1. Introduction
Submesoscale processes, characterized by horizontal scales of 0.1–10 km and Rossby and Richardson numbers of 
The broad significance of these processes is demonstrated in the context of the seasonal cycle of subtropical mode water formation in the North Atlantic. Ocean mode waters are defined by their anomalously low PV, and represent a key pathway for communicating a history of air–sea interaction into the ocean interior, exporting heat and carbon, and influencing the gyre-scale circulation (Hanawa and Talley 2001; Bates et al. 2002; Kwon and Riser 2004). The formation of ocean mode waters is fairly well explained by air–sea buoyancy fluxes; however, their seasonal destruction from late winter through the fall is less thoroughly understood. It has been hypothesized that mode water destruction involves turbulent mixing in the ocean interior (Qiu et al. 2006; Billheimer and Talley 2016), possibly influenced by double diffusion (Wong and Johnson 2003; Tsuchiya and Talley 1998). However, the more complex dynamics of PV relative to temperature, salinity, and velocity raises doubts that turbulent mixing unequivocally acts to destroy mode waters. In particular, turbulent diffusion of heat, salt, and momentum does not necessarily induce a downgradient flux of PV (Thorpe and Rotunno 1989; Keyser and Rotunno 1990) and therefore may not always mix away minima in PV associated with mode waters. Furthermore, following the impermeability theorem of Haynes and McIntyre (1987), interior mixing processes do not directly affect isopycnal PV budgets and hence cannot change the volume-integrated PV anomalies associated with mode waters. The impermeability theorem instead suggests that mode water destruction should involve processes that inject high PV into the mode water isopycnal layer at the ocean surface or bottom.
An important cause of surface PV flux is the air–sea flux of buoyancy (Nurser and Marshall 1991). However, analyses of the mode water seasonal cycle show that surface buoyancy fluxes lead to a net removal of PV, and hence the total mode water PV budget cannot be closed based on air–sea fluxes alone (Maze and Marshall 2011; Forget et al. 2011). A potential candidate for resolving this conundrum comes from considering recent observational and numerical modeling work, which suggests that the mode water formation regions are sites of active submesoscale turbulence (Shcherbina et al. 2013; Gula et al. 2014; Callies et al. 2015; McWilliams 2016). This implies that mode waters, rather than being formed in large, spatially homogeneous, outcropping regions, are the cumulative result of formation, destruction, and subduction over many small outcropping regions between submesoscale fronts, where frontal dynamics can modify the surface flux of PV (Thomas and Ferrari 2008). Here we demonstrate how the TTW circulation acts as a source of PV at submesoscale fronts, modifying the seasonal cycle of mode water formation and destruction, and reducing the total annual mode water PV removal by a factor of ~3, relative to the expectation based on surface buoyancy fluxes alone. Surface TTW PV fluxes in numerical models are also shown to be highly resolution dependent, with important implications for accurate large-scale modeling of the interior ocean circulation.
The manuscript is organized as follows. In section 2 we develop the theory of how the TTW circulation generates a surface PV flux. Simple scalings for the TTW PV flux are then derived and tested in idealized model runs of frontal spindown (section 3). In section 4 these ideas are applied to a realistic high-resolution model of the North Atlantic Subtropical Mode Water (Eighteen Degree Water), and it is demonstrated that these submesoscale processes are a leading-order term in the PV budget. In section 5 we discuss the relation of these nonconservative processes to adiabatic baroclinic mixed layer instabilities and boundary layer restratification, and major findings are summarized in section 6.
