Continuously stratified nonlinear low-mode internal tides

Author Posting. © Sears Foundation for Marine Research, 2008. This article is posted here by permission of Sears Foundation for Marine Research for personal use, not for redistribution. The definitive version was published in Journal of Marine Research 66 (2008): 299-323, doi:10.1357/002224008786176025.A model for hydrostatic, fully nonlinear, low-mode internal tides is extended to continuously stratified conditions. Periodic inertia-gravity solutions of permanent form are shown to exist only for a limited range of amplitudes for a given stratification and frequency. As found in an earlier two-layer model, the solutions fall into two classes. In one, the waves take on a corner-shape as the limiting amplitude is approached. In the other, the waves remain continuous at the limiting amplitude, but have a lobate shape. Numerical investigation using the Euler equations shows that both classes of nonlinear solutions are robust to weak nonhydrostatic effects representative of oceanic conditions. The numerical solutions are also used to explore the evolution of an initial sinusoidal internal tide. It is demonstrated that the presence of the nonlinear solutions may limit the disintegration of the initial tide into shorter solitary-like waves. The nonlinear tide solutions and the disintegration process are briefly explored for conditions of the northeastern South China Sea where large internal tides and solitary waves are observed.This work was supported by ONR Grant N000140610798

Optimal transient growth in thin-interface internal solitary waves

The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered flows with thin interfaces is analyzed 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 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.Comment: 33 pages, 22 figure

2009 program of studies : nonlinear waves

The fiftieth year of the program was dedicated to Nonlinear Waves, a topic with many applications in geophysical fluid dynamics. The principal lectures were given jointly by Roger Grimshaw and Harvey Segur and between them they covered material drawn from fundamental theory, fluid experiments, asymptotics, and reaching all the way to detailed applications. These lectures set the scene for the rest of the summer, with subsequent daily lectures by staff and visitors on a wide range of topics in GFD. It was a challenge for the fellows and lecturers to provide a consistent set of lecture notes for such a wide-ranging lecture course, but not least due to the valiant efforts of Pascale Garaud, who coordinated the write-up and proof-read all the notes, we are very pleased with the final outcome contained in these pages. This year’s group of eleven international GFD fellows was as diverse as one could get in terms of gender, origin, and race, but all were unified in their desire to apply their fundamental knowledge of fluid dynamics to challenging problems in the real world. Their projects covered a huge range of physical topics and at the end of the summer each student presented his or her work in a one-hour lecture. As always, these projects are the heart of the research and education aspects of our summer study.Funding was provided by the National Science Foundation through Grant No. OCE-0824636 and the Office of Naval Research under Contract No. N00014-09-10844

Laboratory study of localized boundary mixing in a rotating stratified fluid

Author Posting. © Cambridge University Press, 2004. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 516 (2004): 83-113, doi:10.1017/S0022112004000473.Oceanic observations indicate that abyssal mixing tends to be localized to regions of rough topography. How localized mixing interacts with the ambient fluid in a stratified, rotating system is an open question. To gain insight into this complicated process laboratory experiments are used to explore the interaction of mechanically induced boundary mixing and an interior body of linearly stratified rotating fluid. Turbulence is generated by a single vertically oscillating horizontal bar of finite horizontal extent, located at mid-depth along the tank wall. The turbulence forms a region of mixed fluid which quickly reaches a steady-state height and collapses into the interior. The mixed-layer thickness, $h_m\,{\sim}\,\gamma ({\omega}/{N})^{1/2}$, is spatially uniform and independent of the Coriolis frequency $f$. $N$ is the initial buoyancy frequency, $\omega$ is the bar oscillation frequency, and $\gamma\,{\approx}\,1$ cm is an empirical constant determined by the bar geometry. Surprisingly, the export of mixed fluid does not occur as a boundary current along the tank perimeter. Rather, mixed fluid intrudes directly into the interior as a radial front of uniform height, advancing with a speed comparable to a gravity current. The volume of mixed fluid grows linearly with time, $V\,{\propto}\,({N}/{f})^{3/2}h_m^3 \textit{ft}$, and is independent of the lateral extent of the mixing bar. Entrainment into the turbulent zone occurs principally through horizontal flows at the level of the mixing that appear to eliminate export by a geostrophic boundary flow. The circulation patterns suggest a model of unmixed fluid laterally entrained at velocity $u_e \,{\sim}\,Nh_m$ into the open sides of a turbulent zone with height $h_{m}$ and a length, perpendicular to the boundary, proportional to $L_f \,{\equiv}\,\gamma ({\omega}/{f})^{1/2}$. Here $L_{f}$ is an equilibrium length scale associated with rotational control of bar-generated turbulence. The model flux of exported mixed fluid $Q\,{\sim}\,h_m L_f u_e$ is constant and in agreement with the experiments.This work was supported by the Ocean Ventures Fund, the Westcott Fund and the WHOI Academic Programs Office. Financial support was also provided by the National Science Foundation through grant OCE-9616949

