THE UNIVERSITY OF MIC H IGAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Department of Engineering Mechanics Final Report NONLINEAR INTERACTIONS IN ROTATING STRATIFIED FLOW S. J. Jacobs ORA'Project 073440 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. Nonr-1224(55), NR 083-204 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1971 Reproduction in whole or in part is permitted for any purpose of the United States Government. Approved for public release; distribution unlimited.

TABLE OF CONTENTS Page SUMMARY iii 1. INTRODUCTION 1 2. FORMULATION 2 5. AN EIGENFUNCTION EXPANSION 6 4. INTERACTIONS FOR PERIODIC WAVES 16 5. CONCLUDING REMARKS 22 REFERENCES 23 DIS'RIBUTION LIST 24 ii

SUMMARY Nonlinear interactions in a rotating, stratified fluid with a free surface are considered. The field variables are expanded in terms of eigenfunctions of the linearized problem, with coefficients dependent on time. Orthogonality relations are developed to obtain evolution equations for the coefficients, and these are simplified through use of the method of averaging, for weakly nonlinear motions. The geostrophic mode, which alone possesses potential vorticity, is found to obeythequasi-geostrophic equations, even though wave modes are present. The phases of the wave modes are affected by the presence of the geostrophic mode, with frequency splitting for steady geostrophic flow, but there is no energy transfer between the geostrophic mode and the wave modes. Resonant interactions between waves are found to occur for a resonant triad consisting of two external waves and one internal wave. When the wave vectors of the external waves are colinear, the internal wave generated by the interaction has a frequency very close to the inertial frequency, but exceeding it slightly. The growth rate for inertial motions generated in this manner is comparatively slow.

INTRODUCTION In recent years extensive interest has been shown in the effect of nonlinearity on wave motions, both for wave motions in the sea (Hasselmann, 1966) and in other branches of physics as well. The interest stems from the fact that nonlinearity, though small, may have an important cumulative effect over long periods of time. In particular, for oceanic motions, the nonlinearity serves to modify the energy spectrum of external gravity waves and perhaps of internal waves. Past attention in this area has been largely confined to periodic waves, either with a continuous or discrete spectrum. In both cases, the point of interest is that secondary waves, generated by the nonlinear interaction, may stay in phase with a forcing wave, thereby allowing a continuous transfer of energy. This phenomenon is called a resonant interaction, and when it takes place the effect of nonlinearity is significant. The purpose of the present study is to develop a formalism for treating nonlinear interaction of nonperiodic waves, such as would occur in a bounded basin. The particular case treated is that of the motion of a rotating, stratified fluid with a free surface on the P-plane. In the linear case, solutions can be obtained, in principle, by expanding in terms of normal modes. Solutions of this type, for fluids confined entirely by a rigid surface, have been obtained by Siegmann and Howard (Howard, 1968). The expansion procedure can be modified to accommodate the nonlinear case by allowing the coefficients in the expansion to be functions of time. It is then Quite easy to set up the interaction equations. A particular point of interest concerns the geostrophic mode, which is an equilibrium solution of the linear equations with P effect neglected. Perhaps not surprisingly, the geostrophic mode, with nonlinearity and 3 effect included, proves to obey the quasi-geostrophic equations. Also of interest is the generation of internal waves due to nonlinear interaction of external waves. It is found in the case of motions periodic in the horizontal that motions with a frequency slightly exceeding the inertial frequency are generated by the interaction of colinear external waves. However, the growth rate of such inertial motions is comparatively slow. We do not attempt in this study the development of a statistical theory, and consequently the results are not directly applicable to situations of goephysical interest. For this reason, no attempts have been made to compare predictions of the theory with observations. 1

2. FORMULATION Consider an inviscid stratified fluid of constant mean depth H on the rotating earth. Let x = (x,y,z) denote the position vector, with x measuring distance to the east, y to the north, and z vertically upwards, and let v (u,v,w) be the particle velocity. The upper boundary of the fluid is a free surface at z = r(x,y,t), with n 0 in the absence of motion. External forcing and surface tension will be neglected. Let p(z) be a basic density distribution, and let the bouyancy b and the gauge pressure p be defined by b = -g(p - p(z))/p, (1) z p = P + gf p(z')dz', (2) where p and y denote the density and pressure, and po = P5(O). The vertical A unit vector is denoted by m. In the P-plane and Boussineq approximations, with Coriolis parameter f = fo + Py and Brunt-Vaisala frequency N(z) = g dp 1/2 () PO dz the equations of motion are V v = 0, (4) Dv A 1 A +f m x v + - Vp = b m, (5) Dt — +N2 w = 0, (6) for -H < z < I, with boundary conditions w(x,y.,-H,t) = 0 (7) and D D (z _- t) = 0, p = go p(z)dz, (8) at z -=. 2

