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  • View in gallery

    The barotropic component of global mean surface geostrophic currents with 1/4° spatial resolution. The color map represents the magnitude of (cm s−1). Red circles show the positions of qualified in situ velocity measurements (n = 632).

  • View in gallery

    Scatter diagram of log10(1/R) vs with least squares fitting results (solid lines). Results from using (a) total surface geostrophic currents and (b) the barotropic component for . The red lines denote fitting in separate intervals, while green lines represent the unitary fitting.

  • View in gallery

    (a) The probability distribution of seafloor roughness. According to the interval of magnitude of roughness (0–180 m2) in the global ocean, the probability distribution is presented in 18 segments. The red line denotes the mean value of probability. (b) The cdξ relationship derived from the inverse method to guarantee the uniform probability distribution of cd based on the segments shown in (a). The red dashed line denotes the median value of the range of cd.

  • View in gallery

    The maps of bottom drag (MW m−2) using 632 in situ measurements derived from (a) the pattern of surface geostrophic velocity () and constant cd (cd = 0.0025), (b) the pattern of barotropic component of surface geostrophic velocity () and constant cd (cd = 0.0025), and (c) the pattern of barotropic component of surface geostrophic velocity () and varying cd [cd = f(ξ)].

  • View in gallery

    (a) The residual of dissipation rates between estimates of bottom drag (MW m−2) from altimetry geostrophic velocity and its barotropic component . The results are computed as the dissipation from and constant cd minus dissipation from and constant cd. (b) Zonal integral of the bottom drag (TW) from the estimates according to (blue) and (red). (c) The difference between maps of the dissipation anomalies (MW m−2) with and without roughness parameter. The results are computed as the dissipation from and varying cd minus dissipation from and constant cd.

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Update on the Global Energy Dissipation Rate of Deep-Ocean Low-Frequency Flows by Bottom Boundary Layer

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  • 1 Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, and Function Laboratory for Ocean and Climate Dynamics, Qingdao National Laboratory for Marine Science and Technology, Qingdao, and University of Chinese Academy of Sciences, Beijing, China
  • 2 Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, and Function Laboratory for Ocean and Climate Dynamics, Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
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Abstract

The global dissipation caused by bottom boundary layer drag is one of the major pathways for the consumption of kinetic energy in the deep ocean. However, the spatial distribution and global integral of the drag dissipation are still debatable. This paper presents an updated estimate of the dissipation rate, using the barotropic component of surface geostrophic currents and 632 in situ velocity measurements. Also, the seafloor roughness is proposed as a parameter of drag efficiency in the parameterized method. The results provide a map of the drag dissipation rate with a global integral of ~0.26 TW. Approximately 66% of this dissipation occurs in the Southern Ocean, which is consistent with the proportion of wind power input into this region. Building upon the work in previous studies on the bottom boundary layer drag, more long-period observations are used, eliminating the influence of the baroclinic contribution to the surface geostrophic currents in the construction of the bottom velocity, and taking topographic roughness into account. The estimates have implications for the maintenance of density structure in the deep ocean and understanding of the kinetic energy budget.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yongsheng Xu, yongsheng.xu@qdio.ac.cn

Abstract

The global dissipation caused by bottom boundary layer drag is one of the major pathways for the consumption of kinetic energy in the deep ocean. However, the spatial distribution and global integral of the drag dissipation are still debatable. This paper presents an updated estimate of the dissipation rate, using the barotropic component of surface geostrophic currents and 632 in situ velocity measurements. Also, the seafloor roughness is proposed as a parameter of drag efficiency in the parameterized method. The results provide a map of the drag dissipation rate with a global integral of ~0.26 TW. Approximately 66% of this dissipation occurs in the Southern Ocean, which is consistent with the proportion of wind power input into this region. Building upon the work in previous studies on the bottom boundary layer drag, more long-period observations are used, eliminating the influence of the baroclinic contribution to the surface geostrophic currents in the construction of the bottom velocity, and taking topographic roughness into account. The estimates have implications for the maintenance of density structure in the deep ocean and understanding of the kinetic energy budget.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yongsheng Xu, yongsheng.xu@qdio.ac.cn

1. Introduction

The wind power input into the ocean surface geostrophic currents is constrained at ~1 TW by both experimental and model investigations (Wunsch 1998; Scott and Xu 2009; von Storch et al. 2007). However, the mechanism of dissipation, which is of great importance to understand the global energy budget, is still a puzzle because of the complexity of physical processes occurring from the large-scale to the Kolmogorov scale. Potential processes that dissipate a significant amount of low-frequency mechanical energy include ageostrophic instabilities in the ocean interior (Müller et al. 2005), nonlinear coupling to internal gravity waves (Bühler and McIntyre 2005), energy scattering into high-wavenumber vertical modes (Zhai et al. 2010; Saenko et al. 2012), lee-wave generation over rough topography (Nikurashin and Ferrari 2010b; Scott et al. 2011; Wright et al. 2014; Nikurashin et al. 2014), and quadratic bottom boundary layer drag (Arbic and Flierl 2004; Sen et al. 2008; Wright et al. 2013).

