A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres

Liao, Xinhao; Zhang, Keke

doi:10.1098/rspa.2009.0582

Cited by 11 publications

(5 citation statements)

References 13 publications

(16 reference statements)

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“…Lastly, it is worth remarking that some of our no-slip calculations are directly comparable to the work of Liao & Zhang 30 , who studied solutions of the same equation at low E using a different body force that also did not satisfy Taylor’s constraint. They confirmed agreement between an asymptotic method for the full problem (not just the geostrophic part) and their numerical method at Ekman numbers no smaller than E = 10 −5 .…”

Section: The Geostrophic Flowsupporting

confidence: 70%

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

Livermore

Bailey

Hollerbach

2016

Sci Rep

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We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth’s core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E1/2, where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1 m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E−1/2 or E−1). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth’s core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤ 10−5.

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Section: The Geostrophic Flowsupporting

confidence: 70%

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

Livermore

Bailey

Hollerbach

2016

Sci Rep

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“…For m = 0 the associated inertial waves propagate eastward (westward) if j > 0 (j < 0). Equatorial symmetries are determined by n − m: the velocity modes are quadrupole symmetric if n − m = 0(mod 2) and dipole symmetric if n − m = 1 (mod 2); and rotational symmetries about Ω are determined by m. The velocities of the geostrophic modes (4.13) agree with Liao & Zhang (2010a) up to normalisation and phase: in terms of Jacobi polynomials their φ-components can be written as Below we label the Coriolis modes using a 3-index Greek letter, e.g. v α ≡ v m α n α ,j α , and denote the degree of the polynomial flow v α by deg α.…”

Section: N+1mentioning

confidence: 94%

Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere

2015

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We consider incompressible flows in the rapid-rotation limit of small Rossby number and vanishing Ekman number, in a bounded volume with a rigid impenetrable rotating boundary. Physically the flows are inviscid, almost rigid rotations. We interpret the Coriolis force, modified by a pressure gradient, as a linear operator acting on smooth inviscid incompressible flows in the volume. The eigenfunctions of the Coriolis operator C so defined are the inertial modes (including any Rossby modes) and geostrophic modes of the rotating volume. We show C is a bounded operator and that −iC is symmetric, so that the Coriolis modes of different frequencies are orthogonal. We prove that the space of incompressible polynomial flows of degree N or less in a sphere is invariant under C. The symmetry of −iC thus implies the Coriolis operator is non-defective on the finite-dimensional space of spherical polynomial flows. This enables us to enumerate the Coriolis modes, and to establish their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spherical polynomial flows under C and completeness, is that the solution of the polynomial Poisson-Neumann problem, i.e. Poisson's equation with a Neumann boundary condition and polynomial data, in a sphere is a polynomial. We also enumerate the Coriolis modes in a sphere, with careful consideration of the geostrophic modes, directly from the known analytic solutions.

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“…where uG,j(r ⊥ ) are the (degenerate) geostrophic solutions (e.g. Liao & Zhang 2010, in spheres) that only depend on the position perpendicular to the rotation axis r ⊥ . They are given by the geostrophic equilibrium…”

Section: Geostrophic Motions and Torsional Alfvén Modesmentioning

confidence: 99%

Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle

Jault

Noir

Vidal

2020

Geophysical Journal International

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We investigate the pressure torque between the fluid core and the solid mantle arising from magnetohydrodynamic modes in a rapidly rotating planetary core. A twodimensional reduced model of the core fluid dynamics is developed to account for the non-spherical core-mantle boundary. The simplification of such a quasi-geostrophic model rests on the assumption of invariance of the equatorial components of the fluid velocity along the rotation axis. We use this model to investigate and quantify the axial torques of linear modes, focusing on the torsional Alfvén modes (TM) in an ellipsoid. We verify that the periods of these modes do not depend on the rotation frequency. Furthermore, they possess angular momentum resulting in a net pressure torque acting on the mantle. This torque scales linearly with the equatorial ellipticity. We estimate that for the TM calculated here topographic coupling to the mantle is too weak to account for the variations in the Earth's length-of-day.

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A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres

Cited by 11 publications

References 13 publications

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere

Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle

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