Hydromagnetic convective instability of a rotating, self-gravitating fluid sphere containing a uniform distribution of heat sources

1977 ◽  
Vol 353 (1673) ◽  
pp. 145-162 ◽  
Keyword(s):  
Cylindrical Shell ◽  
Uniform Distribution ◽  
Critical Rayleigh Number ◽  
Plane Layer ◽  
Fluid Sphere ◽  
Dynamo Model ◽  
Planar Geometry ◽  
Heat Sources ◽  
Curvature Effects ◽  
The Waves

The desire to understand better the magneto-hydrodynamics of the Earth's and planetary interiors has recently motivated a number of studies on convective motions in hydromagnetic rotating systems. These studies have, however, been restricted to planar geometry, the convective layer being confined between two horizontal planes in externally applied uniform gravitational and magnetic fields. This paper takes a step further to the geophysical and astrophysical contexts by restoring curvature effects. The linear stability of a uniformly rotating, self-gravitating fluid sphere in the presence of a co-rotating zonal magnetic field is studied when buoyancy is provided by a uniform distribution of heat sources. The analysis is limited to the case where the Chandrasekhar number, Q , and the Taylor number, λ 2 , are both large. (These are, respectively, dimension-less measures of Lorentz and Coriolis forces relative to the viscous forces.) It is shown that for all values of λ and Q the motions appearing at marginal convection are necessarily time-dependent and associated with a temperature fluctuation which is always symmetric with respect to the equatorial plane. The critical Rayleigh number R c ( λ, Q ), which is a dimensionless measure of the temperature contrast necessary for the onset of convection, is found to be qualitatively the same as for the planar model only when λ ≥ O(Q) , although even in this case certain characteristic curvature effects arise. The motions prevalent at marginal stability, when O ( Q 3/2 ) ≥ λ ≫ Q , occur in the form of a thin cylindrical shell of thickness O (( Q/λ ) 2/3 ) and whose distance from the axis of rotation varies between 0.4 and 0.6 spherical radii depending on the value of q , which is the ratio of the thermal to the magnetic diffusivities. The waves will drift westward or eastward according to whether q ≶ 2.5. (The cause of disagreement in this result with Busse (1975 b ) is explained in an appendix). For λ = O(Q) convection occurs in the whole volume of the sphere and the waves drift westward for all values of q . When λ ≪ Q , not only is R c incorrectly given by that for the plane layer model but also modal degeneracies of convection in the plane layer are removed by the curvature and boundedness of the system. For this range of λ and Q convection again fills the whole sphere but all forms of diffusion are concentrated in multiple boundary layers on the surface of sphere. The waves drift westward. The results are compared with parellel studies, including Braginsky's MAC waves (i. e. Hide's slow magnetohydrodynamic waves) and Busse's recent dynamo model. In particular, it is argued that the last of these may not be representative of planetary magnetism because of a convective growth of field (not considered by Busse) associated with convection patterns occuring in the whole sphere rather than in a cylindrical shell.

scholarly journals The effect of junctions on the propagation of electric waves along conductors

1913 ◽  
Vol 88 (601) ◽  
pp. 103-110
Keyword(s):  
Cylindrical Shell ◽  
Wave Length ◽  
General Character ◽  
Original Form ◽  
Annular Space ◽  
Approximate Methods ◽  
Positive Direction ◽  
Positive Side ◽  
The Waves ◽  
Electric Waves

Some interesting problems in electric wave propagation are suggested by an experiment of Hertz. In its original form waves of the simplest kind travel in the positive direction (fig. 1), outside an infinitely thin conducting cylindrical shell, AA, which comes to an end, say, at the plane z = 0. Co-axial with the cylinder a rod or wire BB (of less diameter) extends to infinity in both directions. The conductors being supposed perfect, it is required to determine the waves propagated onwards beyond the cylinder on the positive side of z , as well as those reflected back outside the cylinder and in the annular space between the cylinder and the rod. So stated, the problem, even if mathematically definite, is probably intractable; but if we modify it by introducing an external co-axial con­ducting sheath CC (fig. 2), extending to infinity in both directions, and if we further suppose that the diameter of this sheath is small in comparison with the wave-length (λ) of the vibrations, we shall bring it within the scope of approximate methods. It is under this limitation that I propose here to consider the present and a few analogous problems. Some considerations of a more general character are prefixed.

