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 Ϯ 1)Њ. 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,
᭧1997 American Meteorological Society
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FIG. 1. A sketch showing the internal wave traveling in the horizontal direction (k, l)atthe
interface between the two layers of uniform densities 1 (upper layer) and 2 (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 ϭϪh. 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. D (١u, (1´A model described in section 2 illustrates the nature of ϭ some of the important effects, and this is expanded and Dt quanti®ed in subsequent sections. where ϭ (x, y, z) is the vorticity and u ϭ (u, , w) is the velocity, implies that the strength of the vor- x induced byץ/uץtices will be modulated by the term x 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 x of the rotational vortex ¯ow. Maximum vor- sity 1 and depth h containing a horizontal array of line ticity will occur over the internal wave crests where the vortices of equal strength ⌫, 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 ϭ 0, is rigid and there is a density increase horizontal compression occurs. If, on the other hand,
2 Ϫ 1 (Ͼ0) at the lower boundary of the mixed layer the internal waves are propagating in the y direction, at mean depth z ϭϪh, below which lies a deep ¯uid (1) implies that the strength of the vortices will be un-
-x ϭ 0). Their separations d will, howץ/ץ of uniform density 2 (Fig. 1). Here, and in following changed (since 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.
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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 ϭ 0atzϭ0. The form chosen for growing disturbances of the vortex array generally also satis®es w ϭ 0atzϭϪhand 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 ϭ (2n ϩ 1)/2, n ϭ to the vortex strength ⌫ and to the inverse square of 0, Ϯ1, Ϯ2,´´´. Thefrequency ⍀ϭA[m2 ϩ (/h)2] the separation d, with typical e-folding growth periods is a measure of the vorticity associated with the vortices of 4d2/7⌫. When the vortices have cores of ®nite and A corresponds to the vortex strength (⌫ in section size, the fastest growing disturbances are three-di- 2). The form chosen for the velocity potential 1 cor- mensional with an x-directed wavenumber that de- responds to an internal wave, 1 ϭ a cos(kx ϩ ly Ϫ pends on the ratio of the vortex separation d and their t), propagating in the horizontal (k, l) direction at the depth in the layer. The presence of long period internal interface, z ϭϪhϩ, where a is the wave amplitude waves will therefore in¯uence the stability of the vor- and B ϭϪa/(K sinh Kh). The velocity potential in tex array; enhanced growth rates will result from the the lower layer is 2 ϭϪBsinh Kh exp[K(z ϩ h)] stretching and consequent increase of vorticity over sin(kx ϩ ly Ϫ t), satisfying the kinematic condition the internal wave crests and from reduced vortex sep- (A11) at z ϭϪhϩ, where aration over wave troughs. The most likely position 2 ϭ ( Ϫ )gK tanh(Kh)/[ ϩ tanh(Kh)] (2) of a breakdown of the ¯ow ®eld and generation of 2 1 1 2 small-scale turbulence relative to the internal wave is the dispersion relation of the internal waves required may depend on the internal wave period and the to satisfy the linearized condition of continuity of pres- growth rate of the vortex instability, and hence on the sure (A13). strength of the vortices. a. Effects of internal waves on the vortices 3. Interaction between vortices and internal waves The x component of vorticity in the upper layer is
In practice, the presence of the vortices may affect given by substituting for 1 and into the x-vorticity the internal waves and perturb the interface, processes equation (A5): ,(zץ/vץy Ϫץ/wץ)⑀␦ ignored in section 2. The model just described is de- ϭ ␦ٌ2 ϩ veloped and made quantitative by taking a streamfunc- x 1 tion (y, z) periodic in y so that the induced velocity is ϭϪ␦⍀{sin my sin z/h Ϫ ⑀(B/) -y) and the vorticity is (ٌ2, 0, 0), repץ/ץ ,zץ/ץϪ ,0) ϫ[{k2sin z/h cosh Kz resenting a general rotational motion in the upper layer.
Velocity potentials 1 and 2, functions of x, y, z, and ϩ K/h cos z/h sinh Kz} t, satisfying the equations of irrotational motion ٌ2 ϭ i ϫsin my cos(kx ϩ ly Ϫ t) 0, i ϭ 1, 2, in the upper and lower layers, respectively, de®ne the ®eld of motion generated by interfacial in- Ϫ ml sin z/h cosh Kz cos my ternal waves. A scheme is adopted in which, for con- ϫsin(kx ϩ ly Ϫ t)]}, (3) venience in distinguishing terms, the motions are sup- posed to be of different orders: ␦ for the rotational mo- where the parameters ␦ and ⑀ are introduced to indicate tion and ⑀ for the irrotational internal wave motions. the ordering of terms. At interfacial wave crests, cos(kx 2 ,(The streamfunction may be chosen so that ٌ ϭ ϩ ly Ϫ t) ϭ 1, and the maximum vorticity, max(x Ϫc, where c is a constant, and then the linearized is at sin my ϭ 1. Substituting for B in terms of the equations of motion are exact, nonlinear terms vanish, wave amplitude a gives and is an exact solution. The upper boundary is sup- max( ) ϭϪ␦⍀[sin z/h ϩ ⑀(a/K sinh Kh)] posed to be rigid, the vortical motion is con®ned to the x y ϭ 0atzϭ0, Ϫh), and motion decays ϫ[k2sin z/h cosh Kzץ/ץ) upper layer to zero as z tends to Ϫϱ. The procedure is to examine K /h cos z/h sinh Kz], the interaction terms of order ␦⑀ in the equations of ϩ motion, which are forced by the ␦ and ⑀ products, as- ϭ ␦⍀{1 ϩ ⑀ak/[2 sinh(Kh/2)͙(1 ϩ [l/k]2 ) ]}, suming that it is at this order that the main effects of interaction become apparent. The effects of nonlinear- at z ϭϪh/2. (4) ities of waves or vortices (represented by terms in ␦2 or It may be shown by differentiating (4) with respect to ⑀2) are presently ignored. The method of solution and z that the maximum vorticity lies in Ϫh Ͻ z ϽϪh/2 the resulting general equations are given in the appen- when kh Ͼ , and in Ϫh/2 Ͻ z Ͻ 0 when kh Ͻ . The dix. greatest vorticity enhancement is proportional to the To proceed, we choose speci®c forms ϭ A sin my slope, ak, of the internal wave, and occurs when Kh is sin z/h and 1 ϭ B cosh Kz sin(kx ϩ ly Ϫ t), where very small [when is also small from (4)] and when l K2 ϭ k2 ϩ l2, and A and B are constants. Each solution ϭ 0, so that the waves are traveling in the direction of
