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Journal ArticleDOI

Numerical study of a rotating fluid in a spheroidal container

01 Jul 2004-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 197, Iss: 2, pp 671-685
TL;DR: In this article, the motion of an incompressible, viscous rotating fluid contained in a spheroidal container is studied by direct numerical simulation in an oblate SpH coordinate system, which allows us to expand any scalar field in spherical harmonics and decompose any vector field into its sphero-poloidal and spherotoroidal scalar parts.
About: This article is published in Journal of Computational Physics.The article was published on 2004-07-01 and is currently open access. It has received 24 citations till now. The article focuses on the topics: Prolate spheroidal coordinates & Spherical harmonics.

Summary (2 min read)

1 - Introduction

  • The role of precession (and convection) for the generation of the geomagnetic field in rotating bodies has also been discussed in detail [10, 11].
  • Most of the numerical studies of rotating fluids have been developed in a spherical geometry, because of its relative simplicity.
  • Section 2 is devoted to the mathematical developments within the oblate spheroidal coordinate system, in particular the Mie-like representation of a vector field and its properties.
  • Perspectives are given in the last section.

2.1 - Definition

  • Within this system, the constant-µ surfaces are ellipsoids with eccentricity !.
  • Surfaces are one-sheet hyperboloids which, for large µ values, become asymptotically cones of revolution around the z-axis with a half-aperture !.

2.3 - Decomposition of a vectorial field in the spheroidal symmetry

  • Some similarities with the expressions for the Mie representation in spherical symmetry may then be emphasized, e.g. about the sphero-poloidal field, but it is not the general case.
  • In particular, the curl of a sphero-poloidal field is not a sphero-toroidal field, nor the opposite assertion.

Spheroidal coordinate singularities

  • A first critical behaviour could be related to the densification of a grid near the poles if a grid was used in real space.
  • As in spherical geometry, this problem could be fixed by considering the spherical harmonic expansion (see Section 2.2), at least for values of µ not too close to zero.
  • This situation will have to be carefully examined if a grid is used (this is not the present case).
  • Another critical behaviour occurs for the expression of vectors around ! = 0 (Eq. 9), as the unit vectors !.

Limit a # 0

  • The limit a # 0 should give again the spherical situation.
  • In that case, µ goes to infinity, and the following approximations can be made : !.

3 – Mathematical formulation

  • 1 - Equation of motion - 9 - As a simplified stage of their work, the viscous correction to the (2,!1,!1) inviscid mode in spheroidal geometry is first considered, by linearizing the full equation of motion.
  • To handle the equation of motion more easily, the pressure was first eliminated by taking the curl of Eq. 14.
  • The main complication arises from their choice of coordinate system.
  • The system of equations can then be reduced to expressions which involve partial derivatives in µ and t only, and are expanded in spherical harmonics.
  • In the present calculation, that implies the vanishing of some quantities like !.

3.3 - Numerical method

  • Numerical calculation has been made by using the finite difference method for the sphero-radial variable µ, with a tanh(n)-grid in order to increase the number of points within the Ekman boundary layer (total number of points has been taken as N=81).
  • For the angular variables, the spherical harmonic expansion has been truncated to lmax = 64.
  • The implicit Crank-Nicolson scheme has been used for the diffusion term (initial value problem).
  • For each time step, the inversion of the big penta-diagonal matrix is carried out by LU factorization and Thomas algorithm, in order to avoid keeping in memory the inverse of this matrix.

4.3 – Numerical results

  • The pseudo-vorticity 'pv can be defined by its magnitude ' and its position.
  • E k 1 2 is approached for low Ekman numbers, while ' d tends to saturate toward a constant which differs from unity as soon as the eccentricity is different from zero.
  • In the spheroidal geometry, the singularity is reported to the focus circle, but the effects of this singular circle seem to be rather limited in the present linear approach where only m=1 terms are included.
  • At last, radial and angular resolution has been checked.

