## 1. Introduction

The importance of physical processes in the upper ocean arise because of their involvement in the transport of passive and active tracers, such as heat, biogeochemical tracers, and buoyant pollutants. Modeling of the air–sea interface requires knowledge about the air–sea fluxes of heat and gas at the water’s surface. The heat and gas fluxes play vital roles in the oceanic and atmospheric circulation (which affect hurricane strength and rainfall patterns): making it an important quantity to measure.

The cool-skin layer is an approximately 1-mm-thick layer at the surface of the water that is formed by net heat loss at the surface of the water to the atmosphere. The multiscale temperature variability at the surface is formed when coherent structures are generated by the disruption of the cool-skin layer by boils, vortices, etc. (Saunders 1967). In the ocean a cool-skin layer is typically present at low wind speeds after solar heating has ceased.

Herlina and Wissink (2019) showed in direct numerical simulations (DNS) that divergence contours had convective cells corresponding to large structures of size ~*L*_{∞} (*L* being the reference length scale). The convective cells had positive surface divergence (up-welling) separated by narrow regions of negative surface divergence (down-welling). In the DNS the strength and sign of divergence was approximately related to the vertical mass transfer. The interplay between near-surface boundary layer processes and the cool skin forms a thermal texture at the water surface observable by an infrared (IR) camera.

Particle image velocimetry (PIV) cross correlates particle laden time series images to extract velocity information from the images in the form of time series vector fields. PIV-like analysis of IR images has been applied to study exchange processes at the atmosphere–ocean boundary (Garbe et al. 2003). IR observations of the microscale breaking waves were carried out in laboratory experiments of (Siddiqui et al. 2001). Oceanographic sea surface IR observations where used by Veron et al. (2008) to show that by using PIV-like correlation of IR images that accurate measurements of surface velocity were possible. Their quantified error of approximately 1 cm s^{−1} was sufficient to derive measurements of air–sea heat flux and surface turbulence. The measurements of Chickadel et al. (2011) used PIV-like analysis of the IR images gathered in a tidal river which were validated with in situ measurements. Their in situ measurements were made with acoustic Doppler velocimeters (ADVs) mounted on a barge with a sampling depth of 2 cm below the surface. Over a 3 min averaging period it was shown both costream and cross-stream velocities from the PIV-like analysis are highly correlated with ADV measurements with *r*^{2} = 0.99 and 0.94, respectively.

The understanding of turbulent kinetic energy (TKE) dissipation rates in the ocean has primarily been limited to vertical profiles (vertical snap shots of how TKE dissipation rates vary with depth) such as Fig. 2 in Oakey (1985). In part, this is due to the difficulty of measuring TKE dissipations rates at the surface. To make inroads in the understanding of near-surface small-scale processes based on particle image velocimetry we have developed a method using the surface features to estimate the flow velocity under the IR camera footprint—the feature image velocity (FIV). FIV as opposed to PIV, uses the multiscale temperature variability of structures present on the surface of water to obtain spatial time series of the flow velocity field over the IR camera footprint. The main sources of the IR observed signal are contributions from the water’s cool skin and underlying turbulent processes.

In our approach the IR imaged thermal texture in the form of orthocorrected images was used as an input to the PIV community software PIVLab 1.43 (Thielicke and Stamhuis 2014) and then with the help of the PIV software measurements of 2D time series velocity fields are made. Based on velocity fields time series we have then calculated the energy dissipation. Our paper seeks to extend applicability of IR PIV-like measurements, FIV, by exploiting several approaches to measure and compare TKE dissipation rates and velocities against those obtained from FIV measurements.

The paper is organized as follows: Section 1a presents and overviews the properties of infrared imaging of water. Section 1b then overviews particle image velocimetry to introduce key concepts which are need for the next section. Section 2 introduces the technique to measure surface velocity fields of water using infrared images and provides an attempt to bound the accuracy of FIV measurements. Section 3 details three models for measuring TKE dissipation rates, how FIV-derived TKE dissipation rate measurements were validated by comparison to in situ shear sensor data, analysis of the three models, and concludes by introducing a simplified field method for measuring TKE dissipation rates with FIV. The paper then finishes with section 4 where the key findings are listed.

