Geophysical Research Letters

A preliminary estimate of the Stokes dissipation of wave energy in the global ocean

Authors


Abstract

[1] The turbulent Reynolds stresses in the upper layers of the ocean interact with the vertical shear of the Stokes drift velocity of the wave field to extract energy from the surface waves. The resulting rate of dissipation of wind waves in the global ocean is about 2.5 TW on the average but can reach values as high as 3.7 TW, making it as important as the dissipation of wave energy in the surf zones around the ocean margins. More importantly, the effect of Stokes dissipation is felt throughout the mixed layer. It also contributes to Langmuir circulations. Unfortunately, this wave dissipation mechanism has hitherto been largely ignored. In this note, we present a preliminary estimate of the Stokes dissipation rate in the global oceans based on the results of the WAVEWATCH III model for the year 2007 to point out its potential importance. Seasonal and regional variations are also described.

1. Introduction

[2] Surface gravity waves are frequently cited as a shining example of the very first successful application of the laws of fluid mechanics to a practical problem. However, for simplicity, most of the work over the past two centuries on oceanic surface gravity waves has considered the ocean to be inviscid [e.g., Kantha and Clayson, 2000a]. The interaction of waves with the turbulent motions in the upper layers has been ignored until recently. A proper treatment of the wave-mean current-turbulence interactions has only been accomplished in the past few years [Ardhuin et al., 2008; Rascle et al., 2008]. The results show conclusively that very similar to the extraction of energy from mean currents by turbulence, turbulence can extract energy from the wave motions by the action of the Reynolds stresses on the vertical shear of the wave-induced Stokes drift.

[3] Wind generated surface gravity waves travel on top of a turbulent and not an inviscid ocean as is commonly assumed. This implies that there is an inevitable interaction between the turbulent motions in the upper layers of the ocean and these gravity waves. Overall, such an interaction leads to extraction of energy from waves by turbulence in the oceanic mixed layer. This interaction is particularly important for the dissipation of the low frequency part of the wave spectrum, swell [Kantha, 2006]. It acts as a source term for turbulent motions [Kantha and Clayson, 2004] and a sink term for waves [Kantha, 2006; Ardhuin and Jenkins, 2006]. This Stokes dissipation of wave energy is comparable to the dissipation of wave energy in the surf zones around the ocean basins. More importantly, unlike wave breaking (white capping), whose effects are confined to the top few meters, Stokes dissipation mechanism enhances upper layer turbulence with its effects felt potentially throughout the mixed layer. It also contributes to Langmuir circulation in the upper ocean. Unfortunately, the importance of this mechanism has not been fully appreciated and hence Stokes dissipation of waves has largely been ignored in wave modeling. In this note, we present a preliminary estimate of the Stokes dissipation in the global ocean based on the results of WAVEWATCH III model for 2007. Seasonal and regional variations are also described. Needless to say that the Stokes mechanism of energy transfer from waves to turbulence constitutes an additional (in addition to the momentum and buoyancy fluxes at the air-sea interface) and important source of turbulent kinetic energy, and consequently, it enhances the intensity of turbulence in the oceanic mixed layer. One consequence of this is enhanced mixing and more uniform profiles in the mixed layer [McWilliams et al., 1997; Carniel et al., 2005].

2. Extraction of Wave Energy by Turbulence in the Mixed Layer

[4] Based on LES simulations of Langmuir cells in the ocean [McWilliams et al., 1997], Kantha and Clayson [2004] have parameterized the extraction of energy from surface gravity waves by turbulence in the oceanic mixed layer. They showed that the rate of change of turbulence kinetic energy (TKE) per unit mass can be written as [see also Kantha, 2006]:

equation image

where q2/2 is the TKE, uS(z) and vS(z) are the components of surface gravity wave-induced Stokes drift velocity, and −ρequation image and −ρequation image are components of the turbulent shear (Reynolds) stress; ρ is water density and z is the vertical coordinate positive upwards. It is the working of the Reynolds stress on the vertical shear of the Stokes drift that extracts energy from the wave motion and transfers it to turbulence. It is rather analogous to working of the Reynolds stress against the mean shear in converting kinetic energy of the mean currents into TKE. The integration of equation (1) with z gives the rate of increase of the total TKE in the water column due to extraction of energy from wave motions. Because of the involvement of Stokes drift in this process, we can characterize this mechanism as Stokes production of TKE. This mechanism leads also to the dissipation of waves with the rate of dissipation of wave energy E given by:

equation image

where equation image(z) is the shear stress vector and equation imageS(z) is the Stokes drift velocity vector. We call this the Stokes dissipation of wave energy. To evaluate equation image, it is necessary to determine equation image(z) and hence it is necessary to appeal to a turbulence closure model [e.g., Kantha and Clayson, 1994, 2004] of the upper layers [see Kantha, 2006].