Unauthenticated | Downloaded 09/30/21 01:49 AM UTC JANUARY 1997 THORPE 65 the vortex cores. The maximum vorticity at z ϭϪh/2 b. Effects of vortices on the density interface is then ␦AL2[1 ϩ ⑀a/h] at the wave crests. A corre- sponding reduction in vorticity occurs in the wave The order ␦⑀ solution for the interfacial displacement troughs. The growth rate of instabilities leading to vor- is given by (A16): 2 ϭ N(l)sin(kx ϩ (l ϩ m)y Ϫ t) tex breakdown is proportional to the vorticity, and the ϩ N(Ϫl)sin(kx ϩ (l Ϫ m)y Ϫ t), with N(l) given by contribution of internal waves will be large when K (A17). This, in general, represents two forced distur- ⍀ (when growth occurs in a timescale much less than bances on the interface. The amplitude of these distur- that of the waves) and the order ⑀ terms in (4) are large bances is proportional to the internal wave amplitude (e.g., when the wave slope, ak, is greatest and the waves and to the ratio of the vortex strength divided by the are long so that Kh K 1). internal wave frequency. When k ϭ 0, the internal waves travel in the y di- If l ϭ 0, the internal waves travel in the x direction parallel to the vortices (see Fig. 1), and is equal to rection through the vortex array (see Fig. 1) and 2 M cos my sin(kx t), where M 2 (a / )mk2( / the location and strength of the vortex cells are mod- Ϫ ϭ 1 ⍀ h){[(2/h 2m2)/K (2/h2 ϩ m2)]coth K h Ϫ (coth kh2)/ ulated. At z ϭϪh/2, the x component of vorticity is ϩ ϩ k}/{⌺2[ (coth kh)/k Ϫ (coth K h)/K ) ϩ (KϪ1 Ϫ ϭ ␦⍀{sin my Ϫ [⑀am/2 sinh(kh/2)]cos my sin(ly Ϫ 1 ϩ ϩ 2 x KKϪ12]} and ϭ k2 ϩ m2. When the internal waves are t)}. The extreme values of the vorticity, equal to ϩϩ long so that kh K 1, M tends to zero as mh tends to ⍀␦ {1 ϩ [⑀am/(2 sinh(lh/2)]2}, occur at the wave ͙ zero or in®nity; has a maximum value of 9(a⍀/)/ nodes. By writing as N(y)sin(my Ϫ (y)), it is seen 2 83when mh ϭ /3and cosmy sin(kx Ϫ t) ϭϪ1. that the dimensions of the vortex cells are reduced from ͙͙ /m by a factor 1 Ϫ ⑀al/[2 sinh(lh/2)] at the wave troughs. 4. Interaction of vortices and internal waves in a The vertical component of vorticity is found by sub- shear ¯ow stituting for 1 and into (A7) and integrating The effects of a uniform shear (U(z), 0, 0) can be
z ϭ ␦⑀(k⍀a/sinh Kh)sin z/h sinh Kz sin my added to the examples examined in section 3. Solutions are more complicated because a y component of velocity ϫsin(kx ϩ ly Ϫ t). (5) has to be added to account for the interaction of the mean shear and the x-independent rotational ¯ow, ϭ Recalling that ϭ ⑀a cos(kx ϩ ly Ϫ t) shows that the .␦ y, at orderץ/ץz, w ϭץ/ץϪ effect of the internal wave is to generate a horizontal x Simple solutions that enable the interaction of arrays gradient of the vertical velocity component at order ⑀, of like-signed and steady vortices in an upper-layer which acts on the y-directed vorticity of order to pro- ␦ shear ¯ow (e.g., neutral solutions of the Rayleigh sta- duce an oscillatory vertical component of the vorticity, bility equation) with interfacial waves might aid un- /2 out of phase with the internal wave. No vertical derstanding of Kelvin±Helmholtz billows and temper- vorticity is generated if the internal wave propagates in ature ramps in the mixed layer. These have proved elu- the y direction so k ϭ 0, and z is zero at the surface sive; general analytical solutions containing both the and foot of the mixed layer. Maximum vertical vorticity vortex ¯ow and the interfacial waves are not known. occurs at the wave nodes following the passage of the The effects described in section 2 (e.g., those of stretch- wave crests. The maximum vertical vorticity occurs at ing vortices) will nevertheless still apply. Progress can a depth z where (5) gives a maximum value; the deriv- be made in particular cases. The ¯ow U(z) ϭ ative of (5) with respect to z is zero where tanh Kz ϭ U0{sin(mh/2) ϩ sin[m(z ϩ h/2)]}, representing a si- Ϫ(Kh/)tan z/h. The signs in this equation are such nusoidal shear with zero speed at z ϭϪh, is known to that the solution is in Ϫh Ͻ z ϽϪh/2, and the maximum be unstable if limited between rigid boundaries at z ϭ vertical vorticity component is therefore in the lower 0 and Ϫh when mh Ͼ , and neutral modes exist with half of the mixed layer. Kelvin cat's eye pattern of streamlines and wavenum- It may similarly be shown from (A6) that an oscil- bers k ϭ ͙(m2h2 Ϫ 2) in the x direction. [In Drazin latory y component of vorticity in phase with the internal and Reid (1981) the fastest growing waves have wave- y ϭ 0). It may beץ/ץ ,.wave is generated by the interaction provided that the numbers in the x direction (i.e internal wave is propagating in neither the x nor the y shown that the range of unstable wavenumbers is in- direction. The ratio y/x is a measure of the angle , creased if the lower boundary is removed and replaced by which the internal waves distort the cell vorticity by a free interface to a lower layer of ¯uid of equal or from its unperturbed x direction. Neglecting terms of greater density with zero mean ¯ow.] Accompanying an- order greater than ␦, tan ϭ [␦akl cosh(Kz)/K alytical solutions for interfacial waves are, however, not sinh(Kh)]cos(kx ϩ ly Ϫ t). The angle is greatest at the known except in the trivial case when they propagate in wave crests and troughs and for internal waves that the y direction and are unaffected by shear; their dis- propagate at 45Њ to the y direction (i.e., when k ϭ l). persion relation is then given by (2). A formal solution At the surface, z ϭ 0, the angle tends to ␦a/h as Kh for the interaction terms can be found with periodic terms tends to zero (i.e., for long interfacial waves). similar in form to those described in section 3.