### a. Thermal imaging of the water surface

An infrared camera integrates the emitted IR radiation of a studied object. The IR radiation emitted by the surface is a function of object surface temperature and emissivity which makes it possible to calculate the surface temperature from the corresponding IR images. The transmittance of the IR radiation from the surface to the camera depends on how much IR energy is absorbed by the atmosphere and it largely depends on the integrated relative humidity over the pathlength from surface to the camera.

The IR energy received by the IR camera consists of the three following components: the reflected emission from ambient sources, emission from the atmosphere and thermal emission from the object carrying information about surface temperature (FLIR 2010). To maximize fidelity of surface IR camera temperature retrievals we need to minimize ambient sources and atmospheric emissions and maximize thermal emissivity. We can minimize the influence of ambient sources and atmospheric emissions if we keep the camera close to the measured surface and minimize reflection from background IR sources. To maximize thermal emissivity we used an IR band wavelength where emissivity is the largest by carrying out measurements in the longwave IR (LWIR) band (8–13 *μ*m). The water emissivity ranges from 0.96 at *λ* = 3 *μ*m to 0.995 at *λ* = 11 *μ*m (approaching blackbody emissivity of 1).

In the work presented here we have used the FLIR T650sc LWIR camera with 20 mK NETD, 640 × 480 resolution, 30 Hz frame rate, and 14 bit dynamic range. For the used IR range of 8–13 *μ*m the penetration depth in water is <20 *μ*m (Wieliczka et al. 1989). The FLIR camera thus captures the spatial temperature distribution integrated over a depth of ≈20 *μ*m (top of the cool-skin layer). In this paper the spatial temperature distribution is used as tracers to infer small-scale properties of the flow.

### b. PIV overview

*W*pixels (px) for window A and 2

*W*px for window B. Each window is cross correlated resulting in a cross-correlation function

*C*(

*n*,

*m*) such that (Thielicke and Stamhuis 2014)

*C*(

*n*,

*m*) exhibits maxima at locations (

*n*,

*m*) which are related to the particles relative displacement between correlated images. The windows can overlap by a set amount of pixels. The amount each window is offset vertically or horizontally is called a step. A step can be expressed as the normalized windows offset given as step = (amount each window is offset from one another)/(the size of the windows). For example, windows offset from each other by 50 px and a window size (

*W*) at 100 px would have a step of 50% and consequently an overlap of 50%. It has been found (Hart 1998) that the step of 50% is a good choice as it minimizes particle pair losses.

Typically in the PIV method the *W* is limited by particle density and selected so that at least 15 particles appear within its area within the region of interest. In general in the PIV approach the window size, determined by particle density, sets the fidelity and the density of the resultant vector field. The application of the PIV technique to measurements at the flow surfaces is not as straightforward as in case of more common PIV measurements away from boundaries.

PIV is typically used for vertical measurements of flow properties. The most basic set up used to make PIV measurements involves a pulsed laser sheet illuminating a 2D plane, orientated streamwise, in the particle seeded flow. The laser pulses are synced with the frame rate of the camera so that images processed with PIV software have well-defined particle position and can provide accurate vertical and streamwise profiles. Horizontal near-surface PIV measurements are feasible but they have their shortcomings. The laser has to be placed below wave troughs precluding realistic surface measurements. In addition for measurements in flumes, there is always a small mean surface slope—the effect of the water being pumped into the tank—so the mean surface tilt depends on the flow rate. To compensate for the tilt, the laser would have to be adjusted to each new flow rate. The near-surface particle dynamics further complicate matters. We thus have used natural surface IR features as source of the PIV input signal.

