## 1. Introduction

Two major components of the Earth system, the ocean and the atmosphere, interact with each other through their own planetary boundary layers (PBLs), where the motions of fluids directly respond to the surface stress and buoyancy flux across the air–sea interface on time scales less than a day (Garratt 1992). Boundary layer processes are naturally relevant in many air–sea interaction problems [e.g., hurricanes (Emanuel 2003), Madden–Julian oscillation (Zhang 2005), El Niño (Dijkstra and Burgers 2002), etc.] and have great implications for accurate predictions of weather and climate. A comprehensive assessment of the role of the ocean and atmosphere in these problems requires a quantitative understanding of PBL dynamics, especially in terms of the vertical transports of momentum, heat and mass. These transports are largely carried by small-scale turbulent motions that must be parameterized in circulation models, due to limitations in computational capacity. The parameterization typically involves building approximations for turbulent fluxes, and using turbulence closure to estimate the effects of unresolved fluctuations upon the evolution of resolved mean fields. Informed by theories of molecular diffusion, early pioneers recognized that turbulent fluxes can be cast as functions of local mean gradients (Taylor 1915; Prandtl 1925; von Kármán 1931), and developing flux–gradient relationships has thus become an important approach in modeling boundary layer flows (Monin and Yaglom 1971).

*U*/

*f*|

*z*|, where

*U*and |

*z*| are characteristic velocity and length scales,

*f*is the Coriolis parameter) and nearly constant turbulent fluxes (Wyngaard 2010). Intrigued by such a well-defined problem, Monin and Obukhov (1954) proposed a similarity theory to delineate the vertical structure of an idealized atmospheric surface layer (ASL). Monin–Obukhov similarity theory (MOST) states that, under horizontally homogeneous and stationary conditions, every dimensionless scaling “group” in the surface layer is a universal function of the dimensionless height

*ζ*= |

*z*|/

*L*, the ratio of height |

*z*| to the Obukhov length (Obukhov 1946)

*u*

_{*}is the friction velocity,

*κ*= 0.4 is the von Kármán constant, and

*θ*, or horizontal velocity

*u*, the dimensionless group for its vertical gradient is exclusively determined by

*ζ*in the framework of MOST:

*z*is the upward vertical coordinate;

*χ*with

*χ*. For instance, the friction velocity

*u*, where the reference seawater density

*ρ*

_{0}= 1025 kg m

^{−3},

*w*is the vertical velocity, and the

*x*axis is chosen to align with the surface wind stress

*τ*

_{w}. The dimensionless gradient, or universal function

*ζ*= 0). Numerous experimental efforts, both over land (e.g., Businger et al. 1971) and sea surface (e.g., Edson et al. 2004), have gone into determining the functional form of

*ϕ*

_{m}and heat

*ϕ*

_{h}is based on the Kansas experiment (Businger et al. 1971), from which particularly homogeneous and steady conditions and well-measured data were optimized to give

*ζ*[as shown in Eqs. (3)]. It also laid a solid foundation for the later establishment of bulk transfer relations (the COARE algorithm; Fairall et al. 1996, 2003), since its formulation implicitly suggests a parameterization for turbulent fluxes:

The ocean surface boundary layer (OSBL) mirrors its atmospheric counterpart in many ways, and its representation in circulation models also shares many features with the ABL. By analogy to the atmosphere, vertical mixing schemes for the OSBL also draw on knowledge from MOST. For example, the widely used K-profile parameterization (KPP; Large et al. 1994) employs the universal function

Not surprisingly, as the community becomes more aware of the dynamical impacts of surface gravity waves, suspicion arises about the validity of Monin–Obukhov scaling in the OSBL (Sullivan and McWilliams 2010). Meanwhile, measurements of fluxes and gradients reported from coastal waters near Martha’s Vineyard (Gerbi et al. 2008) have already shown signs of invalidation.

In fact, the turbulence in the upper layers of the ocean and the consequent vertical mixing are inevitably affected by surface waves. Close to the surface, waves break intermittently as they propagate rapidly. One important effect of wave breaking is the downward injection of kinetic energy as turbulence. The classical law-of-the-wall scaling predicts the turbulence dissipation rate *ε* in the ocean under strong wind and weak buoyancy forcings (Agrawal et al. 1992; Drennan et al. 1996; Terray et al. 1996) repeatedly give much higher magnitude of *ε* than the classical prediction, and exhibit a more complicated, three-layer structure of *ε*: Within about one significant wave height (*H*_{s}) of the surface, *ε* is large and more or less constant; Below that, *ε* decays much faster than the −1 power-law before eventually resuming the classical scaling at deeper depths.