5 - Perspectives

  • The present preliminary results of a numerical study of the spinover mode in spheroidal geometry validate the choice of the method which is developed here.
  • The complexity inherent to the spheroidal symmetry (e.g. the non-separation of µ and ! variables in most of the spheroidal expressions) compared to the relative simplicity of the spherical case has been overcome by using a spheroidal decomposition of the velocity field derived from the spherical Mie representation.
  • The present work needs then to be continued by increasing the spatial resolution, in order to accurately describe the internal shear layers and their eccentricity dependences.
  • For such a non-linear study, m"1 terms will have to be considered in the motion equation and the pole problem carefully examined.
  • The true vorticity will have to be evaluated, giving rise to expressions more complex than in the present study.

Acknowledgments :

  • P. Cardin, H.C. Nataf and the Geodynamo team of the LGIT are gratefully acknowledged for their constant help during this study.
  • Most of the computations presented in this paper were performed at the Service Commun de Calcul Intensif de l'Observatoire de Grenoble (SCCI).

FIGURE CAPTIONS

  • Asymptote of the constant !, also known as Fig. curve; dashed line.
  • Curve for large µ values; note that the spherical colatitude of the asymptote is equal to the spheroidal colatitude of M. Fig.
  • On the spectral solutions of the 3-dimensional Navier- Stokes equations in spherical and cylindrical regions, Computer Physics Communications 90,.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, it was shown that in some parameter ranges, the magnetic field produced by the dynamos enhances the dynamo action in non-spherical bodies of electrically conducting fluid.
Abstract: More than half a century ago, Bullard suggested that the Earth's dynamo might be driven by the motions created in the Earth's core by the luni-solar precession. The precessionally forced motion of the mantle drives core flow through viscous forces and also, because of the electrical conductivity of the deep mantle, through magnetic forces. Both these couplings are thought to be insignificant in comparison with the topographical coupling created by the oblateness of the core-mantle interface. Because of technical difficulties in studying dynamo action in non-spherical bodies of electrically conducting fluid, this is the first serious attempt to study dynamo action by topographically forced flows. It describes the novel numerical methods that were employed, the tests that were devised to validate these methods, and the successful outcome of those tests. Some preliminary results for these dynamos are presented. It is shown that, in some parameter ranges, the magnetic field produced by the dynamos enhances th...

124 citations


Cites methods from "Numerical study of a rotating fluid..."

  • ...As mentioned in section 1, there is no toroidal/poloidal decomposition of u and B for spheroidal boundaries, although an ingenious alternative has been proposed and tested by Schmitt and Jault (2004). It seems to us that computationally this is not significantly more attractive than the alternatives now described, Three different programs were developed for three different types of situation:...

    [...]

Journal ArticleDOI
TL;DR: The principal advances in the design and construction, as well as the static, vibrational, and buckling analysis of thin-walled structures and buildings in the shape of general and axisymmetric ellipsoidal shells are summarized in this article.
Abstract: The principal advances in the design and construction, as well as the static, vibrational, and buckling analysis of thin-walled structures and buildings in the shape of general and axisymmetric ellipsoidal shells are summarized in this review. These shells are particularly useful as internally pressurized vessels or as heads and bottoms of cylindrical tanks and vessels. Reinforced concrete and structural steel domes of buildings, air-supported rubber-fabric shells, and underwater pressure vessels are also made in the form of ellipsoidal, shells. Knowing the geometry of ellipsoids, one can solve various problems in physics, optics, and so on. Basic results of theoretical and experimental investigations of the stress-strain state, buckling, and natural and forced vibrations contained in 209 references are presented in the review. The influence of temperature on the stress-strain state of the shells in question is also discussed. Some parts of the review are also devoted to an analysis of the literature on the stress-strain state of ellipsoidal and torispherical heads of pressure vessels with openings.

58 citations

Journal ArticleDOI
01 Jul 2011-Icarus
TL;DR: The moments of inertia of Titan, as separately deduced from its gravitational field and spin pole orientation, are quite different as discussed by the authors, and this discrepancy can be resolved if Titan is either not precessing as a rigid body (e.g. if the shell is decoupled from the interior by an ocean), or if the spin pole is not fully damped.