## 2. FIV

As noted the FIV images contain *surface features* as opposed to PIV particle-seeded flow. The main issue which FIV faces is that the patterns present in images is the spatial distribution of the temperature gradient on the flow surface rather than a set of discrete points—as typically done in PIV applications. It should be possible for FIV to correlate features using the approach of Eq. (1). However, FIV is more limited than PIV in several ways. There is a significant enhancement of noise (flow features have less contrast) which limits the highest-resolution vector field possible and the consistency of the correlation. To get reliable and consistent correlations using Eq. (1) the gradient must be steep enough for its motion between frames to be distinguishable from background noise. Improvements in contrast are achievable by means of contrast limited adaptive histogram equalization (CLAHE). CLAHE operates by tiling the image. At each location the most frequent intensities are spread to the full range of the dataset, thereby enhancing the gradient between features making them more distinct. This significantly improves the probability of detecting valid vectors (Thielicke and Stamhuis 2014). After applying CLAHE the features present in the images become more pronounced (Fig. 1).

In our testing a small step (defined in PIV overview) approximately less than 25% will lead to a breakdown of the correlation. We suspected that it is due to over sampling which amplifies noise until it dominates the correlation. Steps bigger than 75% lead to information loss which can leave areas of the surface uncorrelated. The reliability and consistency of the correlation can be improved by decreasing the sampling rate of images. Our base sampling rate of 30 Hz did not produce reliable vector fields. Subsampling at a rate of 10 Hz significantly improved the quality of the resultant vector fields (Fig. 2).

Subsampling does not reduce noise but increases the noise to signal ratio by amplifying the signal (signal being feature displacement between the cross-correlated images). The subsampling takes the data sampled at 30 Hz and cross correlates image at time *t*_{i} and image at time *t*_{i} + Δ*t* where Δ*t* is constant at 1/10 s and *t*_{i} is increased sequentially by the sampling period of the IR camera at 1/30 s. The subsequent dataset is then averaged in a similar manner such that the resultant vector field from the first set of correlated images is average in groups of such that the latter is 1/10 s after the first. This sliding average is carried out across the entire dataset. Additionally, a three standard deviation (std) filter was used to eliminate outliers. It has been shown by Xue et al. (2014) that values in the range 2.5% to 97.5% of the distribution have an *R*^{2} of 0.98 when comparing real PIV data with modeled PIV data. The average feature displacement ranged from 0.6 cm at the least to 1.6 cm at the most. The feature displacement was small relative to the streamwise distance across the field of view (FOV) of 17.38 cm and the window size of 4.35 cm for the slowest and 5.43 cm at the fastest flow speed (Fig. 3).

### Testing FIV

The FIV experiments were conducted in the Air-Sea Interaction Saltwater Tank (ASIST) flume at the University of Miami (Fig. 4). The tank is 15 m long with a 1 m × 1 m cross-sectional area. The pump circulates water through the tank with a variable speed. A heater and turbulence generating grid is mounted at the point the flow enters the tank. The grid is square with a thickness of 0.02 m with each cell off set vertically and horizontally by *M* = 0.1 m. A microstructure profiler equipped with two orthogonal shear probes (Lueck et al. 2002) was placed at the tanks cross-stream midpoint and 3.7 m downstream from the turbulence generating grid, and 5 mm below the waters surface. The IR camera was placed 3.7 m downstream from the grid and 0.5 m above the flow surface. The IR camera FOV was orthogonal to the surface with an FOV of approximately 23 cm × 17 cm. That resulted in the ratio of linear surface distance to the pixel width of 0.3622 mm px^{−1}. The depth of water during the experiment remained constant at 0.5 m.

To test the FIV accuracy under varied conditions (window sizes and flow speeds) we have devised the following procedure: a single IR flow image was cut into two square equally sized images separated by a predetermined and variable distance of **v**_{real} px, which serves as a proxy for actual feature displacement between correlating windows (Fig. 5). This type of testing is similar to testing done by Jessup and Phadnis (2005) to verify wave crest displacement measurements via IR based FIV. We acknowledge that in the IR surface flow observations, the near-surface convection causes surface features to change with time. However, the convective flow time scale (average time convective features persist) is greater than its transit time (approximately 1 s) across the IR cameras FOV. Therefore, features (temperature gradients of convective cells in the cool-skin layer observed by the IR camera—see Figs. 1 and 3) remain relatively unchanged as they transit the FOV. There is a reduction in noise which increases the accuracy of this method (elaborated on later). Therefore, the error estimates which follow represent the lower limit of FIV’s error.