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Linear solutions for interfacial waves with wavenum- a. Case 1: Constant buoyancy frequency ber K ϭ (k, l) are found in a two-layer ¯uid with a Consider a ¯uid with buoyancy frequency N ϭ const, uniform shear ¯ow U(z) ϭ U (z ϩ h)/h in the upper 0 bounded above and below by rigid, but free-slip, plane layer and no mean ¯ow in the lower layer with disper- surfaces at z ϭ 0, Ϫh. If the motion is uniform in the sion relation ,(x ϭ 0 (as is usually supposedץ/ץ x direction so that 2 [12ϩ tanh(Kh)] ϭ gK( 21Ϫ )tanh(Kh) then, from (7) and (8), the streamfunction , with ϭ y, satis®esץ/ץz, w ϭץ/ץϪ ϩ10kU tanh(Kh)/Kh, (6) (y2 ϭ 0. (9ץ/2ץ[(t2)ٌ2 ϩ [N2 Ϫ (U u (0)/h2ץ/2ץ) where K ϭ ͦKͦ. This reduces to (2) if k ϭ 0, when the 0 s waves propagate normal to the current. When l ϭ 0 and This has cell-like solutions exponentially growing in 2 2 222 k Ͼ 0, the roots for are all real, so the ¯ow is stable time when N Ͻ U0us(0)/h , ϭ 00say. When N Ͼ [as is the plane Couette ¯ow; Drazin and Reid (1981)] steady internal wave solutions exist with frequency, , 2 2 2 2 2 and no neutral mode exists that is associated with a and y wavenumber, l, related by ϭ [N Ϫ 0]l /[l ϩ disturbance centered in the mixed layer rather than at (n/h)2], n ϭ 1,2,3,...,which is less than N2l2/[l2 ϩ 2 2 the interface. Since, for positive values of the frequency (n/h) ]if0 ϭU0us(0) Ͼ 0; the frequency of the in-
, the right-hand side of (6) exceeds gk(2 Ϫ 1) ϩ tanh ternal waves is less than or greater than that of a wave kh, the positive root for the frequency and the phase in the absence of the effects of Stokes drift, which lead speed of the interfacial waves are increased as a con- to the vortex force in the Craik±Leibovich equations, sequence of the shear with U0 Ͼ 0. [Steady wave so- depending on whether U0us(0) is greater than or less lutions with 0 Ͻ /k Ͻ U0 somewhere in the ¯uid are than zero. For a given ¯ow there are no coexisting in- possible when kh is large. These are not contrary to the ternal waves and unstable modes. y ϭץ/ץ) conclusions reached by Banks et al. (1976) that steady If the motion is uniform in the y direction ,xץ/ץz, w ϭϪץ/ץinternal waves must generally have a phase speed that 0), the streamfunction , with u ϭ lies outside the range of speeds of the mean ¯ow U(z), satis®es (x2) ϭ 0, (10ץ/2ץ)x]2ٌ2 ϩ N2ץ/ץ( t ϩ (U ϩ uץ/ץ] since the Taylor±Goldstein equation on which the theory relies is not singular at the level where /k ϭ U when s the ¯uid is homogeneous and d2U/dz2 ϭ 0.] which is the Taylor±Goldstein equation, but with the mean Eulerian velocity replaced by the total Lagrangian drift. The solutions are known to be stable with ampli- 5. Langmuir circulation and internal waves tude decreasing from initial values as time increases Craik (1977) offers some simple examples of solu- (Eliassen et al. 1953; for more recent references, see tions of the Craik±Leibovich equations for cases in Knobloch 1984) for all positive values of the Richardson which the mean ¯ow (U(z), 0, 0) and the Stokes drift number (here based on the shear in the Lagrangian drift). us ϭ (us(z), 0, 0) have a linear variation with depth and There appear to be no simple cases in which stable the ¯uid is inviscid. As Leibovich (1977) points out, waves and unstable modes coexist. the solutions may then be de®cient as descriptions of Langmuir circulation. When disturbances are indepen- b. Case 2: Two-layer ¯uid dent of x, the boundary condition w ϭ 0 at the surface also implies that u ϭ 0, and this fails to reproduce an Consider now the two-layer ¯uid with homogenous observed feature of Langmuir circulation, namely, that ¯uid in 0 Ͼ z ϾϪhand with a density discontinuity u has a maximum at the surface. Nevertheless, as Lei- at z ϭϪh, as in sections 3 and 4. If the motion is bovich suggests, the inviscid problem will probably be uniform in x, it may be shown from (7) that the upper-
/ץzand w1 ϭץ/ץan adequate description of Langmuir circulation for layer streamfunction , with 1 ϭϪ y, satis®esץ small times. The Craik±Leibovich equations may be written (Tandon and Leibovich 1995a) 2 ץ 22ٌץ 2 ϭ0 , (11) yץt22ץ (١p/01 Ϫ g١z/01, (7 ϫ u) Ϫ ١) ١u ϭ us ϫ´t ϩ uץ/uץ where p is the pressure, (z) is the density, is y ϭ 0atzϭ0. [If the ¯uid is bounded by aץ/ץ 0 ϩ 01 with y ϭ 0, then aץ/ץ the mean (reference) density, and u ϭ (U ϩ u, v, w)is horizontal surface at z ϭϪhwhere the velocity. The continuity of density equation becomes solution is ϭ A sin nz/h exp(1t ϩ ily), n ϭ 1,
١ ϩ w١0 ϭ 0, (8) 2,´´´,with´(t ϩ (us ϩ uץ/ץ
222222 (u ϭ 0. For sim- 10ϭ(lh)/[n ϩ(lh)]. (12´١ and incompressibility implies that plicity of illustration it is supposed that U(z) ϭ U (z ϩ 0 Solutions are exponentially growing, the fastest having h)/h and u ϭ u (0)(z ϩ h)/h (see also Tandon and Lei- s s n 1. Growth rates increase with lh, and the fastest bovich 1995a). We seek cases in which solutions coexist ϭ growing have that represent steady internal waves and growing Lang- 22 muir cells so that their interactions may be examined. 10ϭ (13)
