Relaxation Processes for Internal Waves in Mesoscale Flow (Transport Phenomena/Rossby Waves) KENNETH M
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Proc. NatL Acad. Sci. USA Vol. 80, pp. 1144-1146, February 1983 Applied Physical Sciences Relaxation processes for internal waves in mesoscale flow (transport phenomena/Rossby waves) KENNETH M. WATSON University of California, San Diego, Marine Physical Laboratory of the Scripps Institution of Oceanography, San Diego, California 92152 Contributed by Kenneth M. Watson, October 12, 1982 ABSTRACT A dynamical "test-wave" model has been devel- Here kh is the projection of the wave number k in the horizontal oped to study transport phenomena within oceanic internal wave plane, wkis the angularfrequency of alinearwave ofwave num- fields. This model is extended here to describe effects of a me- ber k, and c.c. stands for complex conjugate. In the absence of soscale flow field on internal wave transport. Previous work with nonlinear interactions among the waves, the Fourier coeffi- weak-interaction perturbation theory has suggested a substantial cients bk are constant. enhancement due to mesoscale currents. Extension to the strong- In the induced diffusion limit [3], the wave field is separated interaction regime in the present paper suggests a relatively small into large-scale waves, designated here with wave number 1, and effect due to mesoscale interactions. small-scale waves, designated here by wave numbers k and m. Systems of nonlinearly interacting waves have various appli- I shall refer to the large-scale waves as the "ambient field. " This cations in plasma physics, geophysics, and solid-state physics. will consist of internal wave (IW) and Rossby wave (RW) com- A "test-wave" model for triad, or three-wave, interactions has ponents. The ambient field can be treated as linear in the in- been described in two publications (1, 2). This model describes duced diffusion limit. The small-scale waves scatter against the the evolution in Fourier space of a system ofwaves of relatively ambient field. shorterwavelengths and higherfrequency interacting withawave A given "wave" of wave number k is studied in the test-wave system of much larger spatial scales and lower frequencies. model. This test wave is permitted to interact with the ambient The test-wave model provides a tractable set of dynamic field and with other small-scale waves ml, min .. The inter- equations at the expense of approximating the physical phe- actions among ml, M2, ... are neglected in the model. nomena being studied. This model is, however, of sufficient generality to reproduce most published results that have used DESCRIPTION OF THE MODEL a "two-time" perturbation theory and, thus, it can provide in- The test-wave model as developed in ref. 2 can be adapted to sight concerning the validity of these mathematical approxi- the present application by merely including a RWfield (8) in the mations. ambient field. A simpler derivation than that of ref. 1 is ob- The test-wave model of refs. 1 and 2 was specifically devel- a transform of the Taylor-Goldstein equa- oped for and applied to the "induced diffusion" mechanism of tained from Fourier McComas and Bretherton (3), which describes a flux of energy tion, which describes IW propagation in a large-scale flowfield. toward shorter vertical wavelengths in the ocean internal wave The details are given in ref. 9 and will not be repeated here. field. This "induced diffusion" regime extends over a vertical The equations for the small-scale wave amplitudes are of the wavelength range from about 1/2 km down to a few meters. It form of equations 2.6 and 2.7 in ref. 2: is suggested (4) that hydrodynamic instabilities convert wave bk = E {h[c18k-m-leit - c1 ik-m+le +] energy into turbulence at scales below a few meters. I nm The Langevin rate constant describing transient response was calculated in ref. 2 for the internal wave field. This exact cal- + h[3l rn eiht- cl 8k-rn-Is I culation by the test-wave model was compared with the results by the two-time perturbation theory of McComas (5) and of X [Zifh X kh]}bm, [2.1] Watson In the I ex, Pomphrey, Meiss, and (6). present paper, bm = ck-rn-i e tend the calculation of ref. 2 to include the interaction of an in- l{h[clIk-m+le Ii+t- ternal wave field with a mesoscale field. A calculation of Mfiller (7) using perturbation theory suggested greatly enhanced trans- + h[ele6km+le-iA+t - cliBk re-n-tl] port due to the mesoscale field. The present calculations are consistent with this in the regime that perturbation theory is X [Zlh X kh]}bk, [2.2] valid, but show a saturation effect in the strong-interaction do- and main. To develop the test-wave model, a Fourier (more generally, cl = 0. [2.3] a linear eigenmode) expansion is performed on the field vari- ables. For example, the vertical (Lagrange) displacement of a In these equations b = db/dt, etc. The amplitude cl refers to fluid element is written in the notation of ref. 1 as large-scale ambient IWs, while 1 refers to RWs. From ref. 2, equations 2.3 and 2.9, = i/(2kh) {bkexp[i(k-r - wkt)] - c.c.