Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow

1987 ◽  
Vol 174 ◽  
pp. 313-326 ◽  
Author(s):  
A. C. Or ◽  
F. H. Busse
Keyword(s):  
Rayleigh Number ◽  
Rossby Waves ◽  
Period Doubling ◽  
Critical Value ◽  
Coriolis Parameter ◽  
Onset Of Convection ◽  
Cylindrical Annulus ◽  
Propagating Waves ◽  
Annular Layer ◽  
Doubling Sequence

The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η*. When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount. Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer. Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves. Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection.

A Balanced Approximation of the One-Layer Shallow-Water Equations on a Sphere

2009 ◽  
Vol 66 (6) ◽  
pp. 1735-1748 ◽  
Author(s):  
W. T. M. Verkley
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Rossby Waves ◽  
Coriolis Parameter ◽  
Beta Plane ◽  
Propagating Waves ◽  
Barotropic Vorticity Equation ◽  
Barotropic Vorticity ◽  
The One

Abstract A global version of the equivalent barotropic vorticity equation is derived for the one-layer shallow-water equations on a sphere. The equation has the same form as the corresponding beta plane version, but with one important difference: the stretching (Cressman) term in the expression of the potential vorticity retains its full dependence on f 2, where f is the Coriolis parameter. As a check of the resulting system, the dynamics of linear Rossby waves are considered. It is shown that these waves are rather accurate approximations of the westward-propagating waves of the second class of the original shallow-water equations. It is also concluded that for Rossby waves with short meridional wavelengths the factor f 2 in the stretching term can be replaced by the constant value f02, where f0 is the Coriolis parameter at ±45° latitude.

Convection in a rotating cylindrical annulus. Part 4. Modulations and transition to chaos at low Prandtl numbers

1997 ◽  
Vol 350 ◽  
pp. 209-229 ◽  
Author(s):  
J. HERRMANN ◽  
F. H. BUSSE
Keyword(s):  
Rossby Waves ◽  
Landau Equation ◽  
Neutral Curve ◽  
Onset Of Convection ◽  
Ginzburg Landau ◽  
Cylindrical Annulus ◽  
Chaotic Convection ◽  
Finite Distance ◽  
Prandtl Numbers ◽  
Rayleigh Numbers

Thermal Rossby waves driven by centrifugal buoyancy in a rotating cylindrical fluid gap become unstable right at the onset of convection when the Prandtl number is small. The Benjamin–Feir–Newell instability leads to modulated thermal Rossby waves which can also be described by a generalized Ginzburg–Landau equation. A resonance instability occurs at a finite distance in Rayleigh number from the neutral curve. It leads to two independent wave patterns propagating past each other and finally gives rise to vacillations of the amplitude of convection. Most of these features can be described to a good approximation by a system of three coupled amplitude equations. Time integrations based on a Galerkin expansion show transitions to chaotic convection at higher Rayleigh numbers.

Asymptotic theory of convection in a rotating, cylindrical annulus

1986 ◽  
Vol 173 ◽  
pp. 545-556 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Equatorial Plane ◽  
Asymptotic Theory ◽  
Rossby Waves ◽  
Numerical Results ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Rotation Rates ◽  
Cylindrical Annulus ◽  
Simple Equations ◽  
Rayleigh Numbers

The problem of convection in a rotating cylindrical annulus heated from the outside and cooled from the inside is considered in the limit of high rotation rates. The constraint of rotation enforces the two-dimensional character of the motion when the angle of inclination of the axisymmetric end surfaces with respect to the equatorial plane is small. Even when the angle of inclination is large only the dependences on the radial and the azimuthal coordinates need to be considered. The dependence on time at the onset of convection is similar to that of Rossby waves. But at higher Rayleigh numbers a transition to vacillating solutions occurs. In the limit of high rotation rates simple equations can be derived which permit the reproduction and extension of previous numerical results.