54 citations


Cites background from "Numerical study of a rotating fluid..."

  • ...The problem of precessional coupling between Earth’s fluid core and solid mantle has long been studied experimentally (Stewartson and Roberts, 1963; Noir et al., 2001), analytically (Noir et al., 2003; Schmitt and Jault, 2004), and observationally (Charlot et al., 1995; Lambert and Dehant, 2007)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically, which corresponds to a tidally deformed spinning astrophysical body.
Abstract: The tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically. Inviscid and viscous analytical models previously developed for the spheroidal geometry by Poincare [Bull. Astron. 27, 321 (1910)] and Busse [J. Fluid Mech. 33, 739 (1968)] are extended to this more complex geometry, which corresponds to a tidally deformed spinning astrophysical body. As confirmed by three-dimensional numerical simulations, the proposed analytical model provides an accurate description of the stationary flow in an arbitrary triaxial ellipsoid, until the appearance at more vigorous forcing of time dependent flows driven by tidal and/or precessional instabilities.

32 citations

Journal ArticleDOI
TL;DR: In this paper, the tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically, which corresponds to a tidally deformed spinning astrophysical body.
Abstract: The tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically. Inviscid and viscous analytical models previously developed for the spheroidal geometry by Poincare [Bull. Astr. 27, 321 (1910)] and Busse [J. Fluid Mech., 33, 739 (1968)] are extended to this more complex geometry, which corresponds to a tidally deformed spinning astrophysical body. As confirmed by three-dimensional numerical simulations, the proposed analytical model provides an accurate description of the stationary flow in an arbitrary triaxial ellipsoid, until the appearance at more vigorous forcing of time dependent flows driven by tidal and/or precessional instabilities.

29 citations

References
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Book
01 Jan 1937

11,054 citations

Journal ArticleDOI

9,185 citations


"Numerical study of a rotating fluid..." refers background in this paper

  • ...where a is the distance between the origin and the circle of foci within the equatorial plane [20]....

    [...]

Book
01 Jul 1968

2,123 citations

Journal ArticleDOI
19 Apr 1968-Science
TL;DR: The observed critical parameters indicate that a laminar flow in the core, due to the earth's precession, would have weak hydrodynamic instabilities at most, but that finite-amplitude hydromagnetic instability could lead to fully turbulent flow.
Abstract: I have proposed that the precessional torques acting on the earth can sustain a turbulent hydromagnetic flow in the molten core. A gross balance of the Coriolis force, the Lorentz force, and the precessional force in the core fluid provided estimates of the fluid velocity and the interior magnetic field characteristic of such flow. Then these numbers and a balance of the processes responsible for the decay and regeneration of the magnetic field provided an estimate of the magnetic field external to the core. This external field is in keeping with the observations, but its value is dependent upon the speculative value for the electrical conductivity of core material. The proposal that turbulent flow due to precession can occur in the core was tested in a study of nonmagnetic laboratory flows induced by the steady precession of fluid-filled rotating spheroids. It was found that these flows exhibit both small wavelike instabilities and violent finite-amplitude instability to turbulent motion above critical values of the precession rate. The observed critical parameters indicate that a laminar flow in the core, due to the earth9s precession, would have weak hydrodynamic instabilities at most, but that finite-amplitude hydromagnetic instability could lead to fully turbulent flow.

442 citations


"Numerical study of a rotating fluid..." refers background in this paper

  • ...Indeed, such simulations are needed to go further into our knowledge of the phenomena and to explain experimental results where fluid instabilities and turbulence are observed and reveal non-linear effects [7-9]....

    [...]

Frequently Asked Questions (15)
Q1. What contributions have the authors mentioned in the paper "Numerical study of a rotating fluid in a spheroidal container" ?

