A note on mixing due to surface wave breaking

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2008jc004758, 2008 A note on mixing due to surface wave breaking Jan Erik H. Weber 1 Received 31 January 2008; revised 14 May 2008; accepted 18 August 2008; published 8 November [1] Breaking of surface gravity waves generates turbulence and causes the entrainment of air at the sea surface, which again may lead to a temperature increase in the breaking region. These processes influence the near-surface buoyancy and generate a thin layer close to the surface which is statically stable. Dynamically, the wave breaking itself (the loss of mean wave momentum) induces an Eulerian mean current near the sea surface through the action of the virtual wave stress. This current is instrumental in mixing the turbulence induced by breaking downward. The mixing is affected by the fact that slightly denser particles are entrained into the stable top layer from below during this process. Utilizing the experimental result that the overall Richardson number is constant in similar mixing problems, we investigate theoretically the growth of the mixed layer due to breaking waves. The results show that the mixing due to a single breaking event penetrates down to a depth which is comparable to the wave height. At this depth the entrainment velocity practically vanishes. This occurs on a timescale that is of the order of the breaking duration time. For multiple breaking events in a steady homogeneous sea state, the breaking frequency is so small that the turbulence due to breaking will not diffuse further into the fluid. Associating the thickness of the breaking mixed layer with a typical roughness length, we find that the roughness length can be expressed in terms of the breaking wave characteristics. Citation: Weber, J. E. H. (2008), A note on mixing due to surface wave breaking, J. Geophys. Res., 113,, doi: /2008jc Introduction [2] It has been known for a long time that shear-induced instability due to the action of the wind could produce turbulence in the oceanic surface water column. The first laboratory experiments that convincingly demonstrated that a surface shear stress generated a turbulent mixed layer which penetrated into a stable density gradient, were performed by Kato and Phillips [1969]. Later, Kantha et al. [1977] repeated this experiment with a two-layer model for the density. Theoretically, Pollard et al. [1973] developed a model for the deepening of the wind mixed layer in the ocean, taking into account the entrainment of dense fluid from below. However, the action of wind inevitably generates surface gravity waves. The wave breaking itself leads to the generation of turbulence and the entrainment of air at the surface (see the comprehensive review by Melville [1996] and the references therein). This entrainment process may change the buoyancy in the surface layer such that the breaking wave mixed layer (hereafter BML) has a smaller overall density than the fluid below. [3] When ocean waves suffer an amplitude reduction due to dissipation (here breaking), the mean wave momentum becomes reduced. Because of the conservation of total momentum, this leads to the generation of an Eulerian 1 Department of Geosciences, University of Oslo, Oslo, Norway. Copyright 2008 by the American Geophysical Union /08/2008JC mean current. It is the virtual wave stress [Longuet-Higgins, 1969], acting at the surface, which drives this Eulerian current in the BML. At the interface between the BML and the water beneath, there is entrainment. This entrainment affects the stress conditions at the bottom of the layer [e.g., Pollard et al., 1973]. [4] The main aim of this paper is to combine experimental data for wave breaking, and classic theory for the mixed layer to determine how the depth of the BML changes with time. Earlier theoretical treatments of the effect of wave breaking [e.g., Craig and Banner, 1994; Kantha and Clayson, 2004], are based on numerical solutions of coupled equations for the turbulence and the mean flow. We here abandon complicating details, and highlight the physics involved through simple applications of the conservation of momentum and mass. The bonus of this simplified approach is that we arrive at a new analytical expression for the mixed layer development. [5] We start out our analysis in section 2 by integrating across the BML to determine the total mean Eulerian momentum due to a breaking wave. This procedure is similar to earlier treatments of wind-induced mixing in the surface layer [e.g., Pollard et al., 1973]. The novel approach here is that the horizontal mean flow that causes mixing with the underlying fluid is driven by the loss of wave momentum due to breaking. In section 3 we apply the same procedure to obtain the total mean density of the mixed layer, which is influenced by the turbulent entrainment of more dense fluid at the bottom. Applying a constant overall 1of6

