scholarly journals Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

2001 ◽  
Vol 435 ◽  
pp. 103-144 ◽  
Author(s):  
M. RIEUTORD ◽  
B. GEORGEOT ◽  
L. VALDETTARO
Keyword(s):  
Lyapunov Exponent ◽  
Velocity Field ◽  
Spherical Shell ◽  
Asymptotic Properties ◽  
Analytical Form ◽  
Shear Layers ◽  
Three Dimensions ◽  
Geometrical Approach ◽  
Steady Flows ◽  
Whole Space

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincaré's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite.We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/√d, d being the distance to the characteristic grazing the inner sphere.We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general reveal an attractor expected at the eigenfrequency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E1/3, E being the Ekman number; for this latter layer, we give its analytical form and show its similarity to vertical 1/3-shear layers of steady flows. Using an inertial wave packet travelling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in [0, 2Ω], contrary to the case of the full sphere (Ω is the angular velocity of the system).Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10−10–10−20), which are out of reach numerically, and this for a wide class of containers.

scholarly journals Axisymmetric inertial modes in a spherical shell at low Ekman numbers

2018 ◽  
Vol 844 ◽  
pp. 597-634 ◽  
Author(s):  
M. Rieutord ◽  
L. Valdettaro
Keyword(s):  
Spherical Shell ◽  
Numerical Solutions ◽  
Asymptotic Properties ◽  
Analytical Formula ◽  
Shear Layers ◽  
Periodic Forcing ◽  
Oscillatory Flows ◽  
Ekman Number ◽  
Critical Latitude

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin (\unicode[STIX]{x03C0}/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

Inertial waves in a rotating spherical shell

1997 ◽  
Vol 341 ◽  
pp. 77-99 ◽  
Author(s):  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Spherical Shell ◽  
Cross Section ◽  
Shear Layers ◽  
Inertial Waves ◽  
Ekman Number ◽  
Distribution Of Eigenvalues ◽  
Simple Rules ◽  
Meridional Cross Section ◽  
Zero Viscosity ◽  
The Web

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

Convection in a rapidly rotating spherical shell with an imposed laterally varying thermal boundary condition

2009 ◽  
Vol 641 ◽  
pp. 335-358 ◽  
Author(s):  
CHRISTOPHER J. DAVIES ◽  
DAVID GUBBINS ◽  
PETER K. JIMACK
Keyword(s):  
Spherical Shell ◽  
Outer Boundary ◽  
Homogeneous Problem ◽  
Steady Flows ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
Thermally Driven ◽  
Layer Flow ◽  
Steady Solutions ◽  
Convection Rolls

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (2) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

scholarly journals A note on parameter estimation for discretely sampled SPDEs

2019 ◽  
Vol 20 (03) ◽  
pp. 2050016 ◽  
Author(s):  
Igor Cialenco ◽  
Yicong Huang
Keyword(s):  
Parameter Estimation ◽  
Bounded Domain ◽  
Asymptotic Properties ◽  
Mesh Size ◽  
Fixed Time ◽  
Estimation Problem ◽  
Joint Estimation ◽  
Heat Equations ◽  
Whole Space ◽  
Stochastic Heat Equations

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

Locus of the Axis of Vibration for a Vibrating System With Variable Location of a Mass Center

2000 ◽  
Author(s):  
Byung Ju Dan ◽  
Yong Je Choi
Keyword(s):  
Vibration Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Analytical Form ◽  
Point Of View ◽  
Information Storage ◽  
Mode Shapes ◽  
Storage Device ◽  
Geometrical Approach ◽  
Mass Center

Abstract A typical approach to a linear vibration analysis of an elastically supported single rigid body is to rearrange a dynamic model into a corresponding eigenvalue problem. From the geometrical point of view, the eigenvectors in the planar vibration analysis can be interpreted as pure rotations about the vibration center or pure translations. In a three dimensional space, they represent repetitive twisting motions about the axes of vibrations. By taking a geometrical approach to the vibration analysis, the vibration mode shapes may be better understood. In this paper, the influence of variable location of a mass center on the locations of the axes of vibrations and the natural frequencies are investigated by means of the locus of the axis of vibration expressed in analytical form, which represents the geometrical locus of the eigenvector. A numerical example is used to clearly illustrate the vibration phenomena of an optical pick-up used in an information storage device.

Plane MHD Steady Flows with Constantly Inclined Magnetic and Velocity Fields

1975 ◽  
Vol 53 (23) ◽  
pp. 2613-2616 ◽  
Author(s):  
O. P. Chandna ◽  
H. Toews ◽  
V. I. Nath
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Velocity Field ◽  
Fluid Flows ◽  
Velocity Fields ◽  
Steady Flows ◽  
Physical Conditions ◽  
Zero Current ◽  
Necessary And Sufficient ◽  
The Magnetic Field

Plane steady state viscous fluid flows, in which the magnetic field and velocity field are constantly inclined to one another, are considered. Necessary and sufficient physical conditions have been derived for flows with zero current density and the general solutions for these flows are obtained. Irrotational flows and flows with straight streamlines are also studied.

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