grid size. This corresponds to the reduction in coverage: while the values at very small grid sizes have a very low required bias correction factor and very small uncertainties, they represent an estimate of the BBL dissipation for only a very small fraction of the World Ocean.
A second key observation is that, interestingly, our results appear very strongly dependent on the grid used. On small scales, both the required bias correction factor and the globally integrated BBL dissipation are very jumpy, with sharp changes in value produced by only very small changes in grid size. In fact, as the left-hand side of each plot clearly shows, we even see large changes in the measured value when there is no change in overall coverage. Separate analyses, omitted for brevity, also show that shifting the origin point used to define the grid leads to similarly large changes in the final globally integrated value computed. Taken together, these results suggest a strong limitation of this analysis method: since the final result obtained is so strongly dependent on the precise grid used, it is difficult to assign a large degree of trust to any individual result.
4) Conclusions
The results for globally integrated BBL dissipation and sampling uncertainty (black line and gray bands) differ strongly between the three analyses, particularly in their absolute value. To compare these absolute values, we would ideally wish to obtain a single number for the globally integrated BBL dissipation, while remaining mindful of the significant range of possible such values. We do so by first excluding regions of each analysis where we believe the results to be unrepresentative and then taking a mean of the remaining values.
The bias correction factor is ≪1 for grid sizes greater than 80°; we therefore omit these values from consideration since they provide a poor estimate of globally integrated De. Interestingly, further analyses, omitted for brevity, show that values remain significantly weak-biased all the way up to a gridsize of 360° (one box covering the whole globe). This is in contrast to Sen et al. (2008), who estimated a strong bias using satellite data in lieu of our bottom current estimates from HYCOM. Conversely, while the bias correction factor at small grid sizes is near unity, the coverage here is too low to consider our result representative of the World Ocean, and consequently we remove from consideration grid sizes for which the coverage is less than ~80%: this leads us to omit results for gridsizes less than 40°. We thus consider the region on each figure between 80° and 40° degrees. This leads us to obtain results for each depth range, as detailed in Table 2. This table shows both the raw results, as shown in Fig. 7, and bias- and area-adjusted values of De, together with uncertainty bounds; we here discus the bias- and area-adjusted values.
De estimates from our analysis. Values presented are means across the range of grid sizes/cluster numbers selected as discussed in the relevant section. Bounds are presented in the order (lower 95%)–(lower 68%)–(mean)–(upper 68%)–(upper 95%).
Table 2 shows a substantive increase in the globally integrated BBL dissipation with increasing analyzed depth range. For the bottom 10% and bottom 20% analyses, these values are fairly similar—0.74 TW and 0.77 TW respectively; for the bottom 30% analysis, the value is slightly higher at 0.83 TW. This agrees with the results presented above in section 3b.
The width of the uncertainty bounds, as a percentage of the mean value, reduces with increasing depth range. This is consistent with the increased number of measurements being used; as the number of measurements per grid box increases, we would expect our value to become closer to the mean for that box. Bias does not show a similar trend; this implies that the additional measurements we add are similarly biased in their distribution to the original ones.
Based upon both Fig. 7 and Table 2, we choose to take the values for depths in the bottom 20% of the ocean as our final result: this provides a compromise between proximity to the BBL and coverage. Accordingly, from our grid analyses, we estimate a globally integrated BBL dissipation, with 68% confidence, in the range 0.60 TW – 0.94 TW, with a central estimated value of 0.77 TW.
6. Hierarchical clustering
a. Method
We next consider a method based upon hierarchical clustering of available current meters, with regions of the globe assigned to the closest cluster based on HYCOM estimates.
Initially, a hierarchical clustering algorithm (Hastie et al. 2009) is applied to the current meter database. The algorithm first computes a weighted distance between each meter pair; this weighted distance is a function of both the geographical separation and the difference in mean BBL dissipation between each meter pair, weighted such that 1 km of physical separation is equivalent to a difference in BBL dissipation of 2.5 × 10−5 W m−2. This weighting is chosen such that the antipodal distance, 20 × 103 km, is equivalent to one tenth of the range covered by the vast majority of the distribution of De, from 0 to 0.05 W m−2; that is to say, the maximum geographic separation is broadly equivalent in weight to one-tenth of the maximum separation in dissipation values. This comparative weighting is chosen since we wish to weight the model-derived De values weakly relative to the precisely known distances.
