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Interactions Between Internal Waves and Boundary Layer Vortices 62 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 Interactions between Internal Waves and Boundary Layer Vortices S. A. THORPE Department of Oceanography, Southampton Oceanography Centre, Southampton, United Kingdom (Manuscript received 12 February 1996, in ®nal form 29 April 1996) ABSTRACT The effect of internal waves on linear vortices in a homogeneous boundary layer is examined by taking a simple model with a horizontal array of line vortices or vortex cells lying in a layer bounded above by a rigid plane and below by a density interface on which interfacial waves are free to propagate. The interfacial waves stretch, compress, and displace the vortices, so changing their vorticity, orientation, and separation by amounts that are estimated. As a consequence, the instability of an array of vortices of alternating signs is enhanced in regions that depend on the local phase of the interfacial waves. The vortices force secondary disturbances on the wave-perturbed density interface. For parameter values typical of the ocean, the potential energy associated with these disturbances may be comparable with the kinetic energy in the vortices. The energy required to drive the vortices is therefore greater than that in the absence of internal waves, and this may affect the growth and development of the vortices. The presence of a density interface at the foot of the mixed layer, however, increases the primary rate of growth of Langmuir circulation in comparison with that found when the lower boundary is rigid. The subsequent instability is also enhanced. In consequence Langmuir cells in mixed layers overlying strati®ed water are expected to grow more rapidly and to be more unstable than those developing in a homo- geneous layer of the same depth overlying a rigid bottom. The effect of codirectional shear and Stokes drift included in the Craik±Leibovich equations is to reduce the phase speed of internal waves that propagate normal to the mean ¯ow. 1. Introduction ature ramps, and the pattern of alternately counterro- The motivation for this study is a desire to understand tating circulating vortices with horizontal axes known better some of the processes contributing to mixing in as Langmuir circulation (see Thorpe 1995). Evidence the upper ocean and, more generally, the interactions of the coexistence of internal waves and Langmuir cir- that occur between internal waves and ocean boundaries. culation during a period of strong winds is provided by It is known that turbulence in the benthic boundary layer Smith (1992; e.g., see Fig. 13). Langmuir cells may may dissipate internal waves (D'Asaro 1982) and that, persist for periods of a few minutes to several tens of particularly on sloping boundaries such as the conti- minutes before amalgamating or otherwise loosing a nental slope, internal waves may break and contribute detectable signature (Thorpe 1992; Farmer and Li to turbulence in the boundary layer (Wunsch 1970; 1995); these periods are similar to those of internal Thorpe 1989; Ivey and Nokes 1989; Taylor 1993). Hol- waves in the thermocline. Tandon and Leibovich ligan et al. (1985) report the occurence of areas of sur- (1995a,b) have examined the nonlinear development of face roughness, about 5 m across, with the appearance Langmuir cells in a shallow mixed layer with a rigid of ``boils'' on a turbulent river surface, which occur on lower boundary. Cell patterns are produced, varying in the generally smooth water surface over the crests of their alignment with the wind direction by (10 6 1)8. internal waves in conditions of fairly low wind speed, Other causes of nonalignment identi®ed by Tandon and apparently indicating the presence of turbulent mixing Leibovich (1995b) are ¯uctuations in wind direction, within the upper mixed layer that is enhanced by the Coriolis effects, and preexisting nonaligned currents. internal waves. No direct observations appear to exist. The existence of a ¯exible density interface separating Turbulence in boundary layers often contains coher- the uniform upper layer from a deep lower layer intro- ent structures that persist for short periods of time. These duces a new degree of freedom into the problem, which are frequently of a vortical nature; examples are Kelvin± will modify the nonlinear development of Langmuir cir- Helmholtz billows, sometimes associated with temper- culation, while internal waves on the interface are a further factor leading to the cell boundaries being non- aligned with the wind. The apparently greater persis- tence of Langmuir cells in shallow, unstrati®ed lakes Corresponding author address: Dr. S. A. Thorpe, Department of Oceanography, The University, High®eld, Southampton S09 5NH, (durations exceeding an hour are reported by B. Kenney United Kingdom. 