[5] Consider a linear monochromatic deep water wave. The Stokes drift velocity can then be written as:

equation image

where k is the wave number, equation image is the unit vector in the direction of wave propagation, σ is the frequency, a is the wave amplitude, c is the phase speed and E is the wave energy; g is the gravitational acceleration. Therefore equation image = equation imageE exp (2kz) equation image and equation (2) becomes:

equation image

For a general wave spectrum, equation (4) has to be integrated over the wavenumber spectrum to compute the rate of decay of the total wave energy in the wave field, or equivalently, equation (2) can be used provided the Stokes drift velocity is the sum of the contributions from the entire wave number spectrum.

[6] Because of the exponential decay of Stokes drift velocity (and hence its shear) with depth, most of the interaction between the wave motions and turbulence takes place in the near surface layers. Since turbulence must be present for this interaction to take place, this also means that the interaction is confined to the depth of the active mixed layer. Therefore the Stokes depth, 1/2k, is an important parameter. Because of the turning of the shear stress vector with depth, the latitude is also an important factor. Equation (2) can be written as

equation image

where equation imagew is the wind stress vector and equation imageS(0) is the Stokes drift velocity vector at the surface. Written this way, the constant of proportionality α becomes a function of the various factors involved in the wave-turbulence interaction: the mixed layer depth, the wavelength, the wave direction and amplitude, as well as the turbulence field in the mixed layer. Simply put, we need to know the vertical profiles of the shear stress vector in the mixed layer and the Stokes drift velocity vector. The use of equation (5) enables a rough estimate of the wave dissipation rate due to wave-turbulence interactions to be made without resorting to equation (2) and hence a second moment closure-based global mixed layer model.

[7] Kantha [2006] has explored the variability in α for a monochromatic wave using a second moment closure-based mixed layer model [Kantha and Clayson, 2004], and has shown that for waves propagating roughly in the same direction as the wind, as is the case with wind-generated waves, its value is between 0.4 and 0.8, with higher values for higher frequency waves and lower latitudes. The average value of α is 0.33 for 15 sec waves, 0.58 for 10 sec waves and 0.87 for 5 sec waves. In this study we use a value of 0.65 appropriate to 8–10 s waves corresponding roughly to the peak of the wind wave spectrum. The choice recognizes the fact that while high frequency waves contribute more to the Stokes drift (as shown by Rascle et al. [2008, Figure 7]), the ratio of the corresponding Stokes depth to the mixed layer depth is smaller. The estimates thus obtained for the wave dissipation rate are accurate to within a factor of about two. If a higher value were to be deemed appropriate for α, the estimates can be simply scaled up. Our intention here is to provide a rough preliminary estimate of the Stokes dissipation rate to point out its importance. We postpone a more accurate estimate and necessarily a more extensive analysis to a later date.

3. WAVEWATCH III Model

[8] As can be expected, observational data on the ocean surface wave field is inadequate to carry out this study. Instead appeal must be made to a global wave model. This is the same approach used by Rascle et al. [2008] to generate a global wave parameter database. Using a version of the popular WAVEWATCH III (WW3) model, they show that on the average, the winds transfer energy into wave motions (which have a global energy content of about 1.52 EJ) at a rate of about 70 TW. Of this energy input, a majority of 67.6 TW goes into generating oceanic turbulence by breaking and other mechanisms, while 2.4 TW is dissipated in the surf zone. They did not however explore in detail the Stokes dissipation rate of the wave energy. Their value for Stokes dissipation of 6 TW, estimated as ∣equation imagew∣ ∣equation imageS(0)∣ is perhaps an overestimate, since it does not allow for the possibility of nonzero angles between the two vectors. In this note, we intend to complement their work by computing the Stokes dissipation rate for the year 2007. Along the way, we also describe its temporal and regional variability.

[9] To estimate the stokes drift over the global ocean, the WW3 Version 2.22 [Tolman, 1991, 2002] was run at 1/2° resolution (approximately 55 km at the equator) globally for the year 2007. This deep ocean wave model has been used at operational weather centers since 2000, and there has been numerous verification studies, using wave buoys and satellite observations, documenting its performance over the years [Rogers et al., 2005]. It is a 3rd generation wave model, solving the non-linear wave interactions using a Discrete Interaction Approximation (DIA). It uses the Tolman-Chalikov wind input formulation and dissipation terms that act separately on sea and swell. The waves are propagated using a 3rd order accurate finite difference method. The wave spectrum was discretized using 24 equally spaced directional bins and 25 logarithmically spaced wave number bins. The model was forced with 12-hourly surface winds from the Navy Operational Global Atmospheric Prediction System (NOGAPS), which have been archived on the Global Ocean Data Assimilation Experiment (GODAE) server. The time step was 3 hours. The Ice edge was updated every 12 hours. It is important to note that currently no wave model, including WW3 incorporates Stokes dissipation mechanism. Instead, the wave dissipation is parameterized by rather ad-hoc means with coefficients tuned to provide accurate estimates of the surface wave field for operational applications. For our purposes here, it does not matter how the wave dissipation is parameterized. What matters is that the resulting wave parameters be accurately reproduced by the model (For a detailed discussion of the skill of the WW3 model, see Rogers et al. [2005]).

