Wave-Breaking Turbulence in the Ocean Surface Layer
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JUNE 2016 T H O M S O N E T A L . 1857 Wave-Breaking Turbulence in the Ocean Surface Layer JIM THOMSON,MICHAEL S. SCHWENDEMAN, AND SETH F. ZIPPEL Applied Physics Laboratory, University of Washington, Seattle, Washington SAEED MOGHIMI Oregon State University, Corvallis, Oregon JOHANNES GEMMRICH University of Victoria, Victoria, British Columbia, Canada W. ERICK ROGERS Naval Research Laboratory, Stennis Space Center, Louisiana (Manuscript received 12 July 2015, in final form 17 March 2016) ABSTRACT Observations of winds, waves, and turbulence at the ocean surface are compared with several analytic formulations and a numerical model for the input of turbulent kinetic energy by wave breaking and the subsequent dissipation. The observations are generally consistent with all of the formulations, although some differences are notable at winds greater than 15 m s21. The depth dependence of the turbulent dissipation rate beneath the waves is fit to a decay scale, which is sensitive to the choice of vertical reference frame. In the surface-following reference frame, the strongest turbulence is isolated within a shallow region of depths much less than one significant wave height. In a fixed reference frame, the strong turbulence penetrates to depths that are at least half of the significant wave height. This occurs because the turbulence of individual breakers persists longer than the dominant period of the waves and thus the strong surface turbulence is carried from crest to trough with the wave orbital motion. 1. Introduction Gemmrich and Farmer 2004; Gemmrich 2010). The dynamic balance assumed is that the wave energy lost Wave breaking at the ocean surface limits wave during breaking S becomes a flux of turbulent ki- growth (Melville 1994), enhances gas exchange (Zappa brk netic energy F into the ocean surface layer that is dis- et al. 2007), and generates turbulence that mixes the sipated at a rate «(z), which varies with depth z ocean surface layer (Burchard et al. 2008; Kukulka and beneath the surface: Brunner 2015). Previous observations of wave- breaking turbulence have shown strong enhancement near the surface, at values that far exceed those pre- Sbrkdf ’ F ’ «(z) dz. (1) dicted by simple ‘‘law of the wall’’ boundary layer ð ð scaling (Agrawal et al. 1992; Terray et al. 1996; Here, the wave energy loss is integrated over a range of frequencies f, and the vertical reference frame for depth z can either be wave following, noted as zw, or fixed, Denotes Open Access content. noted as z. On average, the wave-following surface must, by definition, become the mean sea level, such that zw 5 0 / z 5 0. However, the wave-resolved « may Corresponding author address: Jim Thomson, Applied Physics h i Laboratory, University of Washington, 1013 NE 40th St., Seattle, not be symmetric, such that the average « is likely dif- WA 98105. ferent between the two reference frames. Reconciling E-mail: [email protected] these reference frames is essential to comparing the DOI: 10.1175/JPO-D-15-0130.1 Ó 2016 American Meteorological Society 1858 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 different observations of « reported in the literature, and Gemmrich et al. (1994) relate ce to short waves that it also is essential to obtaining a consistent total energy dominate the roughness of the ocean surface and find 21 dissipation rate when depth integrating «(z)dz. This that ce ; 1ms . This is a bulk value that is meant to simple balance neglects transport of turbulence and Ð represent the range wave speeds that carry the stress buoyancy work by bubbles, both of which may be sig- from the wind. Although the correct ce is still an open nificant in strongly forced conditions. question, the important aspect of this formulation is that Formulations for the total energy flux F typically be- t r 2 the wind stress 5 au* is imparted to the waves, as gin with the momentum flux from a wind stress t 5 r u2 a *a represented by ce, not just a mean sheared current, because wave breaking in deep water (i.e., whitecaps) is which would be represented by another factor of u* [as dominated by the short, wind-driven waves. Here, u*a is in Eq. (2)]. Conceptually, ce should be related to the the air-side friction velocity, which we convert to a speed of the roughness elements (i.e., the short waves). 