• Belcher, S. E., and et al. , 2012: A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett., 39, L18605, https://doi.org/10.1029/2012GL052932.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Goh, G., and Y. Noh, 2013: Influence of the Coriolis force on the formation of a seasonal thermocline. Ocean Dyn., 63, 10831092, https://doi.org/10.1007/s10236-013-0645-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grant, A. L. M., and S. E. Belcher, 2009: Characteristics of Langmuir turbulence in the ocean mixed layer. J. Phys. Oceanogr., 39, 18711887, https://doi.org/10.1175/2009JPO4119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grant, A. L. M., and S. E. Belcher, 2011: Wind-driven mixing beneath the oceanic mixed layer. J. Phys. Oceanogr., 41, 15561575, https://doi.org/10.1175/JPO-D-10-05020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lee, E., Y. Noh, B. Qiu, and S. Yeh, 2015: Seasonal variation of the upper ocean responding to surface heating in the North Pacific. J. Geophys. Res. Oceans, 120, 56315647, https://doi.org/10.1002/2015JC010800.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, Q., A. Webb, B. Fox-Kemper, A. Craig, G. Danabasoglu, W. G. Large, and M. Vertenstein, 2016: Langmuir mixing effects on global climate: WAVEWATCH III in CESM. Ocean Modell., 103, 145160, https://doi.org/10.1016/j.ocemod.2015.07.020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McWilliams, J. C., E. Huckle, J. Liang, and P. P. Sullivan, 2014: Langmuir turbulence in swell. J. Phys. Oceanogr., 44, 870890, https://doi.org/10.1175/JPO-D-13-0122.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Noh, Y., and Y. Choi, 2018: Comments on “Langmuir Turbulence and Surface Heating in the Ocean Surface Boundary Layer.” J. Phys. Oceanogr., https://doi.org/10.1175/JPO-D-17-0135.1, 455458.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pearson, B. C., A. L. M. Grant, J. A. Polton, and S. E. Belcher, 2015: Langmuir turbulence and surface heating in the ocean surface boundary layer. J. Phys. Oceanogr., 45, 28972911, https://doi.org/10.1175/JPO-D-15-0018.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smyth, W. D., E. D. Skyllingstad, G. B. Crawford, and H. Wijesekera, 2002: Nonlocal fluxes and Stokes drift effects in the K-profile parameterization. Ocean Dyn., 52, 104115, https://doi.org/10.1007/s10236-002-0012-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yoshikawa, Y., 2015: Scaling surface mixing/mixed layer depth under stabilizing buoyancy flux. J. Phys. Oceanogr., 45, 247258, https://doi.org/10.1175/JPO-D-13-0190.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilitinkevich, S., 1972: On the determination of the height of the Ekman boundary layer. Bound.-Layer Meteor., 3, 141145, https://doi.org/10.1007/BF02033914.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    Demonstration of the different depth definitions in NC and PGPB using LES profiles of (a) buoyancy flux , (b) TKE transport, and (c) temperature. Profiles are shown for two simulations with f = 1.4 × 10−4 s−1 (solid) and f = 0 (dashed). Gray lines show the length scales hwb in (a), hTKE in (b), and hN in (c) for their respective simulation. Simulations have LL = 93 m and are from the SS and SN simulation sets of PGPB.

  • View in gallery

    Comparison of (left) NC and (right) PGPB scalings for hTKE/h0. The PGPB scaling is [1 + (2β)−1(h0/LL)]−1 with (2β)−1 = 3. Results are shown from the simulation sets of PGPB with varying planetary rotation (SW, SM, SS) and initial layer depth h0 (SH) and are denoted by the same symbols as PGPB. The dashed line represents perfect agreement with the PGPB scaling.

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Reply to “Comments on ‘Langmuir Turbulence and Surface Heating in the Ocean Surface Boundary Layer’”

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  • 1 Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, Rhode Island
  • | 2 Department of Meteorology, University of Reading, Reading, United Kingdom
  • | 3 National Oceanography Centre, Liverpool, United Kingdom
  • | 4 Met Office, Exeter, United Kingdom
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Abstract

The differences between the conclusions of Noh and Choi and of Pearson et al., which are largely a result of defining different length scales based on different quantities, are discussed. This study shows that the layer over which Langmuir turbulence mixes (nominally hTKE) under a stabilizing surface buoyancy flux should be scaled by a combination of the Langmuir stability length LL and initial/nocturnal boundary layer depth h0 rather than by the Zilitinkevich length.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Brodie C. Pearson, brodie_pearson@brown.edu

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-15-0018.1.