It is convenient to scale the variables and deal with nondimensional equations, but there is a difficulty in interpreting the results. For any scaling there will arise nondimensional parameters oa* = ratio of time scale to an internal wave period, B* = ratio of residence time to a Rossby wave period, 7* = ratio of time scale to an external gravity wave period, 6* = ratio of vertical length scale to horizontal length scale, E* = ratio of time scale to residence time, r* = ratio of time scale to an inertial period. In general, it is incorrect to make approximations based on the magnitude of these parameters and to expect such approximations to be valid for all conceivable scales of motion. The only exception involves ~*, a measure of nonlinearity, which we anticipate is small for all scales of motion. The scaling to be used here is appropriate for external gravity waves with a horizontal length scale equal to the depth of the fluid. In this scaling y* = 5* = 1 and all the other parameters are small, but approximations will be based only on the condition e* << 1. With this in mind, we let V be a characteristic velocity, Nm the maximum value of N, and introduce dimensionless variables through the scaling -+ -1/2 -t x = H x*, t = (H/g) t*, = V v*, n = v(H/g) /2, p = V(gH)/ P*, b = V N b*, m N = N *1, p = p P*. (9) With this scaling, the nondimensional parameters are a* = (H/g) N, = H/V, E - v/(gH)1/2 r* = (H/g) f, (10) and have the meanings discussed above. Omitting asterisks, the boundary conditions at z = E~ are w = + ~ v V, ~p = J p(z)dz, (11) where the nondimensional density distribution satisfies 5

dp 2 2 dp (- N2, (0) = 1. (12) dz For ~ small, we can transfer the second of these to the level z = 0, obtaining 12 2 2 = p[1 + + - + N p)] + 0 (21) (13) 2 at z = 0. Transferring the kinematic condition to z = 0 and eliminating I yields p 2 2 2N2 W - [1 (pu) +w(i + ( + a N P) + p 0( W 1 W Z az at - 8 P + 0(E2), (14) where A A v = V- m, u = v-mw. (15) The lower boundary condition is w(x,y,-l,t) = 0, (16) and the other equations governing the flow become v v = 0, (17) v A A A A- - + r m x v + Vp - b m = -e(m P y x v + (v ) V) v U, (18) (>b 2 + a N w -- v - Vb - EB. (19) In addition, we assume either that the fluid is confined in a closed region bounded by a vertical wall or that the motion is periodic in the horizontal. Obviously, the latter case connot be precisely correct, since there must be refraction of waves due to the sphericity of the earth. This will be neglected here. We now introduce the vectors ~ = (v, b, p), ~ = (U, B, P) (20) 4

Then the above equations define an initial value problem for G, with weak nonlinearity as expressed by the presence of ~ 0. For the linear problem, eigensolutions can be obtained by assuming the time dependence of the form exp (-i a t), and the initial value problem can be solved by expanding G in terms of the corresponding eigenfunctions. This expansion can also be employed for the nonlinear problem, but the coefficients in the eigenfunction expansions must be allowed to depend on t. Of course, 0 must also be expanded in terms of the eigenfunctions, and the resulting equations are quite complicated. Nevertheless, there are many advantages to the use of this interaction representation, as will be seen shortly. 5

3. AN EIGENFUNCTION EXPANSION The eigenvalue problem is described by V v = 0, (21) A - A -i a v + r m x v + Vp = a b m, (22) -i a b + a N w = 0, (23) for -1 < z < 0, with w = 0 at z = -1, w = -i a p at z = 0. (24) In addition, the flow is either periodic in the horizontal or has vanishing normal component of velocity at vertical walls. In what follows, the symbol V will stand either for the volume of a periodic cell or for the total volume of the fluid, the symbol S for the upper boundary, and the symbol R for the vertical walls in the second of the cases mentioned above. The geostrophic mode, for which a = 0, must be present in general. Denoting this mode by subscript g, we have r v m 7xV7, ab = (25) where p..P (26) The function t is undetermined at this stage, other than being periodic in the first of the cases mentioned above or being such as to make the normal component of vg vanish at R, in the second of the cases. For arbitrary a, elimination of p between the horizontal momentum equations and substitution from the continuity and energy equations leads to the result A ->+ r 2 C Vx v +- (b/N)) = 0. (27) Also, integration of the continuity equation over the horizontal area of V and substitution from the energy equation yields { //J b d A) = 0, (28) 6