^{−1}. Each set contained 10 still images acquired for that mean current that were hand selected to attempt to cover full breadth of variability in contrast and surface features over the testing period. Each still image in the set was tested with the ranges of feature displacement

**v**

_{real}∈ ⟨4, 80⟩ px and window size

*W*∈ ⟨10, 400⟩ px to obtain error estimates of the PIV-derived velocity measurements. Measurements obtained from the PIV-derived set of spatial distribution of velocity field as (

*i*,

*j*), and for each image (

*k*) resulting in PIV velocity

**v**

_{i,j,k}. Each index vector of the velocity set,

**v**

_{i,j,k}, was compared with the known feature displacement

**v**

_{real}and presented as the normalized mean absolute deviation:

^{−1}case are shown in Fig. 6. As the ratio of simulated feature displacement—

**v**

_{real}over window size—

*W*approaches 0.5 (Fig. 6, inclined lines), the error estimate

**v**

_{real}/

*W*< 0.5 produce error less than 10% (−1 on the log

_{10}bar). Another area with error greater than 10% appears in the Fig. 6, where

*W*< 175 px and velocities <6 pixels per frame (px frame

^{−1}). Here, we suspect that for these relatively small displacements it becomes hard for the correlation to distinguish displacement from background noise.

The most severe limitation of this type of testing stems from the use of still images. Testing was done based on the assumption that features are persistent thus unchanged on the time scales at the sampling frequency (for us the sampling rate was 30 Hz subsampled at 10 Hz). This limits the applicability of the methods proposed here to flows that have features which are present, persistent, and virtually unchanging on the time scale of the sampling (or subsampling) frequency.

From the results of testing we found that the tolerance for errors in velocity measurements to be <10% at a *W* of 120 px, used for both the 6 and 11 cm s^{−1} cases, to be **v** ∈ ⟨12, 60⟩ px frame^{−1}. Similarly, for a *W* of 150 px, used for the 16 cm s^{−1} cases, the tolerance for error to remain <10% was **v** ∈ ⟨8, 75⟩ px frame^{−1}. Almost all velocities in processed data had surface feature displacement of 15–50 px between subsampled images with relative errors at approximately 2%–3%; representing the lower limit of the error on velocity measurements.

To set the upper limit of the error on the velocity measurements we did similar testing except the correlated images used for testing were taken from time series data. Referred to as real-test data. Real-test data are data whose correlating images are from time series data. Hence the “real” part unlike test data that comprise cuts from a single image to simulate displacement. The procedure to obtain real-test data was done in three steps. First, the FIV-measured displacements for the time series data were calculated. Then the measured displacement at each point from the time series data was used to offset the correlating window B (instead of being at the same relative location as window A) such that FIV returned zero displacement between the correlating windows (<1 px due to feature spreading caused by convection). Then window B was shifted from this position by an amount approximately equal to the mean surface velocity for each case. Measured displacements returned using this method had average errors [normalized mean deviations Eq. (2)] of 3% for the 6 cm s^{−1} case, 3% for the 11 cm s^{−1} case, and 7% for the 16 cm s^{−1} case when tested at their mean flow speeds, respectively.

The peak-to-root-mean-squared ratio (PRMSR) given as *C*_{max} is the peak value and *C*_{rms} is the root-mean-squared of the correlation results [Eq. (1)]. The PRMSR is an analytical means of quantifying noise in the correlation in PIV (Xue et al. 2014). We found the PRMSR to be lowest (more noise present in the correlation) for the real-test data, highest (less noise present in the correlation) in the test data, and the real time series data to have an intermediate noise level. Since the real data’s PRMSR can be bounded from above by test data and below from real-test data we have called them the lower and upper limits, respectively. The higher noise in the real-test data is suspected to stem from information loss due to higher in/out-of-plan motion when compared to the real data due to the window shifting. Furthermore, the error is compounded in the real-test data due to error being present in the initial measurement (used to offset window B). We can then threshold the accuracy to be 2%–3% at the lower limit and 3%–7% at the upper limit, dependent upon the case within a displacement of 15–50 px. Velocities outside of these bounds (mostly the smaller cross-stream velocity and <5% of costream velocities) have higher error.

