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Some Three-Dimensional Nonlinear Equatorial Flows

ADRIAN CONSTANTIN* King’s College London, London, United Kingdom

(Manuscript received 28 March 2012, in final form 11 October 2012)

ABSTRACT

This study presents some explicit exact solutions for nonlinear geophysical ocean waves in the b-plane approximation near the equator. The solutions are provided in Lagrangian coordinates by describing the path of each particle. The unidirectional equatorially trapped waves are symmetric about the equator and prop- agate eastward above the thermocline and beneath the near-surface layer to which wind effects are confined. At each latitude the flow pattern represents a traveling wave.

1. Introduction The aim of the present paper is to provide an explicit nonlinear solution for geophysical waves propagating The complex dynamics of flows in the Pacific Ocean eastward in the layer above the thermocline and beneath near the equator presents certain specific features. The the near-surface layer in which wind effects are notice- equatorial region is characterized by a thin, permanent, able. The solution is presented in Lagrangian coordinates shallow layer of warm (and less dense) water overlying by describing the circular path of each particle. Within a deeper layer of cold water. The two layers are separated a narrow equatorial band the flow pattern describes an byasharpthermocline and a plausible assumption is that equatorially trapped wave that is symmetric about the there is no motion in the deep layer—see, for example, equator: at each fixed latitude we have a traveling wave Fedorov and Brown (2009). Typical values in the mid- whose amplitude decays with the meridional distance from Pacific are 80 m for the average depth of the near-surface the equator. In contrast to the investigations of equatorial layer where the effect of wind is noticeable (the trade waves performed by Fedorov and Brown (2009); Moore winds that blow prevailingly westward over the sea surface and Philander (1977); McCreary (1981, 1984) or to the in this region induce a westward current in a near-surface discussion of linear Kelvin waves in Gill (1982), the solu- layer) and 120 m for the mean depth of the thermocline— tions we present are exact solutions to the nonlinear gov- see Fedorov and Brown (2009). The field data examined in erning equations in the b-plane setting. They are obtained Moum et al. (2011) highlight the importance of large am- without restricting one’s attention to the regime of long or plitude internal waves with periods in the range 5–20 min short waves and subsequently linearizing about a stably and wavelengths of order 150–250 m in the dynamics of stratified background state: in our solutions the wavelength the upper-equatorial oceans. There are several ways that appears as a parameter and there are no limitations upon these disturbances may be generated—see the discussion its range. The nonlinear character of the solution di- of linear waves in Smyth et al. (2011). On the other hand, minishes in the short-wave regime, being enhanced with the discussions in Fedorov and Melville (2000) and increasing wavelength (see section 4). In the regime of long Greatbatch (1985) provide ample evidence that nonlinear wavelengths the waves present features that differ from effects are important in the modeling of equatorial waves. those that are specific to the linear Kelvin waves, the most important difference being that our wave solutions are nonhydrostatic. Also, in contrast to the shallow water * Current affiliation: Faculty of Mathematics, University of models investigated in Ripa (1982, 1983b,a); Bouchut et al. Vienna, Vienna, Austria. (2005); Boyd (1980, 1998); Majda et al. (1999), which comprise dynamical fields that are independent of the Corresponding author address: Adrian Constantin, King’s College vertical coordinate (by means of a process of vertical av- London, Strand, WC2R 2LS, London, United Kingdom. eraging), and to the multiple-scale perturbation expansions E-mail: [email protected] thereof investigated in Boyd (1983, 1984); Marshall and