and elementary operations provide t',he additional relations 2 2 a (b)z = - 1, p] at z -, (b = 0. (29) Consequently, the bracketed quantities assume nonzero values only for the geostrophic mode. In particular, identifying the term in (27) as the potential vorticity, we see that the geostrophic mode alone possesses potential vorticity. We next determine an orthogonality relation. Let an and am be eigenvalues, with eigenfunctions 0n and g, and let an asterisk denote the complex conjugate. It is easily shown that i( a* - ) (V v* + b b*/N2) + V (p m* + p ) = o, (30) m n n m n m n m m n and integration over V and use of the boundary conditions leads to (a* - ) (, On) = 0, (31) m n m n where (m, n) = I (v v~ + b b*/N2)dV + pn p* dA, (32).m n n nm n m V S and may be considered to be an inner product. We will call any mode with a / 0 a wave mode. Putting am = 0 in (31), so that m = Qg, we find that the geostrophic mode is orthogonal to all wave modes. Putting m = n and noting that (Gn, Gn) is positive definite, we find that the a's are pure real. Finally, we have the orthogonality condition (Om On) O, m / n (35) Actually, (31) implies orthogonality only if the eigenvalues are different for different modes, and the possibility does exist that an eigenvalue a may possess more than one eigenfunction. If this occurs, we use the Schmidt orthogonalization process to insure the validity of (33). From now on, the subscripts on a will be reserved for wave modes only. Assuming the completeness of the eigenfunctions generated by the above eigenvalue problem, we expand @ in the series 7

o = @~ (x, t) + A (t) ~ (x) e- n (54) in which we now allow 0g to depend on time but still satisfy the geostrophic relations given in (25). To compute the initial values for ~; and the A's, we note that if = QI, a known function, at t = 0, the conditions v = v + A v I g n n n b b + A b I g n n n p + Z A p at z = 0 (35) I n n n must be satisfied at t = O. To find An(O), we compute the inner product (Om, GI) for some m. Invoking the orthogonality condition, we readily obtain the initial condition A (o) = (n OI) (' Qn). (36) n n n n To obtain an initial value for Og, we compute the potential vorticity for OI. Since the potential vorticity vanishes for each of the wave modes, this yields the equation A - r 2 A r b m Vx v7 + - (b/N) = m * Vx v + - (b N ). (37) I a bz I g a z g' In similar manner, using the conditions implied by (28) and (29), we obtain 2 2 b = b at z = -1, b + a N = b + a N PI at z = 0, g I g I (38) and ff b dA ff b dA. (39) g I Together with equation (25), these yield V1 +1 N2 7 z = r [m Vx vI + (b/N)], (h0) 0 N z I 8

to be solved subject to a b at z = - az I i + a N2 b = (bI + a N p) at z = (4) z II the lateral boundary conditions, and - f J t d A = a Jf bI dA. (42) It is of interest to show why the last equation is needed. In the periodic case the condition (42) is satisfied identically, but if the fluid is confined between rigid walls (42) is not identically satisfied and the solution for 4r(x, 0) is apparently not unique. This is because the requirement that the normal conponent of 1 vanishes at R implies only that f = f(z) on R. for arbitrary f(z). To determine f(z) (42) must be used. The simplest method appears to be as follows. Let f = ~1 +,2' where *1 satisfies (40) and (41) and vanishes on R, while o satisfies the homogeneous form of (40) and (41) and assumes the value f(z) on R. Both *1 and t2 are uniquely determined, as is easily shown, and *1 does not depend on f(z). In place of (42), we use v2 ff d A = r Im * Vx vI d A, (43) which is derived by integrating (40) over the horizontal area of V and substituting from (42). Since r1 may be regarded as a known function of x (43) implies that IfS V 2 d A = F(z), (44) where F(z) is a known function of z. We now let \j and hj(z) be the eigenvalues and eigenfunctions satisfying 2 r- d ( 1 dh ) -( - )1 = Xh(z), (45) 2 dz 2 dz' N with h'(-'l) = h'(O) + c2 N (0) h(O) = 0. (46) 9