Unauthenticated | Downloaded 09/30/21 01:49 AM UTC JANUARY 1997 THORPE 67 at zero wavelength. Inclusion of viscosity avoids this unphysical limit (e.g., Leibovich and Paulucci 1981).]
We may de®ne a velocity potential 2 in the lower layer 2 z, whereץ/2ץy, w2 ϭץ/2ץsuch that ٌ12 ϭ 0, 2 ϭ 2 tends to zero as z tends to Ϫϱ. At the density in- terface, z ϭϪhϩ, the linearized kinematic condition zץ/2ץyϭץ/ץt ϭץ/ץ gives the boundary condition at z ϭϪh. If p1 and p2 are the pressures in the upper and lower layers, respectively, continuity of pressure at z ϭϪhϩimplies that the pressure gradient along the interface must be the same in both layers or that pץ ץ pץ pץ ץ pץ 1122ϩϭ ϩ at z ϭϪhϩ. (14) yץ yץ zץ yץ yץ zץ
Recalling that U and us are both zero at z ϭϪh, it may be shown that
(t, (15ץ/1ץy ϭϪ1ץ/p1ץ z ϭϪ1gto zeroץ/p1ץ to ®rst order at z ϭϪhϩ, and order, while Bernoulli's equation gives
ץpץ22ץpץ 222ϭϪ and ϭϪ gϩ 2, (16) zץtץ[] zץ ,tץyץy22ץ to ®rst order. Substitution into (14) then gives the lin- earized boundary condition 2ץ ץ vץ ץ gϩ12ϭgϩ,atzϭϪh. (17) tץyץyץt 22ץyץ11 The functions ϭ F(z)exp(t ϩ ily), ϭ B exp(lz ϩ t ϩ ily), are solutions provided that d2F/dz2 ϭ ␣2F, where FIG. 2. (a) Curves showing the left-hand side (lhs) and right-hand 2 2 2 2 side (rhs) of Eq. (20) as functions of ␣h. (b) The lhs and rhs of Eq. ␣ ϭ l [1 Ϫ 0/ ] (18) (23) as functions of . and
2 2 2 [1␣ coth ␣h ϩ 2l] ϭϪgl(2 Ϫ1). (19) ⌺ ϭ gl(2 Ϫ 1)tanh ␣h/[1(␣/l) ϩ 2tanh ␣h], (21) Substituting for 2 from (18) into (19), we obtain from (19). This is identical to the dispersion relation (2) for ``free'' interfacial internal waves between two ¯uids ␣h coth ␣h ϩ ( / )lh ϭ r2[(␣h)2 Ϫ (lh)2], (20) 2 1 with zero mean ¯ow if ␣ ϭ l. From (18), where r2 ϭ g( Ϫ )/2h , which for simplicity we 2 1 0 1 (␣/l)2 ϭ 1 ϩ 2/⌺2, (22) suppose is Ͼ 0. Without loss of generality we take l Ͼ 0 2 2 0. and so ␣ tends to l as us(0)U0/h ⌺ tends to zero, or as If ␣ is real, the left-hand side (lhs) of (20) is always the shear ¯ow or Stokes drift are small in comparison positive and increases continuously and monotonically with, say, h⌺.If␣kl, then us(0)U01 k gh(2 Ϫ 1) from [1 ϩ (2/1)lh] when ␣h is zero (see Fig. 2a). For and the wave frequency, ⌺, tends to glh(2 Ϫ 1)/ ½ large ␣h, the lhs is proportional to ␣h ϩ (2/1)lh. The 1[us(0)U0] , which is a modi®ed interfacial wave of rhs of (20) also increases monotonically. It is equal to frequency very much less than the free interfacial is negative, so, as ␣ץ/⌺2ץ Ϫr2(lh)2 at ␣h ϭ 0, where its value is less than the lhs, waves. It is easily shown that and is zero at ␣h ϭ lh. It is proportional to (␣h)2 at in case 1, the frequency of the modi®ed interfacial large ␣h and therefore exceeds the lhs at suf®ciently waves is always less than that of the free interfacial large ␣h. A solution of (20) therefore exists for some waves if us(0)U0 Ͼ 0. If ␣ is imaginary, (20) becomes value of ␣h Ͼ lh, whatever the value of r2 Ͼ 0. These  cot  ϩ ( / )lh ϭϪr2[2 ϩ(lh)2], (23) solutions for real values of ␣ have proportional to 2 1 sinh ␣z, and 2 is negative so that is imaginary; the where ␣h ϭ i, with  real. The lhs decreases contin- solution corresponds to a steady interfacial wave of fre- uously and monotonically from [1 ϩ (2/1)lh](Ͼ0) at quency ⌺ propagating in the y direction with wavenum-  ϭ 0, through (2/1)lh at  ϭ /2, and toward minus ber l, where in®nity as  tends to (see Fig. 2b). The rhs also
Unauthenticated | Downloaded 09/30/21 01:49 AM UTC 68 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 decreases monotonically from Ϫ(rlh)2 (Ͻ 0) at  ϭ 0, strength of vortices is also found to be altered when the but is ®nite at  ϭ . There is, therefore, always a root internal waves travel along the array of vortices (in the in /2 Ͻ  Ͻ [and an in®nite set, each root lying in y direction, Fig. 1). The extreme values of the vorticity (2n Ϫ 1)/2 Ͻ  Ͻ n,nϭ2,3,´´´ascotis cyclic]. occur at the wave nodes, and the horizontal cell di- From (18), these roots have [(2n Ϫ 1)/2]22Ͻ  ϭ mensions are reduced in the wave troughs by a factor 222 2 (lh)(0/ Ϫ1) Ͻ (n ) so that f1 ϭ ␦al/[2 sinh(lh/2)]. This will also contribute to a
222 2 2 2 2local increase in the rate of growth of the vortex insta- 00Ͼ Ͼ (lh) /[(lh) ϩ (n) ] ϭ 1(24) Ϫ2 bility, proportional to (1 Ϫ f1) . The presence of in- from (12). These solutions with imaginary ␣ have F(z) ternal waves causes variations in the orientation of vor- ϭ sin(z), corresponding to cell-like motions, and (24) tices. Long interfacial waves traveling at 45Њ to the cell shows that their growth rates are greater than those of direction (taken as the x direction) will change the di- disturbances in a ¯uid bounded by a rigid surface at z rection of the horizontal component of the vorticity ϭϪh. Solutions corresponding to interfacial waves (␣ by 10Њ at the wave crest and troughs if the wave am- real) and to unstable growing cells (␣ imaginary) may plitude is about 18% of the mixed layer depth. This occur in the same parameter ranges and therefore co- nonalignment is comparable to that ascribed to nonlinear exist, unlike case 1. effects at the onset of secondary motions [9Њ for mod- Formal solutions for the interaction terms can be erately forced motions, Tandon and Leibovich (1995a); found when two modes, one representing interfacial 11Њ for stronger forcing, Tandon and Leibovich waves and the other unstable, but relatively slowly (1995b)]. It is therefore important to account for the growing, cells, both independent of x, are taken as order effect of internal waves in assessing the variations in ␦ and ⑀ terms, respectively. Solutions have wavenum- the observed patterns of vortical motions in strati®ed bers in the y direction that are sums and differences of water. the wave and cell wavenumbers, as in section 3b. An In general, the vortices induce forced disturbances on important assumption has, however, been made; the the interface (section 3b). In the particular case when l presence of the internal waves is supposed to have no ϭ 0 and the internal waves are long in comparison with effect that will modify the Stokes drift, us, of the surface the thickness of the upper layer (kh K 1), it is found waves. While this assumption appears valid at the (®rst) that the interfacial disturbance caused by the vortices is order of terms considered here, in practice internal a maximum at mh ϭ /3͙ . The maximum value of 2 waves do modify the surface wave ®eld, making the is 9(a⍀/)/8͙ 3 (ϭ a1, say). Typical speeds of currents signatures of internal waves visible from orbiting sat- (Am) found in Langmuir cells are 0.05 m sϪ1 and typical ellites (e.g., Apel et al. 1975) and sometimes leading to cell length scales (/m) of 10±20 m (e.g., see Leibovich breaking waves on the sea surface (e.g., Thorpe et al. 1983; Weller and Price 1988) give values of ⍀ of about 1987). At the higher order considered in section 3 these (3 to 6) ϫ 10Ϫ2 sϪ1 when mh ϭ /3͙ . Internal wave interactions may affect the terms that should be included periods of 10±40 min (e.g., see Phillips 1966; Holligan in (7) and (8). The interaction of internal waves and et al. 1985) give values of of (3 to 10) ϫ 10Ϫ3 sϪ1, Langmuir circulation is therefore not pursued further and so 9(⍀/)/8͙ 3 ranges from about 2a to 13a. The here. interfacial displacements induced by the vortices may therefore be large. The corresponding time of growth of the instability of vortex arrays, taken as ten times the 6. Conclusions e-folding periods, was found to be between 0.9 and 21 Internal waves have two effects on vortices in bound- min for 10-m cells with 0.05 m sϪ1 current speeds ary layers (section 2). They stretch and compress vor- (Thorpe 1992). This range implies that the instability tices, so enhancing and reducing their vorticity, and they of the vortices will often grow by a signi®cant amount alter their local separation and orientation. Both effects in periods in which the phase of the internal waves does may affect the stability of the vortices. It is shown in not change greatly, so that locally the waves have time section 3a that the maximum enhancement of vorticity to enhance the instability of cells appreciably before occurs when the internal wave travels parallel to the their phase changes. vortices (in the x direction of Fig. 1). The maximum is The mean kinetic energy per unit surface area con- 2 then found in the mixed layer over the wave crests, the tained in vortices with ϭ (⍀/c) sinmy sinz/h is 1⍀ h/ fractional increase of vorticity being equal to f, the wave 8c, where c ϭ m2 ϩ (/h)2, while the mean potential slope divided by 2 sinh(kh/2), which for long internal energy per unit surface area of the interface disturbances 2 waves (kh K 1) is equal to a/2h. The growth rate of of amplitude a1 is g(2 Ϫ 1)a1 /4. The ratio of the po- instabilities of vortex arrays increases linearly with their tential energy of the interface displacements to the en- 2 2 vortex strength, so that, provided that K ⍀ [or, from ergy in the vortices is R ϭ 2g(2 Ϫ 1)a1c/1h⍀ ϭ 2 2 2 the dispersion relation (2) for small kh, gk h(2 Ϫ 1)/ 2a1c(/⍀) (1 ϩ 2tanh Kh)/[1tanhKh)Kh], using (2). 2 2 1 K ⍀ ], a slowly varying ¯ow approximation is jus- It is found that R ϭ (27/8)(am/kh) when l ϭ 0, kh K ti®ed, the effect of internal waves in promoting the in- 1, mh ϭ /3͙͙, and a1 ϭ 9(a⍀/)/8 3.Ifmϭ/(20 stability of vortices will also be proportional to f. The m), kh ϭ 0.1, and the interfacial wave amplitude, a, has
Unauthenticated | Downloaded 09/30/21 01:49 AM UTC JANUARY 1997 THORPE 69 a (small) value of 0.3 m, then R ϭ 0.75. The displace- down and across the wind direction, which may affect, ments resulting from the interaction of the vortices and for example, the possible resonant interactions between the internal waves may therefore contain a large fraction the waves. Large internal waves modulate the surface of the energy in the system. It is observed that the hor- waves and possibly affect their Stokes drift at the order izontal dimensions of Langmuir cells tend to increase ⑀ of the internal waves, so introducing new terms, which with time, both in observations (Smith 1992) and in should be accounted for in the equations of motion de- numerical studies (Li and Garrett 1993). An increase of scribing Langmuir circulation (7). The study emphasizes horizontal cell scale toward mh ϭ /3͙ in the presence the shortcoming of the inviscid analysis