scholarly journals Dynamic instability of a compound nanocomposite shell

2021 ◽  
Vol 27 (5) ◽  
pp. 60-70
Author(s):  
N.H. Sakhno ◽  
◽  
K.V. Avramov ◽  
B.V. Uspensky ◽  
◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Cylindrical Shell ◽  
Supersonic Flow ◽  
Uniform Distribution ◽  
Critical Pressure ◽  
Natural Frequencies ◽  
Dynamic Instability ◽  
Element Analysis

Free oscillations and dynamic instability due to supersonic airflow pressure are investigated in a functional-gradient compound composite conical-cylindrical shell made of a carbon nanotubes-reinforced material. Nanocomposite materials with a linear distribution of the volumetric fraction of nanotubes over the thickness are considered. Extended mixture rule is used to estimate nanocomposite’s mechanical characteristics. A high-order shear deformation theory is used to represent the shell deformation. The assumed-mode technique, along with a Rayleigh-Ritz method, is applied to obtain the equations of the structure motion. To analyze the compound structure dynamics, a new system of piecewise basic functions is suggested. The pressure of a supersonic flow on the shell is obtained by using the piston theory. An example of the dynamic analysis of a nanocomposite conical-cylindrical shell in the supersonic gas flow is considered. The results of its modal analysis using the Rayleigh-Ritz technique are close to the natural frequencies of the shell obtained by finite element analysis. In this case, finite element analysis can only be used for shells made of material with a uniform distribution of nanotubes over the thickness. The dependence of the natural frequencies of a compound shell on the ratio of the lengths of the conical and cylindrical parts is studied. The dependence of the critical pressure of a supersonic flow on the Mach numbers and the type of carbon nanotubes reinforcement is investigated. Shells with a concentration of nanotubes predominantly near the outer and inner surfaces are characterized by higher values of natural frequencies and critical pressure than the shells with a uniform distribution of nanotubes or with a predominant concentration of nanotubes inside the shell.

Radial loading of a cylindrical shell through plate brackets

1977 ◽  
Vol 12 (3) ◽  
pp. 197-207 ◽  
Author(s):  
R Kitching ◽  
J F Hughes ◽  
F Saedi
Keyword(s):  
Cylindrical Shell ◽  
Theoretical Analysis ◽  
Uniform Distribution ◽  
Elastic Behaviour ◽  
Radial Load ◽  
Experimental Stress ◽  
Stress Distributions ◽  
Radial Loading

The elastic behaviour of a cylindrical shell subjected to radial load applied through two circumferential plate bracket attachments positioned diametrically opposite each other at the mid length of the cylinder has been investigated experimentally. An existing theoretical analysis, based on the assumption of a uniform distribution of load across a bracket, was modified in the light of the experimental observations which indicated that the assumption was not valid for the size of bracket used. Experimental stress distributions and maximum values of stress in the cylinder are compared with those calculated from the modified analysis in the text, which showed an improvement on the existing analysis for the proportions of the shell and bracket considered.

Random waves and dynamo action

1975 ◽  
Vol 69 (1) ◽  
pp. 145-177 ◽  
Author(s):  
A. M. Soward
Keyword(s):  
Magnetic Field ◽  
Wave Energy ◽  
Dynamo Model ◽  
Length Scale ◽  
Kinetic Energy Density ◽  
Random Waves ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
The Waves ◽  
Upper Boundary

