3.1 Coastal-zone Langmuir cell alignment

In order to provide some initial, qualitative illustration of the nature of coastal Langmuir turbulence, we show figure 2. Figure 2(a–f) shows instantaneous resolved vertical velocity, $\tilde{w}(x,y,z,t)/u_{\ast }$ , at depths noted in the caption, while figure 2(g,h) shows the associated bottom-bed hydrodynamic (bathymetric) stresses, as per (2.16) (see § 2.2). Results are shown for Case $C_{2221}$ (table 1), which corresponds to a case of coaligned wind and wave forcing and forcing magnitudes, $\unicode[STIX]{x1D6F9}_{1}=1.41$ and $\unicode[STIX]{x1D6F9}_{2}=15$ , imposed at an angle oblique to the streamwise direction. Owing to the oblique wind and wave forcing, relative to the ambient pressure-gradient forcing, the resultant Langmuir cells are also oblique to streamwise. Note, however, that the actual inclination of the cells is less than that of the wind and wave forcing, i.e. $\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D703}_{1}=30^{\circ }$ . This can be deduced from cursory assessment of the figures: since $(L/H)\tan (30^{\circ })\approx 3.5$ , one can infer that the cells are aligned at an angle less than $\unicode[STIX]{x1D703}_{1}$ . In the absence of a coastal pressure-gradient forcing (and bottom-bed shear), the Langmuir cells would be aligned precisely in the direction of the coaligned wind and wave forcing. But, owing to the competing influence of the bottom bed shear and wind and wave shear, the cells align themselves at an intermediate angle, $0\leqslant \unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D703}_{1}$ . In fact, this is a recurrent result that we observe throughout the study. The concluding stages of the results section provide theoretical justification for this result.

Figure 2.

Comparison of structural orientation of the resultant Langmuir cells for Case $C_{2221}$ . (a–f) Instantaneous resolved vertical velocity at column depths $z/H=0.9$ (a), $z/H=0.8$ (b), $z/H=0.7$ (c), $z/H=0.6$ (d), $z/H=0.4$ (e) and $z/H=0.2$ (f). Instantaneous hydrodynamic bed stress (g) $\unicode[STIX]{x1D70F}_{xz,\ast }(z,y)/u_{\ast }^{2}$ and (h) $\unicode[STIX]{x1D70F}_{yz,\ast }(x,y)/u_{\ast }^{2}$ (see also (2.16)). For reference, panels include line denoting the angle $\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}$ .

The signature of the Langmuir cells is evident in figure 2(a–f) by virtue of self-organization into rows of predominant upwelling ( $\tilde{w}(x,y,z,t)/u_{\ast }>0$ , red) and downwelling ( $\tilde{w}(x,y,z,t)/u_{\ast }<0$ , blue), where figures 2(a) and 2(f) correspond to horizontal planes closest to the ocean surface and seafloor, respectively. Moreover, the cells appear to be most coherent at the intermediate depths (i.e. figure 2 d or 2 e). This result is a natural consequence of the lower turbulent kinetic energy, $tke$ , and shear production of turbulence at mid-depths; closer to the wall or ocean surface, where hydrodynamic shear is most pronounced, the presence of vigorous mixing attenuates the largest scales of the cells. We note that the orientation of the cells does not exhibit a strong ‘spiral’ pattern over the depth of the column, and instead the angle, $\unicode[STIX]{x1D6FE}$ , is essentially depth-insensitive. There is a negligible spiral, which can be appreciated by the reference line in figure 2(a–h), but in general the aggregate forcing yields a coherent self-organization to Langmuir cells at angle $\unicode[STIX]{x1D6FE}$ . This result is general to the full range of simulations summarized in table 1, and not specific to Case $C_{2221}$ . Note, also, that in the absence of bottom-bed stress (i.e. open-ocean settings), the cells would be aligned precisely with the coaligned wind and wave forcing – as per the preceding results from Van Roekel et al. (Reference Van Roekel, Fox-Kemper, Sullivan, Hamlington and Haney2012) – but the presence of bottom-boundary stress induces a realignment away from $x^{\prime }$ , toward $x$ (see figure 1 for reference).

Finally, figure 2(g,h) shows the instantaneous components of hydrodynamic bed stress, as imposed during LES with (2.16). Figures 2(g) and 2(h) correspond to the streamwise-vertical and spanwise-vertical momentum fluxes, respectively, and as such correlate closely with figures 2(a–f). Within locations of downwelling, there is a relative excess of streamwise velocity, and as such the zones of elevated $\unicode[STIX]{x1D70F}_{xz,\ast }(x,y,t)$ correlate precisely with regions of $\tilde{w}<0$ . Similarly, at the base of downwelling zones, conservation of mass dictates a lateral outflow, which results in the regions of equal and opposite $\unicode[STIX]{x1D70F}_{yz,\ast }(x,y,t)$ in figure 2(h) (effectively, the $\unicode[STIX]{x1D70F}_{yz,\ast }(x,y,t)$ pattern is identical to $\unicode[STIX]{x1D70F}_{xz,\ast }(x,y,t)$ , but is phase shifted in the $y^{\prime \prime }$ direction, again due to the lateral outflow at the base of downwelling zones).

