Wave Effect on the Ocean Circulations Through Mass Transport and Wave-Induced Pumping

BI Fan, and WU Kejian

Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, P. R. China

Wave Effect on the Ocean Circulations Through Mass Transport and Wave-Induced Pumping

BI Fan, and WU Kejian*

Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, P. R. China

The wave Coriolis-Stokes-Force-modified ocean momentum equations are reviewed in this paper and the wave Stokes transport is pointed out to be part of the ocean circulations. Using the European Centre for Medium-Range Weather Forecasts 40-year reanalysis data (ERA-40 data) and the Simple Ocean Data Assimilation (SODA) version 2.2.4 data, the magnitude of this transport is compared with that of wind-driven Sverdrup transport and a 5-to-10-precent contribution by the wave Stokes transport is found. Both transports are stronger in boreal winter than in summers. The wave effect can be either contribution or cancellation in different seasons. Examination with Kuroshio transport verifies similar seasonal variations. The clarification of the efficient wave boundary condition helps to understand the role of waves in mass transport. It acts as surface wind stress and can be functional down to the bottom of the ageostrophic layer. The pumping velocities resulting from wave-induced stress are zonally distributed and are significant in relatively high latitudes. Further work will focus on the model performance of the wave-stress-changed-boundary and the role of swells in the eastern part of the oceans.

Stokes transport; Sverdrup transport; wave pumping velocity; western boundary currents

1 Introduction

Besides, the wave-driven CSF also alters the mixed layer (ML) structure (Madsen, 1978; Weber, 1983); it has been proved in both Lagrangian and Eulerian frames (Xu and Bowen, 1994). To illustrate the role of wave-induced CSF on the wind-driven ML, Polton et al. (2005) conducted large-eddy simulations of the Ekman-Stokes layer and obtained good agreement with observations. The proposed ‘effective boundary condition’ is therefore proved to be brief and functional, which simplifies the CSF into a boundary stress at the surface.

Based on the work of McWilliams and Restrepo (1999) and Bi et al. (2012), we will further analyze the wavedriven effect on the ocean circulations. The unified long term wind stress and wave reanalysis data provided by ERA-40 from 1957 to 2002 are used. Section 2 focuses on the transport of wave Stokes drift and its contributionto circulations. The recently released long term SODA 2.2.4 (Carton and Giese, 2008) reanalysis ocean velocity data is used to make a comparison. Since the wave CSF can be presented as a boundary condition, the meaning of this ‘boundary’ and its relation to the wave transport effect need clarification. The idea of wave-driven boundary stress and the resultant pumping velocity are explained in section 3. Summary and discussion are following.

2 Wave Stokes Drift-Induced Transport Contribution to Ocean Circulations

What interests us in our Stokes transport analysis in Fig.1 is the average directions of the transport across the open oceans from the southern ocean to the low latitudes, which exhibits a similar pattern to that of circulations. This pattern indicates that the Stokes transport has a component in meridional direction and water mass can be carried along with waves.

This Stokes transport has been pointed out in Eq. (2) of Bi et al. (2012), with the contribution to wind-induced Ekman transport. In the same way we can evaluate the transport contribution to the circulations. Considering the steady momentum equations of ocean currents including the CFS:

Integrating the continuity Eq. (6),

we have

Fig.1 40-year averaged wave Stokes transport (unit: m2s−1). The arrows show the mean wave directions

So Eq. (5) becomes

This is the traditional Sverdrup relation with the wave transport embedded in. Since wave-driven Stokes transport can also be carried by persistent swells and travel to the low latitudes, we want to explore where this widely spread transport goes to. Eq. (8.2) suggests that meridional wave-driven Stokes transport is part of the traditional wind-driven currents. We can examine the wave contribution by comparing the last two terms in Eq. (8.2). This is clearer in the meridional transport comparison between the Stokes transport calculated by wave parameters through Ts=πa2/Peand Sverdrup transport calculated from wind stress.

ary transport and the wave-driven Stokes transport.

Fig.2 Stokes and Sverdrup transports in the meridional direction; a, Stokes transport; b, Sverdrup transport. Positive values represent transport northward (unit: m2s−1).