2. Theory
a. Potential vorticity
Extensive prior work has confirmed that EBF can become a dominant term in both the surface buoyancy and PV budgets at the strong horizontal buoyancy gradients characteristic of the submesoscale (e.g., Thomas and Lee 2005; Thomas and Ferrari 2008; D’Asaro et al. 2011). These strong fronts are however also associated with large thermal wind shears, which implies 
b. Interpretation in terms of frontal dynamics
A detailed discussion of wind-driven PV fluxes can be found in Thomas and Ferrari (2008). However, a point of particular relevance here is that the surface wind stress can alternately inject or remove PV, depending on the alignment of the wind stress with the horizontal buoyancy gradient. Thus, for a wind stress of large horizontal scale, when 
Schematic of the TTW circulation and associated PV fluxes, as discussed in section 2. (left) The basic overturning circulation, where a horizontal buoyancy gradient (thin contours) generates a thermal wind shear, and in the turbulent boundary layer, a TTW overturning circulation in the across-front plane (Wenegrat and McPhaden 2016). (top right) The physical origin of the associated PV fluxes, where the sheared cross-frontal flow is restratifying, generates a positive buoyancy tendency at the surface (shown schematically by the shading). By (10), this indicates a downward flux of PV at the surface, 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
The TTW circulation is also both restratifying and frontogenetic (Thompson 2000; McWilliams et al. 2015), as shown in Fig. 1. This leads to an evolution of the buoyancy field, which in turn affects the efficiency of both vertical and horizontal vertical mixing of buoyancy. This is a critical point, as, through (12), it can be seen that the total PV flux is determined by the difference between the cross-frontal Ekman advection and the turbulent mixing of buoyancy, which will itself be affected by the frontal dynamics and hence may not scale directly with the surface buoyancy flux. This issue is complicated by the fact that, generally, determining the balance between the rate of change, cross-frontal advection, and the turbulent of mixing of buoyancy is intractable to analytical methods, as it will involve both the nonlinear evolution of the buoyancy field caused by TTW advection and the effects of the changes in buoyancy on the resulting turbulence properties of the boundary layer. However, a simplified analytical analysis of this coupling is provided in appendix B, using an asymptotic expansion that assumes a thin Ekman layer relative to the full turbulent boundary layer depth, a limit that is consistent with the deep mixed layers found in the mode water formation regions.
The origin of the last two terms in (8) can thus be understood physically as resulting from a coupled interaction between the cross-frontal advection of buoyancy, by the combined wind-driven and TTW flow, and a partially compensating increase in the turbulent mixing of buoyancy, which appears implicitly in (8) through changes to the convective layer depth h. The term in parentheses in (8) thus acts as an effective buoyancy flux that determines the Lagrangian rate of change of the surface buoyancy following the inviscid flow, consistent with (10) (and Marshall et al. 2001).
c. Scalings for the PV flux
3. Idealized numerical experiments
a. Configuration of experiments
To test these scalings, we run a suite of idealized numerical experiments using the MITgcm (Marshall et al. 1997). The model is run in hydrostatic mode, for a horizontally periodic domain extending 80 km in the alongfront (x) direction, 100 km in the across-front (y) direction, and 300 m in the vertical. The grid resolution is a uniform 500 m in the horizontal, with 50 stretched vertical levels, giving a vertical resolution that ranges from 
The basic cross-frontal structure, and the initial geostrophically balanced velocity field, are shown in Fig. 2, and additional details of the model configuration are given in appendix C. This idealized model configuration is used to test the scalings developed in section 2, by varying the initial horizontal buoyancy gradient 
Basic initial cross-frontal structure, with contours of buoyancy (black) and alongfront velocity (color). Velocity is normalized by 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
b. Example model run
Snapshots of the temperature field for the model run with initial conditions 
Example evolution of the temperature field during frontal spindown for 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Domain-averaged turbulent boundary layer depth 
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(top) Surface vertical relative vorticity 
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The strong frictional PV injection, and enhanced diabatic PV removal, associated with submesoscale fronts can be seen clearly in a snapshot from day 16 of the model run, shown in Fig. 6, with surface fluxes enhanced by two orders of magnitude over a scaling based on the surface buoyancy flux alone, 
Snapshot of the near-surface temperature and vertical PV fluxes, from day 16 of the example model run discussed in section 3b [with 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Composite vertical profiles, conditionally averaged over regions of (left) weak horizontal buoyancy gradients and (right) strong horizontal buoyancy gradients, over ±1 inertial period centered at day 16 of the simulation discussed in section 3b, as shown in Fig. 6. To form these composites, individual profiles are first interpolated to a vertical coordinate that is normalized by the local (in space and time) KPP boundary layer depth and then averaged using the 5% trimmed mean to exclude outliers. In the top row, the value of the surface buoyancy flux 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Away from sharp fronts the flux terms in (7), 
As noted above, the cumulative effect of these submesoscale frontal processes on the PV budget will depend on the relative efficiency factors appropriate for scaling the frictional and diabatic removal, that is, 
c. Comparison to theory
Parameter space is explored using 12 model runs, formed as permutations of a set of four different values for the strength of the initial horizontal buoyancy gradient and a set of three values for the surface heat flux (Table 1). Each run is free to evolve from the initial conditions, and hence transient changes in the PV fluxes form an important part of the overall solutions; however, a useful indication of the relative importance of eddy versus surface effects for a model run is given by 
Comparison of theoretical scalings (y axes) and model output (x axes) for the 12 model runs discussed in section 3c (Table 1). Daily averaged values are plotted for visual clarity, with initial conditions for each run indicated by a combination of marker color and shape (legend). The correlation coefficients for the 2-h output fields are given above each plot. The 1–1 line is indicated by a solid black line in each plot, and the dashed black lines indicate agreement to within a factor of 2.