## 3. TKE dissipation rate measurements via FIV

### a. Divergence model of TKE dissipation rate (ε)

**U**= [

*u*(

**r**),

*υ*(

**r**)] (boldface font denotes vector quantity) with its corresponding gradient:

*u*

_{x}= ∂

*u*/∂

*x*denotes the derivative of

*u*with respect to

*x*. Then the 2D divergence of velocity field

*β*= ∇ ⋅

**U**becomes

*β*) and turbulent mass transfer

*K*

_{L}which in turn is proportional to the TKE dissipation rate (

*K*

_{L}∝

*ε*). At low

*R*

_{T}(turbulent Reynolds number

*R*

_{T}= 2

*Lu*

_{rms}/

*ν*, where

*L*is the integral length scale and

*u*

_{rms}is the velocity rms) the time-averaged spatial correlation

*ρ*(

*K*

_{L},

*β*) (where

*ρ*= 1 if perfectly correlated) obtained by 3D PIV (Turney and Banerjee 2013) produced values of

*ρ*(

*K*

_{L},

*β*) = 0.89, 0.81, and 0.81 for

*R*

_{T}= 84, 195, and 507, respectively. At high

*R*

_{T}of ≈1440 DNS by Herlina and Wissink (2019) had

*ρ*(

*K*

_{L},

*β*) = 0.78. The high-

*R*

_{T}DNS demonstrated that using surface divergence as a proxy for

*ε*estimates is negatively affected by an increasing

*R*

_{T}. However, the decrease in correlation is due an increase in turbulence because of wind induced capillary waves. The upper limit of surface divergence correlation occurs at wind speeds greater than 5 m s

^{−1}due to the breaking of the waves (Turney and Banerjee 2013). Capillary waves are present in our flow but were not observed breaking as relative wind speeds to the surface remained at most 0.2 m s

^{−1}and much less than 5 m s

^{−1}. In our experiment

*R*

_{T}= 2000 to 6000.

As mentioned previously the area of the surface observed by the IR camera was approximately 23 cm × 17 cm. For our studied flow at these scales the kinematic viscosity *υ* and TKE dissipation rate *ε* become the primary quantities in determining flow properties. In our experiment with the aid of FIV, we have carried out TKE dissipation rate estimates, following three literature postulated relationships: *ε*_{1}, *ε*_{2}, and *ε*_{3}, respectively, see Eqs. (8), (10), and (11).

#### 1) Model 1 of the TKE dissipation rate—*ε*_{1}

*ε*

_{1}. The

*ε*

_{1}was obtained experimentally (Okamoto et al. 2016). From the time series of vector velocity

**U**, the fluctuating divergence was derived as

*β*is the divergence and

*ε*any divergence due to waves would need to be removed to only observe divergence due to the TKE dissipation rate at the surface. Any divergence when averaged over a sufficient period of time should be zero provided the waters level has remained constant. This is exactly what we have found the time averaged divergence to be.

However, the spatially averaged divergence produced nonzero values (averaged over three frames). This was unexpected, as a spatially averaged divergence over a larger area should also be zero provided waves longer than the observation area are not present. In exploring whether or not long waves, with wavelengths longer than the observation area, were present we found the mean divergence to have fluctuations that correspond to waves of a few meters in length. We assumed that the small nonzero spatially averaged divergences were due to long waves present at the surface of the tank. For this reason, we removed the spatially averaged divergence as opposed to the time averaged divergence to mitigate its (minuscule) effect on surface TKE dissipation.

*β*′ and the inverse of the Kolmogorov time scale as

*R*

_{T}conditions with a mean surface velocity from 5 to 20 cm s

^{−1}they found for near-surface measurements

*ε*can be given as

*ε*, i.e., Eq. (8), with the spatially varying surface velocity field.