A second effect results as the Lagrangian-mean wave velocity, the Stokes drift, induces the Craik–Leibovich (CL) vortex force (Craik and Leibovich 1976) and additional material transport to generate Langmuir circulations, cells, or turbulence that fill the entire OSBL. Conceptually, they can be viewed as arrays of counterrotating vortices with elongated major axes oriented roughly downwind (Sullivan and McWilliams 2010). Oftentimes, one can identify them by streaks of buoyant debris on the surface (Langmuir 1938), and by bubble clouds trapped beneath these streaks using side-scan sonar (e.g., Thorpe 1984; Zedel and Farmer 1991). Although turbulence measurements in strongly convective OSBL have generally supported Monin–Obukhov scaling (Shay and Gregg 1986), available field observations under strong wind conditions did show some deviations from the ABL (D’Asaro 2014). First, the bulk average of the vertical TKE

As extra physics is introduced by processes related to surface waves, the surface layer of the ocean is expected to be dynamically different from the ASL. The canonical similarity theory with scaling parameters |*z*|, *L*,

In this paper, we evaluate the validity of MOST in a surface wave-affected oceanic surface layer, using data collected from two open ocean sites. The rest of this manuscript is organized as follows. In section 2, we introduce the in situ measurements and data analysis, and compare observations to the predictions by MOST. In section 3, we consider two hypotheses for the theory-observation discrepancies: one attributes the differences to surface wave breaking, and the other, to Langmuir turbulence. Each hypothesis is tested using a simplified turbulence closure model, and the model results are also compared to observations. Finally, a brief summary is presented in section 4.

## 2. Observations

The evaluation of MOST flux–gradient relationships requires accurate measurements of surface fluxes and mean profiles in a quasi-steady surface layer. Datasets considered in this study are from two open ocean taut-line moorings (Fig. 1). The subsurface sensors are assumed to have been sampling upper-ocean properties at their nominal depths. At both mooring sites, oceanic conductivity–temperature measurements are complemented by concurrent recordings of surface waves and a variety of surface observations for estimating air–sea fluxes, including winds, incoming solar and longwave radiation, rain, air temperature, relative humidity, and barometric pressure.

The first dataset (Cronin 2007) is from a long-term time series site, the Ocean Climate Station Papa (OCSP, nominally at 50.1°N, 144.9°W) in the northeastern Pacific. It has long been an attractive natural laboratory for boundary layer studies because of the weak lateral processes there. During active periods, the OCSP mooring is instrumented with sensors to measure subsurface temperature and conductivity once every 10 min. The meteorological sensors mounted above the buoy have higher sampling rates, usually 1 or 2 Hz, but the recorded resolutions vary from 1 to 10 min. Hourly data were created by applying Hanning filters to the original records (Anderson et al. 2018). Since 2010, a Datawell Waverider buoy has also been deployed near the OCSP mooring to report directional surface wave spectrum every 30 min (Thomson 2019). To have simultaneous hydrographic, meteorological and surface wave measurements, data from OCSP are truncated to a time period from June 2010 to November 2019.

The second dataset (Farrar 2015) was collected during the field campaign of the first Salinity Processes in the Upper-Ocean Regional Study (SPURS-I; Lindstrom et al. 2015; Farrar et al. 2015). As part of the monitoring array in the subtropical North Atlantic (approximately 24.5°N, 38°W), a surface buoy was deployed in September 2012 and recovered in September 2013. The SPURS-I mooring carried similar meteorological sensors to measure meteorological conditions once per minute and transmit hourly averages via satellite. Below the surface, the mooring line was equipped with a denser array of sensors (~1 m spacing in upper 10 m) for temperature and conductivity measurements in every 5 min. In addition, the buoy also had an instrument to measure the height and direction of surface waves by recording the angular accelerations of pitch, roll and yaw, as well as the vertical heave (Bouchard and Farrar 2008). These raw data were later processed by J. T. Farrar (2014, unpublished data) to produce hourly records of directional surface wave spectrum.

