Inertial waves in a rectangular parallelepiped

A study of inertial gyroscopic waves in a rotating homogeneous fluid is undertaken both theoretically and numerically. A novel approach is presented to construct a semi-analytical solution of a linear three-dimensional fluid flow in a rotating rectangular parallelepiped bounded by solid walls. The three-dimensional solution is expanded in vertical modes to reduce the dynamics to the horizontal plane. On this horizontal plane the two dimensional solution is constructed via superposition of 'inertial' analogs of surface Poincar\'{e} and Kelvin waves reflecting from the walls. The infinite sum of inertial Poincar\'{e} waves has to cancel the normal flow of two inertial Kelvin waves near the boundaries. The wave system corresponding to every vertical mode results in an eigenvalue problem. Corresponding computations for rotationally modified surface gravity waves are in agreement with numerical values obtained by Taylor (1921), Rao (1966) and also, for inertial waves, by Maas (2003) upon truncation of an infinite matrix. The present approach enhances the currently available, structurally concise modal solution introduced by Maas (2003). In contrast to Maas' approach, our solution does not have any convergence issues in the interior and does not suffer from Gibbs phenomenon at the boundaries. Additionally, an alternative finite element method is used to contrast these two semi-analytical solutions with a purely numerical one. The main differences are discussed for a particular example and one eigenfrequency

On the appearance of internal wave attractors due to an initial or parametrically excited disturbance

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler-Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors

A Note on the Role of Mean Flows in Doppler-Shifted Frequencies

The purpose of this paper is to resolve a confusion that may arise from two quite distinct definitions of "Doppler shifts": both are used in the oceanographic literature but they are sometimes conflated. One refers to the difference in frequencies measured by two observers, one at a fixed position and one moving with the mean flow-here referred to as "quasi-Doppler shifts." The other definition is the one used in physics, where the frequency measured by an observer is compared to that of the source. In the latter sense, Doppler shifts occur only if the source and observer move with respect to each other; a steady mean flow alone cannot create a Doppler shift. This paper rehashes the classical theory to straighten out some misconceptions. It is also discussed how wave dispersion affects the classical relations and their application

First observational evidence of a North Madagascar Undercurrent

<i>In situ</i> observations reveal a southeastward-directed North Madagascar Undercurrent (NMUC) below and opposite to the equatorward-directed North Madagascar Current (NMC) off Cape Amber, at the northern tip of Madagascar. Results show an undercurrent hugging the continental slope with its core at 460 m depth and velocities over 0.7 m s-1. Its volume transport is estimated to be 3.1–3.8 Sv, depending on the velocity extrapolation methods used to fill in the data gaps near the slope (no-slip and full-slip, respectively). The thermohaline characteristics show a saltier and warmer NMUC, compared to the surrounding offshore waters, transporting mainly South Indian Central Water. Also, strong horizontal gradients of density are found in the NMUC domain. An inshore cell of coastal downwelling due to Ekman Transport toward the coast is identified, which can explain, at least in part, the strong baroclinic pressure gradients as well as the NMUC development and possible persistence

Deep-ocean tides in the South-West Indian Ocean: comparing deep-sea pressure to satellite data

Deep ocean pressure measurements in two regions of the South-West Indian Ocean (West and East of Madagascar), covering one to two years of data, are analysed for tidal motions. The pressure data are taken both from Bottom Pressure Recorders as well as from mid-water column instruments. Coherent tides are characterised by fixed amplitudes and phases. Those inferred from bottom measurements compare well to tides obtained from satellite altimetry, and cover up to 99% of the pressure variance in the frequency band having periods shorter than 29 hours. Long-period tides, in the low-frequency band, are regularly overshadowed by (unwanted) eddy-induced mooring motion (‘blow-down’), which events have therefore been eliminated. In the Mozambique Channel, semidiurnal surface tides are stronger than East of Madagascar, and all appear to be near resonance with a basin mode. Away from the bottom, coherent internal tides were determined. Evidence of the presence of incoherent internal tides has been obtained by applying Harmonic Analyses over a moving time window of 1 year duration. East of Madagascar internal tides appear to be very strong, although its source remains unclear

Zwemmers op zoek naar een vast punt.

Archimedes’ uitspraak “Geef mij een vast punt en ik zal de aarde bewegen” geldt ook voor zwemmers. In hun geval om, op meer bescheiden schaal, hun lichaam voort te stuwen. Tijdens mooi weer kan dat vermogen om zich tegen water af te zetten verminderen. Opwarming van oppervlaktewater leidt tot een grensvlak tussen watermassa’s van verschillende dichtheid. Het drukverschil dat door de bewegende hand wordt opgewekt kan dan ‘verspild’ worden doordat het grensvlak uit evenwicht gebracht wordt. Het gevolg is dat de hand zijn vaste punt mist en het water minder ‘zwembaar’ wordt

A comparison of Eulerian and Lagrangian current measurements

During three periods in 1981 and 1982, each lasting 3 to 4 days, Eulerian and Lagrangian currents were simultaneously observed at a moderate number of positions in the southern North Sea. These currents were divided into the ensemble averaged current, first order deformation terms and a turbulent part. The Eulerian and Lagrangian ensemble averaged current fields, except for the low-frequency part, compare well. Stokes' velocity estimates do not significantly improve on the mismatch of the residuals as the Eulerian shear is under sampled. The Eulerian shear matrix terms show a strong semi-diurnal spectral peak, whereas the Lagrangian spectrum is more or less fiat as the drifters sample kinematically induced small-scale spatial velocity differences and therefore smear out this tidal peak. The turbulent fields yield estimates of the effective dispersion rates, which show that shear dispersion due to the tidal current is irrelevant at the time scales concerned. In agreement with the growth of a dye patch, released during one of the experiments, the drifter area grew very slowly or even decreased in time, indicative of anomalous dispersion. It is suggested that this may be due to the regularity of the bottom topography, as the generally highly nonlinear kinematic equations, describing the positions of particles in an idealized, tidally-varying Eulerian velocity field, then become (near-) integrable. In this view the (high) normal values of dispersion rates represent a coarse-grained parameterization of the chaotic processes, that arise from the nonlinearly coupled kinematic equations for a random bottom

A comparison of Eulerian and Lagrangian current measurements

During three periods in 1981 and 1982, each lasting 3 to 4 days, Eulerian and Lagrangian currents were simultaneously observed at a moderate number of positions in the southern North Sea. These currents were divided into the ensemble averaged current, first order deformation terms and a turbulent part. The Eulerian and Lagrangian ensemble averaged current fields, except for the low-frequency part, compare well. Stokes' velocity estimates do not significantly improve on the mismatch of the residuals as the Eulerian shear is under sampled. The Eulerian shear matrix terms show a strong semi-diurnal spectral peak, whereas the Lagrangian spectrum is more or less fiat as the drifters sample kinematically induced small-scale spatial velocity differences and therefore smear out this tidal peak. The turbulent fields yield estimates of the effective dispersion rates, which show that shear dispersion due to the tidal current is irrelevant at the time scales concerned. In agreement with the growth of a dye patch, released during one of the experiments, the drifter area grew very slowly or even decreased in time, indicative of anomalous dispersion. It is suggested that this may be due to the regularity of the bottom topography, as the generally highly nonlinear kinematic equations, describing the positions of particles in an idealized, tidally-varying Eulerian velocity field, then become (near-) integrable. In this view the (high) normal values of dispersion rates represent a coarse-grained parameterization of the chaotic processes, that arise from the nonlinearly coupled kinematic equations for a random bottom