DOI: 10.1175/JPO-D-12-062.1

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Boyd (1987), in the present study the vertical variations of and the equation of mass conservation the fluid velocity play an important role. The motivation for 1 1 1 5 the solutions obtained in the present paper is the fact that rt urx yry wrz 0 (2) for the only available explicit exact solution of the (non- linear) governing equations for periodic two-dimensional with the condition of incompressibility traveling gravity water waves, due to Gerstner (1809) and 1 1 5 rediscovered by Froude (1862); Rankine (1863); Reech ux yy wz 0. (3) (1869), all particles move in circles. Gerstner started with the orbital motion of the particles and arrived at the con- Here t is time, f is the latitude, (u, y, w) is the fluid ve- 2 2 clusion that the surface curve was trochoidal in form, while locity, V57.29 10 5 rad s 1 is the (constant) rotational later British and French interestinshiprollingledtothe speed of the earth (taken to be a sphere of radius R 5 independent development of the same solution, starting 6371 km) round the polar axis toward the east, g 5 2 with the assumption of a trochoidal wave profile. We refer 9.8 m s 2 is the (constant) gravitational acceleration at to Bennett (2006); Constantin (2001b, 2011), and Henry the earth’s surface, r is the water’s density, and P is the (2008a) for modern expositions of Gerstner’s wave. pressure. Gerstner’s solution was modified by Pollard (1970) to de- Under the assumption that the meridional distance scribe deep-water surface waves in a rotating fluid and was from the equator is moderate, the approximations adapted to provide explicit gravity edge waves propagating sinf ’ f and cosf ’ 1 may be used—see the discussion along a sloping beach—see Constantin (2001a) and Weber in Gallagher and Saint-Raymond (2007), Gill (1982), (2012), as well as exact equatorially trapped wind waves and Vallis (2006). This approximation, termed the (cf. Constantin 2012). In our analysis we use the insight equatorial b-plane approximation, captures the most provided by these solutions and we take advantage of the important dynamical effects of the earth’s sphericity. It b-plane effect to investigate the motion induced by the approximates the Coriolis force oscillations of the thermocline. The presented internal flow 0 1 differs from the wave motion obtained recently in w cosf 2 y sinf Constantin (2012), by its generating mechanism and in 2V@ u sinf A other essential aspects. For example, for the flow discussed 2u cosf in the present paper the motion dies out as we ascend above the thermocline, while for the solutions obtained in by the expression Constantin (2012) the motion fades out with depth. 0 1 In section 2 we introduce the governing equations for 2Vw 2 byy geophysical flows and the b-plane approximation. Sec- @ byu A tion 3 is devoted to the presentation of the explicit so- 22Vu lutions, while the last section is devoted to a discussion of the flow patterns. with b 5 2V/R 5 2:28 10211 m21 s21; see Cushman- 2. Preliminaries Roisin and Beckers (2011). The Euler equation [(1)] is thus replaced by Choose a rotating framework with the origin at a point on the earth’s surface: (x, y, z) are Cartesian coordinates, 8 > with the spatial variable x corresponding to longitude, the > 1 1 1 1 V 2 521 > ut uux yuy wuz 2 w byy Px , variable y to latitude, and the variable z to the local vertical, > r <> respectively. The governing equations for geophysical 1 y 1 uy 1 yy 1 wy 1 byu 52 P , (4) ocean waves are, (cf. Gallagher and Saint-Raymond > t x y z r y > 2007; Pedlosky 1979; Vallis 2006), the Euler equation > 8 > 1 :> w 1 uw 1 yw 1 ww 2 2Vu 52 P 2 g. > t x y z r z > 1 > u 1uu 1 yu 1 wu 12Vw cosf22Vy sinf52 P , > t x y z r x <> 1 To visualize this approximation notice that although the y 1 uy 1 yy 1 wy 1 2Vu sinf 52 P , > t x y z r y earth was assumed to be spherical, if the spatial scale of > > 1 motion is moderate enough (this being the case even for > w 1 uw 1 yw 1 ww 2 2Vu cosf 52 P 2 g, :> t x y z r z horizontal ranges of the order of 103 km—cf. the dis- cussion in Majda and Wang 2006), then the region oc- (1) cupied by the fluid can be approximated by a tangent

Unauthenticated | Downloaded 09/27/21 10:05 AM UTC JANUARY 2013 C O N S T A N T I N 167 plane. The linear term of the Taylor expansion yields the 3. Main result b-plane effect by noticeable in (4). We will show that the flow given at time t by specifying Let z 5 h(x 2 ct, y) be the equation of the thermo- the positions cline, under the hypothesis that the oscillations of this interface propagate in the longitudinal direction at 8 > constant speed c. The upper boundary of the region M(t) > 1 2k[r1f (s)] <> x 5 q 2 e sin[k(q 2 ct)], above the thermocline and beneath the near-surface layer k y 5 s, (9) L(t) to which wind effects are confined is given by z 5 > > 1 2k[r1f (s)] h1(x 2 ct, y). Beneath the thermocline the water has :> z 5 r 2 e cos[k(q 2 ct)]. k constant density r1 and is still: at every instant t we have u 5 y 5 w 5 0forz , h(x 2 ct, y). From (4) we infer that of the fluid particles, in terms of the labeling variables 5 2 , 2 (q, r, s), provides us with an explicit solution to the P P0 r1gz throughout z h(x ct, y), governing equations (6)–(8). Here k is the wave- number, the speed of the eastward propagating waves for some constant P0. We are interested in the eastward propagation of geophysical waves with vanishing me- is given by ridional velocity (y [ 0) in the region M(t) above the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thermocline and beneath the near-surface layer L(t) V2 1 kg~1V c 5 . 0, (10) where the effect of wind is noticeable, without addressing k the interaction of geophysical waves and wind waves in the region L(t). Throughout M(t) it is reasonable to while assume that the water has constant density r0 , r1, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi typical value of the reduced gravity V2 V2 1 kg~ c 5 , 0 (11) r 2 r k g~5 g 1 0 (5) r 0 is the speed of the waves propagating westward (these

