2 Preliminaries

We begin this section by first presenting the governing equations of geophysical fluid motion in cylindrical coordinates, and from this framework we derive the appropriate $\unicode[STIX]{x1D6FD}$ -plane approximation. Both systems are highly nonlinear, and the $\unicode[STIX]{x1D6FD}$ -plane approximation utilizes the idea that if the spatial scale of motion on the spherical domain is moderate enough then the horizontal region occupied by the fluid domain can be approximated as a tangent plane. This framework was recently exploited in Constantin & Johnson (Reference Constantin and Johnson2016), where the authors found a new exact solution of the full governing equations which does not vary azimuthally (in spherical coordinates), and this new exact solution was shown to have (more tractable, yet physically illustrative) analogue solutions when transformed to the settings of both cylindrical coordinates and the $\unicode[STIX]{x1D6FD}$ -plane approximation. From our viewpoint, the wave-like terms in the exact solution we present below may be regarded as being in a sense ‘azimuthally periodic’. The cylindrical coordinates $(x,\unicode[STIX]{x1D703},z)$ are chosen such that the origin is located at the centre of the Earth, the generator of the cylinder is the $x$ -axis which represents the ‘straightened-out’ equator (with the positive $x$ -direction going from west to east), $\unicode[STIX]{x1D703}$ is the angle of latitude (and not the typical polar angle) and we set $z=r-R$ to be the variation in the locally vertical direction of the radial variable from the Earth’s surface. The geophysical parameters we employ are as follows: $g=9.8~\text{m}~\text{s}^{-2}$ is the standard gravitational acceleration, $R=6378~\text{km}$ is the Earth’s radius (we make the assumption that the Earth is a perfect sphere), $\unicode[STIX]{x1D6FA}=7.3\times 10^{-5}~\text{rad}~\text{s}^{-1}$ is the (constant) rotational speed of the Earth and $\unicode[STIX]{x1D6FD}=2\unicode[STIX]{x1D6FA}/R\approx 2.28\times 10^{-11}~\text{m}^{-1}~\text{s}^{-1}$ ; cf. Cushman-Roisin & Beckers (Reference Cushman-Roisin and Beckers2011). The Euler equation for fluid motion in these cylindrical coordinates (cf. Constantin & Johnson (Reference Constantin and Johnson2016) for details) assumes the form

together with the equation for incompressibility

Here, $(u,v,w)$ is the fluid velocity field in the $(\boldsymbol{e}_{x},\boldsymbol{e}_{\unicode[STIX]{x1D703}},\boldsymbol{e}_{z})$ directions, $\unicode[STIX]{x1D70C}$ is the water density (which we take to be constant, although see § 4.2 for a discussion on stratified or variable density fluid) and $P$ is the pressure distribution. Terms in (2.1a ) involving $\unicode[STIX]{x1D6FA}$ to the first order of magnitude represent Coriolis forces, whereas terms of the order $\unicode[STIX]{x1D6FA}^{2}$ denote centripetal forces. Typically, in an oceanographic context, centripetal terms are neglected due to their comparably small size. However, we retain them here in our consideration of wave-like solutions, and remarkably it will become apparent that centripetal forces play an important role in ensuring that the exact solution we present below can admit all physically plausible ranges of underlying currents. Indeed, we can see by comparison with Henry (Reference Henry2013) that this robust admissibility property of all physically plausible ranges of underlying currents in our solution hinges completely on the presence of the centripetal forces. We note that this phenomenon whereby Coriolis terms, although negligible in general oceanographic considerations, can perform an important role in specific wave (and wave–current) dynamics, while curious, may be observed in other geophysical fluid dynamical contexts, an example being Rossby waves (see Cushman-Roisin & Beckers Reference Cushman-Roisin and Beckers2011).

The $\unicode[STIX]{x1D6FD}$ -plane approximation results under the assumption that we are close to the equator, namely we are restricted to latitudes in the region $s\in [-s_{0},s_{0}]$ , where $s_{0}=\sqrt{\tilde{c}/\unicode[STIX]{x1D6FD}}\approx 250~\text{km}$ is a typical value for the equatorial radius of deformation (and $\tilde{c}$ is a characteristic geophysical wavespeed; cf. Cushman-Roisin & Beckers (Reference Cushman-Roisin and Beckers2011)). Mathematically, this corresponds to an assumption that $\unicode[STIX]{x1D703}\rightarrow 0$ , and furthermore the radius $R$ is very large relative to the vertical variations $z$ , whence $z/R\rightarrow 0$ . Defining $y=R\unicode[STIX]{x1D703}$ and retaining only terms of linear order $(O(\unicode[STIX]{x1D703}))$ in the expansion of the trigonometric functions in (2.1) we get the $\unicode[STIX]{x1D6FD}$ -plane governing equations,

together with the equation of incompressibility,

Denoting the free surface by $\unicode[STIX]{x1D702}(x,y,t)$ and letting $P_{atm}$ be the (constant) atmospheric pressure, the relevant boundary conditions at the free surface are the kinematic boundary condition

which implies that fluid particles on the free surface remain on the surface for all time, and the dynamic boundary condition

which decouples the water flow from the motion of the air above. Finally, we assume the water to be infinitely deep, with the flow converging rapidly with depth to a uniform zonal current; that is,

The set of (2.2) comprises the governing equations for the modified $\unicode[STIX]{x1D6FD}$ -plane approximation of geophysical ocean waves with a free surface.

