Time-dependent critical layers in shear flows on the beta-plane

1984 ◽  
Vol 142 ◽  
pp. 431-449 ◽  
Author(s):  
Fred J. Hickernell
Keyword(s):  
Evolution Equation ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Critical Layer ◽  
Convolution Integral ◽  
Matched Asymptotics ◽  
Beta Plane ◽  
Neutral Mode ◽  
Critical Layers ◽  
Layer Flow

The problem of a finite-amplitude free disturbance of an inviscid shear flow on the beta-plane is studied. Perturbation theory and matched asymptotics are used to derive an evolution equation for the amplitude of a singular neutral mode of the Kuo equation. The effects of time-dependence, nonlinearity and viscosity are included in the analysis of the critical-layer flow. Nonlinear effects inside the critical layer rather than outside the critical layer determine the evolution of the disturbance. The nonlinear term in the evolution equation is some type of convolution integral rather than a simple polynomial. This makes the evolution equation significantly different from those commonly encountered in fluid wave and stability problems.

The role of nonlinear critical layers in boundary layer transition

1995 ◽  
Vol 352 (1700) ◽  
pp. 425-442 ◽  
Keyword(s):  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Mean Flow ◽  
Layer Transition ◽  
Instability Waves ◽  
Laminar Boundary Layers ◽  
Instability Modes ◽  
Critical Layers

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called ‘critical layer’, with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

scholarly journals Dynamics in coalescing critical layers

2001 ◽  
Vol 449 ◽  
pp. 115-139 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO ◽  
O. M. UMURHAN
Keyword(s):  
Normal Modes ◽  
Critical Layer ◽  
Matched Asymptotic Expansion ◽  
Physical Parameters ◽  
Inviscid Fluid ◽  
Critical Levels ◽  
Governing Equations ◽  
Vortical Structures ◽  
Beta Plane ◽  
Critical Layers

This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cat's eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cat's eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.

Getting Rid of Fast Waves: Slow Dynamics

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Rossby Waves ◽  
Baroclinic Instability ◽  
Nonlinear Effects ◽  
Kelvin Wave ◽  
Beta Plane ◽  
Equatorial Wave ◽  
Flow Over Topography ◽  
Wave Guides ◽  
Rotating Shallow Water ◽  
Fast Waves

After analysis of general properties of horizontal motion in primitive equations and introduction of principal parameters, the key notion of geostrophic equilibrium is introduced. Quasi-geostrophic reductions of one- and two-layer rotating shallow-water models are obtained by a direct filtering of fast inertia–gravity waves through a choice of the time scale of motions of interest, and by asymptotic expansions in Rossby number. Properties of quasi-geostrophic models are established. It is shown that in the beta-plane approximations the models describe Rossby waves. The first idea of the classical baroclinic instability is given, and its relation to Rossby waves is explained. Modifications of quasi-geostrophic dynamics in the presence of coastal, topographic, and equatorial wave-guides are analysed. Emission of mountain Rossby waves by a flow over topography is demonstrated. The phenomena of Kelvin wave breaking, and of soliton formation by long equatorial and topographic Rossby waves due to nonlinear effects are explained.

Finite-amplitude convection in the presence of an unsteady shear flow

1995 ◽  
Vol 287 ◽  
pp. 225-249 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Shear Flow ◽  
Critical Size ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Present Calculation ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boussinesq Fluid

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

Finite-amplitude three-dimensional instability of inviscid laminar boundary layers

1995 ◽  
Vol 290 ◽  
pp. 203-212
Author(s):  
Melvin E. Stern
Keyword(s):  
Three Dimensional ◽  
Streamwise Velocity ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Stream Velocity ◽  
Normal Velocity ◽  
Laminar Boundary Layers ◽  
Vertical Thickness ◽  
Layer Flow ◽  
Dimensional Instability

