18 results
Wave boundary layers in a stratified fluid
- G. M. Reznik
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- Journal:
- Journal of Fluid Mechanics / Volume 833 / 25 December 2017
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- 07 November 2017, pp. 512-537
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We study so-called wave boundary layers (BLs) arising in a stably stratified fluid at large times. The BL is a narrow domain near the surface and/or bottom of the fluid; with increasing time, gradients of buoyancy and horizontal velocity in the BL grow sharply and the BL thickness tends to zero. The non-stationary BL can arise both as a result of linear evolution of the initial perturbation and under the action of an external force (tangential stress exerted on the fluid surface in our case). We analyse both the variants and find that the ‘forced’ BLs are much more intense than the ‘free’ ones. In the ‘free’ BLs all fields are bounded and the gradients of buoyancy and horizontal velocity grow linearly in time, whereas in the ‘forced’ BL only the vertical velocity is bounded and the buoyancy and horizontal velocity grow linearly in time. As to the gradients in the ‘forced’ BL, the vertical velocity gradient grows in time linearly and the gradients of buoyancy and horizontal velocity grow quadratically. In both of the cases we determine exact solutions in the form of expansions in the vertical wave modes and find asymptotic solutions valid at large times. The comparison between them shows that the asymptotic solutions approximate the exact ones fairly well even for moderate times.
Review of the chemistry, metabolism, and dose response of two supplemental methionine sources and the implications in their relative bioefficacy
- M. VÁZQUEZ-AÑÓN, G. BERTIN, Y. MERCIER, G. REZNIK, J-L. ROBERTON
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- Journal:
- World's Poultry Science Journal / Volume 73 / Issue 4 / December 2017
- Published online by Cambridge University Press:
- 17 October 2017, pp. 725-736
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- December 2017
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This review examines the relative bioefficacy of 2-hydroxy-4-(methylthio) butanoic acid (HMTBA) and DL-methionine (DL-Met) which includes chemical, metabolic, nutritional, and statistical aspects of its bioefficacy. The chemical, enzymatic and biological differences and similarities between these two products are explained and the evidence and reasons for HMTBA relative bioefficacy to DL-Met in monogastric animals are discussed. In addition, appropriate statistical methods for comparing the bioefficacy of these two products for successful use of each product are provided. HMTBA is an organic acid precursor of L-Met. The chemical structure differences between HMTBA and DL-Met leads to differences in how and where the two materials are absorbed, enzymatically converted to L-Met and used by the animal. Because of these differences, when the two compounds are supplemented into animal feeds in graded doses, they do not produce dose response curves of the same form due in part to differences in intake and metabolism at the extremes of the dose response curves. At deficient levels of the response curve, HMTBA fed animals may exhibit lower feed consumption and growth than DL-Met while at requirement levels they may have greater feed consumption and growth. This review provides biological evidence for why these differences in growth response occur and demonstrates that lower growth, whether for DL-Met or HMTBA, does not mean that either product is being converted to methionine inefficiently. Since the two products have different dose response curves, statistically valid methods are provided for unbiased determination of relative bioefficacy across tested dose ranges. Field nutritionists typically feed commercial doses of HMTBA or DL-Met at a total sulphur amino acid dietary level capable of achieving maximum performance. At these commercial levels, and based on the evidence, the full relative bioefficacy of HMTBA relative to DL-Met is discussed.
