American Institute of Mathematical Sciences

The paper focusses on some of the recent breakthroughs in the development of models for nonlinear, three-dimensional Equatorial oceanic flows by Constantin and Johnson. The unique character of the formulations is in the systematic approach followed, while making approximations as required, and consequently assessing the implications. These Constantin-Johnson type of models are general enough, as effects such as that of Earth's rotation, Coriolis term, stratification, thermocline, pycnocline, density variations and vertical velocities can be accounted for. Exact solutions based on the use of singular perturbation theory have been obtained for several different cases and situations. The novelty in the models lies in the introduction of a quasi-stream-function which facilitates the derivation of the solutions. Analytical results are supplemented with some numerical illustrations to provide a flavour of the complex flow structures involved. Insights are provided into the velocity field and flow paths, indicating the presence of cellular structures, upwelling/downwelling and subsurface ocean 'bridges'.

The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a nearly Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.

We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.

In this paper, we show by means of a diffeomorphism that when approximating the planet Saturn by a sphere, the errors associated with the spherical geopotential approximation are so significant that this approach is rendered unsuitable for any rigorous mathematical analysis.

In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived.

In this survey article, we provide a mathematical description of oceanic and atmospheric flows, based on the incompressible Navier–Stokes equation (for the ocean), and the compressible version with an equation of state and the first law of thermodynamics for the atmosphere. We show that, in both cases, the only fundamental assumption that we need to make is that of a thin shell on a (nearly) spherical Earth, so that the main elements of spherical geometry are included, with all other attributes of the fluid motion retained at leading order. (The small geometrical correction that is needed to represent the Earth's geoid as an oblate spheroid is briefly described.) We argue that this is the only reliable theoretical approach to these types of fluid problem. A generic formulation is presented for the ocean, and for the steady and unsteady atmosphere, these latter two differing slightly in the details. Based on these governing equations, a number of examples are presented (in outline only), some of which provide new insights into familiar flows. The examples include the Ekman flow and large gyres in the ocean; and in the atmosphere: Ekman flow, geostrophic balance, Brunt–Väisälä frequency, Hadley–Ferrel–polar cells, harmonic waves, equatorially trapped waves.

Liouville links and chains are exact steady solutions of the Euler equation for two-dimensional, incompressible, homogeneous and planar fluid flow, uncovered recently in [11,12,13]. These solutions consist of a set of stationary point vortices embedded in a smooth non-zero and non-uniform background vorticity described by a Liouville-type partial differential equation. The solutions contain several arbitrary parameters and possess a rich structure. The background vorticity can be varied with one of the parameters, resulting in two limiting cases where it concentrates into some point vortex equilibrium configuration in one limit and another distinct point vortex equilibrium in the other limit. By a simple scaling of the point vortex strengths at a limit, a new steady solution can be constructed, and the procedure iterated indefinitely in some cases. The resulting sequence of solutions has been called a Liouville chain [13]. A transformation exists that can produce the limiting point vortex equilibria from a given seed equilibrium. In this paper, we collect together all these results in a review and present selected new examples corresponding to special sequences of 'collapse configurations.' The final section discusses possible applications to different geophysical flow scenarios.

We investigate particle trajectories in equatorial flows with geophysical corrections caused by the earth's rotation. Particle trajectories in the flows are constructed using pairs of analytic functions defined over the labelling space used in the Lagrangian formalism. Several classes of flow are investigated, and the physical regime in which each is valid is determined using the pressure distribution function of the flow, while the vorticity distribution of each flow is also calculated and found to be effected by earth's rotation.

We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.

We study boundedness of solutions to a linear boundary value problem (BVP) modelling a two-layer ocean with a uniform eddy viscosity in the lower layer and variable eddy viscosity in the upper layer. We analyse bounds of solutions to the given problem on the examples of different eddy viscosity profiles in the case of their parameter dependence.

We present an exact solution to the nonlinear governing equations in the \begin{document}$ \beta $\end{document}-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.

In this paper we investigate transients in the oceanic Ekman layer in the presence of time-varying winds and a constant-in-depth but time dependent eddy viscosity, where the initial state is taken to be the steady state corresponding to the initial wind and the initial eddy viscosity. For this specific situation, a formula for the evolution of the surface current can be derived explicitly. We show that, if the wind and the eddy viscosity converge toward constant values for large times, under mild assumptions on their convergence rate the solution converges toward the corresponding steady state. The time evolution of the surface current and the surface deflection angle is visualized with the aid of simple numerical plots for some specific examples.

We prove new existence criteria relevant for the non-linear elliptic PDE of the form \begin{document}$ \Delta_{S^2} u = C-he^{u} $\end{document}, considered on a two dimensional sphere \begin{document}$ S^2 $\end{document}, in the parameter regime \begin{document}$ 2\leq C<4 $\end{document}. We apply this result, as well as several previously known results valid when \begin{document}$ C<2 $\end{document}, to discuss existence of solutions of a particular PDE modelling ocean surface currents.