Nonlinear steady states to Langmuir circulation in shallow layers: an asymptotic study

Hayes, Daniel Thomas; Phillips, W. A.

doi:10.1080/03091929.2016.1263302

Cited by 2 publications

(10 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Not surprisingly, the generic form of profiles for in figures 1 to 3 are closely in accord with the asymptotic findings of Cox & Leibovich (1993) and Hayes & Phillips (2017). Specifically, that the neutral curve is strongly asymmetric, that the curve of maximum growth rate depicted by dots in figure 1 exhibits an almost linear increase in with and, as we see in figure 3, that at for all when (Hayes & Phillips 2016, 2017).…”

Section: Linear Analysissupporting

confidence: 86%

“…Of key interest then was the role played by the extra stress in the instability to LC in the limit and to ascertain that Hayes & Phillips (2016, 2017) carried out an asymptotic study in which they imposed the extra stress only at the free surface. Specifically they impose mixed boundary conditions on the perturbation quantities at and Neumann on , as and in which is an applied stress that gives rise to (Phillips et al.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Irrespective of its origins, however, we need an estimate for , the first being by Cox & Leibovich (1993), based on physical arguments. Phillips & Dai (2014), on the other hand, show formally that may range from an upper bound of unity down to near zero; the latter being utilised in asymptotic studies to isolate the role of in the limit (Hayes & Phillips 2016, 2017). In view of this broad variation we proceed as follows: in much of our study (§§ 4–6), we assume a ‘typical’ value to gain insight into the key dynamical features.…”

Section: Linear Analysismentioning

confidence: 99%

“…Leibovich 1983; Thorpe 2004), in layers separated by a thermocline on deep water (e.g. Cox & Leibovich 1993; Leibovich & Tandon 1993), and in shallow water (Phillips & Dai 2014; Hayes & Phillips 2016, 2017). Also while there are instances where linear instability theory (Phillips 2001, 2002) captures observed (Smith 1992) LC spacings in deep water reasonably well, there are also instances where linear theory is noticeably in error, particularly in shallow waters (Marmorino, Smith & Lindemann 2005).…”

Section: Introductionmentioning

confidence: 99%

“…Instability to LC can then occur when the current driven by wind, in combination with Lagrangian drift arising from the waves, is depth dependent and of the same sense, the simplest case being when depth dependence is linear. We then arrive at a construct for LC in layers employed in the asymptotic studies of Cox & Leibovich (1993) and others (Hayes & Phillips 2016, 2017), whose interest lay with the free surface boundary conditions and whose findings are relevant here.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On Langmuir circulation in 1 : 2 and 1 : 3 resonance

Cui

Phillips

2019

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

This paper is concerned with the nonlinear dynamics of spanwise periodic longitudinal vortex modes (Langmuir circulation (LC)) that arise through the instability of two-dimensional periodic flows (waves) in a non-stratified uniformly sheared layer of finite depth. Of particular interest is the excitation of the vortex modes either in the absence of interaction or in resonance, as described by nonlinear amplitude equations built upon the mean field Craik–Leibovich (CL) equations. Since Y-junctions in the surface footprints of Langmuir circulation indicate sporadic increases (doubling) in spacing as they evolve to the scale of sports stadiums, interest is focused on bifurcations that instigate such changes. To that end, surface patterns arising from the linear and nonlinear excitation of the vortex modes are explored, subject to two parameters: a Rayleigh number ${\mathcal{R}}$ present in the CL equations and a symmetry breaking parameter $\unicode[STIX]{x1D6FE}$ in the mixed free surface boundary conditions that relax to those at the layer bottom where $\unicode[STIX]{x1D6FE}=0$. Looking first to linear instability, it is found as $\unicode[STIX]{x1D6FE}$ increases from zero to unity, that the neutral curves evolve from asymmetric near onset to almost symmetric. The nonlinear dynamics of single modes is then studied via an amplitude equation of Ginzburg–Landau type. While typically of cubic order when the bifurcation is supercritical (as it is here) and the neutral curves are parabolic, the Ginzburg–Landau equation must instead here be of quartic order to recover the asymmetry in the neutral curves. This equation is then subjected to an Eckhaus instability analysis, which indicates that linearly unstable subharmonics mostly reside outside the Eckhaus boundary, thereby excluding them as candidates for excitation. The surface pattern is then largely unchanged from its linear counterpart, although the character of the pattern does change when $\unicode[STIX]{x1D6FE}\ll 1$ as a result of symmetry breaking. Attention is then turned to strong resonance between the least stable linear mode and a sub-harmonic of it, as described by coupled nonlinear amplitude equations of Stuart-Landau type. Both 1 : 2 and 1 : 3 resonant interactions are considered. Phase plots and bifurcation diagrams are employed to reveal classes of solution that can occur. Dominant over much of the ${\mathcal{R}}$-$\unicode[STIX]{x1D6FE}$ range considered are non-travelling pure- and mixed-mode equilibrium solutions that act singly or together. To wit, pure modes solutions alone act to realise windrows with spacings in accord with linear theory, while bistability can realise Y-junctions and, depending upon initial conditions, double or even triple the dominant spacing of LC.

show abstract