Wave adjustment: general concept and examples
- G. M. Reznik
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- Journal:
- Journal of Fluid Mechanics / Volume 779 / 25 September 2015
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- 18 August 2015, pp. 514-543
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We formulate a general theory of wave adjustment applicable to any physical system (not necessarily a hydrodynamic one), which, being linearized, possesses linear invariants and a complete system of waves harmonically depending on the time $t$. The invariants are determined by the initial conditions and are zero for the waves, which, therefore, do not transport and affect the invariants. The evolution of such a system can be represented naturally as the sum of a stationary component with non-zero invariants and a non-steady wave part with zero invariants. If the linear system is disturbed by a small perturbation (linear or nonlinear), then the state vector of the system is split into slow balanced and fast wave components. Various scenarios of the wave adjustment are demonstrated with fairly simple hydrodynamic models. The simplest scenario, called ‘fast radiation’, takes place when the waves rapidly (their group speed $c_{gr}$ greatly exceeds the slow flow velocity $U$) radiate away from the initial perturbation and do not interact effectively with the slow component. As a result, at large times, after the waves propagate away, the residual flow is slow and described by a balanced model. The scenario is exemplified by the three-dimensional non-rotating barotropic flow with a free surface. A more complicated scenario, called ‘nonlinear trapping’, occurs if oscillations with small group speed $c_{gr}\leqslant U$ are present in the wave spectrum. In this case, after nonlinear wave adjustment, the state vector is a superposition of the slow balanced component and oscillations with small $c_{gr}$ trapped by this component. An example of this situation is the geostrophic adjustment of a three-dimensional rotating barotropic layer with a free surface. In the third scenario, called ‘incomplete splitting’, the wave adjustment is accompanied by non-stationary boundary layers arising near rigid and internal boundaries at large times. The thickness of such a layer tends to zero and cross-gradients of physical parameters in the layer tend to infinity as $t\rightarrow \infty$. The layer is an infinite number of wave modes whose group speed tends to zero as the mode number tends to infinity. In such a system, complete splitting of motion into fast and slow components is impossible even in the linear approximation. The scenario is illustrated by an example of stratified non-rotating flow between two rigid lids. The above scenarios describe, at least, the majority of known cases of wave adjustment.
Geostrophic adjustment with gyroscopic waves: stably neutrally stratified fluid without the traditional approximation
- G. M. Reznik
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- Journal:
- Journal of Fluid Mechanics / Volume 747 / 25 May 2014
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- 23 April 2014, pp. 605-634
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We examine nonlinear geostrophic adjustment in a rapidly rotating (small Rossby number Ro) stably neutrally stratified (SNS) fluid consisting of a stratified upper layer with $N>f$ ($N$ is the buoyancy frequency, $f$ the Coriolis parameter) and a homogeneous lower layer, the density and other fields being continuous at the interface between the layers. The angular speed of rotation is non-parallel to gravity; the traditional and hydrostatic approximations are not used. The wave spectrum in the model consists of internal and gyroscopic waves. During the adjustment an arbitrary long-wave perturbation is split in a unique way into slow quasi-geostrophic (QG) and fast ageostrophic components with typical time scales $(Ro\, f)^{-1}$ and $f^{-1}$, respectively. The QG flow is governed by two coupled nonlinear equations of conservation of QG potential vorticity (PV) in the layers. The fast component is a sum of internal waves and inertial oscillations (long gyroscopic waves) confined to the homogeneous layer and modulated by an amplitude depending on coordinates and slow time. On times $t\sim (\, f\, Ro)^{-1}$ the slow component is not influenced by the fast one but the inertial oscillations amplitude is coupled to the QG flow and obeys an equation practically coinciding with that in the barotropic case (Reznik, J. Fluid Mech., vol. 743, 2014, pp. 585–605). A non-stationary boundary layer with large vertical gradients of horizontal velocity develops in the stratified layer near the interface to prevent penetration of the inertial oscillations into the stratified fluid; an analogous weaker boundary layer arises near the upper rigid lid. At large times the internal waves gradually decay because of dispersion and the resulting motion consists of the slow QG component and inertial oscillations confined to the barotropic lower layer.
