Turbulence in the ocean surface layer is generated by time-varying combinations of destabilizing surface buoyancy flux, wind stress forcing, and wave forcing through a vortex force associated with the surface wave field. Observations of time- and depth-averaged vertical velocity variance of full-depth turbulence in shallow unstratified water columns under destabilizing buoyancy forcing are used to determine when process domination can be assigned over a wide range of mixed forcings. The properties of two turbulence archetypes, one representing full-depth Langmuir circulations and the other representing full-depth convection, are described in detail. It is demonstrated that these archetypes lie in distinct regions of the plane of , where and are Langmuir and Rayleigh numbers, respectively, derived from scaling with surface stress velocity and a time scale characteristic of the growth of Langmuir circulation , where and are mean and Stokes velocities, respectively. Situations in which neither process dominates lie between the two end members, with relative dominance given by proximity to one or the other. Cases dominated by direct stress forcing are conspicuous by their absence. In cases of Langmuir domination, surface Stokes velocity is linearly related to , making it impossible to differentiate between scaling depth-averaged vertical velocity variance with , and any other scaling involving both and . A third nondimensional parameter is introduced and used to assess the importance of bottom boundary layer turbulence in a depth-limited system. Questions of time dependence and applicability of results to the open ocean surface boundary layer are considered.
Processes that generate turbulence in the surface layer of the ocean rarely occur in isolation: generally, multiple forcings act simultaneously. The question of which forcing (if any) dominates turbulent production at any given time is of importance to vertical and horizontal transports of heat, salt, momentum, and any materials (plankton, sediment, bubbles, etc.) present within the range of turbulent structures. In shallow coastal oceans, this range can include the entire water column depth .
It is believed that winds generate turbulence in the upper ocean both through direct action of wind stress on the surface and through an indirect process involving surface waves. A term in the wave-averaged momentum equation incorporating the latter mechanism, first derived by Craik and Leibovich (1976), will be here termed the Langmuir vortex force to clearly identify the process represented as that leading to Langmuir circulations (LC; Langmuir 1938). Both effects of wind forcing may be present together, and may be further superimposed on convection forced by destabilizing surface buoyancy flux and, in the case of shallow seas, turbulence associated with a bottom boundary layer.
Because of the complexity of turbulence driven by multiple processes, much of the literature on surface layer turbulence understandably focuses on a single mechanism [e.g., Agrawal et al. (1992), Steffen and D’Asaro (2002), and Plueddemann et al. (1996), for stress-driven, convective, and Langmuir-driven systems, respectively]. Comprehensive reviews of theoretical (Leibovich 1983) and computational (Sullivan and McWilliams 2010) studies that focus on LC and their connection to the surface wave field are available. A few large-eddy simulations (LESs) have treated the combination of stress, convective, and single-wave Langmuir forcings (Skyllingstad and Denbo 1995; Li et al. 2005), while others (Harcourt and D’Asaro 2008; Sullivan et al. 2007) have treated Langmuir forcing by a modeled spectrum of surface waves in the absence of convection: none have yet done both. Although there are exceptions (e.g., D’Asaro 2001; Tseng and D’Asaro 2004), observational studies of LC, reviewed most recently by Thorpe (2004), frequently do not quantify possible contributions from a stress-driven boundary layer and/or from destabilizing surface buoyancy flux, nor is there an accepted method of assessing their importance relative to LC. The present analysis seeks to fill this gap.
We consider interactions among stress, Langmuir vortex, and destabilizing buoyancy forcings, asking whether it is possible to determine when dominance of a single generation process is assured, using basic knowledge of wind, wave, and buoyancy forcings. Post hoc justification will be provided for an initial assumption that bottom boundary layer turbulence is of minor importance in most of the data examined here because of weak currents at the measurement site. In addition, effects of surface wave breaking will not be considered separately from the presence of surface waves. For a broad range of wave age, the majority of wind momentum is converted into waveform stress and drives a mean current (Sullivan and McWilliams 2010). Spatially variable wave breaking plays two roles, feeding momentum to the current and generating vertical vorticity via horizontal variation in the mean current. Along with that generated by other mechanisms, breaking-induced vertical vorticity can be rotated into the downwind direction by mean vertical shear to generate Langmuir circulations: breaking is included in the Langmuir vortex force through its contribution to fluid vertical vorticity.
If a single process is responsible for turbulence generation, dimensional analysis can provide representative length and velocity (or time) scales for large-eddy structures. Boundary layer meteorologists (Holtslag and Nieuwstadt 1986) have documented success of convective scales for turbulence at heights above (below which surface stress must also be considered) in a well-mixed boundary layer. Written with oceanic parameters, assuming that surface buoyancy flux is dominated by destabilizing surface heat flux > 0,
for mean temperature , mean seawater density , and specific heat at constant pressure . Turbulence in a boundary layer driven solely by surface wind stress is well described by stress scaling (Tennekes and Lumley 1972)
where is distance from the boundary. [Parameters associated with rotation do not appear in either of these cases. Rotation can be shown to be unimportant on time scales associated with the large eddies of turbulence (Li et al. 2005; Grant and Belcher 2009) and will not be considered further.]
Because the ocean surface layer involves Langmuir vortex forcing in addition to stress and convection, scalings derived from the atmospheric boundary layer are not automatically relevant nor necessarily adequate. Section 2 presents a new nondimensionalization of the momentum equation that provides two independent nondimensional parameters suitable for determining dominance among the three forcing processes considered. A five-beam vertical-beam acoustic Doppler current profiler (VADCP) operated continuously for several months at the LEO15 cabled observatory on the continental shelf off New Jersey provides an extensive dataset for testing this technique; section 3 describes available data and provides relevant details of processing. Section 4 presents a suite of diagnostics describing distinctly different characteristics of turbulence extending fully over an unstratified water column and demonstrably dominated either by Langmuir forcing or unstable convection, allowing definition of Langmuir supercell (LSC; Gargett et al. 2004) and convection (CVN) archetypes representing these two cases. In section 5, the forcing conditions resulting in each archetype are described. Vertical velocity variances are then presented in the space of the two nondimensional parameters, using the LSC and CVN archetypes to identify characteristic regions in which turbulence generation is dominated by Langmuir and destabilizing buoyancy forcing, respectively. Once identification/sorting has been made in the nondimensional parameter space, in section 6 we check that scaling based on the identified single process is consistent with observed vertical velocity variances, for example, if cases identified as dominated by convection indeed have variances consistent with scaling by . Section 7 provides evidence that neither bottom stress nor time dependence is important for most of the data considered here. Section 8 summarizes significant features of the results and discusses their potential applications.
2. Nondimensionalization of the equations of motion
In cases where all three turbulence-generating processes are present, we seek a nondimensionalization of the momentum equation to identify situations in which only one of the three dominates the production and ensuing characteristics of the turbulent field; by default, this also identifies situations that do not have a dominant forcing, that is, that must be accepted as produced by a mixture of forcings.
