Stability of a potential vorticity front: from quasi-geostrophy to shallow water (original) (raw)
The vorticity dynamics of instability and turbulence in a breaking internal gravity wave
We perform a three-dimensional simulation of a breaking internal gravity wave in a stratified, compressible, and sheared fluid to investigate the vorticity dynamics accompanying the transition from laminar to turbulent flow. Baroclinic sources contribute preferentially to eddy vorticity generation during the initial convective instability of the wave field, yielding counter-rotating vortices aligned with the external shear flow. These vortices enhance the spanwise vorticity of the shear flow via stretching and distort the spanwise vorticity via advective tilting. The resulting vortex sheets undergo a dynamical (Kelvin-Helmholtz) instability which rolls the vortex sheets into tubes which link, in turn, with the original streamwise convective rolls to produce a collection of intertwined vortex loops. Following the formation of discrete vortex loops, the most important interactions are the self-interactions of single vortex tubes and the mutual interactions of adjacent vortex tubes in close proximity. The initial formation of vortex tubes from the roll-up of localized vortex sheets imposes axial vorticity variations having both axisymmetric and azimuthal wavenumber two components. Axisymmetric variations excite axisymmetric twist waves, or Kelvin vortex waves, which propagate along the tubes, drive axial flows, and deplete and fragment the tubes. Azimuthal wavenumber two variations excite m = 2 twist waves on the vortex tubes which amplify and unravel single vortex tubes into pairs of intertwined helical tubes. Other interactions, judged less fundamental to the turbulence cascade, include reconnection among tube fragments, mutual stretching of orthogonal tubes in close proximity, excitation of azimuthal wavenumber one twist waves, and the continual roll-up of weaker vortex sheets throughout the evolution. Collectively, these vortex interactions result in a rapid cascade of energy and enstrophy toward smaller scales of motion.
Quarterly Journal of the Royal Meteorological Society, 2021
A linear wave theory of the Rotating Shallow-Water Equations (RSWE) is developed in a channel on the midlatitude f -plane or 𝛽-plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. Here we show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f -plane similar to the generation of these waves by PV sources such as the 𝛽-effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting Rossby, Poincaré and "Kelvin-like" waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the "no mean-flow" phase speeds does not account for the difference in phase speeds, and the meridional structure is often trapped near one of the channel's boundaries and does not oscillate across the channel. A comparison between the phase speeds of Rossby waves of the present theory and those of the Quasi-Geostrophic Shallow-Water (QG-SW) theory shows that the former can be 2.5 times faster than those of the QG-SW theory. The phase speed of "Kelvin-like" waves is modified by the presence of a mean flow compared to the classical gravity wave speed, and furthermore their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wave number, and the convexity of the dispersion curves also changes with the zonal wave number. These results have implications for the propagation of Rossby wave packets: QG theory overestimates the zonal group velocity.
Nonlinear evolution of ordinary frontal waves induced by low-level potential vorticity anomalies
Quarterly Journal of the Royal Meteorological Society, 1993
Linear, semi-geostrophic (SG) theory reveals the instability of steady fronts with low-level potential vorticity anomalies. Joly and Thorpe (1990) have shown in this context the most unstable normal modes to have sub-synoptic wavelengths. The present study uses a primitive equation (PE) model to construct, at these wavelengths and along the same fronts, the PE normal modes and extends the evolution to the nonlinear regime. It is shown that PE normal modes have a structure similar to the original SG modes at the same given wavenumber. In the nonlinear experiments, two different kinds of behaviour are found, depending on the initial wavelength of the perturbation, the frontal baroclinicity and the width of the potential vorticity anomaly. The first kind, and main finding of this study, is characterized by the inability of a barotropically unstable mode (in the energy sense) to lead to large pressure falls in the vortex. Such a mode, with its wavelength smaller than the Rossby radius, is successful in breaking the frontal flow but saturates within two days. The other occurs when the wavelength is larger than the Rossby radius. Then, it is shown that the initially significant barotropic contribution to the growth vanishes and the wave enters a phase of classical baroclinic growth. It is only when this second phase occurs that the frontal change in structure is accompanied by significant deepening of the surface low. It saturates in a way similar to larger-scale baroclinic waves, by increasing the upper-level jet and shear.
Journal of Fluid Mechanics, 2013
We undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow-water model on the fff-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic form, resulting from the resonance of two frontal waves, to a baroclinic form, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for mu...