Note that ~80% of the wind power input into global ocean circulation occurs in the Southern Ocean (Scott and Xu 2009) and feeds gravitational potential energy in the subsurface layers. Most of the available potential energy is converted into the geostrophic eddy field via baroclinic instabilities (Killworth and Blundell 2007; Smith 2007). Moreover, a simulation of the Southern Ocean suggests that the bulk of the energy is dissipated within the bottom 100 m (Nikurashin et al. 2013). In the deep ocean, the mechanical energy stored in the geostrophic eddy field dissipates either in the ocean interior or through interaction with the bottom boundary by way of internal gravity wave generation, topographic blocking, and bottom boundary layer drag (Nikurashin et al. 2013; Trossman et al. 2013, 2015, 2016). The bottom-sourced internal wave energy flux helps determine the density structure and sustains the diapycnal mixing, which contributes to maintaining the stratification and the associated meridional overturning circulation (MOC; Garrett and St. Laurent 2002; Nikurashin and Vallis 2011; McDougall and Ferrari 2017; Mashayek et al. 2017).

Here, we focus on the energy dissipation rate of low-frequency deep-reaching currents by the bottom boundary layer (i.e., quadratic bottom boundary layer drag; hereinafter, “bottom drag”). Many researchers (e.g., Nikurashin et al. 2013; Trossman et al. 2013, 2016) tend to parameterize the dissipation caused by bottom drag and other dissipative processes (e.g., lee-wave generation and viscosity) using the same geostrophic flow field and add the contributions, though Wright et al. (2013) argued that it was difficult to separate out the contributions because of different processes when estimating the energy consumption due to bottom drag. However, the bottom drag dissipation will stir the bottom few tens of meters of a presumably well-mixed fluid layer, while the lee-wave generation gives rise to the possibility of propagating energy up to higher altitudes, where it can potentially break to mix more stratified fluid at high levels, well above the bottom boundary layer. An estimate of bottom drag dissipation is still important because it reveals that the bottom boundary layer gives rise to the consumption of kinetic energy in geostrophic flow, and the dissipation accounts for about one-fifth or more of the wind power input (Wunsch and Ferrari 2004; Sen et al. 2008; Arbic et al. 2009; Wright et al. 2013). Besides, Wright et al. (2013) used the result of Scott et al. (2011) to determine that lee-wave generation could account for the overwhelming majority of the remainder of the wind power input into the geostrophic circulation. Uncertainties in the magnitude and distribution of bottom drag still remain, although previous studies have attempted to determine the dissipation rate of the geostrophic kinetic energy above the bottom boundary layer. Müller et al. (2005) considered this bottom dissipation as negligible. Sen et al. (2008) calculated a range of ~0.22–0.83 TW, which was close to ~0.14–0.65 TW reported by Arbic et al. (2009) who incorporated the evaluation from model results. Wright et al. (2013) estimated ~0.65 TW using a hierarchical clustering technique with combination of observations and models, while Wunsch and Ferrari (2004) reported a global integral of ~0.2 TW. This broad range of values mainly originates from the statistical and methodology bias arising from extreme data limitations (Wright et al. 2013). Here, we attempt to make a more accurate estimate of the dissipation rate due to bottom drag by using more long-period in situ bottom current measurement than ever before; extracting the barotropic component of satellite-derived surface geostrophic currents to construct the bottom velocity field; and applying a spatially varying roughness parameter in the calculation of bottom drag.

2. Data and methods

In this paper, we employ more long-period in situ velocity data than previous studies to present an update to the estimates of the global distribution and globally integrated bottom boundary layer dissipation rate. There is evidence that there are similarities between the spatial patterns in the surface kinetic energy from altimetry-derived geostrophic currents and patterns from the moored current meter record-derived subsurface kinetic energy (Wunsch 1997). This enables us to compute the bottom velocity field based on scaled surface flows to cover the gaps in the sparse and nonuniform spatial sampling of current meter data. The underlying assumption of our analysis is that the barotropic component of the flow field approximates the bottom flow better than the surface geostrophic flow (Wunsch 1997; Edwards and Seim 2008). This is because the deep ocean’s low-frequency flows are far below the pycnocline and are affected by the mass-loading change in the water column rather than the steric change (Donohue et al. 2010). With the assistance of climatological temperature/salinity data, we calculate the baroclinic component that is associated with observed vertical shear (Edwards and Seim 2008) and then obtain the barotropic component by subtracting this component from the total surface geostrophic velocities. We construct the bottom velocities based on maps of both surface geostrophic currents and the barotropic component of the flow field, using the ratio between in situ bottom currents and surface currents to scale the surface current maps. We also use the seafloor roughness as an index of interface property to describe the energy conversion efficiency in the bottom boundary layer. With the improvements mentioned above and by making use of the parameterization of bottom drag, we find both the spatial pattern and the global integral of the dissipation rate due to bottom drag.