Figure 3.

Numerical experiments on the effect of oblique wind, wave and mean current forcing. Panels (a,c,e,g,i,k,m,o) (i.e. first and third columns) show two-point correlation contour maps, $\unicode[STIX]{x1D70C}_{\tilde{v}^{\prime }\tilde{v}^{\prime }}(\unicode[STIX]{x1D6FF}z,\unicode[STIX]{x1D6FF}y,z)$ (3.1); superimposed upon the colour contours are coloured arrows to denote the direction of forces: red arrow indicates wave direction, blue arrow indicates wind direction and black arrow indicates mean current direction. Panels (b,d,f,h,j,l,n,p) (i.e. second and fourth columns) show instantaneous, resolved vertical velocity at column depth. For all panels, the results are shown at depth $z/H=0.18$ . Results are shown for table 1 cases: $C_{2211}$ (a,b), $C_{2261}$ (c,d), $C_{2221}$ (e,f), $C_{2234}$ (g,h), $C_{2241}$ (i,j), $C_{2225}$ (k,l), $C_{2251}$ (m,n) and $C_{2233}$ (o,p).

Figure 3(b,d,f,h,j,l,n,p) (i.e. the second and fourth columns of figure 3) shows instantaneous, resolved vertical velocity for cases noted in the figure caption at column depth $z/H=0.18$ . Figure 3(a,c,e,g,i,k,m,o) (i.e. the first and third columns of figure 3) shows the corresponding contours of two-point correlation of resolved spanwise velocity, computed as

(3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{\tilde{v}^{\prime }\tilde{v}^{\prime }}(\unicode[STIX]{x1D6FF}x,\unicode[STIX]{x1D6FF}y,z)={\displaystyle \frac{\langle \tilde{v}^{^{\prime }}(x,y,z)\tilde{v}^{^{\prime }}(x+\unicode[STIX]{x1D6FF}x,y+\unicode[STIX]{x1D6FF}y,z)\rangle _{T}}{\unicode[STIX]{x1D70E}_{y}(z)\unicode[STIX]{x1D70E}_{y}(z)}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}x$ and $\unicode[STIX]{x1D6FF}y$ are spatial separation in the streamwise and spanwise direction, respectively, and $\unicode[STIX]{x1D70E}_{y}$ is the root-mean-square value of resolved spanwise velocity. Spanwise velocity provided the most clarity on cell orientation. The time averaging spans ${\approx}10^{2}$ large-eddy turnovers after the turbulence statistics have become stationary. Fluctuations are based upon subtraction of a plane average – where spatial undulation of the Langmuir cells is presumed to impose a modest influence relative to the remaining turbulent fluctuations, thereby enabling the assumption of horizontal homogeneity for the purpose of computing turbulence statistics. In testing performed during preparation of this article, we found that the cells exhibit a negligible ‘spiral’ in the vertical direction (results omitted for brevity), and thus the results at $z/H=0.18$ are representative of the entire column. The original directions of constant wind, monochromatic surface wave and constant mean current are shown with blue, red and black arrows, respectively (this can be cast against the figure 1(a) results, which show the local coordinate systems, $\{x^{\prime },y^{\prime },z\}$ and $\{x^{\prime \prime },y^{\prime \prime },z\}$ ). Note that periodicity is observed for the local maxima and minima in the correlation map of spanwise velocity fluctuation, owing to the characteristic convergence and divergence zones of Langmuir turbulence that increases the strength of spanwise velocity (Leonardi et al. Reference Leonardi, Orlandi, Djenidi and Antonia2004; Tejada-Martínez & Grosch Reference Tejada-Martínez and Grosch2007). It can be understood that the local maxima are separated by the distance between consecutive downwelling limbs between adjacent Langmuir cells.

Characterization of Langmuir cell orientation (i.e. deduction of resultant angle  $\unicode[STIX]{x1D6FE}$ ) is assessed by datafit of a linear ‘best fit’ profile to the two-point correlation contours, where the $\unicode[STIX]{x1D6FE}$ values are summarized in table 1. As expected, in all cases the resultant $\unicode[STIX]{x1D6FE}$ is intermediate to the combined wind, wave and pressure-gradient forcing. The correlation contours agree closely with the instantaneous visualizations, although we emphasize that $\unicode[STIX]{x1D70C}_{\tilde{v}^{\prime }\tilde{v}^{\prime }}(\unicode[STIX]{x1D6FF}x,\unicode[STIX]{x1D6FF}y,z)$ is predicated upon a Reynolds average while the visualizations are instantaneous. The Langmuir cells are transient in nature – their intensity varies, and they undulate in the $y^{\prime \prime }$ direction (see figure 1 a) – but a coherent signature can still be attained from $\unicode[STIX]{x1D70C}_{\tilde{v}^{\prime }\tilde{v}^{\prime }}(\unicode[STIX]{x1D6FF}x,\unicode[STIX]{x1D6FF}y,z)$ .