1995, personal communication) prompts the question, q1997 American Meteorological Society Unauthenticated | Downloaded 09/30/21 01:49 AM UTC JANUARY 1997 THORPE 63 FIG. 1. A sketch showing the internal wave traveling in the horizontal direction (k, l)atthe interface between the two layers of uniform densities r1 (upper layer) and r2 (lower layer), and the vortex ¯ow in the boundary layer with vorticity vector in direction x. The axis z is vertically upward, with the mean position of the interface at z 52h. The effects of a shear ¯ow in the x direction in the upper layer are discussed in section 4; unstable, like-signed Kelvin±Helmholtz vortices may then develop in the upper layer, those with the fastest growth rates having their wavenumber in the x direction. The effects of a shear ¯ow and surface wave Stokes drift, with the generation of Langmuir circulation, are discussed in section 5. ``May the stability of circulation patterns be affected by Two simple cases may be identifed. If the internal the presence of internal waves on an underlying ther- waves are propagating in direction x parallel to the vor- mocline?'' tex lines and so parallel to the vorticity vector in the We have therefore chosen to consider how, in simple upper layer, the effect of the wave-induced motion u in ways, such vortical motions in a mixed layer may be the x direction is to stretch and compress vortex lines. affected by, or may affect, internal waves traveling The vorticity conservation equation along the density discontinuity that marks its boundary. Dv A model described in section 2 illustrates the nature of 5 v´=u, (1) some of the important effects, and this is expanded and Dt quanti®ed in subsequent sections. where v 5 (vx, vy, vz) is the vorticity and u 5 (u, y, w) is the velocity, implies that the strength of the vor- tices will be modulated by the term vx]u/]x induced by 2. The effect of internal waves on line vortices the internal wave x component of current and the vor- For simplicity, consider a homogeneous layer of den- ticity vx of the rotational vortex ¯ow. Maximum vor- sity r1 and depth h containing a horizontal array of line ticity will occur over the internal wave crests where the vortices of equal strength G, but alternating sign, equally result of horizontal stretching in the upper layer is great- spaced at distance d apart in the y direction. The upper est, and minimum over wave troughs where the greatest surface, z 5 0, is rigid and there is a density increase horizontal compression occurs. If, on the other hand, r2 2 r1 (.0) at the lower boundary of the mixed layer the internal waves are propagating in the y direction, at mean depth z 52h, below which lies a deep ¯uid (1) implies that the strength of the vortices will be un- of uniform density r2 (Fig. 1). Here, and in following changed (since ]/]x 5 0). Their separations d will, how- sections, the effect of viscosity is supposed negligible. ever, be altered as they are advected by the wave-in- The motions within the layers are modulated by the duced motion. Vortices separated at distances less than currents produced by an internal wave propagating half the wavelength of the internal wave will be closest along the density interface. The model may be inverted where the effect of convergence brings the vortices clos- to represent a benthic boundary layer. est together, which is over the wave troughs. Unauthenticated | Downloaded 09/30/21 01:49 AM UTC 64 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 These disturbances will affect the stability of the satis®es its respective equation of motion and the bound- vortex array. Thorpe (1992) has shown that the fastest ary condition w 5 0atz50. The form chosen for c growing disturbances of the vortex array generally also satis®es w 5 0atz52hand represents an in®nite have wavelength 2d (so that pairs of vortices interact array of vortices with axes aligned in the x direction of most strongly) and that the growth rate is proportional alternating signs and centers at y 5 (2n 1 1)p/2, n 5 to the vortex strength G and to the inverse square of 0, 61, 62,´´´. Thefrequency V5A[m2 1 (p/h)2] the separation d, with typical e-folding growth periods is a measure of the vorticity associated with the vortices of 4pd2/7G. When the vortices have cores of ®nite and A corresponds to the vortex strength (G in section size, the fastest growing disturbances are three-di- 2). The form chosen for the velocity potential f1 cor- mensional with an x-directed wavenumber that de- responds to an internal wave, h1 5 a cos(kx 1 ly 2 pends on the ratio of the vortex separation d and their st), propagating in the horizontal (k, l) direction at the depth in the layer. The presence of long period internal interface, z 52h1h, where a is the wave amplitude waves will therefore in¯uence the stability of the vor- and B 52as/(K sinh Kh).
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