**u**

^{s}and its vertical shear are estimated from the directional wave spectrum, following the method described in appendix A. Surface wind stress, surface heat, and salt fluxes were estimated using the TOGA COARE 3.0 algorithm (Fairall et al. 2003) from measured meteorological variables (see details about the flux calculations at OCSP on https://pmel.noaa.gov/ocs/flux-documentation). Considering the penetrative effect of shortwave radiation

*I*, the buoyancy flux to a layer of depth |

*z*| is computed following Large et al. (1994):

*B*

^{r}is the radiative component that decays with depth. The sign convention for

*B*

_{f}is positive (negative) when the buoyancy flux is into (out of) the ocean. The radiative component

*B*

^{r}is modeled by a two-band exponential profile (Paulson and Simpson 1977):

*I*

_{0}is the net downward shortwave radiation at the surface,

*α*is the thermal expansion coefficient of seawater,

*g*is the gravitational acceleration, and

*c*

_{p}is the specific heat at constant pressure. Here, the solar spectrum is divided into two wavelength bands, with fraction

*r*

_{s}and 1 −

*r*

_{s}of the total radiation. Each band decays exponentially with a characteristic

*e*-folding length scale

*μ*. The values of

*r*

_{s},

*μ*

_{1}, and

*μ*

_{2}depend on the Jerlov (1976) water type. Based on a global climatology of water optical properties (Simonot and Le Treut 1986), water type II and IB are assumed for OCSP and SPURS-I, respectively (Table 1). The depth-dependent buoyancy flux

*B*

_{f}is used to calculate the Obukhov length, as the buoyancy force experienced by seawater changes with depth. This treatment of the solar radiation is one of the simpler possible formulations, but any limitation of this approach is not likely to have a big impact on our following results, since most data selected are from nighttime (section 2a).

The Jerlov water type and corresponding constants assumed in this study.

### a. Mean temperature profiles and vertical gradients

*H*. Only temperature data are directly used in the validation of MOST. At first glance, the temperature profile may seem smooth, but if one zooms into the surface layer, small-scale serrations are almost ubiquitous (Fig. 2), probably owing to the different thermal responses of sensors. To reduce theses noises and extract the mean thermal stratification, we follow the standard method in atmospheric studies (Businger et al. 1971) to fit every temperature profile with a second-order polynomial function in logarithmic space:

*p*

_{0},

*p*

_{1}, and

*p*

_{2}are the polynomial coefficients. The logic behind this logarithmic fit is that the profile of any mean quantity in the surface layer is expected to vary logarithmically with height (depth) in neutral conditions, while in nonneutral conditions, the profile is usually slightly curved in the logarithmic coordinate (Panofsky 1963). For each temperature profile, multiple second-order polynomial fits are conducted within the depth range of [0, 0.3

*H*], using various numbers of data points. Three criteria are required for a fit to be considered as good: 1) the first- and second-order vertical derivatives are monotonic functions; 2) the root-mean-square error (RMSE) is less than 0.002°C; 3) after adjusting for degrees of freedom, the coefficient of determination (

*r*

^{2}) is larger than 0.5. Among all the good fits for a profile, we pick the best one, if there is any, that has the smallest RMSE. If no best fit can be found, then the profile is excluded from the analysis. In total, about 89% and 54% of the profiles are excluded in OCSP and SPURS-I datasets, respectively, either due to the lack of robustness of fits (17% and 16% in unstable conditions, 12% and 27% in stable conditions), or insufficient data in the surface layer. The best fitting function is used to calculate temperature gradients and

*ϕ*

_{h}at different levels

*z*

_{i}:

### b. Surface layer depth

_{b}reaches a critical value

*z*

_{i}as

*b*= −

*g*(

*ρ*−

*ρ*

_{0})/

*ρ*

_{0}is the buoyancy computed from local potential density

*ρ*, and subscript 1 denotes quantity measured at the first sensor depth. Due to substantial sensor drifts, the salinity measurements from the SPURS-I mooring are calibrated (see appendix B) before calculating potential density. Different values of

^{−3}criterion (de Boyer Montégut et al. 2004). The resulting

### c. Quasi-steady state

*H*and the large-eddy turnover time

*T*

_{H}are regarded as the internal length and time scale of the OSBL. The boundary layer eddies driven by the wind and surface buoyancy loss can be characterized by velocity scales

*T*

_{W}and buoyancy flux

*T*

_{B}exceed 10 times the large-eddy turnover time

*T*

_{H}as the quasi-steady period. An example of such selection is given in Fig. 3. As for the condition of horizontal homogeneity, we argue that the spatial scales of horizontal variability at both sites are much larger than the thickness of the OSBL (~100 m), so that the OSBL turbulence does not feel the effect of horizontal heterogeneity. These boundary layer approximations, including the quasi-steady state and horizontal homogeneity, are also assumed in our later simplification of PBL models (appendix C and section 3).