23 22 waves will later be ruled out as being unphysical), with being 6 10 ms (cf. Fedorov and Brown 2009). Con- the function sequently, we seek solutions, cb f (s) 5 s2 , (12) u(x 2 ct, y, z), w(x 2 ct, y, z), h(x 2 ct, y), 2g~

h1(x 2 ct, y), describing in both cases the dependence of the particle of the following governing equations: the Euler equa- oscillation on the meridional coordinate. The parame- ter q covers the real line, while s 2 [2s , s ], where tions in the form pffiffiffiffiffiffiffiffiffi 0 0 s0 5 c0/b is the equatorial radius of deformation. The 8 2 typical value of c for the tropical ocean is 1.4 m s 1 (cf. > 0 > 1 Cushman-Roisin and Beckers 2011), so that s is about > u 1 uu 1 wu 1 2Vw 52 P , 0 > t x z r x <> 0 250 km. Within the framework of linear theory this is 1 the natural scale over which equatorial disturbances byu 52 P , (6) > r y decay (cf. Vallis 2006). Moreover, if this distance from > 0 > 1 a given meridional position includes the equator, equa- > w 1 uw 1 ww 2 2Vu 52 P 2 g, :> t x z z torial linear dynamics must supersede midlatitude linear r0 dynamics (cf. Cushman-Roisin and Beckers 2011). While the formula for s in Gill (1982) is obtained by 2 , , 2 0 in h(x ct, y) z h1(x ct, y), coupled with the in- replacing b by 2b in the previous expression, in our compressibility condition context the choice is irrelevant since the main purpose of the formula is to give an indication of the order of u 1 w 5 0inh(x 2 ct, y) , z , h (x 2 ct, y), (7) x z 1 magnitude of the distances that are involved. Indeed, while the validity of the wave motion described by (9) and with the boundary condition holds also for distances from the equator in excess of s0, 5 2 5 2 the amplitude decay in the meridional direction (by P P0 r1gz on z h(x ct, y). (8) a factor larger than 100 at s 56s0 in the example

Unauthenticated | Downloaded 09/27/21 10:05 AM UTC 168 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 43 presented in section 4b) shows that at such distances the for some constant r* . 0. The restriction (15) ensures flow can be regarded as a small perturbation of an un- that j . 0 throughout the flow; negative values of j are derlying (linear) flow. Consequently, outside the narrow not allowed since in this case at a fixed latitude s the equatorial band the flow (9) is definitely a secondary parameterization (9) of the oscillations of the thermo- feature of the ocean dynamics and in this region it is cline (with y 5 s fixed) would be a self-intersecting curve appropriate to use a linear approximation to match with (cf. the discussion in Constantin 2011). We now write the the oceanic flow near the tropics. As for the label r, for Euler equation (4) in the form every fixed s, we require r 2 [r0(s), r1(s)], with the two positive numbers to be specified later on. The restriction 8 > on the parameter r is due to the fact that the region M(t) > Du 1 > 1 2Vw 52 P , is relatively narrow in its vertical extent: the choice r 5 > Dt r x <> 0 r0(s) defines the thermocline z 5 h(x 2 ct, y) at the Dy 1 5 5 1 byu 52 P , (16) latitude y s, while r r1(s) marks at this latitude the > Dt r y > 0 lower boundary z 5 h1(x 2 ct, y) of the near-surface > > Dw 2 V 521 2 layer L(t) to which the wind effect is confined. An in- :> 2 u Pz g. Dt r0 dicative value for (r1 2 r0) is 40 m, according to the data provided in Fedorov and Brown (2009). To address the issue of how these solutions connect to the flow above From (9) we compute the velocity and acceleration of the layer M(t), notice that (9) represents an exact so- a particle as lution for any value of r . r1(s), but the amplitude de- cays exponentially fast for increasing r. Therefore, in the 8 upper part of the near-surface layer L(t), where the ef- > > 5 Dx 5 2j > u ce cosu, fect of wind waves is essential, the wave motion (9) can <> Dt be regarded as a small perturbation to the dynamics Dy y 5 5 0, (17) dictated by the wind waves—see the example presented > Dt > > Dz 2 in section 4b. Moreover, according to Smyth et al. (2011), :> w 5 52ce j sinu, strong currents prevail in this region. Consequently the Dt effect of the wave motion (9) is here not of great signifi- cance. For this reason we concentrate on the restriction of respectively, the flow (9) to the layer M(t). To check that (9) is an exact solution, let us first ob- 8 > serve that the Jacobian of the map relating the particle > Du 5 2 2j > kc e sinu, positions to the Lagrangian labeling variables, obtained <> Dt Dy by computing the determinant of the matrix 5 0, (18) > Dt 0 1 > > Dw 2 ›x ›y ›z :> 5 kc2e j cosu. B C 0 1 B C 2 2 Dt B ›q ›q ›q C 1 2 e j cosu 0 e j sinu B ›x ›y ›z C B 2 2 C B C 5 @ f e j sinu 1 f e j cosu A B C s s The change of variables B ›s ›s ›s C 2j 2j @ ›x ›y ›z A e sinu 011 e cosu 0 1 › › › r r r ›x ›y ›z 0 1 B C0 1 B C (13) P B ›q ›q ›q C P B q C B CB x C 5 ›x ›y ›z 2 @ P A B C@ P A, equals 1 2 e 2j, where, to simplify notation, we set s B C y B ›s ›s ›s C P Pr @ ›x ›y ›z A z j 5 k[r 1 f (s)], u 5 k(q 2 ct). (14) ›r ›r ›r

