Leo R. M. Maas

Observations on the vertical structure of tidal and inertial currents in the central North Sea

Tidal and inertial current ellipses, measured at several locations and depths in the central North Sea during a number of monthly periods in 1980, 1981 and 1982, are decomposed into counterrotating, circular components to which Ekman dynamics are applied to determine Ekman layer depths and vertical phase differences, from which are inferred overall values of the eddy viscosity and drag coefficient. Stratification effects produce an additional vertical phase shift of the anticyclonic rotary component, indicative of an inverse proportionality of the eddy viscosity to the vertical density gradient. From the time variations of the Ekman layer depths of the semidiurnal tidal components, as well as from the vertical structure of the inertial current component, we infer variations in the relative vorticity of the low-frequency flow.

Strong inertial currents and marginal internal wave stability in the central North Sea

Solar insolation stabilizes the water column and suppresses vertical exchange. Observations from the central North Sea clearly show that increased heating in summer is accompanied by enhanced de-stabilizing vertical current differences (shear), surprisingly to such extent that the equilibrium state is marginally stable. Under calm weather conditions, the shear is predominantly generated by near-inertial motions while the internal wave spectrum primarily results from nonlinear interaction between the dominating tidal and near-inertial motions. In terms of the associated enhanced vertical mixing across the largest vertical temperature gradients, shelf seas are not different from the abyssal ocean, despite the proximity to energy sources near boundaries in the former. By the lack of sufficiently strong wind- and tidal-mixing this internal mixing is considered to be responsible for the diapycnal transport of nutrients leading to the observed increase in near-surface values and triggering the late-summer phytoplankton bloom.

Observation of an internal wave attractor in a confined, stably stratified fluid

When a container of water is vibrated, its response can be described in terms of large-scale standing waves—the eigenmodes of the system. The belief that enclosed continuous media always possess eigenmodes is deeply rooted. Internal gravity waves in uniformly stratified fluids, however, present a counterexample. Such waves propagate at a fixed angle to the vertical that is determined solely by the forcing frequency, and a sloping side wall of the container will therefore act as a lens, resulting in ray convergence or divergence. An important consequence of this geometric focusing is the prediction that, following multiple reflections, these waves will evolve onto specific paths—or attractors—whose locations are determined only by the frequency. Here we report the results of laboratory experiments that confirm that internal-wave attractors, rather than eigenmodes, determine the response of a confined, stably stratified fluid over a broad range of vibration frequencies. The existence of such attractors could be important for mixing processes in ocean basins and lakes, and may be useful for analysing oscillations of the Earths liquid core and the stability of spinning, fluid-filled spacecraft.

Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids

Rotating fluids support waves. These inertial waves propagate obliquely through the fluid, with an angle that is fixed with respect to the rotation axis. Upon reflection, their wavelength is unchanged only when the wall obeys the local reflectional symmetry, that is, when it is either parallel or perpendicular to the rotation axis. For internal gravity waves in a density-stratified fluid, sloping boundaries thus break the symmetry of ray paths, in a two-dimensional container, predicting their focusing upon attractors: particular paths onto which the wave rays, and hence the energy, converge, and to which the wave energy returns after a small number of refections. Laboratory observations, presented here, show that, despite the intrinsic three-dimensionality of inertial waves, attractors still occur. The intensified wave energy on the attractor encourages centrifugal instabilities, leading to a mean flow. Evidence of this comes from dye spreading, observed to develop most rapidly over the location where the attractor reflects from the sloping wall, being the place where focusing and instabilities occur. This mean flow, resulting from the mixing of angular momentum, accompanying the intensification of the wave field at that location, has geophysical implications, because the ocean, atmosphere and Earths liquid outer core can be regarded as asymmetrically contained. The relevance of wave focusing in a rotating, spherical shell, the modifications due to the addition of radial stratification, and its implications for observed equatorial current patterns and inertial oscillations are discussed. The well-known universality of oceanic, gravito-inertial wave spectra might reflect complementary, divergent (chaotic) wave-ray behaviour, which occurs in containers obeying the reflectional symmetry, but in which symmetry is broken in the horizontal plane. Periodic orbits still exist, but now repell.

