We perform simultaneous coplanar measurements of velocity and density in solitary internal waves with trapped cores, as well as viscous numerical simulations. Our set-up comprises a thin stratified ...layer (approximately 15 % of the overall fluid depth) overlaying a deep homogeneous layer. We consider waves propagating near a free surface, as well as near a rigid no-slip lid. In the free-surface case, all trapped-core waves exhibit a strong shear instability. We propose that Marangoni effects are responsible for this instability, and use our velocity measurements to perform quantitative calculations supporting this hypothesis. These surface-tension effects appear to be difficult to avoid at the experimental scale. By contrast, our experiments with a no-slip lid yield robust waves with large cores. In order to consider larger-amplitude waves, we complement our experiments with viscous numerical simulations, employing a longer virtual tank. Where overlap exists, our experiments and simulations are in good agreement. In order to provide a robust definition of the trapped core, we propose bounding it as a Lagrangian coherent structure (instead of using a closed streamline, as has been done traditionally). This construction is less sensitive to small errors in the velocity field, and to small three-dimensional effects. In order to retain only flows near equilibrium, we introduce a steadiness criterion, based on the rate of change of the density in the core. We use this criterion to successfully select within our experiments and simulations a family of quasi-steady robust flows that exhibit good collapse in their properties. The core circulation is small (at most, around 10 % of the baroclinic wave circulation). The core density is essentially uniform; the standard deviation of the density, in the core region, is less than 4 % of the full density range. We also calculate the circulation, kinetic energy and available potential energy of these waves. We find that these results are consistent with predictions from Dubreil-Jacotin–Long theory for waves with a uniform-density irrotational core, except for an offset, which we suggest is associated with viscous effects. Finally, by computing Richardson-number fields, and performing a temporal stability analysis based on the Taylor–Goldstein equation, we show that our results are consistent with empirical stability criteria in the literature.
Nonlinear oceanic internal solitary waves are considered under the influence of the combined effects of saturating nonlinearity, Earth's rotation, and horizontal depth inhomogeneity. Here the basic ...model is the extended Korteweg–de Vries equation that includes both quadratic and cubic nonlinearity (the Gardner equation) with additional terms incorporating slowly varying depth and weak rotation. The complicated interplay between these different factors is explored using an approximate adiabatic approach and then through numerical solutions of the governing variable depth, i.e., the rotating Gardner model. These results are also compared to analysis in the Korteweg–de Vries limit to highlight the effect of the cubic nonlinearity. The study explores several particular cases considered in the literature that included some of these factors to illustrate limitations. Solutions are made to illustrate the relevance of this extended Gardner model for realistic oceanic conditions.
Internal hydraulic jumps in flows with upstream shear are investigated using two-layer shock-joining theories and numerical solutions of the Navier–Stokes equations. The role of upstream shear has ...not previously been thoroughly investigated, although it is important in many oceanographic situations, including exchange flows. The full solution spaces of several two-layer theories, distinguished by how dissipation is distributed between the layers, with upstream shear are found, and the physically allowable solution space is identified. These two-layer theories are then evaluated using more realistic numerical simulations that have continuous density and velocity profiles and permit turbulence and mixing. Two-dimensional numerical simulations show that none of the two-layer theories reliably predicts the relation between jump height and speed over the full range of allowable solutions. The numerical simulations also show that different qualitative types of jumps can occur, including undular bores, energy-conserving conjugate state transitions, smooth-front jumps with trailing turbulence and overturning turbulent jumps. Simulation results are used to investigate mixing, which increases with jump height and upstream shear. A few three-dimensional simulations results were undertaken and are in quantitative agreement with the two-dimensional simulations.
Abstract
Using a recently developed asymptotic theory of internal solitary wave propagation over a sloping bottom in a rotating ocean, some new qualitative and quantitative features of this process ...are analyzed for internal waves in a two-layer ocean. The interplay between different singularities—terminal damping due to radiation and disappearing quadratic nonlinearity, and reaching an “internal beach” (e.g., zero lower-layer depth)—is discussed. Examples of the adiabatic evolution of a single solitary wave over a uniformly sloping bottom under realistic conditions are considered in more detail and compared with numerical solutions of the variable-coefficient, rotation-modified Korteweg–de Vries (rKdV) equation.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
The evolution of a horizontal shear layer in the presence of a horizontal density gradient is explored by three-dimensional numerical simulations. These flows exhibit characteristics of both free ...shear flows and gravity currents, but have complex dynamics due to strong interactions between the turbulent features of each. Vertical vortices produced by horizontal shear are tilted and stretched by the gravitational adjustment, rapidly enhancing vorticity. Shear intensification at frontal convergences produces high-wavenumber vertical vorticity and the slumping of the density interface produces horizontal Kelvin–Helmholtz vortices typical of a gravity current. The interaction between these instabilities promotes a rapid transition to three-dimensional turbulence. The flow development depends on the relative time scales of shear instability and gravitational adjustment, described by a parameter
$\gamma $
(where the limits
$\gamma \rightarrow \infty $
and
$\gamma \rightarrow 0$
represent a pure gravity current and a pure mixing layer, respectively). The growth rate of three-dimensional instability and the mixing increase for smaller
$\gamma $
. When
$\gamma $
is sufficiently small, there are two distinct regimes: an early period of during which the interface grows rapidly, followed by horizontal diffusive growth. Numerical results are consistent with field observations of tidal separation flows in the Haro Strait (Farmer, Pawlowicz & Jiang, Dyn. Atmos. Oceans., vol. 36, 2002, pp. 43–58), including the magnitude of downwelling vertical currents, horizontal scales of surface vortex features and mixing rate.