2 Richardson number for the turbulent mixing process [Price, 1979; Thompson, 1979], we derive in section 4 a formula for the time dependence of the BML thickness during one single breaking event. In section 5 we consider a saturated sea, and we arrive at a new expression for the nondimensional frequency of wave breaking. Our breaking frequency shows a power 2 dependence of the wind speed, in contrast to the power 3 law derived by Thorpe [1993]. Both field data and semiempirical theoretical predictions indicate that the timescale for the growth of the BML is much smaller than the time between consecutive breakings at the same location. This has consequences for how far the turbulence generated by the breaking process will penetrate downward into the sea. The intermittency of the turbulence in the BML leads us to speculate that there on average must be a residual turbulence due to breaking near the surface for a fully developed sea state. We then relate our results for the BML to the roughness length in a law-of-the-wall distribution for the eddy viscosity. The advantage of the present formulation is that the roughness length can be expressed in terms of parameters that characterize the wave breaking, which is a novel result. Finally, section 6 contains a summary and some concluding remarks. 2. Momentum Considerations [6] If the wave amplitude is (nearly) instantaneously reduced because of breaking, the effect of amplitude reduction is felt immediately through the entire water column by the individual particles. Hence the total wave momentum, i.e., the Stokes flux times the density, is instantly reduced. We let the waves propagate along the x axis. If the wave amplitudes before and after breaking are a 1 and a 2, respectively, the mean momentum lost per unit area for an infinitely long wave train during one breaking event is From (2) we then obtain t 0 ¼ DM t B : ð4þ [7] Breaking occurs at the moving surface, which actually constitutes the upper boundary of the BML. However, the exact position of this surface is not well defined in the case of breaking. Therefore, we shall assume that the top of the BML is situated at the mean water level. Pointing the z axis upward, the mean water level in our case is given by z =0. We consider the development of the BML on a timescale that is much less than the inertial period, so we neglect the effect of the Coriolis force. Furthermore, by neglecting horizontal gradients of the mean quantities, and assuming no mean vertical velocity, the equation for the wave-induced Eulerian mean flow w0 u 0 ; where w 0 u 0 is the Reynolds averaged turbulent vertical flux of horizontal momentum (per unit density). The boundary condition at the top of the BML becomes: t vw r r ¼ w 0 u 0 ; z ¼ 0; ð6þ where t vw is given by (3) [see also Melsom, 1996]. [8] We take that the thickness of the BML is h(t), where t is time. At the bottom of the mixed layer the mean stress goes entirely into accelerating the entrained particles from the value just below the boundary to the velocity at the bottom of the layer [e.g., Pollard et al., 1973]. Hence, per unit density: DM ¼ 1 2 r rw a 2 1 a2 2 rr DU S ð1þ w 0 u 0 ¼ u _ h; z ¼t ðþ; ð7þ where r r is a constant reference density, w the angular wave frequency, and DU S the difference in Stokes flux before and after breaking. The justification for using a constant reference density (r r 10 3 kg m 3 ) for the total horizontal momentum is that the density changes are small in this problem, so we can apply the Boussinesq approximation. It is the virtual wave stress t vw (sometimes written just t w ), acting at the surface, that redistributes the lost wave momentum to Eulerian mean currents [Longuet-Higgins, 1969]. By definition where the dot denotes derivative with respect to time. By integrating (5) across the layer, and applying (6) and (7), we obtain d dt udz ¼ t vw r r : We take that the initial layer depth is h 0. The initial momentum flux then becomes ð8þ Z 1 0 t vw dt ¼ DM: ð2þ 0 u 0 dz ¼ DU S : ð9þ The breaking event typically occurs over a timescale t B which is of the order of the period of the dominating waves [Melville and Rapp, 1985; Farmer and Gemmrich, 1996]. We assume for one breaking event that t vw ¼ t 0 expð t=t B Þ: ð3þ By integrating (8) in time, and applying (9), we find for the Eulerian mean volume flux associated with one single breaking: udz ¼ DU S ð2 expð t=t B ÞÞ: ð10þ 2of6