### d. Comparison of Monin–Obukhov scaling with observations

*z*

_{i},

*z*

_{j}in the surface layer should be

*z*

_{i}| and |

*z*

_{j}| are the nominal sensor depths along the mooring line, and the universal function

*ϕ*

_{h}used here is based on the results [Eqs. (3)] from the Kansas experiment. These theoretical predictions of temperature difference are computed through trapezoidal numerical integration and are compared to those directly calculated from the fitted profiles. The comparison (Figs. 4a,c) shows that the observed temperature gradients are about half of the theoretical predictions, over most of the observed range of

The distribution of the observed dimensionless temperature gradient *ϕ*_{h} in *ζ* is shown in Fig. 5. Again, the smaller observed values of *ϕ*_{h} below the Kansas curve are consistent with the smaller linear regression slopes (less than 1) in Figs. 4a and 4c. The deviation of the observations from the Kansas curve decreases as |*ζ*| increases, indicating that the failure of MOST mainly happens in the forced convection regime (−1 < *ζ* < 0). Looking more closely, the different linear regression slopes (Figs. 4a,c) at these two sites, and the disparity of *ϕ*_{h} values in the same *ζ* bin (Fig. 5) may imply that the *ϕ*_{h} in oceanic surface layer does not only depend on *ζ*, and that other forcing parameters not considered in MOST might be important in setting the near-surface temperature gradients. Moreover, in the surface layer, smaller dimensionless temperature gradients suggest larger thermal diffusivity, as *K*_{h} is inversely proportional to *ϕ*_{h} [Eq. (4)]. These observational evidences clearly show that the classical Monin–Obukhov scaling for temperature is not appropriate for direct application in the unstable oceanic surface layer. This is a major finding of this study and the rest of this paper seeks to identify what process is responsible for the weaker thermal gradients, by considering surface waves.

## 3. Hypotheses testing

The reduced temperature gradients in the unstable oceanic surface layer naturally leads to the question of why the ocean is different from the atmosphere in this regard. As mentioned earlier, surface waves related processes can modify OSBL dynamics significantly, but they are not included in the current form of MOST. Therefore, we consider two hypotheses to explain the discrepancies between observations and MOST: 1) surface wave breaking is responsible for the observed weak temperature gradients, and 2) Langmuir turbulence is responsible for the observed weak temperature gradients. Although our current understanding of these two processes is still incomplete, attempts have been made to incorporate them into models for the PBL. Encouragingly, models with surface wave breaking (Craig and Banner 1994; Burchard 2001) have shown some skill in reproducing the near-surface dissipation measurements under breaking waves, and models including Langmuir turbulence were found to better predict observed vertical TKE profiles (Harcourt 2015). Given the success of existing models in representing these two processes, we will take them as the approach for hypotheses testing.

The PBL models adopted here are commonly termed as second moment closures (SMCs), and use solutions of second-order turbulence statistics to inform the turbulent diffusivity in closing mean equations. In view of the relatively simple dynamics in the surface layer, a full SMC can be accordingly simplified to a superequilibrium version, where turbulence production is locally balanced by dissipation. Classical SMCs (e.g., Mellor and Yamada 1982; Kantha and Clayson 1994) with no explicit consideration of surface wave effects, when reduced to the superequilibrium version, are known to have independent temperature and momentum scalings that are very similar to MOST. An example for the derivation of model intrinsic similarity relations from Kantha and Clayson (1994) is given in appendix C. Also, full SMCs have been shown to approximately follow Monin–Obukhov scaling in steady states when constant forcing is imposed (Burchard et al. 1998). However, for a SMC that includes surface wave effects, we expect that the model-derived temperature and momentum scalings would deviate from MOST.

### a. Similarity relations in surface wave breaking model

*q*

^{2}/2) from the classical superequilibrium SMC [Eq. (C1a)] to get

*q*and length scale

*S*

_{q}= 0.2 (Mellor and Yamada 1982). On the right-hand side are the shear production, buoyancy production (destruction) and dissipation of TKE. Technically, Eq. (13) no longer indicates a superequilibrium state, so we will refer to it as “super-equilibrium,” since it is still a simplified version of the original wave breaking model. Assuming dissipation is constant in the wave breaking layer (|

*z*| <

*z*

_{0},

*z*

_{0}is the roughness scale), the scaling of Terray et al. (1996) argues that about half of the wave energy input is dissipated in the breaking layer; the other half is transported downward via turbulent diffusion. Thus, the upper boundary condition for Eq. (13) is

*α*

_{b}is commonly regarded as a function of sea state (Drennan et al. 1992). However, for well-developed waves, its dependence on sea state is so weak that one can treat it as a constant of order 100. For simplicity, 100 is used here, and we also require the energy flux due to breaking waves approaches zero as depth increases.