The eigenvalues are pure real and positive, and the eigenfunctions form a complete set (Courant and Hilbert, 1953, Chapter V). Expanding f(z) and |2 in the series 00 00 f = E f h(z), 2 = E f. (x, y) hj(z), (47) j=l j=l j we see that for the determination of f2 the terms pj must satisfy V2 c = j (48) 1 j j j with cp = 1 on R. Then, since 00 F(z) = E f. (j V j d A) h.(z), (49) j=l we determine fj by expanding the known function F(z) in terms of the eigenfunctions hi and equating coefficients. To complete the proof of the validity of this method, we must show that the area integrals in (49) do not vanish. To prove this, we note that the maximum principle for elliptic equations (Courant and Hilbert, 1962, Chapter IV) implies that cpj(x,y) > 0 everywhere. Hence J V cp d A = x f cp d A > 0, and we can make this an inequality by using the fact that cp. = 1 on R and a continuity argument. Consequently, the above procedure for determining f(z) is valid. We turn now to the solution of the initial value problem. Substituting the expansion (34) into the equations defining the initial value problem, we obtain ~v -> -i t g + A e n v = E (50) Z~t n n n ~b -t -+ A e n b = B(51) n n n for -1 < z < 0, and 10

+ A e t p = ~ P at z = 0. (2) U fle- p (52) At n: n n Invoking the orthogonality condition, the coefficients An(t) are found to obey the equations A = e1nt( O ) n(G,~). (55) n e n (, n) (n n)' (53) Also, carrying out a calculation similar to that involved in determining x, 0), we find that xr(x,t) solves the partial differential equation + r = E r [m Vx U+ (B/ )], t 1.2 -z 2 az a az a N z subject to 2 aV = a E B at z = -1 (55) t ad z (- + a2 N2 A) = a ~ (B + a N2 P) at z = 0, (56) dt. 6z t - f d A = a e S B d A, (57);o and the lateral boundary conditions. If U, B, and P were known functions, the initial value problem is solved by integration of the above equations. However, 8 is actually a function of ~ and therefore must also be expanded in terms of Gg and n'. The procedure use here thus replaces the original set of partial equations with an infinite set of ordinary differential equations, for An, and a partial differential equation for 4. Substitution of the expansion (34) into the definition of 8 yields (i) -ig *t 8 - G [@ ] + E G [,.] A e gg g gw i where the subscript gg denotes the interaction of the geostrophic mode with 11

itself, gw the interaction of the geostrophic mode with the wave modes, and ww the interaction of the wave modes with themselves. If we let X denote (Og, Al, A2,...), then X satisfies an equation of the form L X= ~ F(X,t), (59) where F is an almost periodic function of t and L is a linear operator with a bounded inverse. The component of F corresponding to the evolution of the geostrophic mode has Fourier exponents 0, -ai, (-vi -aj), and the component of F corresponding to evolution of the wave mode n has Fourier exponents an, (cn - i), and (C - ci - ) Now, it is obvious that an ordinary perturbation expansion in powers of ~ will fail due to the presence of secular terms, which may also be called resonant interactions. For the geostrophic mode, the resonant interaction is due to a gg interaction and a ww interaction involving all modes i and j such that ai + j = 0. For the wave mode n, the resonant interactions are a gw interaction involving the modes i such that ai = an and ww interactions involving modes i and j such that ai + aj= cn. The resonant interactions will cause the 0( ) term in an ordinary perturbation serier to grow like a power of t. To avoid this possibility, we use the method of averaging, which has been justified both for ordinary differential equations (Bogoliubov and Mitropolsky, 1961) and for partial differential equations of elliptic or parabolic type (Khasminskii, 1963). In this method, the 0(~0) term in a perturbation expansion must satisfy L X = ~ F(X), (60) where -T F(X) = Lim I T F(X,t)dt, (61) T o T-tOO with integration over t where it appears explicitly. Thus in the averaged equation for the evolution of X only the terms giving rise to resonant interactions appear on the right side. Turning first to the evolution of the geostrophic mode, we see that equation (57) has the form'tz fD f * d A = -a s~ f V v b d A + [ffv. Vb i zt ~:z g g i. g +v' Vb ] A. e'i +' Z [1(vi Vb g i. 2 12

+ v Vbi)] A A.A ei( + (62) J 1 J Since v is solenoidal, it can be moved inside the gradient operator and the horizontal divergence integrates to zero, by virtue of the lateral boundary conditions. The vertical velocity wg = 0, and consequently there is no gg interaction in this equation. There is a gw interaction, but it does not contribute to a secularity. The integral occuring in the ww terms takes the form ff v ( bj + V bi)dA = -zff (wi bj + w b)dA, (6) and substitution from (25) yields f (w. b. + w b.)dA = ( +.) - N b dA (64) 1 1 i ) 1 j az N i bj dA.3 This vanishes when (ai + c;) = 0, and consequently in this equation there is no resonant interaction. Therefore, the averaged equation is simply (62) with zero on the right side. The calculation for the other equations governing the evolution of the geostrophic mode is more complicated, but the ultimate result is quite simple. The ww terms either vanish identically or have (ai + aj) as a factor as in equation (.64). Consequently, the averaged equations involve only gg interactions. Defining D s + e v - V' (65) it follows from use of the method of averaging that the evolution of the geostrophic mode is governed by D 2 r y 1. D ) V 2 + r ~f -y + 2 — )) = O, (66) Dtg 1 z2 z 2 z to be solved subject to 2 z f dA = 0, (67) D = 0 at z = -1 (8) Dt- g nz15 15

