Critical Azimuthal Wave Number Research Articles

Numerical investigation of atmospherelike flows in a spherical geometry.

Buoyancy-driven convection flows play a crucial role in global heat and momentum transport in atmosphere. Simplified planetary and stellar atmospheres can be described by a spherical gap geometry with special boundary conditions for the temperature. In the spherical gap, the dielectrophoretic effect is used to synthesize the radial gravity field. Lateral thermal boundary conditions are used to model solar radiation at the equator and at the poles. The temperature reaches a maximum value at the equator and becomes colder near the poles. In the case of a rotating gap, the influence of the Coriolis and centrifugal forces are taken into account. Different regimes of the two-dimensional steady basic flow are discussed in dependence on the Taylor number and Rayleigh number and for the radii ratio η=R_{in}/R_{out}, where R_{in},R_{out} are the radii of the inner and outer surfaces, respectively. Linear instability theory is used to study when the basic flow becomes unstable. The critical Rayleigh number at which the steady axisymmetric basic flow becomes time-dependently axisymmetric or three dimensional is found to be a function of the Taylor number. Furthermore, the critical azimuthal wave number m_{c}, which is responsible for the structure of the supercritical three-dimensional flow, and the critical frequency of the perturbation ω_{c} were found. The spatial location of the perturbation helps to understand the origin of the instability.

Oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids

A series of three-dimensional (3D) transient simulations are carried out to study the onset and features of oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids (Pr = 0.01) with unit aspect ratio (As). The critical conditions for different volume ratios on ground and under microgravity are determined. It turns out that the threshold of the second bifurcation is more sensitive to volume ratio than the first bifurcation. The thresholds in deformed liquid bridges are usually higher than those in cylindrical liquid bridges. The critical azimuthal wavenumber increases from 2 to 3 with volume ratio. Gravity affects the flow through three factors. Compared with the hydrostatic deformation and heating orientation, buoyancy is marginal to the flow transition. It only affects the threshold and the deviation caused by buoyancy is within 2%. The effects of gravity on the critical conditions are dependent on volume ratio. Except for the threshold, gravity may also affect the critical azimuthal wavenumber and the critical oscillation mode. 3D oscillatory flow regimes with two different oscillation modes are present. A new type of “Rocking” mode is discovered in the present study.

On three-dimensional flow structures and oscillatory instability due to free surface heat loss in half-floating zones under microgravity

Present work describes a numerical investigation concerning the response of oscillatory thermo-capillary convection to free surface heat loss in a cylindrical half floating zone of high Prandtl number liquid (Pr=67) under microgravity conditions. Unsteady and three-dimensional computations have been carried out in the liquid domain using the finite volume method (FVM) to reveal the oscillatory thermo-fluid dynamic regimes at the onset of instability. The influence of different free surface heat loss conditions on the bifurcation from the basic steady axisymmetric state has been discussed in terms of critical azimuthal wavenumber, m. The dynamic evolution of the temperature disturbances at supercritical conditions are characterized by ‘pulsating’ and ‘rotating’ modes of spatio-temporal convection. The development of counter-propagating hydrothermal waves in azimuthal direction is found to be followed by an even mode of convection characterised by m = 2. The liquid bridge system exhibiting the well-known transition from periodic (m=2) to quasi-periodic behaviour (m=1) with an increase in Biot number, Bi has been analysed from both spatial and temporal points of view. The existence of both symmetric and asymmetric modes of the supercritical state for different Bi are presented. The dominant fluid dynamic information of 3D oscillatory state are extracted from a specific set of disturbance temperature data by employing a data-driven computational technique ‘dynamic mode decomposition’ (DMD). Various neutral, stable and unstable dynamic modes responsible for bifurcation to the 3D supercritical state are depicted with the help of DMD spectrum and the frequency spectrum.

Thermocapillary instabilities in half zone liquid bridges of low Prandtl fluid with non-equal disks under microgravity

Thermocapillary convection and its stability are of particular importance during the floating zone crystal growth process. In this paper, three-dimensional (3D) transient numerical simulations are conducted to study the onset and general features of thermocapillary instabilities in half zone liquid bridges of low Prandtl fluid (Pr = 0.01) with non-equal disks under microgravity. The effect of disk radius ratio on the critical conditions is investigated. The results reveal that both the first critical Reynolds number Rec1 and the second critical Reynolds number Rec2 are sensitive to the disk radius ratio. However, for a liquid bridge with fixed length and volume, varying the disk radius ratio does not change the critical azimuthal wave number of the first bifurcation, as well as the critical oscillation mode of the second bifurcation. Characteristics of the 3D steady thermocapillary convection and the 3D oscillatory thermocapillary convection are presented. For the supercritical state at Re = 1.2Rec1 after the first bifurcation, the flow is 3D steady. The deviations between the amplitudes of azimuthal flow in liquid bridges with different disk radius ratios are insignificant. For the supercritical state at Re = 1.2Rec2 after the second bifurcation, the flow is 3D oscillatory. The oscillating frequency decreases with disk radius ratio. The oscillating frequency in a liquid bridge with a convex free surface is higher than that in one with a concave free surface.