Consider flow in a wind coordinate system with the downwind direction, the crosswind direction to the left of the wind, and positive upward from the surface. Water column depth is finite in the sense that it may be comparable to dominant surface wave wavelength. For simplicity we assume that density is a linear function of deviation from a reference temperature, so , where is the coefficient of thermal expansion. In the absence of rotation, flow is described by the wave-averaged momentum equation (Craik and Leibovich 1976) with added buoyancy term
Here, is the ith component of (total) wave-averaged velocity, is the nth component of fluid vorticity, and is the viscosity of seawater. Here, is the one-dimensional (downwind) Stokes drift velocity expressed as a surface value times a nondimensional function of , , where is surface wavenumber. The pressure term
includes wave-averaged contributions dependent on Stokes drift and any externally imposed pressure gradient ; for now, the latter is assumed zero. Boundary conditions on velocity are
The temperature equation, obtained from energy and mass conservation equations under the Boussinesq approximation, is
where is the coefficient of thermal diffusivity, subject to boundary conditions
where, retaining the atmospheric sign convention, surface heat flux is destabilizing; is thermal conductivity. While precludes exact steady state, temperature changes associated with observed heat flux magnitudes over several time scales of the observed turbulent structures are extremely small, hence steady state is assumed.
For , this set of equations describes the mean plus turbulent flows driven by destabilizing surface buoyancy, wind, and wave forcing. In the scaling that follows, no distinction is made between these equations and those for the turbulent component alone, because the form of the momentum forcing terms examined are identical in both sets.
Appropriate nondimensionalization of the momentum equation, involving specification of any two of time, length, and velocity scales, should yield parameters gauging relative strengths of the three forcing mechanisms. Because surface wind stress drives both “ordinary” and Langmuir turbulence, is usually chosen as a velocity scale. Much of the early literature on LC (Leibovich 1977; Li and Garrett 1993; Li et al. 2005) then specifies a length scale , where is the e-folding depth of the Stokes velocity associated with a single wave of wavenumber . While straightforward in single-wave theoretical or computational studies, the choice of length scale is less clear given the reality of a distributed spectrum. In the finite-depth case, is another possible choice for length scale.
The traditional choice of a length scale is here abandoned by instead specifying a time scale,
the geometric mean of vorticity time scales associated with and , thus deriving a length scale . This time scale is expected to be useful for assessing dominance of LC because Leibovich (1977) demonstrated that , evaluated at the surface, is a characteristic growth rate for LC. Two contributions to growth arise from rotation of vertical vorticity to horizontal through the action of Stokes shear, followed by amplification of downwind velocity by stress-driven mean shear. In the inviscid theory of Leibovich (1977), growth rates increase monotonically from and asymptote to as . In a more realistic (although still simplified) computational study, Gnanadesikan (1996) found that growth rates of LC achieved a maximum at an intermediate wavelength best predicted by averages, weighted by known vertical structure functions, of the two shears over the depth of penetration of LC. For Stokes drift shear, this weighted average approximated the local value of Stokes shear at depth = ~.
It has become generally accepted that the presence of Langmuir circulations decreases mean shear in the bulk of the mixing layer [McWilliams et al. (1997); Gargett and Wells (2007), hereafter GW07; Tejada-Martinez and Grosch (2007), hereafter TMG07, and subsequent LES studies]. We consider this decreased mean shear as an effect rather than a cause, interpreting as a potential time scale for Langmuir growth, that is, at any given time is the growth rate that would result from imposition of the Langmuir vortex force on the stress-driven mean flow that would exist in its absence.
Previous deep-water single-wave scalings involve an additional parameter , the ratio of water depth to surface wave vertical scale . To achieve a two-parameter scaling, is assumed to be large (e.g., Li et al. 2005). A significant additional advantage to the use of is that such a length-scale ratio is an integral part of through the vertical shear of Stokes drift velocity and hence need not be considered as a separate parameter.
Use of requires determination of representative values for the two shears involved. Because surface waves observed at LEO15 are of intermediate type, it is essential to use a measure of Stokes shear calculated with the full-dispersion relationship. Moreover the wave field is frequently neither in equilibrium nor fully developed, requiring a shear estimate based on observations rather than empirical spectra. Assuming a unidirectional surface wave spectrum (see the appendix), the vertical shear of Stokes velocity is the integral over a Stokes shear function defined in
where and are related by the full-dispersion relationship and the displacement spectrum is calculated from , the noise-corrected spectrum of vertical velocity measured by the vertical beam of the VADCP at height above bottom [depth , see the appendix for details].
Because Eq. (11) involves an integral to infinite frequency, surface Stokes shear thus defined is unbounded if the high-frequency surface displacement spectrum is assumed to be proportional to either [the high-wavenumber form of the model spectrum of Phillips (1958); see also Pierson and Moskowitz (1964), hereafter PM] or [that of Toba (1973); see also Alves et al. (2003)] to infinite frequency. This can be seen by considering the deep-water and/or high-frequency/wavenumber limit , denoted by superscript , for which Eq. (11) at reduces to
At a high wavenumber, is flat for and rises as for . Observations exhibit both behaviors as well as periods where neither holds (reinforcing the necessity of using representative values of Stokes shear derived from observed spectra rather than empirical expressions for equilibrium spectra). However, because contributions from very high–frequency waves decrease rapidly with depth, the Stokes shear function becomes bounded when computed at a distance sufficiently below . Figure 1 shows the Stokes shear function as a function of cyclic frequency at a time of prolonged onshore (long fetch) winds when waves are most likely fully developed. Surface shear rises as at frequencies above the displacement peak [until a still stronger rise at frequencies >0.4 cycles per second (cps) associated with inaccuracy in high-frequency noise correction exacerbated by very large response corrections in this range under these measurement conditions; see the appendix]. This rise disappears as the depth of the shear calculation is increased. At the function is flat at high frequencies, while by , it decreases as frequency increases. We have chosen to use a frequency integral of the Stokes shear function at as a characteristic Stokes shear for determination of time scale (for details, see the appendix and Table 1). The depth chosen is similar to that Gnanadesikan (1996) found characterized a “typical” Stokes drift shear responsible for driving LC in his simulations.
The mean shear that appears in Eq. (10) is associated by both Leibovich (1977) and Gnanadesikan (1996) with that of an “ordinary” stress-driven turbulent boundary layer, characterized by an eddy viscosity that appears in a surface boundary condition , where . For such a boundary layer (Pope 2000). Hence, if the eddy viscosity is evaluated at a depth that is a small fraction of , near-surface mean shear associated with the stress-driven boundary layer scales as
Finally, a temperature scale is determined from the surface flux boundary condition of Eq. (9).