Instability of Topographically Forced Rossby Waves in a Two-layer Model
Journal of the Meteorological Society of Japan, 1987
Stability properties of topographically forced baroclinic Rossby waves and zonal flows are investigated by the use of a two-layer, quasi-geostrophic*-channel model. Two kinds of instabilities are found when the vertical shear of the zonal flow exceeds the minimum critical shear for the conventional baroclinic instability of the zonal flow: One is the topographic instability which is identical with that examined by Charney and DeVore (1979) and Mukougawa and Hirota (1986a) in the barotropic model. This instability appears in the near-resonant flow. The other is the baroclinic instability composed of synoptic disturbances with a horizontal modulation by effects of the forced wave. This is found to correspond to the storm-track type instability of free baroclinic Rossby waves investigated by Frederiksen (1978, 1982). Examination of various effects of these unstable modes on the basic flow reveals that the role of the synoptic disturbances on the transition of the weather regime is not so important as suggested by Reinhold and Pierrehumbert (1982). Alternatively, other unstable modes, such as those due to the topographic instability, are expected to cause the regime transition because their effects on the basic flow are completely different from those of the baroclinic instability.
Coupled Kelvin-Wave and Mirage-Wave Instabilities In Semigeostrophic Dynamics
Journal of Physical …, 1998
A weak instability mode, associated with phase-locked counterpropagating coastal Kelvin waves in horizontal anticyclonic shear, is found in the semigeostrophic (SG) equations for stratified flow in a channel. This SG instability mode approximates a similar mode found in the Euler equations in the limit in which particle-trajectory slopes are much smaller than f /N, where f is the Coriolis frequency and N Ͼ f the buoyancy frequency. Though weak under normal parameter conditions, this instability mode is of theoretical interest because its existence accounts for the failure of an Arnol'd-type stability theorem for the SG equations. In the opposite limit, in which the particle motion is purely vertical, the Euler equations allow only buoyancy oscillations with no horizontal coupling. The SG equations, on the other hand, allow a physically spurious coastal ''mirage wave,'' so called because its velocity field vanishes despite a nonvanishing disturbance pressure field. Counterpropagating pairs of these waves can phase-lock to form a spurious ''mirage-wave instability.'' Closer examination shows that the mirage wave arises from failure of the SG approximations to be self-consistent for trajectory slopes տ f /N.
1 P 1 . 12 Simulations of Internal Waves Approaching a Critical Level
2009
Internal gravity waves exist abundantly in our world in stably-stratified fluids, such as the ocean and atmosphere. A stably-stratified fluid is one where the density of the fluid continuously decreases with increasing elevation. When a finite amount of fluid from a stratified environment is removed from its equilibrium density location by moving it upward or downward, buoyancy forces will compel the fluid to return to its original location. If the fluid is forced to oscillate at any frequency below the buoyancy frequency, it may oscillate in the horizontal direction in addition to the vertical direction. These multidimensional oscillations evolve into internal gravity waves, which will propagate throughout the fluid like regular surface water waves, but different in that they may propagate in three directions, and are not restricted to the horizontal plane of a liquid gas interface. The wave phase and group speeds propagate orthogonally.
From Topographic Internal Gravity Waves to Turbulence
Annual Review of Fluid Mechanics, 2017
Internal gravity waves are a key process linking the large-scale mechanical forcing of the oceans to small-scale turbulence and mixing. In this review, we focus on internal waves generated by barotropic tidal flow over topography. We review progress made in the past decade toward understanding the different processes that can lead to turbulence during the generation, propagation, and reflection of internal waves and how these processes affect mixing. We consider different modeling strategies and new tools that have been developed. Simulation results, the wealth of observational material collected during large-scale experiments, and new laboratory data reveal how the cascade of energy from tidal flow to turbulence occurs through a host of nonlinear processes, including intensified boundary flows, wave breaking, wave-wave interactions, and the instability of high-mode internal wave beams. The roles of various nondimensional parameters involving the ocean state, roughness geometry, and...