We remark that the system (2.2a ) requires a very specific pressure distribution in the absence of motion ( $u=v=w=0$ ): if the free surface is a surface of constant atmospheric pressure ( $P_{atm}=1~\text{atm}=1.01325~\text{bar}$ ), then

throughout the fluid, so that the free surface is given by

since $\unicode[STIX]{x1D6FA}^{2}R\approx 3\times 10^{-2}~\text{m}~\text{s}^{-2}\ll g\approx 9.8~\text{m}~\text{s}^{-2}$ . The above distortion from a constant value of $z$ corresponds to a free surface following the curvature of the Earth away from the equator, as the curved surface of the Earth drops below the tangent plane at the equator – this is consistent with, and indeed a consequence of, the $\unicode[STIX]{x1D6FD}$ -plane approximation.

3 Main result

The aim of this section is to explicitly show that, for all physically admissible values of the underlying current $c_{0}$ (taken to be uniform), the fluid motion prescribed by

defines an exact solution of the $\unicode[STIX]{x1D6FD}$ -plane governing equations (2.2). Here, the Eulerian coordinates of fluid particles $(x,y,z)$ are expressed as functions of the Lagrangian labelling variables $(q,r,s)\in (\mathbb{R},(-\infty ,r_{0}),\mathbb{R})$ and time $t$ , where $r_{0}<0$ and $k=2\unicode[STIX]{x03C0}/L$ is the wavenumber with $L$ the wavelength. Regarding the latitudinal $s$ parameter, although we demonstrate below that the formula (3.1) defines a mathematical solution of the system (2.2) for all values of $s\in \mathbb{R}$ , purely geophysical considerations imply that we work in a restricted region $s\in [-s_{0},s_{0}]\subset \mathbb{R}$ for which the $\unicode[STIX]{x1D6FD}$ -plane approximation is applicable. The solution (3.1) prescribes a three-dimensional eastward-propagating steady geophysical wave in the presence of a constant underlying current of magnitude $|c_{0}|$ . For $c_{0}>0$ the underlying current is adverse, while for $c_{0}<0$ the current is following. The wave-like term is periodic in the zonal (azimuthal) direction and it has a constant wave phasespeed $c>0$ which will be prescribed by the dispersion relation (4.18) below. The role of the function $f(s)$ , defined by

is to enforce a strong exponential decay in fluid particle oscillations meridionally, thereby ensuring that the wave is equatorially trapped. We define

to be a modification of standard gravitational acceleration with additional terms due to Coriolis effects and the underlying constant current. We infer that the inequality holds above in (3.3) since otherwise we would have $c_{0}\leqslant -g/2\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D6FA}R/2$ , a scenario we can exclude on physical grounds bearing in mind that $g/2\unicode[STIX]{x1D6FA}\approx 6.7\times 10^{4}~\text{m}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FA}R/2\approx 2.33\times 10^{2}~\text{m}~\text{s}^{-1}$ . We now state the main result of this paper as follows.

We will see in § 4 that one of the primary obstacles in incorporating an underlying current term (and the motivation for condition (3.4)) lies in proving the existence of a unique solution $r(s)<r_{0}$ to (4.13) which prescribes the free surface at each fixed latitude, parameterized by the Langrangian labelling variable $q$ . We note that this method of prescription of the free surface $z=\unicode[STIX]{x1D702}(x,y,t)$ ensures that the kinematic boundary condition (2.2c ) holds by design. In the absence of a current ( $c_{0}=0$ ), at each fixed latitude the free surface is an inverted trochoid (Constantin Reference Constantin2011, Reference Constantin2012; Henry Reference Henry2013) and furthermore particle trajectories take the form of closed circles. It should be noted that closed particle trajectories are encountered beneath periodic travelling surface gravity water waves very rarely in irrotational flow and, when so, only isolated and at specific depths (see the theoretical considerations in Constantin (Reference Constantin2006), Henry (Reference Henry2008) and Constantin & Strauss (Reference Constantin and Strauss2010) confirmed numerically in Nachbin & Ribeiro-Junior (Reference Nachbin and Ribeiro-Junior2014) and experimentally in Umeyama (Reference Umeyama2012)). The fact that this feature is indicative of flows with vorticity will be expanded upon in § 4.