Defining nondimensional variables by and , the resulting nondimensional momentum equation is
is a Reynolds number describing the importance of inertial (nonlinear) forces associated with surface stress driving relative to viscous forces,
is a Rayleigh number describing the importance of destabilizing buoyancy forces relative to the same inertial forces, and
is a Langmuir number describing the relative importance of inertial to Langmuir vortex forces in the case that both exist. Although any power may be used in Eq. (17), including where is the “turbulent Langmuir number” originally defined by McWilliams et al. (1997), it is most straightforward to use . Then, appears in Eq. (14) as a reciprocal, so that implies unimportance of Langmuir vortex forces just as implies unimportance of viscous forces [typical values of Re are O(104) or larger; hence, viscous effects are subsequently ignored].
Determination of La requires a value for . As detailed in the appendix, a spectral Stokes velocity associated with observed surface wave displacement spectrum is given by the integral over frequency of a Stokes function defined in
The surface Stokes velocity can be unbounded for observed or empirical spectral forms extrapolated to frequencies higher than those measured. However a determinate Stokes velocity can be calculated either by evaluating Eq. (18) at a small depth below the surface or by band limiting the surface integral. Because it is desirable to use a surface Stokes velocity to facilitate numerical comparisons with previous observational and computational studies, the latter option has been chosen and is computed as the integral of Eq. (18) at over the range within which surface wave displacement estimates are unaffected by extreme response corrections (see the appendix and Table 1). This band-limited value proves to be well correlated with (although of course larger in magnitude than) Stokes velocity calculated at the depth used for characteristic Stokes shear, as well as with Stokes velocity integrated from this depth to the surface [see the appendix, Fig. A3; the latter value best collapsed LES results of Harcourt and D’Asaro (2008) carried out using surface wave spectra modeled as functions of wind speed and wave age].
3. Observational data
Nearly continuous observations were taken from mid-May through October 2003 at Node B of the LEO15 cabled observatory, a shallow (15 m) site approximately 6 km off of the coast of New Jersey (39°27.69′N, 74°14.68′W). Here, homogeneous water columns are normally associated with downwelling winds and surface buoyancy fluxes that are destabilizing or at most weakly stabilizing, conditions occupying a significant fraction of the annual cycle, from early fall through late spring.
Velocities in instrument coordinates were derived from 5-beam VADCP measurements with vertical resolution of 0.4 m, starting ~1-m-above bottom. Continuously sampled data were returned as sequential records of length ~2.2 h, long enough to contain multiple turbulent structures but short enough that forcing conditions could be considered approximately constant. Continuous recording was broken into sessions of length 4–5 days, after which, for unknown reasons, instrument-identified errors increased rapidly with time. Because of the large number of records and the need to cross-reference individual records between publications, each is given a unique identifier sss.nnn, where nnn is sequential record number within session sss. Sampling frequency (1 Hz) resolved dominant velocities of surface gravity waves, which appear at a fixed site with frequencies higher than those of turbulent eddies advected past by weak tidal currents; thus, turbulent velocities could be separated from wave velocities by low-pass filtering in the time domain. This process requires time-continuous data, hence could only be accomplished at heights less than , the minimum value over the record of instantaneous surface height measured in backscatter of the vertical beam.
Horizontal velocities and turbulent stresses were calculated under respective assumptions of first- and second-order homogeneity over beam spread (GW07). The response function analysis of Gargett et al. (2009) shows that horizontal scales associated with the full-depth turbulent structures considered here are large enough that beam separation effects do not significantly affect first-order estimates of horizontal velocity variances for retained scales. “Mean” flow is defined as a time average over the record length. Because the mean flow thus defined contains slow variation due to changing tides, linear least squares (lls) fits over record length are removed from each bin, leaving fluctuation fields , , and . Vertical velocity was measured directly and unambiguously by the VADCP vertical beam, accurately adjusted to vertical by divers. Data are restricted to times when the water column was effectively unstratified, defined as having top-to-bottom temperature difference less than 0.5°C, corresponding to a Väisälä period longer than the record length. This ensures that fluctuation vertical velocity calculated by removing lls fits over record length will not be contaminated by internal wave vertical velocities. Additional details of VADCP observations at LEO15 and their processing can be found in GW07 and Gargett et al. (2008, 2009).
A fundamental metric used to classify turbulence is , the column average (overbar) of , vertical velocity variance corrected for noise bias (Gargett et al. 2008) and time averaged over record length (angle brackets). The column average is taken from the first measurement bin through the bin just below , a level that varies with sea state.
Atmospheric data were taken at a meteorological tower located on the beach to the west of Node B. Backed by an extensive marsh, the tower has excellent exposure: tower wind speed and direction are highly correlated with those measured at a nearby offshore buoy (Münchow and Chant 2000). Before computing derived quantities such as or , tower variables were averaged over 4 min for consistency with sampling of sea surface temperature at Node B. In the absence of rainfall observations, no corrections were made for possible freshwater effects on surface buoyancy flux; estimates suggest that the additional term is usually small.
Because of the focus on competition among Langmuir, stress, and destabilizing buoyancy forcing, records with destabilizing surface buoyancy (heat) flux were selected from all available periods when the water column was effectively unstratified, as defined above. Records used cover a wide range of forcing, from nearly calm through typical 1–2-day storm conditions. Record-averaged winds range from nearly 0 to 13 m s−1 and wave conditions range from gentle swell to fully developed seas. A final selection criterion required that the peak of the Stokes function should be resolved below the maximum frequency used in the band-limited estimate of . Records fulfilling all these criteria are designated “qualified.”
4. Archetypes of full-depth Langmuir circulation and unstable convection
A suite of large-eddy characteristics is used to define “archetypes” for two end members of mixed forcing, one case in which the large-eddy turbulent structures are LSC, the full-depth Langmuir cells termed Langmuir supercells by Gargett et al. (2004), and another (CVN) in which they demonstrate characteristics of convection in a bounded domain. Although only a single record is presented in detail in each case, records with similar forcing conditions have similar characteristics.
a. Vertical velocity variance and backscatter patterns
Figure 2 depicts fields of fluctuation vertical velocity and backscatter amplitude (corrected for range and spreading effects but not calibrated) from the vertical VADCP beam for two records with obviously different characteristics. Left-hand panels show record 043.024, taken during an episode of strong wind/wave forcing and weakly unstable buoyancy forcing. The turbulent structures in this case have been identified as full-depth Langmuir circulations by the detailed coupled analyses of GW07 and TMG07. Fluctuation vertical velocities in this LSC archetype are much larger than those in record 161.008 (Fig. 2b), taken at a time of weak wind/wave forcing. In addition to quite different magnitudes of , the two cases have strikingly different backscatter fields. The LSC case exhibits distinctive surface-origin clouds of high backscatter correlated with downward vertical velocities, clouds widely attributed to advection of air microbubbles produced by surface wave breaking in the downwelling limbs of Langmuir cells (Zedel and Farmer 1991). In this shallow location, the LSC case also has distinctive bottom-origin backscatter clouds, correlated with upward vertical velocity, that Gargett et al. (2004) suggest are clouds of resuspended sediment moved toward the surface in the upwelling limbs of LSC. In contrast, weak vertical velocities in Fig. 2b are accompanied by middepth backscatter maxima, which may be due to scattering from density microstructure (Thorpe and Brubaker 1983; Goodman 1990) associated with entrainment at the edges of cold downwelling plumes.