Since the determinant is time independent, the flow is enables us to write (16) as volume preserving and (3) holds—see the discussion in Constantin (2011). For the flow to be dynamically pos- 2 P 52r (kc2 2 2Vc 1 g)e j sinu, (19) sible we require that the labeling variables satisfy q 0 1 $ 52 22 V 22j 2 1 2j r f (s) r* (15) Ps r0fs(kc 2 c)e r0(cbs fsg)e cosu, (20)

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2 8 P 52r (kc2 2 2Vc 1 g)e j cosu 2 r g > 1 r 0 0 > P* . if c . 0, <> 0 2 2 2 V 22j 2k 0 1 r0(kc 2 c)e . (21) > 1 kbjcj (25) > . @ 2A , > P* exp s0 if c 0, Choosing : 0 2k g~

b 2 f (s) 5 s , for every s 2 [2s0, s0] we can find a unique solution r 5 2(kc 2 2V) r0(s) . 0 of the equation in (24). By the implicit function theorem, r (s) is a smooth and even function of s. one can easily check that for every constant P~ the 0 0 Evaluating (24) at r 5 r (s) and differentiating the out- gradient of the expression 0 come with respect to s yields

2 2 V 2 kc 2 c 22j 2j 2 5 2 0 bcs e P(q ct, s, r) r0 e r0gr r (s) 5 . (26) 2k 0 g~ 1 2 e22j 2 2 V 1 1 kc 2 c g 2j 1 ~ r0 e cosu P0 (22) k a. Eastward-propagating waves with respect to the labeling variables is given by the In this subsection we discuss the case when the wave 5 right-hand side of (19)–(21). Since r r0 corresponds speed is given by (10). The relation (26) shows that r0(s) to the thermocline z 5 h(x 2 ct), the expression (22) increases as s . 0 increases. From (26) we also infer evaluated at the thermocline matches the boundary bc bcs 1 condition (8) if and only if r0 (s) 1 s 5 . (27) 0 g~ g~ 1 2 e22j r (kc2 2 2Vc 1 g) 5 r g. (23) 0 1 so that the even function 0 1 This leads to (10) or (11), according to whether c . 0or , 1 kbc c 0, respectively. Therefore the choice (10)–(12) yields s1 exp@22kr (s) 2 s2A a solution to (6)–(8), the thermocline being determined 2k 0 g~ by setting r 5 r (s) where r (s) is the unique solution of 0 0 . the equation is strictly decreasing for s 0. (28) 0 1 To complete the solution it remains to specify the 1 kbc P* 5 r 1 exp@22kr 2 s2A (24) boundary delimiting the two layers M(t) and L(t). This 0 2k g~ is obtained by choosing some fixed constant

1 at a fixed value of s 2 [2s , s ], where P . P* . (29) 0 0 0 0 2k ~ P 2 P and setting r 5 r1(s) at a fixed value of s 2 [2s0, s0], P* 5 0 0 . 0 ~ where r1(s) . 0 is the unique solution of the equation r0g 0 1

