scholarly journals On Baroclinic Instability over Continental Shelves: Testing the Utility of Eady-Type Models

2020 ◽  
Vol 50 (1) ◽  
pp. 3-33
Author(s):  
Shih-Nan Chen ◽  
Chiou-Jiu Chen ◽  
James A. Lerczak
Keyword(s):  
Growth Rate ◽  
Phase Locking ◽  
Baroclinic Instability ◽  
Lower Boundary ◽  
Ekman Pumping ◽  
Horizontal Shear ◽  
Bottom Slope ◽  
Mean Flow ◽  
Model Calculations ◽  
The Mean

AbstractThis study examines the utility of Eady-type theories as applied to understanding baroclinic instability in coastal flows where depth variations and bottom drag are important. The focus is on the effects of nongeostrophy, boundary dissipation, and bottom slope. The approach compares theoretically derived instability properties against numerical model calculations, for experiments designed to isolate the individual effects and justified to have Eady-like basic states. For the nongeostrophic effect, the theory of Stone (1966) is shown to give reasonable predictions for the most unstable growth rate and wavelength. It is also shown that the growing instability in a fully nonlinear model can be interpreted as boundary-trapped Rossby wave interactions—that is, wave phase locking and westward phase tilt allow waves to be mutually amplified. The analyses demonstrate that both the boundary dissipative and bottom slope effects can be represented by vertical velocities at the lower boundary of the unstable interior, via inducing Ekman pumping and slope-parallel flow, respectively, as proposed by the theories of Williams and Robinson (1974; referred to as the Eady–Ekman problem) and Blumsack and Gierasch (1972). The vertical velocities, characterized by a friction parameter and a slope ratio, modify the bottom wave and thus the scale selection. However, the theories have inherent quantitative limitations. Eady–Ekman neglects boundary layer responses that limit the increase of bottom stress, thereby overestimating the Ekman pumping and growth rate reduction at large drag. Blumsack and Gierasch’s (1972) model ignores slope-induced horizontal shear in the mean flow that tilts the eddies to favor converting energy back to the mean, thus having limited utility over steep slopes.

On the non-separable baroclinic parallel flow instability problem

1970 ◽  
Vol 40 (2) ◽  
pp. 273-306 ◽  
Author(s):  
Michael E. McIntyre
Keyword(s):  
Flow Instability ◽  
Baroclinic Instability ◽  
Short Wave ◽  
Perturbation Series ◽  
Rate Of Change ◽  
Horizontal Shear ◽  
Mean Flow ◽  
Wave Regime ◽  
Parallel Flows ◽  
The Mean

Perturbation series are developed and mathematically justified, using a straightforward perturbation formalism (that is more widely applicable than those given in standard textbooks), for the case of the two-dimensional inviscid Orr-Sommerfeld-like eigenvalue problem describing quasi-geostrophic wave instabilities of parallel flows in rotating stratified fluids.The results are first used to examine the instability properties of the perturbed Eady problem, in which the zonal velocity profile has the formu=z+ μu1(y,z) where, formally, μ [Lt ] 1. The connexion between baroclinic instability theories with and without short wave cutoffs is clarified. In particular, it is established rigorously that there is instability at short wavelengths in all cases for which such instability would be expected from the ‘critical layer’ argument of Bretherton. (Therefore the apparently conflicting results obtained earlier by Pedlosky are in error.)For the class of profiles of formu=z+ μu1(y) it is then shown from an examination of theO(μ) eigenfunction correction why, under certain conditions, growing baroclinic waves will always produce a counter-gradient horizontal eddy flux of zonal momentum tending to reinforce the horizontal shear of such profiles. Finally, by computing a sufficient number of the higher corrections, this first-order result is shown to remain true, and its relationship to the actual rate of change of the mean flow is also displayed, for a particular jet-like form of profile withfinitehorizontal shear. The latter detailed results may help to explain at least one interesting feature of the mean flow found in a recent numerical solution for the wave régime in a heated rotating annulus.

On the Thickness Ratio in the Quasigeostrophic Two-Layer Model of Baroclinic Instability

2013 ◽  
Vol 70 (5) ◽  
pp. 1505-1511 ◽  
Author(s):  
Noboru Nakamura ◽  
Lei Wang
Keyword(s):  
Growth Rate ◽  
Vertical Structure ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Thickness Ratio ◽  
Layer Model ◽  
Optimal Thickness ◽  
Zonal Wavenumber ◽  
The Mean ◽  
Two Layer Model