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
The success of the constant coefficient scalings in fitting the model results across all of parameter space implies that the results of appendix B, which were derived under an assumption of a thin Ekman layer, holds even for cases where 
d. Isopycnal PV budget
Before considering a realistic model of the North Atlantic in section 4, it is useful to consider an isopycnal layer PV budget in the idealized domain to illustrate how these processes may affect the mode water PV budget during the seasonal transition from winter to spring. To do this we run the same idealized model configuration [with 
Example isopycnal layer PV budget as discussed in section 3d. (top) Bounding isosurfaces (T = 17°C and T = 19°C, in gray) and the surface temperature field (color scale). (middle) Time-varying surface heat flux. (bottom) Cumulative change in the isopycnal-layer PV with components indicated in the legend.
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Considering the cumulative PV budget for this isopycnal layer, it can be seen that between days 7 and 15 there is a rapid frictional injection of PV into the layer, leading to 
4. Realistic model of the North Atlantic Subtropical Mode Water
In this section we evaluate the impact of submesoscale processes in the seasonal PV budget of the North Atlantic Subtropical Mode Water, or Eighteen Degree Water (EDW), using a realistic, submesoscale permitting, model of the Gulf Stream region. The model is implemented using the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005), with configuration details described at length in earlier works such as Gula et al. (2014, 2015). We consider a large region of the northwest Atlantic, run at 1.5-km horizontal resolution, with boundary conditions provided by nesting from a larger domain run at ~5-km resolution (Fig. 10). Surface forcing is provided by climatological heat flux, evaporation and precipitation, and surface wind stress constructed from climatology plus daily winds with variance close to climatological values (see Gula et al. 2015). The use of climatological forcing excludes the possible importance of air–sea feedbacks, although a correction term of 30 W m−2 °C−1 is applied to the surface heat flux based on the difference between the modeled SST and the climatological SST. The model is run for 16 months, and we analyze the mode water formation season over the last 12 months. Given the single realization of the annual cycle, and the other limitations mentioned above, the results of this section are best treated as providing an extension of the findings of section 3 to a model with more complete and realistic physics, rather than a definitive exploration of the importance of these processes for the EDW water in particular (see, e.g., Maze et al. 2013).
Temperature at 400-m depth in the parent simulation of the North Atlantic (color scale). The domain of the 1.5-km run is indicated in green. The heavy black line indicates the 17°C contour, often treated as the lower boundary of the EDW.
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Snapshot of the wintertime density field in the 1.5-km model run along the section indicated in Fig. 12. The isopycnal layer used here to define the mode water is outlined with a heavy black line.
