Predicting Turbulent Vertical Velocity in the Ocean Surface Layer under Mixed Convective and Wind/Wave Forcing

1. Introduction

Turbulence in the surface layer of the open ocean is forced by continuously changing conditions of surface buoyancy flux B_o and wind stress τ. Surface buoyancy forcing usually has strong diurnal variation: buoyancy flux that is stabilizing to the ocean during daylight hours is replaced by destabilizing convective forcing overnight. This basic diurnal cycle can also be modulated on longer time scales, associated with passage of weather systems, that are characteristic of wind stress forcing that drives both mean currents and waves. The challenge addressed here is prediction of the magnitude of mixed layer averaged root-mean-square turbulent vertical velocity rmsw as a function of mixed convective and wind/wave surface forcing conditions. Apart from its role as a key metric for the strength of turbulent motions in the ocean surface layer (OSL), the magnitude of rmsw is a major determinant of the vertical transport of particles such as bubbles, detritus, and plankton. Since such particles affect processes as diverse as air–sea transfers of major gases, phytoplankton productivity, and the export of particulate organic material from the near-surface ocean, a robust predictor of rmsw would have broad application.

Much of our present understanding of OSL turbulence forced by wind and waves is based on computer simulations using large-eddy simulation (LES) techniques. McWilliams et al. (1997) coined the term Langmuir turbulence (LT) to describe turbulence produced by coupled wind/wave forcing, in which vertical vorticity associated with a stress-driven boundary layer is rotated to horizontal by the Stokes drift associated with the surface wave field. This literature, now massive [Li et al. (2004), Tejada-Martínez and Grosch (2007), and Kukulka et al. (2009) are but a few examples], has provided two somewhat different predictions for vertical velocity and vertical length scales characteristic of LT in a well-mixed OSL of depth H. In an initial LES study of an unstratified water column, Polton and Belcher (2007) concluded that the vertical length scale of LT was the mixed layer depth H, provided H was smaller than the Ekman depth ${\delta }_{\text{Ek}}=u_{*}/f$. A subsequent study (Grant and Belcher 2009) found that the turbulent vertical velocity scaled as $w_{*L}={(u_{*}^2{U}_{S0})}^{1/3}$ where U_S0 is the surface Stokes drift velocity. A second set of LES led Harcourt and D’Asaro (2008) to conclude that the vertical depth scale of LT was influenced by the e-folding depth of the surface wave field, and that the vertical velocity scale for eddies of O(H) in the LT limit was described by the form proposed by Grant and Belcher (2009), but computed using a surface-layer Stokes velocity, defined as $U_{\text{SL}}={⟨u^S⟩}_{\text{SL}}-\hspace{0.17em}{u}_{\text{ref}}^S$, in place of U_S0. Here 〈u^S〉_SL is the vertical average of the Stokes velocity from a depth of 0.2H to the surface and $u_{\text{ref}}^S$ is a deep reference Stokes velocity evaluated near H.

Two fundamental issues arise with use of the results of many LES studies focused on LT in the OSL. The first issue is just that focus, i.e., many studies focus on LT alone, regardless of the presence of buoyancy forcing of either sign (e.g., Sullivan et al. 2004; Noh et al. 2004; Harcourt and D’Asaro 2008; Grant and Belcher 2009; Kukulka and Harcourt 2017). The second is that many LES studies of LT over the past two decades report results at steady state, after a minimum of one (McWilliams et al. 1997) to several (Yan et al. 2022) inertial periods, a time extent between ∼1 and several days at midlatitudes. In the actual OSL, steady state over such extended periods is an exception rather than the rule, if for no other reason than the diurnal cycle of the buoyancy forcing that is inevitably present to some extent in conjunction with wind/wave forcing. While some LES studies include mixed, non-steady-state forcing (e.g., Skyllingstad and Denbo 1995; Sullivan et al. 2012; Li et al. 2013), the specific question of predicting OSL rmsw over a typical range of time-varying, mixed surface forcings has not been a focus: it thus seems appropriate to examine direct observational evidence that can be brought to bear on this question.

2. Available data and assumed functional form

A preliminary attempt to derive a predictor for rmsw was made by Gargett (2022, hereafter G22), using direct observations of vertical velocity w from the vertical acoustic beam of a vertical-beam ADCP (VADCP) deployed on a bottom node of the Long-Term Ecological Observatory (LEO) coastal observatory, located ∼6 km off the shore of New Jersey in water of depth H = 15 m, and cabled directly to shore. That analysis involved various assumptions that are either further explored or replaced in this expanded treatment. Specifically, the question of how best to calculate a characteristic (near) surface Stokes velocity for use in various proposed wind/wave scale velocities is explored in section 3, while the assumption made by G22 that extreme locations in forcing space represent “pure” cases dominated by either convection or wind/wave (Langmuir) forcing is removed. The present analysis also expands to include similar direct observations of w from a coastal location much farther from shore (Savidge and Gargett 2017), where a VADCP was deployed on a node cabled to a naval tower (R2) located ∼65 km off the coast of Georgia in water of depth H = 26 m: data were returned to shore by microwave link. At both locations, time-continuous data from a session (typically 4 days at LEO and 7 days at R2) are divided into sequential 2-h records.