The propagation of waves in an inviscid, electrically conducting fluid is considered. The fluid rotates with angular velocity Ω* and is permeated by a magnetic field b* which varies on the length scale L = Ql, where Q = Ω*l2/λ (l is the length scale of the waves, λ is the magnetic diffusivity) is assumed large (Q [Gt ] 1). A linearized theory is readily justified in the limit of zero Rossby number R0 (= U0/Ω*l, where U0 is a typical fluid velocity) and for this case it is shown that the total wave energy of a wave train is conserved and transported at the group velocity except for that which is lost by ohmic dissipation. The analysis is extended to encompass the propagation of a sea of random waves.A hydromagnetic dynamo model is considered in which the fluid is confined between two horizontal planes perpendicular to the rotation axis a distance L0(=O(L)) apart. Waves of given low frequency Ω*0 (= O(R0Q½Ω*)) and horizontal wavenumber l−1 but random orientation are excited at the lower boundary, where the kinetic energy density is 2πρU20. The waves are absorbed perfectly at the upper boundary, so that there is no reflexion. The linear wave energy equation remains valid in the double limit 1 [Gt ] R0Q½[Gt ]Q−½, for which it is shown that dynamo action is possible provided $\Delta = L_0U^2_0/l^3\omega_0^{*2} > 1 $. When dynamo action maintains a weak magnetic field (Δ −1 [Lt ] 1) which only slightly modifies the inertial waves analytic solutions are obtained. In the case of a strong magnetic field (Δ [Gt ] 1) for which Coriolis and Lorentz forces are comparable solutions are obtained numerically. The latter class includes the more realistic case (Δ → ∞) in which the upper boundary is absent.

scholarly journals An Investigation of Spiral Gravity Waves Radiating from Tropical Cyclones Using a Linear, Nonhydrostatic Model

2020 ◽  
Vol 77 (5) ◽  
pp. 1733-1759
Author(s):  
David S. Nolan
Keyword(s):  
Tropical Cyclones ◽  
Gravity Waves ◽  
Static Stability ◽  
Small Scale ◽  
Heat Sources ◽  
Pressure Oscillations ◽  
Nonhydrostatic Model ◽  
Tangential Wind ◽  
Aircraft Observations ◽  
The Waves

Abstract A recent study showed observational and numerical evidence for small-scale gravity waves that radiate outward from tropical cyclones. These waves are wrapped into tight spirals by the radial and vertical shears of the tangential wind field. Reexamination of the previously studied tropical cyclone simulations suggests that the dominant source for these waves are convective asymmetries rotating along the eyewall, modulated in intensity by the preferred convection region on the left side of the environmental wind shear vector. A linearized, nonhydrostatic model for perturbations to a balanced vortex is used to study the waves. Forcing the linear model with rotating and pulsing asymmetric heat sources generates radiating gravity waves with multiple vertical and horizontal structures. The pulsation of the rotating heat source generates two types of waves: fast, deep waves with larger radial wavelengths, and slower, secondary waves with shorter radial and vertical wavelengths. The deeper waves produce surface pressure oscillations that have time scales consistent with surface observations, whereas the shorter waves have little surface indication but produce oscillations in vertical velocity with shorter radial wavelengths that are consistent with aircraft observations. Convective forcing that is either not pulsing or not rotating produces gravity waves but they are not as similar to the observed or simulated waves. The effects of varying the intensity of the cyclone, the asymmetry of the forcing, and the static stability of the surrounding atmosphere are explored.

scholarly journals Functionally gradient isotropic cylindrical shell locally heated by heat sources

2019 ◽  
Vol 6 (2) ◽  
pp. 367-373
Author(s):  
Musii Musii ◽  
◽  
U. V. Zhydyk ◽  
O. Ya. Mokryk ◽  
N. B. Melnyk ◽  
...  
Keyword(s):  
Cylindrical Shell ◽  
Heat Sources ◽  
Functionally Gradient ◽  
Isotropic Cylindrical Shell

Subcritical convective instability Part 1. Fluid layers

1966 ◽  
Vol 26 (4) ◽  
pp. 753-768 ◽  
Author(s):  
Daniel D. Joseph ◽  
C. C. Shir
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Source ◽  
Linear Theory ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Heat Sources ◽  
Fluid Layers ◽  
Rayleigh Numbers