[10] The wave directional spectrum was integrated to estimate the Stokes drift velocity at the surface needed to compute the Stokes dissipation rate using equation (5):

equation image

where f is the frequency (in Hz), E(f, θ) is the directional spectrum and θ is the propagation angle. Large and Pond [1981] formulation is used to convert the winds at 10 m to the surface wind stress. The wind stress and the Stokes drift velocity vectors at the surface are stored every 6 hours along with other wave parameters such as the significant wave height (SWH) and the peak period for further analysis. The energy in the wave motions is computed using the SWH.

4. Analysis and Interpretation

[11] The wave energy and the dissipation rate are summed up over the globe as well as the northern and southern hemispheres. In addition, values are computed for the longitudinal band 110° to 285° designated as the Pacific sector, 285° to 25° (Atlantic sector) and 25° to 110° (Indian sector). Figures 1a and 1b show (see also Table 1) the temporal variability of the energy in the global surface wave field and the Stokes dissipation rate of this wave energy. Black curves in Figures 1a and 1b denote the global values; the red curve in Figure 1a corresponds to the northern and the blue curve to the southern hemisphere; the blue curve in Figure 1b corresponds to the Pacific, the red curve to the Atlantic and the green curve to the Indian sectors. The average value of global wave energy of 1.68 EJ (Table 1) is consistent with the 3-year average of 1.52 EJ estimated by Rascle et al. [2008]. Compare this to the total energy in barotropic ocean tides of only 0.6 EJ. The average Stokes dissipation rate of 2.5 TW is more than the 2.4 TW rate of dissipation of the wave energy in the surf zones at the ocean margins [Rascle et al., 2008], but can reach values as high as 3.7 TW. Comparatively, the dissipation rate of tidal energy in the global oceans and hence of the gravitational energy of the Earth-Moon-Sun system, is only about 3.75 TW [Kantha and Clayson, 2000b].

Figure 1a.

Time series of the energy in the global surface gravity wave field (in EJ) and the rate of dissipation of that energy by Stokes dissipation (in TW). Global average (black), Northern Hemisphere (red), Southern Hemisphere (blue).

Figure 1b.

Time series of the energy in the global surface gravity wave field (in EJ) and the rate of dissipation of that energy by Stokes dissipation (in TW). Global average (black), Pacific (blue), Atlantic (red), Indian Ocean (green). Note the broad seasonal variability superimposed on the synoptic scale variability. The average global wave energy is consistent with the estimate of 1.52 EJ by Rascle et al. [2008].

Table 1. Wave Energy and Stokes Dissipation Rate Statistics
RegionStokes Dissipation Rate (TW) (Minimum/Average/Maximum)Wave Energy (EJ) (Minimum/Average/Maximum)
Global1.6/2.5/3.70.9/1.7/2.1
North Hemisphere0.2/0.7/2.50.2/0.5/1.0
South Hemisphere0.9/1.8/3.40.4/1.2/1.9
Pacific sector0.7/1.3/2.30.4/0.9/1.2
Atlantic sector0.2/0.7/1.40.3/0.4/0.7
Indian sector0.2/0.6/1.70.2/0.4/0.8

[12] The Stokes dissipation rate of 2.5 TW is likely to be a slight underestimate due to issues related to the resolution of the high frequency end of the wave spectrum in wave models (The use of 24 frequency bands implied maximum frequency of 0.4 Hz instead of 0.7 Hz, which might have reduced the surface Stokes drift value by a few tens of percent (F. Ardhuin, personal communication, 2008). Because of the complicated dependence of the constant α in equation (5) on various factors, this estimate should be regarded as preliminary; more accurate estimates await the use of equation (2).

[13] While the global energy and dissipation rates show very little seasonal variability, the hemispheric values show a prominent seasonal variability, with values in the northern hemisphere being higher during the boreal winter than the boreal summer but exactly opposite behavior is evident in the southern hemisphere.

