Evolution of a turbulent cloud under rotation

2014 ◽  
Vol 756 ◽  
pp. 488-509 ◽  
Author(s):  
A. Ranjan ◽  
P. A. Davidson
Keyword(s):  
Rotation Axis ◽  
Spatial Filter ◽  
Dimensionless Time ◽  
Inertial Waves ◽  
Lagrangian Particle ◽  
Rotating Turbulence ◽  
Negative Helicity ◽  
The Waves ◽  
Group Velocities ◽  
Quiescent Fluid

AbstractLocalized patches of turbulence frequently occur in geophysics, such as in the atmosphere and oceans. The effect of rotation, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\boldsymbol{\Omega}$, on such a region (a ‘turbulent cloud’) is governed by inhomogeneous dynamics. In contrast, most investigations of rotating turbulence deal with the homogeneous case, although inhomogeneous turbulence is more common in practice. In this paper, we describe the results of $512^3$ direct numerical simulations (DNS) of a turbulent cloud under rotation at three Rossby numbers ($\mathit{Ro}$), namely 0.1, 0.3 and 0.5. Using a spatial filter, fully developed homogeneous turbulence is vertically confined to the centre of a periodic box before the rotation is turned on. Energy isosurfaces show that columnar structures emerge from the cloud and grow into the adjacent quiescent fluid. Helicity is used as a diagnostic and confirms that these structures are formed by inertial waves. In particular, it is observed that structures growing parallel to the rotation axis (upwards) have negative helicity and those moving antiparallel (downwards) to the axis have positive helicity, a characteristic typical of inertial waves. Two-dimensional energy spectra of horizontal wavenumbers, $k_{\perp }$, versus dimensionless time, $2 \varOmega t$, confirm that these columnar structures are wavepackets which travel at the group velocities of inertial waves. The kinetic energy transferred from the turbulent cloud to the waves is estimated using Lagrangian particle tracking to distinguish between turbulent and ‘wave-only’ regions of space. The amount of energy transferred to waves is 40 % of the initial at $\mathit{Ro}=0.1$, while it is 16 % at $\mathit{Ro}=0.5$. In both cases the bulk of the energy eventually resides in the waves. It is evident from this observation that inertial waves can carry a significant portion of the energy away from a localized turbulent source and are therefore an efficient mechanism of energy dispersion.

scholarly journals A physical conjecture for the dipolar–multipolar dynamo transition

2019 ◽  
Vol 874 ◽  
pp. 995-1020 ◽  
Author(s):  
B. R. McDermott ◽  
P. A. Davidson
Keyword(s):  
Numerical Simulations ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Low Frequency ◽  
Rossby Number ◽  
Length Scale ◽  
Inertial Waves ◽  
Rotation Rates ◽  
Rotating Turbulence ◽  
Planetary Dynamos

In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of ${\sim}0.4$. Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value $Ro^{crit}\sim 0.4$, consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos ($Ro^{crit}\sim 0.1$), is a result of a different choice of the length scale used to define the local $Ro$. We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at $Ro^{crit}\sim 0.5$. Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for $Ro>Ro^{crit}$.

scholarly journals The spin-up of a linearly stratified fluid in a sliced, circular cylinder

2016 ◽  
Vol 806 ◽  
pp. 254-303
Author(s):  
R. J. Munro ◽  
M. R. Foster
Keyword(s):  
Circular Cylinder ◽  
Rossby Waves ◽  
Rotation Axis ◽  
A Priori ◽  
Propagation Speed ◽  
Stratified Fluid ◽  
Decay Rates ◽  
The Waves ◽  
Western Half ◽  
Good Agreement

