The limiting form of inertial instability in geophysical flows

2008 ◽  
Vol 605 ◽  
pp. 115-143 ◽  
Author(s):  
STEPHEN D. GRIFFITHS
Keyword(s):  
Growth Rate ◽  
Asymptotic Series ◽  
Normal Modes ◽  
Perturbation Expansion ◽  
Vertical Component ◽  
Phase Speed ◽  
Coriolis Parameter ◽  
Uniform Shear ◽  
Inertial Instability ◽  
Schrödinger Perturbation

The instability of a rotating, stratified flow with arbitrary horizontal cross-stream shear is studied, in the context of linear normal modes with along-stream wavenumber k and vertical wavenumber m. A class of solutions are developed which are highly localized in the horizontal cross-stream direction around a particular streamline. A Rayleigh–Schrödinger perturbation analysis is performed, yielding asymptotic series for the frequency and structure of these solutions in terms of k and m. The accuracy of the approximation improves as the vertical wavenumber increases, and typically also as the along-stream wavenumber decreases. This is shown to correspond to a near-inertial limit, in which the solutions are localized around the global minimum of fQ, where f is the Coriolis parameter and Q is the vertical component of the absolute vorticity. The limiting solutions are near-inertial waves or inertial instabilities, according to whether the minimum value of fQ is positive or negative.We focus on the latter case, and investigate how the growth rate and structure of the solutions changes with m and k. Moving away from the inertial limit, we show that the growth rate always decreases, as the inertial balance is broken by a stabilizing cross-stream pressure gradient. We argue that these solutions should be described as non-symmetric inertial instabilities, even though their spatial structure is quite different to that of the symmetric inertial instabilities obtained when k is equal to zero.We use the analytical results to predict the growth rates and phase speeds for the inertial instability of some simple shear flows. By comparing with results obtained numerically, it is shown that accurate predictions are obtained by using the first two or three terms of the perturbation expansion, even for relatively small values of the vertical wavenumber. Limiting expressions for the growth rate and phase speed are given explicitly for non-zero k, for both a hyperbolic-tangent velocity profile on an f-plane, and a uniform shear flow on an equatorial β-plane.

scholarly journals The Quasi-Nondispersive Regimes of Long Extratropical Baroclinic Rossby Waves over (Slowly Varying) Topography

2006 ◽  
Vol 36 (1) ◽  
pp. 104-121 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Rossby Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Dispersive Wave ◽  
Mean Flow ◽  
Coriolis Parameter ◽  
Zonal Wavenumber ◽  
Topographic Parameter ◽  
Ray Paths

Abstract Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio τ, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with τ, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f /hαc = a constant (αc is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

scholarly journals Ekman-inertial instability

2021 ◽  
Author(s):  
Nicolas Grisouard ◽  
Varvara E Zemskova
Keyword(s):  
Growth Rate ◽  
Stokes Problem ◽  
Initial Perturbation ◽  
Initial Response ◽  
Initial Growth ◽  
Mean Flow ◽  
Viscous Effects ◽  
Coriolis Parameter ◽  
Initial Flow ◽  
Inertial Instability

<p>We report on an instability arising in sub-surface, laterally sheared geostrophic flows. When the lateral shear of a horizontal flow in geostrophic balance has a sign opposite to the Coriolis parameter and exceeds it in magnitude, embedded perturbations are subjected to inertial instability, albeit modified by viscosity. When the perturbation arises from the surface of the fluid, the initial response is akin to a Stokes problem, with an initial flow aligned with the initial perturbation. The perturbation then grows quasi-inertially, rotation deflecting the velocity vector, which adopts a well-defined angle with the mean flow, and viscous stresses, transferring horizontal momentum downward. The combination of rotational and viscous effects in the dynamics of inertial instability prompts us to call this process “Ekman-inertial instability.” While the perturbation initially grows super-inertially, the growth rate then becomes sub-inertial, eventually tending back to the inertial value. The same process repeats downward as time progresses. Ekman-inertial transport aligns with the asymptotic orientation of the flow and grows exactly inertially with time once the initial disturbance has passed. Because of the strongly super-inertial initial growth rate, this instability might compete favourably against other instabilities arising in ocean fronts.</p>

Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

2017 ◽  
Vol 47 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Robert D. Hetland
Keyword(s):  
Growth Rate ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Buoyancy Frequency ◽  
Bottom Slope ◽  
Coriolis Parameter ◽  
Plume Front ◽  
Instability Theory ◽  
Instability Growth ◽  
Flow Configurations

AbstractBaroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SH ≡ SRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

scholarly journals Heat transfer in the seabed boundary layer

2021 ◽  
Vol 928 ◽  
Author(s):  
S. Michele ◽  
R. Stuhlmeier ◽  
A.G.L. Borthwick
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Free Surface ◽  
Surface Waves ◽  
Multiple Scales ◽  
Perturbation Expansion ◽  
Phase Speed ◽  
Maximum Heat ◽  
Practical Applications ◽  
Free Surface Waves

We present a theoretical model of the temperature distribution in the boundary layer region close to the seabed. Using a perturbation expansion, multiple scales and similarity variables, we show how free-surface waves enhance heat transfer between seawater and a seabed with a solid, horizontal, smooth surface. Maximum heat exchange occurs at a fixed frequency depending on ocean depth, and does not increase monotonically with the length and phase speed of propagating free-surface waves. Close agreement is found between predictions by the analytical model and a finite-difference scheme. It is found that free-surface waves can substantially affect the spatial evolution of temperature in the seabed boundary layer. This suggests a need to extend existing models that neglect the effects of a wave field, especially in view of practical applications in engineering and oceanography.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