*κ*|

*z*|, and applying the surface layer approximations [Eqs. (C3a), (C3g), (C3i)], Eq. (13) can be reorganized into

*ϕ*

_{m}−

*ζ*) in the surface layer is quite small, the analytical solution for Eq. (16) can be approximated as

*ξ*= |

*z*|/

*z*

_{0}, and constants

*ϕ*

_{h}as a function of

*ζ*and

*ξ*(Fig. 6a). This model considers extra TKE input from surface wave breaking, so its temperature scaling

*ϕ*

_{h}is additionally regulated by the distance from the depth of energy injection

*z*

_{0}. However, the influence of surface wave breaking is mainly confined in a shallow layer of thickness about 6

*z*

_{0}. Compared to the scaling from the classical superequilibrium SMC (or MOST), the reduction of

*ϕ*

_{h}in the forced convection regime can be up to 60%, while the deviation from MOST gradually diminishes as the magnitude of

*ξ*or

*ζ*becomes larger. Enhancement of

*ϕ*

_{h}can also happen in stronger unstable conditions, and this model behavior is traced back to the reduced vertical TKE ratio

*ϕ*

_{h}is physically realistic, it does not impact our test results for this wave breaking model (Fig. 8).

To assess the reliability of the approximate solution [Eq. (17)] used in the calculation of model *ϕ*_{h}, numerical solution of Eq. (16) is investigated with a fourth-order finite difference method (Kierzenka and Shampine, 2001), after substituting *ϕ*_{m} with an expression derived from Eqs. (18b) and (18c), and rewriting *ζ* as *e*^{−κy}*z*_{0}/*L*. Figure 6b shows an example of the numerically solved *z*_{0}/*L* = −0.2, along with solutions from Eq. (17), from the neutral limit (Craig 1996) and from the classical superequilibrium SMC without wave breaking. The approximate solution exactly coincides with the numerical solution near the surface, but gently converges to the classical solution at depth. Compared to the neutral solution of Craig (1996), the approximate solution agrees better with the numerical solution as buoyancy gradually becomes important at depth, and its advantage is even more prominent at larger |*z*_{0}/*L*|. Since the subtle differences between the numerical and approximate solution are found to have negligible impact on the resulting *ϕ*_{h}, and, given that the major parameter values observed here have |*z*_{0}/*L*| smaller than 0.2, we may conclude that Eq. (17) is a sound approximation for the TKE under breaking waves, as modeled by Eq. (13). Therefore, the temperature scaling *ϕ*_{h} based on Eq. (17) should largely reflect the surface layer temperature predicted by the full model of Craig and Banner (1994).

### b. Similarity relations in Langmuir turbulence model

*K*

_{m}is the conventional eddy viscosity, and

*u*

^{s}and

*υ*

^{s}, Stokes drift in the downwind and crosswind directions, respectively. Angle

*γ*denotes the orientation of the Langmuir cells (Van Roekel et al. 2012) relative to the

*y*coordinate in a clockwise sense;

*γ*. A detailed expression for

*x*coordinate is still chosen to align with the surface wind stress, though the surface wave direction may be misaligned. In the case of nonzero crosswind Stokes drift

*υ*

^{s}, although crosswind stress

*z*coordinate. Note that

*η*

^{x}shares the sign convention of forcing parameter

*ζ*, not of dimensionless shear

*ϕ*

_{m}. Following the procedure in appendix C, Eqs. (20) are nondimensionalized and reduced to