Abstract It is shown that the classical quasigeostrophic two-layer model of baroclinic instability possesses an optimal ratio of layer thicknesses that maximizes the growth rate, given the basic-state shear (thermal wind), beta, and the mean Rossby radius. This ratio is interpreted as the vertical structure of the most unstable mode. For positive shear and beta, the optimal thickness of the lower layer approaches the midheight of the model in the limit of strong criticality (shear/beta) but it is proportional to criticality in the opposite limit. For a set of parameters typical of the earth’s midlatitudes, the growth rate maximizes at a lower-layer thickness substantially less than the midheight and at a correspondingly larger zonal wavenumber. It is demonstrated that a turbulent baroclinic jet whose statistical steady state is marginally critical when run with equal layer thicknesses can remain highly supercritical when run with a nearly optimal thickness ratio.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

scholarly journals Moisture Modes and the Eastward Propagation of the MJO

2013 ◽  
Vol 70 (1) ◽  
pp. 187-192 ◽  
Author(s):  
Adam Sobel ◽  
Eric Maloney
Keyword(s):  
Growth Rate ◽  
Moisture Gradient ◽  
Moist Static Energy ◽  
Mean Flow ◽  
Considerable Uncertainty ◽  
Low Level ◽  
Zonal Advection ◽  
The Mean ◽  
Frictional Convergence ◽  
Static Energy

Abstract The authors discuss modifications to a simple linear model of intraseasonal moisture modes. Wind–evaporation feedbacks were shown in an earlier study to induce westward propagation in an eastward mean low-level flow in this model. Here additional processes, which provide effective sources of moist static energy to the disturbances and which also depend on the low-level wind, are considered. Several processes can act as positive sources in perturbation easterlies: zonal advection (if the mean zonal moisture gradient is eastward), modulation of synoptic eddy drying by the MJO-scale wind perturbations, and frictional convergence. If the sum of these is stronger than the wind–evaporation feedback—as observations suggest may be the case, though with considerable uncertainty—the model produces unstable modes that propagate weakly eastward relative to the mean flow. With a small amount of horizontal diffusion or other scale-selective damping, the growth rate is greatest at the largest horizontal scales and decreases monotonically with wavenumber.

scholarly journals How Does the Vertical Profile of Baroclinicity Affect the Wave Instability?

2011 ◽  
Vol 68 (4) ◽  
pp. 863-877 ◽  
Author(s):  
Toshiki Iwasaki ◽  
Chihiro Kodama
Keyword(s):  
Growth Rate ◽  
Kinetic Energy ◽  
Wave Energy ◽  
Vertical Profile ◽  
Baroclinic Instability ◽  
Zonal Flow ◽  
Instability Waves ◽  
P Flux ◽  
The Mean ◽  
Mean Kinetic Energy

Abstract The growth rate of baroclinic instability waves is generalized in terms of wave–mean flow interactions, with an emphasis on the influence of the vertical profile of baroclinicity. The wave energy is converted from the zonal mean kinetic energy and the growth rate is proportional to the mean zonal flow difference between the Eliassen–Palm (E-P) flux convergence and divergence areas. Mass-weighted isentropic zonal means facilitate the expression of the lower boundary conditions for the mass streamfunctions and E-P flux. For Eady waves, intersections of isentropes with lower/upper boundaries induce the E-P flux divergence/convergence. The growth rate is proportional to the mean zonal flow difference between the two boundaries, indicating that baroclinicity at each level contributes evenly to the instability. The reduced zonal mean kinetic energy is compensated by a conversion from the zonal mean available potential energy. Aquaplanet experiments are carried out to investigate the actual characteristics of baroclinic instability waves. The wave activity is shown to be sensitive to the upper-tropospheric baroclinicity, though it may be most sensitive to baroclinicity near 800 hPa, which is the maximal level of the E-P flux. The local wave energy generation rate suggests that the increased upper-tropospheric zonal flow directly enhances the upper-tropospheric wave energy at the midlatitudes. Note that the actual baroclinic instability waves accompany a considerable amount of the equatorward E-P flux, which causes extinction of wave energy in the subtropical upper troposphere.

On the mean structure of sharp-fin-induced shock wave/turbulent boundary layer interactions over a cylindrical surface

2019 ◽  
Vol 865 ◽  
pp. 212-246 ◽  
Author(s):  
J. D. Pickles ◽  
B. R. Mettu ◽  
P. K. Subbareddy ◽  
V. Narayanaswamy
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Growth Rate ◽  
Cylindrical Surface ◽  
Separated Flow ◽  
Oblique Shock Wave ◽  
Mean Flow ◽  
Downstream Distance ◽  
Separation Shock ◽  
The Mean