In Fig.3, the two transports show similar seasonal variations and the results are similar to those in previous studies (Kagimoto and Yamagata, 1997; Qiu and Lukas, 1996). The tranport is largest in boreal winter and smallest in late summer. The mean ratio of the Stokes transportto the Sverdrup transport is about 5% to 10%. That is to say, in some time, as large as 10% of the western boundary current transport is from the surface waves. The negative values of Stokes transport in Fig.3b represent poleward transport and have a canceling effect. Many scientists argue about whether the Sverdrup relation holds in real oceans, so we utilize SODA2.2.4 current data to do a comparison for the averaged status in 40 years. The meridional velocities along 27.5°N from 122.25°E to 129.75°E though all the water depth are integrated to calculate the Kuroshio transport. The seasonal signal of Kuroshio transport is not as strong as that predicted by Sverdrup relation, but is still significant. The averaged meridional transport is 39.7 Sv; the current is relatively stronger in winter than in summer. The western boundary current transport relies not only on the wind stress; there are more physical processes taking effect. Since the annual cycle of Stokes transport is the leading order signal, the ratio of it to the Kuroshio transport also shows a seasonal variation in Fig.4b. The Hilbert-Huang transform (HHT) (Huang et al., 1998) analysis of the 40-year monthly time series of both the transports confirms the 10–14 months significant period (figures are not shown). The effect of wave transport can reach 5% and there are cancellations caused by negative values as deduced from the Sverdrup transport (Fig.3) in summer months. This is the result of the wave transport direction shift from going southward to northward in summer.

Fig.3 Seasonal variations of the meridional transport integrated along three latitude lines in North Pacific (unit: Sv). a, Sverdrup transport; b, Stokes transport.

Fig.4 Monthly transport comparisons. a, SODA reanalysis Kuroshio transport and Stokes transport (unit: Sv); b, ratio of Stokes transport to reanalysis Kuroshio transport.

3 Wave-Induced Stress and Wave-Induced Pumping Velocity

For a monochromatic wave with wave number k, the depth of the Stokes drift is about 1/2k. Although this is not as deep as Ekman depth in most oceans, the wave- driven effect can penetrate into the whole Ekman layer (Polton et al., 2005). The wave-induced Stokes transport contribution to the main circulations can reach at least 5%; what is the physical process to drive this transport? The traditional Sverdrup balance takes effect through the pumping velocity driven by the surface wind stress. If waves can play a role in forming an effective boundary, what role does this boundary plays in the whole circulations?

where σ is the wave frequency; z is the water depth, the momentum equation can be rearranged into

in deep water with the surface wave-induced stress

in the direction perpendicular to the wave propagation direction. Here the CSF is transformed into a wave-induced boundary stress term when the Ekman layer depth is much larger than the wave Stokes layer depth. If so, the depth dependence of the wave-induced stress is not obvious and the stress can be expressed as the surface wave-induced stress. Since the wave-induced CSF is effectively considered as the boundary condition, the governing equations depict the layer below the Stokes layer and the Stokes drift does not appear in the continuity equation.

For steady ageostrophic motion equations,

Integrating Eq. (12) from z=−d (the Ekman-Stokes layer depth) to z = 0, we get

And also

Since at the surface w(0)=0, Eq. (13) becomes

Now it is clear that the wave-induced Stokes transport is part of the circulation because the wave takes effect together with the Coriolis force in the form of a waveinduced boundary stress. It is part of the wind stress whose curl sucks or presses the water and leads to motions of the geostrophic layer. The pumping velocity induced by surface-wave-induced stress is named wave pumping velocity. Its relative importance in the pumping velocity can be scaled by the stresses themselves. To roughly estimate the ratio of the two stresses, we define

The numerator on the right side represents wave pumping velocity and the denominator is the traditional wind pumping velocity. If the simplifications and assumptions below are made:

Then we get a simple form of Res(Fig.5b):

i.e.,

At relatively high latitudes or when wind shear is large while wind stress is not too strong (so that wave stress can be felt), wave pumping is likely to be more important. For example, at the edge of westerlies or wind convergence zones, both | f | and vorticity of wind are large while wind stress is weak, then wave pumping velocity will be more obvious.

Fig.6 shows the average pumping velocity distribution calculated by ERA-40. The wind pumping velocity is obviously large in subtropical areas to cause the pressure gradient. But the wave pumping velocity is relatively small with the magnitude 0.5 to 2 cm d−1. Negative values mean velocities downward and form a press; on the contrary, positive values mean the water being sucked up to the surface. There is a rough line at about 30° of latitude in the Pacific and Atlantic Ocean to separate the positive and negative wind pumping velocities as shown in Fig.6a. To further clarify the differences with wind pumping, we integrate the pumping velocities within several latitude bands. Fig.7 shows the integration range and the comparisons. The wind pumping velocity dominates between 20°–40° in both hemispheres as expected. The wave pumping is nonneglegible between 35°N–55°N and 40°S–60°S. The dividing lines change with the seasons but not much. The expression of wave stress offers a view to explain the wave effect on ocean circulations. From this point of view, waves are more important in higher latitudes to drive the circulations initially.

