scholarly journals Nonlinear evolution of a baroclinic wave and imbalanced dissipation

2014 ◽  
Vol 756 ◽  
pp. 965-1006 ◽  
Author(s):  
Balasubramanya T. Nadiga
Keyword(s):  
Large Scale ◽  
Nonlinear Evolution ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Base Flow ◽  
Vertical Shear ◽  
Baroclinic Wave ◽  
Set Up ◽  
Secondary Instabilities

AbstractWe consider the nonlinear evolution of an unstable baroclinic wave in a regime of rotating stratified flow that is of relevance to interior circulation in the oceans and in the atmosphere: a regime characterized by small large-scale Rossby and Froude numbers, a small vertical to horizontal aspect ratio and no bounding horizontal surfaces. Using high-resolution simulations of the non-hydrostatic Boussinesq equations and companion integrations of the balanced quasi-geostrophic (QG) equations, we present evidence for a local route to dissipation of balanced energy directly through interior turbulent cascades. That is, analysis of simulations presented in this study suggest that a developing baroclinic instability can lead to secondary instabilities that can cascade a small fraction of the energy forward to unbalanced scales whereas the bulk of the energy is confined to large balanced scales. Mesoscale shear and strain resulting from the hydrostatic geostrophic baroclinic instability drive frontogenesis. The fronts in turn support ageostrophic secondary circulation and instabilities. These two processes acting together lead to a quick rise in dissipation rate which then reaches a peak and begins to fall slowly when frontogenesis slows down; eventually balanced and imbalanced modes decouple. A measurement of the dissipation of balanced energy by imbalanced processes reveals that it scales exponentially with Rossby number of the base flow. We expect that this scaling will hold more generally than for the specific set-up we consider given the fundamental nature of the dynamics involved. In other results, (a) a break is seen in the total energy (TE) spectrum at small scales: while a steep $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k^{-3}$ geostrophic scaling (where $k$ is the three-dimensional wavenumber) is seen at intermediate scales, the smaller scales display a shallower $k^{-5/3}$ scaling, reminiscent of the atmospheric spectra of Nastrom & Gage and (b) at the higher of the Rossby numbers considered a minimum is seen in the vertical shear spectrum, reminiscent of similar spectra obtained using in situ measurements.

Baroclinic instability of large-amplitude geostrophic flows

1993 ◽  
Vol 251 ◽  
pp. 501-514 ◽  
Author(s):  
E. S. Benilov
Keyword(s):  
Large Amplitude ◽  
Large Scale ◽  
Hamiltonian Structure ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Vertical Shear ◽  
Governing Equations ◽  
Zonal Flows ◽  
Wave Disturbances ◽  
Geostrophic Flows

This paper examines the large-scale dynamics of a layer of stratified fluid on the β-plane. A three-dimensional asymptotic system is derived which governs geostrophic flows with large displacement of isopycnal surfaces. This is then reduced to a two-dimensional set of equations which describe the interaction of a baroclinic ‘quasi-mode’ with arbitrary vertical profile and barotrophic motion. The baroclinic instability of large-amplitude zonal flows with vertical shear is studied within the framework of these equations. In the case where the displacement of isopycnal surfaces is small, the results obtained should overlap with the ‘traditional’ baroclinic instability of quasi-geostrophic (small-amplitude) flows. In order to compare the two types of instability, the quasi-geostrophic boundary-value problem is solved asymptotically for the case of long-wave disturbances and weak β-effect (the latter limit of quasi-geostrophic theory has not been considered previously). The instability that is found is linked to the Hamiltonian structure of the governing equations. The equations derived are generalized for the case of more than one baroclinic quasi-mode.

scholarly journals Nonlinear Equilibration of Baroclinic Instability: The Growth Rate Balance Model

2014 ◽  
Vol 44 (7) ◽  
pp. 1919-1940 ◽  
Author(s):  
T. Radko ◽  
D. Peixoto de Carvalho ◽  
J. Flanagan
Keyword(s):  
Numerical Simulations ◽  
Large Scale ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Vertical Shear ◽  
Balance Model ◽  
Mesoscale Variability ◽  
Beta Effect ◽  
Eddy Transport ◽  
Secondary Instabilities

