dorsal/arxiv
View SchemaInertial waves in a rotating spherical shell: attractors and asymptotic spectrum
| Authors | M. Rieutord, B. Georgeot, L. Valdettaro |
|---|---|
| Categories | |
| ArXiv ID | physics/0007007 |
| URL | https://arxiv.org/abs/physics/0007007 |
| DOI | 10.1017/S0022112001003718 |
Abstract
We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincar\'e's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.
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"date_created": "2026-03-02T18:00:32.093000Z",
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"abstract": "We investigate the asymptotic properties of inertial modes confined in a\nspherical shell when viscosity tends to zero. We first consider the mapping\nmade by the characteristics of the hyperbolic equation (Poincar\\\u0027e\u0027s equation)\nsatisfied by inviscid solutions. Characteristics are straight lines in a\nmeridional section of the shell, and the mapping shows that, generically, these\nlines converge towards a periodic orbit which acts like an attractor.\n We then examine the relation between this characteristic path and\neigensolutions of the inviscid problem and show that in a purely\ntwo-dimensional problem, convergence towards an attractor means that the\nassociated velocity field is not square-integrable. We give arguments which\ngeneralize this result to three dimensions. We then consider the viscous\nproblem and show how viscosity transforms singularities into internal shear\nlayers which in general betray an attractor expected at the eigenfrequency of\nthe mode. We find that there are nested layers, the thinnest and most internal\nlayer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an\ninertial wave packet traveling around an attractor, we give a lower bound on\nthe thickness of shear layers and show how eigenfrequencies can be computed in\nprinciple. Finally, we show that as viscosity decreases, eigenfrequencies tend\ntowards a set of values which is not dense in $[0,2\\Omega]$, contrary to the\ncase of the full sphere ($\\Omega$ is the angular velocity of the system).\nHence, our geometrical approach opens the possibility of describing the\neigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers\n($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide\nclass of containers.",
"arxiv_id": "physics/0007007",
"authors": [
"M. Rieutord",
"B. Georgeot",
"L. Valdettaro"
],
"categories": [
"physics.flu-dyn",
"astro-ph",
"cond-mat",
"nlin.PS"
],
"doi": "10.1017/S0022112001003718",
"title": "Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum",
"url": "https://arxiv.org/abs/physics/0007007"
},
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