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P. Ripa

Abstract

There are pairs of resonant triads with two common components. Analytic solutions describing the evolution of a system with such a double resonant triad are presented and compared with the resonant three-wave problem. Both solutions for constant energies (and shifted frequencies) and for maximum energy exchange (and unshifted frequencies) are discussed. The latter problem is integrable; a subclass of solutions can be written in terms of those of the one-triad system.

Unlike problems of mid-latitude quasi-geostrophic flow and internal gravity waves in a vertical plane, there are resonant triads of equatorial waves with the same speed which have a finite interaction coefficient. This includes the case of second-harmonic resonance or, more generally, a chain of resonant harmonies (a finite number of them in the case of Rossby waves, but an infinite number for inertia–gravity modes). Some analytic and numerical solutions describing the evolution of different chains of resonant harmonies are presented and compared with the (resonant) three-wave problem. Both solutions for constant energies (and shifted frequencies) and for maximum energy exchange (and unshifted frequencies) are presented. The evolution of a chain of resonant harmonies with more than five components is aperiodic, chaotic and unstable.

The derivation of the equations of long-short wave resonances and Korteweg-deVries is straightforward from the evolution equations in phase-space, i.e., there is no need of the usual and cumbersome perturbation expansion in physical space. These equations govern the interaction of a packet of Rossby and inertia–gravity waves with a long Rossby mode of the same group velocity and the self-interaction of long Rossby waves, respectively.

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P. Ripa

Abstract

It is feasible to construct vertical structure functions for the analysis of oceanic data, from the gravest scales sustained by the water column to fine-structure scales. The short scales need not be treated separately by Fourier transforms in a WKB stretched coordinate. The structure functions—with depth as coordinate—are numerically sound and can be defined for the entire water column (normal modes) or a selected part of it. It is shown how to derive a numerical algorithm from a variational principle in such a way that the orthogonality of the eigen-solutions is guaranteed. The errors introduced by the discrete algorithm are discussed, for both the linear eigenvalues (separation constants) and the overlapping integrals (used in the evaluation of nonlinear coupling coefficients). The uncertainty of the modal amplitudes, calculated from experimental data, is also discussed. The method is illustrated with some preliminary applications to PEQUOD data.

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P. Ripa

Abstract

Dispersive equatorial waves are labeled by the zonal slowness s, the meridional quantum number n and the vertical separation constant c. The slowness (reciprocal of phase speed) is a variable more useful than the wavenumber to relate the interactions among equatorial waves. For instance, frequency is a simpler function of slowness than it is of wavenumber, and the four classes of equatorial waves are separated in s-space; viz., Rossby (R): sc ≤ −2n − 1, mixed Rossby-gravity (M): sc < 1, gravity (G): −1 < sc < 1, and Kelvin (K): sc = 1. Moreover, total energy and pseudo-momentum conservation require for the component with intermediate slowness of each triad to gain (loose) energy from (to) the other two. (If the triad is resonant, the wave with intermediate s must also have maximum absolute frequency.)

Nonlinear effects are parameterized by a single variable, the interaction coefficient γ for each resonant triad (RT). The interaction and resonance conditions are reduced to finding the zeros of a polynomial of, at most, sixth degree is s; allowing for classification of all possible resonant triads: There are three types of RT for n > 0: RRR, GGR, and GGG; resonant triads with M (n = 0) and/or K (n = −1) components have the properties of one of these three classes, depending on the frequency of the wave(s) with n < 1 (namely, the M and K may be taken as an R for ω2 ≤ βc/2 or as a G otherwise).

Non-local resonant triads in frequency space include: the packets of Rossby or inertia–gravity waves interacting with a long Rossby mode; short Rossby or inertia–gravity waves with different meridional quantum numbers interacting with a long Rossby or Kelvin mode (geostrophic flow); and the scattering of a short westward propagating inertia–gravity wave into a short eastward propagating inertia–gravity, mixed Rossby–gravity or Kelvin wave, by a short Rossby (or a mixed Rossby–gravity) wave with twice the wavenumber.

Unlike the problems of quasi-geostrophic flow at midlatitude and internal gravity waves in a vertical plane, there are resonant triads of equatorial waves with the same speed, which have a finite interaction coefficient.

