Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence field

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1 GEOPHYSICAL RESEARCH LETTERS, VOL. 40, , doi: /grl.50919, 2013 Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence field Andrey Y. Shcherbina, 1 Eric A. D Asaro, 1 Craig M. Lee, 1 Jody M. Klymak, 2 M. Jeroen Molemaker, 3 and James C. McWilliams 3 Received 21 June 2013; revised 24 August 2013; accepted 27 August 2013; published 13 September [1] A detailed view of upper ocean vorticity, divergence, and strain statistics was obtained by a two-vessel survey in the North Atlantic Mode Water region in winter Synchronous Acoustic Doppler Current Profiler sampling provided the first in situ estimates of the full velocity gradient tensor at O(1 km) scale without the usual mix of spatial and temporal aliasing. The observed vorticity distribution in the mixed layer was markedly asymmetric (skewness 2.5), with sparse strands of strong cyclonic vorticity embedded in a weak, predominantly anticyclonic background. Skewness of the vorticity distribution decreased linearly with depth, disappearing completely in the pycnocline. Statistics of divergence and strain rate generally followed the normal and χ distributions, respectively. These observations confirm a high-resolution numerical model prediction for the structure of the active submesoscale turbulence field in this area. Citation: Shcherbina, A. Y., E. A. D Asaro, C. M. Lee, J. M. Klymak, M. J. Molemaker, and J. C. McWilliams (2013), Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence field, Geophys. Res. Lett., 40, , doi: /grl Introduction [2] The submesoscale band of oceanic turbulence occupies length scales O( km) [Thomas et al., 2008] and is believed to play a critical role in upper ocean mixing. Submesoscale dynamics are characterized by equally strong influences of planetary vorticity, lateral, and vertical shears (Rossby and Richardson numbers O(1)). At these scales, a breakdown of the gradient wind balance and a transition between quasi two-dimensional mesoscale and fully threedimensional small-scale turbulence takes place. The resulting forward energy cascade provides a means to dissipate largeand mesoscale geostrophic turbulence [McWilliams et al., 2001]. [3] Theoretically, an asymmetry in the distribution of the vertical component of relative vorticity ζ v x u y is expected at the submesoscale when the Rossby number Additional supporting information may be found in the online version of this article. 1 Applied Physics Laboratory, University of Washington, Seattle, Washington, USA. 2 University of Victoria, Victoria, Canada. 3 Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California, USA. Corresponding author: Andrey Shcherbina, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105, USA. (ashcherbina@apl.washington.edu) American Geophysical Union. All Rights Reserved /13/ /grl Ro ζ /f = O(1) (here u and v are zonal and meridional velocity components, x and y subscripts represent zonal and meridional derivatives, respectively, f is planetary vorticity) [e.g., Hoskins and Bretherton, 1972; Rudnick, 2001; Klein et al., 2008]. Since flows with strong anticyclonic vorticity ζ < f are unstable and flows with strong cyclonic vorticity are not [Hoskins and Bretherton, 1972], stronger eddies with ζ =O(f) are expected to be predominantly cyclonic. Such an asymmetry is indeed found in high-resolution numerical simulations whenever the signature of strong submesoscale turbulence is not overshadowed by the mesoscale features [e.g., Mahadevan and Tandon, 2006; Capet et al., 2008; Klein et al., 2008]. These conditions can be expected in the interiors of subtropical gyres (Mode Water regions, e.g., Figure 1a) in wintertime due to the instability of deep mixed layers in the presence of lateral density gradients [Boccaletti et al., 2007; Thomas et al., 2008]. [4] Observationally, Rudnick [2001, R01 hereafter] used underway Acoustic Doppler Current Profiler (ADCP) measurements to demonstrate a slight but statistically significant positive skewness of a vorticity proxy, an along-track derivative of cross-track velocity, in the wintertime subtropical North Pacific. We will refer to this technique as the one-ship method and designate vorticity proxy thus obtained as ζ 1 to distinguish it from the full vertical vorticity ζ. Griffa et al. [2008] used global drifter observations to show the prevalence of cyclonic rotation of Lagrangian velocities in the interiors of most subtropical gyres. These estimates also cannot capture the full vertical vorticity ζ and may be biased by trapping in strong eddies [Middleton and Garrett, 1986; Veneziani et al., 2005]. Radar-based observations of submesoscale surface currents [Parks et al., 2009; Kim, 2010] offer excellent resolution and synopticity but are presently limited to coastal regions only. Radiator pattern surveys [e.g., Rudnick, 1996] allow mapping of the full velocity field and estimation of full vertical vorticity but can suffer from severe time space aliasing due to rapid variations of submesoscale velocity field. [5] Here we present statistics of vorticity, divergence, and strain rate of submesoscale turbulence obtained via synchronous velocity measurements by a pair of vessels running on parallel tracks. To our knowledge, this is the first case of consistent sampling of the full gradient tensor of the horizontal velocity field at O(1 km) scale in the open ocean. 2. Observations [6] Upper ocean vorticity statistics in the North Atlantic Mode Water (NAMW) region just offshore of the Gulf Steam were surveyed in February March 2012 as a part of the Lateral Mixing (LatMix) experiment. R/Vs Atlantis and Knorr occupied parallel tracks separated by 1 km for about

2 SHCHERBINA ET AL.: SUBMESOSCALE TURBULENCE STATISTICS Figure 1. Submesoscale turbulence in the North Atlantic Mode Water region in winter. Sea surface temperature (SST) produced in a high-resolution (0.5 km) numerical simulation (see supporting information for model details). Detailed view of simulated normalized relative vertical vorticity, ζ /f; area covered is marked in Figure 1a. Track of the synchronous Lateral Mixing (LatMix) survey 11 to 13 March 2012 superimposed on Moderate Resolution Imaging Spectroradiometer (MODIS) SST image taken 22 March (d) Enlarged fragment of the survey, showing measured near-surface velocity vectors and an example of a circle used to calculate velocity and tracer gradients at a reference point marked with a cross. Aqua MODIS SST imagery courtesy of National Aeronautics and Space Administration Goddard Space Flight Center [Feldman and McClain, 2012]. 500 km through an area of intense submesoscale turbulence (Figure 1c). Both vessels were equipped with 300 and 75 khz underway ADCPs and towed conductivity-temperature-depth profilers (Moving Vessel Profiler and TRIAXUS, respectively). Vertical sampling of the two vessels ADCPs was identical, spanning the range between 15 and 87 m with 4 m bin size for 300 khz instruments and between 21.5 and 570 m with 8 m bin size for 75 khz ADCPs. One minute ensemble averages were used, producing along-track resolution of about 0.2 km. Careful alignment of ADCP measurements was performed to minimize aliasing of ship speed into the measured velocities [Firing and Hummon, 2010]. Additionally, the tracks were crossed at the midpoint to reveal any biases that may have remained in the data. [7] The horizontal velocity gradient tensor " vðx0 ; y0 Þ ¼ ux uy vx vy # ¼ a11 a12 a21 a22 was calculated at 0.2 km intervals along the mean path of the pair of the ships. Derivatives were obtained by fitting linear functions of the form u = u0 + a11x + a12y, v = v0 + a21x + a22y to sets of velocity component measurements within a certain search p ffiffiffi radius of a reference point (x0, y0). A search radius of 2 km was chosen, so that the selected measurement locations formed a consistent, symmetric 1 1 km square pattern comprising of 12 points on average (Figure 1d). Fitting was performed independently for each pair of matching ADCPs (300 and 75 khz) and at each depth bin. We did not impose a nondivergence constraint on the gradients of the horizontal velocity component, as is often done in mesoscale velocity mapping [e.g., Rudnick, 1996]. At the submesoscale, where the Rossby number may approach O(1), lateral divergence ux + vy (matched by the vertical convergence wz) can be expected to be of the same order of magnitude as some of the velocity gradient terms. [8] Statistics of vertical vorticity ζ vx uy, lateral divergence h 2 2 i = δ ux + vy, and lateral strain rate α ux vy þ vx þ uy are shown in Figure 2. The vertical vorticity distribution in the mixed layer (0 50 m) was markedly asymmetric (Figure 2a), with the mode at ζ /f = 0.32, a long positive tail, and a resulting skewness of 2.52 ± 0.10 (the bootstrap standard deviation is given hereafter). Below the mixed layer, the observed vorticity distribution became nearly Gaussian (Figure 2d). [9] Vorticity skewness decreased linearly to zero at 300 m (Figure 3a), which roughly corresponded to the maximum mixed layer depth observed during the survey. Note that vorticity estimates based on 75 khz ADCP observations have higher standard deviations and lower skewness than those based on the 300 khz measurements because of higher