Effects of radial distribution of entropy diffusivity on critical modes of anelastic thermal convection in rotating spherical shells

Linear stability analysis of anelastic thermal convection in a rotating spherical shell with entropy diffusivities varying in the radial direction is performed. The structures of critical convection are obtained in the cases of four different radial distributions of entropy diffusivity; (1) κ is constant, (2) κT0 is constant, (3) κρ0 is constant, and (4) κρ0T0 is constant, where κ is the entropy diffusivity, T0 is the temperature of basic state, and ρ0 is the density of basic state, respectively. The ratio of inner and outer radii, the Prandtl number, the polytropic index, and the density ratio are 0.35, 1, 2, and 5, respectively. The value of the Ekman number is 10-3 or 10-5. In the case of (1), where the setup is same as that of the anelastic dynamo benchmark (Jones et al., 2011), the structure of critical convection is concentrated near the outer boundary of the spherical shell around the equator. However, in the cases of (2), (3) and (4), the convection columns attach the inner boundary of the spherical shell.A rapidly rotating annulus model for anelastic systems is developed by assuming that convection structure is uniform in the axial direction taking into account the strong effect of Coriolis force. The annulus model well explains the characteristics of critical convection obtained numerically, such as critical azimuthal wavenumber, frequency, Rayleigh number, and the cylindrically radial location of convection columns.The radial distribution of entropy diffusivity, or more generally, diffusion properties in the entropy equation, is important for convection structure, because it determines the distribution of radial basic entropy gradient which is crucial for location of convection columns.

First results of the high-resolution multibeam ULF wave experiment at the Ekaterinburg SuperDARN radar: Ionospheric signatures of coupled poloidal Alfvén and drift-compressional modes

A continuous experiment was carried out at the Ekaterinburg (EKB) stereoradar of the Russian segment of SuperDARN in order to examine the spatio-temporal characteristics of radar-detected magnetospheric ULF waves. The study of magnetospheric oscillations is based on analysis of scattering from field-aligned F-layer irregularities. Their E×B drift Doppler velocity at F-layer heights is associated with the background electric field in the ionosphere. During the experiment one of the radar channels operates in 0–2 beam scanning, with an integration time of 6s, which corresponds to the total 18-s time resolution at each beam. This allows detecting magnetospheric ULF waves with periods of 40s and up. Beam 0 is along the 132 magnetic meridian, so the registered velocity oscillations correspond to the wave electric field azimuthal component. Operation of the radar in this mode was started in December 2013.The first ULF wave events observed in the experiment and presented here occurred on 14 December 2013 and 2 January 2014 in the nightside magnetosphere during two geomagnetic disturbances classified as small magnetic storms and associated with high speed streams from coronal holes. Both the ULF events occurred after substorm-like auroral disturbances. The ULF waves observed during these events are classified as Pc5 geomagnetic pulsations. Two oscillation branches were observed, the higher and the lower frequency ones. As the azimuthal wave numbers m increase, the branches converge and merge into a single oscillation branch at some critical azimuthal wave number value m⋆. This ω(m) dependence is characteristic of the coupled Alfvén and drift-compressional waves which according to theory merge if the azimuthal wave number exceeds some critical value. This merged single oscillation branch represents an unstable drift ballooning coupling mode. Thus, the following interpretation of the observed events can be suggested; at m<m⋆ the higher and lower frequency branches can be identified with the Alfvén and drift-compressional modes, respectively, and at m>m⋆ the single branch can be identified with the drift ballooning coupling mode.

Primary oscillatory instability in a rotating disk–cylinder system with aspect (height/radius) ratio varying from 0.1 to 1

Three-dimensional instability of axisymmetric flow in a rotating disk–cylinder configuration is studied numerically for the case of low cylinders with the height/radius aspect ratio varying between 1 and 0.1. A complete stability diagram for the transition from steady axisymmetric to oscillatory three-dimensional flow regime is reported. A good agreement with the experimental results is obtained. It is shown that the critical azimuthal wavenumber grows with the decrease of the aspect ratio, reaching the value of 19 at the aspect ratio 0.1. It is argued that the observed instability cannot be described as resulting from a boundary layer only. Other reasons that can destabilize the flow are discussed.