A linearly stratified fluid contained in a circular cylinder with a linearly sloped base, whose axis is aligned with the rotation axis, is spun-up from a rotation rate $\unicode[STIX]{x1D6FA}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}$ to $\unicode[STIX]{x1D6FA}$ (with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}\ll \unicode[STIX]{x1D6FA}$) by Rossby waves propagating across the container. Experimental results presented here, however, show that if the Burger number $S$ is not small, then that spin-up looks quite different from that reported by Pedlosky & Greenspan (J. Fluid Mech., vol. 27, 1967, pp. 291–304) for $S=0$. That is particularly so if the Burger number is large, since the Rossby waves are then confined to a region of height $S^{-1/2}$ above the sloped base. Axial vortices, ubiquitous features even at tiny Rossby numbers of spin-up in containers with vertical corners (see van Heijst et al.Phys. Fluids A, vol. 2, 1990, pp. 150–159 and Munro & Foster Phys. Fluids, vol. 26, 2014, 026603, for example), are less prominent here, forming at locations that are not obvious a priori, but in the ‘western half’ of the container only, and confined to the bottom $S^{-1/2}$ region. Both decay rates from friction at top and bottom walls and the propagation speed of the waves are found to increase with $S$ as well. An asymptotic theory for Rossby numbers that are not too large shows good agreement with many features seen in the experiments. The full frequency spectrum and decay rates for these waves are discussed, again for large $S$, and vertical vortices are found to occur only for Rossby numbers comparable to $E^{1/2}$, where $E$ is the Ekman number. Symmetry anomalies in the observations are determined by analysis to be due to second-order corrections to the lower-wall boundary condition.

The generation of waves by wind

1952 ◽  
Vol 215 (1122) ◽  
pp. 299-328 ◽  
Keyword(s):  
Ocean Waves ◽  
Irish Sea ◽  
North Coast ◽  
Empirical Relationships ◽  
The North ◽  
Component Wave ◽  
Wave Trains ◽  
The Irish Sea ◽  
The Waves ◽  
Group Velocities

This paper describes an investigation of the height and length of ocean waves and swell in relation to the strength, extent and duration of the wind in the generating area, and the subsequent travel of the swell through calm and disturbed water. The investigation is based on records of waves made on the north coast of Cornwall, in the Irish Sea and in Lough Neagh. It is a practical continuation of the work of Barber & Ursell (1948), who showed that the waves leaving the generating area behave as a continuous spectrum of component wave trains which travel independently with the group velocities appropriate to their periods. The spectral distribution of energy in the storm area is considered, and the relative amplitudes of the different components are deduced empirically under various wind conditions. The results indicate that the wave characteristics become practically independent of fetch after 200 to 300 miles, and that in the equilibrium condition the steepness of the highest waves is inversely proportional to the square root of the wind speed. Some theoretical foundation can be found for the form of the empirical relationships if it is assumed that the wind acts on each wave component independently, and that the sheltering coefficient used by Jeffreys is proportional to the wave steepness. The results provide a basis for making reasonably accurate predictions of waves and swell from meteorological charts and forecasts.

Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids

2001 ◽  
Vol 437 ◽  
pp. 13-28 ◽  
Author(s):  
LEO R. M. MAAS
Keyword(s):  
Wave Energy ◽  
Rotation Axis ◽  
Outer Core ◽  
Internal Gravity Waves ◽  
Inertial Waves ◽  
Mean Flow ◽  
Rotating Fluids ◽  
Wave Focusing ◽  
Wave Ray ◽  
Equatorial Current

Rotating fluids support waves. These inertial waves propagate obliquely through the fluid, with an angle that is fixed with respect to the rotation axis. Upon reflection, their wavelength is unchanged only when the wall obeys the local reflectional symmetry, that is, when it is either parallel or perpendicular to the rotation axis. For internal gravity waves in a density-stratified fluid, sloping boundaries thus break the symmetry of ray paths, in a two-dimensional container, predicting their focusing upon attractors: particular paths onto which the wave rays, and hence the energy, converge, and to which the wave energy returns after a small number of refections. Laboratory observations, presented here, show that, despite the intrinsic three-dimensionality of inertial waves, attractors still occur. The intensified wave energy on the attractor encourages centrifugal instabilities, leading to a mean flow. Evidence of this comes from dye spreading, observed to develop most rapidly over the location where the attractor reflects from the sloping wall, being the place where focusing and instabilities occur. This mean flow, resulting from the mixing of angular momentum, accompanying the intensification of the wave field at that location, has geophysical implications, because the ocean, atmosphere and Earth's liquid outer core can be regarded as asymmetrically contained. The relevance of wave focusing in a rotating, spherical shell, the modifications due to the addition of radial stratification, and its implications for observed equatorial current patterns and inertial oscillations are discussed. The well-known universality of oceanic, gravito-inertial wave spectra might reflect complementary, divergent (chaotic) wave-ray behaviour, which occurs in containers obeying the reflectional symmetry, but in which symmetry is broken in the horizontal plane. Periodic orbits still exist, but now repell.