3 We note that (8) has the same form as in Pollard et al., where the right-hand side is the constant wind stress per unit density. In their case, neglecting the Coriolis force, this leads to an Eulerian volume flux in the mixed layer that increases linearly in time. In our case, the source term on the right-hand side is the virtual wave stress associated with wave breaking. Since this event has such a short duration, e.g., (3), the Eulerian mean volume flux in this case suffers a rapid change in time. 3. Mass Conservation [9] In order to keep this discussion as general as possible, we consider the development of the density, or the buoyancy, in the BML. Then, in specific examples we can consider the effects of air entrainment, temperature increase, rainfall etc. through their respective contributions to the mean density r 0 of the initial layer of thickness h 0. Assuming zero mean horizontal density gradients, and zero mean vertical velocity, the conservation of mean density r in the BML r0 w 0 : Here r 0 w 0 is the Reynolds averaged turbulent vertical buoyancy flux. For wave breaking, we assume that the entrainment of air etc. through the surface is an instantaneous process, which determines the density r 0 of the initial layer h 0. Hence, for the subsequent mixing process we assume that r 0 w 0 ¼ 0; z ¼ 0: ð12þ At the bottom of the BML we take that the upward buoyancy flux equals the density increase due to entrainment. Mathematically Here u av is the vertically averaged velocity given by u av ¼ 1 h udz; ð16þ and Dr is the density difference across the BML based on the average density, i.e., Dr ¼ r I 1 h rdz: ð17þ It appears that the overall Richardson number is constant and close to 0.6 in similar mixing experiments [Price, 1979; Thompson, 1979]. A constant Ri is an appropriate closing condition for this problem. Inserting from (10) and (14) into (15), we finally obtain h ¼ ðri Þ1=2 DU S ð2 expð t=t B ÞÞ : ð18þ ðgh 0 B 0 Þ 1=2 Here we have defined the initial BML buoyancy B 0 as B 0 ¼ r I r 0 r 0 : ð19þ From (18) we obtain for the initial depth of the BML: h 0 ¼! Ri ð DU SÞ 2 1=3 : ð20þ gb 0 This relation yields the initial BML thickness in terms of the initial density, and the breaking wave characteristics. Utilizing (20), we realize that (18) can be written: r 0 w 0 ¼ ðr I rþ h; _ z ¼t ðþ; ð13þ h ¼ 2 expð t=t B Þ; ð21þ h 0 where r I is the constant density in the water just below the mixed layer (see also Niiler [1975] in the case of temperature). Integrating (11) across the BML, and applying the boundary conditions (12) and (13), we obtain [e.g., Kantha et al., 1977]. 4. Result for a Single Breaking rdz ¼ r I ðh h 0 Þþr 0 h 0 ; ð14þ [10] We now introduce the overall Richardson number Ri defined by Ri ¼ ghdr r 0 u 2 : ð15þ av i.e., the initial mixed layer doubles in thickness during one single breaking. [11] It is of particular interest to relate h 0 to the typical amplitude of the breaking wave. To do so, we define the fractional energy loss De during one breaking event as De ¼ a2 1 a2 2 a 2 ; ð22þ where a is the average amplitude of the breaking sea. Furthermore, we introduce the average wave steepness e = ka. Then (20) can be written h 0 a ¼ Ri ð De 4B 0 Þ2 e! 1=3 : ð23þ Observations indicate that De lies between 10 2 and 10 1 [Melville and Rapp, 1985]. In our estimate we take that De = 1/30. A middle of the road value for the steepness e in breaking waves is 0.3 (Stokes limiting value is ). The 3of6

4 The results (21) and (24) show that wave breaking only influences the turbulence in a surface layer which has a thickness of the order of the wave height. This conforms to the results from field measurements [e.g., Farmer and Gemmrich, 1996], and to more comprehensive model studies of turbulent mixing due to breaking waves [Kantha and Clayson, 2004]. 