Note, finally, that measures of coherence demonstrate an attenuation in the coherence of the Langmuir cells with changing obliquity. Consider, for example, a measure of the integral length based upon integration in the resultant cell direction:

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(z)=\int _{-\infty }^{\infty }\unicode[STIX]{x1D70C}_{\tilde{v}^{\prime }\tilde{v}^{\prime }}(x^{\prime \prime },0,z)\,\text{d}x^{\prime \prime }, & & \displaystyle\end{eqnarray}$$

where the two-point correlation has been mapped into the $\{x^{\prime \prime },y^{\prime \prime },z\}$ coordinate system. It is clear that for Cases $C_{2211}$ (figure 3 a,b), $C_{2261}$ (figure 3 e,f) and $C_{2221}$ (figure 3 i,j) – that is, cases for which the wind and wave forces are coaligned, and not normal to the streamwise direction – the coherence is strongest and, thus, $\unicode[STIX]{x1D706}(z)$ would be greatest. In contrast, when the wind and wave forcing is imposed at an angle normal (Case $C_{2251}$ , figure 3 m,n) or greater than normal (Case $C_{2261}$ , figure 3 c,d), or when the wind and wave forcing is not coaligned (Cases $C_{2234}$ , $C_{2225}$ and $C_{2233}$ , shown in figures 3 g,h, 3 k,l and 3 o,p, respectively), the correlation weakens. Even when the cell coherence is weakest – for example, Case $C_{2251}$ (figure 3 m,n) – some record of Langmuir cells is nevertheless present. Although the focus of this article is deduction of the resultant angle, $\unicode[STIX]{x1D6FE}$ , for variable coastal forcing conditions, it is notable that the self-organization process leading to Langmuir cells is robust for such complex conditions.

Development of prognostic models for a priori prediction of resultant Langmuir cell orientation angle is a major component of this work. In § 3.1, we showed how this angle varied for a range of cases (figure 3 and accompanying text), while table 1 includes $\unicode[STIX]{x1D6FE}$ from all the cases considered in this article. Fundamentally, the production and sustenance of Langmuir circulations are associated with the tilting of spanwise and vertical vorticity by Lagrangian shear and, thus, the resultant cells must be aligned with the direction of mean Lagrangian shear. As a preliminary effort, we consider an a priori prediction model involving wind, wave and mean current, inspired by the model developed by Van Roekel et al. (Reference Van Roekel, Fox-Kemper, Sullivan, Hamlington and Haney2012). They considered the scenario of wind and wave forcing in open-ocean conditions, with the wind directed in the streamwise direction; their model for resultant Langmuir cell orientation, $\unicode[STIX]{x1D6FC}$ , is

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=\arctan \left({\displaystyle \frac{\left\langle {\displaystyle \frac{\text{d}v_{s}}{\text{d}z}}\right\rangle _{z}}{\left\langle {\displaystyle \frac{\text{d}u_{s}}{\text{d}z}}\right\rangle _{z}+\left\langle {\displaystyle \frac{\text{d}u|_{w}}{\text{d}z}}\right\rangle _{z}}}\right), & & \displaystyle\end{eqnarray}$$

where $u_{s}$ and $v_{s}$ are the horizontal components of Stokes drift, $u|_{w}$ is the streamwise velocity profile due to the imposed wind shear (approximated by the law of wall) and $\langle \rangle _{z}$ denotes vertical averaging from the surface to twice the e-folding depth of the Stokes drift.

The present coastal ocean context is a generalized scenario of the above, for which the mean current forcing affects the dynamics and the resultant cell alignment. However, since shear is the underlying driver of Langmuir cell intensity and generation, it is self-evident that the arguments upon which (3.3) were developed are also valid in coastal settings. To this extent, equation (3.3) can be reformulated to obtain resultant coastal-zone Langmuir cell orientation angle, $\unicode[STIX]{x1D6FE}_{1}$ , as follows:

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{1}=\arctan \left({\displaystyle \frac{\left\langle {\displaystyle \frac{\text{d}v_{s}}{\text{d}z}}\right\rangle _{z}+\left\langle {\displaystyle \frac{\text{d}v|_{w}}{\text{d}z}}\right\rangle _{z}}{\left\langle {\displaystyle \frac{\text{d}u_{s}}{\text{d}z}}\right\rangle _{z}+\left\langle {\displaystyle \frac{\text{d}u|_{w}}{\text{d}z}}\right\rangle _{z}+\left\langle {\displaystyle \frac{\text{d}u_{b}}{\text{d}z}}\right\rangle _{z}}}\right), & & \displaystyle\end{eqnarray}$$

where $\text{d}u|_{w}/\text{d}z$ and $\text{d}v|_{w}/\text{d}z$ are the horizontal components of the Eulerian shear induced by surface wind shear – again defined here with the logarithmic law for the purpose of developing a prognostic model – and $\text{d}u_{b}/\text{d}z$ is the Eulerian shear induced by the mean current (also defined with the logarithmic law). It is to be noted here that the direction of the mean current is fixed in the streamwise direction and the direction of the wind and wave varies. Substitution of the definitions of the Stokes drift profile (2.12) and parameters $\unicode[STIX]{x1D6F9}_{1}$ and $\unicode[STIX]{x1D6F9}_{2}$ (2.8) and (2.9) yields the reduced form of (3.4):