Geostrophic adjustment with gyroscopic waves: barotropic fluid without the traditional approximation
- G. M. Reznik
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- Journal:
- Journal of Fluid Mechanics / Volume 743 / 25 March 2014
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- 10 March 2014, pp. 585-605
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We study geostrophic adjustment in rotating barotropic fluid when the angular speed of rotation $\boldsymbol{\Omega}$ does not coincide in direction with the acceleration due to gravity; the traditional and hydrostatic approximations are not used. Linear adjustment results in a tendency of any localized initial state towards a geostrophically balanced steady columnar motion with columns parallel to $\boldsymbol{\Omega}$. Nonlinear adjustment is examined for small Rossby numbers Ro and aspect ratio $H/L$ ($H$ and $L$ are the layer depth and the horizontal scale of motion), using multiple-time-scale perturbation theory. It is shown that an arbitrary perturbation is split in a unique way into slow and fast components evolving with characteristic time scales $(\mathit{Ro}f)^{-1}$ and $f^{-1}$, respectively, where $f $ is the Coriolis parameter. The slow component does not depend on depth and is close to geostrophic balance. On times $O(1/f\, \mathit{Ro})$ the slow component is not influenced by the fast one and is described by the two-dimensional fluid dynamics equation for the geostrophic streamfunction. The fast component consists of long gyroscopic waves and is a packet of inertial oscillations modulated by an amplitude depending on coordinates and slow time. On times $O(1/f\, \mathit{Ro})$ the fast component conserves its energy, but it is coupled to the slow component: its amplitude obeys an equation with coefficients depending on the geostrophic streamfunction. Under the traditional approximation, the inertial oscillations are trapped by the quasi-geostrophic component; ‘non-traditional’ terms in the amplitude equation provide a meridional dispersion of the packet on times $O(1/f\, \mathit{Ro})$, and, therefore, an effective radiation of energy from the initial perturbation domain. Another important effect of the non-traditional terms is that on longer times $O(1/f\, \mathit{Ro}^{2})$ a transfer of energy between the fast and the slow components becomes possible.
Resonant excitation of trapped waves by Poincaré waves in the coastal waveguides
- G. M. REZNIK, V. ZEITLIN
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- Journal:
- Journal of Fluid Mechanics / Volume 673 / 25 April 2011
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- 24 February 2011, pp. 349-394
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After having revisited the theory of linear waves in the rotating shallow-water model with a straight coast and arbitrary shelf/beach bathymetry, we undertake a detailed study of resonant interaction of free Poincaré waves with modes trapped in the coastal waveguide. We describe and quantify the mechanisms of resonant excitation of waveguide modes and their subsequent nonlinear saturation. We obtain the modulation equations for the amplitudes of excited waveguide modes in the absence and in the presence of spatial modulation and analyse their solutions. Different saturation regimes are exhibited, depending on the nature of the modes involved. The excitation is proved to be efficient, i.e. the saturated amplitudes of the excited waves considerably exceed the amplitude of the generator waves. Back-influence of the excited waveguide modes onto the open ocean results in a phase shift of the reflected Poincaré waves and possible energy redistribution between them. A comparison of rotating and non-rotating cases displays substantial differences in excitation mechanisms in the two cases.
An analytical theory of distributed axisymmetric barotropic vortices on the β-plane
- G. M. Reznik, W. K. Dewar
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- Journal of Fluid Mechanics / Volume 269 / 25 June 1994
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- 26 April 2006, pp. 301-321
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An analytical theory of barotropic β-plane vortices is presented in the form of an asymptotic series based on the inverse of vortex nonlinearity. In particular, a solution of the initial value problem is given, in which the vortex is idealized as a radially symmetric function of arbitrary structure. Motion of the vortex is initiated by its interaction with the so-called ‘β-gyres’ which, in turn, are generated by the vortex circulation. Comparisons of analytical and numerical predictions for vortex motion are presented and demonstrate the utility of the present theory for times comparable to the ‘wave’ timescale. The latter exceeds the temporal limit derived from formal considerations. The properties of the far-field planetary wave radiation are also computed.
This theory differs from previous calculations by considering more general initial vortex profiles and by obtaining a more complete solution for the perturbation fields. Vortex trajectory predictions accrue error systematically by integrating vortex propagation rates which are too strong. This appears to be connected to higher-order planetary wave radiation effects.
