JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. C9, PAGES 16,501-16,507, SEPTEMBER 15, 1993 Three-DimensionalLangmuir Circulation Instability in a Stratified Layer SIDNEY LEIBOVICH AND AMIT TANDON1 Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York Some reports of Langmuir circulations describe windrows at systematic angles to the local wind direction. Other observersfind windrows in close alignment to the wind direction, but with a systematic drift sidewaysto the wind. These effectsprobably result from more than one physical cause. Here it is shown, by an analysis of the linear stability of the surface layer, averaged to remove surface wave fluctuations, that persistent small windrow angles can result from weak stable density stratification. In these cases, the linearly most unstable modes are found to be weakly three dimensional for the range of parameters considered. Possible surface windrow patterns include rolls mostly parallel to the wind but of finite length, with branching and merging, as well as parallel rolls inclined to the wind and drifting laterally with respect to the wind direction. For unstratified flow, steady two-dimensional rolls are preferred. INTRODUCTION Streaks may be formed on the sea surface by a variety of causes,including convergences due to internal waves, possi- bly instability of the Ekman layer, Langmuir circulation, as well as other convective motions. The hallmark of Langmuir circulation is windrows approximately parallel to the wind direction, especially when the water column is unstratified or stably stratified. Most reports of streaks attributed to Langmuir circulations do not indicate any systematic bias in windrow direction relative to the wind. Nevertheless, there are observations (specifically, Fatter [1964] and Katz et at. [1965]) that haverevealed a systematic angular deviation in a system of parallel windrows. In the more commonly reported cases, individual windrows nearly parallel to the wind terminate after a fi- nite distance or split into two, or two windrows coalesce into one. This branching process appears to result, at least some of the time, from a secondary instability of the system of parallel rolls. This origin of the branching process is a non- linear effect not requiring density variations and has been exploredby Thorpe[1992], Tandon[1992], and A. Tandon andS. Leibovich (in preparation). The problem treated in this paper is more appropriate as a model for experiments in a wind-wave tank than as a model for phenomena in the mixed layer, since it assumes a rigid no-slip bottom, and the effects of the Coriolis accel- eration are ignored. Nevertheless, the analysis showsthat a small stable density stratification can lead either to persis- tent (small) angles of parallel windrows to the wind direc- tion, or to the branching and merging of windrows otherwise aligned with the wind. Both patterns are time-periodic in a stationary reference frame, but each pattern results from a travelling wave, and so is steady in a frame of reference moving eitheracross the wind (former case) or with it (latter ca..•o_ Tho•o are offoct.• of hnnva.ncv. a.nd a.ri•e a.f, f,ho ]inoa.r ...... i .................... .• ---.•; ................ level in a stability analysis of the current and its wind-wave interactionin the Craik-Leibovich (CL) theory [Craik and • Now at Centre for Earth and Ocean Research, University of Victoria, Victoria, British Columbia, Canada. Copyright 1993 by the American GeophysicalUnion. Paper number 93JC01234. 0148-0227/93/93JC-01234505.00 Leibovich, 1976] of Langmuir circulation as formulated by Leibovich [1977b]. Which pattern is described by the lin- ear analysis depends on the choice of linear eigenfunctions combined, and so depends on initial conditions. A more complete analysis of which form is physically realizable re- quires a nonlinear (secondary stability) analysis not done here. The angular deviation of windrows observed by Fatter [1964] averaged about 13 ø to the right of the wind, and Katz et at. [1965]report a similar figure (with a smaller data set). The tendency to lie to the right of wind suggests, as FMler emphasizes, that the deviation is due to Corio- lis effects. This is plausible, since the surface current on which Langmuir circulation is imposed can exhibit Ekman spiralling. This effect is not included in the present anal- ysis, which shows that the angular deviation possible from the linear stabihty analysis is maximum at an intermediate value of the stratification. The largest angle found for the rangeof stratificationinvestigated hereis 5.3 ø , but windrows have an equal likelihood of being on either side of the wind. Furthermore, the speed of lateral drift associated with the angular deviation increaseswith stratification. As noted by Leibovich [1983], in two dimensions the CL theory of Langmuir circulations is mathematically analo- gousto more extensively studied double diffusive convection problems(with unit Prandtl number and with a possibly nonlinear temperature profile). This is no longer true for the CL theory in three dimensions. The equations controlling the stability problem are more complex, and the parame- ter space is enlarged. For example, given the form of the depth variation of the Stokes drift and assuming eddy coef- ficients for diffusivity of momentum and heat, the problem depends on four dimensionless parameters R, Re., $, and r. Even when we restrict consideration to the molecular value of the f' • .......... ( ) rl:itlLklb! IttlllLUCl: in water, v rutverse) = 0.14, we are able to examine only a small corner of the remaining three-dimensionM parameter space. The parameter R is a relative measure(analogous to a Rayleighnumberin ther- mal convection) of the destabilizing effect of the vortexforce due to wave-current interaction [Leibovich, 1983]; Re, is a Reynolds number based on the friction velocity, layer depth, and eddy viscosity; and S is a measure of the stabilization due to buoyancy relative to viscous effects. If c•is the phaseanglemade by disturbancemodesasmea- sured fromthe winddirection (soc•-- tan -• m/k, where 16,501