b. Horizontal velocity variances and horizontal anisotropy
The two turbulent flows seen in Fig. 2 have very different associated mean flows, velocity variances and stresses, and degrees of horizontal anisotropy. Defining total velocity as the sum of mean (upper case) and fluctuating (prime) parts and using subscripts from 1 to 3 to denote velocities in the wind coordinates of section 2, the LSC archetype is characterized by strong downwind-mean velocity , with a much smaller (predominantly tidal) crosswind velocity (Fig. 3a). The turbulent part of the flow is also characterized by strong horizontal anisotropy. Figure 3b illustrates a typical near-bottom downwind maximum in . Crosswind variance near the bottom, increasing to similar magnitude only in the middle of the water column. Vertical velocity variance typically reaches a maximum value near the upper extent of the observationally accessible part of the water column. As seen in Fig. 3c, shear stress is typically large, positive, and relatively constant with depth in the upper two-thirds of the water column, while is approximately zero. The third shear stress is more variable and can change sign within the water column. However, when several LSC records are averaged together, the resulting ensemble average is near zero (GW07).
In contrast, record 161.008 has small mean velocities (Fig. 3d) and turbulent velocity variances (Fig. 3e) that are much weaker than those of LSC (note scale changes). All three shear stresses are near zero and the turbulence is approximately horizontally isotropic throughout the water column. Vertical velocity variance is smaller than horizontal variances.
c. Profile traces in the Lumley triangle
Measurement of profiles of the three-dimensional turbulent velocity field enables calculation of the depth map of Lumley invariants [Lumley 1978; Simonsen and Krogstad 2005; the form shown in Fig. 4 is that of Pope (2000)]. As seen in TMG07, stress-driven turbulent boundary layers have a depth trajectory that begins (at the wall, where = 0) along the upper-curved boundary, moves first toward the upper-right corner, then down along the right-hand boundary of the triangle toward the bottom “isotropic” vertex as distance from the wall increases, returning toward the upper-right-hand vertex as the free surface is approached. When Langmuir forcing is included, LES (TMG07) instead show the characteristic “C shaped” near-bottom trajectory seen in the observational LSC record in Fig. 4a.
The CVN record is characterized by a very different Lumley map (Fig. 4b). The trajectory begins well away from the upper two-dimensional boundary, indicating persistence (relative to stress or Langmuir-dominated flows) of significant vertical velocity variance near the bottom, and subsequently remains within the interior of the triangle. Although the trace “jumps” from one side to the other of the central axis of the triangle as a result of noise in the weak shear stresses, there is a left-hand () bias consistent with turbulence that has nearly equal horizontal velocity variances that are larger than vertical velocity variance.
All of the characteristics described above for record 043.024 are consistent with identification as full-depth Langmuir circulations (as detailed further in the coupled observational and computational studies of GW07 and TMG07). Identification of record 161.008 as an archetype for convective dominance (i.e., CVN) rests on a number of features. First, the large-eddy structures are horizontally isotropic, as in the convective atmospheric boundary layer. Vertical velocity variance that is smaller than horizontal, unlike the atmospheric case where it is of the same order of magnitude, Lenschow et al. (1980), is nevertheless consistent with the continuity relationship if the vertical length scale becomes constrained by water column depth while the horizontal scale remains determined dynamically. Similar features were obtained in a finite-depth convective LES (Zikanov et al. 2002), where turbulent vertical velocity variance approached a limit while horizontal variances continued to grow to a maximum set by the horizontal scale of the simulation. Second, Fig. 5 shows the relationship between and the ratio , where with von Kármán’s constant = 0.41 is the Monin–Obukov length scale used to characterize the importance of buoyancy forcing in the atmospheric boundary layer. The value of = ~5 deemed sufficient to drive the atmospheric boundary layer to the convective state (Holtslag and Nieuwstadt 1986) corresponds to a value of . Records 161.006–161.009 lie well above this threshold, consistent with an assumption of convective dominance, while LSC records 043.024–043.030 lie well below the threshold, consistent with irrelevance of convection for . Finally, as will be shown in section 6, vertical velocity variances of qualified session 161 records scale with convective parameters as does convective atmospheric turbulence.
The horizontal scale of turbulent structures in each case can be estimated using known features, wavelet analysis, and measurements of an appropriate advection velocity to predict apparent periods as a function of time. GW07 showed that apparent period in the LSC event of Fig. 6 was well predicted by an assumption of structures with large downwind scale and fixed crosswind scale, advected by mean crosswind velocity. Crosswind scale thus predicted was ~(4–6)H, in rough agreement with horizontal scales of found in a theoretical study [Cox and Leibovich (1993) for Langmuir forcing in a nearly unstratified, depth-limited system], and from a laboratory study (Faller and Caponi 1978) with waves long compared to fluid depth. A similar calculation for the CVN cases of Fig. 7 (using total mean velocity, due to the horizontal isotropy of these structures) suggests horizontal scale of ~4H.
5. Identification of dominant forcing
In this section, forcing regimes giving rise to the two archetypes defined above are described using a suite of derived variables. Turbulence in these fundamentally different regimes is then shown to occupy distinctive regions in the space of and .
a. Characteristics of forcing fields associated with LSC and CVN archetypes
Figure 6 displays a suite of parameters derived from record-averaged forcing fields for session 043. Series are plotted versus record number to simplify reference to specific records, such as the LSC archetype 043.024 that appears within a group of records 043.021–043.034 that all match its descriptive features. Surface heat flux (Fig. 6c) illustrates diurnal variation between destabilizing heat loss (Q > 0) at night and stabilizing gain during the day; characteristically, daytime heat gains are minimal during the storm event marked by arrows, presumably the result of cloud cover. Filled circles denote qualified records used in subsequent analyses. Figure 6d shows various velocity scales plotted on a (discontinuous) common scale. Bottom stress velocity
is calculated as (Sternberg 1968; Sherwood et al. 2006) with representing the mean speed from the lowest VADCP bin; and are as defined in section 1 (see also Table 1). Although there is some variation, , , and are of the same order of magnitude throughout the session. In contrast, the surface Stokes velocity begins with similar magnitude but rises precipitously in record 043.013 with onset of the wind event, attains a maximum (0.13 m s−1 in record 043.022) that is an order of magnitude larger than , , and , and returns to comparable values only near the end of the session. The LC potential growth rate (Fig. 6e) rises with onset of the wind/wave event, decreasing relatively abruptly in record 043.033. The water column becomes effectively unstratified by record 043.020 (Fig. 6f) and remains so through the remainder of the session. Values of (Fig. 6g) are near zero during the storm event, but rise to exceed ~5 in record 043.035. The LSC archetype 043.024 is associated with , large , near-zero , and wave age (not shown) characteristic of fully developed seas.