For every fixed s 2 [2s0, s0], the function 1 kbc P 5 r 1 exp@22kr 2 s2A. (30) 0 1 0 2k g~ kbc r1r 1 [1/(2k)] exp@22kr 2 s2A g~ The previous considerations show that P0 determines a unique number r1(s) . r0(s). The function s 1 r1(s)pre- sents the same features as the function s 1 r (s), namely, is a strictly increasing diffeomorphism from (0, ‘) onto 0 it is even, smooth, and strictly increasing for s . 0, with 0 0 1 1 0 1 @ @ kbc 2A A [1/(2k)] exp 2 s , ‘ . 1 kbc g~ s1 exp@22kr (s) 2 s2A 2k 1 g~

This ensures that, given strictly decreasing for s . 0. (31)

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We see that (15) holds with r* 5 r0(0). This completes the proof that (9) is an exact solution to (6)–(8). For

fixed values of r0 , r1, we have constructed a family of solutions with three degrees of freedom: the param- . P eter k 0 and the constants 0 and P0*, subject to the constraint (29). b. Westward-propagating waves If the wave speed is given by (11), then (26) shows that r0(s) decreases as s . 0 increases, and (27) yields that the even function

0 1 FIG. 1. Depiction of the eastward-propagating fluctuations of the thermocline. The figure shows (on its short side) a range of lati- 1 1 @2 2 kbc 2A s exp 2kr0(s) ~ s tudes within 250 km from the equator. The longitudinal wave- 2k g length of the equatorially trapped wave is 100 m and its amplitude is 10 m. (One unit in the vertical and in the longitudinal scales stands for 20 m, and one meridional unit stands for 30 km.) is strictly increasing for s . 0. We now specify the joint boundary of the layers M(t) and L(t) by choosing some fixed constant oscillations of the thermocline—see Fig. 1. The motion 5 2 0 1 of the upper boundary z h1(x ct, y) of this layer also presents interest since it determines the near-surface 1 kbjcj P . P* . exp@ s2A layer L(t) to which wind effects are confined. In this 0 0 2k g~ 0 paper we do not explore the interaction of geophysical waves and wind waves in the layer L(t). 5 in accordance with (25). For s 2 [2s0, s0], let r1(s) . r0(s) At every latitude y s, the flow (9) describes vertical be the unique solution of the Eq. (30). The previous oscillations of the cross section of the layer M(t), de- considerations show that the function s 1 r1(s) presents limited by the traveling waves x 1 h(x 2 ct, y) and x 1 2 the same features as the function s 1 r0(s), namely, it is h1(x ct). The wave crests of both waves correspond to even, smooth, and strictly decreasing for s . 0, with u 5 (2n 1 1)p with n an integer, so that the positions of 0 1 the crests are xcrests 5 [(2n 1 1)p/k] 1 ct. The wave- 5 1 kbc length is L 2p/k, with wave amplitudes s1 exp@22kr (s) 2 s2A 2k 1 g~ 0 1 1 kbc a(s) 5 exp@2kr (s) 2 s2A, k 0 2g~ strictly increasing for s . 0. The validity of (15) is en- 0 1 sured by the choice r* 5 r0(s0), so that (9) is an exact solution to (6)–(8). However, the amplitude of these 5 1 @2 2 kbc 2A a1(s) exp kr1(s) ~ s , (32) westward-propagating waves grows exponentially with k 2g the distance from the equator, and therefore they are merely mathematical solutions that must be disregarded respectively, while kc is the frequency of the particle physically. oscillation. For a fixed s 2 [2s0, s0], the curve z 5 h(x, s), given parametrically by 4. Discussion 8 0 1 > > z 1 @ kbc 2A In this section we perform a qualitative analysis of > x 5 2 exp 2kr (s) 2 s sinz, <> k k 0 2g~ the flow pattern (9). We also present some quantitative 0 1 considerations to show the practical relevance of the > > 1 1 kbc theoretical results. :> z 5 2 exp@2kr (s) 2 s2A cosz, k k 0 2g~ a. Qualitative aspects of the solution The three-dimensional flow pattern (9) describes the represents a trochoid—see the discussion in Constantin eastward propagation within the layer M(t) at constant (2011): this is the path traced by a fixed point at a dis- speed c, its lower boundary z 5 h(x 2 ct, y) capturing the tance

