Karl R. Helfrich

Internal solitary wave breaking and run-up on a uniform slope

Laboratory experiments have been conducted to study the shoaling of internal solitary waves of depression in a two-layer system on a uniform slope. The shoaling of a single solitary wave results in wave breaking and the production of multiple turbulent surges, or boluses, which propagate up the slope. Significant vertical mixing occurs everywhere inshore of the breaking location. The kinematics of the breaking and bolus runup are described and a breaking criterion is found. The energetics of the breaking are investigated.

The formation and fate of internal waves in the South China Sea

Internal gravity waves, the subsurface analogue of the familiar surface gravity waves that break on beaches, are ubiquitous in the ocean. Because of their strong vertical and horizontal currents, and the turbulent mixing caused by their breaking, they affect a panoply of ocean processes, such as the supply of nutrients for photosynthesis, sediment and pollutant transport and acoustic transmission; they also pose hazards for man-made structures in the ocean. Generated primarily by the wind and the tides, internal waves can travel thousands of kilometres from their sources before breaking, making it challenging to observe them and to include them in numerical climate models, which are sensitive to their effects. For over a decade, studies have targeted the South China Sea, where the oceans’ most powerful known internal waves are generated in the Luzon Strait and steepen dramatically as they propagate west. Confusion has persisted regarding their mechanism of generation, variability and energy budget, however, owing to the lack of in situ data from the Luzon Strait, where extreme flow conditions make measurements difficult. Here we use new observations and numerical models to (1) show that the waves begin as sinusoidal disturbances rather than arising from sharp hydraulic phenomena, (2) reveal the existence of >200-metre-high breaking internal waves in the region of generation that give rise to turbulence levels >10,000 times that in the open ocean, (3) determine that the Kuroshio western boundary current noticeably refracts the internal wave field emanating from the Luzon Strait, and (4) demonstrate a factor-of-two agreement between modelled and observed energy fluxes, which allows us to produce an observationally supported energy budget of the region. Together, these findings give a cradle-to-grave picture of internal waves on a basin scale, which will support further improvements of their representation in numerical climate predictions.

ON LONG NONLINEAR INTERNAL WAVES OVER SLOPE-SHELF TOPOGRAPHY

An experimental and theoretical study of the propagation and stability of long nonlinear internal waves over slope-shelf topography is presented. A generalised Korteweg-de Vries (KdV) equation, including the effects of nonlinearity, dispersion, dissipation and varying bottom topography, is formulated and solved numerically for single and rank-ordered pairs of solitary waves incident on the slope. The results of corresponding laboratory experiments in a salt-stratified system are reported. Very good agreement between theory and experiment is obtained for a range of stratifications, topography and incident-wave amplitudes. Significant disagreement is found in some cases if the effects of dissipation and higher-order (cubic) nonlinearity are not included in the theoretical model. Weak shearing and strong breaking (overturning) instabilities are observed and found to depend strongly on the incident-wave amplitude and the stratification on the shelf. In some cases the instability of the lowest-mode wave leads to the generation of a second-mode solitary wave. The application of these findings to the prediction and interpretation of field data is discussed.

Transcritical two-layer flow over topography

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Kortewegae Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the correspondmg homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here. Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