Flows produced by the combined oscillatory rotation and translation of a circular cylinder in a quiescent fluid

2014 ◽  
Vol 764 ◽  
pp. 148-170 ◽  
Author(s):  
Christopher Koehler ◽  
Philip Beran ◽  
Marcos Vanella ◽  
Elias Balaras
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Bluff Body ◽  
Stokes Equations ◽  
Time Windows ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Immersed Boundary ◽  
Dimensionless Time ◽  
Quiescent Fluid

AbstractFlows produced by a circular cylinder undergoing oscillatory rotation and translation in a quiescent fluid have been studied via direct numerical simulations. The incompressible Navier–Stokes equations were solved for large dimensionless time windows using an immersed boundary method with adaptive Cartesian grid refinement. Parametric studies were conducted in two dimensions on the Reynolds number, Keulegan–Carpenter number and phase shift. In addition to the previously reported net thrust case (Blackburn et al., Phys. Fluids, vol. 11, 1999, pp. 4–6), the study catalogued the appearance of several streaming jet regimes with varying deflection angles, deflected and horizontal vortex shedding regimes, and a double mirrored jet regime with varying inter-jet angles, as well as several chaotic cases. Visualizations are presented to clarify each observed flow regime and to illustrate the parameter space. Connections are drawn between these canonical bluff-body deflected wakes and a similar phenomenon observed in aerofoils oscillating at high reduced frequencies in a cross-flow. Also, the discovery of the streaming jet regimes with varying deflection angles opens the door for using these flows as a low-Reynolds-number propulsive mechanism requiring only a two-degree-of-freedom actuator. Simulation results suggest that the flow phenomena observed in two dimensions persist in three dimensions, despite spanwise fluctuations.

scholarly journals Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment

2015 ◽  
Vol 91 (4) ◽  
Author(s):  
Antoine Campagne ◽  
Basile Gallet ◽  
Frédéric Moisy ◽  
Pierre-Philippe Cortet
Keyword(s):  
Inertial Waves ◽  
Rotating Turbulence

The theory of waves travelling on the core in a swirling liquid

1951 ◽  
Vol 205 (1083) ◽  
pp. 530-540 ◽  
Keyword(s):  
Surface Tension ◽  
Cross Section ◽  
Centrifugal Force ◽  
High Speed ◽  
Critical Length ◽  
The Core ◽  
Angular Velocities ◽  
The Waves ◽  
Minimum Velocity ◽  
Group Velocities

An examination is made of waves moving under centrifugal force and surface tension along the oore in a swirling liquid. The waves may be of varicose form in which the cross-section of the core remains circular, or they may be helical, giving the core the shape of a multithreaded screw. The relation is obtained between the lengths of the waves and their axial and angular velocities; at a critical length the waves possess a minimum velocity. The group velocities are determined, and are shown to be negative under certain conditions. It is found that waves can exist which move so slowly that they should be readily visible although the core may be revolving at high speed.

scholarly journals Nonlinear Effects on the Precessional Instability in Magnetized Turbulence

Atmosphere ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 14 ◽  
Author(s):  
Abdelaziz Salhi ◽  
Amor Khlifi ◽  
Claude Cambon
Keyword(s):  
Magnetic Field ◽  
Solid Body ◽  
Magnetic Energy ◽  
Rotation Axis ◽  
Base Flow ◽  
Dimensionless Time ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Saturation Stage ◽  
The Impact

