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

Ivers, D. J.; Jackson, Andrew; Winch, D. E.

doi:10.1017/jfm.2015.27

Cited by 27 publications

(27 citation statements)

References 47 publications

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“…It is sensible to assume that the response to a body force (per unit volume), f , of frequency ω can be decomposed, at least partially, onto the (infinite) set of inertial modes { u α }:

bold-italicv \sim \sum_{α} \frac{< bold-italicf false| u_{α} >}{false| λ_{α} - i ω false|} u_{α},

where <·|·> denotes the projection operator over pairs of vector fields. Ivers et al () and Backus and Rieutord () have demonstrated the completeness of the above modal expansion for an inviscid fluid inside a spherical or ellipsoidal container (with no inner core). In analogy with those works, we define the projection operator as

< bold-italicv false| bold-italicw > \equiv Re \int_{V} v^{*} \cdot bold-italicw

.…”

Section: Discussionmentioning

confidence: 85%

“…where < ·|· > denotes the projection operator over pairs of vector fields. Ivers et al (2014) and Backus and Rieutord (2017) have demonstrated the completeness of the above modal expansion for an inviscid fluid inside a spherical or ellipsoidal container (with no inner core). In analogy with those works, we define the projection operator as < v|w >≡ Re∫  v * · w. A resonance takes place when the factor multiplying one or more u becomes large, which occurs when the distance between the forcing frequency and the eigenvalue approaches zero and/or < f|u > is large.…”

Section: The Role Of Inertial Modesmentioning

confidence: 90%

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Internal Energy Dissipation in Enceladus's Subsurface Ocean From Tides and Libration and the Role of Inertial Waves

et al. 2019

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Enceladus is characterized by a south polar hot spot associated with a large outflow of heat, the source of which remains unclear. We compute the heat generated via viscous dissipation resulting from tidal and (longitudinal) libration forcing in the moon's subsurface ocean using the linearized Navier‐Stokes equation in a three‐dimensional spherical model. We conclude that libration is the dominant cause of dissipation at the linear order, providing up to ∼0.001 GW of heat to the ocean, which remains insufficient to explain the ∼10 GW observed by Cassini. We also illustrate how resonances with inertial modes can significantly augment the dissipation. Our work is an extension to Rovira‐Navarro et al. (2019, https://doi.org/10.1016/j.icarus.2018.11.010) to include the effects of libration and the presence of the icy crust. The model developed here is readily applicable to the study of other moons with a subsurface ocean and planets with a liquid core.

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“…It is sensible to assume that the response to a body force (per unit volume), f , of frequency ω can be decomposed, at least partially, onto the (infinite) set of inertial modes { u α }:

bold-italicv \sim \sum_{α} \frac{< bold-italicf false| u_{α} >}{false| λ_{α} - i ω false|} u_{α},

< bold-italicv false| bold-italicw > \equiv Re \int_{V} v^{*} \cdot bold-italicw

.…”

Section: Discussionmentioning

confidence: 85%

Section: The Role Of Inertial Modesmentioning

confidence: 90%

Internal Energy Dissipation in Enceladus's Subsurface Ocean From Tides and Libration and the Role of Inertial Waves

et al. 2019

View full text Add to dashboard Cite

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“…The present work is therefore a follow up of the work of Ivers et al [24] who obtained a first set of mathematical results when the problem is restricted to the sphere and when the velocity fields are supposed to be oncecontinuously differentiable. The two works share many common results, but hopefully they complete one another and offer the broadest view of the Poincaré problem.…”

Section: Introductionmentioning

confidence: 83%

Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid

Backus

Rieutord

2017

Phys. Rev. E

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Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid and contained in a rigid container, these modes satisfy Poincaré's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. In this paper we investigate the mathematical side of this problem. We first show that the Poincaré problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincaré operator is bounded and self-adjoint and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric then the base can be identified with the set of Poincaré modes, first obtained by Bryan (1889) [1], and completed with the geostrophic modes.

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“…A special class of slow, equatorially-symmetric, inertial modes, that we refer to as QG modes (Zhang et al 2001;Busse et al 2005;Maffei et al 2017), have been shown to efficiently describe rotating flow in a sphere at the onset of convection (Zhang & Liao 2004;Zhang et al 2007), and when combined with the geostrophic mode can also describe weakly-nonlinear convection (Zhang & Liao 2004;Liao et al 2012). More generally, geostrophic and inertial modes can be used to describe the transient response of rotating spherical systems to a forcing (Liao & Zhang 2010) and in principle they provide a complete basis for representing flows in a rotating sphere (Cui et al 2014;Ivers et al 2015;Backus & Rieutord 2017). Of particular interest here is that they are well-suited to describing motions at low latitudes.…”

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confidence: 99%

Time-dependent low-latitude core flow and geomagnetic field acceleration pulses

Kloss

Finlay

2019

Geophysical Journal International

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We present a new model of time-dependent flow at low latitudes in the Earth's core between 2000 and 2018, derived from magnetic field measurements made on board the Swarm and CHAMP satellites and at ground magnetic observatories. The model, called CoreFlo-LL.1, consists of a steady background flow without imposed symmetry plus a time-dependent flow that is dominated by geostrophic and quasi-geostrophic components but also allows weak departures from equatorial symmetry. Core flow mode amplitudes are determined at 4-month intervals by a robust least squares fit to ground and satellite secular variation data. The l 1 norm of the square root of geostrophic and inertial mode enstrophies, and the l 2 norm of the flow acceleration, are minimised during the inversion procedure. We find that the equatorial region beneath the core-mantle boundary is a place of vigorous, localised, fluid motions; time-dependent flow focused at low latitudes close to the core surface is able to reproduce rapid field variations observed at non-polar latitudes at and above Earth's surface. Magnetic field acceleration pulses are produced by alternating bursts of non-zonal azimuthal flow acceleration in this region. Such bursts are prominent in the longitudinal sectors from 80-130 • E and 60-100 • W throughout the period studied, but are also evident under the equatorial Pacific from 130 • E to 150 • W after 2012. We find a distinctive interannual alternation in the sign of the non-zonal azimuthal flow acceleration at some locations, involving a rapid cross-over between flow acceleration convergence and divergence. Such acceleration sign changes can occur within a year or less, and when the structures involved are of large spatial scale they can give rise to geomagnetic jerks at the Earth's surface. For example, in 2014, we find a change in the sign of the non-zonal azimuthal flow acceleration under the equatorial Pacific, as a region of flow acceleration divergence near 130 • E changes to a region of flow acceleration convergence. This occurs at a maximum in the amplitude of the time-varying azimuthal flow under the equatorial Pacific and corresponds to a geomagnetic jerk at the Earth's surface.

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

Cited by 27 publications

References 47 publications

Internal Energy Dissipation in Enceladus's Subsurface Ocean From Tides and Libration and the Role of Inertial Waves

Internal Energy Dissipation in Enceladus's Subsurface Ocean From Tides and Libration and the Role of Inertial Waves

Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid

Time-dependent low-latitude core flow and geomagnetic field acceleration pulses

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