On Slow Inertial Waves in the Solar Convection Zone

Busse, Friedrich H.; Zhang, K.; Liao, Xinhao

doi:10.1086/497300

Cited by 15 publications

(18 citation statements)

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“…where ω si = 2σ Nmn and this approximation is exact for the case N = 1. All such slow modes propagate eastwards and are none other than the well known Rossby modes in a full sphere (Busse et al, 2005).…”

mentioning

confidence: 89%

“…It has long been recognized that Rossby modes in a closed container are no more than a special class of inertial modes, in which ∂u/∂t is much smaller than the pressure and Coriolis forces, so that related motions are very close to geostrophic (Longuet-Higgins, 1964;Greenspan, 1968;Busse et al, 2005). Although we present a comparison here of purely hydrodynamic modes, it is interesting to note that in the case of an imposed Malkus magnetic field (30), then a very similar comparison can also be made for hydromagnetic modes -see Malkus (1967); Zhang et al (2003).…”

Section: Appendix a Comparison Between Quasi-geostrophic Modes And Smentioning

confidence: 99%

“…Nonetheless, there is evidently a poor agreement between our QG model and the threedimensional theory for the modes with largest radial wavelengths, particularly when the azimuthal wavenumber is very high. Examination of the associated eigenfunctions shows that, for a given radial mode, as m increases the eigenfunction is increasingly pushed towards the outer boundary; in this region the boundary slope is large and the three-dimensional solutions show more prominent departures from axial invariance that are incompatible with the QG assumption, compare for example Figures 1 and 2 from Busse et al (2005) for m = 6. The failure of the QG model manifests itself in terms of an incorrect slope of the dispersion curve at very large m, see for example the mode with larges radial length scale (the blue curve 30 10 1 10 2 10 3 10 0 10 1 Azimuthal wavenumber Period (in units of 2π/Ω) -1st QG mode -3rd QG mode -6th QG mode -10th QG mode -15th QG mode --3D modes in Figure A.11); physically this implies that the azimuthal group velocity of such modes is incorrect in the QG model.…”

mentioning

confidence: 99%

“…Considering the special case of slow inertial modes for which σ si = σ Nmn O(m/(N + m), (A.5) that are close to geostrophy and so possess relatively simple structures in the axial z direction, then an analytic approximation to (A.4) is available (Zhang et al, 2001;Busse et al, 2005)…”

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confidence: 99%

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Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

Canet

Finlay

Fournier

2014

Physics of the Earth and Planetary Interiors

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The core of a terrestrial-type planet consists of a spherical shell of rapidly rotating, electrically conducting, fluid. Such a body supports two distinct classes of quasi-geostrophic (QG) eigenmodes: fast, primarily hydrodynamic, inertial modes with period related to the rotation time scale and slow, primarily magnetic, magnetostrophic modes with much longer periods. Here, we investigate the properties of these hydromagnetic quasi-geostrophic modes as a function of non-dimensional parameters controlling the strength of the background magnetic field, the planetary rotation rate, and the amount of magnetic dissipation.We first present analytic solutions that illustrate the essential parameter dependences of the modes and provide a convenient benchmark for our numerical scheme. A comparison between known three-dimensional inertial modes in a sphere and our axially invariant QG modes shows encouraging agreement at low azimuthal wavenumbers, particularly for the slowest modes. The container geometry and background magnetic field structure are found to influence the radial structure of the modes, but not the scaling of their frequency with the control parameters. When the background magnetic field decreases toward the outer boundary in a spherical shell, QG modes tend to be compressed towards the outer boundary. Including magnetic dissipation, we find a continuous transition from diffusionless slow magnetic modes into quasi-free decay magnetic modes. During that transition (which is controlled by the magnitude of the Elsasser number), we find that slow magnetic modes weakly modified by diffusion exhibit a distinctive spiralling planform. When magnetic diffusion is significant (Elsasser number much smaller than unity), we find quasi-free decay slow magnetic modes whose decay time scale is comparable to, or shorter than, their oscillation time scale.Based on our analysis, we expect Mercury to be in a regime where the slow magnetic modes are of quasi-free decay type. Earth and possibly Ganymede, with their larger Elsasser numbers, may possess slow modes that are in the transition regime of weak diffusion, depending on the details of their poorly known internal magnetic fields. Fast QG modes, that are almost unaffected by the background magnetic field, are expected in the cores of all three bodies.

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confidence: 89%

Section: Appendix a Comparison Between Quasi-geostrophic Modes And Smentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

Canet

Finlay

Fournier

2014

Physics of the Earth and Planetary Interiors

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“…Analytical solutions for these are available in a full sphere geometry (Zhang et al 2001;Zhang & Liao 2017). 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).…”

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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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Long Period (0.9–5.5 Year) Oscillations in Surface Spherical Harmonics of Sunspot Longitudinal Distributions

Juckett

2006

Sol Phys

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The temporal changes in the two-dimensional patterns of sunspot groups, spanning 125 years (1650 Carrington Rotations), were previously analyzed using surface spherical harmonics (SSHs) (Juckett, 2003) in an attempt to quantify properties of the active longitudes among sunspot distributions. Common trends in the oscillations of both the amplitudes and spatial phases of sectoral combinations of SSHs were examined. The amplitude analysis revealed strong evidence for the second and third harmonics of the 11-year cycle across all SSHs, plus evidence for other structured variations spanning both longer and shorter time scales. In this report, temporal oscillations above the second harmonic reveal a dispersion relationship with respect to order, m, in the m = l and m = l − 1 SSH modes. Furthermore, the relationship between amplitude and abrupt spatial phase transitions for these oscillations is consistent with the behavior of standing waves. Under this assumption, each standing-wave half-cycle is identified by spatial phase transitions between 3 4 π and π . This was used to convert the SSH amplitude series for each mode from a rectified version of the standing wave to an estimate of the full cycle. Spectral analysis yielded a dispersion relation over the SSH order range m = 1 to m = 18 spanning the cycle periods from the 11-year solar cycle down to that of the well-documented, but ill-understood 1.3 and 1.8 year quasi-periodic cycles of the quasi-biennial oscillation. Examination of the spatial phase patterns of the SSH modes suggests that the longitudinal variations in sunspot clustering are a complex phenomena with patterns occurring in several time scales. The standing wave trait of the SSH modes may offer evidence uniting the dynamo waves in the convective zone to interfacial oscillations in the tachocline.

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On Slow Inertial Waves in the Solar Convection Zone

Cited by 15 publications

References 18 publications

Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

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

Long Period (0.9–5.5 Year) Oscillations in Surface Spherical Harmonics of Sunspot Longitudinal Distributions

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