In the surface layer, we use the classical mixing length argument that *ϕ*_{h} as a function of *ζ*, *η*^{x}, *η*^{y}, and *η*^{y} = 0), a map of the model-derived *ϕ*_{h} in *ζ* and *η*^{x} space (Fig. 7a) with *ϕ*_{h} than the classical superequilibrium SMC (or MOST), when the Stokes drift shear is in downwind direction (*η*^{x} < 0). The reduction of *ϕ*_{h} generally increases as the magnitude of *η*^{x} gets larger, but the modification of *ϕ*_{h} by Langmuir turbulence only seems to be significant when convection is weak. The smaller *ϕ*_{h} in this model is physically related to the enhanced vertical TKE and the additional contribution to vertical heat flux from horizontal heat flux [Eqs. (20d), (20j)], both due to the CL vortex force. This model also predicts larger *ϕ*_{h} than the classical superequilibrium SMC (or MOST) when *η*^{x} > 0, implying intensified unstable thermal gradient when Stokes drift is in the opposite direction of the wind, however, this rarely occurs. Note that the model prediction of *ϕ*_{h} is also set by the surface proximity function, and such dependence of *ϕ*_{h} on *η*^{x} = −0.5 is demonstrated in Fig. 7b. As the surface proximity function effects a local redistribution between components of Reynolds stress production, higher values normally decrease the contribution of the CL vortex production to the vertical TKE and vertical fluxes, leading to weaker reduction of *ϕ*_{h} by Langmuir turbulence. Similarly, the model-derived *ϕ*_{h} is sensitive to

### c. Comparison of model results with observations

The analyses of model behaviors under idealized forcing (Figs. 6a and 7) clearly show that both surface wave breaking and Langmuir turbulence can modify the temperature scaling, and the temperature gradient is additionally regulated by wave forcing parameters. As a further comparison with observations needs realistic wave forcing parameters, we estimate them from available observations.

The calculation of *ξ* assumes the roughness length scale *z*_{0} = 0.6*H*_{s} (Terray et al. 1996), where the coefficient 0.6 is admittedly uncertain, but the proportionality with *H*_{s} is supported by previous studies (Duncan 1981; Zippel et al. 2018). Stokes stability parameters *η*^{x} and *η*^{y} are specified by the vertical shear of Stokes drift according to Eqs. (22). The surface proximity function *l*^{s} that is not easily quantifiable. Hence in Eq. (21), we use a linear profile of momentum flux, *l*_{s} are about 0.3 and 0.7 m at OCSP and SPURS-I, respectively, generally smaller than the effective *e*-folding depth scale [approximately

The computed forcing parameters are laid out in Figs. 6a and 7. On the *ζ* axis, most of the data are in the forced convection regime, and the data from SPURS-I span a wider range than OCSP. On the *ξ* axis, measurements mostly occurred within a distance of 7*z*_{0} below the wave breaking layer. Though highly scattered, linear relationships with varying slopes (*z*_{0}/*L*) between *ζ* and *ξ* stand out prominently, owing to the nearly constant Obukhov length *L* in unstable conditions. In general, OCSP has smaller *z*_{0}/*L* values than SPURS-I (Fig. 6a). The Stokes drift shear is mostly in the destabilizing downwind direction, indicated by negative *η*^{x} and much smaller *η*^{y} (not shown). Given a similar depth for the shallowest sensor at both sites, the higher tip of SPURS-I data in Fig. 7b suggests relatively larger decay lengths for

Model predictions of *ϕ*_{h} in the case of wave breaking and Langmuir turbulence are computed with realistic forcing parameters. The aggregated results are displayed in Fig. 8, together with the observations from section 2. The super-equilibrium wave breaking model basically gives *ϕ*_{h} similar to the Kansas curve, except at relatively small |*ζ*|. This is consistent with the example given in Fig. 6c, where *ϕ*_{h} are integrated to show the difference between temperature profiles predicted by the super-equilibrium wave breaking model and by MOST. The peak in the model-derived *ϕ*_{h} curve occurs because the observed forcing parameters (*ζ*, *ξ*) go through a ridge in the *ϕ*_{h} contours (Fig. 7a). Different constants in the roughness length formula, from 0.3 to 1.2, have been tested, yet the resulting *ϕ*_{h} curves have essentially the same shape, although the location of inflection differs a bit. In all, it seems that the simplified wave breaking model predicts reduced *ϕ*_{h} only in near-surface region, where |*ζ*| is relatively small. Considering the *ϕ*_{h} reduction is observed over a broader *ζ* range, there is insufficient evidence to support the first hypothesis that surface wave breaking is the main cause of the observed weak temperature gradients.