and reveals the of internal waves will involve a greater amount of en- inadequacy of representing Langmuir cells by inviscid ergy being transferred into the potential energy of the vortices. Both fail to reproduce the observed pattern of interfacial displacement, an effect that may limit the cell downwind velocity variation across the Langmuir cells, development at cell scales below /m ϭ h͙3. a feature associated with the retention of the viscous These effects are, however, secondary, depending on terms (see section 5). Further work is required to ac- interactions between waves and vortices. When the count for the complexity of internal wave modulation mixed layer in which wind-wave-driven ``primary'' of Stokes drift and of retaining viscosity. The latter will Langmuir cells develop is bounded below by a density result in a problem involving the simultaneous growth interface, the rates of cell growth are greater than of Langmuir cells and the damping of the internal those found at the same wavenumbers when the bound- waves. The simple theory developed here falls far short ary is rigid, 1, at least at small times when the effects of providing an explanation of, for example, the scale of viscosity may be neglected (section 5). The size of or form of turbulent eddies observed by Holligan et al. the increase depends on the wavenumber l of the cells, (1985). Although it does show that eddy stretching will 2 2 and on a parameter r ϭ g(2 Ϫ 1)/0h1, which is the be greatest over the internal wave crests, the position ratio of the square of the speed of long interfacial waves, at which the surface ``boils'' are reported, further study 2 c ϭ gh(2 Ϫ 1)/1, to the product of the Eulerian and is needed to consider, for example, the higher-order ef- Lagrangian drift at the surface U0us(0). For example, if fects that affect the shape of internal waves (resonances 2 2 lh ϭ 0.2 and r ϭ 2, /1 ϭ 1.058, while if lh ϭ 1 and appear to be possible at order ␦ ⑀) and the coexistence 2 r ϭ 1, /1 ϭ 1.086; the increase in growth rate that of waves over a broad band of frequencies and direc- may occur in the ocean is typically less than 10%. This tions. The instability of vortices usually develops in a study provides a limit on the validity of those in which three-dimensional manner [e.g., as commonly observed it is assumed that the lower boundary is a density in- in contrail vortices; see Crow (1970)], and further study terface with a density difference suf®cient to prevent will require a methodology that allows for three-di- vertical motion (e.g., Moroz and Leibovich 1985). In mensional effects to be represented. accordance with the view that Langmuir circulation is a transient process, contributing to the turbulence and Acknowledgments. I am grateful to a referee for draw- dispersion in the upper ocean and in lakes, it is shown ing my attention to a paper by Leibovich that helped that slower growth rates, and hence less vorticity at a clarify the role of viscosity in Langmuir circulation. given time after the onset of growth, will also imply longer periods before the cells become unstable. This helps account for the observed shorter lifetimes of cells APPENDIX in deep strati®ed waters (up to a few tens of minutes) The Order ␦⑀ Equations (§ 3). compared with shallow lakes, where periods are re- ported to exceed one hour. We consider a two-layer ¯uid. The upper layer of
The effects of the Stokes drift terms represented in density 1 is rotational, containing vortices. The lower the Craik±Leibovich equations (7) is to reduce the phase layer below z ϭϪhϩhas density 2 and is supposed speed of interfacial waves propagating normal to a mean irrotational, its primary motion being that forced by ¯ow, when this ¯ow has the same direction as the Stokes interfacial waves of order ⑀ at the density interface and drift, as is normally the case for wind-driven waves and the vortex motions of order ␦ producing no interfacial drift currents (section 5). The reduction depends on r. disturbance. The velocity and vorticity vectors are de- For r2 ϭ 2 and lh ϭ 0.2, where l is the wavenumber scribed to order ␦⑀ by (A1)±(A4). The forcing of the of the interfacial wave (so that the wavelength is here ␦⑀ terms through the interactions represented in the vor- equal to 31.4h), the reduction in wave speed from that ticity equation (1) are given in (A5)±(A7), and these, in the absence of Stokes drift terms is 8% when 1 ഠ together with the incompressibility equation (A8), are 2 2. A 17% reduction is found when r ϭ 1 and lh ϭ 1. manipulated to give equations for the x and z compo- This decrease contrasts with the effect of a mixed layer nents of the rotation parts of the motion (A10) and (A9) shear ¯ow in the direction of wave propagation, which respectively. The order ␦⑀ boundary conditions at the increases the phase speed of interfacial waves (section interface, (A11)±(A15), are then applied to ®nd the or-
4), and a substantial difference may therefore be ex- der ␦⑀ interfacial disturbance 2 (A16), given a suitable pected between the speeds of internal waves traveling choice of the order ␦ and ⑀ solutions for the motion.