Next, a hierarchical “tree” of the separation between each individual meter is generated, where the leaves of the tree are the individual current meters, and are linked by branches to other meters close to them in weighted-distance. The linkage used for this tree is Ward’s linkage: this is a method in which the distance between each cluster is computed by minimizing the error sum of squares at each analysis step.
These branches themselves link hierarchically upward until a tree is formed between all the current meters. This tree is then subdivided into individual clusters of meters by continually removing the longest branch until a chosen number of separate clusters has been reached. This number of clusters (cluster number) is varied across the range 1–300. The geographic mean, or cluster center, of each cluster is then calculated. Ideally, these clusters should be close geographically and exhibit similar levels of BBL dissipation, but at small cluster numbers this may not be true; accordingly, we expect larger numbers of clusters to give more accurate results.
We then assign each point on the HYCOM grid to the nearest cluster center using the same weighted-distance-BBL dissipation metric. In principle, this method could be used to assign all points on the World Ocean to the nearest cluster center, but we impose a maximum weighted distance criterion of the equivalent in weighted space of 2 × 103 km at constant De to prevent regions being assigned in places too remote or too different in BBL dissipation from any cluster. This is highly conservative at low cluster numbers, as large parts of the globe will remain unfilled.
Finally, we ensure that at least one current meter remains in each region generated. Owing to the regions being generated from the geographic center of the cluster, it is possible, although rare, for regions to be generated which do not actually contain any of the meters used to generate them. This can occur, for example, when a region has a steep peak in BBL dissipation, leading to a cluster being generated from the low-dissipation meters on either side and then a region generated by filling from the geographic mean point of these meters which lies in the peak region. Accordingly, we check how many meters fall in each region, and remove any empty regions, refilling the space they took up with the next-nearest meter-containing region, provided this does not exceed our previously specific maximum distance criterion. Figure 8 shows an example set of regions generated using this method.
Examples of regions generated using the clustering method. Each color represents the area associated with an individual cluster of current meters; 40 clusters have been used. Note the greater cluster density in regions of high coverage such as the western boundary of the North Atlantic, and the disconnected areas of the red region in the South Atlantic, where the dissipation weighting element of the weighting dominates over distance.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Examples of regions generated using the clustering method. Each color represents the area associated with an individual cluster of current meters; 40 clusters have been used. Note the greater cluster density in regions of high coverage such as the western boundary of the North Atlantic, and the disconnected areas of the red region in the South Atlantic, where the dissipation weighting element of the weighting dominates over distance.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Examples of regions generated using the clustering method. Each color represents the area associated with an individual cluster of current meters; 40 clusters have been used. Note the greater cluster density in regions of high coverage such as the western boundary of the North Atlantic, and the disconnected areas of the red region in the South Atlantic, where the dissipation weighting element of the weighting dominates over distance.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
An advantage for regions determined via this method is that each region is guaranteed to contain at least one current meter, since they are defined outwards from the clusters. Additionally, we expect the meters in any given cluster to be exhibit similar results. However, the regions themselves, as opposed to the meters within them, are inherently reliant on the HYCOM BBL estimates, imposing a model dependence on our results which does not exist in the case of the grid-based analysis above.
b. Results
1) Coverage
Coverage in all three cases starts low, as we would expect, and then increases with increasing cluster number, with a few exceptions. Values stabilize in the high 80s percentile in all three cases for cluster numbers >~170.
2) Bias correction factor
Bias is also typically consistent across the analyzed range, to a much greater degree than with the grid-based analyses. Values are initially high at very small cluster numbers, but settles out fairly quickly to a value of around 0.75–0.8 in all three cases. It should be noted that, even though the bias is more variable in the lower two panels, in all three cases the full range of bias values once we exclude the very low cluster numbers is less than the difference we often observe in the grid based analyses between consecutive grid sizes.
