[4] In a rotating framework with the origin at a point on the Earth's surface, with the *x*-axis chosen horizontally due east, the*y*-axis horizontally due north and the*z*-axis upward, let*z* = −*d* be the upper boundary of the centre layer and *z* = −*η*(*x*, *y*, *t*) be the thermocline. In the region −*η*(*x*, *y*, *t*) ≤ *z* ≤ −*d*, the governing equations in the *f*-plane approximation near the Equator are [cf.*Gallagher and Saint-Raymond*, 2007] the Euler equation

and the equation of mass conservation

Here *t* represents time, (*u*, *v*, *w*) is the fluid velocity, *ω* = 73 · 10^{−6} rad/s is the (constant) rotational speed of the Earth (taken to be a perfect sphere of radius 6371 km) round the polar axis towards the east, *g* = 9.8 m/s^{2} is the (constant) gravitational acceleration at the Earth's surface, and *P* is the pressure. Beneath the thermocline *z* = −*η*(*x*, *y*, *t*) the motionless colder water has a slightly higher density *ρ* + Δ*ρ* (the typical value of Δ*ρ*/*ρ* for the equatorial Pacific being 0.006 [cf. *Fedorov and Brown*, 2009]). In this region (2) is not altered and (1) holds with *ρ* + Δ*ρ* replacing *ρ*, so that here *u* = *v* = *w* = 0 and

for some constant *P*_{0}. A peculiar feature of (1)–(2) can be inferred from the vorticity equation [see *Pedlosky*, 1979]: for two-dimensional flows, independent upon the*y*-coordinate and with*v* ≡ 0, the vorticity *γ* = (0, *u*_{z} − *w*_{x}, 0) is preserved. By abuse of notation we identify *γ* with the scalar *u*_{z} − *w*_{x} and assume that it is a positive constant throughout the layer −*η*(*x*, *t*) ≤ *z* ≤ −*d*. We seek traveling waves, with the velocity field, the pressure and the thermocline exhibiting in this layer an (*x*, *t*)-dependence of the form (*x − ct*), where *c* > 0 is the propagation speed of the oscillations of the thermocline. The boundary conditions are therefore the kinematic boundary conditions

expressing the fact that a particle on this surface remains confined to it (see the discussion by *Constantin* [2011]), together with the dynamic boundary condition

which in view of (3) ensures the continuity of the pressure across the thermocline. The system governing the motion becomes

in the frame of reference (*x*, *z*) moving at the constant wave speed *c* > 0: we performed the change of variables *x* ↦ *x* − *ct*. Using (2) we now introduce the stream function *ψ*(*x*, *z*) determined up to an additive constant by

This permits us to re-formulate(6) as

where *m* is the relative mass flux, given by

Note that the first two equations in (8) merely state the fact that the expression

is constant throughout the layer −*η*(*x*) < *z* < −*d*, in analogy with Bernoulli's law for gravity water waves [see *Constantin*, 2011]. Therefore (8) takes the simpler form

for some constant *Q*. Here

is the reduced gravity [cf. *Fedorov and Brown*, 2009]. Under the assumption of no stagnation points of the flow, that is, provided

throughout the fluid, the hodograph transform

converts the free-boundary problem(10) into the nonlinear boundary problem

for the unknown function

(representing the level beneath the upper flat boundary of the centre layer) in the fixed strip 0 < *p* < |*m*|. Note that (9) and (12) yield *m* < 0 so that |*m*| = −*m*. The laminar flow solutions to (14), representing parallel shear flows for which every particle moves horizontally with a speed that depends only on the depth, are given by

the parameter *λ* > 0 being related to the speed on the flat thermocline *p* = 0 by

We linearize the problem (14) by setting

where *ε* > 0 is a small parameter. Seeking solutions that are even and periodic in the *q*-variable with principal period*L* > 0, we obtain at order *ε*the weighted Sturm-Liouville problem

where is the wavenumber and . The substitution transforms the differential equation in (16) into *F*″ = *k*^{2}*F*. Taking into account the boundary condition at *p* = |*m*| in (16), we obtain

The boundary condition at *p* = 0 holds if and only if

The solution *λ*^{∗} of (17) yields the bifurcating laminar flow

where *z* = −*D* is the thermocline. The corresponding relative mass flux

can be computed by means of (9). Viewing this as an equation for (*D − d*), we get

since the alternative expression is eliminated by the requirement (12). The above permits us to write (17) as

Solving (18) for *λ*^{∗} > 0, we obtain

At the thermocline *u* = 0 so there. Consequently the previous relation leads us to a polynomial equation of degree two in the unknown *c*, the only positive root being

This is the exact dispersion relation, presenting a slight resemblance to the dispersion relation for gravity water waves with constant vorticity discussed by *Constantin and Strauss* [2004] and *Constantin and Varvaruca* [2011].