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ץץץ 2ץץץ At order ␦⑀, the motion in the upper layer is divided into a rotational part (u, ,w) and irrotational part with (ٌ2w)ϭٌ2 Ϫ1 ٌ2 211 (y(A9ץyץ[]yץyץxץtץ velocity potential ⌽ . The motion in the lower layer is 1 irrotational with velocity potential ⌽ . Up to order ␦⑀, ץץץ 2 .the velocity and vorticity in the upper layer are then Ϫٌ1 2 z1ץzץ[]zץ ,(xץ/⌽ 1ץx ϩ ␦⑀(u ϩץ/ ץ⑀] u11ϭ This can be integrated to ®nd the rotational solution w. -y), A solution for u may be found similarly by differentiץ/⌽ץzϩ␦⑀(v ϩץ/ץ␦yϪץ/11ץ⑀ ating (A6) with respect to z, subtracting (A7) differ- (z), (A1ץ/⌽ץyϩ␦⑀(w ϩץ/ץ␦zϩץ/ץ⑀ 11entiated with respect to y, and adding (A8) differentiated and with respect to x, giving
ץץץץץץ ( 1 ϭ (xyz, , (2u)ϭٌ112Ϫٌ2, (A10ٌ) yץzץz11ץyץxץtץ 2 [] ,(zץ/vץy Ϫץ/wץ)⑀␦ϭ[␦ٌ1ϩ (y)], (A2ץ/uץx Ϫץ/vץ)⑀␦ ,(xץ/wץz Ϫץ/uץ)⑀␦ from which the rotational solution u can be found by respectively, and in the lower layer integration. The boundary conditions at the interface are x), used to ®nd the order ␦⑀ disturbance 2. The kinematicץ/⌽ 2ץ⑀␦ x ϩץ/ ץ⑀] u22ϭ boundary condition applied at z ϭϪhϩ⑀1 ϩ ␦⑀2 y, givesץ/ ⌽ץ⑀␦yϩץ/22ץ⑀ (z], (A3ץ/ ⌽ץ⑀␦ z ϩץ/ ␦⑀ (z, (A11ץ/2ץzϭץ/1ץt ϭץ/1ץ 22 and at order ⑀, and
2 ϭ [0, 0, 0], (A4) (z), (A12ץ/ץy( 1ץ/ץzϩwϩץ/⌽ 1ץzϭץ/ ⌽ץt ϭץ/22ץ z2,ٌ2⌽ ϭٌ2⌽ץ/2ץy2 ϩץ/2ץrespectively, where ٌ2 ϭ 1 1 2 at order ␦⑀, both equations being evaluated at z ϭϪh. z2, and the equationץ/2ץy2 ϩץ/2ץx2 ϩץ/2ץϭ0 with ٌ2 ϭ The condition that the pressure is continuous at the in- for the interface is z ϭϪhϩ⑀ ϩ ␦⑀ . The vorticity 1 2 terface leads to equation (1), at order ␦⑀, then gives the rate of change of vorticity in the upper layer: (t, (A13ץ/2ץt Ϫ 2ץ/1ץ2 Ϫ 1)1 ϭ 1) i) the x-directed vorticity: at order ⑀, and from ٌץץ 22ץ vץ wץץ pץ ץ pץ pץ ץp1122ץ 2111 (Ϫϭٌ1 Ϫ ϩϭ ϩ at z ϭϪhϩ. (A14 xץ xץ zץ xץ xץ zץ yץyץ x2ץ zץyץ[]tץ 2ٌץץ Ϫ11, (A5) See also (14) to give zץ zץ ץץ uץ ⌽22ץץ ii) the y-directed vorticity: (Ϫ)21ϭϩϪ 1 yץxץzץtץxץtץ]x 1ץ21 2 1ץ wץ uץץ 2 22 ⌽ץץץ (Ϫϭٌ1, and (A6 y 12ץxץxץzץtץ [] ϩϪ2 , (A15) xץtץ [zץxץ yץ iii) the vertical component of vorticity: at order ␦⑀, both equations being evaluated at z ϭϪh. 2 [Either the pressure gradient along the interface in theץ uץ vץץ (Ϫϭٌ21. (A7 ,(z x direction, as in (A14), or in the y direction, as in (14ץxץ y1ץxץ[]tץ may be used as the boundary condition. The two may The incompressibility equation is be shown to be equivalent]. w Using the chosen order ␦ and order ⑀ solutions ϭץ vץ uץ ϩϩ ϭ0. (A8) A sinmy sinz/h, ϭ B cosh Kz sin(kx ϩ ly Ϫ t), z 1ץ yץ zץ 2 ϭϪBsinhKh exp[K(z ϩ h)]sin(kx ϩ ly Ϫ t), 1 By differentiating (A5) with respect to y, subtracting ϭ a cos(kx ϩ ly Ϫ t) where B ϭϪa/(K sinhKh), K2 (A6) differentiated with respect to x and adding (A8) ϭ k2 ϩ l2 and A and B are constants, to solve(A9) and differentiated with respect to z and t, and adding all (A10) for w and u, respectively, and substituting these 2 t(ٌ w) in terms of solutions in (A12) and (A14) to solve for ⌽2, we ®ndץ/ץ three, we obtain an equation for and 1, 2 from (A12):
