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.

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.

scholarly journals Investigation of dominant traveling 10-day wave components using long-term MERRA-2 database

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Chunming Huang ◽  
Wei Li ◽  
Shaodong Zhang ◽  
Gang Chen ◽  
Kaiming Huang ◽  
...  
Keyword(s):  
Flow Instability ◽  
Transmission Condition ◽  
Wave Number ◽  
Mean Flow ◽  
Excited Wave ◽  
Propagating Waves ◽  
Zonal Wave Number ◽  
The Mean

AbstractThe eastward- and westward-traveling 10-day waves with zonal wavenumbers up to 6 from surface to the middle mesosphere during the recent 12 years from 2007 to 2018 are deduced from MERRA-2 data. On the basis of climatology study, the westward-propagating wave with zonal wave number 1 (W1) and eastward-propagating waves with zonal wave numbers 1 (E1) and 2 (E2) are identified as the dominant traveling ones. They are all active at mid- and high-latitudes above the troposphere and display notable month-to-month variations. The W1 and E2 waves are strong in the NH from December to March and in the SH from June to October, respectively, while the E1 wave is active in the SH from August to October and also in the NH from December to February. Further case study on E1 and E2 waves shows that their latitude–altitude structures are dependent on the transmission condition of the background atmosphere. The presence of these two waves in the stratosphere and mesosphere might have originated from the downward-propagating wave excited in the mesosphere by the mean flow instability, the upward-propagating wave from the troposphere, and/or in situ excited wave in the stratosphere. The two eastward waves can exert strong zonal forcing on the mean flow in the stratosphere and mesosphere in specific periods. Compared with E2 wave, the dramatic forcing from the E1 waves is located in the poleward regions.

scholarly journals Sediment Transport Processes during Barrier Island Inundation under Variations in Cross-Shore Geometry and Hydrodynamic Forcing

2019 ◽  
Vol 7 (7) ◽  
pp. 210
Author(s):  
Anita Engelstad ◽  
Gerben Ruessink ◽  
Piet Hoekstra ◽  
Maarten van der Vegt
Keyword(s):  
Sediment Transport ◽  
Morphological Changes ◽  
Barrier Island ◽  
Wave Model ◽  
Short Wave ◽  
Bedload Transport ◽  
Transport Processes ◽  
Mean Flow ◽  
Flow Transport ◽  
The Mean

Inundation of barrier islands can cause severe morphological changes, from the break-up of islands to sediment accretion. The response will depend on island geometry and hydrodynamic forcing. To explore this dependence, the non-hydrostatic wave model SWASH was used to investigate the relative importance of bedload transport processes, such as transport by mean flow, short- (0.05–1 Hz) and infragravity (0.005–0.05 Hz) waves during barrier island inundation for different island configurations and hydrodynamic conditions. The boundary conditions for the model are based on field observations on a Dutch barrier island. Model results indicate that waves dominate the sediment transport processes from outer surfzone until landwards of the island crest, either by transporting sediment directly or by providing sediment stirring for the mean flow transport. Transport by short waves was continuously landwards directed, while infragravity wave and mean flow transport was seaward or landward directed. Landward of the crest, sediment transport was mostly dominated by the mean flow. It was forced by the water level gradient, which determined the mean flow transport direction and magnitude in the inner surfzone and on the island top. Simulations suggest that short wave and mean flow transport are generally larger on steeper slopes, since wave energy dissipation is less and mean flow velocities are higher. The slope of the island top and the width of the island foremost affect the mean flow transport, while variations in inundation depth will additionally affect transport by short-wave acceleration skewness.

Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer

2010 ◽  
Vol 665 ◽  
pp. 209-237 ◽  
Author(s):  
J. GULA ◽  
V. ZEITLIN ◽  
F. BOUCHUT
Keyword(s):  
Shallow Water ◽  
Linear Stability ◽  
Nonlinear Evolution ◽  
Lower Layer ◽  
Short Wave ◽  
Finite Volume Scheme ◽  
Coastal Currents ◽  
Mean Flow ◽  
The Mean ◽  
Rotating Shallow Water

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linear stability problem for coastal currents and study the nonlinear evolution of the instabilities with the help of high-resolution direct numerical simulations. We show how nonlinear saturation of the ageostrophic instabilities leads to reorganization of the mean flow and emergence of coherent vortices. We follow the same lines as in Part 1 and, first, perform a complete linear stability analysis of the baroclinic coastal currents for various depths and density ratios. We then study the nonlinear evolution of the unstable modes with the help of the recent efficient two-layer generalization of the one-layer well-balanced finite-volume scheme for rotating shallow water equations, which allows the treatment of outcropping and loss of hyperbolicity associated with shear, Kelvin–Helmholtz type, instabilities. The previous single-layer results are recovered in the limit of large depth ratios. For depth ratios of order one, new baroclinic long-wave instabilities come into play due to the resonances among Rossby and frontal- or coastal-trapped waves. These instabilities saturate by forming coherent baroclinic vortices, and lead to a complete reorganization of the initial current. As in Part 1, Kelvin fronts play an important role in this process. For even smaller depth ratios, short-wave shear instabilities with large growth rates rapidly develop. We show that at the nonlinear stage they produce short-wave meanders with enhanced dissipation. However, they do not change, globally, the structure of the mean flow which undergoes secondary large-scale instabilities leading to coherent vortex formation and cutoff.

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.

scholarly journals Eastward-propagating planetary waves in the polar middle atmosphere

2021 ◽  
Vol 21 (23) ◽  
pp. 17495-17512
Author(s):  
Liang Tang ◽  
Sheng-Yang Gu ◽  
Xian-Kang Dou
Keyword(s):  
Middle Atmosphere ◽  
Flow Instability ◽  
Planetary Waves ◽  
Background Wind ◽  
Mean Flow ◽  
Zonal Wavenumber ◽  
Critical Layers ◽  
The Mean ◽  
Modern Era ◽  
Research Analysis

Abstract. According to Modern-Era Retrospective Research Analysis for Research and Applications (MERRA-2) temperature and wind datasets in 2019, this study presents the global variations in the eastward-propagating wavenumber 1 (E1), 2 (E2), 3 (E3) and 4 (E4) planetary waves (PWs) and their diagnostic results in the polar middle atmosphere. We clearly demonstrate the eastward wave modes exist during winter periods with westward background wind in both hemispheres. The maximum wave amplitudes in the Southern Hemisphere (SH) are slightly larger and lie lower than those in the Northern Hemisphere (NH). Moreover, the wave perturbations peak at lower latitudes with smaller amplitudes as the wavenumber increases. The period of the E1 mode varies between 3–5 d in both hemispheres, while the period of the E2 mode is slightly longer in the NH (∼ 48 h) than in the SH (∼ 40 h). The periods of the E3 are ∼ 30 h in both the SH and the NH, and the period of E4 is ∼ 24 h. Despite the shortening of wave periods with the increase in wavenumber, their mean phase speeds are relatively stable, ∼ 53, ∼ 58, ∼ 55 and ∼ 52 m/s at 70∘ latitudes for E1, E2, E3 and E4, respectively. The eastward PWs occur earlier with increasing zonal wavenumber, which agrees well with the seasonal variations in the critical layers generated by the background wind. Our diagnostic analysis also indicates that the mean flow instability in the upper stratosphere and upper mesosphere might contribute to the amplification of the eastward PWs.

scholarly journals The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation

2012 ◽  
Vol 42 (3) ◽  
pp. 459-475 ◽  
Author(s):  
Alain Colin de Verdière
Keyword(s):  
Kinetic Energy ◽  
Internal Waves ◽  
Relative Vorticity ◽  
Rotation Vector ◽  
Vertical Shear ◽  
Earth’S Rotation ◽  
Horizontal Shear ◽  
Mean Flow ◽  
Earth's Rotation ◽  
The Mean