(3.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{1}=\left\{\begin{array}{@{}ll@{}}\arctan (\unicode[STIX]{x1D712})\quad & \text{for }\unicode[STIX]{x1D712}>0,\\ \unicode[STIX]{x03C0}+\arctan (\unicode[STIX]{x1D712})\quad & \text{for }\unicode[STIX]{x1D712}\leqslant 0,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where substitution of terms yields the definition for $\unicode[STIX]{x1D712}$

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}={\displaystyle \frac{{\displaystyle \frac{\unicode[STIX]{x1D6F9}_{2}\unicode[STIX]{x1D705}}{\langle z^{-1}\rangle _{z}}}\left\langle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{s}(z)}{\text{d}z}}\right\rangle _{z}\sin (\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2})+\unicode[STIX]{x1D6F9}_{1}\sin (\unicode[STIX]{x1D703}_{1})}{{\displaystyle \frac{\unicode[STIX]{x1D6F9}_{2}\unicode[STIX]{x1D705}}{\langle z^{-1}\rangle _{z}}}\left\langle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{s}(z)}{\text{d}z}}\right\rangle _{z}\cos (\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2})+\unicode[STIX]{x1D6F9}_{1}\cos (\unicode[STIX]{x1D703}_{1})+1}}, & & \displaystyle\end{eqnarray}$$

where

(3.7a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}u_{s}}{\text{d}z}}=\cos (\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{s}(z)}{\text{d}z}},\quad {\displaystyle \frac{\text{d}v_{s}}{\text{d}z}}=\sin (\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{s}(z)}{\text{d}z}}, & \displaystyle\end{eqnarray}$$

(3.8a,b ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}u|_{w}}{\text{d}z}}={\displaystyle \frac{u_{w}}{\unicode[STIX]{x1D705}z}}\cos (\unicode[STIX]{x1D703}_{1}),\quad {\displaystyle \frac{\text{d}v|_{w}}{\text{d}z}}={\displaystyle \frac{u_{w}}{\unicode[STIX]{x1D705}z}}\sin (\unicode[STIX]{x1D703}_{1}) & & \displaystyle\end{eqnarray}$$

and

(3.9) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}u_{b}}{\text{d}z}}={\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}z}}. & & \displaystyle\end{eqnarray}$$

Substitution of the gradients summarized in (3.7), (3.8) and (3.9), and modest algebra, yields (3.6). The development leading to (3.6) is predicated upon the assumption of logarithmic scaling due to imposed aerodynamic shear, and we must emphasize that this differs from numerical treatment of this boundary within the LES code (see § 2.2).

Figure 4.

Study of the dependence of resultant Langmuir cell orientation, $\unicode[STIX]{x1D6FE}$ , on parameters $\unicode[STIX]{x1D6F9}_{1}$ , $\unicode[STIX]{x1D6F9}_{2}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ as defined by (2.14). Panel (a) shows $\unicode[STIX]{x1D6FE}$ for changing $\unicode[STIX]{x1D703}_{1}$ , with fixed $\unicode[STIX]{x1D6F9}_{2}=15$ and $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ , and increasing $\unicode[STIX]{x1D6F9}_{1}$ values (Sets 1, 3 and 5, which are denoted by squares, circles and ‘+’ signs, respectively). Panel (b) shows $\unicode[STIX]{x1D6FE}$ for changing $\unicode[STIX]{x1D703}_{1}$ , with fixed $\unicode[STIX]{x1D6F9}_{1}=1.41$ and $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ , and increasing $\unicode[STIX]{x1D6F9}_{2}$ values (Sets 2, 3 and 4, which are denoted by squares, circles and ‘+’ signs, respectively). Panel (c) shows $\unicode[STIX]{x1D6FE}$ for changing $\unicode[STIX]{x1D703}_{2}$ , with fixed $\unicode[STIX]{x1D6F9}_{1}=1.41$ and $\unicode[STIX]{x1D6F9}_{1}=15$ , and increasing $\unicode[STIX]{x1D703}_{1}$ (Sets 3, 6 and 7, which are denoted by filled circles, open squares and open circles, respectively). In (a), linear profiles of $\unicode[STIX]{x1D6FE}\sim \unicode[STIX]{x1D703}$ , with increasing slopes, are shown for perspective.