a. In situ bottom velocity

The in situ velocity data are collected from the Global Multiarchive Current Meter Database (GMACMD) (Scott et al. 2010, 2011), a cluster of current meters and ADCPs from available projects in past decades. Before constructing the global bottom velocity field, we select the velocity time series satisfying the following criteria: 1) The seafloor depth of the moored site should be larger than a threshold value H. Considering the sensitivity of the dissipation rate to the threshold and the effects of shallow water (Sen et al. 2008; Wright et al. 2013), we select 3000 m as the threshold depth (H = 3000 m) in this study. 2) The depth of the device falls within the range of 10 m above the seabed to the bottom 10% of the full depth. The lower limit corresponds to the thickness of the turbulent Ekman layer (Armi and Millard 1976) with typical values for strong currents (Wright et al. 2013), while the upper limit is set to exclude the measurements involving other physical processes beyond the position where the bottom drag occurs. 3) If there are several records within the aforementioned depth range at the same site, only the deepest time series is selected. We believe this is the best available representation of the bottom velocity field because the bottom drag represents the dissipation of the kinetic energy just above the bottom boundary layer. 4) The minimal length of the time series that we require in the current meter records to use in our analysis is 180 days. This is a compromise between temporal coverage and the number of available records based on the database. 5) We do the ellipse fitting based on velocity vectors for each current meter record. To make sure that the bias caused by the effect of topography-induced strong currents is excluded (e.g., channeling flows), we eliminate those records in which the ratio of the major axis and minor axis is larger than 5 (representing “limited in direction”), and of which the magnitude of velocity is more than 20 times higher than the average computed from all the records. After applying the above criteria, the number of observations that we used (n = 632) is much larger than the number of observations Sen et al. (2008) used (n = 266), although the measurements are still sparse and unevenly distributed throughout the global ocean. Since we use the mean values of both the surface geostrophic and the integrated baroclinic velocities in parameterizations of the bottom drag dissipation over years, the in situ measurements should also be averaged within a long period (at least one year) to capture the seasonal cycle. However, the sites of observations become too sparse to estimate the global dissipation if we only use the records longer than 365 days. Besides, uncertainty may arise if we loosen the cutoff criterion of duration to a month because signals with short periods cannot be canceled and seriously corrupt the construction of the bottom velocity field. Figure 1 illustrates the sites of 632 records (red circles) that are mainly concentrated in particular regions such as the Kuroshio Extension and the North Atlantic basin. Hence, the incorporation of surface geostrophic currents is indispensable when scaling bottom velocities from in situ observations to the global scale. By plotting a similar figure (not shown) to the one Wright et al. (2013, see their Fig. 2) showed, we observe no obvious effect of the arrest of the bottom boundary layer in our in situ data.

Fig. 1.
Fig. 1.

The barotropic component of global mean surface geostrophic currents with 1/4° spatial resolution. The color map represents the magnitude of (cm s−1). Red circles show the positions of qualified in situ velocity measurements (n = 632).

Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-16-0287.1

b. Deriving global bottom velocities from sea surface height, stratification, and a barotropic correction

We apply daily mean absolute dynamic topography (MADT) data (from 2003 to 2014) from the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO, http://aviso.altimetry.fr/) with horizontal resolution in 0.25 Mercator degrees to calculate the surface geostrophic currents in the zonal and meridional direction:
e1
Equation (1), derived from hydrostatic equilibrium (Cushman-Roisin 1994), displays the zonal and meridional geostrophic velocities except in the equatorial regions (within ±5° latitude θ, because f → 0 when θ → 0°), where g = 9.8 m s−1 is gravitational acceleration, f = 2ω sin(θ) is the Coriolis parameter, ω = 7.292 × 10−5 rad s−1 is the rotational angular velocity of the earth, and ζ is the MADT field. The dissipation rate due to bottom drag in the equatorial regions accounts for only a tiny portion of the globally integrated dissipation rate (Sen et al. 2008; Wright et al. 2013), so its omission does not severely bias our estimates. The following calculations exclude the equatorial regions.
Note that the satellite altimeters monitor the sea surface heights arising from the combination of the mass-loading (barotropic) and the steric (baroclinic) contributions. The satellite surface geostrophic velocity is the sum of barotropic and baroclinic components, . With respect to flows in the deep ocean, the isopycnic surfaces tend to be parallel with the isobaric surfaces, which leaves very little steric change (Donohue et al. 2010). It is plausible that the bottom velocities have more coherence with the barotropic component of surface geostrophic currents than with the total surface geostrophic currents used in previous studies. The barotropic component is obtained by subtracting from . Therefore, we utilize climatological temperature and salinity data from the World Ocean Atlas 2013 (WOA13, Locarnini et al. 2013; Zweng et al. 2013; http://www.nodc.noaa.gov/OC5/indprod.html), with a 0.25° spatial resolution and 102 levels in the vertical direction down to a depth of 5500 m, to evaluate the baroclinic velocities induced by the steric change. The annually averaged data from 2005 to 2012 are used to calculate the three-dimensional density profiles of the oceanic interior, preventing high-frequency variations. According to the conservation of potential density and vorticity, the geostrophy, and the Boussinesq approximation, the relative velocities between the adjacent depth levels can be calculated from thermal wind (Cushman-Roisin 1994):
e2
where ρ represents the water density and ρ0 = 1.03 × 103 kg m−3. Equation (2) (Chu et al. 1998) yields the baroclinic velocities relative to the reference plane. We select 3000 m below the surface as the reference layer that is consistent with the threshold depth for in situ velocity records. The WOA13 data have 77 layers between the sea surface and a depth of 3000 m. Summing the velocities derived from Eq. (2) in the water column for each grid point, we obtain the global surface baroclinic currents relative to the reference plane (3000 m). Though the baroclinic velocity is nonzero at 3000-m depth, these results can also be approximately regarded as the baroclinic component of the surface geostrophic currents because they account for the vast majority of the steric change [Figs. 2b and 2c in Donohue et al. (2010)]. Thereupon, we have the map of barotropic geostrophic currents , Fig. 1 by removing this baroclinic velocity component from satellite altimetry results .