D) ( -+c +2 2 ) = 0 at z = 0, (69) Dt g and the lateral boundary conditions. These are simply the quasi-geostrophic equations' Hence the quasi-geostrophic equations can be derived, even when wave modes are present, simply by requiring that a perturbation expansion remain uniformly valid in time. In discussing evolution of the wave modes, we must consider the inner product (en, 0) occurring on the right side of (53). Substitution of (34) into the definition of this inner produce leads to complicated expansions which may be simplified by considering only those terms which have nonzero average in the sense of equation (61). For equation (553), the gg terms have zero average, the gw terms have a nonzero average involving wave modes i such that an = ai,: and the ww terms have a nonzero average involving modes i and j such that oi + aj = an. The resulting averaged equation is A = ~' E I A + i a.E. H. A. A.) (e, @ ), (70) n ^ iin i ni,j ij-n i j n n where Iin and Hij-n are functionals of g and of the eigenfunctions for the wave modes. It may appear paradoxical that a gw resonant interaction takes place, since the geostrophic mode evolves by itself. However, this interaction leads only to phase modulation of the wave modes. The interaction coefficient for the gw interaction is I. = fffv* x V i) (g x m Pi y) + v. ( x V*) + in n g g I n R (. b + v bi) -V (b/ )dV- If p* ([2 V i g g i ndV n 1 ~1.53~S. ~w. +- _3 (v pi + ui ) + ~ ( - i Pi)])dA, (71) 1 for ci = cn, ana Iin = 0 otherwise. Here X = V xv (72) is the vorticity and u is the horizontal velocity defined earlier. It can be shown that Iin is skew-Hermitian, I* = -Ii' (73) 14

and this implies that if there are no ww resonant interactions and if the geostrophic mode is independent of time, the gw interaction serves only to modify the frequencies of the wave modes. In the general case, we temporarily normalize the eigenfunctions so that (On, =n) =1 and the energy of any wave mode n is proportional to iAn|2. The rate of change of this energy due to the gw interaction is d IA I ) = Z (I A. A* + I* A. A ) (74) dt ni in i n in i n and the sum of (74) over all modes having a common frequency an vanishes, by virtue of (75). Consequently, there is no net energy transfer between the geostrophic mode and the wave modes. The ww interaction coefficient is more easily written in terms of a symmetric expression Hijn. By a change of n to -n we mean changing ~n to en and an to -n. Also, let P(i,j,n) denote cyclic permutation over (i,j,n). Then Hijn vanishes for ai + aj + an / 0, and otherwise is given by 1 = P1 2 2'H' = 2 P (i,j,n) - (a + a. + a a ) 7f (Pi P. P )dA n -5 1 -+ - i f/f I u. u. +- w. u - V. w i Jz i jU a. i n 1 j R n 1 1 -+ +- wa u V_7 w.} dV. (75) a_., n 1 i 15

4. INTERACTIONS FOR PERIODIC WAVES To solve the eigenvalue problem, we use the method of separation of variables. Let u - (i a V1F + r m x V1F), w = -i C c F,.k2 1 2 2 ~21/b21~pF2 2 2 b = -a rp F, p [(a- r )/k2] F, (76) where F =F(x,y), p = p(z), and k is a separation constant.. Then F solves 2 2 ( V +.k )F = 0, (77) with F either periodic or with u having vanishing normal component at R, and CD solves'' (z) + (xh - k)F = 0, (78) 2 cp (-1) = vp (0) - ao' (0) = 0, (79) where 22 2 2 2 2 = k2 a/( - r ), h(z) = N (z) - (r/a). (80) Substitution of (76) into (75) gives the interaction coefficient, and solution of the eigenvalue problem determines whether resonant ww interactions can take place. Some interesting effects are produced by the presence of walls, particularly the interaction of Kelvin waves (-Saylor, 1970), but we will limit ourselves to the study of periodic waves. Let F = exp(i k. x), where k is a horizontal wave vector. Then (77) is solved, with k = and we are left with the problem of finding the vertical eigenfunctions and evaluating the interaction coefficients. It is seen that any subscript i for a wave mode must denote four numbers, the two components of k, the number of zeros of cp in [-1,0], and the sign of a.- Denote dependence on k by a subscript, and let a superscript s be a positive or negative integer, with |s| being the number of zeros of cR. The sign convention is that sgn C = sgn s, with o invariant under the 16