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FIG. 2. Depiction of a large-amplitude oscillation of the thermocline in a vertical plane at latitude s, with wavelength L, crest level c2(s), mean level l2(s), and trough level t2(s). The eastward-propagating trochoidal wave presents sharper troughs and flatter crests than the si- nusoidal waves typical of linear theory. 2 3 2k[r0(s)1f(s)] c2(s) 5 r0(s) 1 (1/k)e ,whilefrom(9)weget 4 kbc 25 (1/k) exp 2kr (s) 2 s , 1/k that the mean level l2(s) of these oscillations is 0 2g~ ð   L 1 1 2 1 5 2 k[r0(s) f (s)] 2 l2(s) r0(s) e cos[k(q ct)] dx from the center of a circle of radius 1/k rolling without L 0 k slipping along a horizontal line. The curve z 5 h1(x, s), given parametrically by ð 0 1 1 kL 8 5 2 2j u 2 2j u u r0(s) 2 e cos (1 e cos ) d > z 1 kbc k L 0 > x 5 2 exp@2kr (s) 2 s2A sinz, <> k k 1 2g~ 0 1 1 22k[r (s)1f (s)] 5 r (s) 1 e 0 . > 0 2k > 1 1 kbc :> z 5 2 exp@2kr (s) 2 s2A cosz, k k 1 2g~ In the second step we differentiated the x component of (9) with respect to the q variable. 5 also represents a trochoid. At every fixed value of y s, According to (9), each particle in the layer M(t) de- the relations in (9) and the above considerations show scribes in time counterclockwise a circular path in the 5 2 that the thermocline z h(x ct, y) is given by vertical plane determined at a fixed latitude, with the diameter (2/k)e2j of the circular paths increasing with 1 2 5 2 1 2 depth (see Fig. 3). Note that in irrotational gravity water h(x ct, y) r0(s) h(x ct, s), k waves, whether in deep water or for flows with a flat bed, the particle paths are not closed—over a wave period, while for the wave z 5 h1(x 2 ct, y) we have each particle performs a backward/forward and upward/ 1 downward movement, with the path an elliptical-like h (x 2 ct, y) 5 r (s) 2 1 h (x 2 ct, s). 1 1 k 1 loop, not closed but with a forward drift (its average being the Stokes drift) (see Constantin 2006; Constantin

Since r1(s) . r0(s), the curve h1(, s) lies above h(, s). and Strauss 2010; Henry 2008b). Notice that while the The crest–trough asymmetry of trochoidal wave pat- vertical oscillation of the particles in an irrotational terns, in contrast to the sinusoidal wave profiles spe- traveling periodic gravity water wave and in the equato- cific to linear theory, is a hallmark of the nonlinear rially trapped wind waves obtained recently in Constantin character of the solution (see Fig. 2). As an illustra- (2012) decreases with increasing depth, in the layer M(t) tion of the asymmetry, notice that at a given latitude the opposite occurs. s 2 [2s0, s0], the thermocline oscillates between the The explicit form of the matrix ›(x, y, z)/›(q, s, r)in 2k[r0(s)1f (s)] horizontal levels t2(s) 5 r0(s) 2 (1/k)e and (13) easily yields its inverse

0 1 0 1 ›q ›s ›r B 1 1 e2j cosu e2j sinu C B C B 0 2 C B ›x ›x ›x C B 1 2 e22j 1 2 e22j C B C B 2 2 2 C B ›q ›s ›r C B e j sinu e j cosu 2 e 2j C B C 5 B 2f 1 2f C. (33) B ›y ›y ›y C B s1 2 e22j s 1 2 e22j C B C B 2 2 C @ ›q ›s ›r A @ e j sinu 1 2 e j cosu A 2 0 ›z ›z ›z 1 2 e22j 1 2 e22j

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FIG. 3. The motion at a fixed latitude: in the vertical plane with the x axis chosen horizontally due east, along a line of latitude, each particle located above the thermocline describes a circle. The circular motions are all in the same phase, with the diameter of the circles decreasing exponentially with the distance to the thermocline (shown by the dashed curve). Depicted is a wave of small amplitude, a feature that diminishes the nonlinear character highlighted in Fig. 2.