Abstract

The differences between the conclusions of Noh and Choi and of Pearson et al., which are largely a result of defining different length scales based on different quantities, are discussed. This study shows that the layer over which Langmuir turbulence mixes (nominally hTKE) under a stabilizing surface buoyancy flux should be scaled by a combination of the Langmuir stability length LL and initial/nocturnal boundary layer depth h0 rather than by the Zilitinkevich length.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Brodie C. Pearson, brodie_pearson@brown.edu

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-15-0018.1.

Noh and Choi (2018, hereinafter NC) recently submitted a comment on the published work of Pearson et al. (2015, hereinafter PGPB). In their comment, Noh and Choi suggest that the depth of the thermocline should be scaled by the Zilitinkevich scale LZ (Zilitinkevich 1972), as opposed to PGPB who suggested that the mixed and boundary layer depths scale as a combination of the Langmuir stability length LL (Belcher et al. 2012) and initial ocean surface boundary layer (OSBL) depth h0. In this reply, we first summarize the different depth scales used in NC and PGPB. We then reexamine the results of NC and PGPB to identify the root of any discrepancies. Finally, we demonstrate that PGPB’s main conclusions are unchanged, discuss how NC’s work has encouraged us to reconsider elements of PGPB, and present a summary and outlook for this work.

Noh and Choi’s comment encourages clarity around different depths within the OSBL. In Fig. 1, we demonstrate these depths along with the LES profiles used to derive them. The first hwb is the depth at which a linear fit to the near-surface buoyancy flux reaches zero. The second hTKE is the depth at which turbulence kinetic energy (TKE) transport tends to zero. Finally, hN is the depth of maximum stratification N2. Conceptually, hwb estimates the depth over which turbulence homogeneously mixes temperature θ because this “θ-mixing layer” must have constant with depth [salinity is constant in these large-eddy simulations (LES), so θ is proportional to buoyancy], and hTKE is the depth over which turbulence mixes TKE. Both hwb and hTKE relate to the processes (turbulence) driving the evolution of the OSBL and can be diagnosed from LES but are difficult to measure in observations. Meanwhile, hN is a function of stratification, an emergent property of the turbulent OSBL, and is easy to diagnose in observations but offers less insight into turbulent processes within the OSBL. Each depth has different utility; to parameterize the evolution of the OSBL, the depths relating to the processes driving this evolution, hTKE and hwb, are most useful, but for comparison between LES and generic observations, hN is most useful because of the ease of observing N2. It should be noted that PGPB did not use the above notation and instead called hTKE and hwb the boundary layer and mixed layer depths, respectively. This seemed reasonable terminology for large f, where hTKE > hwb, but is not suitable as f → 0 and hwb > hTKE (Figs. 1a,b).

Fig. 1.
Fig. 1.

Demonstration of the different depth definitions in NC and PGPB using LES profiles of (a) buoyancy flux , (b) TKE transport, and (c) temperature. Profiles are shown for two simulations with f = 1.4 × 10−4 s−1 (solid) and f = 0 (dashed). Gray lines show the length scales hwb in (a), hTKE in (b), and hN in (c) for their respective simulation. Simulations have LL = 93 m and are from the SS and SN simulation sets of PGPB.

Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0176.1

PGPB proposed that hTKE does not vary significantly with f (Fig. 1b) and can be estimated as
e1
where , is the velocity scale of Langmuir turbulence (Grant and Belcher 2009), B0 is the surface buoyancy flux, h0 is the OSBL depth before heating is applied, and (2β)−1 = 3. In contrast, NC highlight the important point that hN (which PGPB did not consider) and hwb vary with f; specifically, they scale well with , where is the velocity scale of shear turbulence and in the present simulations. PGPB did propose Eq. (1), with (2β)−1 = 3.5, to estimate hwb but also stated the caveat that as f decreases, hwb is no longer a good estimate of the θ-mixing layer. This is because hwb is based on the assumption that varies from its surface value to zero at the base of the θ-mixing layer. As a result, hwb is a good estimate of the θ-mixing layer when the buoyancy flux into the thermocline is small, such as for an infinitely thin thermocline. However, as f → 0 the thermocline becomes thicker, the buoyancy flux into the thermocline increases (Fig. 1, dashed lines), and hwb overestimates the depth of the θ-mixing layer (hwb > hN and hTKE). The mixing of heat within the thermocline is driven by shear turbulence (Grant and Belcher 2011). Previous work demonstrated several scenarios where hwb would be a good (PGPB, their Figs. 2, 4b; NC, their Figs. 1a,b) or poor (Grant and Belcher 2011, their Figs. 3, 6c; NC, their Figs. 1c,d) estimate of the θ-mixing layer. The dependence of hwb on f shown by NC could therefore be attributed to hwb capturing more of the thermocline as f decreases as seen in Fig. 1, where we compare LES with f = 0 and f = 1.4 × 10−4 s−1, and in NC (their Figs. 1a,b vs 1c,d).

In contrast to hwb, the TKE transport depth hTKE (Fig. 1b) diagnoses the depth over which turbulence mixes TKE in both rotating (Grant and Belcher 2009) and nonrotating (Grant and Belcher 2011) scenarios. PGPB compared LES values of hTKE against Eq. (1) in their Fig. 10b, but they did not demonstrate whether this scaling performs better than the LZ scaling proposed by NC. To test this, Fig. 2 shows hTKE/h0 from LES as a function of LZ/h0 and as a function of the PGPB scaling. The scaling proposed by PGPB with hTKE = f(LL, h0) performs better than hTKE = f(LZ), even without considering LES with LZ = ∞. This figure is analogous to Fig. 3 of NC, but here we show hTKE rather than hwb, and we contrast LZ scaling with the PGPB scaling rather than with the Monin–Obukhov length. There is good agreement between the PGPB scaling and LES (dashed line signifies perfect agreement). When the LES results are scaled by LZ, they show more spread in hTKE than the PGPB scaling. Data are only shown from simulation sets where f or h0 were varied, TKE transport profiles were available (hence no NC simulations), and the data could be plotted on both figures. The nonrotating simulations of PGPB have LZ = ∞ but plot close to the PGPB scaling (not shown). The dependence of hTKE on a length scale other than LL or LZ was anticipated by the nonlinear relationship between hTKE and LL for LZ = ∞ (PGPB, their Fig. 9b, black crosses).

Fig. 2.
Fig. 2.

Comparison of (left) NC and (right) PGPB scalings for hTKE/h0. The PGPB scaling is [1 + (2β)−1(h0/LL)]−1 with (2β)−1 = 3. Results are shown from the simulation sets of PGPB with varying planetary rotation (SW, SM, SS) and initial layer depth h0 (SH) and are denoted by the same symbols as PGPB. The dashed line represents perfect agreement with the PGPB scaling.

Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0176.1

Following NC, we have been encouraged to reconsider PGPB, and we believe that more care should have been taken to discuss the physical meaning of each layer, as we have done above, and the conditions under which their physical interpretations are no longer appropriate. Because hTKE robustly measures the TKE mixing layer in all simulations, while hwb diverges from the θ-mixing layer for small f, PGPB should have emphasized more strongly the collapse of hTKE in their Fig. 9b rather than the (less complete) collapse of hwb in their Fig. 9a.

NC commented that the scaling of the seasonal thermocline by LZ is supported observationally (Lee et al. 2015; Yoshikawa 2015), however, these studies typically measure hN. Calculating hTKE directly from observational data would require measurement of turbulent velocities and their correlations, which is beyond the scope of present global observations. It should also be noted that the time and depth scales of diurnal variability are smaller than seasonal variability, which could preclude different physical balances.

PGPB and Goh and Noh (2013) argue that scalings that depend on h0 and LZ, respectively, could appear from energetic budget arguments. However, to the extent of our knowledge no extant work has provided a robust physical justification of why LZ or h0 should be important scalings for the Langmuir turbulence mixing layer under diurnal cycling. In particular, two key assumptions made by Zilitinkevich (1972) in justifying LZ scaling for shear-driven turbulence have both been shown to be poor approximations in Langmuir turbulence, namely, that turbulence mixes momentum along the Eulerian current shear (Smyth et al. 2002; McWilliams et al. 2014) and that there is a standard Richardson number criterion for stability (Li et al. 2016).