Dynamics of singular vortices on a beta-plane
- G. M. Reznik
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- Journal of Fluid Mechanics / Volume 240 / July 1992
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- 26 April 2006, pp. 405-432
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A new singular-vortex theory is presented for geostrophic, beta-plane dynamics. The stream function of each vortex is proportional to the modified Bessel function Ko(pr), where p can be an arbitrary positive constant. If p−1 is equal to the Rossby deformation scale Rd, then the vortex is a point vortex; for p−1 ≠ Rd the relative vorticity of the vortex contains an additional logarithmic singularity. Owing to the β-effect, the redistribution of the background potential vorticity produced by the vortices generates a regular field in addition to the velocity field induced by the vortices themselves. Equations governing the joint evolution of singular vortices and the regular field are derived. A new invariant of the motion is found for this system. If the vortex amplitudes and coordinates are set in a particular way then the regular field is zero, and the vortices form a system moving along latitude circles at a constant speed lying outside the range of the phase velocity of linear Rossby waves. Each of the systems is a discrete two-dimensional Rossby soliton and, vice versa, any distributed Rossby soliton is a superposition of the singular vortices concentrated in the interior region of the soliton. An individual singular vortex is studied for times when Rossby wave radiation can be neglected. Such a vortex produces a complicated spiral-form regular flow which consists of two dipoles with mutually perpendicular axes. The dipoles push the vortex westward and along the meridian (cyclones move northward, and anticyclones move southward). The vortex velocity and trajectory are calculated and applications to oceanic and atmospheric eddies are given.
Non-conservation of ‘geostrophic mass’ in the presence of a long boundary and the related Kelvin wave
- G. M. REZNIK, G. G. SUTYRIN
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- Journal:
- Journal of Fluid Mechanics / Volume 527 / 25 March 2005
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- 09 March 2005, pp. 235-264
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The evolution of a localized flow in a half-plane bounded by a rigid wall is analysed when the total mass is not conserved within the equivalent-barotropic quasi-geostrophic (QG) approximation. A simple formula expressing the total geostrophic mass in terms of the QG potential vorticity is derived and used to estimate the range of the geostrophic mass variability. The behaviour of the total mass is analysed for a system of two point vortices interacting with a wall. Distributed localized perturbations are examined by means of numerical experiments using the QG model. Two types of time variability of the total geostrophic mass are revealed: oscillating (the mass oscillates near some mean value) and limiting (the mass tends to some constant value with increasing time).
In the framework of a rotating shallow-water model, the QG model is known to describe the slow evolution of the geostrophic vorticity, assuming the Rossby number to be small. Consideration of the next-order dynamics shows that conservation of the total mass and circulation is provided by a compensating jet taking away the surplus or shortage of mass from the localized geostrophic disturbance. The along-wall jet expands with the fast speed of Kelvin waves to the right of the initial perturbation. The slow time-dependent amplitude determines the jet sign and intensity at each instant. The dynamics of the compensating jet are discussed for both oscillating and limiting regimes revealed by the QG analysis.
The role of Kelvin waves in establishing the usual Phillips condition for conservation of circulation of the along-wall QG velocity is discussed. In the case of periodic motion or motion in a finite domain, the approximation of an infinitely long boundary can be used if (i) the typical basin scale greatly exceeds the typical size of the localized perturbation and the Rossby scale; and (ii) the time does not exceed the typical time required for the Kelvin wave to travel the typical basin scale. Both these conditions are typical of synoptic variability in the ocean.
Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane
- J. LE SOMMER, G. M. REZNIK, V. ZEITLIN
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- Journal:
- Journal of Fluid Mechanics / Volume 515 / 25 September 2004
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- 09 September 2004, pp. 135-170
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We study the nonlinear response of the equatorial shallow-water system at rest to a localized long-wave perturbation with small meridional to zonal aspect ratio. An asymptotic theory of such a response (adjustment) for small Rossby numbers is constructed. Possible scenarios of nonlinear adjustment are classified depending on the relation between the Rossby number and the aspect ratio. The calculations show that slow, geostrophically balanced Rossby and Kelvin waves and the fast inertia–gravity waves are dynamically split off. The fast component of motion exerts no drag on the slow one, which is proved by direct computation. Evolution equations are derived for both components confirming earlier results which were obtained by ad hoc filtering of one of the components of motion. A well-defined initialization procedure is developed for each component.
Due to the breaking of non-dispersive Kelvin waves, the asymptotic theory has obvious limits of validity. In order to go beyond these limits and to study strongly nonlinear effects during the adjustment process we undertook high-resolution shock-capturing numerical simulations based on recent progress in finite-volume numerical methods. The simulations confirm theoretical results but also reveal new effects such as fission of a strongly nonlinear Rossby-wave packet into a sequence of equatorial modons or jet formation in the wake of a breaking Kelvin wave.
Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations
- V. ZEITLIN, G. M. REZNIK, M. BEN JELLOUL
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- Journal:
- Journal of Fluid Mechanics / Volume 491 / 25 September 2003
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- 27 August 2003, pp. 207-228
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This paper continues the work started in Part 1 (Reznik, Zeitlin & Ben Jelloul 2001) and generalizes it to the case of a stratified environment. Geostrophic adjustment of localized disturbances is considered in the context of the two-layer shallow-water and continuously stratified primitive equations in the vertically bounded and horizontally infinite domain on the $f$-plane. Using multiple-time-scale perturbation expansions in Rossby number $\hbox{\it Ro}$ we show that stratification does not substantially change the adjustment scenario established in Part 1 and any disturbance of well-defined scale is split in a unique way into slow and fast components with characteristic time scales $f_0^{-1}$ and $(f_0 \hbox{\it Ro})^{-1}$ respectively, where $f_0$ is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times $t\,{\gg}\, (f_0 \hbox{\it Ro})^{-1}$). In the FG case the fast component is a packet of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schrödinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.
Nonlinear geostrophic adjustment in the presence of a boundary
- G. M. REZNIK, R. GRIMSHAW
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- Journal:
- Journal of Fluid Mechanics / Volume 471 / 25 November 2002
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- 05 November 2002, pp. 257-283
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The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.
The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.
The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.
On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.
Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model
- G. M. REZNIK, V. ZEITLIN, M. BEN JELLOUL
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- Journal:
- Journal of Fluid Mechanics / Volume 445 / 25 October 2001
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- 16 October 2001, pp. 93-120
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We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.
The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.
For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.
Ageostrophic dynamics of an intense localized vortex on a β-plane
- G. M. REZNIK, R. GRIMSHAW
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- Journal:
- Journal of Fluid Mechanics / Volume 443 / 25 September 2001
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- 25 September 2001, pp. 351-376
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We consider the non-stationary dynamics of an intense localized vortex on a β-plane using a shallow-water model. An asymptotic theory for a vortex with piecewise-continuous potential vorticity is developed assuming the Rossby number to be small and the free surface elevation to be small but finite. Analogously to the well-known quasi-geostrophic model, the vortex translation is produced by a secondary dipole circulation (β-gyres) developed in the vortex vicinity and consisting of two parts. The first part (geostrophic β-gyres) coincides with the β-gyres in the geostrophic model, and the second (ageostrophic β-gyres) is due to ageostrophic terms in the governing equations. The time evolution of the ageostrophic β-gyres consists of fast and slow stages. During the fast stage the radiation of inertia–gravity waves results in the rapid development of the β-gyres from zero to a dipole field independent of the fast time variable. Correspondingly, the vortex accelerates practically instantaneously (compared to the typical swirling time) to some finite value of the translation speed. At the next slow stage the inertia–gravity wave radiation is insignificant and the β-gyres evolve with the typical swirling time. The total zonal translation speed induced by the geostrophic and ageostrophic β-gyres tends with increasing time to the speed of a steadily translating monopole exceeding (not exceeding) the drift velocity of Rossby waves for anticyclones (cyclones). This cyclone/anticyclone asymmetry generalizes the well-known finding about the greater longevity of anticyclones compared to cyclones to the case of non-stationary evolving monopoles. The influence of inertia–gravity waves upon the vortex evolution is analysed. The main role of these waves is to provide a ‘fast’ adjustment to the ‘slow’ vortex evolution. The energy of inertia–gravity waves is negligible compare to the energy of the geostrophic β-gyres. Yet another feature of the ageostrophic vortex evolution is that the area of the potential vorticity patch changes in the course of time, the cyclonic patch contracting and the anticyclonic one expanding.