3 (d) (e) (f) Figure 2. Histograms of normalized (a, d) vorticity, (b, e) divergence, and (c, f) strain rate in the mixed layer (0 50 m, 300 khz ADCP, top row) and upper pycnocline ( m, 75 khz ADCP, bottom row). Red curves show corresponding distributions produced in a 0.5 km numerical model. Blue dashed curves in Figures 2a and 2d show one-ship LatMix vorticity distributions. All distributions are scaled by their maximum value. Parameters of observation-based distributions are shown on the insets. measurement noise levels arising from the necessarily slower ping rate of the low-frequency ADCP (see the instrument noise discussion in supporting information). [10] The divergence distribution in the mixed layer was close to Gaussian with slight negative skewness of 0.20 ± 0.13 and standard deviation of 0.47 ± 0.01 (Figure 2b); it did not change much with depth (Figure 2e). Lateral normalized strain rate statistics were well described by a χ distribution with 2 degrees of freedom (Figures 2c and 2f) with standard deviations of 0.56 in the mixed layer and 0.27 in the pycnocline. 3. Discussion 3.1. Comparison With Numerical Simulations [11] A set of nested numerical models was developed in support of the LatMix experiment (see supporting information for details). The large-scale model domain covered most of the Atlantic basin with 5 7 km resolution. Within it, two progressively smaller domains were embedded, with the innermost one focusing on the Gulf Stream region with 0.5 km resolution. This hierarchy allowed realistic forcing Depth (m) khz 75 khz (one-ship) 300 khz 300 khz (one-ship) model towed profiler Skew.[ζ/f] St. dev.[ζ/f] N/f Figure 3. Skewness and standard deviation of the normalized vorticity distribution as a function of depth. Shown are the statistics of LatMix observations with 75 and 300 khz ADCPs (solid and dashed black lines), of the same statistics derived from one-ship LatMix observations (solid and dashed blue lines), and of the numerical model (red line). Mean normalized buoyancy frequency based on Moving Vessel Profiler observations during LatMix survey (green) and the numerical model (red). 4708

4 (d) Figure 4. Power spectral density of (a, c) horizontal velocity and (b, d) normalized vorticity, ζ /f, measured in the mixed layer (0 50 m, top row) and in the upper pycnocline ( m, bottom row) during LatMix experiment with 300 khz ADCPs (solid black line) and produced in a numerical model (red line). Horizontal velocity spectra based on 75 khz ADCP data are shown in Figures 4a and 4c with dashed black lines; 300 khz ADCP was not available below 80 m. The Garret-Munk internal wave spectra (GM75) are shown in Figures 4c and 4d with dashed grey lines. Spectra of one-ship LatMix vorticity estimates are shown in Figures 4b and 4d with blue lines. Grey shading represents relative scale of the 95% confidence interval of the LatMix spectra. of submesocale motions in the innermost domain by larger scales. We take these results as representative of multiple similar model predictions of submesoscale turbulence [e.g., Mahadevan and Tandon, 2006; Capet et al., 2008; Klein et al., 2008] and compare them with the LatMix observations in order to test these predictions. [12] The model predictions matched the mixed layer observations of distributions of vorticity, divergence, and strain rate well (Figures 2a c). However, below the mixed layer observed distributions were substantially wider than their model counterparts (Figures 2d f). The observed gradual decrease of relative vorticity skewness with depth (Figure 3) was qualitatively similar to the numerical results and consistent with Klein et al. [2010], but the change of sign of vorticity skewness at m depth predicted by the models was not present in our observations. [13] Discrepancies between the model and observations in the upper pycnocline can be attributed to the underrepresentation of inertia-gravity waves (IGW) by the model and to the reduced signal-to-noise ratio in the deeper ADCP data (see the instrument noise discussion in supporting information). IGW deficit in the model can be illustrated by comparing the observed and model spectra of horizontal velocity and vertical vorticity (Figure 4). In the mixed layer, the model reproduced the observed spectra well (Figures 4a and 4b), suggesting adequate simulation of submesoscale dynamics and a weak IGW signal. This is consistent with the expected low IGW signal in a deep mixed layer (see supporting information). In the upper pycnocline, the observed spectra were close to those