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P. Ripa

Abstract

A one-layer reduced-gravity model in the unbounded equatorial β plane is the simplest way to study nonlinear effects in a tropical ocean. The free evolution of the system is constrained by the existence of several conserved quantities, namely, potential vorticity, zonal momentum, zonal pseudomomentum and energy; only the last two are quadratic, to lowest order, in the departure from the equilibrium state.

The eigenfunctions of the linear problem are found to span an orthogonal and complete basis; this is used to expand the dynamical variables, without making any assumption on their magnitude. Thus, the state of the system is fully described, at any time, by the set of expansion amplitudes; their evolution is controlled by a system of equations (with only quadratic nonlinearity) which are an exact representation of the original ones. A straightforward formula is obtained for the evaluation of the coupling coefficients.

As a first example for the use of this formalism, the Kelvin modes self-interaction (which is the most effective, due to the lack of dispersion) is considered, arbitrarily neglecting the contribution of all other components to the nonlinear term. With this approximation, the evolution of the Kelvin part of the system is controlled by the one-dimensional advection equation, as found independently by Boyd (1980a) using a perturbation expansion and strained coordinate technique. Noticeable nonlinear effects are found using this result in two “realistic” problems: the relaxation of the tropical Pacific zonal pressure gradient and the Kelvin wave signal proposed by Ripa and Hayes (1981) in order to explain subdiurnal variability of the surface elevation at the Galápagos.

The one-dimensional advection equation is known to develop unphysical (multivalued) solutions. In order to overcome this difficulty, for the problem considered here, the (off-resonant) excitation of non-Kelvin modes must be taken into account. Long Rossby components are, mainly, forced if the initial Kelvin wave is very long. Short eastward-propagating gravity modes are excited otherwise; this may lead to the formation of a front, both in the zonal velocity and density fields.

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P. Ripa

Abstract

The annual component of the horizontal heat flux F x is calculated from the temperature advection by the geostrophic velocity. This estimate of F x is in good agreement, in amplitude and phase and as a function of the distance x to the head, with that calculated from the difference between the surface heat flux Q and the local heating, that is, from ∂F x/∂x = Q − ∂H/∂t, where H denotes the heat content.

Sea level η variations are well correlated with those of H, while the surface velocity u surf (which can be calculated from the difference of η between both coasts) is well correlated with F x. The proportionality coefficients between (η, H) and between (u surf, F x) correspond to what is expected for a dominance of the first baroclinic mode, in spite of the inhomogeneity of the gulf’s topography.

A linear one-dimensional two-layer model is enough to reproduce the observations of the transversely averaged (η, H, u surf, F x) fields at the annual frequency. Most of the dynamics and thermodynamics are controlled by the Pacific Ocean, which excites a baroclinic Kelvin wave at the mouth of the gulf. Wind drag produces a slight slope in η, whereas Q causes a local heating of the upper layer; both surface forcings have a small effect on u surf and F x.

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P. Ripa

Abstract

A disk over the frictionless surface of the earth shows an interaction between the center of mass and internal motions. At low energies, the former is an “inertial oscillation” superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity ω (as measured by a terrestrial observer).

The dynamics is understood best in a stereographic frame following the secular drift. The center of mass has a circular but not uniform motion; its meridional displacement induces the variations of the orbital and internal rotation rates. On the other hand, the temporal mean of the Coriolis forces due to both rotations produces the secular drift.

In spherical terrestrial coordinates geometric distortion complicates the description. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities 〈u〉, as a result of the earth’s curvature. The drift c and the temporal means U and u are all three different. In addition, ω differs from the local vertical angular velocity σ (as measured by an observer following the disk). The classical“β plane” approximation predicts correctly the value of c but makes order-one errors in everything else (e.g., it makes U = u = c and ω = σ).

The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Part II.

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P. Ripa

Abstract

Zonally propagating solutions of the primitive equations for an isolated volume of fluid are considered. In a moving stereographic projection (from the antipode of the center of mass) geometric distortion enters at O(R −2), with R the radius of the earth, whereas planet curvature effects are O(R −1). The imbalance between the centrifugal force and the poleward gravitational force, due to the drift c, is equilibrated by the average Coriolis force, proportional to β. The results are valid for both homogeneous and stratified cases and the lowest-order solution need not be an axisymmetric vortex. The classical β-plane approximation predicts correctly the leading order of c/β, but makes large errors in the O(R −1) term of the vortex structure.