1/2 water-side friction velocity u*w 5 (ra/rw) u*a, using the Although it has not been adopted in the subsequent ratio of the air and water densities. Dropping the sub- literature, Phillips (1985) also considered an adjust- script w, the water-side u* is used throughout the for- ment to the energy fluxes based on wave scales. Phillips mulations that follow. The wind stress acts upon the (1985,p.522)wrote,‘‘theenergysourcefornear- waves, which are roughness elements at the free surface, surface turbulence represented by wave breaking is, and thus some formulations for total energy flux include in mean, distributed over a wide range of scales, rather the speeds or scales of the waves in prescribing the work than being concentrated at the energy-containing done by the wind stress. The subsequent dissipation of scales as it is in shear-generated turbulence.’’ His for- energy happens in a region of depth z where waves are a 3 mulation considers the available flux, defined as u*, dominant process, which is a working definition of the that can be carried by wavenumbers bounding the ocean surface layer in any of the respective vertical ‘‘equilibrium range’’ k0 and k1. The result is a formu- reference frames. lation for a balance of total dissipation and total TKE One common scaling for turbulence in the ocean input given by surface layer is based on the shear production of an applied stress acting on a surface with a characteristic k F 5 2gb3I(3p)u3 ln 1 , (4) speed (e.g., Craig and Banner 1994). In this approach, 3 * k 0 the input of turbulent kinetic energy (TKE) increases with the cube of the friction velocity where the combined constants gb3I(3p) are approxi- mately 1023. The wavenumber limits are defined by a F 5 au3 , (2) 1 * lower bound that is twice the peak wavenumber k0 5 2kp and an upper bound that is caused by suppression of 2 where a is a constant, typically O(10 ). Although the waves that are slower than the drift speed of the surface, actual values exceed the log-layer estimates, the physical which can be related to the wind friction velocity by scaling for the TKE input is a generic boundary layer 2 k1 5 g/u*. In practice, the factor ln(k1/k0) in Eq. (4) scaling because the additional velocity scale multiplied limits the increase in energy flux at high winds by nar- with t is u* (as opposed to a velocity scale associated rowing the range of scales on which the wind acts. A 3 with the waves). Craig and Banner (1994) apply this u* wave spectrum with a smaller equilibrium range will input in a diffusive model for the vertical structure of the have a smaller ratio of k1/k0 and thus a smaller input rate turbulent dissipation rate «(z), which is now in regular F3. This dependence on the short waves is loosely similar usage (e.g., Gerbi et al. 2013). to using an effective speed that remains tied to short Turbulence generation during wave breaking, how- (and relatively slow) waves in Eq. (3). ever, creates a boundary layer that may not scale as Phillips (1985) argued that these wavenumbers were simply as the cube of the friction velocity. Wave the scales in which wave-breaking dissipation was in breaking is an intermittent process that injects turbu- balance with wind input and nonlinear transfers within lence at scales related to the waves themselves, rather the spectrum, hence the term equilibrium range. Equi- than the wind alone. Recognizing the importance of librium also was proposed by Kitaigorodskii (1983), al- wave scales, Gemmrich et al. (1994) proposed a differ- though with a different derivation. The equilibrium ent form for the TKE input, which uses an effective hypothesis has been shown to be consistent with wave- speed ce to transfer the wind stress t into the surface as breaking turbulence measurements under moderate TKE, such that wind speeds (Thomson et al. 2013). Assuming equilib- rium, and using a wind input formulation from Plant F 5 c t 5 c u2 . (3) 2 e e * (1982), the TKE input is JUNE 2016 T H O M S O N E T A L . 1859 u2 the wave crests, above the still water level. He fit the F 5 S df 5 * cos(u)(2pf )E(f ) df , (5) 4 wind c2 vertical decay of the turbulence using Eq. (7) and ð ð obtained 21.5 , l ,21, with a deep log layer below where u is the relative difference between the wind and one Hs. Similarly, wave tank measurements of spilling wave direction at every frequency f of the wave energy waves on a sloping beach suggest that 80% of the total spectrum E, and the factor of 0.04 in Plant (1982) is dissipation occurs above the still water level (Govender 1/2 omitted because the adjustment of (ra/rw) has already et al. 2004). Gemmrich (2010) converted the wave- been made in determining the water-side u*. In deep following measurements to a fixed vertical coordinate, water, the phase speed is c 5 g/(2pf), and thus the net which resulted in maximum dissipation at z ’ 0.3Hs.