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Holligan, P. M., R. D. Pingree, and G.T. Mardell, 1985: Oceanic 2 ϭN(l)sin[kx ϩ (l ϩ m)y Ϫ t] solitons, nutrient pulses and phytoplankton growth. Nature, 314, ϩ N(Ϫl)sin[kx ϩ (l Ϫ m)y Ϫ t], (A16) 348±350. Ivey, G. N., and R. I. Nokes, 1989: Vertical mixing due to the breaking where of critical internal waves on sloping boundaries. J. Fluid Mech., 204, 479±500. N(l) Knobloch, E., 1984: On the stability of strati®ed plane Couette ¯ow. Geophys. Astrophys. Fluid Dyn., 29, 105±116. a(⍀/)[X(l)/K)coth Kh Ϫ (Y(l)/K )coth Kh] Leibovich, S., 1977: Convective instability of stably strati®ed water ϭ1 ϩϩ, 2c[(coth Kh)/K Ϫ (coth Kh)/K ϩ (KϪ1Ϫ KϪ1] in the ocean. J. Fluid Mech., 82, 561±581. 1 ϩϩ2 ϩ , 1983: The form and dynamics of Langmuir circulation. Annu. (A17) Rev. Fluid Mech., 15, 391±427. , and S. Paulucci, 1981: The instability of the ocean to Lang- 2 2 2 2 2 2 with K ϭ k ϩ l , Kϩ ϭ k ϩ (l ϩ m) , X(l) ϭ (2/ muir circulations. J. Fluid Mech., 102, 141±167. h⌺2){2l[cl(l ϩ m) ϩ k22/h2] ϩ k2mc}, Y(l) ϭ (2/ Li, M., and C. Garrett, 1993: Cell merging and the jet/downwelling h⌺2){2(m ϩ l){(k/h)2 ϩ l2m2 ϩ lmc)] Ϫ mck2}, c ϭ ratio in Langmuir circulation. J. Mar. Res., 51, 737±769. 2 2 2 2 2 Moroz, I. M., and S. Leibovich, 1985: Oscillatory and competing (/h) ϩ m , ⍀ϭAc, and ⌺ ϭ (2k/h) ϩ (c ϩ 2lm) . instabilities in a nonlinear model for Langmuir circulations. The denominator in (A17) is nonzero, and resonances Phys. Fluids, 28, 2050±2061. are therefore excluded at this order. Phillips, O. M., 1966: The Dynamics of the Upper Ocean. Cambridge University Press, 261 pp. Smith, J. A., 1992: Observed growth of Langmuir circulations. J. REFERENCES Geophys. Res., 97, 5651±5664. Tandon, A., and S. Leibovich, 1995a: Secondary instabilities in Lang- Apel, J. R., H. M. Byrne, J. R. Proni, and R. L. Charnell, 1975: muir circulations. J. Phys. Oceanogr., 25, 1206±1217. Observations of oceanic internal and surface waves from the ,and , 1995b: Simulations of three-dimensional Lang- Earth Resources Technology satellite. J. Geophys. Res., 80, 865± muir circulation in water of constant density. J. Geophys. Res., 881. 100, 22 613±22 623. Banks, W. H. H., P. G. Drazin, and M. B. Zaturska, 1976: On the Taylor, J. R., 1993: Turbulence and mixing in the boundary layer normal modes of parallel ¯ow of inviscid strati®ed ¯uid. J. Fluid generated by shoaling internal waves. Dyn. Ocean±Atmos., 19, Mech., 75, 149±171. 233±258. Craik, A. D. D., 1977: The generation of Langmuir circulations by Thorpe, S. A., 1989: The distortion of short internal waves by a long an instability mechanism. J. Fluid Mech., 81, 209±223. wave, with application to ocean boundary mixing. J. Fluid Crow, S. C., 1970: Stability theory for a pair of trailing vortices. Mech., 208, 395±415. AIAA J., 8, 2172±2179. , 1992: The break-up of Langmuir circulation and the insta- D'Asaro, E. A., 1982: Absorption of internal waves by the benthic bility of an array of vortices. J. Phys. Oceanogr., 22, 350±360. boundary layer. J. Phys. Oceanogr., 12, 323±336. , 1995: Dynamical processes of transfer at the sea surface. Drazin, P. G., and W. H. Reid, 1981: Hydrodynamic Stability. Cam- Progress in Oceanography, Vol. 35, Pergamon, 315±352. bridge University Press, 525 pp. , M. B. Belloul, and A. J. Hall, 1987: Internal waves and Eliassen, A., E. Hoiland, and E. Riis, 1953: Two-dimensional per- whitecaps. Nature, 330, 740±742. turbations of a ¯ow with constant shear of a strati®ed ¯uid. Weller, R. A., and J. F. Price, 1988: Langmuir circulation within the Institute Weather Climate Research, Oslo, Publ. No. 1, 30 pp. oceanic mixed layer. Deep-Sea Res., 35, 711±747. Farmer, D. M., and M. Li, 1995: Patterns of bubble clouds organised Wunsch, C., 1970: On oceanic boundary mixing. Deep-Sea Res., 17, by Langmuir circulation. J. Phys. Oceanogr., 25, 1426±1440. 293±301.
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