predicted by Garrett and Munk [1975, GM75 hereafter] with buoyancy frequency N = s 1 =46f, while the model spectra were 5 10 times lower at scales 20 km (Figures 4c and 4d), indicating a deficiency of short IGW. Since the model forcing does not include high-frequency wind stress variations or tides, the two major sources of internal wave energy, and has a relatively coarse grid resolution (with respect to the GM spectrum), model underrepresentation of IGW should not be surprising Comparison With Previous Field Observations [14] The pioneering R01 study found statistically significant but much weaker skewness of near-surface submesoscale vorticity (0.36). Differences between R01 and this study could result from several factors Scale [15] Vorticity skewness is expected to decrease at larger scales as the Rossby number decreases [Polvanietal., 1994] (although other mechanisms producing vorticity asymmetry may come into play at planetary scales [Cushman-Roisin et al., 1990; Theiss, 2004]). R01 derived vorticity by finite differentiation of 3 km (12 min) averages of ADCP velocities. In the present study, much finer averaging (1 min or 0.5 km) was used, allowing 1 km vorticity analysis. When vorticity was low passed with a 3 km wide boxcar filter, skewness was reduced only slightly (from 2.5 to 2.2). Therefore, the discrepancy between our results and those of R01 can only be attributed partially to the scale difference Seasonal and Regional Variability [16] Development of submesoscale turbulence is closely related to the mixed layer structure and surface forcing [Thomas et al., 2008] and therefore can be expected to vary 4709

5 (d) Figure 5. Joint probability distribution functions (JPDFs) of (a, b) vorticity and strain rate and (c, d) vorticity and divergence in the mixed layer (0 50 m) based on LatMix 300 khz ADCP observations (Figures 5a and 5c) and the numerical model (Figures 5b and 5d). Black 45 lines in Figures 5a and 5b correspond to one-dimensional shear flow (α = ζ ); horizontal axes (α = 0) correspond to solid body rotation. All JPDFs are normalized by their maximum values. regionally and temporally. Although both the LatMix 2012 and R01 surveys were conducted in wintertime with typically strong surface forcing, the maximum mixed layer depth in the NAMW region during LatMix (250 m) was greater than in R01 (150 m), which is consistent with the climatological means [de Boyer Montégut et al., 2004]. Additional enhancement of submesoscale turbulence during LatMix may have been due to a gale with wind speed reaching 20 m s 1 and heat loss of 700 W m 2 two days prior to the survey (10 11 March). In contrast, during the summertime LatMix survey in the same area in June 2011, with light winds (<10 m s 1 ) and a mixed layer depth 20 m, virtually no vorticity asymmetry was observed: The skewness of one-ship vorticity was Mesoscale Activity [17] Submesoscale turbulence is an intermediate step in the energy cascade from mesoscale flows to small-scale dissipation [McWilliams et al., 2001]. The R01 study area in the Eastern Pacific is remote from major western boundary current systems and therefore characterized by a weak mesoscale eddy field [Chelton et al., 2007]. The relative unimportance of mesoscale forcing in R01 is indirectly supported by Boccaletti et al. [2007], who reproduce the vorticity distribution observed in R01 in a numerical model of free instability of a laterally stratified mixed layer. In contrast, the NAMW region is affected by the strong mesoscale eddies produced by the Gulf Stream (Figure 1); their nonlinear interaction is likely a strong source of submesoscale turbulence One- Versus Two-Ship Vorticity Survey Methodology [18] The only option for estimating vorticity from a singletrack velocity survey is the one-ship method. Our study gives a rare opportunity to quantify the discrepancies introduced by this simplification. [19] LatMix observations show that the one-ship method distorts the shape of the inferred vorticity distribution (Figure 2a): The ζ 1 distribution is substantially more symmetric and has a zero mode. Studies of small-scale isotropic turbulence show distinct differences in the shape of the wave number spectra of vorticity terms [Antonia et al., 1984]. Similarly, our observations show that the one-ship method severely underestimates vorticity variance at the larger scales (Figure 4b). As a result, only 49% of the actual near-surface vorticity variance would be resolvable with the one-ship method in LatMix data (corresponds to a 30% RMS vorticity underestimation). [20] R01 estimated bounds on the one-ship observable vorticity by