Turbulent vertical velocity w is extracted by first low-pass filtering total vertical beam velocity to remove velocities associated with the passage of surface waves (spatial scales remaining after this filtering are much larger than horizontal diameters of the vertical beam spreads at the surface, 0.7 m at LEO and 1.7 m at R2), then removing a time-local (∼2 h) mean from each bin of velocity data. An observational value of $\text{wvar}\equiv \overline{⟨w^2⟩}$ is computed as w² averaged over the time (∼2 h) of a standard record length (angle brackets) and over depth (overbar: because the vertical beam measurement is unaffected by sidelobes, the depth average is to the minimum value over the record length of the surface as identified in vertical beam backscatter amplitude). Correction for observational noise variance $w_n^2$¹ determines the value used for rmsw as $\text{rmsw}\equiv \sqrt{\text{wvar}}=\sqrt{\overline{⟨w^2⟩}-w_n^2}$. As defined, rmsw becomes zero when $\overline{⟨w_{}^2⟩}$ falls to its noise level.

Associated surface forcings are derived from direct tower-based observations of wind speed and of the terms involved in estimating surface buoyancy flux B_o (defined as positive when destabilizing to the ocean) from net heat flux Q. Full descriptions of the observational sites, derivation of τ and B_o, Doppler instruments and their deployments, the vertical beam measurement of vertical velocity and its subsequent processing are found in Gargett and Wells (2007) for LEO and Savidge and Gargett (2017) for R2.

Data used from both locations have two major qualifications. First is restriction to times when the water column is “unstratified” over water column depth H. Since salinity is not available at either location, stratification $\mathrm{\Delta }\rho /\mathrm{\Delta }z\sim \alpha \mathrm{\Delta }T/H$ is defined in terms of ΔT, the change in temperature over H, available at both sites from nearby thermistor chain moorings. The value of ΔT < 0.5°C used to define “unstratified” at LEO is essentially the same at R2, due to nearly compensating changes in H and α. Second, although not a problem with the weak tides typical of LEO, stronger tides at R2 raise the possibility of contamination from periods of low convective and wind/wave forcing but maximum tidal flows, periods characterized by a near-bottom maximum in rmsw. Thus, a second major qualification is that the record(time)-averaged profile of rmsw must have its maximum in the upper half of the water column.

Some assumptions are made in the following analysis. First it is assumed that the dominant waves are aligned with the wind. Although various LES studies have found significant effects of nonalignment (e.g., Van Roekel et al. 2012), observational studies (D’Asaro et al. 2014, online supplemental material; Wang et al. 2022, their Fig. 4a) indicate a strong tendency for wind-driven waves to be at least approximately wind aligned. Given these results, plus a recent conclusion by Fan et al. (2020) that LES solutions in cases with large wind/wave misalignment are sensitive to the subgrid models used and need to be considered with caution, the simplifying assumption of wind-aligned waves is reasonable for an exploration of combined surface forcings. This is particularly the case if, as argued by Csanady (1994), Teixeira and Belcher (2002), and Sullivan et al. (2007), wave breaking supplies a major source of seed vorticity for the Craik–Leibovich (CL2; Leibovich 1983) instability that generates Langmuir circulations (LC), since wave breaking occurs primarily in the downwind direction. Another underlying assumption made here is that in the real world there is only rarely a case of pure wind forcing, if that is defined as wind forcing that results only in a classical stress-driven boundary layer. Imposition of wind stress τ on the ocean surface is accompanied by surface wave generation, hence the presence of forces generating Langmuir circulations in addition to mean flow. Exceptions to this co-occurrence are conditions of short fetch, such as those reported in Wang et al. (2022) and G22 or rapidly changing wind direction (Gargett and Savidge 2020; Wang and Kukulka 2021).

A convective scale velocity $w_{*}\equiv {(B_{o}H)}^{1/3}$, where B_o > 0 is buoyancy flux from the ocean to the atmosphere and H is mixed layer depth, has a long history from the convective atmospheric boundary layer (Holtslag and Nieuwstadt 1986). Here B_o = (gαQ/ρ_oC_υ) is calculated from observed total heat flux Q using g = 9.81 m s⁻² and reference seawater density ρ_o = 1024 kg m⁻³. Values of C_υ, the specific heat of seawater at constant volume, and α, the thermal expansion coefficient of seawater, are those appropriate to average ocean surface temperatures of 10°C at LEO (C_υ = 4.19 × 10³ J kg⁻¹ K⁻¹, α = 1.7 × 10⁻⁴ K⁻¹) and 25°C at R2 (C_υ = 4.00 × 10³ J kg⁻¹ K⁻¹, α = 2.98 × 10⁻⁴ K⁻¹). The term H is water column depth, since only unstratified records are analyzed.

In Eqs. (2b) and (2c), U_S is a characteristic Stokes velocity. While this has most frequently been taken as the surface value, it is desirable to determine a “best” value to use for U_S in (2b) and (2c) before proceeding to examine their use in Eq. (1).

3. Determination of a best value for U_S

The three possibilities considered for U_S are

1. the surface value

   $U_{S0}=U_S\hspace{0.17em}(x_3=0);$
2. the value of U_S at a fixed depth below the surface, here taken as

   $U_{S3}=U_S(x_3=-3\mathrm{m})\text{;\hspace{1em}and}$
3. a near-surface-integral Stokes drift velocity defined as

   $\mathrm{int}{U}_{S3}=\frac{1}{3\mathrm{m}}{\displaystyle {\int }_{x_3=-3\mathrm{m}}^0{U}_S(x_3)d{x}_3},$

   the integral of the Stokes drift velocity from x₃ = −3 m to the surface.