Abstract A theoretical model is developed, which attempts to predict the lateral transport by mesoscale variability, generated and maintained by baroclinic instability of large-scale flows. The authors are particularly concerned by the role of secondary instabilities of primary baroclinically unstable modes in the saturation of their linear growth. Theory assumes that the fully developed equilibrium state is characterized by the comparable growth rates of primary and secondary instabilities. This assumption makes it possible to formulate an efficient algorithm for evaluating the equilibrium magnitude of mesoscale eddies as a function of the background parameters: vertical shear, stratification, beta effect, and bottom drag. The proposed technique is applied to two classical models of baroclinic instability—the Phillips two-layer model and the linearly stratified Eady model. Theory predicts that the eddy-driven lateral mixing rapidly intensifies with increasing shear and weakens when the beta effect is increased. The eddy transport is also sensitive to the stratification pattern, decreasing as the ratio of upper/lower layer depths in the Phillips model is decreased below unity. Theory is successfully tested by a series of direct numerical simulations that span a wide parameter range relevant for typical large-scale currents in the ocean. The spontaneous emergence of large-scale patterns induced by mesoscale variability, and their role in the cross-flow eddy transport, is examined using a suite of numerical simulations.

scholarly journals Vibration characteristics of triple-gear-rotor system in compressed air energy storage under variable torque load

2021 ◽  
Vol 104 (1) ◽  
pp. 003685042098705
Author(s):  
Xinran Wang ◽  
Yangli Zhu ◽  
Wen Li ◽  
Dongxu Hu ◽  
Xuehui Zhang ◽  
...  
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Element Model ◽  
Rotor System ◽  
Dynamic Responses ◽  
System Response ◽  
Rotating Speed ◽  
Torque Load ◽  
Set Up ◽  
Two Stages

This paper focuses on the effects of the off-design operation of CAES on the dynamic characteristics of the triple-gear-rotor system. A finite element model of the system is set up with unbalanced excitations, torque load excitations, and backlash which lead to variations of tooth contact status. An experiment is carried out to verify the accuracy of the mathematical model. The results show that when the system is subjected to large-scale torque load lifting at a high rotating speed, it has two stages of relatively strong periodicity when the torque load is light, and of chaotic when the torque load is heavy, with the transition between the two states being relatively quick and violent. The analysis of the three-dimensional acceleration spectrum and the meshing force shows that the variation in the meshing state and the fluctuation of the meshing force is the basic reasons for the variation in the system response with the torque load. In addition, the three rotors in the triple-gear-rotor system studied show a strong similarity in the meshing states and meshing force fluctuations, which result in the similarity in the dynamic responses of the three rotors.

scholarly journals Properties of Steady Geostrophic Turbulence with Isopycnal Outcropping

2012 ◽  
Vol 42 (1) ◽  
pp. 18-38 ◽  
Author(s):  
G. Roullet ◽  
J. C. McWilliams ◽  
X. Capet ◽  
M. J. Molemaker
Keyword(s):  
High Resolution ◽  
Large Scale ◽  
Mechanical Energy ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Primary Source ◽  
Eddy Diffusivity ◽  
Energy Cycle ◽  
Geostrophic Turbulence ◽  
Pv Gradient

Abstract High-resolution simulations of β-channel, zonal-jet, baroclinic turbulence with a three-dimensional quasigeostrophic (QG) model including surface potential vorticity (PV) are analyzed with emphasis on the competing role of interior and surface PV (associated with isopycnal outcropping). Two distinct regimes are considered: a Phillips case, where the PV gradient changes sign twice in the interior, and a Charney case, where the PV gradient changes sign in the interior and at the surface. The Phillips case is typical of the simplified turbulence test beds that have been widely used to investigate the effect of ocean eddies on ocean tracer distribution and fluxes. The Charney case shares many similarities with recent high-resolution primitive equation simulations. The main difference between the two regimes is indeed an energization of submesoscale turbulence near the surface. The energy cycle is analyzed in the (k, z) plane, where k is the horizontal wavenumber. In the two regimes, the large-scale buoyancy forcing is the primary source of mechanical energy. It sustains an energy cycle in which baroclinic instability converts more available potential energy (APE) to kinetic energy (KE) than the APE directly injected by the forcing. This is due to a conversion of KE to APE at the scale of arrest. All the KE is dissipated at the bottom at large scales, in the limit of infinite resolution and despite the submesoscales energizing in the Charney case. The eddy PV flux is largest at the scale of arrest in both cases. The eddy diffusivity is very smooth but highly nonuniform. The eddy-induced circulation acts to flatten the mean isopycnals in both cases.

