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Register a new userInertial Modes in Near-Spherical Geometries
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We propose a numerical method to compute the inertial modes of a container with near-spherical geometry based on the fully spectral discretisation of the angular and radial directions using spherical harmonics and Gegenbauer polynomial expansion respectively. This allows to solve simultaneously the Poincare equation and the no penetration condition as an algebraic polynomial eigenvalue problem. The inertial modes of an exact oblate spheroid are recovered to machine precision using an appropriate set of spheroidal coordinates. We show how other boundaries that deviate slightly from a sphere can be accommodated for with the technique of equivalent spherical boundary and we demonstrate the convergence properties of this approach for the triaxial ellipsoid.
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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 Poincare 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 $sin(pi/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.
Oscillations have been detected in a variety of stars, including intermediate- and high-mass main sequence stars. While many of these stars are rapidly and differentially rotating, the effects of rotation on oscillation modes are poorly known. In this communication we present a first study on axisymmetric gravito-inertial modes in the radiative zone of a differentially rotating star. These modes probe the deep layers of the star around its convective core. We consider a simplified model where the radiative zone of a star is a linearly stratified rotating fluid within a spherical shell, with differential rotation due to baroclinic effects. We solve the eigenvalue problem with high-resolution spectral simulations and determine the propagation domain of the waves through the theory of characteristics. We explore the propagation properties of two kinds of modes: those that can propagate in the entire shell and those that are restricted to a subdomain. Some of the modes that we find concentrate kinetic energy around short-period shear layers known as attractors. We characterise these attractors by the dependence of their Lyapunov exponent with the BV frequency of the background and the oscillation frequency of the mode. Finally, we note that, as modes associated with short-period attractors form dissipative structures, they could play an important role for tidal interactions but should be dismissed in the interpretation of observed oscillation frequencies.
The relative importance of the helicity and cross-helicity electromotive dynamo effects for self-sustained magnetic field generation by chaotic thermal convection in rotating spherical shells is investigated as a function of shell thickness. Two distinct branches of dynamo solutions are found to coexist in direct numerical simulations for shell aspect ratios between 0.25 and 0.6 - a mean-field dipolar regime and a fluctuating dipolar regime. The properties characterising the coexisting dynamo attractors are compared and contrasted, including differences in temporal behavior and spatial structures of both the magnetic field and rotating thermal convection. The helicity $alpha$-effect and the cross-helicity $gamma$-effect are found to be comparable in intensity within the fluctuating dipolar dynamo regime, where their ratio does not vary significantly with the shell thickness. In contrast, within the mean-field dipolar dynamo regime the helicity $alpha$-effect dominates by approximately two orders of magnitude and becomes stronger with decreasing shell thickness.
In this paper, we study the inertial and gravity wave transmissions near the radiative-convective boundaries in the {it f}-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or In this paper, we study inertial and gravity wave transmissions near radiative-convective boundaries on the {it f}-plane. Two configurations have been considered: waves propagate from the convective layer to the radiative stratified stable layer, or the other way around. It has been found that waves prefer to survive at low latitudes when the stable layer is strongly stratified ($N^2/(2Omega)^2>1$). When the stable layer is weakly stratified ($N^2/(2Omega)^2<1$), however, waves can survive at any latitude if the meridional wavenumber is large. Then we have discussed transmission ratios for two buoyancy frequency structures: the uniform stratification, and the continuously varying stratification. For the uniform stratification, we have found that the transmission is efficient when the rotation is rapid, or when the wave is near the critical colatitude. For the continuously varying stratification, we have discussed the transmission ratio when the square of buoyancy frequency is an algebraic function $N^2propto z^{ u} ( u >0)$. We have found that the transmission can be efficient when the rotation is rapid, or when the wave is near the critical colatitude, or when the thickness of the stratification layer is far greater than the horizontal wave length. The transmission ratio does not depend on the configurations (radiative layer sits above convective layer, or vice versa; wave propagates outward or inward), but only on characteristics of the wave (frequency and wavenumber) and the fluid (degree of stratification).
Vortices play an unique role in heat and momentum transports in astro- and geo-physics, and it is also the origin of the Earths dynamo. A question existing for a long time is whether the movement of vortices can be predicted or understood based on their historical data. Here we use both the experiments and numerical simulations to demonstrate some generic features of vortex motion and distribution. It can be found that the vortex movement can be described on the framework of Brownian particles where they move ballistically for the time shorter than some critical timescales, and then move diffusively. Traditionally, the inertia of vortex has often been neglected when one accounts for their motion, our results imply that vortices actually have inertial-induced memory such that their short term movement can be predicted. Extending to astro- and geo-physics, the critical timescales of transition are in the order of minutes for vortices in atmosphere and ocean, in which this inertial effect may often be neglected compared to other steering sources. However, the timescales for vortices are considerably larger which range from days to a year. It infers the new concept that not only the external sources alone, for example the solar wind, but also the internal source, which is the vortex inertia, can contribute to the short term Earths magnetic field variation.
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