For the superequilibrium Langmuir turbulence model, predictions generally follow the trend of observations. At large |*ζ*|, the model agrees well with observations, but as |*ζ*| becomes smaller, its prediction gradually deviates from observations, shifting toward the Kansas curve. Considering uncertainties in the estimate of the surface proximity function, we test the model with two bounding values, 0 and 1, of *ϕ*_{h} caused by the uncertainties of *ζ* is consistent with the fact that the variation of *ϕ*_{h} much at larger |*ζ*| (Fig. 7b). In all, the simplified Langmuir turbulence model can predict reduced *ϕ*_{h} over a *ζ* range similar to that observed, and makes robust *ϕ*_{h} predictions in relatively strong unstable conditions. However, its predictions in relatively weak unstable conditions are obscured by the uncertainties of the

*κ*|

*z*| (e.g., Harcourt 2013). The fact that we prescribed

*E*

_{4}is traditionally lumped together with dissipation to ensure

*E*

_{1}=

*E*

_{3}= 1.8,

*E*

_{2}= 1,

*E*

_{4}= 1.33. Various values have been suggested for

*E*

_{6}(Kantha and Clayson, 2004; Carniel et al. 2005; Kantha et al. 2010), and Harcourt (2015) settled at

*E*

_{6}= 6. Applying the same scaling arguments as those in section 3b, we can nondimensionalize Eq. (24) to

*ϕ*

_{h}without invoking the assumption for

*ϕ*

_{h}sensitive to the coefficient

*E*

_{6}that regulates the CL vortex production of

*E*

_{6}= 6) actually gives too small

*ϕ*

_{h}over most of the observed

*ζ*range (Figs. 9a,b), due to seemingly too large

*E*

_{6}that produces results most similar to observations is about 2.5. In this simplified model, the necessity of using a

*E*

_{6}different than that of the full model (Harcourt 2015) may be linked to the negligence of turbulence transport in the TKE equation and the assumption of weak depth dependence of

*q*

^{2}in the

With *E*_{6} = 2.5, the corresponding *κ*|*z*| and the enhancement of *κ*|*z*| scaling. These are broadly consistent with the fact that Langmuir cells are indeed large-scale structures, and in line with the speculation that vigorous convection outcompetes Langmuir turbulence in strong cooling events (Li and Fox-Kemper 2017). Despite the empiricism involved here, the success of this enhanced superequilibrium Langmuir model in reproducing the mean scaling behavior of observations, and the reasonable underlying length scale variations, both serve to increase confidence in Langmuir turbulence as the major process responsible for reducing the near-surface temperature gradients. Note that our results for Langmuir model generally contradict previous findings from tuning model to LES solutions (Harcourt 2013, 2015). Here we need to either use fixed length scale (*κ*|*z*|) and turn off surface proximity function (*E*_{6} constant and retain default surface proximity function to achieve better consistency with observations.

## 4. Summary

In this study, we find that in the unstable oceanic surface layer, MOST fails to quantitatively predict the mean thermal stratification from surface fluxes. Observations consistently present temperature gradients smaller than those suggested by MOST. As the thermal diffusivity *K*_{h} is related to the dimensionless temperature gradient *ϕ*_{h} through Eq. (4), smaller *ϕ*_{h} also means larger *K*_{h} and more efficient vertical heat transport.

To further investigate the cause of the theory–observation discrepancies, two hypotheses are considered. The first one attributes the weak temperature gradients to the effects of surface wave breaking, while the second one considers Langmuir turbulence as the major contributor. Although imperfect, PBL models that include the effects of surface wave breaking (Craig and Banner 1994) and Langmuir turbulence (Harcourt 2015) are taken to represent these two hypotheses, respectively. Each is tested in the framework of superequilibrium SMC (“level 2” in Mellor and Yamada 1982), where model equations are reduced to give predictions of *ϕ*_{h}. It appears that the simplified surface wave breaking model can only give results partially matching with observations in weakly unstable conditions. In contrast, predictions from the simplified Langmuir turbulence model are very similar to observations across a broad stability range. When supplemented by a length scale equation, the simplified Langmuir turbulence model can quantitatively reproduce the mean scaling behavior of observations if a model constant in the length scale equation is appropriately tuned. Hence we conclude first, that there is not enough evidence to support the surface wave breaking as the main cause of the weak temperature gradients observed; second, that the observed weak temperature gradients are more likely due to Langmuir turbulence.

We have evaluated several approaches to combine the wave breaking and Langmuir turbulence parameterizations in one model, but none of these combined models gives better results than the Langmuir model alone. However, it is of significant concern that the downgradient diffusion assumption for TKE flux that leads to the *S*_{q} term in Eq. (13) is of uncertain validity when the CL vortex force interacts with the elevated TKE from wave breaking (see, e.g., Sullivan et al. 2007). These are issues that merit much more detailed study.

In the end, we allow that the data here may be insufficient to constrain all of the parameters in the models examined, and the conclusion of our hypotheses testing is by no means definitive. However, a combination of the data shown here and other observations will provide powerful constraints on models and theory designed to gain a physically realistic and quantitatively predictive understanding of the upper-ocean boundary layer.