The structure of weakly compressible grid-generated turbulence

2001 ◽  
Vol 432 ◽  
pp. 219-283 ◽  
Author(s):  
G. BRIASSULIS ◽  
J. H. AGUI ◽  
Y. ANDREOPOULOS
Keyword(s):  
Mach Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Three Dimensional ◽  
High Amplitude ◽  
Time Dependent ◽  
Wide Range ◽  
Dynamical Flow ◽  
Set Up ◽  
Decay Exponent

A decaying compressible nearly homogeneous and nearly isotropic grid-generated turbulent flow has been set up in a large scale shock tube research facility. Experiments have been performed using instrumentation with spatial resolution of the order of 7 to 26 Kolmogorov viscous length scales. A variety of turbulence-generating grids provided a wide range of turbulence scales with bulk flow Mach numbers ranging from 0.3 to 0.6 and turbulent Reynolds numbers up to 700. The decay of Mach number fluctuations was found to follow a power law similar to that describing the decay of incompressible isotropic turbulence. It was also found that the decay coefficient and the decay exponent decrease with increasing Mach number while the virtual origin increases with increasing Mach number. A possible mechanism responsible for these effects appears to be the inherently low growth rate of compressible shear layers emanating from the cylindrical rods of the grid. Measurements of the time-dependent, three dimensional vorticity vectors were attempted for the first time with a 12-wire miniature probe. This also allowed estimates of dilatation, compressible dissipation and dilatational stretching to be obtained. It was found that the fluctuations of these quantities increase with increasing mean Mach number of the flow. The time-dependent signals of enstrophy, vortex stretching/tilting vector and dilatational stretching vector were found to exhibit a rather strong intermittent behaviour which is characterized by high-amplitude bursts with values up to 8 times their r.m.s. within periods of less violent and longer lived events. Several of these bursts are evident in all the signals, suggesting the existence of a dynamical flow phenomenon as a common cause.

Three-dimensional instabilities in steady and unsteady non-parallel boundary layers, including effects of Tollmien-Schlichting disturbances and cross flow

1987 ◽  
Vol 409 (1837) ◽  
pp. 229-248 ◽  
Keyword(s):  
Boundary Layers ◽  
Large Scale ◽  
Three Dimensional ◽  
Steady Motion ◽  
Cross Flow ◽  
Parallel Flows ◽  
Roughness Elements ◽  
Tollmien Schlichting ◽  
Flow Effects ◽  
Secondary Instabilities

Three-dimensional (3D) linear stability properties are considered for steady and unsteady 2D or 3D boundary layers with significant non-parallelism present. Two main examples of such non-parallel flows whose stability is of interest are, firstly, steady motion, over roughness elements, in cross flow, or in large-scale separation and, secondly, unsteady 2D Tollmien-Schlichting (TS) motion, with its associated question of secondary instabilities. A high-frequency stability analysis is presented here. It is found that, for 2DTS or steady boundary layers, there is a swing in the direction of maximum TS spatial growth rate, from 0° for parallel flow towards 64.68° away from the free-stream direction, as the nonparallel flow effects increase. These effects then depend principally on, and indeed are proportional to, the local slope of the boundary-layer displacement. Cross flow can also have a profound impact on TS instabilities. Further implications for higher-amplitude and/or fasterscale disturbances, their secondary instability, and nonlinear interactions, are also discussed.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

Baroclinic Instability and Loss of Balance

2005 ◽  
Vol 35 (9) ◽  
pp. 1505-1517 ◽  
Author(s):  
M. Jeroen Molemaker ◽  
James C. McWilliams ◽  
Irad Yavneh
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Baroclinic Instability ◽  
Boussinesq Equations ◽  
Approximate Model ◽  
Slow Manifold ◽  
Energy Cascades ◽  
Centrifugal Instability ◽  
Forward Energy ◽  
Balanced Dynamics