[14] Majority of the contribution to Stokes dissipation (1.8 TW) is from the southern hemisphere, with the roaring fifties contributing significantly. During the boreal summer, the Stokes dissipation is mostly from the southern hemisphere. The average dissipation rate is 1.3 TW in the Pacific, 0.7 TW in the Atlantic and 0.6 TW in the Indian sectors. The Atlantic and Indian sectors contribute roughly equally to Stokes dissipation, with the Pacific sector contributing significantly more.

[15] Figure 2 shows the distribution of the wave energy (in TJ) and Stokes dissipation rate (in MW) in each equation image° × equation image° box over the global oceans averaged over 2007. High dissipation rate regions are well correlated with high wave energy regions; for example the roaring fifties in the southern hemisphere, and the Gulf Stream and Kuroshio extension regions as well as the Arabian Sea in the northern hemisphere. The average dissipation rates reach as high as 70 MW and average wave energies as high as 36 TJ (in equation image° × equation image° box) in the southern hemisphere. The southern latitudes, the Gulf Stream and Kuroshio extension regions contribute heavily to Stokes dissipation. In the North Indian Ocean, the summer monsoon-affected regions along the Arabian coast display high wave energy and dissipation rates.

Figure 2.

Spatial variability of the annual average of (top) the wave energy (in TJ) and (bottom) the Stokes dissipation rate (in MW) in each equation image° × equation image° grid. Note the high dissipation rate regions are well correlated with high wave energy regions. The high dissipation rates in the southern latitudes are noteworthy.

[16] Figures 3a and 3b show the seasonal variability. Figure 3a shows the boreal winter (January–March) and Figure 3b the boreal summer (July – September) averages. The seasonal contrasts are rather striking. For example, during the boreal winter, high dissipation rates are evident in the northern hemisphere, while during the boreal summer, the values there are very low. In contrast, the values remain high in the roaring fifties region of the southern hemisphere during both boreal summer and winter, although the boreal summer values are significantly higher. In the North Indian Ocean, summer monsoon winds cause high dissipation rates in the Arabian Sea region.

Figure 3a.

Spatial variability of the boreal winter (January–March) average of (top) the wave energy (in TJ) and (bottom) the Stokes dissipation rate (in MW) in each equation image° × equation image° grid. Note the high dissipation rates in the northern hemisphere.

Figure 3b.

Spatial variability of the boreal summer (July–September) average of (top) the wave energy (in TJ) and (bottom) the Stokes dissipation rate (in MW) in each equation image° × equation image° grid. Note the high dissipation rates in the roaring fifties in southern hemisphere and the very low values in the northern.

[17] The above findings are consistent with what could be expected based on what we know about the regional and seasonal characteristics of surface gravity waves in the global oceans. High wave regions tend also to be regions of high Stokes dissipation rates.

[18] It is worth reminding that what we have presented above are preliminary estimates using equation (5). But the trends and patterns are expected to be reproduced when more accurate estimates are made using equation (2) and a global mixed layer model. Nevertheless, the study highlights the importance of the Stokes dissipation mechanism of wind waves in the global oceans.

5. Concluding Remarks

[19] We have provided a preliminary estimate of the Stokes dissipation rate of the surface gravity waves in the global oceans. The average for the year 2007 is about 2.5 TW, more than the 2.4 TW dissipation rate of wave energy in the surf zones around the ocean margins. More importantly, unlike wave breaking (white capping), whose effects are confined to the top few meters, Stokes dissipation mechanism enhances upper layer turbulence with its effects felt potentially throughout the mixed layer. It also contributes to Langmuir circulation in the upper ocean. Although it is small compared to the deep ocean dissipation rate by white capping of 68 TW, to improve the physical basis of parameterization of wave dissipation, it may be important to incorporate this mechanism into operational wave models.

[20] The energy so extracted from waves is deposited in the mixed layer and unlike energy injected by breaking waves, which affects only the top few meters of the mixed layer, it affects deeper parts of the mixed layer leading to enhanced turbulence, more uniform velocity profiles and higher deepening rate of the mixed layer [Kantha and Clayson, 2004; Carniel et al., 2005].

[21] Needless to say that this study needs to be followed up by a more thorough in-depth study using a reliable mixed layer model [e.g., Kantha and Clayson, 1994, 2004]. The primary objective of this note is to merely point out the potential importance of the Stokes dissipation of surface gravity waves in the global ocean.

Acknowledgments

[22] This research was supported by the US Office of Naval Research under its N00014-05-1-0759 grant to LK and the Italian CNR-Short Term Mobility Programme 2008. MS and SC were partially supported by the Project “EquiMar, Equitable Testing and Evaluation of Marine Energy Extraction Devices in terms of Performance, Costs and Environmental Impact” (FP7-Energy-2007-213380), and CNR-RSTL “MOM.”

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