5. Application to a Saturated Sea [12] In a fully developed sea breaking occurs fairly regularly. However, the time between breaking events at the same location is much larger than the breaking duration time. This can be seen if we introduce the nondimensional frequency of breaking, f, as the number of waves breaking at a fixed position in a given time, divided by the number of waves of the dominant wave frequency that pass in the same period of time [Thorpe, 1993]. If T B is the average time from one breaking to the next, and T is the period of the dominating wave, we have that Figure 1. The number of breaking waves per wave, f, from (29), versus wind speed divided by the speed of the dominant waves U 10 /C, both plotted on log scales (solid line). The broken lines are from Thorpe [1993], his equation (1). Data points for deep water breaking are closed squares [Longuet-Higgins and Smith, 1983], star [Weissmann et al., 1984], open circles [Thorpe and Humphries, 1980], closed circles [Katsaros and Atatürk, 1992], and open squares [Thorpe, 1992]. (Figure 1 is adapted from Thorpe s [1993] Figure 1.) initial buoyancy B 0 is more difficult to assess. According to Melville [1996] warm pools in the western equatorial Pacific Ocean are as much as 3 K warmer than the water below, while direct measurements at 0.17 m below the surface by Gemmrich [2000] yield values of the order 0.1 K. Let us use the latter value, together with a thermal expansion coefficient of K 1. Then, from (19), the initial buoyancy becomes B 0 = Also the content of entrained air during the breaking process may create a thin stable layer near the surface. From laboratory and field measurements one finds a sharp increase in the air fraction during a breaking event [e.g., Melville, 1996; Gemmrich and Farmer, 1999, 2004; Gemmrich, 2000]. In fact, the air fraction g found near the surface is probably the most important source of initial buoyancy in the BML. While larger bubbles tend to disappear fairly quickly, smaller ones are present for many seconds after breaking. From the measurements of Gemmrich and Farmer [2004] at 0.85 m below the surface, average air fractions after breaking are typically g If air is the only source of buoyancy, we find from our definition (19) that B 0 g. From these considerations, we choose B 0 = in a numerical example, which should cover heat and air as sources of initial buoyancy in the MBL. Taking that Ri = 0.6 in order to have sustained mixing, as suggested by Price [1979] and Thompson [1979], we find from (23) that h 0 a ¼ 1:2 ð24þ f ¼ T=T B : ð25þ Thorpe [1993] presents a formula for f, where f / (U 10 /C) 3. We here suggest an alternative formulation. By assuming that the wave amplitude grows approximately linearly in the time interval from post breaking to the next breaking, the fractional energy loss (22) during one breaking event can be written [Melsom, 1996]: De ¼ a2 1 a2 2 a 2 2bT B ; ð26þ where b is the growth rate of the fastest growing waves. From experimental data [Plant, 1982] we find that b w ¼ K U 2 * C 2 ; ð27þ where U * is the friction velocity in the air. Plant [1982] suggests that the constant K is typically of the order Utilizing that U 2 * = c D U 2 10, where c D is a drag coefficient and U 10 is the wind speed, we find from (25), (26), and (27) that f ¼ 2bT De ¼ 4pKc D U 2 10 : ð28þ De C With K =10 2, c D =2 10 3, De = 1/30, (28) yields f ¼ 7: ðu 10 =CÞ 2 : ð29þ In Figure 1 we have plotted (29) together with the deepwater data and the graphs from Thorpe s [1993] Figure 1. Our curve is less steep in the log-log plot, and fits the data fairly well. We note that the breaking duration time generally is much smaller than the average time between breaking events at the same location (f 1). For the 4of6

5 entrainment velocity we find from (21) for one single breaking that _h ¼ h 0 expð t=t B Þ: ð30þ t B Accordingly the deepening of the BML vanishes on a timescale t B. It is then obvious that the turbulence will decay in time before the next breaking occurs [Gemmrich, 2000]. Hence, persistent breaking will introduce a transient, or intermittent state of turbulence in the BML very close to the sea surface. This turbulence will not penetrate further into the fluid below. Although it varies in time, it will on average (over the timescale T B ) keep this layer well mixed. [13] In the open ocean surface gravity waves normally break only in the presence of wind, and the wind is the main source of turbulence in the surface layer. For the equilibrium range in wind waves, where the dissipation due to breaking is balanced by the work of the form stress [e.g., Phillips, 1985], a bulk value for the eddy viscosity due to breaking waves has been assessed by Weber et al. [2006]. However, in the literature there is compelling evidence [see Craig and Banner, 1994, and references therein] that the joint effect of wind and waves