considering two limiting cases: independence of vorticity terms or their perfect correlation (achieved with solid body rotation). Our observations were generally consistent with the former conjecture. Applicability of this assumption, however, varies with the scale considered (as evident from Figure 4b) as well as the local character of the flow, e.g., for positive and negative vorticity regions. Based on the model analysis, the correlation of vorticity terms was 0.25 for ζ > 0.2f and 0.58 for ζ < 0.2f (see also section 3.3). This explains the distortions of the vorticity distribution shape by the one-ship method. [21] The relationship of skewness of the ζ 1 distribution to that of the full vertical vorticity is more uncertain. Despite the marked difference in distribution shapes (Figure 2a), skewness of ζ 1 in the mixed layer (20 50 m) is only slightly lower than that of ζ (2.48 versus 2.52). This small discrepancy is not universal: At 80 m, the skewness of ζ 1 is only 63% of full vorticity skewness (Figure 3a). In the model, ζ 1 skewness is actually 1% greater than that of ζ, even though 4710

6 the distribution of ζ appears more asymmetric. This illustrates thefactthatpearson s skewness used by R01 and in this study may not be the most robust measure of distribution asymmetry, as it may be excessively affected by the outliers [Brys et al., 2003]. We show skewness values for consistency with the previous studies, but it should be considered in conjunction with the general shapes of the distributions (Figure 2). [22] In light of the LatMix observations, caution is warranted regarding the use of the one-ship method to infer vorticity. If the goal of the observations is to validate or discriminate theoretical or model predictions, then a more productive approach is to focus on metrics that can be estimated reliably from the observations. One example of such an approach is the recent study by Callies and Ferrari [2013], who used along-track spectra of along- and cross-track velocities to infer the governing dynamics Submesoscale Turbulence Structure [23] Additional insight into the structure of submesoscale turbulence is gained from the pairwise joint probability distribution functions (JPDFs) of vorticity, divergence, and strain rate in both the model and the data (Figure 5). The JPDFs of vorticity and strain rate (Figures 5a and 5b) show the pronounced structural differences of cyclonic and anticyclonic vorticity regions, especially at the highest values of ζ. Strong positive vorticity was typically associated with high strain rate and approached a pure shear relationship, α = ζ, for large ζ /f, indicating that the cyclonic vorticity occurred predominantly in fronts. In contrast, negative vorticity features were weaker and had a higher probability of nearly solid body rotation, α ζ, and thus a more eddy-like structure. This confirms visual interpretation of vorticity maps (such as Figure 1b) obtained in highresolution numerical models [e.g., Mahadevan and Tandon, 2006; Capetetal., 2008; Klein et al., 2008]. The model predicts weakly positive divergence associated with negative vorticity (Figure 5d). LatMix observations do not support this pattern, but due to large scatter, neither do they clearly refute it (Figure 5c). 4. Conclusion [24] Synchronous two-ship ADCP sampling during LatMix experiment provided the first in situ estimates of the full velocity gradient tensor at O(1 km). The observed near-surface vorticity distribution in the NAMW region in winter was substantially more asymmetric than that found in previous submesoscale observations. LatMix observations were in excellent agreement with numerical model predictions for active submesoscale turbulence in this region. [25] Acknowledgments. We are grateful to the captains and crews of R/Vs Atlantis and Knorr, who made these observations possible. We thank Jules Hummon and Eric Firing (University of Hawaii) for their help calibrating and processing the shipboard ADCP. Many thanks to the entire LatMix team for continuous stimulating discussion. We thank Amala Mahadevan and an anonymous reviewer for their suggestions to improve the manuscript. This work was supported by ONR grants N , N , and N under the Scalable Lateral Mixing and Coherent Turbulence Departmental Research Initiative. [26] The Editor thanks Amala Mahadevan and an anonymous reviewer for assistance evaluating this manuscript. References Antonia, R. A., L. W. B. Browne, and A. J. Chambers (1984), On the spectrum of the transverse derivative of the streamwise velocity in a turbulent flow, Phys. 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