The depth of 3 m is chosen in (2) and (3) primarily because that is the depth at which GG14 found that the Stokes shear critical to LC generation becomes bounded within the range of measured frequency at LEO, hence completely determined by the observations [note that for typical wave fields, the Kenyon (1969) form produces infinite Stokes shear at the surface, as a result of contributions from small-wavelength, high-frequency waves: see discussion in appendix of GG14]. GG14 used Stokes shear at 3-m depth in calculating a value for Leibovich’s (1977) growth rate for Langmuir circulations, subsequently used by G22 to define a forcing space that allows qualitative determination of the mixture of convective and wind/wave forced turbulence from associated surface forcing. The depth of 3 m is also shallow enough that this value of Stokes drift velocity can be considered a characteristic near-surface value, without the disadvantages of U_S0 that will be examined below. While using 3-m depth here retains continuity with previous published results, another near-surface depth could be used without difference in outcome, save for the value of the constant a_S.

Figure 1 cartoons the behavior of the above 3 possibilities for waves of wavelength long (Fig. 1a) and short (Fig. 1b) compared with the depth of 3 m.

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Cartooned profiles of measured Stokes velocity ($U_S^M$, light solid curve), and Stokes velocity corrected for missing variance at frequencies higher than the upper limit of measurements (U_S, heavy solid curve), normalized to 1 at the surface value U_S0 = U_S (x₃ = 0), for cases of dominant waves of length (a) long and (b) short compared to the depth of 3 m. Associated values of U_S3 = U_S (x₃ = −3 m) (solid vertical line) and intU_S3, defined as the integral of U_S from x₃ = −3 m to the surface (dotted vertical line) are also cartooned for each case. In (b), U_S3 is assumed effectively coincident with the vertical axis.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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While the surface value U_S0 has been most widely used as a characteristic Stokes velocity in LES, it has issues that make it less than ideal for use with observations. Foremost is the necessity of correcting $U_{S0}^M$, the value derived from surface wave displacement spectra measured to some upper frequency f_M (usually of order 0.5 Hz), for variance due to missing higher-frequency waves in order to produce a value for U_S0. This correction has been done in various ways, all based on (different) assumed spectral forms of wind-forced waves (e.g., Belcher et al. 2012; Clarke and Van Gorder 2018): the corrected values shown in Fig. 2a use the correction method described in the appendix of G22. While each method that has been defined produces a slightly different result, a more serious concern is the lack of an accepted correction technique when waves are determined not to be actively wind-forced waves [using the ECMWF criterion (Bidlot 2020), as discussed in G22]. Such situations, which include those of swell domination and/or decaying wind waves, are not uncommon in measurements. There is no present basis for correction of such records, leading to the discontinuities in U_S0 seen in Fig. 2a.

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Values of (a) U_S0 and (b) intU_S3 as functions of record number for LEO session 043. In both panels, the heavy line shows values originally derived from surface wave displacement spectra measured to an upper frequency f_M = 0.4 Hz, and the light line shows those values corrected for missing variance at frequencies f_M < f < f_B, where f_B is the wave breaking frequency of Forristall (1981). Periods where corrected and uncorrected curves coincide are those when the surface wave field is determined (by the ECMWF criterion) to be not wind forced.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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A final problem with determining U_S0 occurs in cases where the peak of the Stokes spectrum lies at frequencies very near or beyond the maximum observed frequency f_M: in such cases U_S0 is indeterminate.

Because Stokes velocities associated with high-frequency waves drop off rapidly with depth, correction for missing high-frequency variance is much smaller in U_S3 (not shown), than in U_S0: hence in this regard U_S3 is preferable to U_S0. However, as cartooned in Fig. 1b, U_S3 can fall to near zero while finite Stokes drift velocities due to high-frequency, short-wavelength waves remain at shallower depths. The best characteristic Stokes velocity would seem to be one that retains a nonzero (even if small) value under these conditions, but remains relatively unaffected by a missing variance correction.

The third form above fulfills both of these conditions. The missing variance correction for wind-forced waves in intU_S3 (see the appendix) is much smaller than that in U_S0 and as a result, discontinuities in intU_S3 across wind-forced/not-wind-forced wave boundaries are much less severe, as seen in Fig. 2b. As well, the value of intU_S3 remains finite in cases dominated by high-frequency waves. Thus, for use in the scaling velocity forms (2b) and (2c), I will use U_S = intU_S3 as the best estimate of a characteristic near-surface Stokes velocity. Note that x₃ = −3 m = −0.2H_LEO at LEO, hence this near-surface-integral value, when corrected for near-bottom Stokes drift, equals the surface-layer value suggested by the LES analysis of Harcourt and D’Asaro (2008). For R2, intU_S3 remains the near-surface-integral value, while the surface-layer value is $\mathrm{int}{U}_{S5}-u_{\text{ref}}^S$, defined as the integral from x₃ = −5.2 m = −0.2H_R2 to the surface, corrected for bottom Stokes drift. This surface-layer value is also computed for the R2 data and will be assessed relative to intU_S3 in section 5.

4. Scaling rmsw at LEO

Testing Eq. (1) requires determination of the constants a_C and a_S that multiply respectively convective and wind/wave scale velocities. To determine these constants, all qualified data are used in MATLAB fits using two variables, the convective scale velocity $w_{*}$ and one of the three expressions (2a), (2b), or (2c) for w_S. Records are considered qualified if they fulfill the following conditions:

• the Stokes spectrum peaks below the maximum observed frequency f_M;

• the water column is unstratified;

• surface buoyancy forcing is convective, i.e., B_o > 0;

• $\overline{⟨w_{}^2⟩}>w_n^2$; and

• the record(time)-averaged profile of rmsw has its maximum in the upper half of the water column.