#### 2) Model 2 of the TKE dissipation rate—*ε*_{2}

*ε*

_{2}, was proposed by Okamoto et al. (2016) as

*u*

^{1},

*u*

^{2}, and

*u*

^{3}correspond to

*u*,

*υ*, and

*w*and the directions

*x*

^{1},

*x*

^{2}, and

*x*

^{3}correspond to

*x*,

*y*, and

*z*, respectively. For simplicity we would like to introduce the following notation for derivatives: ∂

*u*/∂

*x*→

*u*

_{x}, ∂

*υ*/∂

*y*→

*υ*

_{y}, etc. Through the continuity equation

*u*

_{x}+

*υ*

_{y}+

*w*

_{z}= 0, under low Froude number condition

*u*

_{z}=

*υ*

_{z}= 0, and when

*w*= 0,

*w*

_{x}=

*w*

_{y}= 0, which can be assumed for a free surface, Eq. (9) simplifies to (Okamoto et al. 2016)

#### 3) Model 3 of the TKE dissipation rate—*ε*_{3}

*ε*

_{3}, was presented in Woods (2010) and was obtained with assumption of local isotropy,

### b. Validation of the FIV-derived TKE dissipation estimates with the shear sensor submerged 5 mm below surface

To validate FIV measurements of *ε* we used a vertical microstructure profiler (VMP) (Lueck et al. 2002) to directly measure near-surface TKE dissipation rates. The VMP is an instrument to measure velocity shear on the scales of less than a millimeter. The VMP has two shear probes which can measure the shear velocity on the perpendicular direction to each other. The VMP sensor tip was carefully placed 5 mm below the water surface. Values of *ε* obtained from VMP measurements are denoted as *ε*_{VMP} and were compared with FIV-derived *ε*_{1}, *ε*_{2}, and *ε*_{3}. We present the result in terms of relative deviation as (*ε*_{i} − *ε*_{VMP})/*ε*_{VMP} × 100% for results see Table 1. Two distributions of *ε*_{VMP} for selected 11 and 16 cm s^{−1} cases can be seen in Fig. 7.

Comparison of FIV-derived surface TKE with the shear probe located 5 mm below surface. The results are presented as percentage difference (% diff.) given as (*ε*_{i} − *ε*_{VMP})/*ε*_{VMP} × 100% for the (*i* = 1, 2, 3) corresponding *ε*_{i}–TKE model. The TKE comparison was carried out for two current speeds 11 and 16 cm s^{−1}. The shear probe data are in column *ε*_{VMP}.

There are wind stresses on the surface of the flow present in our experiment. The air above the flow was still. The larger value of the FIV-derived *ε* when compared to the *ε*_{VMP}, obtained at 5 mm below the surface, is suspected to be caused by the wind stress present at the surface. To verify our suspicion that this would lead to higher TKE dissipation rates at the surface we have carried out a “classical” PIV measurements in the plane spanned by the vertical axis and the axis parallel to the flow. The results of the vertical PIV measurements is presented in the Fig. 8 for a selected case of the current speed of 11 cm s^{−1}. We noticed an enhanced TKE dissipation rate as we approached the surface such that the surface dissipation was 15% larger than the dissipation at the depth of 5 mm (Fig. 8). Comparing this estimated difference with the ones obtained from the VMP and the three modeled cases of *ε* (Table 1), we note *ε*_{1} appears to give the correct increase from a depth of 5 mm. In our experiments, the deviation of the *ε*_{1} from the *ε*_{VMP} was likely due to the presence of the small capillary waves and wind stress we observed in our experiment and was not documented in Okamoto et al. (2016).

The two other analyzed relationships, *ε*_{2} and *ε*_{3}, are based on an idealized assumptions about velocity change as we approach the surface (normal velocity component → 0) and with assumption of local isotropy, assumed for *ε*_{3} [Eq. (11)]. However, a general feature of boundary layers is flows become anisotropic as the flow approaches a surface. Based on our observations the surface flow is anisotropic (in a three dimensional sense) and it becomes more anisotropic as the surface current increases.

### c. Analysis of three models: ε_{1}, ε_{2}, and ε_{3} to evaluate the surface TKE dissipation

Our analyzed data encompassed datasets with mean surface velocity of approximately 6, 11, and 16 cm s^{−1}. From the time series images we observed that small-scale features become less distinguished as the current speed increases (Fig. 3). This is consistent with increase in turbulent mixing as the current speeds increases.

