Troposphere-Stratosphere Coupling and the Role of Critical Layer Nonlinearity

2020 ◽  
Author(s):  
Imogen Dell
Keyword(s):  
Rossby Waves ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Critical Layer ◽  
Coupling Mechanism ◽  
Mathematical Framework ◽  
Mean Flow ◽  
Weather And Climate ◽  
Dynamical Coupling

<p>There exists a coupling mechanism between the troposphere and the stratosphere, which plays a fundamental role in weather and climate. The coupling is highly complex and rests upon radiative and chemical feedbacks, as well as dynamical coupling by Rossby waves. The troposphere acts as a source of Rossby waves which propagate upwards in to the stratosphere, affecting the zonal mean flow. Rossby waves are also likely to play a significant role in downward communication of information via reflection from the stratosphere in to the troposphere. A mechanism for this reflection could be from a so-called critical layer. A shear flow exhibits a critical layer where the phase speed equals the flow velocity, where viscous and nonlinear effects become important. A wave incident upon a critical layer may be absorbed, reflected or overreflected, whereby the amplitude of the reflected wave is larger than that of the incident wave. In the case of troposphere-stratosphere coupling, the concept of critical layer overreflection is key to understanding atmospheric instability.</p><p>Motivated by this, a mathematical framework for understanding the coupling will be presented together with an investigation in to the role of nonlinearity versus viscosity inside the critical layer.</p>

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 Physics of tidal dissipation in early-type stars and white dwarfs: hydrodynamical simulations of internal gravity wave breaking in stellar envelopes

2020 ◽  
Vol 495 (1) ◽  
pp. 1239-1251 ◽  
Author(s):  
Yubo Su ◽  
Daniel Lecoanet ◽  
Dong Lai
Keyword(s):  
Wave Breaking ◽  
White Dwarfs ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Early Type ◽  
Mean Flow ◽  
Layer Width ◽  
Early Type Stars ◽  
Hydrodynamical Simulations

ABSTRACT In binaries composed of either early-type stars or white dwarfs, the dominant tidal process involves the excitation of internal gravity waves (IGWs), which propagate towards the stellar surface, and their dissipation via non-linear wave breaking. We perform 2D hydrodynamical simulations of this wave breaking process in a stratified, isothermal atmosphere. We find that, after an initial transient phase, the dissipation of the IGWs naturally generates a sharp critical layer, separating the lower stationary region (with no mean flow) and the upper ‘synchronized’ region (with the mean flow velocity equal to the horizontal wave phase speed). While the critical layer is steepened by absorption of these waves, it is simultaneously broadened by Kelvin–Helmholtz instabilities such that, in steady state, the critical layer width is determined by the Richardson criterion. We study the absorption and reflection of incident waves off the critical layer and provide analytical formulae describing its long-term evolution. The result of this study is important for characterizing the evolution of tidally heated white dwarfs and other binary stars.

The critical layer in linear-shear boundary layers over acoustic linings

2012 ◽  
Vol 710 ◽  
pp. 545-568 ◽  
Author(s):  
E. J. Brambley ◽  
M. Darau ◽  
S. W. Rienstra
Keyword(s):  
Boundary Layers ◽  
Phase Speed ◽  
Sound Field ◽  
Critical Layer ◽  
Uniform Density ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Sheared Flow ◽  
Linear Shear ◽  
Critical Layers

AbstractAcoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as $O(1/ {x}^{4} )$, where $x$ is the distance downstream of the source. For sources within the sheared flow, the contribution from the critical layer is found to consist of a non-modal disturbance of constant amplitude and a disturbance decaying algebraically as $O(1/ {x}^{3} )$. For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

A theoretical model analysis of the sudden narrow temperature-layer formation observed in the ALOHA-93 Campaign

2002 ◽  
Vol 80 (12) ◽  
pp. 1543-1558 ◽  
Author(s):  
H Hur ◽  
T Y Huang ◽  
Z Zhao ◽  
P Karunanayaka ◽  
T F Tuan
Keyword(s):  
Energy Dissipation ◽  
Gravity Wave ◽  
Temperature Rise ◽  
Critical Level ◽  
Inversion Layer ◽  
Phase Speed ◽  
Temperature Inversion ◽  
Critical Layer ◽  
Mean Flow ◽  
Wind Profiles