Circulation around islands and ridges

The circulation in an ocean basin containing an island is studied under nearly geostrophic, beta plane dynamics. The model is a fluid of uniform density driven by wind forcing or sources and sinks of mass at the upper boundary of the flow. The circulation is studied analytically, numerically, as well as in the laboratory through the device of the “sliced cylinder” model for the ocean circulation. Of particular interest is the estimate of the transport between the island and the oceanic basin’s boundary. The model is conceived as relevant to both the wind-driven circulation as well as the circulation of abyssal waters around deep topographic features such as mid-ocean ridge segments. Godfrey’s Island Rule for the transport is rederived in general form and the validity of the original approximation of Godfrey (1989) is examined in a variety of circumstances. In particular, the role of dissipative boundary layers and inertial effects such as vortex shedding are scrutinized to determine their role in determining the net transport around the island. Linear theory in many cases predicts a recirculation on the eastern side of the island, provided the meridional extent of the island is large enough. The existence of the recirculation, containing trapped fluid, is confirmed in both laboratory and numerical experiments and the evolution of the structure of the recirculation is examined as a function of the boundary layer Reynolds number. In both the laboratory and numerical studies, the recirculation predicted by linear theory is joined and then superseded by an inertial recirculation springing from boundary layer separation as the Reynolds number increases past a critical value. Even in the linear limit it is shown that the recirculation region, which is closed in quasigeostrophic theory, is subject to a small leak due to planetary geostrophic effects, which prediction is confirmed in the laboratory. The original island rule of Godfrey yields an estimate of the transport which is surprisingly robust and generally within 75% of the values measured in our numerical experiments. Agreement is moderately good when island western boundary layer transport is used as a basis for comparison. Several cases are discussed, however, in which the assumptions made by Godfrey are violated. One occurs when the frictional boundary layers of the island and the basin boundary overlap. We derive a threshold width for the gap for the case where the island is close to a northern or southern boundary of the basin and show how the transport is increasingly blocked as the gap is reduced. A second case occurs when the island is thin and zonally elongated so that the dissipative effects on the northern and southern boundaries of the island become important. Here the vorticity balance assumed in the simple Island Rule is fundamentally altered, and we extend the Island Rule to account for the new dissipation.

Buoyant gravity currents along a sloping bottom in a rotating fluid

Author Posting.

Time-Dependent Two-Layer Hydraulic Exchange Flows

Abstract A theory is presented for time-dependent two-layer hydraulic flows through straits. The theory is used to study exchange flows forced by a periodic barotropic (tidal) flow. For a given strait geometry the resulting flow is a function of two nondimensional parameters, γ = (g′H)1/2T/L and qb0 = ub0/(g′H)1/2. Here g′, H, L, T, and ub0 are, respectively, the reduced gravity, strait depth and length scales, the forcing period, and the barotropic velocity amplitude; γ is a measure of the dynamic length of the strait and qb0 a measure of the forcing strength. Numerical solutions for both a pure contraction and an offset sill-narrows combination show that the exchange flow, averaged over a tidal cycle, increases with qb0 for a fixed γ. For fixed qb0 the exchange increases with increasing γ. The maximum exchange is obtained in the quasi-steady limit γ→∞. The minimum exchange is found for γ→0 and is equal to the unforced steady exchange. The usual concept of hydraulic control occurs only in these two limit...

Finite-amplitude evolution of two-layer geostrophic vortices

The finite-amplitude evolution of circular two-layer quasi-geostrophic vortices with piecewise uniform potential vorticity in each layer (also termed ‘heton’ clouds by Hogg & Stommel 1985 a and Pedlosky 1985) is studied using the contour dynamics method. The numerical investigations are preceded by a linear stability analysis which shows the stabilizing influence of deepening the lower layer. Net barotropic flow may be either stabilizing or destabilizing. The contour dynamics calculations for baroclinic vortices show that supercritical (i.e. linearly unstable) conditions may lead to explosive break up of the vortex via the generation of continuous hetons at the cloud boundary. The number of vortex pairs is equal to the azimuthal mode number of the initial disturbance. An additional weakly supercritical regime in which amplitude vacillation occurs, but not explosive growth, is identified. Vortices with net barotropic circulation behave similarly except that the layer with vorticity opposite to the barotropic circulation will break up first. Strong barotropic circulation can inhibit the development of hetons. The stronger layer may eject thin filaments, but remain mostly intact. Calculations for initial conditions composed of several unstable modes show that the linearly most unstable mode dominates at finite amplitude.

On interfacial solitary waves over slowly varying topography

The propagation of long, weakly nonlinear interfacial waves in a two-layer fluid of slowly varying depth is studied. The governing equations are formulated to include cubic nonlinearity, which dominates quadratic nonlinearity in some parametric neighbourhood of equal layer depths. Numerical solutions are obtained for an initial profile corresponding to either a single solitary wave or a rank-ordered pair of such waves incident in a monotonic transition between two regions of constant depth. The numerical solutions. supplemented by inverse-scattering theory, are used to investigate the change of polarity of the incident waves as they pass through a ‘turning point ’ of approximately equal layer depths. Our results exhibit significant differences from those reported by Knickerbocker & Newell (1980), which were based on a model equation. In particular, we find that more than one wave of reversed polarity may emerge.

Nonlinear disintegration of the internal tide

Abstract The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of d...