The motion of an incompressible, viscous rotating fluid contained in a spheroidal container is studied by a direct numerical simulation in an oblate spheroidal coordinate system. 

the order of the !-derivatives can be reduced down to first order by using Eq. 10 as many times as needed, the L2 operator introduced being again well adapted to the spherical harmonic expansion. 

In the spheroidal geometry, the singularity is reported to the focus circle, but the effects of this singular circle seem to be rather limited in the present linear approach where only m=1 terms are included. 

In spheroidal symmetry, expanding the scalar fields of any vector field u in spherical harmonics and integrating their angular parts leads to the existence of only two contributions, which can be written, in the cartesian basis, as:!" pv1 = #6 $% st1(µ)( ) ˆ ex + & st1(µ)( ) ˆ ey[ ]0µc ' sinh(2µ)dµ" pv2 = 2 6#5 & sp2(µ)( ) ˆ ex + % sp2(µ)( ) ˆ ey[ ]0µc ' dµ()* *+* *(24)It turns out that this pseudo-vorticity 'pv, apart a constant prefactor, is nothing else but the projection of the vector field u on the two orthogonal spinover modes!u 211and!u 21 1whichcorrespond to the inertial vortical flows around!ˆ e x and!ˆ e y , respectively:!"pv = #coth(2µc ) (u211 * $u)spheroid% ˆ ex + (u21 1 * $u)spheroid% ˆ ey &' ( () * + +(25)The evolution of the system will then be followed by investigating the time-variation of this vector 'pv as well as its dependence on viscosity and eccentricity. 

Within this system, an extension- 3 -of the spherical Mie representation of a vector field [19] is first considered: a solenoidal vector field (here the velocity field) is decomposed into the sum of two unique vectors which are themselves derived from two scalar fields, namely the sphero-poloidal and the spherotoroidal ones. 

In the present work, equation of motion of a rotating, viscous, incompressible fluidwithin a spheroidal container is directly treated in real space by using an appropriate set of coordinates, namely the oblate spheroidal coordinate system. 

2: Calculated variation of the decay factor , (left scale) and eigenfrequency 'd (right scale) of the spinover mode for Ekman numbers Ek ranging from 10 -9 to 10-4 and eccentricities e = 0.02 (left hand side) and 0.266 (right hand side); lines are least square fits; horizontal dotted lines are the theoretical limits for Ek = 0. 

4: Radial structure of the viscous boundary layer, seen through the scaling of the first four sphero-toroidal radial components for e = 0.266; successive curves correspond to Ek = 7. 10 -5 to 7. 10-9 (factor 10 between two successive values); the scaled radial coordinate is (µc - µ).(Ek) -1/2 (µc = 2 for e = 0.266).- 26 -REFERENCES[1] 

The spheroidal geometry of the rigid boundary of the latter was explicitely considered, giving rise to specific effects on the Earth's nutation compared to the case of a spherical container, but viscosity was neglected. 

It is worth noting that the inner core, which is often included in spherical symmetry in order to avoid a singularity at the origin, is not considered here. 

It is the eigen mode of rotation for an inviscid flow, which is excited when a fluid is in rapid rotation around the z-axis and its rotation axis is suddenly tilted by a slight quantity [4]. 

As for the spherical symmetry, this Mie-like representation allows one to work with only two scalar fields instead of the three velocity components, once the pressure has been eliminated from the equations. 

For the azimutal part, only m = 1 terms are needed here (see Sect. 4), since the (m = -1) terms can be deduced through the general relations:!st l"m (µ ) = (-1) m stlm (µ)( )*sp l"m (µ ) = (-1) m splm (µ)( )*# $ % & %(18)The corresponding matrices consist in a penta-diagonal matrix associated with theradial grid, the elements of which are band-matrices of order lmax, arising from the coupling- 12 -of scalar field components having different l values. 

At last, realistic geophysical applications would require to include an inner boundary with a possibly- 17 -different eccentricity, which would demand to extend the present coordinate system but would simplify the inner boundary condition. 

Using a more realistic coordinate system appears however highly desirable, because natural bodies or cavities are often ellipsoidal rather than spherical.