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Annual averaged surface fields from the 1.5-km run. The J vectors are evaluated from the scalings given in the text and are multiplied by 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
The seasonal budget of the mode water volume-integrated PV is calculated over a subdomain (indicated in top-left panel of Fig. 12), encompassing the regions of the largest surface fluxes, shown in Fig. 13. The mode water outcrops in this domain in early November, with PV being removed until March, followed by a brief period of PV injection until the mode water is subducted in May. The scalings developed in section 3 follow the modeled changes in PV closely, indicating that the reduction of PV loss in the model can be attributed to PV injection by TTW circulations. Notably, the rate of wintertime PV removal is reduced by approximately 50%, and the formation season shortened by approximately 3 weeks, relative to the expectation from surface fluxes alone. The cumulative PV budget (Fig. 13, bottom) indicates that there is a net loss of mode water PV in this subdomain, although the total PV loss is only approximately 30% of the loss that would be expected based on surface buoyancy fluxes alone. These findings are consistent with the results of Lévy et al. (2010), who observed an increase in stratification of the mode water in an idealized gyre when the submesoscale was resolved. Likewise, while volume-integrated PV does not uniquely determine the mode water volume (Deremble et al. 2014), the reduction of mode water PV loss relative to surface fluxes echoes observational analyses that find the volume of subducted mode water is greatly reduced from that formed by surface fluxes (Kelly and Dong 2013).
Seasonal PV budget for the subdomain indicated in Fig. 12. (top) The rate of change, with 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
The volume-integrated PV varies by a factor of 2 over the seasonal cycle, with a seasonal minimum in late February. After this seasonal minimum, advective fluxes through the edges of the subdomain replenish the volume-integrated mode water PV, and the final cumulative loss of PV through the surface is balanced entirely by the cumulative advective fluxes. This is consistent with the analysis of Deremble and Dewar (2013); however, here the cumulative nonconservative PV loss is 30% of the average mode water PV in the model domain, indicating a renewal time scale of only ~3 years, as opposed to the ~100-yr time scale they find. These differences may be partly due to Deremble and Dewar (2013) considering annually averaged PV, excluding the seasonal cycle of mode water PV creation and destruction, over a control volume that encompassed less of the formation region than the domain considered here (Fig. 10). A full comparison between our results and prior work on the relationship between mode water PV and volume would be confounded by differences in model resolution, our single realization of an annual cycle, and our model domain covering an incomplete portion of the total mode water volume. However, the results of our analysis suggest that surface PV fluxes may induce large seasonal variability of the mode water PV and lead to a cumulative annual PV flux that represents a significant portion of the total mode water PV. Submesoscale processes greatly modify these surface PV fluxes and hence may exert a strong control on the gyre-scale circulation through their effects on the mode water PV budget.
One of the implications of the above analysis, and (23), is that the potential vorticity budget of the mode water in numerical models will be strongly dependent on the strength of the resolved surface buoyancy fronts. An example of the effect of decreased resolution can be found by averaging the model fields to a 10.5-km grid, equivalent to the resolution of what would currently be considered a very high-resolution global ocean model, and then recalculating the PV fluxes from the scalings. Figure 14 shows 
Comparison of (left) surface buoyancy PV flux and (right) TTW PV flux evaluated (top) at high resolution (HR) and (bottom) using fields smoothed to lower resolution (LR). The ratio of the area-integrated scalings is indicated in the bottom plots.
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
5. Turbulent thermal wind restratification
One of the principal topics driving interest in the ocean submesoscale is the tendency of submesoscale processes to restratify the near surface, and it is now widely recognized that submesoscale restratification is likely important to a wide range of physical, and biogeochemical, processes. As such, important steps have been taken toward parameterizing the effects of these processes within models that do not explicitly resolve the relevant length scales (Fox-Kemper and Ferrari 2008; Fox-Kemper et al. 2008, 2011; Bachman et al. 2017). The theory developed in section 2, and tested in section 3, suggests that, beyond submesoscale instabilities, turbulent nonconservative processes at the submesoscale likely also modify the boundary layer in critical ways. This is further supported by application of these ideas to a high-resolution submesoscale-resolving simulation of the mode water formation region in the North Atlantic (section 4).