change k.- -.k. In addition, q. i. invariant under a. change of' dign ot' elitler k or s, and hence -S S (81) A", = (A) (81) -k k is a reality condition. Letting subscript i denote k., s., and A the area of a periodic cell, we have s.-s H.. = A T n (82) ij-n - k k -k j n where s.s.s k+kk i j n i j n i j n 13 n = -2P(i,j,n) { 3(c. + a. + ai aCT)p(o ) (p )—' kj n 2 2 2 153 3 1 O Cp.Cp Cpt 2 1i n i n j n 3 n k.' k 2 + - nP _i jk 2 2 3 2 2 ki 12 + k k k+ k2 k 1 j n k) p! pt: p') (a. a* 2 2 2 3. k. k k a n (83) otherwise. Returning to the eigenvalue problem for cp, we assume that the unsealed Brunt-Vaisala frequency exceeds fo for all z. Then h(z) is positive, and x = D[c]/H[cpl, (84) 17 17

D[p,]. f (p2 + k p ) dz, 2 2 0 2 H[rp] = (o)/2 + 1 h p dz. (85) It is convenient to regard X as the eigenvalue with a to be determined by the first of the relations (80). It is then seen that the eigenvalue problem is characterized by a variational principle, with D/H the Rayleigh quotient. The largest value of |cl, corresponding to the least value of', is the frequency of the external mode, and the other frequencies are for the internal modes. The following results can be obtained without the aid of an explicit solution. (i) From (84), |o| > r, with equality for k = 0. (86) (ii) The phase velocity c = |al/k and the group velocity U = d la|/dk satisfy dc 2 k3 d < r/ck, U < c, 0 < U, for k # 0. (87) dk Since D/H is an extremum when cp is an eigenfunction, the derivative of D/H with respect to any parameter involves only the explicit appearance of the parameter. Therefore, \ is an increasing function of k, which implies the first inequality, the first implies the second, and x/k2 is a decreasing function of k which implies the third. (iii) For the external mode 2 1 >jc > (r + k tanh k [1 + 2 h sinh [k(l + z)] d(88 -1..... sinh k This is obtained by taking p = sinh [k(l + z)] as a trial function. (iv) For the internal modes, 18

12 2 < ( -2 — 2 —- < c. (.) 2 k + n If h is replaced by its maximum value, (1 - r /a ), the eigenvalues are decreased. The altered eigenvalues can be found by integrating (78) and using a graphical solution to solve a transcendental equation. The values of the altered eigenvalues imply the first inequality of (89), which implies the second. (v) For the external mode, there exists a k* such that dU.~ dk < 0 for k > k. (90) dk * Equation (88) implies \/k << 1 when (k tanh k) >> a. Then p = sinh [k(l+z)] is the first term in an asymptotic expansion, and this implies (90). For small a, k << i. For a resonant three-wave external mode interaction to take place, the equations 1 1 1 - + + ~ y = a -k + k k k. + k., (91) k k - k. n i j n. i. j must be satisfied simultaneously. Using equations (87) and (90), we can rule out this possibility if the wave numbers are sufficiently large. There is no possibility of an interaction involving two internal modes and one external mode, if the wave number of the external mode is sufficiently large, as can be seen by use of (88) and (89). Interactions between three internal modes are possible, but have not been investigated here. Finally, as regards interactions between one internal mode and two external modes, it can be seen that the interaction 1 1 I 1, c' + cr -k~k k k - k k = ki s >1, (92) n i j is impossible for the sum interaction, if ki and kj are sufficiently large, but is possible for the difference interaction if kli > kj, and if (ki - kj) is sufficiently small. 19

This last possibility has been investigated for no rotation, r = 0, both for a two-layer model (Ball, 1964) and more generally (Thorpe, 1966). It is of interest to study the effect of rotation for this interaction. We note first that (ki - kj) must be very small, since in general a << 1 and the internal modes have frequencies which do not exceed a. If ki and kj are colinear, then kn is also very small, and the frequencies of the internal modes so generated is very close to the inertial frequency, r, but exceeding it slightly. If the angle between the wave vectors of the two external modes is increased, then kn increases and for fixed s the frequency of any internal mode which can be created by this mechanism also increases. The following case serves as an example. Consider two external modes with wave vectors ko and k1, ko > kl, and let Ai = 0, so that these vectors are the directions of wave propagation. Also, normalize the vertical eigenfunctions by taking D[p] = (k/.)2. (93) 1 1 With this normalization, (2n, 9n) = 2A, and p k (0) =1, so that the A's are effectively the scaled amplitudes of surface gravity waves. Letting Isil Isil A. = A1 =, |s = |sJi = 1, (94) i i and 1 121 -1 T _T lI2 -l (95) k - k 1 2 o the interaction equations are found to reduce to A i oi TA A, o o 1 2 A i ~ a1 T* A A*, 1 1 o 2' A2 = i 6 2 T* A A*, (96) 2- 2 T o 1' 96 where k k = k+ = (97) 1 2 o'1 2 o 20

