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Downloaded 09/27/21 10:05 AM UTC 166 JOURNAL of PHYSICAL OCEANOGRAPHY VOLUME 43 JANUARY 2013 C O N S T A N T I N 165 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 Ó 2013 American Meteorological Society Unauthenticated | Downloaded 09/27/21 10:05 AM UTC 166 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 43 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 Á 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 Á 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).
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