Linear Stability of Viscous Zonal Jet Flows on a Rotating Sphere

Linear stability of viscous zonal jet flows on a rotating sphere is investigated. This problem setting is similar to the Kolmogorov problem on the point that the basic solutions are expressed by a single eigenfunction of Laplacian of each manifold, a sphere and a torus. In the non-rotating case, as the number of jets increases the critical Reynolds number increases monotonically with the critical azimuthal wavenumber being \(m_{\text{c}}=2\). In the rotating cases, the zonal flows are stable for large rotation rates while the effect of small rotation does not always stabilize the zonal flows. We find that the critical rotation rate in the inviscid limit does not coincide with the inviscid one. This seeming contradiction between the inviscid limit stability and the inviscid stability is resolved by an observation that, as the Reynolds number increases, the growth rates of viscous unstable mode converge to zero in the regions where the inviscid zonal flow is stable while the viscous zonal flow is unstable.

Stability of Thermocapillary Convection in Rotating Shallow Annular Pool of Silicon Melt

In order to understand the effect of pool rotation on stability of thermocapillary convection, the critical conditions for the incipience of oscillatory flow in rotating shallow annular pools of silicon melt (Pr = 0.011) are investigated by means of linear stability analysis (LSA) under different pool rotation rates ranging from Ta = 0 to 1,513.9 and under microgravity condition. The results indicate that with increase of Ta numbers, the critical Marangoni for the incipience of hydrothermal wave (HTW) increases, i.e., pool rotation stabilizes the steady axisymmetric thermocapillary convection of silicon melt. The critical azimuthal wave number mc decreases linearly with increase of Ta number when Ta ≤ 378.5. However, when Ta > 378.5, mc almost keeps constant with mc = 34. When Ta 550.

Effect of volume ratio on thermocapillary flow in liquid bridges of high-Prandtl-number fluids

In present study, the transition of thermocapillary convection from the axisymmetric stationary flow to oscillatory flow in liquid bridges of 5cst silicon oil (aspect ratio 1.0 and 1.6) is investigated in microgravity conditions by the linear instability analysis. The corresponding marginal instability boundary is closely related to the gas/liquid configuration of the liquid bridge noted as volume ratio. With the increasing volume ratio, the marginal instability boundary consists of the increasing branch and the decreasing branch. A gap region exists between the branches where the critical Marangoni number of the corresponding axisymmetric stationary flow increases drastically. Particularly, a unique axisymmetric oscillatory flow (the critical azimuthal wave number is m=0 ) in the gap region is reported for the liquid bridge of aspect ratio 1.6. Moreover, the energy transfer between the basic state and the disturbance fields of the thermocapillary convection is analyzed at the corresponding critical Marangoni number, which reveals different major sources of the energy transfer for the development of the disturbances in regimes of the increasing branch, the gap region and the decreasing branch, respectively.

Influence of Buoyancy Force on Thermocapillary Convection Instability in the Differentially Heated Annular Pools of Silicon Melt

The influence of buoyancy force on the thermocapillary convection instability in the annular pools (Ri = 20 mm, Ro = 40 mm, and depth d ranging from 1 to 10 mm) of silicon melt (Pr = 0.011), differentially heated at the outer wall and cooled at the inner wall, is investigated numerically. The critical Marangoni numbers (Mac) for the incipience of oscillatory flow are determined by linear stability analysis (LSA) under both microgravity and normal gravity conditions. The results indicate that the buoyancy force destabilizes the thermocapillary convection under different liquid layer depths from 3 to 10 mm. With increasing the layer depth, the critical Ma number, critical azimuthal wave number and critical phase velocity decrease. Some of 3-D simulation results are compared with those of LSA. 3-D results are found consistent with the LSA results except for a case of D = 0.05 where 3-D simulation gives a stationary 3-D flow under a large Ma.

Azimuthal modulational instability of vortices in the nonlinear Schrödinger equation

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize such solutions.

On the onset of low-Prandtl-number convection in rotating spherical shells: non-slip boundary conditions

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E &lt; 10−5, the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ~ E−4/3, as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ, E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ≳0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability.