The behavior of temperature and wind profiles observed on 21 October 1993 in the ALOHA-93 Campaign is theoretically and numerically analyzed. A sudden temperature rise took place in a very narrow vertical region (3–4 km) at about 87 km. Simultaneously observed radar wind profiles and mesospheric airglow wave structures that show a horizontal phase speed of 35 m/s and a period of about half an hour strongly suggest that a critical level may occur in the proximity of that altitude and that the energy dissipation due to the interaction of the gravity wave with the critical level causes the temperature rise. The numerical model used is a solution to the gravity wave – mean-flow interaction in the critical layer, including a simple cooling mechanism and a wave-energy dissipation simulated by the "optical model" technique. The solutions for the temperature variations so obtained show good agreement with the observed temperature profiles at different times, providing a quantitative explanation for the temperature inversion layer as a phenomenon of gravity wave – critical layer interaction. PACS Nos.: 91.10V, 94.10D

Nonlinear spatial evolution of an externally excited instability wave in a free shear layer

1988 ◽  
Vol 197 ◽  
pp. 295-330 ◽  
Author(s):  
M. E. Goldstein ◽  
Lennart S. Hultgren
Keyword(s):  
Nonlinear Effects ◽  
Numerical Results ◽  
Critical Layer ◽  
Instability Wave ◽  
Neutral Stability ◽  
Spatial Evolution ◽  
Mean Flow ◽  
Nonlinear Convection ◽  
Wave Growth ◽  
Stability Point

We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.

Laboratory Observations of Gravity Wave, Critical Layer Flows Using Single and Double Wave Forcing

1994 ◽  
Vol 47 (6S) ◽  
pp. S113-S117
Author(s):  
Donald P. Delisi ◽  
Timothy J. Dunkerton
Keyword(s):  
Gravity Wave ◽  
Wave Breaking ◽  
Critical Level ◽  
Phase Speed ◽  
Vertical Shear ◽  
Critical Layer ◽  
Small Scale ◽  
Mean Flow ◽  
Wave Forcing ◽  
Internal Mixing

Laboratory measurements of gravity wave, critical layer flows are presented. The measurements are obtained in a salt-stratified annular tank, with a vertical shear profile. Internal gravity waves are generated at the floor of the tank and propagate vertically upward into the fluid. At a depth where the phase speed of the wave equals the mean flow speed, defined as a critical level, the waves break down, under the right forcing conditions, generating small scale turbulence. Two cases are presented. In the first case, the wave forcing is a single, monochromatic wave. In this case, the early wave breaking is characterized as Kelvin-Helmholtz breaking at depths below the critical level. Later wave breaking is characterized by weak overturning in the upper part of the tank and regular, internal mixing regions in the lower part of the tank. In the second case, the wave forcing is two monochromatic waves, each propagating with a different phase speed. In this case, the early wave breaking is again Kelvin-Helmholtz in nature, but later wave breaking is characterized by sustained overturning in the upper part of the tank with internal mixing regions in the lower part of the tank. Mean velocity profiles are obtained both before and during the experiments.

Quantifying Residual, Eddy, and Mean Flow Effects on Mixing in an Idealized Circumpolar Current

2017 ◽  
Vol 47 (8) ◽  
pp. 1897-1920 ◽  
Author(s):  
Phillip J. Wolfram ◽  
Todd D. Ringler
Keyword(s):  
High Performance ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Regime ◽  
Linear Waves ◽  
Parameter Ratio ◽  
Low Pass ◽  
Circumpolar Current

AbstractMeridional diffusivity is assessed for a baroclinically unstable jet in a high-latitude idealized circumpolar current (ICC) using the Model for Prediction across Scales Ocean (MPAS-O) and the online Lagrangian in Situ Global High-Performance Particle Tracking (LIGHT) diagnostic via space–time dispersion of particle clusters over 120 monthly realizations of O(106) particles on 11 potential density surfaces. Diffusivity in the jet reaches values of O(6000) m2 s−1 and is largest near the critical layer supporting mixing suppression and critical layer theory. Values in the vicinity of the shelf break are suppressed to O(100) m2 s−1 because of the presence of westward slope front currents. Diffusivity attenuates less rapidly with depth in the jet than both eddy velocity and kinetic energy scalings would suggest. Removal of the mean flow via high-pass filtering shifts the nonlinear parameter (ratio of the eddy velocity to eddy phase speed) into the linear wave regime by increasing the eddy phase speed via the depth-mean flow. Low-pass filtering, in contrast, quantifies the effect of mean shear. Diffusivity is decomposed into mean flow shear, linear waves, and the residual nonhomogeneous turbulence components, where turbulence dominates and eddy-produced filamentation strained by background mean shear enhances mixing, accounting for ≥80% of the total diffusivity relative to mean shear [O(100) m2 s−1], linear waves [O(1000) m2 s−1], and undecomposed full diffusivity [O(6000) m2 s−1]. Diffusivity parameterizations accounting for both the nonhomogeneous turbulence residual and depth variability are needed.