Note that (3.5) is undermined by the presence of $\langle z^{-1}\rangle _{z}$ , which is infinity for $z/H=0$ . In order to mitigate the influence of this, we consider a slightly narrower range of values, $10^{-3}\leqslant z/H\leqslant 10^{0}$ . In (3.5), some deductions on the changing role of different parameters can be made. First, for $\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D703}_{2}=0$ , $\unicode[STIX]{x1D6FE}_{1}=0$ , which corresponds to the case of coaligned Langmuir cells aligned in the prevailing transport direction ( $x$ ). Similarly, for $\unicode[STIX]{x1D703}_{2}=0$ – i.e. coaligned wind and wave forcing – one can deduce that $\unicode[STIX]{x1D6FE}_{1}$ always lags $\unicode[STIX]{x1D703}_{1}$ , which is consistent with the LES results. Here $\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}$ for all input arguments, which is justified through theoretical explanations in § 4.

Comparisons of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FE}_{1}$ for different misalignment set-ups can be made in table 1, which shows their good agreement (within $\pm 6^{\circ }$ ). The only exception is the simulation set with $\unicode[STIX]{x1D6F9}_{1}=2$ (higher wind stress) and $\unicode[STIX]{x1D6F9}_{2}=15$ , and for $\unicode[STIX]{x1D703}_{1}\geqslant 90^{\circ }$ (Cases $C_{2351}$ and $C_{2361}$ ). When $\unicode[STIX]{x1D703}_{1}\geqslant 90^{\circ }$ , Langmuir turbulence attenuates and diminishes the role of waves in the generation and sustenance of Langmuir cells, which further diminishes the interaction with the ambient bathymetric drag turbulence. Moreover, if the large obliquity is accompanied with a higher wind stress condition (as in the above case), the stronger wind stress dominates in orientating the direction of resultant weaker Langmuir cells. Thus, Langmuir cells align closer to the wind direction, inducing larger deviation from the $\unicode[STIX]{x1D6FE}_{1}$ estimation. Similar deviations from the actual Langmuir cell orientation angle by $\unicode[STIX]{x1D6FC}$ (3.3) in large obliquity cases have been reported in open-ocean conditions by Van Roekel et al. (Reference Van Roekel, Fox-Kemper, Sullivan, Hamlington and Haney2012). Note that the model proposed by Van Roekel et al. (Reference Van Roekel, Fox-Kemper, Sullivan, Hamlington and Haney2012) is based on averaging over twice the $e$ -folding depth, which is designed to capture the region of the layer affected by shear-driven mixing (associated with Langmuir turbulence and hydrodynamic shear). For the cases chosen here, the Stokes profile is non-zero at the seafloor (see also figure 1 b); to this extent, the column depth – which acts to constrain the growth of shear-driven structures – is the most plausible length over which averaging should be performed.

In order to associate the dependence of resultant Langmuir cell orientation angle ( $\unicode[STIX]{x1D6FE}$ ) on the misalignment angles ( $\unicode[STIX]{x1D703}_{1}$ , $\unicode[STIX]{x1D703}_{2}$ ) and various forcing scenarios ( $\unicode[STIX]{x1D6F9}_{1}$ , $\unicode[STIX]{x1D6F9}_{2}$ ) through LES, the data points in table 1 are summarized in figure 4. The figure presents the results of Sets 1, 3 and 5 (figure 4 a), Sets 2, 3 and 4 (figure 4 b) and Sets 3, 6 and 7 (figure 4 c), where inspection of the table demonstrates that this is the only plotting approach that allows isolation of the influence of all four parameters.

Figure 4(a) indicates a linear relationship, $\unicode[STIX]{x1D6FE}\sim \unicode[STIX]{x1D703}_{1}$ , where linear profiles with different slopes have been superimposed for perspective. We also added an annotation for the direction of increasing $\unicode[STIX]{x1D6F9}_{1}$ for the three sets considered. From this figure, we can conclude that both $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D6F9}_{1}$ impose a strong influence on $\unicode[STIX]{x1D6FE}$ , and we may conclude that $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6F9}_{1},\unicode[STIX]{x1D703}_{1};\unicode[STIX]{x1D6F9}_{2};\unicode[STIX]{x1D703}_{2})=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6F9}_{1}\unicode[STIX]{x1D703}_{1}$ , where $\unicode[STIX]{x1D706}<1$ , and is a non-dimensional constant of proportionality. Since $\unicode[STIX]{x1D6F9}_{1}$ defines imposed atmospheric drag, its role in ‘turning’ the cells can be understood simply by the decomposition of the imposed stress with increasing angle. Note that in all cases, $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D703}_{1}<1$ – that is, the cells never align precisely in the direction of the imposed wind and wave forcing, with the exception of the case for which all forces are aligned in $x$ – this result having been reported in the preceding qualitative results (figure 3 and accompanying text). Note that these data points are all summarized in table 1.