Effect of Coriolis force on edge waves (I) Investigation of the normal modes

2020 ◽  
Vol 78 (4) ◽  
pp. 229-261
Author(s):  
Robert O. Reid
Keyword(s):  
Coriolis Force ◽  
Wind Stress ◽  
Formal Solution ◽  
Wave Length ◽  
Normal Modes ◽  
Low Frequency ◽  
Phase Speed ◽  
Free Edge ◽  
Edge Wave ◽  
Edge Waves

Essentially two classes of free edge waves can exist on a sloping continental shelf in the presence of Coriolis force. For small longshore wave length, fundamental waves of the first class behave like Stokes edge waves. However, for great wave lengths (of several hundred kilometers or more) the characteristics of the first class are significantly altered. In the northern hemisphere the phase speed for waves moving to the right (facing shore from the sea) exceeds the speed for waves which move to the left. Also, the group velocity for a given edge wave mode has a finite upper limit. Waves of the second class are essentially quasigeostrophic boundary waves with very low frequency and, like Kelvin waves, move only to the left (again facing shore from the sea). Unlike Stokes edge waves, those of the quasigeostrophic class are associated with large vorticity. Examination of the formal solution for forced edge waves indicates that those of the second class may be excited significantly by a wind stress vortex. Also, in contrast to the conclusion of Greenspan (1956), it is proposed that a hurricane can effectively excite the higher order edge wave modes in addition to the fundamental if wind stress is considered.

scholarly journals Instabilities of continuously stratified zonal equatorial jets in a periodic channel model

2002 ◽  
Vol 20 (5) ◽  
pp. 729-740 ◽  
Author(s):  
S. Masina
Keyword(s):  
Channel Model ◽  
Baroclinic Instability ◽  
Phase Speed ◽  
Maximum Velocity ◽  
Equatorial Region ◽  
Eastern Pacific ◽  
Vertical Shear ◽  
Coriolis Parameter ◽  
Vertical Scale ◽  
Surface Jet

Abstract. Several numerical experiments are performed in a nonlinear, multi-level periodic channel model centered on the equator with different zonally uniform background flows which resemble the South Equatorial Current (SEC). Analysis of the simulations focuses on identifying stability criteria for a continuously stratified fluid near the equator. A 90 m deep frontal layer is required to destabilize a zonally uniform, 10° wide, westward surface jet that is symmetric about the equator and has a maximum velocity of 100 cm/s. In this case, the phase velocity of the excited unstable waves is very similar to the phase speed of the Tropical Instability Waves (TIWs) observed in the eastern Pacific Ocean. The vertical scale of the baroclinic waves corresponds to the frontal layer depth and their phase speed increases as the vertical shear of the jet is doubled. When the westward surface parabolic jet is made asymmetric about the equator, in order to simulate more realistically the structure of the SEC in the eastern Pacific, two kinds of instability are generated. The oscillations that grow north of the equator have a baroclinic nature, while those generated on and very close to the equator have a barotropic nature.  This study shows that the potential for baroclinic instability in the equatorial region can be as large as at mid-latitudes, if the tendency of isotherms to have a smaller slope for a given zonal velocity, when the Coriolis parameter vanishes, is compensated for by the wind effect.Key words. Oceanography: general (equatorial oceanography; numerical modeling) – Oceanography: physics (fronts and jets)

scholarly journals Exponential asymptotics and Stokes lines in a partial differential equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2385-2421 ◽  
Author(s):  
S. Jonathan Chapman ◽  
David B Mortimer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Second Generation ◽  
Saddle Points ◽  
Asymptotic Series ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Stokes Line ◽  
Partial Differential ◽  
Stokes Lines

A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al . (Berk et al . 1982 J. Math. Phys. 23 , 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al . (Aoki et al . 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.

scholarly journals A Global Climatology of Tropospheric Inertial Instability

2018 ◽  
Vol 75 (3) ◽  
pp. 805-825 ◽  
Author(s):  
Callum F. Thompson ◽  
David M. Schultz ◽  
Geraint Vaughan
Keyword(s):  
North Atlantic ◽  
North Atlantic Ocean ◽  
Western Pacific Subtropical High ◽  
Synoptic Scale ◽  
Coriolis Parameter ◽  
Local Maxima ◽  
The North ◽  
Inertial Instability ◽  
The North Atlantic ◽  
The Tropics

Abstract A climatology of tropospheric inertial instability is constructed using the European Centre for Medium-Range Weather Forecasts interim reanalysis (ERA-Interim) at 250, 500, and 850 hPa. For each level, two criteria are used. The first criterion is the traditional criterion of absolute vorticity that is opposite in sign to the local Coriolis parameter. The second criterion, referred to as the gradient criterion, is the traditional criterion with an added term incorporating flow curvature. Both criteria show that instability, on all pressure levels, occurs most frequently in the tropics and decreases toward the poles. Compared to the traditional criterion, the gradient criterion diagnoses instability much more frequently outside the tropics and less frequently near the equator. The global distribution of inertial instability also shows many local maxima in the occurrence of instability. A sample of these local maxima is investigated further by constructing composites of the synoptic-scale flow associated with instability. The composites show that instability occurs in association with cross-equatorial flow in the North Atlantic Ocean, the Somali jet, tip jets off northern Madagascar, the western Pacific subtropical high, gap winds across Central America, upper-level ridging over western North America, and the North Atlantic polar jet. Furthermore, relatively long-lived synoptic-scale regions of instability are found within the midlatitude jet streams.

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