AVISO daily MADT data are multisatellite-merged products. Different repeat periods of these satellites may yield a short time interval between two observations, implying that the merged MADT data may contain high-frequency signals. In addition, Sen et al. (2008) reported a 20% increasing in the estimate of the dissipation due to bottom drag with the current meter records including high-frequency variation. Since we focus on the low-frequency geostrophic flows, of which the typical time scale is longer than several days, both the in situ velocity time series and surface geostrophic velocity data have a 72-h low-pass filter applied to them (Sen et al. 2008).

Here, we set up the ratio of the mean value of cubic surface velocity to the mean value of cubic bottom velocity , , as a function [Eq. (3)] of and latitude θ, indicating the relationship of deformation (Sen et al. 2008), where the overbar represents the mean value over the time period of each particular data point, and ai denotes the coefficient for the fitting result:
e3
Cushman-Roisin (1994) showed the Coriolis parameter is involved in the expression of the barotropic currents. Meanwhile, Carl Wunsch (1997) argued that the vertical partition of kinetic energy may depend on latitude. The latitude term (a2θ2) in Eq. (3) is proposed as a general correction, which is reasonable to be taken into account when describing the relationship between surface and bottom velocities. Two types of velocity fields are used as the surface velocity to establish the functional relationship [Eq. (3)]. One is the surface geostrophic flow derived from satellite altimetry, while the other is the barotropic component of . The constructed bottom velocity field can be derived from the ratio of to or to . The case that construction of using the barotropic component (hereinafter, “barotropic correction”) yields the velocity caused by the mass-loading change in the water column.

c. Bottom drag

The dissipation rate due to bottom drag is estimated following the parameterized method introduced by Taylor (1920),
e4
where D′(t) is the dissipation rate (W m−2) representing the energy consumption in identity time and area, Vb(t) is the time-dependent bottom velocity, ρ = 1035 kg m−3 is the average marine water density, and cd is the constant coefficient weighting the drag efficiency. The value of cd varies from 0.50 × 10−3 to 8.00 × 10−3 (Shearman and Lentz 2003; Perlin et al. 2005; Brink and Lentz 2010; Döös et al. 2004) and has been used as a constant in parameterizations by earlier studies. One’s choice of the value of cd can change the estimates of the dissipation rate by several times. Wright et al. (2013) claimed the value of cd was one of major uncertain factors in this parameterized method. In the scope of classical mechanics, the friction dissipation occurring in the interface is in connection with the roughness of interfaces. Here, by analogy, we treat the bottom boundary layer as an interface and consider the effect of the topographic roughness. Although it is well accepted that the complexity of physical processes in the bottom boundary extends far beyond friction, the overall effect that the bottom boundary layer acting on the kinetic energy of the deep-reaching geostrophic flows turns out to be dissipation. Note that the key mechanisms, including topographic blocking, lee-wave generation, and friction, in the bottom boundary layer are related to topographic characteristics (Scott et al. 2011; Nikurashin et al. 2013, 2014). In this study, we establish a variable cd = f(ξ) to assess the effect of seafloor roughness ξ on maps of bottom drag (details in section 3). The dissipation due to the bottom drag occurs at scales of millimeters which are much smaller than the scales of the available topographic data in characterizing the roughness (at dissipation scales) relevant to the bottom drag. We assume the following: 1) the roughness at dissipation scale is uniform; 2) for a given region, the large-scale topographic fluctuation increases the contact area between flows and the seabed; and 3) the frictional dissipation at a given region increases with the contact area. Then, the ETOPO1 seafloor topography data, with 1-min gridded resolution, from NOAA (http://ngdc.noaa.gov/mgg/global/global.html; Amante and Eakins 2009) is applied to compute the roughness of the topography on the surface velocity database’s grid (1/4° grid). The topographic roughness is defined as the variance of the topographic height in a 1/4° square.