Baroclinic topographic modons
- GREGORY M. REZNIK, GEORGI G. SUTYRIN
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- Journal:
- Journal of Fluid Mechanics / Volume 437 / 25 June 2001
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- 22 June 2001, pp. 121-142
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The theory of solitary topographic Rossby waves (modons) in a uniformly rotating two-layer ocean over a constant slope is developed. The modon is described by an exact, form-preserving, uniformly translating, horizontally localized, nonlinear solution to the inviscid quasi-geostrophic equations. Baroclinic topographic modons are found to translate steadily along contours of constant depth in both directions: either with negative speed (within the range of the phase velocities of linear topographic waves) or with positive speed (outside the range of the phase velocities of linear topographic waves). The lack of resonant wave radiation in the first case is due to the orthogonality of the flow field in the modon exterior to the linear topographic wave field propagating with the modon translation speed, that is impossible for barotropic modons. Another important property of a baroclinic topographic modon is that its integral angular momentum must be zero only in the bottom layer; the total angular momentum can be non-zero unlike for the beta-plane modons over flat bottom. This feature allows modon solutions superimposed by intense monopolar vortices in the surface layer to exist. Explicit analytical solutions for the baroclinic topographic modons with piecewise linear dependence of the potential vorticity on the streamfunction are presented and analysed.
On the long-term evolution of an intense localized divergent vortex on the beta-plane
- G. M. REZNIK, R. GRIMSHAW, E. S. BENILOV
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- Journal:
- Journal of Fluid Mechanics / Volume 422 / 10 November 2000
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- 03 November 2000, pp. 249-280
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The evolution of an intense barotropic vortex on the β-plane is analysed for the case of finite Rossby deformation radius. The analysis takes into account conservation of vortex energy and enstrophy, as well as some other quantities, and therefore makes it possible to gain insight into the vortex evolution for longer times than was done in previous studies on this subject. Three characteristic scales play an important role in the evolution: the advective time scale Ta (a typical time required for a fluid particle to move a distance of the order of the vortex size), the wave time scale Tw (the typical time it takes for the vortex to move through its own radius), and the distortion time scale Td (a typical time required for the change in relative vorticity of the vortex to become of the order of the relative vorticity itself). For an intense vortex these scales are well separated, Ta [Lt ] Tw [Lt ] Td, and therefore one can consider the vortex evolution as consisting of three different stages. The first one, t [les ] Tw, is dominated by the development of a near-field dipolar circulation (primary β-gyres) accelerating the vortex. During the second stage, Tw [les ] t [les ] Td, the quadrupole and secondary axisymmetric components are intensified; the vortex decelerates. During the last, third, stage the vortex decays and is destroyed. Our main attention is focused on exploration of the second stage, which has been studied much less than the first stage. To describe the second stage we develop an asymptotic theory for an intense vortex with initially piecewise-constant relative vorticity. The theory allows the calculation of the quadrupole and axisymmetric corrections, and the correction to the vortex translation speed. Using the conservation laws we estimate that the vortex lifetime is directly proportional to the vortex streamfunction amplitude and inversely proportional to the squared group velocity of Rossby waves. For open-ocean eddies a typical lifetime is about 130 days, and for oceanic rings up to 650 days. Analysis of the residual produced by the asymptotic solution explains why this solution is a good approximation for times much longer than the expected formal range of applicability. All our analytical results are in a good qualitative agreement with several numerical experiments carried out for various vortices.