3) Globally integrated BBL dissipation and sampling uncertainty
We next observe that, as with bias and coverage, our globally integrated BBL dissipation results are generally stable, especially when compared to our results for grid analyses. Whereas small changes in grid size led to changes of as much as 20% in the final value of globally integrated BBL dissipation for all three depth ranges, this is not the case for the smaller two depth ranges when analyzed with the clustering method. Sampling uncertainty also varies little across the range of cluster numbers considered.
4) Conclusions
As with the grid analyses, we wish to reduce our analysis down to a single value. We again select a range from each graph to average over; in this case, results remain fairly consistent for all variables in the first two panels above a cluster number of ~200 in all three cases, and accordingly we average over the range 200–300 clusters. The resulting values are described in Table 2 for our three depth ranges.
In all three cases, the values obtained are lower than for the equivalent grid-based analysis. We again consider the bottom 20% as our final result. For the clustering method, therefore, we estimate a globally integrated BBL dissipation, with 68% confidence, in the range 0.44–0.63 TW, with a central estimated value of 0.54 TW.
7. Conclusions
While our two methods show some difference in their final result at 68% confidence—0.8 ± 0.2 TW for the grid-based method and 0.55 ± 0.1 TW for the cluster-based method—given the significant differences in how they were analyzed, the results are statistically consistent with each other, since the confidence intervals overlap substantially. These error bars are broad, but take account of consequently more sources of possible uncertainty than previous studies, and accordingly form a much more conservative estimate of our knowledge. The dominant uncertain variable remaining in the calculation is the value of cd; however, the value (1.0 × 10−3) we have taken for this is conservative compared to many in the literature, and accordingly we conclude that at least half of the 0.90 ± TW wind input to the general circulation is dissipated via quadratic bottom boundary layer drag.
Of the two sets of results, we believe that the cluster analysis results are the more accurate—the analysis method takes better account of known ocean dynamics, and the results, as shown in Fig. 9, are much less sensitive to changes in the initial conditions of the analysis. Nevertheless, the grid-based method is also a sound method of computing this parameter and has the possible advantage of the regions being determined independently of HYCOM, and accordingly our conclusions should take account of this. Accordingly, we conclude that the globally integrated BBL dissipation is 0.65 ±0.15 (stat.) ± 0.15 (meth.) TW, where stat. refers to the statistical uncertainty and meth. to the additional methodological uncertainty. This is significantly larger than has been previously assumed (e.g., Wunsch and Ferrari 2004; Müller et al. 2005; Ferrari and Wunsch 2009), and consequently suggests that this process is much more important to the global oceanic energy budget than had previously been assumed.
Results for the cluster-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) the bottom 30%.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Results for the cluster-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) the bottom 30%.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
Results for the cluster-based analyses. (top) Results for the bottom 10% of the ocean, (middle) the bottom 20%, and (bottom) the bottom 30%.
Citation: Journal of Physical Oceanography 43, 2; 10.1175/JPO-D-12-082.1
It would be tempting to simply take the result of this analysis and, combined with the lee wave estimates of Scott et al. (2011), suggest that the global oceanic energy budget could largely be closed by these results. However, as discussed in the introduction above, the values of cd in the literature have been obtained empirically, and accordingly should include the contribution due to lee waves. The relative contribution of the involved processes has important ramifications: if the dissipation is primarily in the BBL itself, this will drive localized mixing at the bottom of the ocean, whereas if it is due to lee wave generation, this will allow energy and momentum to propagate vertically, driving mixing much higher in the column (Ferrari and Wunsch 2009; Nikurashin and Ferrari 2009; Garrett 2003). Accordingly, future work will focus on elucidating the relative contribution of wave-generation processes, and hence determining what proportion of the estimated power input remains available to drive other oceanic processes.
Acknowledgments
CJW was funded by Stratégie d’Attractivité Durable from the Région Bretagne awarded to RBS. RBS acknowledges funding provided by the CNRS and by NSF Grants OCE-0851457 and OCE-0960834, NASA Grant NNX10AE93G, a contract with the National Oceanography Centre, Southampton, United Kingdom. DF was funded by a Marie Curie Career Integration Grant awarded to RBS.
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