An inviscid laminar boundary layer flow Û(ŷ) with vertical thickness λy, and free stream velocity U is disturbed at time $\tcirc$ = 0 by a normal velocity $\vcirc$ and by a spanwise velocity ŵ([xcirc ],ŷ, $\zcirc$, 0) of finite amplitude αU, with spanwise ($\zcirc$) scale λz, and streamwise ([xcirc ]) scale λx = λz/α; the streamwise velocity û([xcirc ],ŷ,$\zcirc$,$\tcirc$) is initially undisturbed. A long wave λy/λz → 0) expansion of the Euler equations for fixed α and time scale $\tcirc$s = U−1λz/α results in a hyperbolic equation for Lagrangian displacements ŷ. Within the interval $\tcirc$ > $\tcirc$s of asymptotic validity, finite parcel displacements (O(λy)) with finite (O(U)) û fluctuations occur, independent of α no matter how small; the basic flow Û is therefore said to be unstable to streaky (λx [Gt ] λz) spanwise perturbations. The temporal development of the ('spot’) region in the (x,z) plane wherein inflected û profiles appear is computed and qualitatively related to observations of ‘breakdown’ and transition to turbulence in the flow over a flat plate. The maximum $\vcirc$([xcirc ],ŷ,$\zcirc$,$\tcirc$) increases monotonically to infinity as $\tcirc$ → $\tcirc$s.

scholarly journals The transformation of an interfacial solitary wave of elevation at a bottom step

2009 ◽  
Vol 16 (1) ◽  
pp. 33-42 ◽  
Author(s):  
V. Maderich ◽  
T. Talipova ◽  
R. Grimshaw ◽  
E. Pelinovsky ◽  
B. H. Choi ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Study ◽  
Nonlinear Effects ◽  
Ocean Model ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
Layer Flow ◽  
The One ◽  
Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

Effect of Destabilizing Heating on Go¨rtler Vortices

1985 ◽  
Vol 107 (4) ◽  
pp. 877-882 ◽  
Author(s):  
Y. Kamotani ◽  
J. K. Lin ◽  
S. Ostrach
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Nonlinear Effects ◽  
Experimental Investigations ◽  
Wave Form ◽  
Sinusoidal Wave ◽  
Lateral Direction ◽  
Vortex Instability ◽  
Nonlinear Region ◽  
Layer Flow

Experimental investigations of the effect of destabilizing heating on the vortex instability in a laminar boundary-layer flow of air along a concave surface are reported. The ranges of the parameters studied herein are Gr (Grashof number) from 0 to 70 and G (Go¨rtler number) from 0.46 to 9.0. The wavelength of the vortices remains unchanged with heating but the strength of the vortices is enhanced by heating. The amplitude of the vortices increases almost exponentially with the combined parameter (G2 + f Gr)1/2, where f is found to be between 0.3 and 0.4, until the nonlinear effects become important. In the nonlinear region the original sinusoidal wave form of the vortices becomes distorted and they meander in the lateral direction.

Trapped Lee Wave Interference in the Presence of Surface Friction

2011 ◽  
Vol 68 (4) ◽  
pp. 918-936 ◽  
Author(s):  
Ivana Stiperski ◽  
Vanda Grubišić
Keyword(s):  
Boundary Layer ◽  
Nonlinear Effects ◽  
Surface Friction ◽  
Boundary Layer Separation ◽  
Separation Distance ◽  
Finite Amplitude ◽  
Destructive Interference ◽  
Wave Interference ◽  
Lee Wave ◽  
The Impact

Abstract Trapped lee wave interference over double bell-shaped obstacles in the presence of surface friction is examined. Idealized high-resolution numerical experiments with the nonhydrostatic Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) are performed to examine the influence of a frictional boundary layer and nonlinearity on wave interference and the impact of interference on wave-induced boundary layer separation and the formation of rotors. The appearance of constructive and destructive interference, controlled by the ratio of the ridge separation distance to the intrinsic horizontal wavelength of lee waves, is found to be predicted well by linear interference theory with orographic adjustment. The friction-induced shortening of intrinsic wavelength displays a strong indirect effect on wave interference. For twin peak orography, the interference-induced variation of wave amplitude is smaller than that predicted by linear theory. The interference is found to affect the formation and strength of rotors most significantly in the lee of the downstream peak; destructive interference suppresses the formation and strength of rotors there, whereas results for constructive interference closely parallel those for a single mountain. Over the valley, under both constructive and destructive interference, rotors are weaker compared to those in the lee of a single ridge while their strength saturates in the finite-amplitude flow regime. Destructive interference is found to be more susceptible to nonlinear effects, with both the orographic adjustment and surface friction displaying a stronger effect on the flow in this state. “Complete” destructive interference, in which waves almost completely cancel out in the lee of the downstream ridge, develops for certain ridge separation distances but only for a downstream ridge smaller than the upstream one.

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