Inertial waves in a rotating spherical shell

1997 ◽  
Vol 341 ◽  
pp. 77-99 ◽  
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.

scholarly journals Inertial waves in a differentially rotating spherical shell

2013 ◽  
Vol 719 ◽  
pp. 47-81 ◽  
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.

scholarly journals Axisymmetric inertial modes in a spherical shell at low Ekman numbers

2018 ◽  
Vol 844 ◽  
pp. 597-634 ◽  
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.

scholarly journals Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

2001 ◽  
Vol 435 ◽  
pp. 103-144 ◽  
Author(s):  
M. RIEUTORD ◽  
B. GEORGEOT ◽  
L. VALDETTARO
Keyword(s):  
Lyapunov Exponent ◽  
Velocity Field ◽  
Spherical Shell ◽  
Asymptotic Properties ◽  
Analytical Form ◽  
Shear Layers ◽  
Three Dimensions ◽  
Geometrical Approach ◽  
Steady Flows ◽  
Whole Space

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.

scholarly journals The creation of a computational model of corrugated beams using the author’s program “GOPRO”

2018 ◽  
Vol 251 ◽  
pp. 04007 ◽  
Author(s):  
Vadim Alpatov ◽  
Alexey Lukin ◽  
Irina Laguta
Keyword(s):  
Finite Element ◽  
Computational Model ◽  
Cross Section ◽  
Geometrical Shape ◽  
File Format ◽  
Element Mesh ◽  
Automated Generation ◽  
Numerical Studies ◽  
Geometrical Scheme ◽  
The Web

The program «Gofro» is intended for the automated generation of data on the geometrical scheme of the beam with corrugated or plane web for further use in design complexes. The program has got a window interface, and it consists of one module for the input of feed da-ta, for calculation and for the display of its results in txt file format. The program offers a possibility to choose the outline of the structure, the profile of the web, the type of cross-section, and to set other parameters of the structure. When building up the model with the help of the author program «Gofro» and GMSH preprocessor for the automatic genera-tion of finite element mesh, the correctness of geometrical shape of elements is monitored by the algorithms that are input in the preprocessor. The author compares the time re-quired to create the models using the author program and GMSH preprocessor and using the standard resources of «Lira» software system. The authors performed numerical studies of various I-beams created in the program «GOPRO».

Experimental survey of linear and nonlinear inertial waves and wave instabilities in a spherical shell

2016 ◽  
Vol 789 ◽  
pp. 589-616 ◽  
Author(s):  
Michael Hoff ◽  
U. Harlander ◽  
C. Egbers
Keyword(s):  
Spherical Shell ◽  
Experimental Studies ◽  
Vertical Shear ◽  
Meridional Plane ◽  
Inertial Waves ◽  
Supercritical Regime ◽  
Laser Plane ◽  
Linear And Nonlinear ◽  
Wave Instabilities ◽  
Time Periodic

We experimentally study linear and nonlinear inertial waves in a spherical shell with a radius ratio of ${\it\eta}=1/3$. The shell rotates with a mean angular velocity ${\it\Omega}_{0}$ around its vertical axis. This rotation is overlaid by a time-periodic libration of the inner sphere in the range $0<{\it\omega}_{lib}<2{\it\Omega}_{0}$ to excite inertial waves with a defined frequency. In the first part, we investigate linear inertial waves. The influence of the libration amplitude and the libration frequency on the waves and further the efficiency of the forcing to excite linear inertial waves will be discussed. For this, qualitative data from Kalliroscope visualisation in a meridional laser plane, as well as quantitative particle image velocimetry (PIV) data in a horizontal plane, have been analysed. A simple two-dimensional ray-tracing model is applied for the meridional plane to interpret the visualisations with respect to energy focusing and wave attractors. For sufficiently high/low libration amplitudes/frequencies, the Stewartson layer, a vertical shear layer tangential to the inner sphere’s equator, becomes unstable. This so-called ‘supercritical’ regime, where centrifugal and shear instabilities occur, allows for nonlinear wave coupling. PIV analyses in the horizontal laser plane in the corotating frame show low-frequency structures that correspond to Rossby-wave instabilities of the Stewartson layer. Some of these are travelling retrograde and are trapped near the Stewartson layer, others are travelling prograde filling the whole gap outside the Stewartson layer. Since libration can be viewed as a time-periodic variation of differential rotation, we assume that these two different structures are related to either the retrograde $(Ro_{d}<0)$ or the prograde $(Ro_{d}>0)$ phase of the libration cycle. The experimental results confirm theoretical, numerical as well as other experimental studies on Stewartson-layer instabilities.

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