Abstract The interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth’s rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth’s rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2Ω/N as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean–atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.

scholarly journals New Perspectives on the Generation and Maintenance of the Kuroshio Large Meander

2019 ◽  
Vol 49 (8) ◽  
pp. 2095-2113 ◽  
Author(s):  
Yang Yang ◽  
X. San Liang
Keyword(s):  
Baroclinic Instability ◽  
Mesoscale Eddy ◽  
Low Frequency ◽  
Analysis Tool ◽  
Mean Flow ◽  
Large Meander ◽  
Kuroshio Large Meander ◽  
The Mean ◽  
The Kuroshio ◽  
Canonical Transfer

AbstractThe internal dynamical processes underlying the Kuroshio large meander are investigated using a recently developed analysis tool, multiscale window transform (MWT), and the MWT-based canonical transfer theory. Oceanic fields are reconstructed on a low-frequency mean flow window, a mesoscale eddy window, and a high-frequency synoptic window with reference to the three typical path states south of Japan, that is, the typical large meander (tLM), nearshore non-large meander (nNLM), and offshore non-large meander (oNLM) path states. The interactions between the scale windows are quantitatively evaluated in terms of canonical transfer, which bears a Lie bracket form and conserves energy in the space of scale. In general, baroclinic (barotropic) instability is strengthened (weakened) during the tLM state. For the first time we found a spatially coherent inverse cascade of kinetic energy (KE) from the synoptic eddies to the slowly varying mean flow; it occupies the whole large meander region but exists only in the tLM state. By the time-varying multiscale energetics, a typical large meander is preceded by a strong influx of mesoscale eddy energy from upstream with a cyclonic eddy, which subsequently triggers a strong inverse KE cascade from the mesoscale window to the mean flow window to build up the KE reservoir for the meander. Synoptic frontal eddies are episodically intensified due to the baroclinic instability of the meander, but they immediately feed back to the mean flow window through inverse KE cascade. These results highlight the important role played by inverse KE cascades in generating and maintaining the Kuroshio large meander.

scholarly journals The transport of vorticity and heat through fluids in turbulent motion

1932 ◽  
Vol 135 (828) ◽  
pp. 685-702 ◽  
Keyword(s):  
Horizontal Plane ◽  
Vertical Axis ◽  
Rate Of Change ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Values ◽  
Plane Element ◽  
The Mean ◽  
Coefficient Of Viscosity ◽  
Change In Mean

In Reynolds’ well-known theory of turbulent flow the effect of turbulence on the mean flow of a fluid is conceived as the same as that of a system of stresses which, like those due to viscosity, may have tangential as well as normal components across any plane element. Taking the case of laminar mean flow, that is when the mean flow is, say, horizontal and constant in direction and magnitude at any given height, the components of stress over a horizontal plane at height z are F x and F y where F x = — ρ uw — , F y = — ρ vw — , and u , v , w are the components of turbulent velocity parallel to two horizontal axes x and y and the vertical axis z . The bar denotes that mean values have been taken over a large horizontal area and ρ is the density of the fluid. The stress F x , is therefore due to the existence of a correlation between u and w. In the extension of Reynolds’ theory due to Prandtl this correlation depends on the rate of change in mean velocity. In its most simplified form the theory may be expressed as follows. A portion of fluid possessing the mean velocity of a level z 0 may be conceived to move upwards to a layer of height z 0 + l preserving the mean velocity U 0 of the layer from which it originated. At this height it is conceived to mix with its surroundings. If l is small the mean velocity of this layer is U 0 + l d U/ dz , U being the mean velocity at height z , so that u = — l d U/ dz , and hence F x = ρ wl — d U/ dz . The quantity ρ wl — is therefore of the same dimensions as viscosity and in Prandtl’s theory it is treated as though it were in fact a coefficient of viscosity, though not necessarily as one which has the same value at all points in the field.

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