Stability of thermal convection in two superimposed miscible viscous fluids

2002 ◽  
Vol 471 ◽  
pp. 339-363 ◽  
Author(s):  
MICHAEL LE BARS ◽  
ANNE DAVAILLE
Keyword(s):  
Rayleigh Number ◽  
Linear Theory ◽  
Thermal Convection ◽  
Chemical Diffusion ◽  
Critical Value ◽  
Marginal Stability ◽  
Layer Depth ◽  
Onset Of Convection ◽  
Density Anomaly ◽  
Wide Range

The stability of two-layer thermal convection in high-Prandtl-number fluids is investigated using laboratory experiments and marginal stability analysis. The two fluids have different densities and viscosities but there is no surface tension and chemical diffusion at the interface is so slow that it is negligible. The density stratification is stable. A wide range of viscosity and layer depth ratios is studied. The onset of convection can be either stationary or oscillatory depending on the buoyancy number B, the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly: when B is lower than a critical value (a function of the viscosity and layer depth ratios), the oscillatory regime develops, with a deformed interface and convective patterns oscillating over the whole tank depth; when B is larger than this critical value, the stratified regime develops, with a flat interface and layers convecting separately. Experiments agree well with the marginal stability results. At low Rayleigh number, characteristic time and length scales are well-predicted by the linear theory. At higher Rayleigh number, the linear theory still determines which convective regime will start first, using local values of the Rayleigh and buoyancy numbers, and which regime will persist, using global values of these parameters.

scholarly journals Quasi-geostrophic approximation of anelastic convection

2014 ◽  
Vol 751 ◽  
pp. 216-227 ◽  
Author(s):  
Friedrich H. Busse ◽  
Radostin D. Simitev
Keyword(s):  
Compressible Fluid ◽  
Reasonable Agreement ◽  
Rossby Waves ◽  
Three Dimensional ◽  
Azimuthal Direction ◽  
Onset Of Convection ◽  
Cylindrical Annulus ◽  
Spherical Fluid ◽  
Density Scale ◽  
Onset Location

AbstractThe onset of convection in a rotating cylindrical annulus with parallel ends filled with a compressible fluid is studied in the anelastic approximation. Thermal Rossby waves propagating in the azimuthal direction are found as solutions. The analogy to the case of Boussinesq convection in the presence of conical end surfaces of the annular region is emphasised. As in the latter case, the results can be applied as an approximation for the description of the onset of anelastic convection in rotating spherical fluid shells. Reasonable agreement with three-dimensional numerical results published by Jones, Kuzanyan & Mitchell (J. Fluid Mech., vol. 634, 2009, pp. 291–319) for the latter problem is found. As in those results, the location of the onset of convection shifts outwards from the tangent cylinder with increasing number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}N_{\rho }$ of density scale heights until it reaches the equatorial boundary. A new result is that at a much higher number $N_{\rho }$ the onset location returns to the interior of the fluid shell.

Chaotic transitions of convection rolls in a rapidly rotating annulus

1994 ◽  
Vol 261 ◽  
pp. 1-19 ◽  
Author(s):  
A. C. Or
Keyword(s):  
Prandtl Number ◽  
Period Doubling ◽  
Mode Locking ◽  
Initial Value ◽  
Coriolis Parameter ◽  
Chaotic Flows ◽  
Cylindrical Annulus ◽  
High Prandtl Number ◽  
Prandtl Numbers ◽  
Convection Rolls

Drifting convection rolls in a rapidly rotating cylindrical annulus with conical endwalls exhibit different transitional modes to chaotic flows at different Prandtl numbers. Three transition sequences for Prandtl numbers 0.3, 1.0 and 7.0 are studied for a moderately large Coriolis parameter and a wavenumber near the critical value using an initial-value code. As the Rayleigh number increases, each transition sequence first leads to a vacillating flow, and then to an aperiodic flow, the route of which is Prandtl-number dependent. From the low Prandtl number to the high Prandtl number, the transitions take different routes of torus folding, period doubling, and mode-locking intermittency.

scholarly journals Conduction and convection heat transfer characteristics of water-based au nanofluids in a square cavity with differentially heated side walls subjected to constant temperatures

2014 ◽  
Vol 18 (suppl.1) ◽  
pp. 189-200 ◽  
Author(s):  
Primoz Ternik ◽  
Rebeka Rudolf
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Base Fluid ◽  
Volume Fraction ◽  
Heat Transfer Characteristics ◽  
Square Cavity ◽  
Onset Of Convection ◽  
Transfer Characteristics ◽  
Wide Range ◽  
Water Based