Note that this procedure for determining a_C and a_S, unlike that of G22, makes no assumptions about particular records being pure convection or pure LC. It is expected that the coefficient fit for a_C will be constant across the three fits involving the three different versions of U_S, while the coefficient a_S will depend on the assumed form of U_S.

For all three fits to the LEO data, residuals approach normality (hence error bounds are reliable) and coefficients approach approximately constant values when the number of records N > ∼100, more than fulfilled by the set of N = 149 qualified records.

For the LEO dataset, Table 1 shows fit parameters with error bounds in square brackets for each of the three candidates for characteristic wind/wave velocity, calculating both $w_{S2}={(u_{*}{U}_S)}^{1/2}$ and $w_{S3}={(u_{*}^2{U}_S)}^{1/3}$ with U_S = intU_S3. Also shown for each are values of root-mean-square error $\text{rmse}=\text{sqrt}\left[{\displaystyle {\sum }_{i=1}^N{(\text{rmsw}-{\text{rmsw}}^{\text{fit}})}^2}/(N-3)\right]$, where rmsw and rmsw^fit are observed and fitted values, respectively, as well as the R² value of the fit. In Table 1, parameters in parentheses for w_S2 and w_S3 result from separate fits using the surface-layer value $U_{\text{SL}}=\mathrm{int}{U}_{S3}-u_{\text{ref}}^S$, where $u_{\text{ref}}^S$ is taken as the Stokes drift velocity calculated at x₃ = −H (using the value at x₃ = −0.8H (not shown) makes no noticeable difference to the results), in place of the near-surface integral value intU_S3. Figures 3a–c show scatterplots of observed values of rmsw and residuals (rmsw–rmsw^fit) versus rmsw^fit for all records used in the fits. Figure 3d shows the same for the fit using the surface-layer value to calculate w_S2.

Table 1.

LEO values of the coefficients in Eq. (1), with error bounds in square brackets, as determined from a fit to observed values of rmsw using the two parameters $w_{*}$ and w_Si calculated from observed surface forcing fields. Here $w_{S1}=u_{*}$, $w_{S2}={(u_{*}{U}_S)}^{1/2}$, and $w_{S3}={(u_{*}^2{U}_S)}^{1/3}$, the latter two calculated with the near-surface-integral value U_S = intU_S3. Entries in parentheses () are results of fits using w_S2 and w_S3 calculated with $U_S=U_{\text{SL}}=\mathrm{int}{U}_{S3}-u_{\text{ref}}^S$, the surface-layer value at LEO.

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The upper panels in each group show values of rmsw observed at LEO as a function of values predicted using observational values of $w_{*}$ and either (a) $w_{S1}=u_{*}$, (b) $w_{S2}={(u_{*}{U}_S)}^{1/2}$, or (c) $w_{S3}={(u_{*}^2{U}_S)}^{1/3}$, with the latter two calculated using the near-surface-integral value U_S = intU_S3. The solid line is a 1/1 relationship, a line with slope 1 through the origin. The lower panels in each group show the associated residuals (rmsw − rmsw^fit). (d) For comparison with (c), results for the alternate fit using w_S2 calculated with the LEO surface-layer value of $U_S=U_{\text{SL}}=\mathrm{int}{U}_{S3}-u_{\text{ref}}^s$.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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Table 1 shows that the fit associated with w_S1 has the largest value of rmse and smallest value of R² of the three, suggesting that it is indeed useful to add information about wind/wave forcing via use of a Stokes drift velocity in either w_S2 or w_S3. Thus, first restricting comments to features of these latter two choices:

1. For both fits, the offset a_o is approximately the same magnitude and negative, consistent with the fact that the observations being fitted are corrected by subtraction of a noise level, so the prediction of total rmsw by $a_C{w}_{*}+a_S{w}_S$ should be decreased by some (small) amount.

2. The convective scale velocity coefficient a_C is the same, within error bounds, for both fits, consistent with expectation that the convective and wind/wave scale velocities are independent, provided they are chosen properly.

3. The fits using w_S2 and w_S3 are associated with only minor difference in goodness of fit parameters. This feature would result if $U_S\propto u_{*}$, i.e., if the characteristic Stokes velocity were nearly linearly proportional to the surface stress velocity. However, the proportionality cannot be exact, or else the stress velocity itself would work equally well as a wind/wave scaling velocity, which is not the case.

4. There is very little change in either the a_S coefficient or goodness of fit parameters when $U_S=\mathrm{int}{U}_{S3}-u_{\text{ref}}^S$ is used in place of U_S = intU_S3 when calculating w_S2 or w_S3.

In contrast to (2) above, when using $w_{S1}=u_{*}$ as the wind/wave scale velocity, the fit coefficient a_C obtained for the convective scale velocity is significantly different from the average value of a_C obtained using w_S2 and w_S3. Moreover the offset a_o has positive rather than the expected negative sign. These features, plus decreased goodness of fit and increased rmse, indicate that among those considered, $u_{*}$ is the least desirable choice for a wind/wave scaling velocity at this site.