The field data examined in Moum, Nash & Smyth (Reference Moum, Nash and Smyth2011) highlight the importance of waves with relatively short wavelengths (in the range 150–250 m) for the dynamics of the upper-equatorial oceans. It should be noted that the vanishing of the meridional component of the Coriolis force at the equator has the effect that the equator works as a (fictitious) natural boundary, facilitating azimuthal flow propagation. Moreover, equatorial field data (see Johnson, McPhaden & Firing (Reference Johnson, McPhaden and Firing2001)) confirm the fact that meridional speeds near the equator are much smaller than the zonal speeds, and neglecting them, as the prescribed fluid motion (3.1) does, therefore has an insignificant dynamical effect. All of these observations show that wave patterns of the type predicted by our considerations are relevant for the ocean dynamics in the equatorial Pacific. We point out that while our theoretical considerations were restricted to roughly $2^{\circ }$ of latitude from the equator (since further away the underlying current structure starts to present significant changes), wave patterns of the type described in this paper quite accurately match observations recorded within 50–100 km of the equator.

4 Fluid kinematics

To examine aspects of the fluid motion prescribed by the exact solution (3.1) we take advantage of working in the Lagrangian framework, a characteristic of which is that the fluid kinematics can often be ascertained explicitly and with relative ease – a nice exposition of general characteristics of the Lagrangian approach to fluid dynamics can be found in Bennett (Reference Bennett2006). For ease of notation we denote $\unicode[STIX]{x1D709}=k(r-f(s))$ and $\unicode[STIX]{x1D703}=k(q-ct)$ , and the Jacobian matrix of the transformation (3.1) is computed as

The determinant of the Jacobian is $1-\text{e}^{2\unicode[STIX]{x1D709}}$ , which is non-zero (and hence the transformation (3.1) is well defined) if

A consequence of (4.2) is that we must have $c>0$ in order for our flow to exhibit both meridional and vertical oscillatory decay, a fact that has further implications when we derive the dispersion relation in (4.18) below. Furthermore, as $1-\text{e}^{2\unicode[STIX]{x1D709}}$ is time-independent, the flow defined by (3.1) must be volume preserving, ensuring that (2.2b ) holds in the Eulerian setting (Bennett Reference Bennett2006; Constantin Reference Constantin2011). Since the solution (3.1) is explicit in the Lagrangian formulation, by direct calculation we may immediately discern some qualitative properties of the fluid kinematics:

where $\text{D}/\text{D}t$ is the material derivative. The fact that $v\equiv 0$ throughout the fluid is in keeping with (and indeed necessary for) the equatorially trapped nature of the flow. The vorticity of the flow prescribed by (3.1) is determined by way of computing

and accordingly the vorticity takes the form $\unicode[STIX]{x1D714}=(w_{y}-v_{z},u_{z}-w_{x},v_{x}-u_{y})$

We note that since the current is constant it does not impact on the vorticity (4.5) of the flow directly, yet it has an indirect influence through the formulation of the wavespeed $c$ given by the dispersion relation (4.18). Furthermore, the vorticity (4.5) is (weakly) three-dimensional away from the equator, and it becomes one-dimensional either at the equator ( $s=0$ ) or by neglecting Coriolis effects (letting $\unicode[STIX]{x1D6FD}\rightarrow 0$ ). Using (4.3) we can express (2.2a ) as

with $\mathfrak{g}$ defined by (3.3) above. By multiplying both sides of (4.6) by the Jacobian matrix (4.1) we obtain an expression for the pressure gradient in terms of the Lagrangian variables:

The next stage in proving that (3.1) represents an exact solution of (2.2) is to construct a suitable pressure function $P$ such that (4.7), and also (2.2d ), holds. Choosing

gives

To satisfy (2.2d ), which enforces a time independence in the pressure function at the surface, it is necessary to eliminate terms containing $\unicode[STIX]{x1D703}$ in (4.8) by setting

The first implication of relation (4.10) is that the function $f(s)$ , defined by (3.2), has an alternative form which will be useful in calculations, namely

It now follows directly from (4.11) that the pressure gradient expression (4.9) matches (4.7). Using (4.10) in relation (4.8), we infer that the choice of pressure function

together with the flow determined by (3.1), satisfies the governing equations (2.2a ). The constant terms on the right-hand side of (4.12) have been chosen bearing in mind the last step in our process, which is to ensure that conditions (2.2c ) and (2.2d ) hold on the free surface. This is achieved if we show that, for each fixed latitude $s$ , there exists a unique solution $r(s)\leqslant r_{0}<0$ such that $P(r(s),s)=P_{atm}$ in (4.12), which is equivalent to

where

At the equator, for $s=0$ , the choice $r(0)=r_{0}$ works in (4.13). For $|s|>0$ , we infer that the last term on the right-hand side above is negative for physically plausible values $c_{0}$ such that (3.4) holds, bearing in mind (3.3), and so $\mathscr{P}(r,s)$ decreases as $|s|$ increases. It then follows that, for each fixed $s\neq 0$ , there exists a unique $r(s)$ such that the equilibrium (4.13) holds, since

implies that $\mathscr{P}(r,s)$ is a monotonically decreasing function of $r$ , and furthermore $\lim _{r\rightarrow -\infty }\mathscr{P}(r,s)=\infty$ . Finally, differentiating (4.13) with respect to $s$ we have

which by way of (3.4) gives us

and so the even function $s\mapsto r(s)$ is decreasing whenever condition (3.4) holds. This completes the proof of Proposition 1.