scholarly journals The Interaction of a Baroclinic Mean Flow with Long Rossby Waves

2005 ◽  
Vol 35 (5) ◽  
pp. 865-879 ◽  
Author(s):  
A. Colin de Verdière ◽  
R. Tailleux
Keyword(s):  
Doppler Shift ◽  
Rossby Waves ◽  
Numerical Solutions ◽  
Phase Speed ◽  
Standard Theory ◽  
Buoyancy Frequency ◽  
Implicit Assumption ◽  
Constant Sign ◽  
Mean Flow ◽  
The Mean

Abstract The effect of a baroclinic mean flow on long oceanic Rossby waves is studied using a combination of analytical and numerical solutions of the eigenvalue problem. The effect is summarized by the value of the nondimensional numberwhen the mean flow shear keeps a constant sign throughout the water column. Because previous studies have shown that no interaction occurs if the mean flow has the shape of the first unperturbed mode (the non–Doppler shift effect), an implicit assumption in the application of the present work to the real ocean is that the relative projections of the mean flow on the second and higher modes remain approximately constant. Because R2 is large at low latitudes between 10° and 30° (the southern branches of subtropical gyres or the regions of surface westward shear), the phase speed of the first mode is very slightly decreased from the no mean flow standard theory case. Between 30° and 40° latitudes (the northern branches of the subtropical gyres or the regions of surface eastward shear), R2 is O(10) and the westward phase speed can increase significantly (up to a factor of 2). At still higher latitudes when R2 is O(1) a critical transition occurs below which no discrete Rossby waves are found that might explain the absence of observations of zonal propagations at latitudes higher than 50°. Our case studies, chosen to represent the top-trapped and constant-sign shear profiles of observed mean flows, all show the importance of three main effects on the value of the first baroclinic mode Rossby wave speed: 1) the meridional gradient of the quantity N2/f (where N is the buoyancy frequency) rather than that of the potential vorticity fN2; 2) the curvature of the mean flow in the vertical direction, which appears particularly important to predict the sign of the phase speed correction to the no-mean-flow standard theory case: increase (decrease) of the westward phase speed when the surface-intensified mean flow is eastward (westward); and 3) a weighted vertical average of the mean flow velocity, acting as a Doppler-shift term, which is small in general but important to determine the precise value of the phase speed.

Mean flow generation by topographic Rossby waves

1979 ◽  
Vol 94 (1) ◽  
pp. 39-64 ◽  
Author(s):  
Alain Colin De Verdiere
Keyword(s):  
Rossby Waves ◽  
Phase Speed ◽  
Zonal Flow ◽  
Travelling Wave ◽  
General Tendency ◽  
Primary Wave ◽  
Mean Flow ◽  
Transfer Energy ◽  
Flow Strength ◽  
Topographic Rossby Waves

This paper makes use of the ease of modelling topographic Rossby waves in a laboratory context to investigate the ability of these waves to generate strong zonal mean flows when the geostrophic (f/H) contours are closed. A zonally travelling wave is forced in a narrow latitude band of a ‘polar beta plane’. Stronger signals occur when the motion of the driving is retrograde and at the phase speed of the gravest free modes. An important zonal westward mean flow occurs in the free interior while a compensating eastward jet is found at forced latitudes. The dependence of the mean flow strength upon the wave steepness indicates that genuine rectification processes are indeed taking place when the fluid is stirred by purely oscillating devices.This general tendency for topographic Rossby waves to transfer energy to zonal components is first analysed theoretically by investigating a side-band instability mechanism within an unforced fluid. Among the products of the interactions between a primary wave of wavenumberkand its side bands of wavenumberk± δk, the zonal flow is prominent. Wave steepnesses of order (|δk|/|k|)½only are required for zonal energy to grow whereas non-zonal components of scale longer or shorter than the primary wave need huge steepnesses [of order (|δk|/|k|−3/2] for amplification. This supplements the earlier notion that ‘nearly zonal’ waves may be generated by weak resonant interaction.For gentle driving certain classical aspects of Rossby wave propagation can be checked against the experiments. The linear theory provides also a convenient framework to discuss the meridional structure of the wave-induced Reynolds stress. For more energetic driving, a test of the potential vorticity mixing theory can be carried out and sheds further light upon the rectification mechanisms.

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.

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