5. Scaling rmsw at R2

I now turn to the question of whether the constants that characterize Eq. (1) at LEO are the same as those determined with a similar dataset from R2, the location in deeper water, farther from shore. First, it is useful to consider similarities/differences of the two datasets in the forcing space defined by G22, using surface buoyancy flux B_o and the growth rate $g_{*}={[(d{U}_S/d{x}_3)(dU/d{x}_3)]}^{1/2}$, evaluated near the surface, that Leibovich (1977) demonstrated is a characteristic growth rate for Langmuir circulations. Following GG14, $g_{*}$ is evaluated using Stokes shear at x₃ = −3 m and approximating mean shear as $u_{*}/H$. G22 demonstrates the connection between location in the upper half of this space and the dominance of either buoyancy (moderate B_o, small $g_{*}$) or wind/wave (small to moderate B_o, large $g_{*}$) forcing of surface layer turbulent structures. Distributions of data in this space are seen in Fig. 4 for both LEO and R2: both signs of surface buoyancy forcing are shown, although here the focus is on convective records with B_o > 0.

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All qualified records, color coded by magnitude of wvar, from (a) LEO and (b) R2 are shown in the forcing space of near-surface Langmuir circulation growth rate calculated as $g_{*}\equiv {[{(d{U}_S/d{x}_3)}_{x_3=-3\mathrm{m}}(u_{*}/H)]}^{1/2}$ and surface buoyancy flux B_o, as defined by G22. Using $\log (g_{*})$ expands resolution of low values of $g_{*}$: similarly, color coding log(wvar) reveals increased detail of signal strength.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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As seen in Fig. 4, overall magnitudes of wvar = rmsw² are similar in the regions where the distributions overlap. However, there are significant differences. The R2 dataset, despite being larger than that from LEO, has no situations with very low $g_{*}$. This is consistent with the fact that the few measurements of very low $g_{*}$ at LEO were associated with a rare condition of weak winds with very short offshore fetch. While R2 has some records with slightly higher $g_{*}$ than LEO, the most noticeable difference is the presence of records with much higher buoyancy flux at large values of $g_{*}$, filling in one of the “holes” pointed out in G22. These larger buoyancy fluxes are associated with the higher surface layer temperature at R2. Finally, R2 also has many more records with large stratifying buoyancy flux, although these aren’t considered in the present analysis.

For a set of R2 data qualified by the same criteria as those for LEO, residuals approach normality (hence error bounds are reliable) and coefficients approach approximately constant values when the number of records N > ∼200 (the fact that this number is larger than that determined for LEO is likely associated with larger scatter of observed rmsw at R2 relative to LEO, discussed further below). The N = 288 qualified R2 records are adequate to determine stable coefficients.

Carrying out the procedure outlined in section 4 on qualified R2 data using the near-surface-integral value U_S = intU_S3 in calculation of w_S2 and w_S3 results in the fits shown on the upper lines of Table 2, while results using the surface-layer value $U_{\text{SL}}=\mathrm{int}{U}_{S5}-u_{\text{ref}}^S$ are shown in brackets on the lower lines. Figure 5 shows the same fits seen in Fig. 3 for LEO.

Table 2.

Values of the coefficients in Eq. (1), with error bounds in square brackets, as determined from a linear fit to values of rmsw observed at R2 using the two parameters $w_{*}$ and w_Si determined from observed surface forcing fields. Here, $w_{S1}=u_{*}$, $w_{S2}={(u_{*}{U}_S)}^{1/2}$ and $w_{S3}={(u_{*}^2{U}_S)}^{1/3}$, the latter two calculated with the near-surface-integral value U_S = intU_S3. Entries in parentheses () are results of fits using w_S2 and w_S3 calculated with $U_S=U_{\text{SL}}=\mathrm{int}{U}_{S3}-u_{\text{ref}}^S$, the surface-layer value at R2.

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The upper panels in each group show values of rmsw observed at R2 as a function of values predicted using observational values of $w_{*}$ and either (a) $w_{S1}=u_{*}$, (b) $w_{S2}={(u_{*}{U}_S)}^{1/2}$, or (c) $w_{S3}={(u_{*}^2{U}_S)}^{1/3}$, with the latter two calculated using the near-surface-integral value U_S = intU_S3 in Eq. (1). The solid line is a 1/1 relationship, a line with slope 1 through the origin. The lower panels in each group show associated residuals (rmsw − rmsw^fit). (d) For comparison with (c), results for an alternate fit using w_S2 calculated with the R2 surface-layer value $U_S=U_L=\mathrm{int}{U}_{S5}-u_S^{\text{ref}}$. Ellipses highlight a region of consistent underestimation of rmsw by this alternate fit.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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At R2, the fit using $u_{*}$ again has the largest value of rmse and smallest value of R², although the differences are smaller in magnitude than those at LEO. Unlike LEO, all 3 fits now have comparable (small, negative) values of a_o and equal (within error bounds) values of a_C, i.e., the major differences in these two coefficients in the LEO fit using $w_{S1}=u_{*}$ have disappeared. There are, however, additional major differences from LEO. First, as seen in the plots of Fig. 5, the scatter for all of the R2 fits is nearly twice that at LEO, hence values of R² are smaller. Second, the wind/wave coefficients determined from the fits using w_S2 and w_S3 are both roughly half the comparable values at LEO.