Effect of pool rotation on thermocapillary convection in shallow annular pool of silicone oil

The influence of pool rotation on the thermocapillary flow in a shallow annular pool ( R i=20 mm, R 0=40 mm, and depth d=1 mm) of silicone oil (0.65 cSt, Pr=6.7), heated from the outer wall and cooled at the inner wall, is investigated by numerical simulations and linear stability analysis (LSA). The increase in Ta from 0 to 0.322 and 0.806 destabilizes the basic steady axisymmetric flow field against hydrothermal waves (HTW) and the critical Marangoni number Ma c decreases from 8.396×10 3 to 8.096×10 3 and 7.999×10 3, while the critical azimuthal wave number m c increases from 27 to 30 and 33. In rotating annular pool, the HTW propagates in the direction opposite to the pool rotation, when observed from a view point in the rotating coordinate. At Ma=1.34×10 4, the azimuthal wave number m increases up to 54 and branching of spiral arms occurs near the inner wall. This branching phenomenon occurs only at Ma=1.34×10 4 and disappears at larger values of Ma. However, at Ma=2×10 4, two groups of HTWs ( m=48 and m=5) coexist and the HTW with m=5 propagates in the same azimuthal direction as that of the pool rotation. Both the numerical simulation and LSA results indicate that the pool rotation destabilizes the basic steady axisymmetric thermocapillary flow in the present Ta range ( Ta⩽0.806). The influence of the pool rotation on the instability of the thermocapillary flow is distinguishable even for such a small rotating rate of the shallow annular pool. This can be explained by the additional energy supply from the azimuthal flow field induced by the Coriolis force.

Three-dimensional numerical simulation of Marangoni instabilities in non-cylindrical liquid bridges in microgravity

Instability of Marangoni convection in non-cylindrical (convex or concave) liquid bridges of low Prandtl number fluids is investigated by direct three-dimensional and time-dependent simulation of the problem. Body-fitted curvilinear coordinates are adopted; the non-cylindrical original physical domain in the ( r, z, φ) space is transformed into a cylindrical computational domain in a ( ξ, η, φ) space. The geometry of the domain is transformed using a coordinate transformation method by surface fitting technique. The field equations are numerically solved explicitly in time and with a finite difference technique in a staggered grid. The numerical results are analyzed and interpreted in the general context of the bifurcation's theory. The computations show that for semiconductor melts the first bifurcation is characterized by the loss of spatial symmetry rather than by the onset of oscillatory flow and that it is hydrodynamic in nature. The flow field azimuthal organization related to the critical wave number, depends on the geometrical aspect ratio A= L/ D of the liquid bridge and on the shape factor S (convex S>1, concave S<1) of the free surface. The critical azimuthal wave number increases when the geometrical aspect ratio of the bridge is decreased and, for a fixed aspect ratio, can be shifted to higher values by increasing the volume (convex bridges) or to lower values by decreasing the volume (concave bridges). This behavior is explained on the basis of the relation between the typology of the azimuthal disturbances and the structure of the fluid-dynamic field. A generalized law is found to correlate the critical azimuthal wave number of the instability to the geometrical aspect ratio and to the shape factor. A second oscillatory (Hopf) bifurcation occurs when further increasing the Marangoni number. Experimental results available in literature on this second bifurcation are considered for comparison. The experimental and numerical results show a good agreement.

Influence of buoyancy forces on Marangoni flow instabilities in liquid bridges

The influence of buoyancy forces on oscillatory Marangoni flow in liquid bridges of different aspect ratio is investigated by three‐dimensional, time‐dependent numerical solutions and by laboratory experiments using a microscale apparatus and a thermographic visualisation system. Liquid bridges heated from above and from below are investigated. The numerical and experimental results show that for each aspect ratio and for both the heating conditions the onset of the Marangoni oscillatory flow is characterized by the appearance of a standing wave regime; after a certain time, a second transition to a travelling wave regime occurs. The three‐dimensional flow organization at the onset of instability is different according to whether the bridge is heated from above or from below. When the liquid bridge is heated from below, the critical Marangoni number is larger, the critical wave number (m) is smaller and the standing wave regime is more stable, compared with the case of the bridge heated from above. For the critical azimuthal wave number, two correlation laws are found as a function of the geometrical aspect ratio A.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

The influence of Ekman boundary layers on rotating convection

Abstract The influence of the boundary conditions on convective instability in rapidly (10−5 < E < 10−3) rotating spherical shells is investigated. For the purpose of comparison, both stress-free and rigid boundaries are studied at exactly the same parameter range of the problem. In the rigid boundary case an Ekman boundary layer is formed in middle latitudes where columnar convective rolls meet the outer spherical surface. There is also a boundary layer at the equatorial region of the inner sphere where the rolls are attached. When the Prandtl number Pr ≥ O(1), the Ekman boundary layer produced by the rigid boundaries has a destabilizing effect, the critical Reyleigh number being substantially reduced. The value of the critical azimuthal wavenumber and the drifting rate of the convection rolls are also significantly lower compared wilh the free boundary case. When the Prandtl number Pr < O(1), so thermal dissipation dominates over the viscous dissipation, the Ekman boundary layers exhibit a stabilizing e...