} [1.1] ~3(r,t) Ik (q = [f2 + N2k/k 2]1/2, A+ = -Ok W(m ± (.1O, The publication costs of this article were defrayed in partby page charge payment. This article must therefore be hereby marked "advertie- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. Abbreviations: 1W, internal wave; RW, Rossby wave. 1144 Downloaded by guest on September 24, 2021 Applied Physical Sciences: Watson Proc. Natl. Acad. Sci. USA 80 (1983) 1145 and spective IW and RW fields. It is also convenient to write h =fkh B/2. [2.4] fd2lhI(lnlz) = FI(lz) [2.8] Here No is the Vaisala frequency (assumed constant), fis the in- fd2ln4)M(lnlZ) = (M(lz) [2.9] ertial frequency, and B (1,200 m) is the Garrett-Munk scale Expression 2.8 is given by the Garrett-Munk equation (see length. ref. 2, equation 2.11). To provide a model for numerical cal- For the RW field (8), there is an angular frequency fl, that culation, I shall take is sufficiently low that it can be set to equal zero here. Then, we have = (a/ir) [2.10] FM(lz) + a')' A+ = Ok (mn ±I- °Wk -( [2.5] (1=z a = irIB. The ambient field amplitudes cl and cl are related to the spectra [2.11] 4)I(lh,lz) and FM(lhlz) RESPONSE TO THE TEST WAVE of the respective 1W and mesoscale fields as follows (see ref. 2, As before (2), I now wish to calculate the ensemble averaged 2. 10): amplitude (bk(t)), given that bk(O) is fixed for all members of the equation ensemble. In performingthe ensemble average (... .), I shall again > (IC12) = (Bf)-2fd31 (Dl Il) assume that the bm(O) are Gaussian variables. The integration of Eqs. 2.1 and 2.2 by a Laplace transform technique, which follows that of ref. 2 and need not be repeated = (Bf)-2V2 [2.6] here, gives = fd3 4Mflhlz) 1 2 1/2(l31i2) (Bf)2 = Pt (bk(t)) (2iri)-' Idpe (~p) [3.1] = (Bf)-2V2. [2.7] Here C is a contour parallel to the imaginary p axis and to the Here (... .) represents an ensemble average over realizations of right of any singularities of (D)-1. the ambient field. For this ensemble, the cl and cl are assumed The quantity has the form to be uncorrelated Gaussian variables. The quantities VI and VM (D) represent the root-mean-square horizontal velocities of the re- (D(p)) = DI(p) + DM(p), [3.2] 10-2 1 0-4 k B FIG. 1. Dependence of relaxation rate on mesoscale velocity. Downloaded by guest on September 24, 2021 1146 Applied Physical Sciences: Watson Proc. Natl. Acad. Sci. USA 80 (1983) where DI is given by equation 3.10 of ref. 2 and of the Langevin form 0x (bk(t)bk(°)) ==e-"tlbk(0)1', [3.9] DM(p) = iU2mk2/(4IVz(k)l) ddZ1FM(lZ) except for a possible oscillatory factor. X - - The quantity v is shown as a function of (k1B) in Fig. 1 for co [(1z + ip/lVz(k)l)-' (lz ip/IVz(k)I)-']. [3.3] = 0.05 Noand 0.3 No. The curves are labeled by the root-mean- In obtaining this expression, I have used the assumed scale sep- square mesoscale velocity VM. The case VM = 0 corresponds to aration between the small-scale and ambient fields to write the calculation (2) forwhich only the IW ambientfieldwas taken into account. A+ +zlVz(k)l In Fig. 1, the rapidly rising portion of the curves at the left corresponds to the regime of validity of two-time perturbation where =Wk theory, as has been shown (2). In this regime, the response rate Vl(k) vincreases rapidly with VM, a result consistent with Muller's (7) Use of Eq. 2.10 leads to the expression calculations. For (kzB) > 40, which is outside the range of validity of per- DM(p) = 1/2Vmk2/(P + q/3), [3.4] turbation theory, the mesoscale flowhas much less effect on the Langevin relaxation rate. My physical interpretation of this is where, in the notation of ref. 2, that in the regime of fast relaxation rates, the IW field contin- ually adjusts to the local RW flow and does not sense the shear. q = 31TIVz(k)I/B. [3.5] This research was supported in part by the Office of Naval Research The quantity (D(q)) is a rational function of p with four zeros, under Contract N00014-79-C-0472.