Interestingly, figure 4(b) reveals a far weaker influence of $\unicode[STIX]{x1D6F9}_{2}$ upon the resultant cell alignment, indicating that $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6F9}_{2},\unicode[STIX]{x1D703}_{1};\unicode[STIX]{x1D6F9}_{1};\unicode[STIX]{x1D703}_{2})=\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}_{1}$ , where $\unicode[STIX]{x1D706}<1$ , and is a non-dimensional constant of proportionality. Thus, this result suggests that $\unicode[STIX]{x1D6F9}_{2}$ – a measure of the influence of the Stokes drift profile – has a virtually negligible effect on the resultant inclination of the cells. This result, however, makes no measure of the intensity of the cells – which could, for example, be measured by circulation in the $y^{\prime \prime }$ – $z$ plane. Indeed, inspection of (2.13) confirms that increasing $\unicode[STIX]{x1D6F9}_{2}$ corresponds to stronger Langmuir forcing. We again observe that $\unicode[STIX]{x1D6FE}$ never attains the value of $\unicode[STIX]{x1D703}_{1}$ , which is consistent with all previous results and is a consequence of the competing shear from the pressure gradient, wind and wave forcing (see also (3.5)). Others have observed that $U_{s}$ has a relatively modest influence on alignment (Polonichko Reference Polonichko1997).

Finally, figure 4(c) provides insight on how the cell alignment results to wind and wave forcing both imposed at differing obliquity angles (see also figure 1 a for perspective). The presentation format, $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2};\unicode[STIX]{x1D6F9}_{1};\unicode[STIX]{x1D6F9}_{2})$ , provides rich insight on how the variability of the angles associated with each force affects the result inclination. For Set 3 (filled circles), there is a systematic increase in $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2};\unicode[STIX]{x1D6F9}_{1};\unicode[STIX]{x1D6F9}_{2})$ for each $\unicode[STIX]{x1D703}_{1}$ , while the data points for Sets 6 and 7 suggest a relatively weaker influence of $\unicode[STIX]{x1D703}_{2}$ for the given physically motivated wind and wave forcing conditions. This is consistent with the results from figure 3(a,b), where it was shown that $\unicode[STIX]{x1D6F9}_{2}$ , again under forcing regimes listed in table 1, imposes a virtually negligible influence on the resultant inclination. In the following section, we will further these results by demonstrating how the resultant Langmuir cell streamwise vorticity changes as the imposed forcing angle deviates from $\unicode[STIX]{x1D6FE}$ . These results provide a theoretical basis for the results.

3.2 Turbulence statistics

Having established that Langmuir cells with varying levels of coherence remain present under a range of realistic forcing conditions in coastal environments, it is prudent now to assess how turbulence statistics vary within the column. This analysis will provide yet more insight on how the intensity of mixing affects levels of large-scale coherence within the column. For the purpose of this article, first- and second-order turbulence statistics are sufficient, which are shown in figure 5. Figures 5(a,b) and 5(c–h) show first- and second-order statistics, respectively. Note that the stresses shown in figure 5(c–h) are the ‘total’, or Reynolds, stresses, which are recovered from LES as a post-processing exercise via addition of the resolved and subgrid-scale components, $\langle \boldsymbol{u}^{\prime }\otimes \boldsymbol{u}^{\prime }\rangle _{T}=\langle \tilde{\boldsymbol{u}}^{\prime }\otimes \tilde{\boldsymbol{u}}^{\prime }\rangle _{T}+\boldsymbol{\unicode[STIX]{x1D70F}}$ . Specific details of the cases are provided in the figure caption, and in table 1.

Figure 5.

Vertical profiles of (a,b) first-order and (c–h) second-order flow quantities of different misalignment cases. Axis labels denote quantities, where stresses in (c–h) represent ‘total’, or Reynolds-averaged, which are the sum of the subgrid-scale and resolved stresses. Red: $\unicode[STIX]{x1D703}_{1}=0^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2211}$ ); blue: $\unicode[STIX]{x1D703}_{1}=30^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2221}$ ); green: $\unicode[STIX]{x1D703}_{1}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2241}$ ); magenta: $\unicode[STIX]{x1D703}_{1}=90^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2251}$ ); cyan: $\unicode[STIX]{x1D703}_{1}=120^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2261}$ ); black: $\unicode[STIX]{x1D703}_{1}=45^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=45^{\circ }$ (Case $C_{2234}$ ); dark grey: $\unicode[STIX]{x1D703}_{1}=30^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=60^{\circ }$ (Case $C_{2225}$ ); light grey: $\unicode[STIX]{x1D703}_{1}=45^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=30^{\circ }$ (Case $C_{2233}$ ). Note that the profiles shown correspond to all of Set 3 ( $C_{22i1}$ , red, blue, green, magenta and cyan), all of Set 7 ( $C_{223i}$ , light grey and black) and a single case from Set 6 ( $C_{2225}$ , dark grey).