From (17) and (33), by means of the chain rule, we can in the interval (0, 1), we deduce that the nonlinear char- express the relative vorticity acter and the amplitude both increase with increasing wavelength. The limit k / 0 is not permissible as this 5 2 2 2 g (wy yz, uz wx, yx uy) would lead to unbounded amplitudes at the equator. Therefore the equatorially trapped Kelvin waves arising of the fluid flow in Lagrangian variables as in the linear shallow-water regime, and described in Gill 0 1 (1982), are not limiting cases of the solutions (9). Notice 2 2j 22j 2 2j 22j that the maximal amplitude of the wave motion is kc b e sinu 2kce kc b e cosu 2 e 2 g 5 @s , , s A. (1/k)e kr0 , so that for a given wavelength L 5 2p/k,the ~ 2 22j 2 22j ~ 2 22j g 1 e 1 e g 1 e permissible amplitudes range between (L/2p)e2Ld/(2p) (34) and L/2p,whered is the average depth of the thermo- cline. An important feature of the solution in (9) is that the Note that the middle component of g is always strictly vertical and horizontal fluid velocities have similar mag- positive, while the first and third component vanish at nitudes, as seen from (17). This allows for a more elabo- the equator but are both nonzero for s 6¼ 0. Because of rate vertical structure of the flow pattern and shows that (28) and (31), the wave amplitude decays in the merid- the shallow-water limit does not apply to these waves. ional direction (see Fig. 1). Let us now highlight a fundamental difference to the To investigate the nonlinear character of the flow equatorial wave solutions obtained within the frame- pattern, (9), (17), and (33) yield work of linear or shallow water theory: in addition to the planetary vorticity, the flow (9) also presents a nonzero 8 > 2 2 2 horizontal component of the wave-induced relative > e j sinu e 2j 2 e j cosu < u 52cu 5kc2 , u 5 kc2 , vorticity. This salient feature is the hallmark of a non- t x 2 22j z 2 22j 1 e 1 e uniform underlying current. To see this, let us fix the > 22j 2j 2j > e 1 e cosu e sinu 5 2 : w 52cw 5 kc2 , w 5 kc , latitude y s 2 [ s0, s0]. The Lagrangian mean velocity t x 2 22j z 2 22j 1 e 1 e vanishes: (35) ð T 5 1 2 5 huiL u(q ct, s, r) dt 0 with the notation (14), so that in the Euler equation (6) T 0 the terms (ut: uux: wuz) and (wt: uwx: wwz) have orders in 2 2 a ratio (1: e j: e j). For a given wavelength L 5 2p/k,at since the time average of the horizontal fluid velocity u, the equator the thermocline oscillates with amplitude given by (17), over a wave period T 5 L/c is clearly zero. 22pr0(0)/L (L/2p)e , with r0(0) . 0 in view of (15). Since the Let us now investigate the Eulerian mean velocity at a 22pr0(0)/L function L1e is strictly increasing with values fixed depth z0 2 [c2(s), t1(s)], where

Unauthenticated | Downloaded 09/27/21 10:05 AM UTC JANUARY 2013 C O N S T A N T I N 173   1 kbs2 so that the Eulerian mean flow is westward. Therefore t (s) 5 r (s) 2 exp 2kr (s) 2 1 1 k 1 2(kc 2 2V) the Stokes drift, defined by Longuet-Higgins (1969) as the difference between the Lagrangian and the Eulerian is the trough level of the oscillation of the upper mean velocities, is eastward (the Lagrangian mean flow boundary of the layer M(t), and being zero). This contrasts with the fact that for linear   Kelvin waves the Eulerian mean flow is zero and the 1 kbs2 c (s) 5 r (s) 1 exp 2kr (s) 2 Stokes drift is eastward, as shown by Moore (1970). 2 0 k 0 2(kc 2 2V) Differentiating (36) with respect to z0, we get is the crest level of the fluctuations of the thermocline. 5 1 2j 1 az az e cosu, The relation 0 0

1 2 so that z 5 r 2 e j cosu (36) 0 k 1 a 5 . 0. z 1 2j imposes a functional dependence r 5 a(q, z0). Differ- 0 1 e cosu entiating (36) with respect to the q variable, we get This relation in combination with (14) and (38) leads to 2 2 0 5 a 1 a e j cosu 1 e j sinu ð q q L 2kc 2 1 5 2k[a(q,z0) f (s)] . ›z huiE(z0) az e dq 0. so that 0 L 0 0

2 e j sinu Therefore, while the Lagrangian mean velocity huiL a 52 . (37) q 1 1 e2j cosu vanishes, the Eulerian mean velocity huiE depends monotonically on the depth (the westward flow increases