l:quation (9') was solved as follows. First, adopting the tractible but unrealistic model of constant N, the eigenvalue problem (78) was solved using a perturbation approach, for small r and a. Next, values of r, a, k0, Is2|, and X, the angle between o and il, were chosen. Then (97), regarded an an equation for determining kl, was solved numerically. The procedure was repeated for different values of X and of the other parameters. As anticipated, the calculation showed a continuous increase of a2 with X. Comparison with a seperate calculation for r = 0 showed that the effect of the earth's rotation is felt only for fairly small values of X, that is, when the wave vectors ko and kl are colinear or almost so. When X = 0 2 r, as anticipated; when IXI > 15~, very little effect of r is seen. Returning to (96), it can be seen that if A and A2 are infinitesimal, 0 then A1 is slowly varying, and the equations for Ao and A2 are effectively linear with constant coefficients. The solutions prove to be neutrally stable. However, if A1 and A2 are initially infinitesimal, they grow exponentially with a growth rate R =(a1/C2 Q, (98) where aQ = ~ C A0 TI. This is in accord with a general result (Hasselman, 1967a) and indicates that a short external wave tends to lose energy to a longer external wave and an internal wave. From the numerical calculations, Q, a measure of the initial growth rate for the internal mode, proves to increase monotonically with X, for small X, and hence is favorable to the growth of short internal waves with frequencies large compared to the inertial frequency. The exponential growth rate, on the other hand, has a sharp local maximum at X = 0. For a variety of conditions typical for the oceanic case, the e-folding time for the growth of internal waves with inertial frequency is of the order of a few days. 21

5. CONCIITING REMARKS For application to realistic situations, the work reported above must be modified in a number of ways. The effect of variable depth must be treated and, under certain circumstances, it is necessary to provide a realistic treatment of refraction due to the sphericity of the earth. When such effects are included, the low frequency oscillations are strongly affected, and the treatment of the geostrophic mode is somewhat different. Also, it would be desirable to include higher order nonlinear effects. These modifications could be made in the context of the formalism used here, but it would be extremely difficult to solve the linear eigenvalue problem. Much more serious is the problem of constructing a statistical theory. For motions in an unbounded fluid, with constant mean depth, this can be accomplished formally either by introducing a Gaussian approximation for the Fourier coefficients and passing to the limit of a continuous spectrum (Hasselmann, 1966) or by a formal multiple time approach in which a continuous spectrum is assumed from the start (Davidson, 1967). An important point is that the statistical theories which have appeared to date are based on the assumption that the field variables are homogeneous random functions of the horizontal spatial coordinates. This assumption is built into the multiple time approach, and appears in Hasselmann's work also, particularly in his proof of the approximately Gaussian character of the wave amplitudes in the linear case (Hasselmann, 1967b). If the field variables are not homogeneous random functions, the multiple time approach must be modified considerably or some other method must be found to effect a closure of the moment equations. The difficulty is that for motions in bounded basins, and in particular for the longer waves, the assumption of statistical homogeneity is untenable. Though statistical homogeneity is not needed to prove the Gaussian character of linear wave fields, or rather of the wave amplitudes, one does need a mixing condition which expresses the asymptotic independence of field variables at two different points for large separation between these points (Volkonskii and Rozanov, 1959). It is hard to see how such a condition could be satisfied for long waves in a bounded basin, when the wavelength is comparable to the dimension of the basin. A similar problem arises for systematic changes in depth. For such cases, creation of a statistical theory is an important and difficult problem, as yet unsolved. 22