Planetary waves in a stratified ocean of variable depth. Part 2. Continuously stratified ocean
- A. V. BOBROVICH, G. M. REZNIK
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- Journal:
- Journal of Fluid Mechanics / Volume 388 / 10 June 1999
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- 10 June 1999, pp. 147-169
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Linear Rossby waves in a continuously stratified ocean over a corrugated rough-bottomed topography are investigated by asymptotic methods. The main results are obtained for the case of constant buoyancy frequency. In this case there exist three types of modes: a topographic mode, a barotropic mode, and a countable set of baroclinic modes. The properties of these modes depend on the type of mode, the relative height δ of the bottom bumps, the wave scale L, the topography scale Lb and the Rossby scale Li. For small δ the barotropic and baroclinic modes are transformed into the ‘usual’ Rossby modes in an ocean of constant depth and the topographic mode degenerates. With increasing δ the frequencies of the barotropic and topographic modes increase monotonically and these modes become close to a purely topographic mode for sufficiently large δ. As for the baroclinic modes, their frequencies do not exceed O(βL) for any δ. For large δ the so-called ‘displacement’ effect occurs when the mode velocity becomes small in a near-bottom layer and the baroclinic mode does not ‘feel’ the actual rough bottom relief. At the same time, for some special values of the parameters a sort of resonance arises under which the large- and small-scale components of the baroclinic mode intensify strongly near the bottom.
As in the two-layer model, a so-called ‘screening’ effect takes place here. It implies that for Lb<Li the small-scale component of the mode is confined to a near-bottom boundary layer (Lb/Li)H thick, whereas in the region above the layer the scale L of motion is always larger than or of the order of Li.
Planetary waves in a stratified ocean of variable depth. Part 1. Two-layer model
- G. M. REZNIK, T. B. TSYBANEVA
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- Journal:
- Journal of Fluid Mechanics / Volume 388 / 10 June 1999
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- 10 June 1999, pp. 115-145
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Linear Rossby waves in a two-layer ocean with a corrugated bottom relief (the isobaths are straight parallel lines) are investigated. The case of a rough bottom relief (the wave scale L is much greater than the bottom relief scale Lb) is studied analytically by the method of multiple scales. A special numerical technique is developed to investigate the waves over a periodic bottom relief for arbitrary relationships between L and Lb.
There are three types of modes in the two-layer case: barotropic, topographic, and baroclinic. The structure and frequencies of the modes depend substantially on the ratio Δ = (Δh/h2)/(L/a) measuring the relative strength of the topography and β-effect. Here Δh/h2 is the typical relative height of topographic inhomogeneity and a is the Earth's radius. The topographic and barotropic mode frequencies depend weakly on the stratification for small and large Δ and increase monotonically with increasing Δ. Both these modes become close to pure topographic modes for Δ>1.
The dependence of the baroclinic mode on Δ is more non-trivial. The frequency of this mode is of the order of f0L2i/aL (Li is the internal Rossby scale) irrespective of the magnitude of Δ. At the same time the spatial structure of the mode depends strongly on Δ. With increasing Δ the relative magnitude of motion in the lower layer decreases. For Δ>1 the motion in the mode is confined mainly to the upper layer and is very weak in the lower one. A similar concentration of mesoscale motion in an upper layer over an abrupt bottom topography has been observed in the real ocean many times.
Another important physical effect is the so-called ‘screening’. It implies that for Lb<Li the small-scale component of the wave with scale Lb is confined to the lower layer, whereas in the upper layer the scale of the motion L is always greater than or of the order of, Li. In other words, the stratification prevents the ingress of motion with scale smaller than the internal Rossby scale into the main thermocline.