Circulation around a thin zonal island

Author Posting. © Cambridge University Press, 2001. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 437 (2001): 301-323, doi:10.1017/S0022112001004402.Laboratory and numerical experiments are used to study flow of a uniform-density fluid on the [beta]-plane around a thin zonally elongated island (or ridge segment in the abyss). This orientation is chosen specifically to highlight the roles of the zonal boundary layer dynamics in controlling the circulation around the island. There are examples of deep ocean topography that fall into this category which make the work directly applicable to oceanic flows. Linear theory for the transport around the island and the flow structure is based on a modification of the Island Rule (Pedlosky et al. 1997; Pratt & Pedlosky 1999). The linear solution gives a north–south symmetric flow around the island with novel features, including stagnation points which divide the zonal boundary layers into eastward and westward flowing zones, and a western boundary layer of vanishing length, and zonal jets. Laboratory experiments agree with the linear theory for small degrees of nonlinearity, as measured by the ratio of the inertial to Munk boundary layer scales. With increasing nonlinearity the north–south symmetry is broken. The southern stagnation point (for anticyclonic forcing) moves to the eastern tip of the island. The flow rounding the eastern tip from the northern side of the island now separates from the island. Time-dependence emerges and recirculation cells develop on the northern side of the island. Mean transport around the island is relatively unaffected by nonlinearity and given to within 20% by the modified Island Rule. Numerical solutions of the shallow water equations are in close agreement with the laboratory results. The transition from zonal to meridional island orientation occurs for island inclinations from zonal greater than about 20°.This work was supported by the National Science Foundation (Grant Number OCE96-16949)

Laboratory experiments and simulations for solitary internal waves with trapped cores

Author Posting. © The Author(s), 2014. This is the author's version of the work. It is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 757 (2014): 354-380, doi:10.1017/jfm.2014.501.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.Funding from NSF grant OCE-1029672 is gratefully acknowledged. P.L.F. is thankful for support from the Postdoctoral Scholar program at the Woods Hole Oceanographic Institution, and for funding from the Devonshire Foundation

Internal hydraulic jumps in two-layer flows with upstream shear

Author Posting. © The Author(s), 2015. This is the author's version of the work. It is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 789 (2016): 64-92, doi:10.1017/jfm.2015.727.Internal hydraulic jumps in ows 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 ows. 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 identi ed. These two-layer theories are then evaluated using more realistic numerical simulations that have continuous density and velocity pro les 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 di erent 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.This work was supported by National Science Foundation grant OCE-1029672 and the Natural Sciences and Engineering Research Council of Canada.2016-07-1

Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid

Author Posting. © Cambridge University Press, 2002. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 468 (2002): 179-204, doi:10.1017/S0022112002001520.A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g[prime prime or minute], Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H1 is uniform in [minus sign][infty infinity] < x [less-than-or-eq, slant] xnose(t), 0 < y [less-than-or-eq, slant] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H0, which is located at L < y < [infty infinity]. If L0 = L([minus sign][infty infinity], t), the nose point with L(xnose(t), t) = 0 moves with velocity Unose = [surd radical]g[prime prime or minute]H0 [phi], where [phi] is a function of H1/H0, f2L20/g[prime prime or minute]H0. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of Unose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.This work is funded by National Science Foundation grants OCE-9726584 & OCE-0092504 (M. E. S.) and OCE-9810599 (K. R. H.)