Interactions between an oblique shock wave generated by a sharp fin placed on a cylindrical surface and the incoming boundary layer are investigated to unravel the mean features of the resulting shock/boundary layer interaction (SBLI) unit. This fin-on-cylinder SBLI unit has several unique features caused by the three-dimensional (3-D) relief offered by the cylindrical surface that noticeably alter the shock structure. Complementary experimental and computational studies are made to delineate both the surface and off-body flow features of the fin-on-cylinder SBLI unit and to obtain a detailed understanding of the mechanisms that dictate the mean flow and wall pressure features of the SBLI unit. Results show that the fin-on-cylinder SBLI exhibits substantial deviation from quasi-conical symmetry that is observed in planar fin SBLI. Furthermore, the separated flow growth rate appears to decrease with downstream distance and the separation size is consistently smaller than the planar fin SBLI with the same inflow and fin configurations. The causes for the observed diminution of the separated flow and its downstream growth rate were investigated in the light of changes caused by the cylinder curvature on the inviscid as well as separation shock. It was found that the inviscid shock gets progressively weakened in the region close to the triple point with downstream distance due to the 3-D relief effect from cylinder curvature. This weakening of the inviscid shock feeds into the separation shock, which is also independently impacted by the 3-D relief, to result in the observed modifications in the fin-on-cylinder SBLI unit.

scholarly journals The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

2018 ◽  
Vol 48 (11) ◽  
pp. 2779-2797 ◽  
Author(s):  
Megan A. Stamper ◽  
John R. Taylor ◽  
Baylor Fox-Kemper
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Initial Conditions ◽  
Boussinesq Equations ◽  
Small Region ◽  
Computational Domain ◽  
Mean Flow ◽  
The Mean

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

Linear stability and energetics of rotating radial horizontal convection

2016 ◽  
Vol 795 ◽  
pp. 1-35 ◽  
Author(s):  
Gregory J. Sheard ◽  
Wisam K. Hussam ◽  
Tzekih Tsai
Keyword(s):  
Boundary Layer ◽  
Potential Energy ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Baroclinic Instability ◽  
Mean Flow ◽  
Thermal Boundary Layer Thickness ◽  
Available Potential Energy ◽  
Horizontal Convection ◽  
The Mean

The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number $\mathit{Ra}=10^{9}$ and rotation parameters up to $Q=60$ (where $Q$ is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number $\mathit{Pr}=6.14$ representative of water and enclosure height-to-radius ratio $H/R=0.4$. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at $Q\approx 1$ in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in $Q$ sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest $Q$ the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with $Q$. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over $5\lesssim Q\lesssim 30$, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

Ageostrophic instabilities of fronts in a channel in a stratified rotating fluid

2009 ◽  
Vol 627 ◽  
pp. 485-507 ◽  
Author(s):  
J. GULA ◽  
R. PLOUGONVEN ◽  
V. ZEITLIN
Keyword(s):  
Baroclinic Instability ◽  
Water Model ◽  
Small Scale ◽  
Rotating Fluid ◽  
Shallow Water Model ◽  
Mean Flow ◽  
Nonlinear Saturation ◽  
The Mean ◽  
Rotating Shallow Water ◽  
Stability Results

It is known that for finite Rossby numbers geostrophically balanced flows develop specific ageostrophic instabilities. We undertake a detailed study of the Rossby–Kelvin (RK) instability, previously studied by Sakai (J. Fluid Mech., vol. 202, 1989, pp. 149–176) in a two-layer rotating shallow-water model. First, we benchmark our method by reproducing the linear stability results obtained by Sakai (1989) and extend them to more general configurations. Second, in order to determine the relevance of RK instability in more realistic flows, simulations of the evolution of a front in a continuously stratified fluid are carried out. They confirm the presence of RK instability with characteristics comparable to those found in the two-layer case. Finally, these simulations are used to study the nonlinear saturation of the RK modes. It is shown that saturation is achieved through the development of small-scale instabilities along the front which modify the mean flow so as to stabilize the RK mode. Remarkably, the developing instability leads to conversion of kinetic energy of the basic flow to potential energy, contrary to classical baroclinic instability.

The effects of turbulence on the mean flow past square rods

1983 ◽  
Vol 137 ◽  
pp. 331-345 ◽  
Author(s):  
Y. Nakamura ◽  
Y. Ohya
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Cross Section ◽  
Large Scale ◽  
Small Scale ◽  
Mean Flow ◽  
Flow Past ◽  
The Mean ◽  
Scale Turbulence ◽  
Main Effects

There are two main effects of turbulence on the mean flow past rods of square cross-section aligned with the approaching flow. Small-scale turbulence increases the growth rate of the shear layer, while large-scale turbulence enhances the roll-up of the shear layer. The consequences of these depend on the length of a square rod. The mean base pressure of a square rod varies considerably with turbulence intensity and scale as well as with its length.

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