The session containing the convective archetype 161.008 has very different forcing characteristics (Fig. 7). Although has a magnitude comparable to that in session 043, is mostly much smaller as a result of both weak wind speeds and short fetch (wind directions were weak and variable during this session, but usually had a small offshore component). The CVN archetype (161.008) is associated with , small LC growth rate, and > 5. During these quiet conditions, backscatter (Fig. 7b) reveals a biological diurnal migration signal. However estimated vertical velocities are less than the noise level of the VADCP vertical velocity measurement and will not affect results.
b. Location of archetypes in the plane
Figure 8 illustrates how the space of the nondimensional parameters and defined in section 2 clearly separates the two archetypes (solid dots) and records with similar characteristics, differentiating between classes of turbulent flows dominated by Langmuir and convective forcing. All qualified LSC records of session 043 (043.024–043.030) lie within the circle (labeled A) at the lowest values of and , while qualified records of session 161 with characteristics similar to the 161.008 CVN archetype lie within the circle (labeled E) at the highest values of both nondimensional parameters.
The remaining qualified records in each session lie at intermediate values, occupying a continuum between LSC and CVN end members in terms of both location in the plane and the large-eddy properties described in section 4. These records will be referred to as “transitional” and, because of space limitations, their features will be described but not documented (visualizations of all LEO15 records can be found at http://www.ccpo.odu.edu/TOO/cgi-bin/processedplot.cgi, while full details of the supercell event of session 043, 17–18 May 2003 will appear as a separate paper). In session 043, the end of the supercell event occurs within record 043.035, accompanied by falling magnitude of and the first occurrence of after the storm peak. Event end is marked by cessation of full-depth backscatter and return of large mean shear between a bottom layer in which mean current changes in both magnitude and direction and a surface layer that retains the weakly sheared, largely downwind, mean velocities typical of LSC (Fig. 3a). In records 043.035–043.037, surface-origin backscatter clouds still exist but are restricted to this surface layer. At this time, horizontal scales implied by the apparent periodicities in vertical velocities are too small for resolution of horizontal velocity components and, hence, for computation of the diagnostics of section 4. During a subsequent short period of marginally increased Stokes growth rate (Fig. 6d), horizontal spatial scales again increase and records 043.038–043.039 exhibit some characteristics of LSC (full-depth backscatter, predominantly downwind flow over the entire water column, near-bed downwind variance exceeding crosswind variance and significantly positive ). However phase relationships among the three velocity components are not as clear as they are for LSC [see Fig. 2, Gargett et al. (2004)] and Lumley traces start within the triangle interior, suggesting an influence of convection. In Fig. 5, records 043.038–043.039 lie in the vicinity of the threshold, consistent with the mixed large-eddy characteristics they display. Records 43.045–043.049 (in circle C) near the end of session 043 have still larger values of and characteristics tending to those of the marginally convective records 161.004–161.006 (in circle D). These latter have some “convective” properties (near-zero shear stresses, small normal stresses and mean velocities, and Lumley traces within the triangle interior), but horizontal variances differ by up to a factor of 2 in some parts of the water column, unlike the equal horizontal variances of the CVN archetype.
c. The complete dataset in the plane
Figure 9 shows the distribution of 170 qualified records in the plane; color codes the magnitude of for each record. As described previously, additional records falling near the LSC and CVN archetypes exhibit similar characteristics, while those lying between them have mixed characteristics. It is clear that LSC are associated with larger values of than convection. Another striking feature of Fig. 9 is the absence of observational data points within the lower-right-hand region of small and where, if anywhere, one might expect to find cases of pure stress forcing. Similar results were obtained by Li et al. (2005), who plotted stress-normalized vertical velocity variance from LES run under different combinations of forcings in a plane of , where and is broadly related to . However, their results are not directly comparable because of their assumption of fixed Stokes e-folding depth and the fact that is a ratio of buoyancy to Langmuir terms rather than buoyancy to inertial terms.
Also of note in Fig. 9 is the relative paucity of points near the CVN archetype, relative to those near the LSC archetype. Dependence of on implies that in addition to large , must be also be large (i.e., LC potential growth rate small) for convective dominance. Sufficiently small happens only rarely at LEO15 during the unstratified conditions selected, which frequently coincide with strong wind/wave forcing.
6. Scaling of turbulent vertical velocity variance
We have used the parameters La and Ra that result from the scaling of Eq. (14) to identify records that should be dominated by a single type of turbulence forcing. Once that identification has been made, one expects that velocity and length scales will be those appropriate to the single process thus identified. We now consider whether such a single process scaling velocity indeed provides acceptable normalization of observed vertical velocity variance in those cases identified as dominated by either convection or Langmuir forcings (in the absence of any identified as dominated by stress forcing).
and (Fig. 10c) convective velocity scale . The form chosen for (among many possible combinations of and ) is that suggested by both theoretical (Smith 1996) and numerical-modeling (Min and Noh 2004; Grant and Belcher 2009; Harcourt and D’Asaro 2008) studies. It is chosen because of underlying assumptions that accord with observed properties of LSC. The Smith (1996, p. 619) derivation of requires that “a constant fraction of the total momentum transport through the mixed layer is accounted for by the circulation (for example, 100%).” An alternate derivation by Grant and Belcher (2009) requires that in the stress-shear source term for TKE, turbulent stress scales as . Observational support for these hypotheses comes from GW07, who report that directly measured in-water stress associated with LSC is roughly constant with depth and roughly equal to the imposed surface stress, that is, when the turbulent large eddies are demonstrably LC, they indeed carry nearly 100% of the surface wind stress.
A successful scaling is one that produces constant scaled variance with minimal scatter. The color-coded results of Fig. 10, where values of scaled variance of O(1) are blue, suggest that convective scaling with is indeed the most appropriate for records with large values of and , while the smaller values of and that characterize LSC are more successfully scaled by than by alone. The former conclusion is reinforced in Table 2, where has means of 0.2 and 0.28 for records within the smaller and larger boxes in Fig. 10c, respectively (the smaller box requires records with “pure” CVN properties, the larger box relaxes these requirements somewhat), values near that (0.3) found for the convective atmosphere (and post hoc justification for definition of these records in section 4 as convectively dominated). Moreover the standard deviation associated with convective scaling is minimal among the three scalings for these records.