Fig.5 Schematic, showing the wave stress and wind stress directions.

Fig.6 The 40-year averaged wind (a) and wave (b) pumping velocity (unit: cm d−1). Negative values represent velocity downward.

Fig.7 Pumping velocity averaged over several latitude bands (unit: cm d−1). a, 35°N–55°N; b, 20°N–40°N; c, 40°S–60°S; d, 20°S–40°S. Wind pumping velocity plotted with cross and wave pumping velocity plotted with dot.

4 Summary and Discussion

Through the comparison between wave-induced Stokes transport and wind-driven Sverdrup transport, the waveinduced effect is evaluated in this paper. The wave transport is largest in the westerlies as a result of strong winds and relatively young waves, but the significant meridional component of the ocean waves that reside in the subtropical oceans shows the potential to contribute to or cancel the wind-driven transport. The seasonal variation is similar between the Sverdrup transport and the Stokes transport. The Stokes transport can reach 5% to 10% of the traditional transport. The comparison with the SODA-based Kuroshio transport shows a slightly smaller contribution, but can also reach 5% at most. That means the storms in the high latitudes generate waves of all frequencies and some of them propagate out of the source areas and carry some water mass persistently toward low latitudes. This mass transport will not disappear but is part of the local currents. When wave transport is in the same direction as wind-driven transport, it means that the surface winds not only drive currents but also drive wave Stokes drift. The opposite direction means that the wave drift causes actually a partial cancellation of the currents and the pure wind-driven transport should be larger to balance this discrepancy.

From the wave-induced stress analysis, the physical process is clarified. Wave effect can be expressed as part of the surface driving force and takes effect down to the whole Ekman layer. CSF can be considered as a boundary layer stress assuming that the Ekman layer is much deeper than the Stokes layer. The resultant vertical velocity, i.e., wave pumping velocity is the way connecting the surface wave-induced stress and the geostrophic currents. The distribution of this wave pumping is also zonal in open oceans except in the Indian Ocean. Its effect is obvious in relatively high latitudes. We can also find locally opposite directions between wind pumping and wave pumping, which means the same process as the direct analysis from the transport comparison. There are contribution and cancellation by the wave-induced effect. Although not directly effective to the geostrophic flows, the wave-induced transport and stress plays a part in the whole upper layer motions.

As for the effectiveness of wave-induced boundary stress, further work will be done in designing a simple circulation model experiment to test the wave effect by directly changing the boundary conditions. The wave effect on ocean circulations is reflected both vertically and horizontally. From the view point of sea water mixing, wave-induced turbulence is important in deepening the mixed layer (Qiao et al., 2008). The Stokes layer depth 1/2k is a rough evaluation of the wave penetration, which results in a deeper layer in the eastern oceans where swells dominate. The deepening is mainly of the order of 10 m. The HYCOM model results show that mixed layers are about 50 m in the eastern oceanic areas (personal communications), so the wave effect can not be ignored. The recent work on CSF effect in ocean models by including it as surface wave-induced stress shows progress in reproducing the tropical SST (Deng, 2010; Zhang, 2012). The wave-induced effects in other physical processes and performance in coupled ocean models are still under discussion.

Hanley et al. (2010) reported the use of inverse wave age to separate the ocean into regions of wind-driven wave regime or wave-driven wind regime. This also indicates that the remotely generated swell-induced upward momentum can be important in the global ocean. In their Fig.6 the areas with the inverse wave age lower than 0.15, which indicates the wave-driven wind regime, are locatedin the eastern subtropical oceans. Here are also the areas where wave Stokes transport caused by swells is obvious in meridional direction. Besides, the wave-induced pumping are most effective in high latitudes and the winddriven wave regime defined by Hanley et al. (2010) are mainly in the same latitude bands. If we consider the ocean waves as a link in connecting the atmosphere, the ocean surface and the interior ocean body, the high latitude bands may be the main sources of the wave energy and the swell transport scatter this energy out to the lower latitudes. The energy are partially given back to the atmosphere and partially transmitted down to the deep ocean at eastern subtropical areas. The potential effect of regular swells is thus another interesting point being worth working on.

Acknowledgements

We appreciate the European Centre for Medium-Range Weather Forecasts (ECMWF, the Centre) for providing the 40-year reanalysis wind and wave data. This work is funded by the National Science Foundation of China (40976005 and 40930844).

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(Edited by Xie Jun)

(Received January 18, 2012; revised June 11, 2012; accepted November 20, 2012)

© Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2014

* Corresponding author. Tel: 0086-532-66782270

E-mail: kejianwu@ouc.edu.cn