Before discussing these features, it is first necessary to emphasize that they do not result from the choice of U_S = intU_S3 for calculation of the scale velocities w_S2 and w_S3. For LEO, Table 1 showed the coefficients resulting from use of intU_S3 (upper values) and $\mathrm{int}{U}_{S3}-u_{\text{ref}}^S$ (lower values, in brackets) in calculation of both w_S2 and w_S3. Within error bounds, there is no significant difference between the two values of a_S because, at LEO, values of $u_{\text{ref}}^S$ although increasing with wind stress, remain a small fraction of intU_S3 throughout the range of observed values (see Fig. 7b). Table 2 shows similar coefficients for R2. Here the a_S coefficients obtained using the surface-layer value (i.e., integrating the Stokes velocity from −5.2 m to surface) are only slightly larger than those obtained using the near-surface-integral value, i.e., where the integration is over the same (3 m) layer depth as at LEO (also note that error bounds of the two estimates still overlap). This insensitivity arises partly because intU_S5 is only ∼20%–30% smaller than intU_S3 over the observational range of $u_{*}$, plus $u_{\text{ref}}^S$ is effectively zero for all conditions at R2 (see Fig. 7a), and partly because w_S2 and w_S3 involve respectively the square and cube root of the expression used for U_S. Use of the surface-layer value at R2 does appear to make slight improvement in the reported goodness-of-fit parameters rmse and R². However, these parameters are related to the actual fit which, while constrained to be linear, is not constrained to be a 1/1 fit, i.e., a line through zero with slope of 1. Indeed, the R2 results of Fig. 5d reveal definite underestimation of observed rmsw at large values (ellipses in Fig. 5d), i.e., a departure from a 1/1 fit when using the surface-layer value rather than the near-surface-integral value. This latter result suggests that when computing a characteristic Stokes velocity for use in evaluating wind/wave scaling velocities, a near-surface integral value of Stokes velocity, computed as an integral over a fixed distance from the surface, is preferable to a surface-layer value where integration depth varies with mixed layer depth.

The above results at R2 thus support the conclusion of Grant and Belcher (2009) that the appropriate scale velocity for LT in a well-mixed layer is calculated with a (near) surface value unaffected by mixed layer depth relative to the wavelength of dominant surface waves. The results from both sites indicate that fits using w_S2 and w_S3 have nearly identical values of rmse and R², hence do not suggest a preference between these two forms as a scale velocity for Langmuir turbulence. Finally, the coefficient values in Tables 1 and 2 show that the observed difference between fitted values of a_S at LEO and R2 is statistically significant.

6. Discussion and conclusions

A final analysis, using fitted data, reinforces a basic premise of the present study, i.e., that a mixture of forcings, rather than pure convective or wind/wave forcing, is more the norm than the exception in the OSL. Figure 6 shows the convective (B_o > 0) half-plane of the forcing space defined by G22. Color codes values of $\log (R_F\equiv a_{S2}{w}_{S2}/a_C{w}_{*})$, where the forcing ratio R_F is the ratio of the vertical velocity of the wind/wave component relative to the convective component, using fits to both LEO and R2 data made with U_S = intU_S3 in calculation of w_S2.² Using the logarithm of R_F results in a [−1, 1] range that covers cases where wind/wave forcing dominates (logR_F ≫ 0) and where convection dominates (logR_F ≪ 0). The region of white in the middle of the color bar highlights values within ±20% of the value of R_F = 1 that corresponds to equal contributions from the two turbulence sources. The shift in the distribution of the ratio toward higher values of $\log (g_{*})$ at R2 relative to LEO arises from the difference of approximately a factor of 2 between the values of a_S, hence magnitude of wind/wave forcing, at the two sites as documented in Tables 1 and 2 (a possible explanation for this difference will be discussed below). Despite this difference, in both locations there are clear bands of white-coded records, those in which buoyancy and wind/wave forcings make approximately equal contributions to observed rmsw. These bands trend upward as $\log (g_{*})$ increases, i.e., the buoyancy flux required for the convective component to equal the wind/wave component increases with $\log (g_{*})$. While there certainly are situations in which wind/wave forcing dominates (log > 0.5 corresponds to wind/wave forcing more than ∼3 times convective forcing) and those in which convective forcing dominates (log < −0.5 corresponds to convective forcing more than ∼3 times wind/wave forcing), the number of records in which the two have comparable magnitudes is striking.

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(a) LEO and (b) R2 data in the upper half of the forcing space of near-surface Langmuir circulation growth rate $g_{*}\equiv {[{(d{U}_S/d{x}_3)}_{x_3=-3\mathrm{m}}(u_{*}/H)]}^{1/2}$ and surface buoyancy flux B_o. Color codes $\log (R_F\equiv a_{S2}{w}_{S2}/a_C{w}_{*})$, where R_F is the ratio of the contribution to rmsw from wind/wave forcing to that from buoyancy forcing, and the wind/wave scale velocity w_S2 is calculated using U_S = intU_S3.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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The present demonstration that observed ocean surface layer turbulence frequently results from a mixture of convective and wind/wave forcing is consistent with analysis by Belcher et al. (2012) of ERA reanalysis data of surface forcing in the Southern Ocean in winter. In a space of regime parameters defined by Belcher et al. (2012), the peak of a joint pdf of observations lies midway between regime regions defined as dominated by buoyancy forcing and wind/wave (in present terminology) forcing. The conclusion is also consistent with LES results of Walker et al. (2016), who found that a large heat flux (560 W m⁻², corresponding to B_o ∼ 2.2 × 10⁻⁷ W kg⁻¹ at LEO) was required for convection to exceed the (storm force) wind/wave forcing contribution to computed turbulent kinetic energy. Using parameter values in Walker et al. (2016) and an appropriate single-wave version of the Stokes drift profile, their LES is found to be characterized by a value of $\log (g_{*})\sim -2.6$, a value at the high end seen for that parameter at LEO. While none of the direct observations near this value of $\log (g_{*})$ have convective fluxes in excess of ∼2 × 10⁻⁷ W kg⁻¹, given the upward slope of the white band in Fig. 7a, it is clear that buoyancy flux in excess of this value would indeed be required for convection to make a contribution to observed rmsw that is comparable to the computational wind/wave contribution.