Notably, (10) and (23) together imply that the restratification associated with the TTW flow might be considered in terms of an effective buoyancy flux 
Plots of these quantities are shown in Fig. 15, for the model run discussed in section 3b. In the upper 75 m it can be seen that restratification is driven by a combination of both FRONT and JNC, with the periods of the largest rates of restratification (e.g., day 
Bulk stratification budget, as given by (25) and discussed in section 5, for the model run discussed in section 3b [
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
Beyond the implications for restratification, the TTW circulations are also associated with significant PV fluxes (sections 3 and 4), which differ from adiabatic overturning circulations, which rearrange but do not change the volume-integrated PV. These processes are, however, closely coupled, as frontogenesis during the baroclinic instability process enhances the TTW PV flux, in turn providing a boundary source of PV for the eddy equilibration process (Nakamura 1994). Models that do not resolve the submesoscale are thus likely to overestimate the net surface PV removal, with far-reaching implications, including for the seasonal cycle of mode water formation and destruction, as shown in section 4. Given the strong constraint PV provides on the general circulation, parameterization of these processes may be at least as important as parameterizing the restratifying effects of adiabatic submesoscale baroclinic instabilities. Implementation of a parameterization for the nonconservative effects discussed here would be simplified by the fact that the surface TTW flow is always down the buoyancy gradient, hence of the sense to inject PV, unlike wind-driven PV fluxes [(18)], which have a directional dependence, and hence require resolved submesoscale fronts.
6. Summary
In this manuscript we considered the flux of potential vorticity at the ocean surface and the importance of submesoscale processes in the surface boundary layer. Particular focus was given to the PV flux associated with the turbulent thermal wind, whereby boundary layer turbulence at a horizontal buoyancy gradient generates an ageostrophic cross-frontal circulation, leading to a downgradient flux of buoyancy, the geostrophic Ekman buoyancy flux [(9)]. The TTW flow thus leads to a source of PV at the ocean surface 
The frictional and diabatic PV fluxes are also shown to be coupled, with enhanced frictional PV fluxes largely offset by enhanced diabatic fluxes. This coupling between nonconservative terms in the momentum and buoyancy equations is consistent with realistic modeling studies that have noted that enhanced resolution of submesoscale processes does not always result in enhanced boundary layer restratification, as vertical buoyancy fluxes are partially compensated for by enhanced turbulent mixing (Capet et al. 2008). In appendix B we show this coupling of nonconservative PV fluxes formally using an asymptotic expansion, the results of which suggest that 
Applying these scalings to a realistic model of the North Atlantic suggests that 
These findings thus serve to emphasize the role of the submesoscale in providing a direct connection between turbulence in the boundary layer and the properties of the gyre interior. Here we have focused on the modifications to the volume-integrated mode water PV, which has consequences for the dynamics of the gyre-scale circulation through changes in the distribution, variability, and mixing of interior PV (Qiu et al. 2007). Important modifications to other physical and biogeochemical properties of the mode water by these processes are also anticipated. For example, in simulations of an idealized ocean gyre, Lévy et al. (2010) noted an increase in stratification of the mode water layer and a decrease of meridional heat transport when the submesoscale was resolved. The effect of these changes on biological productivity, and carbon export, will involve local changes to productivity in the boundary layer (Lévy et al. 2012a), gyre-scale responses to thermocline and nutricline depths (Lévy et al. 2012b), and preconditioning of the western boundary current source waters (Iudicone et al. 2016).
An important potential limitation to the generality of these findings arises from uncertainty in the properties of boundary layer turbulence at frontal systems, which will alter the effect of surface buoyancy forcing and 
Finally, while adiabatic restratification by submesoscale baroclinic instabilities has been the subject of extensive work (Fox-Kemper et al. 2008; Fox-Kemper and Ferrari 2008; Fox-Kemper et al. 2011), nonconservative processes have received less attention. Here we show that boundary layer turbulence at submesoscale fronts can dominate the restratification of the near-surface layer (Fig. 15) and further lead to significant PV fluxes, crucial for setting the interior circulation. These frictional fluxes are not independent of the adiabatic baroclinic instabilities, but rather they are intertwined with the baroclinic instability process, being intensified by MLI frontogenesis, and in turn affecting the eddy life cycle by changing the PV of the boundary layer (Nakamura 1994). The PV source, and restratification, from the TTW circulation will be largely absent in coarsely resolved models, leading to a deficit of PV in the ocean interior.