For records within the lower set of boxes shown in Figs. 10a and 10b, the conclusion is less clear. Scalings with both and produce approximately constant scaled variances with comparable standard errors, even when records are restricted to those (small box, LC Region) with “pure” LSC characteristics. In this region of forcing space, is functionally as well normalized by , a parameter relating solely to the wind field, as it is by the more complex scaling velocity involving a wave parameter, despite the evidence presented here that the turbulent large eddies are demonstrably Langmuir circulations, hence mechanistically connected to a wave state. This result is consistent with demonstrations by D’Asaro (2001) and Tseng and D’Asaro (2004) that vertical velocity variance from mixed-layer Lagrangian floats under moderate wind forcing can be successfully scaled by alone. It is also consistent with a linear relationship (Fig. 11a) observed between and in our LSC records. The visual fit
exhibits a threshold below which LSC are not observed (30 qualified records have mean wind stress below this level, so the threshold is unlikely an artifact). The threshold value of = ~0.002 m s−1 is associated with = ~2 m s−1, in reasonable agreement with an anecdotal value of 3 m s−1 necessary for generation of LC [Pollard (1976) and references therein]. The slope agrees with that found in the Lake Ontario measurements of Kitaigorodskii et al. (1983) and with the relationship = ~ derived from the PM spectrum for fully developed seas by Li and Garrett (1993). However as seen in Fig. 11b, LSC records form an indistinguishable subset of the larger set of records in the large box of Figs. 10a and 10b that includes both growing and decaying seas. Thus, the observed relationship applies to more than fully developed seas.
When a linear relationship like (21) exists, scaling with (or any other dimensionally correct combination of and , such as suggested by Plueddemann et al. 1996) cannot be functionally distinguished from scaling with . Can the difference in scaled magnitudes documented in Table 2 be considered evidence of the significance of rather than ? Atmospheric boundary layer observations have long established that in a stress-driven (neutral) boundary layer, vertical velocity variance , where C = 1.4–1.7 (Panofsky 1973). The observed value of C = ~3 (Table 2, LC Region) might thus imply velocities significantly in excess of stress driven. However while present results are suggestive, error bounds (~50%) are sufficiently large that C = 1.4–1.7 cannot be ruled out. The question of the relevance of , independent of , to scaling of wind-/wave-driven turbulence must await observations of LC in which a linear relationship like Eq. (22) does not hold, or in which error bounds are sufficiently reduced.
Note that implies that is approximately constant. Records in the “Langmuir forcing” regime (small and large boxes in Figs. 10a,b; where convection is not a player) indeed occur over a limited range of , or , that is, values of are absent as well as values >1.
Before concluding, it is necessary to justify the assumption that bottom stress forcing of turbulence at LEO15 is unimportant relative to those forcings retained. Consider a pressure gradient–driven flow , such as that due to the tides, taken for simplicity to have the same direction as the mean stress-driven flow considered until now. Because stress in the water column at equals the applied surface stress, which is zero for a pressure gradient–driven flow, as . Near-surface mean shear is thus unchanged by addition of a pressure gradient–driven flow, hence time scale and are also unchanged. If the velocity scale used to nondimensionalize the momentum equation were changed from to the bottom stress velocity defined earlier, the only change would be that the Langmuir-forcing term would be multiplied by the reciprocal of a nondimensional parameter
describing the importance of inertial forces associated with turbulence driven by a pressure gradient–driven flow relative to Langmuir forces. In the absence of pressure gradient–driven flow, because the magnitude of bottom stress associated with surface stress forcing alone equals that of the surface stress. On the other hand, if turbulence driven by a pressure gradient–driven flow were much larger than that driven by wind stress, and the Langmuir-forcing term which is multiplied by the reciprocal of would tend to zero, as required.
All qualified data are presented in the three-dimensional space of in Fig. 12a. A base plane of , where buoyancy forcing is certainly unimportant, is shaded to suggested process dominance, assuming for now that values of both and roughly equal to unity separate regions of dominance by Langmuir forcing and the relevant stress forcing. Most of the records fall in the “Langmuir” region, as seen more clearly in Fig. 12b where points are projected onto the base plane. In particular, LSC records (the dense cluster circled in Fig. 12b) lie only very slightly above the line = ~ expected for a stress-driven flow, with or without Langmuir forcing (TMG07), providing strong evidence that pressure gradient–induced bottom stress is indeed negligible for these cases. Transitional and/or CVN records at larger are only affected by “extra” bottom boundary layer turbulence, that is, that associated with an additional pressure gradient–driven flow, if they lie significantly above the line that defines surface stress–driven flow. With present data, “significantly” cannot be defined with any precision. Some of the records lying above the dashed line shown in Fig. 12b exhibit characteristics that might be expected if bottom boundary layer turbulence is prominent: bottom-intensified backscatter, dominance of (when the axis is taken in the direction of the mean current, not the wind), and Lumley trace confined to the upper-right-hand boundary of the triangle. However, these features are not all consistently present. Moreover, given the smaller spatial scales of boundary-confined turbulence, quantitative measures may be affected by limits on both the vertical and horizontal resolution of the observations. Despite these caveats, it appears that a condition for the necessity of including bottom boundary layer turbulence associated with pressure gradient–driven flow may be of the form , where , rather than the condition that was initially assumed. For the present observations, n = ~2.
A second, implicit, assumption made in the present analysis is that the observed turbulence is steady state, that is, adjusts instantly to changes in forcings (instantly is defined here as within the record period of ~2.2 h over which both are averaged). It is widely accepted that turbulence adjusts to forcing change within one to a few large-eddy overturning time scales; for full-depth turbulent structures, this time scale = ~ where is a characteristic turbulent vertical velocity. For LSC, = ~, hence a typical = ~0.01 m s−1 for these events yields = ~0.14 h. Because the 2.2-h record length contains many such periods, the steady-state assumption is well supported. For CVN, = ~: with typical = ~0.01 m s−1, = ~1.5 h and the averaging period is marginal. Not surprisingly then, it is in situations dominated by convection that suggestions of time dependence are observed, noticeably when convective turbulence persists for a time after the surface heat flux switches sign after sunrise (not shown). Assuming that growth occurs with similar time scales, CVN records might thus best be associated with a previous, rather than simultaneous, averaging period of forcing. Because the CVN archetype occurs during near constant (low) wind stress and surface heat flux, this will not change the present results significantly, but should be considered in future studies, particularly with detailed observations of convection near changes in sign of surface heat flux, or of LC associated with moderate wind forcing, where large-eddy turnover times may be larger than those estimated above.
Using an extensive set of data characterizing the large eddies of full-depth turbulence under a wide range of surface-forcing conditions, it has been demonstrated that situations dominated by either Langmuir vortex forcing or destabilizing surface heat flux can be identified by distinct locations in the plane of , where and are Langmuir and Rayleigh numbers, respectively, derived from a scaling that uses and characteristic of the growth of Langmuir circulation. Situations in which neither process dominates, including those in which turbulence is transitioning from domination by one to the other, lie between the two end members, with relative dominance given by proximity to one or the other. Identification of process domination via the parameters defined here requires knowledge of only meteorological-forcing fields and the surface wave frequency spectrum (as well as some indication of water column stratification). If the resulting parameters lie within either the “Langmuir region” or the “convection region” defined here, many turbulence characteristics can be considered as reliably known without direct in-water measurements.