In summary, LES provides information about turbulence, which can be used to define depths that are directly related to the processes driving the OSBL evolution but may be difficult to diagnose in observations. TKE transport provides a robust definition of the TKE mixing depth hTKE across a wide range of parameter space. We showed that hTKE agrees with the scaling proposed by PGPB. Meanwhile, NC showed that hwb is affected by variations in f and scales reasonably with LZ. We suggest that the latter result is because as the thermocline becomes thicker and more heat is mixed down into it, hwb captures more of the (f dependent) thermocline. There are several interesting avenues of research on the stable, wave-driven OSBL suggested by PGPB, NC, and the present work. These include understanding the relationships between depth scales that are easily observed (e.g., hN) and those that are calculated directly from OSBL turbulence (e.g., hTKE), investigating what determines the buoyancy flux into the thermocline (and hence the deviations of hwb from the θ-mixing layer), and robustly diagnosing the mechanisms by which LZ and h0 affect the vertical structure of the OSBL.

Acknowledgments

Feedback from two anonymous reviewers improved this work. This work was supported by a grant from the Office of Naval Research N00014-17-1-2963 and from the National Environment Research Council under the OSMOSIS project NE/I020083/1.

REFERENCES

  • Belcher, S. E., and et al. , 2012: A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett., 39, L18605, https://doi.org/10.1029/2012GL052932.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Goh, G., and Y. Noh, 2013: Influence of the Coriolis force on the formation of a seasonal thermocline. Ocean Dyn., 63, 10831092, https://doi.org/10.1007/s10236-013-0645-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grant, A. L. M., and S. E. Belcher, 2009: Characteristics of Langmuir turbulence in the ocean mixed layer. J. Phys. Oceanogr., 39, 18711887, https://doi.org/10.1175/2009JPO4119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grant, A. L. M., and S. E. Belcher, 2011: Wind-driven mixing beneath the oceanic mixed layer. J. Phys. Oceanogr., 41, 15561575, https://doi.org/10.1175/JPO-D-10-05020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lee, E., Y. Noh, B. Qiu, and S. Yeh, 2015: Seasonal variation of the upper ocean responding to surface heating in the North Pacific. J. Geophys. Res. Oceans, 120, 56315647, https://doi.org/10.1002/2015JC010800.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, Q., A. Webb, B. Fox-Kemper, A. Craig, G. Danabasoglu, W. G. Large, and M. Vertenstein, 2016: Langmuir mixing effects on global climate: WAVEWATCH III in CESM. Ocean Modell., 103, 145160, https://doi.org/10.1016/j.ocemod.2015.07.020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McWilliams, J. C., E. Huckle, J. Liang, and P. P. Sullivan, 2014: Langmuir turbulence in swell. J. Phys. Oceanogr., 44, 870890, https://doi.org/10.1175/JPO-D-13-0122.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Noh, Y., and Y. Choi, 2018: Comments on “Langmuir Turbulence and Surface Heating in the Ocean Surface Boundary Layer.” J. Phys. Oceanogr., https://doi.org/10.1175/JPO-D-17-0135.1, 455458.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pearson, B. C., A. L. M. Grant, J. A. Polton, and S. E. Belcher, 2015: Langmuir turbulence and surface heating in the ocean surface boundary layer. J. Phys. Oceanogr., 45, 28972911, https://doi.org/10.1175/JPO-D-15-0018.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smyth, W. D., E. D. Skyllingstad, G. B. Crawford, and H. Wijesekera, 2002: Nonlocal fluxes and Stokes drift effects in the K-profile parameterization. Ocean Dyn., 52, 104115, https://doi.org/10.1007/s10236-002-0012-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yoshikawa, Y., 2015: Scaling surface mixing/mixed layer depth under stabilizing buoyancy flux. J. Phys. Oceanogr., 45, 247258, https://doi.org/10.1175/JPO-D-13-0190.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zilitinkevich, S., 1972: On the determination of the height of the Ekman boundary layer. Bound.-Layer Meteor., 3, 141145, https://doi.org/10.1007/BF02033914.

    • Crossref
    • Search Google Scholar
    • Export Citation
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