Buoyant gravity currents along a sloping bottom in a rotating fluid

Author Posting. © Cambridge University Press, 2002. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 464 (2002): 251-278, doi:10.1017/S0022112002008868.The dynamics of buoyant gravity currents in a rotating reference frame is a classical problem relevant to geophysical applications such as river water entering the ocean. However, existing scaling theories are limited to currents propagating along a vertical wall, a situation almost never realized in the ocean. A scaling theory is proposed for the structure (width and depth), nose speed and flow field characteristics of buoyant gravity currents over a sloping bottom as functions of the gravity current transport Q, density anomaly g[prime prime or minute], Coriolis frequency f, and bottom slope [alpha]. The nose propagation speed is cp [similar] cw/ (1 + cw/c[alpha]) and the width of the buoyant gravity current is Wp [similar] cw/ f(1 + cw/c[alpha]), where cw = (2Qg[prime prime or minute] f)1/4 is the nose propagation speed in the vertical wall limit (steep bottom slope) and c[alpha] = [alpha]g/f is the nose propagation speed in the slope-controlled limit (small bottom slope). The key non-dimensional parameter is cw/c[alpha], which indicates whether the bottom slope is steep enough to be considered a vertical wall (cw/c[alpha] [rightward arrow] 0) or approaches the slope-controlled limit (cw/c[alpha] [rightward arrow] [infty infinity]). The scaling theory compares well against a new set of laboratory experiments which span steep to gentle bottom slopes (cw/c[alpha] = 0.11–13.1). Additionally, previous laboratory and numerical model results are reanalysed and shown to support the proposed scaling theory.This research was supported by NSF grant OCE-0095059

A model for large-amplitude internal solitary waves with trapped cores

© The Authors, 2010. This article is distributed under the terms of the Creative Commons Attribution 3.0 License. The definitive version was published in Nonlinear Processes in Geophysics 17 (2010): 303-318, doi:10.5194/npg-17-303-2010.Large-amplitude internal solitary waves in continuously stratified systems can be found by solution of the Dubreil-Jacotin-Long (DJL) equation. For finite ambient density gradients at the surface (bottom) for waves of depression (elevation) these solutions may develop recirculating cores for wave speeds above a critical value. As typically modeled, these recirculating cores contain densities outside the ambient range, may be statically unstable, and thus are physically questionable. To address these issues the problem for trapped-core solitary waves is reformulated. A finite core of homogeneous density and velocity, but unknown shape, is assumed. The core density is arbitrary, but generally set equal to the ambient density on the streamline bounding the core. The flow outside the core satisfies the DJL equation. The flow in the core is given by a vorticity-streamfunction relation that may be arbitrarily specified. For simplicity, the simplest choice of a stagnant, zero vorticity core in the frame of the wave is assumed. A pressure matching condition is imposed along the core boundary. Simultaneous numerical solution of the DJL equation and the core condition gives the exterior flow and the core shape. Numerical solutions of time-dependent non-hydrostatic equations initiated with the new stagnant-core DJL solutions show that for the ambient stratification considered, the waves are stable up to a critical amplitude above which shear instability destroys the initial wave. Steadily propagating trapped-core waves formed by lock-release initial conditions also agree well with the theoretical wave properties despite the presence of a "leaky" core region that contains vorticity of opposite sign from the ambient flow.This work is supported as part of the Office of Naval Research NLIWI and IWISE program grants N00014-06-1- 0798 and N00014-09-1-0227