The helpful comments and suggestions from John Marshall, George Nurser, and an anonymous reviewer are gratefully acknowledged. J.O.W. and L.N.T. were supported by NSF Grant OCE-1459677. J.C.M. and J.G. were supported by ONR N000141410626. ROMS simulations were performed using HPC resources from GENCI-TGCC (Grant 2017-A0010107638).
APPENDIX A
PV Flux Near the Surface
In section 2 the importance of the surface flux of PV was highlighted in light of the impermeability theorem [(6)]. However, a fundamental issue of concern is that the PV equation is higher order in spatial derivatives than the Navier–Stokes, indicating the potential difficulty of properly defining 
It is also useful to note that since 
APPENDIX B
Asymptotic Analysis of TTW PV Fluxes
As discussed in section 2, the TTW circulation is restratifying, and hence is expected, through (8) and (10), to lead to a net downward flux of PV at the surface. The magnitude of the total TTW PV flux will however be the residual of partial cancellation between the frictional PV injection (through TTW advection of the horizontal buoyancy gradient) and the diabatic PV removal as a result of the turbulent mixing of buoyancy. In this appendix we provide an asymptotic analysis of an idealized frontal configuration, to illustrate the coupling between across-front advection and turbulent mixing, explaining the scalings used in (19) and (20).
a. Momentum equations
b. Buoyancy equations
(top) Nondimensional surface PV flux and (bottom) time-integrated cumulative surface PV flux, for the simplified model discussed in appendix B, as a function of Ekman number (legend). Analytical solutions for a semi-infinite domain [solid lines; (B28)] and numerical solutions for a finite depth layer, including 
Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-17-0219.1
The analysis stated above can easily be extended to a layer of finite depth by adding no-flux boundary conditions on momentum and buoyancy at 
APPENDIX C
Configuration of the MITgcm
The most relevant aspects of the model configuration are documented in section 3; however, in the interest of reproducibility, further details of the model configuration are documented here.
For all runs, horizontal mixing of momentum is parameterized using a biharmonic operator, with a Smagorinsky coefficient of 3, and Leith and modified Leith coefficients of 1 (as in Brannigan et al. 2015). Horizontal diffusion of temperature is implemented using a uniform biharmonic diffusivity of 
As noted above, the KPP scheme is used for vertical mixing; however, it is worth documenting that in the pure convective conditions considered here, we found it necessary to ensure that the package configuration option for horizontal smoothing of diffusivity/viscosity was disabled and that vertical smoothing of the Richardson number was enabled. Without these choices, the strong Ekman restratification at sharp fronts would occasionally lead to very thin KPP boundary layer depths overlying unstable density profiles that could persist for long periods of time. A possible interpretation of this is that the horizontal smoothing of turbulence properties across multiple grid cells is physically inconsistent in the presence of sharp frontal features with horizontal scales approaching the grid scale.
The potential vorticity budgets were constructed offline, using time-averaged diagnostics output at 2-h time steps. The nonconservative J vectors were constructed using the built-in MITgcm diagnostics for the temperature [(4)] and momentum budgets [(5)]. The exception to this being the advection terms in each budget, which were reconstructed offline using velocity diagnostics and a second-order accurate derivative operator. This approach allows numerical diffusion and viscosity to be approximated as the difference between the online and offline advection diagnostic in each budget. These terms are generally small, with some transient exceptions at particularly sharp frontal features.
Practically, for evaluating the vertical PV fluxes in gridded numerical models, it is necessary to evaluate the vertical fluxes at a finite depth, 
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In pure convective conditions, as considered in section 3, the friction velocity 
Parameterization of entrainment fluxes is itself the subject of a large body of literature (see, e.g., Kraus and Businger 1994; Deremble and Dewar 2012); however, the simple approach we take here is supported by both empirical and numerical results for pure convective conditions (Large et al. 1994; Taylor and Ferrari 2010), and further was found to give good agreement with numerical model results in testing both simplified 1D models, as well as the 3D models discussed in section 3.