The above conclusions hold only if bottom boundary layer turbulence associated with pressure gradient forcing like tides is negligible. A third nondimensional number has been suggested as a metric for the relative strength of pressure gradient– to Langmuir-forced turbulence. Based on this metric, Langmuir forcing is expected to dominate tidal forcing at the measurement site for all but a small fraction of the present data.
For values of there are no records with . Although computations can define its properties, the purely stress-driven state that might be expected at values of when convective forcing is minimal is apparently not frequent in the surface ocean, presumably because imposition of surface stress is quickly accompanied by generation of surface waves and thus, of wave-induced Langmuir vortex forces.
Examination of possible normalizations of turbulent vertical velocity variance shows that data from the “convective region” are well normalized by the velocity scale associated with atmospheric convection (Table 2). Records in the “Langmuir region” are found to have similar values of standard deviation when normalized by surface stress velocity or by a Langmuir velocity (Table 2), the consequence of a linear relationship observed between and for such records (Fig. 11). The observed threshold value is roughly consistent with an anecdotal wind speed threshold for generation of LC, while the slope is roughly consistent with that predicted from the PM model spectrum for fully developed waves, even though the present data includes cases that are not well described by this model. The physics underlying the observed linear relationship between and during supercell events remains undetermined.
Similar observations in locations with different depths, tidal magnitudes, et cetera, are required to discover whether the present results at an inner-shelf location characterized by weak tides are applicable to other depth-limited systems. Preliminary results from a deeper (26 m) site with stronger tides (Savidge et al. 2008) show a similar distribution of data in space, but less frequent occurrence of “pure” LSC features (D. Savidge 2013, personal communication).
Another question is whether results obtained in a depth-limited system are applicable to the deep-water ocean surface layer. In a depth-limited system, coherent full-depth LC (LSC) are obtained both in steadily forced LES (TMG07) and in observations (GW07). LES of this case show that steady state is achieved rapidly because a depth-limited stress-driven flow contains large scale coherent but weak structures called Couette cells (Papavassiliou and Hanratty 1997; TMG07) that transition quickly to larger, stronger and highly coherent Langmuir structures upon imposition of Langmuir vortex forcing.
In deep-water LES (Skyllingstad and Denbo 1995; McWilliams et al. 1997; Tejada-Martínez et al. 2009), surface wind stress is balanced primarily by the Coriolis force and “mean” and turbulent velocities achieve a statistical steady state only when averaged over several inertial periods (Tejada-Martínez et al. 2009). Moreover while deep-water LES typically show early development of distinctly structured small-scale LC, under steady forcing these eventually transition to less structured “Langmuir turbulence” while increasing in horizontal and (to a lesser extent) vertical scale (Skyllingstad and Denbo 1995; McWilliams et al. 1997; Tejada-Martínez et al. 2009).
Tejada-Martínez et al. (2009) suggest that the differences in both coherence and developmental time scales between deep- and shallow-water LES result from the different base structures present in the underlying stress-driven flows at the time Langmuir forcing is turned on. However, differences in boundary conditions may also contribute. Using the terminology of Pope (2000), the free-surface boundary condition is near wall resolved for the depth-limited case, but near wall modeled for the deep-water case, resulting in major differences in the role played by subgrid stress, hence potentially in characteristics of the computed turbulence. At the bottom boundary, turbulence in the depth-limited case is bounded by a solid, stress-supporting surface for which a well-established modeling framework exists. In the deep-water case, it interacts with a stratified interior below the mixing layer base, an interaction parameterized in a variety of ways. Skyllingstad and Denbo (1995), McWilliams et al. (1997), and Li et al. (2005) impose an internal wave radiation condition at the top of the stratified region, while Tejada-Martínez et al. (2009) set normal velocity to zero and tangential stresses to zero at the depth of the Ekman layer, preventing internal waves from reflecting back into the domain by a Rayleigh-damping term at the simulation base. However the actual stress at the bottom of deep ocean surface boundary layers, involving interaction with an existing internal wave field as well as local generation of internal waves, is essentially unknown. We conclude that the present results should not be applied to the deep-water case without further modeling and observational studies.
Stress exerted on the ocean is dominated by high events. The present data show that during such events, the energy-containing eddies of turbulence in an unstratified surface layer of depth 15 m are full-depth Langmuir circulations. The analysis presented here strongly suggests that at least in shallow coastal oceans, stress transmission is integrally related, through generation of Langmuir turbulence, to the presence of the surface waves that inevitably accompany the imposition of surface stress on the real ocean.
Research support from the National Science Foundation (OCE0136403 and OCE0927724) and NOAA (NA06RU0139) is gratefully acknowledged. The observations owe much to the skills of Christopher Powell and Shuang Huang. We gratefully acknowledge the substantial contributions made by the reviewers of this paper and those of an earlier version.
Stokes Velocity and Stokes Shear Functions
For a single inviscid, incompressible wave with amplitude , wavenumber , frequency , dispersion relationship , where is water column depth, and surface displacement , displacement variance is related to the variance of vertical velocity measured at height above bottom h by
In a multiwave case, it will be assumed that the surface wave spectrum depends only on frequency (wavenumber). Storm wind conditions at LEO15 usually involve winds from the NE, more rarely from the SE; in both conditions, locally generated seas are superimposed on swell propagating roughly westward toward the nearby shore, producing roughly symmetrical directional spectra. Although it can be argued that the geometric aspect of a directional surface wave spectrum decreases the downwind Stokes drift velocity, other mechanisms that could potentially increase it (Craik and Leibovich 1976; Garrett 1976) remain observationally unproven. Until uncertainties about the net effect of a directional spectrum are resolved, a unidirectional assumption is the most straightforward. This assumption also provides results more easily compared with LES driven by single-wave characteristics, results usefully compared with observations in previous identification of full-depth Langmuir circulations.
It is further assumed that can be converted to an operator, so that , the vertical velocity spectrum (corrected for noise by removal of an average white noise level determined from a number of low wind/wave records) is related to that of surface displacement by
Figure A1 shows as a function of cyclic frequency . Because the time-continuous record of required for the vertical velocity spectrum is available only below the minimum surface displacement within a record period, is smaller (i.e., the distance of the measurement bin from the surface is larger) when surface waves are larger, with major effects on measurements at both low and high frequencies. Figure A1a uses values of and for a record (043.024) during high waves, when data used are at . Figure A1b is for a record (161.008) taken when waves are small, hence data are at . In both cases, the response function rises steeply at low frequencies because of the small vertical velocities associated with low-frequency waves. In Fig. A1a, the response also rises steeply at frequencies above ~0.4 cps, as increasingly short vertical e-folding scales reduce the associated vertical velocity present at the necessarily deeper instrument depth. The magnitude of response corrections at low and high frequencies means that any underestimation of the noise level removed from the vertical velocity spectrum could potentially lead to large erroneous values in both regions, a point to which we return below.