REFERENCES Ball, F. K., J. Fluid Mech., 19, 465, 1964. Bogoliubov, N. N., and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. Courant, R. and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York, 1953. Courant, R. and D. Hilbert, Methods of Mathematical Physica, Vol. II, Interscience, New York, 1962. Hasselmann, K., Rev. Geophys., 4, 1, 1966. Hasselmann, K., J. Fluid Mech., 30, 737, 1967a. Hasslemann, K., Proc. Roy. Soc. (London), Series A, 299, 77, 1967b. Howard, L., W.H.O.I. Geophys. Fluid Dynamics Lecture Notes, 1968. Khasminskii, R.A., Theory Prob. Applications, 8, 1, 1963. Saylor, J. H. On the Generation of Internal Waves in Lake Michigan By Nonlinear Wave Resonance, U. of Mich. Thesis, 1970. Thorpe, S. A., J. Fluid Mecy., 24, 737, 1966. Volkonskii, V. A. and Y. A. Rozanov, Theory Prob. Applications, 4, 178. 1959. 23

DISTR![ UTION LIST (One copy unless otherwise noted) Director of Defense Research Director and Engineering Naval Research Laboratory Office of the Secretary of Defense Washington, D.C. 20390 Washington, D.C. 20301 Attn: Library, Code 2029 Attn: Office, Assistant Director (ONRL) 6 (Research) Attn: Library, Code 2000 6 Office of Naval Research Commander Department of the Navy Naval Oceanographic Office Washington, D.C. 20360 Washington, D.C. 20390 Attn: Ocean Sciences and Tech- Attn: Code 1640, Library nology Division, Code 480 2 Attn: Code 70 Attn: Naval Applications and Analysis Division, Code 480 Defense Documentation Center 20 Attn: Earth Sciences Division Cameron Station Code 410 Alexandria, Virginia 22314 Office of Naval Research Director Branch Office National Oceanographic Data Center 5536 South Clark Street Building 100 Chicago, Illinois 60605 Navy Yard Washington, D.C. 20390 Mr. Fred 0. Briggson ONR Resident Representative The University of Michigan 121 Cooley Building Ann Arbor, Michigan 48104 24

Unclassi ited S.cisttt Cl. ttkutPnc DOCUMENT CONTROL DATA. R & D (.Serri#ry ri,,irti n of i te, Od. "I fferr.nd fr i rf Weft eninn m^fln "eaot e efrntr d whnn he0 owvrall reportf f cln#lfled) I. QRGI~NA T1NG ACTIVITY V (Cporportaof e.hot) 2-. REPORT SECUnITV CLASSIFICATION Th,:e Regents of Tile University of Michigan Unclassified Ann Arbor, Michigan 48104 h. GROUp 3. REPOTt TITLKt Nonlinear Interactions in Rotating Stratified Flow 4. oE9C sIPTI v trNO.TS (Type of report and Inctrle-ve date.) Final Report S. AU TH-,, firat n",me, ml dl. Inl. -,, F name) S. J. Jacobs 4. REPORT oATSI To. TOTAL NO. OF PAGES 7b. NO. OF REFS January 1971 23 | 12 Sc. CONTRACT OR GRANT NO. O9. ORIGINATOR'S REPORT NUMBERIS) Nonr-1224(55) b. PROJCT NO. 07341i-5-F NR 083-204 c b. OTHER REPOIRT NO(S (Any other numbere Ihat may be eoligned Ihl report) d. 10. OISTRItUTION STATEMENT Reproduction in whole or in part is permitted for any purpose of the United States Government. Approved for public release; distribution unlimited. it. SUPPLEMEW.TARY NOTES 12. ISPONSORING MILITARY ACTIVITY Department of the Navy Office of Naval Research Washington, D.C. 13. ABSTRACT Nonlitnv.r interactions in a rotating, stratified fluid with a free surface are considered. The fiie'i. variables are expanded in terms of eigenfurl.ctions of the linearized problem, with coefficients dependent on time. Orthogonality relations are developed to obtain evolution equations for the coefficients, and these are simplified through use of the method of averaging for weakly nonlinear motions. The geostrophic mode, which alone possesses potential vorticity, is found to obey the quasi-geostrophic equations, even though wave modes are present. The phases of the wave modes are affected by the presence of the geostrophic mode, with frequency splitting for steady geostrophic flow, but there is no energy transfer between the geostrophic mode and the wave modes. Resonant interactions between waves are found to occur for a resonant triad consisting of two external waves and one interval wave. When the wave vectors of the external waves are colinear, the internal wave generated by the interaction has a frequency very close to the inertial frequency, but exceeding it slightly. The growth rate for inertial motions generated in this manner is comparatively slow. DD,NoV. 1473 Unclassified'" Security Classifcation

Unciassiriea __ Sec.urit CIaslt sication- _ 14. LINK A L LINI * KtH C nOL a WT nOLG WT' IM'O L' - Wr - Unclassified Security C'lkwilhcuIlion