On Laplace's tidal equations

Abstract

The parametric limit process for Laplace's tidal equations (LTE) is considered, starting from the full equations of motion for a rotating, gravitationally stratified, compressible fluid. The boundary-value problem for free oscillations of angular frequency σ is not well posed if σ2<N2+ 4ω2, whereNis the Väisälä frequency and ω is the rotational speed of the Earth, and the governing partial differential equation is elliptic/hyperbolic on the polar/equatorial sides of the inertial latitudes given by ± σ =f(vertical component of 2ω) if σ < 2ω [Lt ]N. The solution of this ill-posed problem is considered for a global ocean of uniform depth, with the effects of ellipticity, the ‘traditional’ approximation and stratification measured by the small parametersm= ω2a/g, δ =h/aands=hN2/g(g= acceleration due to gravity,h= depth of ocean,a= radius of Earth). LTE represent the joint limitm, δ,s↓ 0 and yield bounded solutions for all latitudes. It is argued that the parametric expansion inmis regular. The joint expansion in δ andswith LTE as the basic approximation is singular at the inertial latitudes if σ < 2ω, which difficulty is traced to the failure of LTE to provide an adequate description of the characteristics in the hyperbolic domain. It is shown that an alternative formulation, in which the buoyancy force is retained in the basic equations in the joint limits↓0, δ↓0 withN[Gt ] 2ω, yields solutions that are uniformly valid in the neighbourhoods of the inertial latitudes. The resulting representation comprises a barotropic mode, which satisfies LTE, and an infinite discrete set of baroclinic modes, each of which has Airy turning points at the inertial latitudes and is trapped between them. The barotropic and baroclinic modes are coupled by the Coriolis acceleration associated with the horizontal component of the Earth's rotation. The relative effects of this coupling are uniformlyO(δ) if σ > 2ω, but it induces currentsO(δ/s1/4) and vertical displacementsO(δ/s3/4) between the inertial latitudes if σ < 2ω [Lt ]N. It appears that resonant amplification of the baroclinic modes forced by the barotropic modes could imply internal displacements that dominate those of the basic motion.