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Relationship between the near-surface-integral Stokes drift velocity intU₃ and $u_{*}$ for binned records from (a) R2 and (b) LEO. The first-order (thin straight line) and second-order (heavy curve) fits to the R2 data are shown in both panels. Diamonds on each panel are binned values of $u_{\text{ref}}^S$.

Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-22-0213.1

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Having reiterated the necessity of a predictive algorithm for rmsw that incorporates both buoyancy and wind/wave forcing, it is necessary to understand the statistically reliable differences between wind/wave scalings derived for LEO and R2, as well as the increased scatter in the R2 results. For any such understanding, it first needs to be recognized that the physical environment at R2 differs from that at LEO in a number of possibly significant ways.

1. Because R2 is much farther from a coast than LEO, it seemed possible that the horizontal pressure gradient produced by Ekman transport in the presence of a nearby coast, a pressure gradient that rapidly produces steady shore-parallel flow at LEO (Grosch and Gargett 2016), would take longer at R2, implying stronger effects of rotation, disrupting the organized Langmuir cells typical of LEO and producing the less organized, weaker structures characteristic of Langmuir turbulence (McWilliams et al. 1997) at R2. However, examination of mean current hodographs (not shown) reveals that, within uncertainty associated with definition of the “start” of a wind event, mean current response times are ∼6–8 h at both locations, removing differences in inertial effects as a major factor in observed differences in turbulence structure and strength.

2. Tidal flows at R2 have typical magnitude of ∼0.4 m s⁻¹, much larger than the 0.1 m s⁻¹ typical of LEO. Although records that were possibly contaminated by strong tidal bottom boundary layers were rejected, it is possible that other effects of tides may remain, as will be discussed below.

3. The dominant storm-driven surface waves at R2 are close to deepwater waves (Fig. 14b of Gargett et al. 2014), while those at LEO during strong wind events are of intermediate type (Fig. 14a of Gargett et al. 2014).

4. Observations clearly show that during storms, bottom stress at LEO rises rapidly to equal the applied surface wind stress in magnitude (Gargett and Savidge 2020, their Fig. 12a). In contrast, bottom stress at R2 is typically much smaller than the surface stress, both in similar storm events and in the rapidly shifting surface conditions of a tropical storm (Gargett and Savidge 2020, their Figs. 12c and 12b, respectively).

Discounting item 1 above, items 2–4 are potential contributors to observed differences between scatter and/or wind/wave scaling at LEO and R2.

First, I consider the larger scatter observed in the R2 data relative to LEO, and suggest that this results primarily from the stronger tides at R2 (item 2 above). Since the wind/wave scale velocity w_S is considered predictive of pure LC, disruption of pure cells by the effects of clockwise tidal turning with time would produce effects similar to those associated with inertial turning, leading to the less organized structures characteristic of LT and decreasing correlation between measured variance and that predicted using w_S. Although much of the LES literature treating the effects on LC of crosswind mean flow due to tides report results at steady-state (Martinat et al. 2011; Shrestha et al. 2019), among steady-state results Kukulka et al. (2011) discuss one case of the early time-dependent stages of imposition of a crosswind pressure gradient. Initial LC cell structures first weakened as crosswind flow increased from zero, then after ∼6 h, the two-cell structure of the initial computational state transitioned to a single weaker cell. Presumably (although not part of the reported computations) the structure would revert to the original when the crosswind flow subsequently decreased to zero again as part of the tidal cycle. Such temporal evolution under continuously varying tidal flows would provide a source of scatter in the 2 h observational averages which would be expected to be larger for the larger tidal velocities at R2.

Turning now to the lower overall rmsw associated with wind/wave forcing at the deeper site, i.e., the difference in a_S values between the two sites, a potential explanation may lie in the differences 3 and 4 between the physical settings of LEO and R2, particularly when these two are combined. Gargett et al. (2014) first suggested how differences in wave type and/or bottom stress magnitude could lead to different levels of forcing of Langmuir features in different coastal settings. Storm wind-generated waves at LEO and R2 have roughly the same wavelength (about 90 m; Gargett and Wells 2007; Gargett et al. 2014), but move in water of different depth. If the depth is great enough that the wave is effectively in deep water, Stokes shear near the bottom is very small, whereas the same wave in shallower water will have larger near bottom Stokes shear (a single wave of wavelength 90 m has Stokes shear 2 m above the bottom at LEO that is nearly 6 times that at R2 for the same wave amplitude). In the presence of significant near-bottom Stokes shear, vertical vorticity associated with bottom boundary layer turbulence will be rotated into the horizontal, essentially duplicating the CL2 vortex force that is operative near the surface. Thus, waves of intermediate type, i.e., those with significant Stokes shear near the bottom, have two sources of energy forcing Langmuir circulations, rather than the single near-surface source characteristic of deepwater waves. Larger storm-driven bottom stress at LEO will contribute to this additional source by increasing the supply of boundary layer vorticity available for rotation by Stokes shear. Gargett and Savidge (2020, their Fig. 13) demonstrate that LEO indeed has near-bottom LC growth rates (defined in analogy with the surface growth rate $g_{*}$) that are much larger than values at R2.

In summary, consider first the effectively deepwater wave case of R2, where the fits shown in Table 2 and Fig. 5 indicate that scaling with the convective scale velocity $w_{*}$ and any one of three suggested forms for a wind/wave scale velocity w_S produces the same value for a_C, within its error bounds. Constancy of a_C reaffirms that the convective scale velocity determined in the atmospheric boundary layer is equally applicable to convection in the OSL, as found in other observational results from situations dominated by convection (Steffen and D’Asaro 2002) or strongly influenced by convection (Lombardo and Gregg 1989).