Abstract Under the influences of stable density stratification and the earth’s rotation, large-scale flows in the ocean and atmosphere have a mainly balanced dynamics—sometimes called the slow manifold—in the sense that there are diagnostic hydrostatic and gradient-wind momentum balances that constrain the fluid acceleration. The nonlinear balance equations are a widely successful, approximate model for this regime, and mathematically explicit limits of their time integrability have been identified. It is hypothesized that these limits are indicative, at least approximately, of the transition from the larger-scale regime of inverse energy cascades by anisotropic flows to the smaller-scale regime of forward energy cascade to dissipation by more nearly isotropic flows and intermittently breaking inertia–gravity waves. This paper analyzes the particular example of an unbalanced instability of a balanced, horizontally uniform, vertically sheared current, as it occurs within the Boussinesq equations. This ageostrophic, anticyclonic, baroclinic instability is investigated with an emphasis on how it relates to the breakdown of balance in the neighborhood of loss of balanced integrability and on how its properties compare with other examples of ageostrophic anticyclonic instability of rotating, stratified, horizontally sheared currents. It is also compared with the more familiar types of instability for a vertically sheared current: balanced (geostrophic) baroclinic instability, centrifugal instability, and Kelvin–Helmholtz instability.

Small-Scale Models for Testing Masonry Structures

2010 ◽  
Vol 133-134 ◽  
pp. 497-502 ◽  
Author(s):  
Alvaro Quinonez ◽  
Jennifer Zessin ◽  
Aissata Nutzel ◽  
John Ochsendorf
Keyword(s):  
Large Scale ◽  
Low Cost ◽  
Three Dimensional ◽  
Model Testing ◽  
Scale Model ◽  
Small Scale ◽  
Material Strength ◽  
Masonry Structures ◽  
Scale Models ◽  
Set Up

Experiments may be used to verify numerical and analytical results, but large-scale model testing is associated with high costs and lengthy set-up times. In contrast, small-scale model testing is inexpensive, non-invasive, and easy to replicate over several trials. This paper proposes a new method of masonry model generation using three-dimensional printing technology. Small-scale models are created as an assemblage of individual blocks representing the original structure’s geometry and stereotomy. Two model domes are tested to collapse due to outward support displacements, and experimental data from these tests is compared with analytical predictions. Results of these experiments provide a strong understanding of the mechanics of actual masonry structures and can be used to demonstrate the structural capacity of masonry structures with extensive cracking. Challenges for this work, such as imperfections in the model geometry and construction problems, are also addressed. This experimental method can provide a low-cost alternative for the collapse analysis of complex masonry structures, the safety of which depends primarily on stability rather than material strength.

An Exact Thickness-Weighted Average Formulation of the Boussinesq Equations

2012 ◽  
Vol 42 (5) ◽  
pp. 692-707 ◽  
Author(s):  
William R. Young
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Weighted Average ◽  
Boussinesq Equations ◽  
Weighted Averaging ◽  
Buoyancy Frequency ◽  
Averaged Equations ◽  
Induced Velocity

Abstract The author shows that a systematic application of thickness-weighted averaging to the Boussinesq equations of motion results in averaged equations of motion written entirely in terms of the thickness-weighted velocity; that is, the unweighted average velocity and the eddy-induced velocity do not appear in the averaged equations of motion. This thickness-weighted average (TWA) formulation is identical to the unaveraged equations, apart from eddy forcing by the divergence of three-dimensional Eliassen–Palm (EP) vectors in the two horizontal momentum equations. These EP vectors are second order in eddy amplitude and, moreover, the EP divergences can be expressed in terms of the eddy flux of the Rossby–Ertel potential vorticity derived from the TWA equations of motion. That is, there is a fully nonlinear and three-dimensional generalization of the one- and two-dimensional identities found by Taylor and Bretherton. The only assumption required to obtain this exact TWA formulation is that the buoyancy field is stacked vertically; that is, that the buoyancy frequency is never zero. Thus, the TWA formulation applies to nonrotating stably stratified turbulent flows, as well as to large-scale rapidly rotating flows. Though the TWA formulation is obtained by working on the equations of motion in buoyancy coordinates, the averaged equations of motion can then be translated into Cartesian coordinates, which is the most useful representation for many purposes.

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