The Reynolds-averaged streamwise (figure 5 a) and spanwise (figure 5 b) velocity profiles provide major insights into the bulk transport of the column under changing conditions. In figure 5(a), the limiting maximum profile of Case $C_{2211}$ (red) corresponds with coaligned wind, wave and column pressure-gradient forcing, while the limiting minimum profile of Case $C_{2261}$ (cyan) corresponds with wind and wave forcing imposed at $\unicode[STIX]{x1D703}_{1}=120^{\circ }$ and $\unicode[STIX]{x1D703}_{2}=0$ (i.e. opposing the ambient column pressure gradient). Between these limiting profiles, the velocities vary monotonically as the role of wind and wave forcing alters the balance of streamwise momentum. For the limiting maximum profile, this case also imposes the largest hydrodynamic drag (evidenced by the bottom-boundary-layer wall-normal gradient). The profiles all exhibit a pronounced gradient at $z/H\approx 0.9$ : this gradient is induced by atmospheric surface layer drag, which is imposed at the sea surface level (and defined via the $\unicode[STIX]{x1D6F9}_{1}$ parameter). As expected, the vertical profiles for spanwise velocity are effectively a reverse (in terms of order, not magnitude) of the streamwise profiles. Although every effort was made to attain a true Reynolds average, it is apparent from figure 5(b), for Case $C_{2211}$ (red), that some transience is retained within the averages (for this case, since the imposed forcing is all aligned in the streamwise direction, Reynolds-averaged spanwise transport should be zero). Nevertheless, a modest value remains. As the wind and wave forcing is rotated through a range of obliquity angles, the value of $\langle \tilde{v}\rangle _{xyT}(z)/u_{\ast }$ increases, which occurs as a momentum balance develops in the spanwise direction. Note, too, that for the oblique cases (where the asymptotic example in figure 5 b corresponds to Case $C_{2261}$ ), a pronounced bottom boundary layer develops, even though the pressure gradient – which would ordinarily be responsible for bottom-boundary-layer shear – is aligned in the $x$ direction.

Figures 5(c–e) and 5(f–h) correspond to the normal and shear components of the Reynolds stress tensor, as denoted by the abscissa labels. Beginning firstly with the streamwise-normal component of the stress tensor (figure 5 c), it is clear, firstly, that the largest bottom-boundary-layer value occurs for $C_{2211}$ – the case for which all forces are coaligned in the streamwise direction and for which the bottom-bed shear is greatest (figure 5 a). The bottom-boundary-layer streamwise-normal stresses decrease monotonically until Case $C_{2261}$ . In the upper regions, however, there is a seeming reversal in the values of $\langle \tilde{u} ^{\prime }\tilde{u} ^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ , with the largest values being associated with Cases $C_{2221}$ (blue) and $C_{2225}$ (dark grey). Recall, however, that figure 3(k,l) provides instantaneous and two-point correlation contours for these cases, which indicated that these exhibited the lowest $\unicode[STIX]{x1D6FE}$ (also summarized in table 1). With this, it is clear that the largest $\langle \tilde{u} ^{\prime }\tilde{u} ^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ values occur, in general, when the cells are aligned with the streamwise direction. Likewise, the spanwise stresses show similar patterns, where elevated mixing in the upper layer is most pronounced due to fluctuations superimposed upon the counter-rotating large-scale Langmuir cells (see profiles for Cases $C_{2211}$ , red; and $C_{2221}$ , blue).

For all cases, figure 5(e) shows elevated $\langle \tilde{w}^{\prime }\tilde{w}^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ within the central regions of the column, $0.4\lesssim z/H\lesssim 0.8$ , where the streamwise- and spanwise-normal stresses were smallest. This is a product of the vertical transport associated with convergence and divergence at the surface, which directs high- $tke$ fluid in the vertical (figure 6, indeed, confirms that shear production of $tke$ is dominant at the surface and bottom-boundary-layer regions (McWilliams et al. Reference McWilliams, Sullivan and Moeng1997; Harcourt Reference Harcourt2013; Sinha et al. Reference Sinha, Tejada-Martínez and Akan2015; Deng et al. Reference Deng, Yang, Xuan and Shen2019).

Figure 5(f–h) shows the Reynolds shear stress profiles, $\langle \tilde{u} ^{\prime }\tilde{w}^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ (figure 5 f), $\langle \tilde{u} ^{\prime }\tilde{v}^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ (figure 5 g) and $\langle \tilde{v}^{\prime }\tilde{w}^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ (figure 5 h). Here $\langle \tilde{u} ^{\prime }\tilde{w}^{\prime }\rangle _{xyT}/u_{\ast }^{2}$ represents the column streamwise–wall-normal momentum fluxes, and at the domain bottom and top represents the bottom-bed shear and imposed atmospheric drag, respectively, for the case of coaligned forcing. Indeed, the Case $C_{2211}$ profile (red) indicates the largest bottom-bed stresses, which corresponds to linear momentum balance against the imposed pressure-gradient forcing ( $\boldsymbol{e}_{x}\unicode[STIX]{x1D6F1}(H/u_{\ast }^{2})$ in (2.7), and accompanying text). As the forces are imposed at larger obliquity angles, we observe a monotonic decrease in the imposed bottom-bed stress.

Figure 6.