Several things became apparent upon studying the results of *ε* as a distribution. The distributions are composed of values of *ε* which were calculated per vector to observe the full spread of *ε* over the frame. First, for all cases the measure distributions were approximately lognormal. Lozovatsky et al. (2017) noted that probability distributions of *ε* have been found to be close to lognormal in boundary layers in accordance with refined similarity hypothesis first proposed by Kolmogorov (1962).

Second, it was observed that the removal of all vectors on the border of the vector field had a significant effect on the distribution (Fig. 9). The border vectors appeared to be amplifying a *ε* higher than the mean value. We note that features on the border of the vector field can have error in correlation caused by out of or into plane motion of the features. For that reason we removed (masked) all vectors on the border of the vector field.

However, there remained a departure from lognormality in the distribution of *ε*_{2} and *ε*_{3}. The departure was weakest in 6 cm s^{−1} cases and strongest in 16 cm s^{−1} cases (Fig. 10). Upon investigating the departure we found that the nondivergent (vortical) components of ∇**U** [Eq. (3)] are picking up turbulence being injecting into flow from a nonuniformity in the tank wall, located near the bottom right corner of the IR camera’s FOV (Fig. 11), in the form of rotational energy. Which is amplified due to the multiplication of the vortical components in Eqs. (10) and (11) and seen as the departure noted in their respective distributions (Fig. 10).

Third, the value of *ε* increases with the surface velocity (Table 2). All reported values of *ε* were calculated over space [Eqs. (5), (10) and (11)] then averaged over observation period. Reported values of *ε* are denoted by the time (UTC) in which the observation started and mean surface velocity (cm s^{−1}). One of the datasets from Table 2 is marked with an asterisk and exhibits larger deviation from the lognormal TKE dissipation likely due to closer proximity of the IR camera FOV to the tank wall.

The summary of the FIV-derived TKE for three TKE models *ε*_{1}, *ε*_{2}, and *ε*_{3}, i.e., Eqs. (8), (10), and (11). Each is denoted by a case, surface velocity is measured in cm s^{−1}, and time is the start of the observation. The asterisk denotes poor-quality data.

From Table 2 it can be seen that the general trend of increasing values of *ε* relative to surface velocity holds true for all *ε*. Near-surface turbulence is dominated by convection and wind stress (Babanin and Haus 2009). If we consider the frame of reference to be on the surface, with the air above the surface being still, then *ε* can be seen to increase with apparent wind speed. A corresponding graph accompanied with the lower limit error estimates of each measurement can seen in Fig. 12. The error estimates of *ε* were derived from the error propagation of the lower limit velocity error of approximately 2%–3%. Error estimates of *ε*_{2} and *ε*_{3} had a range from 20% to 33%. While error estimates of *ε*_{1} had a range from 41% to 50%.

Noted in section 3b, we found from PIV measurements linearly extrapolated to the surface (Fig. 8) that there is a 15% increase in TKE dissipation from 5 mm below the surface. This was in accordance with the difference in *ε*_{1} (Okamoto et al. 2016) measured surface TKE dissipation rates versus VMP-measured TKE dissipation rates (Table 1). If we assume the distributions of *ε* to well approximated by a lognormal curve, then we can quantify its mean (*μ*_{logε}) and variance (*Y* = log_{10}*ε*. From which the mean is defined as *ε*_{1}. In comparing the VMP- and FIV-measured distributions of *ε* we find that the variance (

In our experiment with the FIV approach, we could measure all components of the surface current velocity, which is not always possible during oceanic experiments. Below we will use the FIV estimated *ε*_{1} to be correct estimate of the near-surface TKE dissipation rates and with that assumption we will derive a more suitable methods of the TKE dissipation estimation in the field-generated IR data.

