The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.

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[1] In studies on geophysical fluid dynamics, it is common practice to take the Coriolis force only partially into account by neglecting the components proportional to the cosine of latitude, the so-called traditional approximation (TA). This review deals with the consequences of abandoning the TA, based on evidence from numerical and theoretical studies and laboratory and field experiments. The phenomena most affected by the TA include mesoscale flows (Ekman spirals, deep convection, and equatorial jets) and internal waves.Abandoning the TA produces a tilt in convective plumes, produces a dependence on wind direction in Ekman spirals, and gives rise to a plethora of changes in internal wave behavior in weakly stratified layers, such as the existence of trapped short low-frequency waves, and a poleward extension of their habitat. In the astrophysical context of stars and gas giant planets, the TA affects the rate of tidal dissipation and also the patterns of thermal convection.

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 Earth's 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.

Reflecting internal gravity waves in a stratified fluid preserve their frequency and thus their angle with the gravitational direction. At boundaries that are neither horizontal nor vertical, this leads to a focusing or defocusing of the waves. Previous theoretical and experimental work has demonstrated how this can lead to internal wave energy being focused onto ‘wave attractors’ in relatively simple geometries. We present new experimental and theoretical results on the dynamics of wave attractors in a nearly two-dimensional trapezoidal basin. In particular, we demonstrate how a basin-scale mode forced by simple mechanical excitation develops an equilibrium spectrum. We find a balance between focusing of the basin-scale internal wave by reflection from a single sloping boundary and viscous dissipation of the waves with higher wavenumbers. Theoretical predictions using a simple ray-tracing technique are found to agree well with direct experimental observations of the waves. With this we explain the observed behaviour of the wave attractor during the initial development, steady forcing, and the surprising increase of wavenumber during the decay of the wave field after the forcing is terminated.

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Abstract.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 nearinertial 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.

Environmental context. The low concentration of iron in the oceans limits growth of phytoplankton. Dissolved organic molecules, called ligands, naturally present in seawater, bind iron thereby increasing its solubility and, consequently, its availability for biological uptake by phytoplankton. The characteristics of these ligands are determined indirectly with various mathematical solutions; we critically evaluate the underlying method and calculations used in these determinations.Abstract. The determination of the thermodynamic characteristics of organic Fe binding ligands, total ligand concentration ([L t ]) 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 ligand characteristics is used. The quality of the results strongly depends on the amount of titration points or Fe additions in 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 [L t ]. As a result estimating the individual concentration of two ligands, although perhaps present, might not always be justified.