Linear instabilities of a two-layer geostrophic surface front near a wall
Journal of Marine Research, 2004
The development of linear instabilities on a geostrophic surface front in a two-layer primitive equation model on an f-plane is studied analytically and numerically using a highly accurate differential shooting method. The basic state is composed of an upper layer in which the mean flow has a constant potential vorticity, and a quiescent lower layer that outcrops between a vertical wall and the surface front (defined as the line of intersection between the interface that separates the two layers and the ocean's surface). The characteristics of the linear instabilities found in the present work confirm earlier results regarding the strong dependence of the growth rate ( i ) on the depth ratio r (defined as the ratio between the total ocean depth and the upper layer's depth at infinity) for r Ն 2 and their weak dependence on the distance L between the surface front and the wall. These earlier results of the large r limit were obtained using a much coarser, algebraic, method and had a single maximum of the growth rate curve at some large wavenumber k. Our new results, in the narrow range of 1.005 Յ r Յ 1.05, demonstrate that the growth rate curve displays a second lobe with a local (secondary) maximum at a nondimensional wavenumber (with the length scale given by the internal radius of deformation) of 1.05. A new "fitting function" 0.183 r Ϫ0.87 is found for the growth rate of the most unstable wave ( imax ) for r ranging between 1.001 and 20, and for L Ͼ 2 R d (i.e. where the effect of the wall becomes negligible). Therefore, imax converges to a finite value for ͉r Ϫ 1͉ Ͻ Ͻ 1 (infinitely thin lower layer). This result differs from quasi-geostrophic, analytic solutions that obtain for the no wall case since the QG approximation is not valid for very thin layers. In addition, an analytical solution is derived for the lower-layer solutions in the region between the wall and the surface front where the upper layer is not present. The weak dependence of the growth rate on L that emerges from the numerical solution of the eigenvalue problem is substantiated analytically by the way L appears in the boundary conditions at the surface front. Applications of these results for internal radii of deformation of 35-45 km show reasonable agreement with observed meander characteristics of the Gulf Stream downstream of Cape Hatteras. Wavelengths and phase speeds of (180 -212 km, 39 -51 km/day) in the vicinity of Cape Hatteras were also found to match with the predicted dispersion relationships for the depth-ratio range of 1 ϩ Ͻ r Ͻ 2.
The role of forcing in the local stability of stationary long waves. Part 1. Linear dynamics
Journal of Fluid Mechanics, 2007
The local linear stability of forced, stationary long waves produced by topography or potential vorticity (PV) sources is examined using a quasigeostrophic barotropic model. A multiple scale analysis yields coupled equations for the background stationary wave and lowfrequency (LF) disturbance field. Forcing structures for which the LF dynamics are Hamiltonian are shown to yield conservation laws that provide necessary conditions for instability and a constraint on the LF structures that can develop. Explicit knowledge of the forcings that produce the stationary waves is shown to be crucial to predicting a unique LF field. Various topographies or external PV sources can be chosen to produce stationary waves that differ by asymptotically small amounts, yet the LF instabilities that develop can have fundamentally different structures and growth rates. If the stationary wave field is forced solely by topography, LF oscillatory modes always emerge. In contrast, if the stationary wave field is forced solely by PV, two LF structures are possible: oscillatory modes or non-oscillatory envelope modes. The development of the envelope modes within the context of a linear LF theory is novel.
Instabilities of a two-layer coupled front
Deep Sea Research Part A. Oceanographic Research Papers, 1987
We consider the linear instability of a two-layer fluid, whose mean state consists of a motionless lower layer and a surface layer confined between two parallel fronts. An inverted form occurs at many locations in the deep ocean, notably in the Denmark Strait overflow. Because of the vanishing surface layer depth, quasigeostrophy cannot hold, and primitive equations must be used. Two modes of long wave instability are found. The first, valid for intermediate values of the ratio of total fluid depth to surface layer depth, is analogous to a mode found for an isolated front in an otherwise similar geometry. The second mode is the extension to two layers of the mode already discovered for the same geometry but with an infinitely deep lower layer. Numerical extensions of these long wave results to shorter waves show that the former mode would be observed in practice. The theory is applied to laboratory results, and is in excellent agreement with observations.
INTERNAL GRAVITY WAVES: From Instabilities to Turbulence
Annual Review of Fluid Mechanics, 2002
▪ We review the mechanisms of steepening and breaking for internal gravity waves in a continuous density stratification. After discussing the instability of a plane wave of arbitrary amplitude in an infinite medium at rest, we consider the steepening effects of wave reflection on a sloping boundary and propagation in a shear flow. The final process of breaking into small-scale turbulence is then presented. The influence of those processes upon the fluid medium by mean flow changes is discussed. The specific properties of wave turbulence, induced by wave-wave interactions and breaking, are illustrated by comparative studies of oceanic and atmospheric observations, as well as laboratory and numerical experiments. We then review the different attempts at a statistical description of internal gravity wave fields, whether weakly or strongly interacting.
On the stability of gravity waves on deep water
Journal of Fluid Mechanics, 1990
This note presents numerical results on the stability of large-amplitude gravity waves on deep water. The results are then used to predict new two-dimensional superharmonic instabilities. They are due to collisions of eigenvalues of opposite signatures, confirming the recent condition for instability of .
