Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows

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 (Poincaré'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 (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite.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. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/√d, d being the distance to the characteristic grazing the inner sphere.We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general reveal an attractor expected at the eigenfrequency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E1/3, E being the Ekman number; for this latter layer, we give its analytical form and show its similarity to vertical 1/3-shear layers of steady flows. Using an inertial wave packet travelling 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, 2Ω], contrary to the case of the full sphere (Ω 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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Inertial waves in a rotating spherical shell

Journal of Fluid Mechanics ◽

10.1017/s0022112097005491 ◽

1997 ◽

Vol 341 ◽

pp. 77-99 ◽

Cited By ~ 108

Author(s):

M. RIEUTORD ◽

L. VALDETTARO

Keyword(s):

Spherical Shell ◽

Cross Section ◽

Shear Layers ◽

Inertial Waves ◽

Ekman Number ◽

Distribution Of Eigenvalues ◽

Simple Rules ◽

Meridional Cross Section ◽

Zero Viscosity ◽

The Web

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

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Inertial waves in a differentially rotating spherical shell

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.605 ◽

2013 ◽

Vol 719 ◽

pp. 47-81 ◽

Cited By ~ 30

Author(s):

C. Baruteau ◽

M. Rieutord

Keyword(s):

Spherical Shell ◽

Solid Body ◽

Shear Layers ◽

Inviscid Limit ◽

Order Partial Differential Equation ◽

Body Rotation ◽

Inertial Waves ◽

Visible Absorption ◽

Solid Body Rotation ◽

Short Period

AbstractWe investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

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Non-axisymmetric Shear Layers in a Rotating Spherical Shell

Solar and Planetary Dynamos ◽

10.1017/cbo9780511662874.020 ◽

1994 ◽

pp. 145-152 ◽

Cited By ~ 7

Author(s):

R. Hollerbach ◽

M.R.E. Proctor

Keyword(s):

Spherical Shell ◽

Shear Layers ◽

Axisymmetric Shear

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Axisymmetric inertial modes in a spherical shell at low Ekman numbers

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.201 ◽

2018 ◽

Vol 844 ◽

pp. 597-634 ◽

Cited By ~ 7

Author(s):

M. Rieutord ◽

L. Valdettaro

Keyword(s):

Spherical Shell ◽

Numerical Solutions ◽

Asymptotic Properties ◽

Analytical Formula ◽

Shear Layers ◽

Periodic Forcing ◽

Oscillatory Flows ◽

Ekman Number ◽

Critical Latitude

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 Poincaré 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 $\sin (\unicode[STIX]{x03C0}/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

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Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell

Journal of Fluid Mechanics ◽

10.1017/s0022112099007776 ◽

2000 ◽

Vol 408 ◽

pp. 239-274 ◽

Cited By ~ 6

Author(s):

ANDREW M. SOWARD ◽

RAINER HOLLERBACH

Keyword(s):

Magnetic Field ◽

Free Boundary ◽

Boundary Conditions ◽

Spherical Shell ◽

Shear Layers ◽

Numerical Results ◽

Free Boundary Conditions ◽

Viscous Forces ◽

Inner Sphere ◽

Stress Free Boundary

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.

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The Structures of the Shear Layers in Box Shear Test of Soil

Landslides ◽

10.3313/jls1964.31.10 ◽

1994 ◽

Vol 31 (1) ◽

pp. 10-20_1

Author(s):

Ryojiro KISHIMOTO

Keyword(s):

Shear Test ◽

Shear Layers

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Analysis of shear bands in slow granular flows using a frictional Cosserat model

MRS Proceedings ◽

10.1557/proc-627-bb4.3 ◽

2000 ◽

Vol 627 ◽

Author(s):

Prabhu R. Nott ◽

K. Kesava Rao ◽

L. Srinivasa Mohan

Keyword(s):

Scaling Laws ◽

Shear Bands ◽

Shear Layers ◽

Theoretical Description ◽

Field Variable ◽

Cosserat Continuum ◽

Momentum Balance ◽

Rigid Plastic ◽

Independent Field ◽

Cosserat Model

ABSTRACTThe slow flow of granular materials is often marked by the existence of narrow shear layers, adjacent to large regions that suffer little or no deformation. This behaviour, in the regime where shear stress is generated primarily by the frictional interactions between grains, has so far eluded theoretical description. In this paper, we present a rigid-plastic frictional Cosserat model that captures thin shear layers by incorporating a microscopic length scale. We treat the granular medium as a Cosserat continuum, which allows the existence of localised couple stresses and, therefore, the possibility of an asymmetric stress tensor. In addition, the local rotation is an independent field variable and is not necessarily equal to the vorticity. The angular momentum balance, which is implicitly satisfied for a classical continuum, must now be solved in conjunction with the linear momentum balances. We extend the critical state model, used in soil plasticity, for a Cosserat continuum and obtain predictions for flow in plane and cylindrical Couette devices. The velocity profile predicted by our model is in qualitative agreement with available experimental data. In addition, our model can predict scaling laws for the shear layer thickness as a function of the Couette gap, which must be verified in future experiments. Most significantly, our model can determine the velocity field in viscometric flows, which classical plasticity-based model cannot.

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