Vertical profiles of normalized $tke$ shear production, ${\mathcal{P}}$ (a), dissipation, $\unicode[STIX]{x1D716}$ (b), and production by Stokes drift (c); see also (3.10) and accompanying text. The $tke$ terms are normalized by $u_{\ast }^{3}/H$ . Red: $\unicode[STIX]{x1D703}_{1}=0^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2211}$ ); blue: $\unicode[STIX]{x1D703}_{1}=30^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2221}$ ); green: $\unicode[STIX]{x1D703}_{1}=60^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2241}$ ); magenta: $\unicode[STIX]{x1D703}_{1}=90^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2251}$ ); cyan: $\unicode[STIX]{x1D703}_{1}=120^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=0^{\circ }$ (Case $C_{2261}$ ); black: $\unicode[STIX]{x1D703}_{1}=45^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=45^{\circ }$ (Case $C_{2234}$ ); dark grey: $\unicode[STIX]{x1D703}_{1}=30^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=60^{\circ }$ (Case $C_{2225}$ ); light grey: $\unicode[STIX]{x1D703}_{1}=45^{\circ }$ , $\unicode[STIX]{x1D703}_{2}=30^{\circ }$ (Case $C_{2233}$ ). Note that the profiles shown correspond to all of Set 3 ( $C_{22i1}$ , red, blue, green, magenta and cyan), all of Set 7 ( $C_{223i}$ , light grey and black) and a single case from Set 6 ( $C_{2225}$ , dark grey).

Figure 6 shows vertical profiles of constituent right-hand-side terms in the Reynolds-averaged $tke$ transport equation:

(3.10) $$\begin{eqnarray}\displaystyle \tilde{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\langle tke\rangle _{T}=\underbrace{{\mathcal{T}}}_{\mathit{Transport}}-\underbrace{\langle \boldsymbol{u}^{\prime }\otimes \boldsymbol{u}^{\prime }\rangle _{T}\boldsymbol{ : }\unicode[STIX]{x1D735}\langle \tilde{\boldsymbol{u}}\rangle _{T}}_{\mathit{Production},\,{\mathcal{P}}}-\underbrace{\langle \unicode[STIX]{x1D749}\rangle _{T}\boldsymbol{ : }\langle \unicode[STIX]{x1D64E}\rangle _{T}}_{\mathit{Dissipation},\,\unicode[STIX]{x1D716}}+\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D6F9}_{2}}{\unicode[STIX]{x1D6F9}_{1}}}\;\unicode[STIX]{x1D753}_{s}\times \langle \tilde{\unicode[STIX]{x1D74E}}^{\prime }\otimes \tilde{\boldsymbol{u}}^{\prime }\rangle _{T}}_{\mathit{CL\,vortex\,forcing}},\quad & & \displaystyle\end{eqnarray}$$

where all quantities have been normalized by $u_{\ast }^{3}/H$ . Other researchers have used this equation to establish the contribution to $tke$ in different conditions (e.g. Harcourt Reference Harcourt2013; Sinha et al. Reference Sinha, Tejada-Martínez and Akan2015; Deng et al. Reference Deng, Yang, Xuan and Shen2019). Figure 6(a) plots production of $tke$ by mechanical shear, which indicates – as expected – that shear production is most pronounced in the lower and upper regions. This is a product of bottom-boundary-layer shear and Stokes shear. For all cases, the upper regions ( $0.6\lesssim z/H\lesssim 0.9$ ) indicate the potential for negative production, which is a product of the negative vertical gradient in streamwise velocity due to the Langmuir cells (i.e. figure 5 a,f; observed also by Kukulka et al. (Reference Kukulka, Pleuddemann and Sullivan2012)). Suzuki & Fox-Kemper (Reference Suzuki and Fox-Kemper2016) have developed a ‘momentum sorting’ mechanism to explain the $tke$ production in this region of the column, although attribution of the present results to this mechanism is beyond the scope of the present work. Since $tke$ is a scalar, there is no directional dependence, which simplifies comparison between cases. It is known that $tke$ production due to Stokes drift shear decreases with increasing misalignment (Van Roekel et al. Reference Van Roekel, Fox-Kemper, Sullivan, Hamlington and Haney2012). Figure 6(b) plots $tke$ dissipation, where the profiles are similar to those reported by others for coastal-zone Langmuir turbulence (Kukulka et al. Reference Kukulka, Pleuddemann and Sullivan2012; Tejada-Martínez et al. Reference Tejada-Martínez, Grosch, Sinha, Akan and Martinat2012). Finally, figure 6(c) shows contributions to $tke$ due to the vortex forcing (‘CL vortex forcing’ in (3.10)). For the very high $Re_{\unicode[STIX]{x1D70F}}$ values considered for this article, the transport term contribution is negligible with the exception of locations closest to the bed and surface that are not resolved during LES (Kim, Moin & Moser Reference Kim, Moin and Moser1987). Note that the figure panels have been slightly truncated in the vertical direction at the top and bottom. The values for production and dissipation are extremely large in the bottom boundary layer and upper regions of the Stokes layer, while Stokes drift production is largest near the top of the domain. The figures show that these quantities also do not vary significantly in the interior with respect to misalignment. The figures are consistent with observations from figure 2, where we showed that the large-scale cells were least correlated in the upper and lower regions, due to strong shear and vigorous mixing.