3. Results

Because the locations of the in situ bottom velocity measurements are limited in their geographical coverage (e.g., sampling is particularly sparse in the midocean and Southern Ocean regions), the maps of surface geostrophic currents and the barotropic component help us estimate the bottom velocity field. First, we define Rg () using for the surface velocity , and examine the relationship between Rg and . This is an update to the estimate made by Sen et al. (2008) in terms of the number of in situ measurements, with our records being 2.4-fold (n = 632) higher than theirs (n = 266). Figure 2a is the scatter diagram of log10(1/Rg) versus , with two kinds of least squares fitting results (solid lines). However, two different trends can be seen from the scattered points with a demarcation line (dashed line) at = 0.03 distinctly. Here, we apply a partitioned fitting (red lines) to describe the local features of the relationship. Two kinds of simulations for links between Rg and behave quite differently in the scale of large surface velocities, where the residual decreases by 16.5% with the usage of partitioned fitting.

Fig. 2.
Fig. 2.

Scatter diagram of log10(1/R) vs with least squares fitting results (solid lines). Results from using (a) total surface geostrophic currents and (b) the barotropic component for . The red lines denote fitting in separate intervals, while green lines represent the unitary fitting.

Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-16-0287.1

It is possible that the mechanisms could vary with flow speed, for example, the blocking and splitting effects occur at a relatively low flow speed for a particular topographic height (Nikurashin et al. 2014). Nevertheless, the jump of local trends in the scatter relationship may demonstrate potential shortcomings of this method when using to describe the connection between R and surface velocity. We infer the baroclinic component of surface geostrophic velocity, which is included in the model when is used, as one of the most important factors to cause the inaccuracy of the construction of bottom velocity, because the baroclinicity changes with locations and has very limited influence on bottom flows in the deep ocean.

Then, we replace the surface geostrophic velocities with the barotropic contribution and obtain a new scatter relationship (). Figure 2b shows the linear relationship clearly rather than different local characteristics in this circumstance. Because the logarithms of velocities are normally distributed, the r2 statistic can be employed to assess the goodness of fitting results (Barlow 1989). The coefficient of the determination (r2 value) is only 30% when we use the surface geostrophic velocity to build the relationship; however, it increases to 60% in the fitting result with the barotropic velocity . This means that the barotropic component of surface geostrophic currents, , has a tighter relationship with in situ bottom currents than the total surface geostrophic velocity, , does. As the barotropic component is depth independent and the baroclinic component has little correlation with , we get a better correction of fitting showed in Fig. 2b than that in Fig. 2a by removing the irrelevant component (i.e., the baroclinic component). Once the ratio R () is settled, the bottom velocity field can be depicted on the basis of the pattern of surface velocity Vs. The maps of the bottom velocity computed from and are not shown here, since their spatial distributions are the same as the maps of bottom drag derived from Eq. (4) and constant cd (Figs. 4a and 4b).

With the help of the mapping relationship, we substitute the near-global-covered surface currents divided by ratio R in the dissipation function [Eq. (4)]. The dissipation rate due to bottom drag can be computed in the form of the parameterized expression [Eq. (5)], where represents integral over grids where the seabed depth is greater than H, A is the area corresponding to the position of the grid points, and is the time-mean rate of energy consumption caused by the effect of the bottom boundary layer in area A:
e5
Here, we establish a functional relationship between the drag efficiency cd and seafloor roughness ξ to evaluate the influence of seabed fluctuation on dissipative processes in the bottom boundary layer. This operation is motivated in part by the features in the energy-converting processes occurring at the air–sea interface where the dissipation efficiency is proportional to sea surface roughness (Large and Pond 1981). Although the seabed is a solid interface making the seafloor roughness unchanged by flow speed (ignoring processes on geological time scales), the spatial variations of seafloor roughness should be taken as a parameter of drag efficiency instead of selecting a fixed value of cd. Also, Wunsch (1997) mentioned that the topographic roughness may have influence on the vertical partition of kinetic energy, implying that the energy dissipation near the seafloor relies on the topographic roughness. The magnitude of topographic roughness ξ, computed as the root-mean-square (RMS) from ETOPO1 in each 1/4° grid (Li and Xu 2014) corresponding to the surface current data, varies from ~0 to 180 m2 in the global ocean. The dissipation caused by the interaction between the bottom flows and the bottom boundary layer is determined by the stress multiplying by the contact area. Compared to the smooth plane, the topographic fluctuations increase the area of the contact area. Therefore, we present the RMS as a proxy of the area in grid point to describe this variation used as a roughness parameter in construction of the bottom velocity field. The probability distribution of seafloor roughness (Fig. 3a) illustrates that the regions with smooth topography (ξ < 40) make up only a small portion of the global seabed. Because the drag efficiency cd is a multiplication term in Eq. (5), the magnitude of cd is proportional to the estimate of bottom drag (). Therefore, compared with using a median value of cd, the linear uniform correspondence in ascending order according to the numeric values between the intervals of cd and ξ would inherently increase the estimate of the global integral of bottom drag, since the percentage of rough topography (ξ > 90) is above average (red line, 0.055) in the probability distribution (Fig. 3a). Indeed, there is no well-accepted method to calculate the drag coefficient. Here, our goal is to show a possible variation of the dissipation rate with roughness instead of an exclusive estimate, since the bottom drag dissipation (at extreme small scales) still has a large uncertainty. We use an inverse method by deducing the cdξ relationship to form a uniform probability distribution of cd(ξ) over the entire global ocean to show the modulation due to topographic roughness without altering the estimate of the globally integrated dissipation rate apparently. The space-dependent cd(ξ), of which the probability distributes uniformly, help to estimate the dissipation implying the equity of each value in cd interval rather than the dominance of particular values (e.g., expectation of the normal distribution). We select cd = 0.0025, a popular value used in previous studies (Sen et al. 2008; Arbic et al. 2009; Wright et al. 2013; Trossman et al. 2013, 2016) as the median value of the interval used for correspondence. The range of cd is from 0.5 × 10−3 to 4.5 × 10−3, where the minimum is the lower limit of the range of values used in previous studies while the maximum is a symmetrical choice around the median value. We believe this interval of cd is reasonable since values used in most literature fall within this scope. The values of cd are picked uniformly within the range of cd according to the number of grids in the map of the constructed bottom velocity field . We sort the roughness of grids mentioned above from small to large in magnitude. Then, these two arrays match with each other in the ascending order. The cdξ relationship (Fig. 3b) illustrates a linear feature approximately when the topography is rough; however, it behaves differently if the roughness is less than 30 m2. According to this one-to-one correspondence, we obtain the global map of cd, in which large values represent a strong bottom dissipation efficiency in regions with rough topography. The space-dependent cd indicates the modulation of the topographic features, which is a key factor of the interface that has been ignored in previous studies of bottom drag. However, Nikurashin and Ferrari (2010a,b, 2011) noted that the topographic roughness is a determinant in the dissipation of the kinetic energy from the large-scale flows in terms of lee-wave generation. Since the dissipation due to bottom drag also represents the energy consumption of flow by the bottom boundary layer, we infer the roughness as a parameter with the intention to supply an addition to the mechanism of bottom drag.
Fig. 3.
Fig. 3.

