JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 93, NO. C4, PAGES 3570-3582, APRIL 15, 1988 Diffusion by Internal Waves BRIAN G. SANDERSON Newfoundland Institute of Cold Ocean Science and Department of Physics, Memorial University of Newfoundland, St.John's AKIRA OKUBO Marine Sciences Research Center,State University of New York, Stony Brook A perturbation analysis of the Lagrangian equations of motionis used to examine the diffusion induced by a random field of internal waves. At second order there is a random field of shearing motion in the horizontal plane. We calculate the single-particle, two-particle, and patch diffusivities that result from thismotion. Patchand two-particle diffusivity both increase as scale raised to the power of 4/3 for an internal wave field in which thespectrum of horizontal velocity varies as o•-2 INTRODUCTION Horizontal diffusion in the ocean is probably caused by many dynamical processes. Usually, the dispersion of material is caused by relative velocities that are small compared with the mean flow. Such relative velocities are therefore difficult to measure and describein detail. Descriptions of diffusion usu- ally focus on relating statistical properties of the flow field (and sometimes averagedforms of the dynamical equations)to the statistics of tracer distribution. There are a range of em- pirical results describing horizontal mixing in the ocean from the kinematic viewpoint [Okubo, 1971; Stommel, 1949]. More specifickinematic properties such as velocity shearshave also been related to dispersion through the shear-diffusion effect. For example, Young et al. [1982] have studied the interaction of vertical diffusion with the vertical shear of the horizontal velocity field of internal waves. This kinematic process dis- perses material in the horizontal plane. The model of Young et al. is referred to as a kinematic model sinceeven though the vertical shear is an analytic solution of the dynamical equa- tions, there is no analytic solution of the dynamical equations that adequately describes the motion causingthe vertical dif- fusion. The work outlined in this paper is fundamentally dif- ferent from the above kinematic models of horizontal eddy diffusion because we find solutions of the momentum equa- tions that of their own accord will disperse material in the horizontal plane. The existence of turbulence in stratified fluids has been un- derstood, through stability analysis, to be related to the gradi- ent Richardson number for some time [Taylor, 1931]. Ana- lytic solutions for such turbulent motion are not available, and there appears to be little hope of finding such solutions in the near future. Measurements by Grant et al. [1968] indicate that in and below the main thermocline, turbulence occurs in patches.The present analysis assumes that internal waves in- teract sufficientlyweakly that they do not break into turbu- lence as such. However, we will show that such interactions can still result in dispersive motion in the horizontal plane, which may be important for redistributing material when the fluid is not fully turbulent. This dispersive motion may also play a role by advecting patchesof turbulencethat result from sporadicinstabilities. Copyright 1988 by the American GeophysicalUnion. Paper number 7C0941. 0148-0227/88/007C-0941 $05.00 Pierson[1962] was the first to suggest that it might be possible to obtain analytical solutions to the Lagrangian form of the equations of motion that cause mixing.Okubo [1967] usedsimilar solutions and a perturbation analysis of the La- grangian diffusion equation to examine the mixing caused by small-scale decaying turbulence. One of the earliest attempts to find solutionsof the equa- tions of motion that would cause diffusion due to the interac- tions of waves was by Tamai [1972]. Tamai used a pertur- bation analysis of the equations of motion expressed in La- grangian coordinates. He restricted his analysis to the dynam- icsof surface gravitywaves and at a crucial stage chose only to consider two-dimensional motion in the vertical plane. As a result,he observed no eddy diffusionat second order in his perturbation analysis. Later,Herterich and Hasselmann [1982] consideredsecond-orderinteractions of surface gravity waves in a three-dimensional Eulerian coordinate system. The solu- tion for velocity was transformed to Lagrangian coordinates in orderto evaluate eddydiffusivities as a function of sea state. This transformation results in a Stokes drift that can be inter- pretedas a field of randomshears in the horizontal plane. Thus theyobtained horizontal diffusion, but not vertical diffu- sion. Our work is similar to that of Herterich and Hasselmann, but it differs in two respects. First, Herterichand Hasselmann are concerned with surface waves, while we are concerned with internal waves.Second, our approachis Lagrangian and therefore leads directlyto a description of particle motion and diffusion. A transformation is requiredto obtain particle mo- tions, and thereforediffusion,from the Eulerian solutions of Herterich and Hasselmann. The Lagrangian equations of motionappear more difficult to solve. In fact,a comparison of the Lagrangiansolutionfor two-dimensional internal waves [Sanderson, 1985], with the Euleriansolution [Neumann and Pierson, 1966] indicates that the fundamental Lagrangian equations consist of three nonlinear equations and one trivial linear equation, whereas the fundamental Eulerian equations consist of three nonlinear equationsand one nontrivial linear equation. Admittedly, the nonlinear terms takedifferent forms in the different coordinate systems, but when perturbation techniques are used to obtain a solution, thereis nothing to suggest that one coordinate system is intrinsically morediffi- cult to work in than the other. A variety of arguments sup- portingthe studyof the Lagrangian equations havebeenad- vanced by Pierson [1962], Okubo [1967], and Atterrneyer [1974]. 3570