This paper elaborates on the assertion that energy methods provide an always mathematically rigorous and a sometimes physically precise theory of sub-critical convective instability. The general theory, without explicit solutions, is used to deduce that the critical Rayleigh number is a monotonically increasing function of the Nusselt number, that this increase is very slow if the Nusselt number is large, and that a fluid layer heated from below and internally is definitely stable when $RA < \widetilde{RA}(N_s) > 1708/(N_s + 1)$ where Ns is a heat source parameter and $\widetilde{RA}$ is a critical Rayleigh number. This last problem is also solved numerically and the result compared with linear theory. The critical Rayleigh numbers given by energy theory are slightly less than those given by linear theory, this difference increasing from zero with the magnitude of the heat-source intensity. To previous results proving the non-existence of subcritical instabilities in the absence of heat sources is appended this result giving a narrow band of Rayleigh numbers as possibilities for subcritical instabilities.

scholarly journals Probing latitudinal variations of the solar magnetic field in cycles 21–23 by Parker's Two-Layer Dynamo Model with meridional circulation

2013 ◽  
Vol 31 (11) ◽  
pp. 2023-2038 ◽  
Author(s):  
E. Popova ◽  
V. Zharkova ◽  
S. Zharkov
Keyword(s):  
Magnetic Field ◽  
Meridional Circulation ◽  
Dynamo Model ◽  
Cycle Number ◽  
Odd Cycle ◽  
Latitudinal Variations ◽  
Magnetic Poles ◽  
The Waves ◽  
Dipole Sources ◽  
Dynamo Waves

Abstract. Principle component analysis (PCA) of the solar background magnetic field (SBMF) measured from Wilcox Solar Observatory (WSO) magnetograms revealed the following principal components (PCs) in latitudes: two main symmetric components, which are the same for all cycles 21–23, and three pairs of asymmetric components, which are unique for each cycle. These SBMF variations are assumed to be those of poloidal magnetic field travelling slightly off-phase from pole to pole while crossing the equator. They are assumed to be caused by a joint action of dipole and quadruple magnetic sources in the Sun. In the current paper, we make the first attempt to interpret these latitudinal variations in the surface magnetic field with Parker's two-layer dynamo model. The latitudinal distributions of such waves are simulated for cycles 21–23 by the modified Parker's dynamo model taking into account both α and ω effects operating simultaneously in the two (upper and lower) layers of the solar convective zone (SCZ) and having opposite directions of meridional circulation. The simulations are carried out for both dipole and quadruple magnetic sources with the dynamo parameters specifically selected to provide the curves fitting closely the PCs derived from SBMF variations in cycles 21–23. The simulations are optimised for matching the positions of maximums in latitude, the number of equator crossings and the phase difference between the two dynamo waves operating in the two layers. The dominant pair of PCs present in each cycle is found to be fully asymmetric with respect to the magnetic poles and produced by a magnetic dipole. This pair is found to account for the two main dynamo waves operating between the two magnetic poles. There are also three further pairs of waves unique to each cycle and associated with multiple magnetic sources in the Sun. For the odd cycle 21 the simulated poloidal field fits the observed PCs, only if they are produced by magnetic sources with a quadruple symmetry in both layers, while for the even cycle 22 the fit to the observed PCs is achieved only in the case of quadruple magnetic sources in the upper layer and dipole sources in the inner layer. For the other odd cycle 23 the fit to observation is obtained for the quadruple magnetic sources in the inner layer and the dipole sources in the upper layer. The magnitudes of dynamo numbers D defining the conditions (depth and latitude) of a magnetic flux formation and the numbers N of zeros (equator crossings by the waves) are found to increase and the meridional circulation speed to decrease with a cycle number increase (D = −700, N = 3 for cycle 21 and D = −104, N = 9 for cycle 23). The phase delays between the waves in each unique pairs are also found to increase with the cycle number from ~9° in cycle 21 to ~13° in cycle 23.

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