(a) The probability distribution of seafloor roughness. According to the interval of magnitude of roughness (0–180 m2) in the global ocean, the probability distribution is presented in 18 segments. The red line denotes the mean value of probability. (b) The cdξ relationship derived from the inverse method to guarantee the uniform probability distribution of cd based on the segments shown in (a). The red dashed line denotes the median value of the range of cd.

Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-16-0287.1

According to Eq. (5), we choose both the satellite surface geostrophic current and its barotropic component as the mapping frame () to construct the bottom velocity field () and compute the spatial distribution and global integral of bottom drag (Fig. 4). Differences in these patterns (Fig. 4) originate from the introduction of barotropic correction and/or a roughness parameter. First, Fig. 4a delineates the pattern of bottom drag derived from and constant cd (cd = 0.0025), which has much similarity with the map of surface geostrophic currents with strong bottom drag occurring in the major flow systems. Second, Fig. 4b is the global map computed from and constant cd (cd = 0.0025), showing an extreme enhancement of bottom drag in the domain of the Antarctic Circumpolar Current (ACC), which is consistent with the strong bottom-reaching currents in this area (Nikurashin and Ferrari 2010a,b). Compared to the map shown in Fig. 4a, there is no remarkable energy consumption occurring at mid- and high latitudes in the Northern Hemisphere, because the western boundary currents and their extensions have only a small barotropic component. The global integral of bottom drag from is 0.24 TW, with a 20% increase over the estimate based on (0.20 TW). The differences between these two estimates are shown in Fig. 5a. According to the map of the residual, the usage of barotropic correction increases the value of the dissipation rate due to bottom drag in most of the global ocean, especially in the Southern Ocean. Finally, the estimate of bottom drag with both the barotropic correction and varying cd according to the cdξ relationship (Fig. 4c) shows a pattern similar to that shown in Fig. 4b. The usage of the cdξ relationship changes the map of the dissipation rate by spatial adjustment due to the topographic roughness (Fig. 5c). Compared with the order of bottom drag (~several MW m−2), the differences between Figs. 4b and 4c reach 1.4 MW m−2 on average, which implies the effect of topographic roughness on bottom drag. The global sum of the dissipation rate by quadratic bottom boundary layer drag derived from the combined effect of the barotropic correction and roughness parameter is 0.26 TW accounting for an important portion of the consumption of power input (~1 TW), which supports the notion that boundary mixing plays an important role in maintaining the abyssal stratification (Munk and Wunsch 1998). We believe the method in which the surface barotropic velocity field is taken as the mapping frame to construct bottom currents can yield a more accurate estimate of bottom drag because the baroclinic component of surface currents that has little association with the bottom velocities is excluded.

Fig. 4.
Fig. 4.

The maps of bottom drag (MW m−2) using 632 in situ measurements derived from (a) the pattern of surface geostrophic velocity () and constant cd (cd = 0.0025), (b) the pattern of barotropic component of surface geostrophic velocity () and constant cd (cd = 0.0025), and (c) the pattern of barotropic component of surface geostrophic velocity () and varying cd [cd = f(ξ)].

Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-16-0287.1

Fig. 5.
Fig. 5.

(a) The residual of dissipation rates between estimates of bottom drag (MW m−2) from altimetry geostrophic velocity and its barotropic component . The results are computed as the dissipation from and constant cd minus dissipation from and constant cd. (b) Zonal integral of the bottom drag (TW) from the estimates according to (blue) and (red). (c) The difference between maps of the dissipation anomalies (MW m−2) with and without roughness parameter. The results are computed as the dissipation from and varying cd minus dissipation from and constant cd.

Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-16-0287.1

4. Discussion

The number of in situ records has a significant impact on the estimate of bottom drag (Sen et al. 2008), and we calculate the global dissipation rates caused by bottom boundary layer drag with more long-period moored measurements. Since the arrangement of measurements focuses on particular scientific objectives such as strong currents and channel effects, the increasing observations cannot guarantee the coverage of the global ocean yet, especially in the midocean and the Southern Hemisphere. Previous studies (Sen et al. 2008; Wright et al. 2014) resolved this problem with the help of a map of surface geostrophic currents . However, according to the scatter relationship between R and Vs (Figs. 2a and 2b), inaccuracies may arise if the map of surface geostrophic velocity derived from satellite altimetry is used as the mapping frame to construct the bottom velocity field because of the existence of the baroclinic component of surface currents. The steric change caused by seawater density contributes to the dynamic height of the sea surface and subsequently the surface geostrophic velocities, which have little correlation with the tilting of the pressure surface in the deep ocean. Hence, we synthesize the climatology data and to obtain the barotropic component of the surface geostrophic currents by deducing the integral of the relative baroclinic velocity in the vertical direction from a 3000-m reference plane. The fitting results (Fig. 2b) show the barotropic correction can improve the correlation (r2 statistic) between R and Vs, implying a possible advance in the assessment of the map of bottom velocity. In addition, since the bottom boundary layer acts as a barrier on the flows above itself, the effect of seafloor roughness should be considered in the mechanism of bottom drag. The seafloor roughness is proposed as a parameter of drag efficiency, cd = f(ξ), to modify the mechanism of energy dissipation in the bottom boundary layer. We choose cd = f(ξ) so that cd varies within the range suggested by previous studies, and so that the global integral of drag dissipation is not changed.

The outcome suggests the global integral of bottom drag is ~0.26 TW (Fig. 4c), accounting for a quarter of wind power input. The remarkable boost of energy dissipation in the Southern Ocean (south of 40°S) is ~0.17 TW, as a result of the combined effects of the rough topography and the robust deep-reaching ACC (Nikurashin and Ferrari 2010a,b). This component of dissipation accounts for approximately 66% of the global integral, yet it is underestimated in previous studies (e.g., 18%; Sen et al. 2008). This proportion is comparable to the ratio, 70% (Wunsch 1998), of global wind power input in that region. It is logical that the quasi-frictional dissipation converts the mechanical energy of fluid into the internal energy locally instead of radiating at a distance. Our estimate of bottom drag is an addition to the understanding of the bottom energy budget, which maintains the structure of MOC (Marshall and Speer 2012). We also test the sensitivity of our estimates of bottom drag to the values of the threshold depth H. The global integrals of the dissipation due to bottom drag are 0.23, 0.26, 0.30, and 0.46 TW when the values of H are selected as 4000, 3000, 2000, and 1000 m, respectively. However, the proportions of dissipation occurring in the Southern Ocean fluctuate slightly between 63% and 68% and do not change as significantly as the global integrals.