### d. Field method of measuring surface TKE dissipation with IR camera

*ε*at the surface we first compared the best measurement of

*ε*,

*ε*

_{1}[Eq. (8)], with the squared components of ∇

**U**[Eq. (3)]. We took two datasets from the least and most turbulent cases to compare

*R*

^{2}values in Table 3. From Table 3,

*ε*

_{1}. A crude measurement of

*ε*using only the component

**v**and

*y*are in costream direction. There should then be some constant such that

*A*has values which are given in Table 4 [the model for obtaining the constant

*A*can be found in the appendix; Eqs. (A1) and (A2)]. To obtain the best estimate of the value of

*A*we took the ensemble average and obtained the equation (velocity

**v**expressed in meters per second and

*y*in meters)

**U**. However, it is not clear what other component will cause a higher correlation given that for low surface velocity, as observed in the 6 cm s

^{−1}case, the component

*υ*

_{x}shows higher correlation than the

*u*

_{x}component. While the opposite is true for the 16 cm s

^{−1}case (Table 3). Like the single term measurement of

*ε*there should be constants

*B*and

*C*such that either

*R*

^{2}values of

**U**[Eq. (3)] which comprise

*β*[Eq. (4)] produce the highest correlation for all cases. Then

*ε*can be approximated by the components of the surface divergence (

*β*) encompassing as much as 89% of the variance of values obtained form

*ε*

_{1}[Eq. (8)]. Values of

*B*and

*C*are given in Table 4 [the model for obtaining the constants

*B*and

*C*can be found in the appendix; Eqs. (A3)–(A5)].

*R* values of the comparison of the squared components of ∇**U** [Eq. (3)] vs *ε*_{1} [Eq. (8)]. The asterisk denotes poor-quality data.

*R* values of the comparison between *ε*_{I} [Eq. (12)] and *ε*_{II} [Eq. (15)] vs *ε*_{1} [Eq. (8)] accompanied by values of *A*, *B*, and *C* for *ε*_{I} and *ε*_{II}. Each is denoted by a case, surface velocity is measured in cm s^{−1}, and the time is the time series from which the images were taken. Values of *A*, *B*, and *C* have units of m^{2} s^{−3} and are the constant per case from Eq. (12) and Eq. (15). The asterisk denotes poor-quality data.

## 4. Conclusions and discussion

The FIV method was found to be capable of determining the near-surface TKE dissipation rate and is also much easier to use than the traditional microstructure instruments (such as shear probes) and horizontal PIV measurements. We found FIV to be capable of measure surface velocity to within 2%–3% at the lower limit and 3%–7% at the upper limit, dependent upon case. By comparing the FIV-derived data with the in situ dissipation measurements we have validated our FIV measurements and found variance (*ε* to be higher at the near-surface boundary layer when compared to the variance at a depth of 5 mm below the surface.

To be able to use FIV in the field were conditions may not always permit measurements of both velocity components we have constructed the following equations: The TKE dissipation rate can be approximated by *one* FIV-measured current component—e.g., ^{2} s^{−3}] with *R*^{2} = 0.8—with velocity **v** expressed in meter per second and *y* in meters. Using *two* surface current components improves the above estimate by increasing the correlation to *R*^{2} = 0.9 and resulting in the relationship ^{2} s^{−3}]—with velocity **U** expressed in meters per second with *x* and *y* in meters.

We observed in our tank experiment that the surface TKE dissipation increases linearly with current speed and we confirm that the surface TKE dissipation increases linearly with the apparent wind (for observed wind speeds from 6 to 16 cm s^{−1}), in accordance with the Kolmogorov/Obukhov theory (Kolmogorov 1962): which fundamentally links fluctuations in velocity with energy dissipation (Birnir 2013). This observation has a direct impact on observed global air–sea fluxes as even the weakest wind, or strong currents (such as the Gulf Stream), can impart the near-surface with increased values of turbulent mixing and speed up the gas transfer—a function of the *ε*(*z* = 0) (Kitaigorodskii et al. 1983).

## Acknowledgments

We thank Nathan Laxague and Sanchit Mehta for help with preparation of the experiment and its execution. We also would like to thank the reviewer Andrea Cimatoribus for his contribution during the review process. This work was supported by the Gulf of Mexico Research Initiative grant and the National Science Foundation Grant 1434670.

## APPENDIX

### Obtaining Constants of the Model

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