Therefore the Eulerian mean velocity huiE(z0) at the downward), indicating a nonuniform wave-induced cur- fixed depth z0 can be expressed by means of rent. Recall from Longuet-Higgins (1969) that huiL is ð sometimes called the mass-transport velocity, being the L/c 1 5 c 1 2 mean velocity of a marked particle. c huiE(z0) [c u(x ct, y, z0)] dt L 0 To further elucidate the flow (9), we investigate the ð mass transport past a fixed point at a fixed latitude y 5 1 L 2 5 5 1 2 s 2 [ s0, s0]. The mass flux past x x0,withx0 fixed, [c u(x ct, y, z0)] dx L 0 representing the instantaneous zonal transport across ð a fixed longitude, is given by 1 L ›x 5 fc 1 u(q 2 ct, s, a(q))g dq ð h 2 L ›q 1(x0 ct) 0 2 5 2 m(x0 ct) u(x0 ct, y, z) dz. (39) ð h(x 2ct) 1 L 0 5 fc 1 u(q 2 ct, s, a(q))g L 0 From (9) and (17) we deduce that 3 (1 1 e2ja sinu 2 e2j cosu) dq ð q r1(s) 2 5 2j 1 2j 1 2j ð m(x0 ct) c [e cosu](1 e cosu bre sinu) dr L r (s) 1 2 0 5 c 1 e j u ð (1 cos ) r (s) 22j L 0 1 2 1 2 e 5 c [e j cosu] dr, (40) 2 2j   r (s) 1 e cosu e22j sin2j 0 3 1 2 e2j cosu 2 dq 1 2j 1 e cosu where q 5 b(r, t, s) is the function determined by the ð L constraint 1 2 5 c(1 2 e 2j) dq, L 0 1 2 1 x 5 q 2 e k[r f (s)] sin[k(q 2 ct)]. (41) 0 k due to (9), (17), and (37). Consequently ð The formula (40) is obtained by differentiating the z L c 2 1 5 52 2k[a(q,z0) f (s)] 2 component of (9), evaluated at q b(r, t, s), with respect huiE(z0) e dq 2 ( c, 0), (38) L 0 to r, and subsequently using the relation

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FIG. 4. At any fixed latitude within a narrow equatorial band, due to a nonuniform wave- induced current, mass is moved forward near the troughs and backward near the crests, with zero total mass flux.

2j ~ 2 ’ : 2.3 ’ e sinu (kbc/2g)s0 2 3ande 10, at 250 km from the equator b 52 , r 1 2 e2j cosu the amplitude of the oscillations of the thermocline is about 1 m, while that of the upper boundary of the layer obtained by differentiating (41) with respect to r. When M(t) is reduced to about 0.08 m. Notice that as we ap- the wave troughs/crests lie on the line x 5 x0, we have proach the water’s free surface the flow (9) fades out: al- cosu 561, and from (40) we infer that mass is carried ready half way between the upper boundary of the layer forward/backward, respectively (see Fig. 4). However, M(t) and the water’s free surface, at a depth of about the time average of the mass flux over a wave period T is 40 m, the amplitude decay with respect to the oscilla- zero. This can be seen at once from (40) by noticing that tions of the thermocline is more than hundredfold 2 2 2 2 2 2 (41) yields since [(1/k)e k[2r 1 r0]]/[(1/k)e kr0 ] 5 e 2k(r 1 r0) ’ e 5 and e5 ’ 148. In the equatorial ocean layer within 40 m from 2 ce j cosu the surface, and at distances in excess of 250 km from b 52 t 1 2 e2j cosu the equator, the flow (9) represents a small-amplitude wave motion, the corresponding amplitude parameter for the T-periodic function t 1 b(r, t, s). Therefore, the (amplitude/depth ratio) being throughout these regions time average of the mass flux over a wave period T is less than 1/40 5 0.025. This example supports our con- 5 zero. This was expected since huiL 0. Since in the tention that dominant effects of the flow (9) occur in oceans trace elements are transported from place to a region confined in the y direction to a narrow equa- place by the Lagrangian displacement of water particles— torial band and extending vertically from the thermo- see e.g., Li et al. (1996)—the presented wave motion has cline to roughly half way between the surface and the essentially a stirring character. thermocline. b. Some quantitative considerations Acknowledgments. Theauthorhasthepleasureto 5 ’ 21 The wavelength L 100 m corresponds to k 1/16 m express his deep appreciation and gratitude to Prof. ’ 21 and (10) yields the wave speed c 0.31 m s ,sothat W. Kessler for extremely valuable comments. He is also ’ 5 the wave period is L/c 5 min. With kr0(0) ln(1.6), indebted to both referees for useful suggestions. The the amplitude of the equatorial fluctuations of the support of the ERC Advanced Grant ‘‘Nonlinear stud- 2 kr0(0) ’ thermocline is (1/k)e 10 m, while for the width ies of water flows with vorticity’’ is acknowledged. r1 2 r0 of the layer equal to 40 m, the amplitude of the equatorial oscillations of the upper boundary of the 2 2 2 2 layer is (1/k)e kr 1(0) 5 (1/k)e kr0(0)e k[r 1(0) r0(0)] ’ 0:8m REFERENCES 2.5 as e ’ 12. The trapped character of the waves is reflected Bennett, A., 2006: Lagrangian Fluid Dynamics. Cambridge Uni- in an amplitude decay in the meridional direction—since versity Press, 308 pp.

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