We perform residual analysis between two estimates with and without barotropic correction (Fig. 5a) and the zonal integral of bottom drag (Fig. 5b) to check the differences induced by the barotropic correction. The zonal integral shows that the Southern Ocean plays an important role in bottom drag under both circumstances. In general, the barotropic correction further increases the dissipation rate in the Southern Ocean (Fig. 5b, blue line) with modulation of the spatial distribution (Fig. 5a). Although the extreme large values in the pattern without barotropic correction are suppressed to the south of Africa, lee of Kerguelen Plateau, south of New Zealand and Drake Passage, the regional sum of bottom drag, however, increases by 42% (from 0.12 to 0.17 TW), which corresponds to the relative enhancement of constructed bottom velocity derived from the barotropic component of surface geostrophic currents. The high energy dissipation level within the bottom boundary layer along the ACC area is consistent with the Southern Ocean’s ability to consume momentum related to the vertical structure of the ACC (Peña-Molino et al. 2014). In the Northern Hemisphere, the high dissipation rates around 40°N (Fig. 5b, red line) disappear in the estimate with barotropic correction, while the dissipation computed from the (Fig. 5b, blue line) exhibits elevations at low latitude (within ±30°). The decreases in the dissipation rate emerging in regions near ±40° latitude accompany warm currents, such as the North Pacific Current, North Atlantic Current, Agulhas Current, East Australian Current, and Brazil Current. Since the roughness parameter is set to illustrate the potential function of topography and does not change the magnitude of the global integral, we calculate the anomalies of bottom drag in estimates with varying and fixed drag efficiency [cd = f(ξ) and cd = 0.0025] to analyze the effects of roughness. The differences between the two maps of anomalies are shown in Fig. 5c. The topographic roughness can modify the bottom drag with respect to the topographic features in the scope of the world’s oceans. The most drastic changes can be seen in the Southern Ocean, with a variation of over ~2 MW m−2 in amplitude. Weakened dissipation accompanies smooth topography, such as that west of Kerguelen Plateau, south of Australia, and the southeast Pacific sector of the Southern Ocean. Meanwhile, the drag efficiency is high in other areas of ACC due to rough fluctuations of the seabed, producing a regional sum of the estimate of the dissipation rate at the same level as the estimate with fixed cd. In the Northern Pacific, the significant increases in dissipation rate due to the roughness parameter are located to the west of Mariana Islands and along the Aleutian Islands, while decreases exist in the open ocean. In the Atlantic and Indian Oceans, the bottom drag strengthens along the midocean ridge and weakens near the continent. The strong dissipation in regions with rough topography could help explain the bottom stratification and the enhanced mixing (Nikurashin and Vallis 2011; Mashayek et al. 2017). The spatially varying cd(ξ) may be important to represent in models to simulate the oceanic energy pathways (Nikurashin et al. 2013). The errors in our estimates may mainly originate from the following three aspects: 1) the residual of the baroclinic component in surface geostrophic velocities, 2) the bias in constructed bottom currents because of the limited quantity of bottom in situ measurements, and 3) the defective drag efficiency parameterization and the incomplete understanding of the drag mechanism.

5. Summary

We suggest a new approach to the near-global distribution of the bottom velocity and contribute to determining the map and amount of bottom drag, which is of great importance to delineate the balance of the deep-sea energy budget. We make improvements by using an increased number of long-period in situ measurements and introducing a barotropic correction and a seafloor roughness-dependent drag coefficient. It is noteworthy that the assessment of bottom drag may involve several dissipative processes including the generation of lee-wave energy (Wright et al. 2013). However, there is no direct evidence showing that the dissipation rate derived from the parameterized method (Sen et al. 2008; Wright et al. 2013) contains the whole power of lee-wave generation (Scott et al. 2011; Wright et al. 2014; Trossman et al. 2013, 2015, 2016), and these studies on the energy dissipation, including both bottom drag, topographic blocking, and lee-wave generation in the deep ocean, do not mention that these processes have overlap. Future projects are needed to distinguish the details of processes in the bottom boundary layer. In addition, the meaning of the estimate depends on the definition of drag efficiency cd, which is unclear. One can determine this energy conversion rate as friction or the overall effect of the bottom boundary layer artificially. In this paper, we tend to take cd(ξ) as the efficiency of topographic friction. Since the magnitudes of drag dissipation are sensitive to cd, in situ measurements to investigate the exact relationship cd = f(ξ) are required in the future. On one hand, although there is controversy over the present methods in the estimate of bottom drag, it is still important to investigate the spatial distribution of bottom drag. Our estimates clarify the dominant role of the Southern Ocean in dissipation of the eddy kinetic energy by the bottom boundary layer. On the other hand, there are also defects in the present estimates of lee-wave generation. Nikurashin and Ferrari (2011) showed a 20%–30% reduction in lee-wave energy radiation if taking into account the presence of a turbulent bottom boundary layer. Scott et al. (2011) suggested that using linear theory with corrections for finite-amplitude topography causes a decrease of approximately 10% in the magnitude of lee-wave energy generation. Combined with the dissipation due to lee-wave generation (0.75 ± 0.19 TW; Wright et al. 2014), our estimate could almost compensate the wind power input. We plan to estimate lee-wave dissipation using the bottom velocity obtained from the methods described herein, which is beyond the scope of this paper. Combining the locally consumed portion of the kinetic energy caused by bottom drag with other dissipative processes, such as lee-wave generation (Nikurashin and Ferrari 2011), future research can improve the understanding of the global oceanic energy budget (Ferrari and Wunsch 2010) and eddy diffusivity (Griesel et al. 2014).

Acknowledgments

We thank Dr. R. B. Scott very much for supplying GMACMD data and reviewing our manuscript. Thanks are given also to two anonymous reviewers and Prof. K. J. Heywood for their great suggestions to improve this paper. This work was supported by Aoshan S&T Innovation Project from Qingdao National Laboratory for Marine Science and Technology, the National Key Research and Development Program (Grants 2016YFC1401004 and 2016YFC1401008), the National Natural Science Foundation of China (Grants 41676168 and 41376028), the NSFC-Innovation research group of Sciences Fund (Grant 41421005), the NSFC-Shandong Joint Fund for Marine Science Research Centers (Grant U1406402), National Science Foundation for Young Scientists of China (Grant 41606200), and the Natural Science Foundation of Shandong Province, China (Grant ZR2014DQ027).

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