Observations of inertial waves in a rectangular basin with one sloping boundary

Inertial waves in a homogeneous rotating fluid travel along rays that are inclined with respect to the rotation axis. The angle of inclination depends solely on the ratio of the wave frequency and twice the angular frequency. Because of this fixed angle, the waves can become focused when reflected at a sloping wall. In an infinitely long channel with a sloping wall, the repeated action of focusing may lead to the approach to a limit cycle, the so-called wave attractor, where the energy is concentrated. This effect is studied in the laboratory in a rectangular tank with one sloping wall, placed excentrically on a rotating table. The waves are excited by modulation of the background rotation. Several frequency ratios are used to study different wave attractors and one standing wave. The observations consist mainly of particle image velocimetry data in horizontal and vertical cross-sections in one half of the basin. The attractors are observed in the vertical cross-sections. They show continuous phase propagation, which distinguishes them from the standing wave where the phase changes at the same time over the whole cross-section. However, particle motion of inertial waves is three-dimensional and the actual basin is not an ideal twodimensional channel but is of finite length. This implies that the waves must adapt to the vertical endwalls, although a prediction of the nature of these adaptations and the structure of the three-dimensional wave field is at present lacking. For critical waves, whose rays are parallel to the slope, clear three-dimensional behaviour is observed. The location of most intense motion along this critical slope attractor changes in the horizontal direction and horizontal phase propagation is observed, with a wavelength between 1/5 and 1/4 of the basin length. For the other attractors there is little evidence of phase propagation in the horizontal direction. The motion along the attractor is however stronger near the vertical endwalls for attractors with wave rays of slopes close to 1 or larger. The standing wave and the other attractors are more clearly visible near 1/5 of the tank length.

On the nonlinear Helmholtz response of almost-enclosed tidal basins with sloping bottoms

Short relatively deep tidal basins, which are connected to the open sea by a narrow strait, may exhibit either an amplified (resonant), or damped (choked) response to the tide at the entrance. Here particular attention is given to the lowest mode of response, which is the Helmholtz or pumping mode, for which the sea level within the basin executes a spatially uniform oscillation. When the basins sidewalls slope, the restoring process of this oscillator is nonlinear. Possible consequences of this nonlinearity are that the basin may exhibit either long-lasting high-waters and short, peaked low-waters or long-term regular modulation of its tidal amplitude, or a chaotic modulation of its tidal amplitude, or ‘bent resonance horns’, implying that resonant and choked tidal responses may exist simultaneously for the same parameter regime. Related field observations will be discussed.

WAVE ATTRACTORS: LINEAR YET NONLINEAR

A number of physical mechanisms give rise to confined linear wave systems whose spatial structure is governed by a hyperbolic equation. These lack the discrete set of regular eigenmodes that are found in classical wave systems governed by an elliptic equation. In most 2D hyperbolic cases the discrete eigenmodes are replaced by a continuous spectrum of wave fields that possess a self-similar spatial structure and have a (point, line or planar) singularity in the interior. These singularities are called wave attractors because they form the attracting limit set of an iterated nonlinear map, which is employed in constructing exact solutions of this hyperbolic equation. While this is an inviscid, ideal fluid result, observations support the physical relevance of wave attractors by showing localization of wave energy onto their predicted locations. It is shown that in 3D, wave attractors may co-exist with a regular kind of trapped wave. Wave attractors are argued to be of potential relevance to fluids that are density-stratified, rotating, or subject to a magnetic field (or a combination of these) all of which apply to geophysical media.

Tide-topography interactions in a stratified shelf sea II. Bottom trapped internal tides and baroclinic residual currents

Abstract The generation of topographically bounded internal tides and baroclinic residual currents is discussed for a linearly stratified fluid in the limit of small-amplitude topography and weak damping. It is shown that quasi-nonlinear advection by the barotropic tide generates a residual and higher harmonics, besides the ever present fundamental driving frequency in the internal motions. At all frequencies the along-isobath currents and the cross-isobath circulation are bottom intensified, the intensification being a function of the internal Rossby deformation radius. For a step topography the weak cross-isobath residual circulation has a characteristic three-cell structure with a down-slope bottom current at the position of the steepest slope, which is also the center of the along-isobath residual jet. In contrast to barotropic currents in the quasi-nonlinear regime, the baroclinic currents are exponentially bounded to the topography, rather than extending over a finite interval, in which the e-foldin...

Inertial waves in rotating grid turbulence

Using liquid helium, liquid nitrogen, and water as test fluids, we attempt to generate homogeneous turbulence in a steadily rotating system. We create turbulence by pulling a grid in rotating channels with both square and round cross sections, and observe large-scale inertial waves in the flow. These inertial waves quickly sense the boundaries, and resonate at frequencies characteristic of the container. We describe some of their properties and argue that the resultant inhomogeneity is a feature of any real system.

A critical look at the calculation of the binding characteristics and concentration of iron complexing ligands in seawater with suggested improvements

The determination of the thermodynamic characteristics of organic Fe binding ligands, total ligand concentration ((Lt)) and conditional binding constant (K 0 ), by means of titration of natural ligands with Fe in the presence of an added known competing ligand, is an indirect method. The analysis of the titration data including the determination of the sensitivity (S) and underlying model of ligand exchange is discussed and subjected to a critical evaluation of its underlying assumptions. Large datasets collected during the International Polar Year, were used to quantify the error propagation along the determination procedure. A new and easy to handle non-linear model written in R to calculate the ligandcharacteristicsisused.ThequalityoftheresultsstronglydependsontheamountoftitrationpointsorFeadditionsin a titration. At least four titration points per distinguished ligand group, together with a minimum of four titration points where the ligands are saturated, are necessary to obtain statistically reliable estimates of S, K 0 and (Lt). As a result estimating the individual concentration of two ligands, although perhaps present, might not always be justified.