## Acknowledgments

We thank the OCS Project Office of NOAA/PMEL for collecting and distributing the OCSP mooring data. The surface wave data collection at OCSP was operated by Dr. Jim Thomson’s team from APL-UW, with funding support from the National Science Foundation. Coastal Data Information Program (CDIP) provided the surface wave data curation for OCSP. CDIP is primarily supported by the U.S. Army Corps of Engineers. Data from the SPURS-I surface mooring were made available by Dr. J. Tom Farrar of the Woods Hole Oceanographic Institution. Support for the SPURS-I project was provided by NASA. This research was funded by the National Science Foundation (OCE1558459). Ramsey Harcourt was also supported in part by the Office of Naval Research (N00014-15-1-2308).

## Data availability statement

Datasets analyzed in this article are openly available at locations cited in the reference section, with the exception of surface wave data at SPURS-I. A copy of the original data and analysis scripts can be found in the Zenodo repository (http://doi.org/10.5281/zenodo.3988503).

## APPENDIX A

### Stokes Drift

*ω*

^{2}=

*gk*) is calculated by integrating the directional surface wave spectrum

*S*(

*f*,

*λ*) (Kenyon 1969),

*λ*is the direction the wave is coming from (clockwise from true north);

*f*and

*ω*are the wave frequency in hertz and radian. The directional wave spectrum

*S*(

*f*,

*λ*) is estimated from archived records as

*a*

_{1},

*b*

_{1},

*a*

_{2},

*b*

_{2}are the normalized coefficients of directional Fourier series, and

*E*(

*f*) is the nondirectional wave spectrum. Therefore, the east and north components of the Stokes drift velocity are

*f*

_{c}, we use an analytical form that is consistent with the Phillips spectrum (Harcourt and D’Asaro 2008; Breivik et al. 2014),

*μ*= −8

*π*

^{2}

*z*/

*g*. The vertical shear of the Stokes drift is computed by adding the vertical derivatives of the resolved part and the tail contribution, thus

## APPENDIX B

### Calibration of Salinity in SPURS-I Dataset

Although salinity profiles are not directly used to test the classical scaling, they are still important in setting upper ocean stratification. Unfortunately, salinity measurements from the SPURS-I mooring have shown consistent unrealistic variations in the OSBL. This is most evident during nighttime convection, as exampled by the raw profiles in Fig. B1. The upper ocean salinity variation is so large that it dominates the density structure, while the nearly homogeneous temperature profile indicates that the convective plume has resulted in a well-mixed layer. Therefore, we think these salinity jumps are probably due to biofouling or instrumental drift. To get a reasonable estimate of the boundary layer depth, it is necessary to adjust the raw salinity profiles to make the temperature–salinity structures consistent with expectations from convective mixing.

^{−1}deviation bands of drift curves indicates that the calibrations are well constrained.

## APPENDIX C

### Similarity Relations in the Classical Second Moment Closure

*θ*′ on buoyancy, though an extension to include salinity can be made easily. After applying the boundary layer approximations and directing the

*x*coordinate to the direction of surface wind stress, the model equations can be simplified to

*u*′,

*υ*′, and

*w*′ are the three-dimensional velocity fluctuations in Cartesian coordinate system

*x*,

*y*,

*z*;

*A*

_{1}and

*C*

_{1}were introduced in the approximations for the slow return to isotropy, and the rapid distortion components of the pressure–strain rate covariances, respectively (Rotta 1951; Crow 1968); constants

*A*

_{2},

*C*

_{2}, and

*C*

_{3}come from the approximations for the pressure–scalar gradient covariances (Mellor 1973; Andrén and Moeng 1993; Moeng and Wyngaard 1986; Launder 1975); constants

*B*

_{1}and

*B*

_{2}originate from the small-scale local isotropy hypothesis for the dissipation of TKE and temperature variance, respectively (Kolmogorov 1941). Standard values for these constants are listed below:

*r*= 1/3 − 2

*A*

_{1}/

*B*

_{1}. The length scale (

*ϕ*

_{m}and

*ϕ*

_{h}as a function of

*ζ*. These predictions of dimensionless gradients are referred to as the “model intrinsic similarity relations” in this paper. Previous studies have shown that the prediction of

*ϕ*

_{h}in this classical SMC matches the Kansas experimental data pretty well, especially under unstable conditions (see Fig. 2 of Kantha and Clayson 1994).

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