A method is developed to construct the correct O(R −1) term, starting from any steady solution of the f-plane equations, as the O(R 0) term. The expansion is exemplified starting with a homogeneous fluid, solid body rotating at an anticyclonic rate −νf 0, with 0 < ν < 1. To O(R −1) particle orbits and isobaths belong to different families of nonconcentric circles. A water column moves faster and becomes taller the farther away it is from the equator. In order to keep its potential vorticity, the water column experiences changes of relative vorticity equal to −(2 − ν)/(3 − 3ν) times the variations of the ambient vorticity (Coriolis parameter). The physics of this solution is compared with that of a circular and rigid disk, studied in Part I.

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P. Ripa

Abstract

The general solution for the motion of a particle in the frictionless surface of a rotating planet is reviewed and a physical explanation of asymptotic solutions is provided. In the rotating frame at low energies there is a well-known quasi-circular oscillation superimposed to a weak westward drift; the latter is shown to be due to three different contributions, in the relative proportion of 1:tan2 ϑ 0:− tan2 ϑ 0, where ϑ 0 is the mean latitude. The first contribution is due to the “β effect,” that is, the variation of the Coriolis parameter with latitude. The other two contributions are geometric effects, due to the tendency to move along a great circle and the change of the distance to the rotation axis with the latitude. The mean zonal velocity is produced by the first two contributions, and therefore is underestimated by the classical β-plane approximation [by a factor of (1 +tan2 ϑ 0)−1 = cos2 ϑ 0] because of the lack of geometric effects in such a system. Correct first-order approximations are derived and found to belong to a one-parameter family, whose optimum element is obtained.

The key to develop a consistent approximation, with the right conservation laws, is to redefine three geometric coefficients by means of an expansion in a meridional coordinate, up to a fixed order. Making the expansion directly in the equations of motion, as done by other authors, leads to undesirable consequences for the conservation laws. This is true not only for particle dynamics but also for fields, as illustrated with the shallow-water equations. Correct approximations developed for this system are found to have the same integrals of motion as the exact one (angular momentum, energy, volume, and the potential vorticity of any fluid element). In the quasigeostrophic approximation, the geometric corrections cancel out in the potential vorticity law, and therefore the classical β plane gives the right prognostic equation, even though the (diagnostic) momentum equations are incorrect.

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E. Beier and P. Ripa

Abstract

The circulation pattern in the northern Gulf of California, based on drifting buoys and hydrographic observations, can be explained using the results of a linear two-layer primitive equations model forced, at the annual frequency, by the Pacific Ocean, wind stress, and heat flux through the surface. The modeled surface circulation consists of a cyclonic gyre from June to October and an anticyclonic gyre from December to April, both located in the central region of the northern Gulf of California, which includes Ángel de la Guarda Island. The maximum intensities of the gyres occur in August and February, respectively, with values of surface velocities of 65 cm s−1 (in agreement with the observations) and very low opposite velocities in the bottom layers. May and November are transition months in which both gyres can be observed. Finally, in June/July or December/January the growing gyre is still connected with the rest of the Gulf of California, through the narrows between Tiburón Island, San Esteban Island, and the Baja California coast, whereas from August through October and from February through April the respective gyre is isolated. The vertical structure of the model results indicates a mainly baroclinic signal both in the southern and central regions of the Gulf of California. In the northern gulf, however, the velocities in the annual signal are a combination of barotropic and baroclinic movements, with similar intensities, coupled by topography effects. Thus, only part of the dynamics is associated to great movements of the interface, which shows maximum values of 40 m.

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P. Ripa and Sergio Jiménez

Abstract

Elongaged anticyclonic vortices are modeled by the reduced gravity equations: a finite volume of water on top of an inert heavier fluid, with allowance for horizontal divergence and Coriolis effects. Stable eddies experience pulsations in shape and size, rotations of its orientation, and inertial oscillations of their center of mass. In addition to these motions, unstable vortices also have a tendency to sharpen in the extremes; a process that might lead to filament formation, as the nondivergent case does.

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