yields a turbulent eddy diffusion coefficient A that increases linearly with depth, i.e., the law-of-the-wall, in a neutral log layer region below the surface. We can write A ¼ ku * ðjjþ z z 0 Þ; ð31þ where k 0.4 is von Karman s constant, and u * is the friction velocity in the water. Furthermore, z 0 is the socalled roughness length. The surface eddy value becomes A 0 ¼ ku * z 0 ; ð32þ In the boundary layer above a rigid wall, the roughness length, representing the scale of the smallest turbulence elements is usually very small. Traditionally, we have that z 0 = bu * /g, where b is a constant [Charnock, 1955]. Stacey [1999] estimated the roughness length from data from the Knight Inlet, and found that z 0 O(H s ), where H s is the significant wave height. Appropriate values for the Charnock constant b in this case was large (b O(10 5 )). These findings are consistent with results from Craig and Banner [1994], assuming that the roughness length is proportional to the wave amplitude. They obtained good fit with observational data for z 0 = 1 m. Furthermore, Kantha and Clayson [2004] use that z 0 =1.6H s in their analysis. [14] For a saturated sea it is tempting to associate z 0 with the final thickness h m =2h 0 of our BML, where h 0 is given by (20), or alternatively, by (23). This relates the roughness length to the breaking wave characteristics, i.e., z 0 ¼ 2! Ri ð DU SÞ 2 1=3 : ð33þ gb 0 [15] This formulation has the appealing property that z 0 becomes larger when the breaking is more vigorous (DU s large), and smaller when the stability due the surface entrainment of air or heat/rainwater increases. From our previous example at the end of section 4, we find that z 0 = 1.2H, where H(= 2a) is the wave height. This result for the roughness length is not inconsistent with the values cited above. 6. Summary and Concluding Remarks [16] In the present paper we have studied mixing in a thin surface layer where wave breaking is the mixing agency. Utilizing that the loss of wave momentum due to breaking must induce a horizontal Eulerian flow driven by the virtual wave stress at the mean surface, we apply classic theory for shear flow mixing to obtain the thickness of the mixed layer due to a single breaking event. This is a novel approach. The main result is that the time development of the BML occurs on a scale comparable to the breaking duration time. For multiple breaking we derive a new formula for the breaking frequency. The result supports earlier findings and observational data that the time between consecutive breakings at the same location is much longer than the breaking duration time. Thus, we conclude that for each breaking, the generation of the BML starts afresh, and the turbulence due to wave breaking will not penetrate further down than a distance comparable to the wave height. This result is in accordance with field observations and earlier model studies. Since the wave mixing is confined to a thin surface layer, we speculate that this layer scale can be related to the roughness length in a law-of-the-wall distribution for the eddy viscosity. The advantage of this approach is that the roughness length is determined by quantities that characterize surface wave breaking. [17] The final thickness of the BML depends on the initial buoyancy (19) that is induced by the breaking process. The initial buoyancy depends on the entrainment of air, heat, rainwater etc. at the sea surface. We here take this entrainment to occur instantaneously as part of the breaking process. Furthermore, in the present study we have used averaged values at a given depth from field observations for the air fraction and heat content. But this is clearly unsatisfactory from a modeling point of view. Future research on wave mixing definitely needs to focus on how to model realistically the development of the near-surface buoyancy during the breaking process. References Charnock, H. (1955), Wind stress on a water surface, Q. J. R. Meteorol. Soc., 81, , doi: /qj Craig, P. D., and M. L. Banner (1994), Modeling wave-enhanced turbulence in the ocean surface layer, J. Geophys. Res., 24, Farmer, D. M., and J. R. Gemmrich (1996), Measurements of temperature fluctuations in breaking waves, J. Phys. Oceanogr., 26, , doi: / (1996)026<0816:motfib>2.0.co;2. Gemmrich, J. R. (2000), Temperature anomalies beneath breaking waves and the decay of wave-induced turbulence, J. Geophys. Res., 105, , doi: /1999jc Gemmrich, J. R., and D. M. 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