Abstract
Deep ocean passages are advantageous sites for long-term monitoring of deep transport and other physical properties relevant to climate. Rotating hydraulic theory provides potential for ...simplifying monitoring strategy by reducing the number of quantities that need to be measured. However, the applicability of these theories has been limited by idealizations such as restriction to zero or uniform potential vorticity (pv) and to channels with rectangular cross sections. Here the relationship between the flow characteristics in a canonical sea strait and its upstream condition is studied using uniform pv rotating hydraulic theory and a reduced-gravity shallow-water numerical model that allows for variation in pv. The paper is focused mainly on the sensitivity of the hydraulic solution to the strait geometry. We study the dynamics of channels with continuously varying (parabolic) cross sections to account for the rounded nature of sea-strait topographies and potentially improve monitoring strategies for realistic channel geometries. The results show that far enough from the channel entrance, the hydraulically controlled flow in the strait is insensitive to the basin circulation regardless of parabolic curvature. The controlled transport relation is derived for the case of uniform pv theory. Comparing the model to theory, we find that the measurement of the wetted edges of the interface height at the critical section can be used to estimate the volume flux. Based on this finding, we suggest three monitoring strategies for transport estimation and compare the estimates with the observed values at the Faroe Bank Channel. The results showed that the estimated transports are within the range of observed values.
Significance Statement
The paper investigates the relationship between the flow characteristics in an idealized sea strait and its upstream condition using rotating hydraulic theory and numerical modeling. We study the dynamics of channels with continuously varying (parabolic) cross sections to account for the rounded nature of sea-strait topographies and potentially improve monitoring strategies for realistic channel geometries. We suggest three monitoring strategies for transport estimation and apply the methods to the Faroe Bank Channel. Our estimates of dense water transport are within the range of observed values. This is significant, because the suggested monitoring strategies only require 1–3 measurements to estimate the transport at a given passage and can be used to guide observing systems.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
The dynamics of perturbations to large-amplitude internal solitary waves (ISWs) in two-layered flows with thin interfaces is analysed by means of linear optimal transient growth methods. Optimal ...perturbations are computed through direct–adjoint iterations of the Navier–Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin–Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity
$c$
(alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough
$c$
) of potentially unstable Richardson number,
$Ri<0.25$
. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with
$c$
. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local Wentzel–Kramers–Brillouin (WKB) approximation for spatially growing Kelvin–Helmholtz (K–H) waves through the
$Ri<0.25$
zone. The WKB approach is able to capture properties (e.g. carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K–H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to non-normal effects that cause absorption of disturbance energy into the leading face of the wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance packets leads to the generation of large-amplitude K–H billows that can emerge on the leading face of the wave and that break down into turbulence in the lee of the wave. The nonlinear calculations are used to derive a slowly varying model of ISW decay due to repeated encounters with optimal or free wave packets. Field observations of unstable ISW by Moum et al. (J. Phys. Oceanogr., vol. 33 (10), 2003, pp. 2093–2112) are consistent with excitation by optimal disturbances.
Swimming organisms may actively adjust their behavior in response to the flow around them. Ocean flows are typically turbulent and are therefore characterized by chaotic velocity fluctuations. While ...some studies have observed planktonic larvae altering their behavior in response to turbulence, it is not always clear whether a plankter is responding to an individual turbulence fluctuation or to the time-averaged flow. To distinguish between these two paradigms, we conducted laboratory experiments with larvae in turbulence. We observed veliger larvae of the gastropod Crepidula fornicata in a jet-stirred turbulence tank while simultaneously measuring two components of the fluid and larval velocity. Larvae were studied at two different stages of development, early and late, and their behavior was analyzed in response to different characteristics of turbulence: acceleration, dissipation and vorticity. Our analysis considered the effects of both the time-averaged flow and the instantaneous flow, around the larvae. Overall, we found that both stages of larvae increased their upward swimming speeds in response to increasing turbulence. However, we found that the early-stage larvae tended to respond to the time-averaged flow, whereas the late-stage larvae tended to respond to the instantaneous flow around them. These observations indicate that larvae can integrate flow information over time and that their behavioral responses to turbulence can depend on both their present and past flow environments.
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 disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
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