By means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid-body rotation around the vertical ( x 3 ) axis (with rate Ω ) and a plane shear (with rate S = 2 ε Ω ) viewed in a frame rotating (with rate Ω p = ε Ω ) about an axis normal to the plane of shear and to the solid-body rotation axis and under an imposed magnetic field that aligns with the solid-body rotation axis ( B ‖ Ω ) . While rotation rate and Poincaré number are fixed, Ω = 20 and ε = 0.17 , the B intensity was varied, B = 0.1 , 0.5 , and 2.5 , so that the Elsasser number is about Λ = 0.1 , 2.5 and 62.5 , respectively. At the final computational dimensionless time, S t = 2 ε Ω t = 67 , the Rossby number Ro is about 0.1 characterizing rapidly rotating flow. It is shown that the total (kinetic + magnetic) energy ( E ) , production rate ( P ) due the basic flow and dissipation rate ( D ) occur in two main phases associated with different flow topologies: (i) an exponential growth and (ii) nonlinear saturation during which these global quantities remain almost time independent with P ∼ D . The impact of a "strong" imposed magnetic field ( B = 2.5 ) on large scale structures at the saturation stage is reflected by the formation of structures that look like filaments and there is no dominance of horizontal motion over the vertical (along the solid-rotation axis) one. The comparison between the spectra of kinetic energy E ( κ ) ( k ⊥ ) , E ( κ ) ( k ⊥ , k ‖ = 1 , 2 ) and E κ ) ( k ⊥ , k ‖ = 0 ) at the saturation stage reveals that, at large horizontal scales, the major contribution to E ( κ ) ( k ⊥ ) does not come only from the mode k ‖ = 0 but also from the k ‖ = 1 mode which is the most energetic. Only at very large horizontal scales at which E ( κ ) ( k ⊥ ) ∼ E 2 D ( κ ) ( k ⊥ ) , the flow is almost two-dimensional. In the wavenumbers range 10 ≤ k ⊥ ≤ 40 , the spectra E ( κ ) ( k ⊥ ) and E ( κ ) ( k ⊥ , k ‖ = 0 ) respectively follow the scaling k ⊥ − 2 and k ⊥ − 3 . Unlike the velocity field the magnetic field remains strongly three-dimensional for all scales since E 2 D ( m ) ( k ⊥ ) ≪ E ( m ) ( k ⊥ ) . At the saturation stage, the Alfvén ratio between kinetic and magnetic energies behaves like k ‖ − 2 for B k ‖ / ( 2 ε Ω ) < 1 .

scholarly journals Could hydrodynamic Rossby waves explain the westward drift?

2018 ◽  
Vol 474 (2213) ◽  
pp. 20180119 ◽  
Author(s):  
O. P. Bardsley
Keyword(s):  
Rossby Waves ◽  
Rotation Axis ◽  
Outer Core ◽  
Momentum Equation ◽  
Energy Propagation ◽  
Initial Value ◽  
Core Mantle Boundary ◽  
Novel Theory ◽  
The Waves ◽  
Westward Drift

A novel theory for the origin of the westward drift of the Earth’s magnetic field is proposed, based upon the propagation of hydrodynamic Rossby waves in the liquid outer core. These waves have the obscure property that their crests always progress eastwards—but, for a certain subset, energy can nevertheless be transmitted westwards. In fact, this subset corresponds to sheet-like flow structures, extended in both the axial and radial directions, which are likely to be preferentially excited by convective upwellings in the Earth’s rapidly rotating outer core. To enable their analysis, the quasi-geostrophic (QG) approximation is employed, which assumes horizontal motions to be independent of distance along the rotation axis, yet accounts for variations in the container height (i.e. the slope of the core–mantle boundary). By projecting the momentum equation onto flows of a QG form, a general equation governing their evolution is derived, which is then adapted for the treatment of two initial value problems—in both Cartesian and spherical geometries—which demonstrate the preference for westward energy propagation by the waves in question. The merits of this mechanism as an explanation for westward drift are discussed.

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