Fits using $u_{*}$ and the two suggested forms of w_S calculated with U_S = intU_S3 all provide 1/1 relationships between observed and predicted values of rmsw. As mentioned previously, similar linearity using any of the three wind/wave scale velocities (albeit with different values of a_S) would result if $U_S\propto u_{*}$, i.e., if the Stokes velocity were at least approximately linearly proportional to $u_{*}$. The degree to which such a linear relationship is achieved by the R2 data is shown in Fig. 7a. While the linear fit lies (just) within the error bars of the data, the second-order fit is clearly preferable. It is this inaccuracy in the assumption of $U_S\propto u_{*}$ which leads to the decreased accuracy (i.e., higher rmse and lower R²) of the fit using only $u_{*}$, relative to those incorporating the slightly nonlinear relationship seen in Fig. 7a. Note, however, that while the fit using $u_{*}$ has the lowest R² of the three considered at R2, even this fit provides R² > 0.8, not insignificant predictive skill. This result is in general agreement with early analysis of rmsw from neutrally buoyant floats at a deepwater site (D’Asaro 2001).

Use of the surface-layer value in the R2 computations results in a fit that, while linear, is not a line through zero with slope of 1, leading to clear underestimate of large values of observed rmsw (Fig. 5d). This result supports use of a near-surface-integral value of Stokes velocity, computed as an integral over a fixed distance from the surface, rather than a surface-layer value that varies with mixed layer depth, in calculation of wind/wave scaling velocity.

I now return to the anomalous nature of coefficients found for the fit made using $w_{S1}=u_{*}$ at LEO, where the value obtained for a_C is significantly different from all of the other fits, including LEO fits made using either w_S2 and w_S3, both of which yield values of a_C that agree within error bounds with all of those from R2. The root of this single anomalous fit appears to lie with a different relationship between $u_{*}$ and U_S at LEO, as seen in Fig. 7b. The second-order fit to the R2 data provides a reasonable fit to the LEO data at low values of $u_{*}$: however, the two data points at the highest values of $u_{*}$ lie clearly above the R2 curve. A possible interpretation of this observation is that the relationship between $u_{*}$ and U_S depends on whether the waves at LEO are effectively deepwater waves (waves short compared to water depth at low wind forcing) like all observations at R2, or intermediate waves, as documented at higher LEO wind speeds by Gargett et al. (2014). The anomalous fit found at LEO when using $u_{*}$ alone would then presumably be associated with absence of a continuous quasi-linear relationship (such as that shown in Fig. 7a for R2) between $u_{*}$ and U_S over the entire range of $u_{*}$. While this conclusion is tentative, depending as it does on only two binned data points in the present LEO dataset, it suggests that measurement of the wave field, hence observational determination of U_S, is more important in shallow coastal environments, where wave character can change during a wind event, than it is in locations where surface waves retain deepwater character throughout such an event.

Leaving aside only this case (i.e., the fit at LEO using $w_{S1}=u_{*}$), the analysis presented here enables prediction of rmsw, the root-mean-square vertical velocity fluctuation associated with turbulence generated by a mixture of forcing by wind/wave processes and convection at both shallow and deep coastal sites, within reported errors associated with determination of the coefficients in Eq. (1). The predictor is a linear combination of scale velocities appropriate for convective and wind/wave (Langmuir) turbulence. The coefficient of the convective scale velocity a_C is constant, within error bounds, independent of the form chosen for the wind/wave scale velocity, affirming the assumption that the two forcings are independent, hence that their results can be combined linearly. The value of a_C ∼ 0.5 lies within the range of values found by other observations. The coefficient of the wind/wave scale velocity a_S depends on the form used for w_S. In both locations, overall goodness-of-fit parameters (rmse and R²) are nearly identical for fits using w_S2 and w_S3 calculated with U_S = intU_S3, hence the present analysis provides no guidance as to which should be preferred as a wind/wave scale velocity. However, it does indicate that in calculating such scale velocities, a near-surface-integral value of the Stokes velocity is preferable to a surface-layer value, i.e., neither removal of a deep reference value nor use of a mean over a depth that depends on mixed layer depth H is supported by the present data.

Both LEO and R2 datasets inevitably include situations with directional spread of the wave spectrum about the wind direction, wind-wave directional misalignment, as well as a range of wave ages. The linearity of the predictions, plus the tightness of the results at LEO, where tidal effects should be minimal, indicate that none of these factors produce effects large enough to preclude a useful prediction of rmsw based solely on surface forcing fields.

The observational analysis of Belcher et al. (2012) and the present results both strongly suggest that relevant models of OSL turbulence must treat a realistic combination of both wind/wave effects and convection (either nocturnal or quasi-continuous over longer storm periods). However, an assumption inherent to the present analysis is that OSL turbulence adapts reasonably rapidly to changes in surface forcing, hence turbulence characteristics averaged over a 2-h time period can be characterized by surface forcing averaged over that same time period. Observations reported in G22 suggest that this is true for changes in wind/wave forcing, but that the adjustment time for changes in buoyancy flux may be somewhat longer. Although beyond the scope of the present investigation, determination of time scales of buoyancy response in situations of mixed forcings will be particularly important for applications such as the plankton/particulate distributions mentioned in section 1.

Data availability statement.

Data used are available on Zenodo: LEO data doi: 10.5281/zenodo.5048343, R2 data doi: 10.5281/zenodo.7191195.