Journal of Geophysical Research, 1994
A nonlinear, compressible, spectral collocation code is employed to examine gravity wave breaking in two and three spatial dimensions. Twodimensional results exhibit a structure consistent with previous efforts and suggest wave instability occurs via convective rolls aligned normal to the gravity wave motion (uniform in the spanwise direction). Three-dimensionM results demonstrate, in contrast, that the preferred mode of instability is a series of counterrotating vortices oriented along the gravity wave motion, elongated in the streamwise direction, •nd confined to the region of convective instability within the wave field. Comparison of the two-dimensional results (averaged spanwise) for both two-and three-dimensional simulations reveals that vortex generation contributes to much more rapid wave field evolution and decay, with rapid restoration of near-adiabatic lapse rates and stronger constraints on wave energy and momentum fluxes. These results also demonstrate that two-dimensional models are unable to describe properly the physics or the consequences of the wave breaking process, at least for the flow parameters examined in this study. The evolution and structure of the three-dimensional instability, its influences on the gravity wave field, and the subsequent transition to quasi-isotropic small-scale motions are the subjects of companion papers by Fritts et M. (this issue) and Isler et al. (this issue). Introduction Atmospheric gravity waves were first studied in connection with airflow over orography and atmospheric fluctuations at greater altitudes several decades ago. More recently, they have enjoyed a resurgence of interest with the recognition of their major role in the transports of energy and momentum throughout the atmosphere. This interest has focused on the wave-wave and wave-mean flow interactions accompanying wave propagation as well as the processes acting to limit wave amplitudes and control their spectral character. An important component of this latter work was the attempt to understand the effects of wave field instability on wave amplitudes, turbulent diffusion, and the convergences of wave energy and momentum fluxes accompanying dissipation. Initial studies addressed the potential for gravity wave momentum transports and their effects due to wave dissipation [Bretherton, 1969; Holton and Lindzen,
Journal of Fluid Mechanics, 1995
There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generaliz...
Fluid-Dynamic Models of Geophysical Waves
2018
Geophysical waves are waves that are found naturally in the Earth’s atmosphere and oceans. Internal waves, that is waves that act as an interface between fluids of different density, are examples of geophysical waves. A fluid system with a flat bottom, flat surface and internal wave is initially considered. The system has a depth-dependent current which mimics a typical ocean set-up and, as it is based on the surface of the rotating Earth, incorporates Coriolis forces. Using well established fluid dynamic techniques, the total energy is calculated and used to determine the dynamics of the system using a procedure called the Hamiltonian approach. By tuning a variable several special cases, such as a current-free system, are easily recovered. The system is then considered with a non-flat bottom. Approximate models, including the small amplitude, long-wave, Boussinesq, Kaup-Boussinesq, Korteweg-de Vries (KdV) and Johnson models, are then generated using perturbation expansion technique...
Breaking of standing internal gravity waves through two-dimensional instabilities
Journal of Fluid Mechanics, 1995
The evolution of an internal gravity wave is investigated by direct numerical computations. We consider the case of a standing wave confined in a bounded (square) domain, a case which can be directly compared with laboratory experiments. A pseudo-spectral method with symmetries is used. We are interested in the inertial dynamics occurring in the limit of large Reynolds numbers, so a fairly high spatial resolution is used (1292 or 2572), but the computations are limited to a two-dimensional vertical plane. We observe that breaking eventually occurs, whatever the wave amplitude: the energy begins to decrease after a given time because of irreversible transfers of energy towards the dissipative scales. The life time of the coherent wave, before energy dissipation, is found to be proportional to the inverse of the amplitude squared, and we explain this law by a simple theoretical model. The wave breaking itself is preceded by a slow transfer of energy to secondary waves by a mechanism of resonant interactions, and we compare the results with the classical theory of this phenomenon: good agreement is obtained for moderate amplitudes. The nature of the events leading to wave breaking depends on the wave frequency (i.e. on the direction of the wave vector); most of the analysis is restricted to the case of fairly high frequencies. The maximum growth rate of the inviscid wave instability occurs in the limit of high wavenumbers. We observe that a well-organized secondary plane wave packet is excited. Its frequency is half the frequency of the primary wave, corresponding to an excitation by a parametric instability. The mechanism of selection of this remarkable structure, in the limit of small viscosities, is discussed. Once this secondary wave packet has reached a high amplitude, density overturning occurs, as well as unstable shear layers, leading to a rapid transfer of energy towards dissipative scales. Therefore the condition of strong wave steepness leading to wave breaking is locally attained by the development of a single small-scale parametric instability, rather than a cascade of wave interactions. This fact may be important for modelling the dynamics of an internal wave field.