Figure A2a presents results of Eq. (A2) for several records chosen to span the range of observed sea state, from strongly wind-forced states associated with LSC (top) to nearly calm conditions associated with CVN (bottom); Table A1 provides values of significant forcing parameters. In all panels, spectral values derived from vertical velocity spectral estimates below the average noise level are set to zero. As seen in the top two panels of Fig. A2a, the assumption involved in Eq. (A2) is verified observationally. Provided wave amplitude is significantly larger than the 0.4-m Doppler bin size, agrees well with , a displacement spectrum calculated independently from simultaneous measurement of surface displacement available by tracking the surface in backscatter amplitude of the fifth beam. This agreement at large wave amplitudes provides confidence in Eq. (A2) at lower wave amplitudes (bottom two panels) when the independent surface-tracked measure is not resolved because wave amplitude becomes comparable to VADCP bin size.
For a single wave, Stokes drift velocity is given by
(Phillips 1977). For a spectrum of waves, it has been assumed since Kenyon (1969) that what we will call a spectral Stokes drift velocity may be obtained by converting in Eq. (A3) to an operator, an assumption similar to that made in deriving displacement spectra from vertical velocity spectra, but lacking the independent verification available in that case. The usage is nonetheless widespread and is continued here. Thus, spectral Stokes drift velocity is given by
where the Stokes response function . For , this response function rises with frequency from zero at zero frequency, as seen in Fig. A1. The integrand will be termed the Stokes function, rather than the Stokes spectrum, because its integral is the Stokes velocity, not Stokes velocity variance. Surface Stokes velocity is given by the integral over the surface Stokes function .
The deep-water limit () of Eq. (A4), denoted by superscript , is
in agreement with previous results (Kenyon 1969; McWilliams and Restrepo 1999; Webb and Fox-Kemper 2011). For , the integrand in Eq. (A5) varies as for the high-frequency form of Phillips (1958), see also Pierson and Moskowitz (1964). For the more recently accepted (Alves et al. 2003) form of (Toba 1973; Donelan et al. 1985), the integrand varies as . Because high-frequency waves are all effectively in deep water, the deep-water limit describes the high-wavenumber limit of , hence is unbounded for the latter form. In fact, observations upon which modeled deep-water displacement spectra are based extend only to = ~0.4, where is displacement peak frequency, hence it must be assumed that the spectral shape eventually becomes steep enough to provide convergence. While various choices for high-wavenumber extensions have been made in studies using modeled wave spectra [e.g., Harcourt and D’Asaro (2008) assume a transition from to in the vicinity of ], observed surface Stokes functions vary considerably, as seen in Fig. A2b. Above the displacement peak, sometimes varies as (043.024), sometimes as (152.001), while at other times resembles neither.
However Stokes velocities of high-frequency (wavenumber) waves drop off rapidly with depth, hence are unlikely to contribute significantly to the generation of full-depth LC. We thus assume that a determinate Stokes velocity appropriate for assessing generation of the large-scale structures considered here can be calculated for arbitrary Stokes functions by low-pass filtering, either explicitly by band limiting the surface integral or implicitly either by evaluating Eq. (A4) at a small depth below the surface or by averaging Eq. (A4) from some small depth to the surface. Because it is desirable to use a surface Stokes velocity to facilitate numerical comparisons with most previous (single wave) observational and computational studies, the former option has been chosen. The value of used in analysis (see Table 1) is the integral of Eq. (A4) for , over a standard-frequency range determined by considering possible contamination of results arising through the extreme response corrections at low and high frequency shown in Fig. A1. Obvious in the higher-wave cases (upper two panels) in Fig. A2a is the rapid rise at low frequencies that would be expected from the composite response if the vertical velocity noise level determined during low-wave conditions underestimated the actual noise level under these conditions. The associated rise expected at high frequencies is not obvious in the displacement spectra because of their dynamic range, but becomes visible with the smaller dynamic range of the Stokes functions in Fig. A2b. The standard integration range chosen avoids both of these observational artifacts. The missing contribution due to very low frequencies is unimportant because it is associated with large e-folding depths, small Stokes shears, and small potential growth rates. While we assumed above that the missing contribution from high frequencies is unlikely to be important for generation of the large-scale LC considered here, the contribution to Langmuir forcing by very small–scale waves is presently not well determined by either theory or observation. However any potential contribution should scale with the band-limited value, because small-scale waves adjust rapidly to changes in forcing. It is encouraging that as we calculate it proves to be well correlated with other proposed “Stokes velocities” (corresponding to different effective low-pass filters), calculated at a (small) fixed depth (Gnanadesikan 1996) or formed as an average from some (small) fixed depth to the surface (Harcourt and D’Asaro 2008). Figure A3 illustrates these correlations for ; results for different choices of depth are similarly linear, with different slopes. Either of these values could thus be used in place of in nondimensional parameters that incorporate Stokes velocity, the result being only a multiplicative change in magnitude.
The standard integration range described above is modified only in very low wind/wave conditions, when the Stokes function may consist of two peaks, a lower due to swell and a higher associated with locally generated seas, separated by a range of frequencies where = 0 (e.g., record 161.008 in Fig. A2b). In such situations, the few at LEO15 in which different frequency waves may be traveling in substantially different directions, Stokes velocity and Stokes shear (see below) associated with the low swell frequencies are smaller than those associated with the high-frequency wind seas, which are thus expected to dominate LC formation. Taking the lower bound of the Stokes surface integral as the frequency of the lowest nonzero value of Stokes function below its peak (filled circle on the frequency axis of the bottom panel of Fig. A2b) removes the swell contribution, hence effectively results in wind-aligned Stokes velocity and shear in such cases. Finally, we reject records (such as 043–011 in Fig. A2) in which the peak of the Stokes function is not reached before the upper limit () of the standard integration range.
The Stokes shear function associated with Eq. (A4) is given by
In the deep-water/high-frequency limit
hence the surface shear function, for which the ratio in square brackets equals 1, exhibits an even steeper rise at high frequencies as result of either high-wavenumber form of modeled displacement spectra than does the surface Stokes function. However the ratio in square brackets has a limit of 0 as for nonzero , hence the Stokes shear function at any nonzero depth is (eventually) bounded. A bounded value for Stokes shear can thus be computed either as a frequency integral (denoted by curly brackets and taken over the same frequency range used for calculation of ) of Eq. (A6) at a (sufficiently deep) fixed depth or as a vertical integral from some fixed depth to the surface. As seen in Fig. A2c (and Fig. 1) for the steepest surface spectral rise observed (, seen in record 152.001), is the shallowest depth at which Stokes shear is bounded, in the sense of decreasing at high frequency. As shown in Fig. A4, , the frequency integral of Stokes shear at this depth is strongly correlated with an alternate choice of , the vertical integral from to the surface, hence either could be used in defining a characteristic Stokes shear: we have chosen .