The present work deals with the natural convection in a square cavity filled with the water-based Au nanofluid. The cavity is heated on the vertical and cooled from the adjacent wall, while the other two horizontal walls are adiabatic. The governing differential equations have been solved by the standard finite volume method and the hydrodynamic and thermal fields were coupled together using the Boussinesq approximation. The main objective of this study is to investigate the influence of the nanoparticles? volume fraction on the heat transfer characteristics of Au nanofluids at the given base fluid?s (i.e. water) Rayleigh number. Accurate results are presented over a wide range of the base fluid Rayleigh number and the volume fraction of Au nanoparticles. It is shown that adding nanoparticles in a base fluid delays the onset of convection. Contrary to what is argued by many authors, we show by numerical simulations that the use of nanofluids can reduce the heat transfer rate instead of increasing it.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Waves, jets and vortices: Regime transitions in shallow water turbulence

2021 ◽  
Author(s):  
Nikos Bakas
Keyword(s):  
Shallow Water ◽  
Large Scale ◽  
Rossby Waves ◽  
Phase Speed ◽  
Critical Value ◽  
Turbulent Regime ◽  
Wave Regime ◽  
Frequency Spectra ◽  
Spectra Analysis ◽  
Zonal Jets

<p>Forced-dissipative beta-plane turbulence in a single-layer shallow-water fluid has been widely considered as a simplified model of planetary turbulence as it exhibits turbulence self-organization into large-scale structures such as robust zonal jets and strong vortices. In this study we perform a series of numerical simulations to analyze the characteristics of the emerging structures as a function of the planetary vorticity gradient and the deformation radius. We report four regimes that appear as the energy input rate ε of the random stirring that supports turbulence in the flow increases. A homogeneous turbulent regime for low values of ε, a regime in which large scale Rossby waves form abruptly when ε passes a critical value, a regime in which robust zonal jets coexist with weaker Rossby waves when ε passes a second critical value and a regime of strong materially coherent propagating vortices for large values of ε. The wave regime which is not predicted by standard cascade theories of turbulence anisotropization and the vortex regime are studied thoroughly. Wavenumber-frequency spectra analysis shows that the Rossby waves in the second regime remain phase coherent over long times. The coherent vortices are identified using the Lagrangian Averaged Deviation (LAVD) method. The statistics of the vortices (lifetime, radius, strength and speed) are reported as a function of the large scale parameters. We find that the strong vortices propagate zonally with a phase speed that is equal or larger than the long Rossby wave speed and advect the background turbulence leading to a non-dispersive line in the wavenumber-frequency spectra.</p>

scholarly journals Instability of coupled gravity-inertial-Rossby waves on a β-plane in solar system atmospheres

2009 ◽  
Vol 27 (11) ◽  
pp. 4221-4227 ◽  
Author(s):  
J. F. McKenzie
Keyword(s):  
Rossby Waves ◽  
Narrow Band ◽  
Boussinesq Approximation ◽  
The Other ◽  
The Earth ◽  
Propagating Waves ◽  
Inertial Wave ◽  
The One ◽  
Fifth Order ◽  
Critical Latitude

Abstract. This paper provides an analysis of the combined theory of gravity-inertial-Rossby waves on a β-plane in the Boussinesq approximation. The wave equation for the system is fifth order in space and time and demonstrates how gravity-inertial waves on the one hand are coupled to Rossby waves on the other through the combined effects of β, the stratification characterized by the Väisälä-Brunt frequency N, the Coriolis frequency f at a given latitude, and vertical propagation which permits buoyancy modes to interact with westward propagating Rossby waves. The corresponding dispersion equation shows that the frequency of a westward propagating gravity-inertial wave is reduced by the coupling, whereas the frequency of a Rossby wave is increased. If the coupling is sufficiently strong these two modes coalesce giving rise to an instability. The instability condition translates into a curve of critical latitude Θc versus effective equatorial rotational Mach number M, with the region below this curve exhibiting instability. "Supersonic" fast rotators are unstable in a narrow band of latitudes around the equator. For example Θc~12° for Jupiter. On the other hand slow "subsonic" rotators (e.g. Mercury, Venus and the Sun's Corona) are unstable at all latitudes except very close to the poles where the β effect vanishes. "Transonic" rotators, such as the Earth and Mars, exhibit instability within latitudes of 34° and 39°, respectively, around the Equator. Similar results pertain to Oceans. In the case of an Earth's Ocean of depth 4km say, purely westward propagating waves are unstable up to 26° about the Equator. The nonlinear evolution of this instability which feeds off rotational energy and gravitational buoyancy may play an important role in atmospheric dynamics.

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