18 research outputs found
Axisymmetric inertial modes in a spherical shell at low Ekman numbers
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 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 , and
control the shape of the shear layers that are associated with these
modes. These scales point out the key role of the small parameter in
these oscillatory flows. With a simplified model of the viscous Poincar\'e
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 and
explain why a fluid inside a spherical shell cannot respond to any periodic
forcing at this frequency when viscosity vanishes.Comment: 38 pages, 25 figures, to appear in J. Fluid Mechanic
Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm
An efficient way of solving 2D stability problems in fluid mechanics is to
use, after discretization of the equations that cast the problem in the form of
a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This
method preserves the banded structure sparsity of matrices of the algebraic
eigenvalue problem and thus decreases memory use and CPU-time consumption.
The errors that affect computed eigenvalues and eigenvectors are due to the
truncation in the discretization and to finite precision in the computation of
the discretized problem. In this paper we analyze those two errors and the
interplay between them. We use as a test case the two-dimensional eigenvalue
problem yielded by the computation of inertial modes in a spherical shell. This
problem contains many difficulties that make it a very good test case. It turns
out that that single modes (especially most-damped modes i.e. with high spatial
frequency) can be very sensitive to round-off errors, even when apparently good
spectral convergence is achieved. The influence of round-off errors is analyzed
using the spectral portrait technique and by comparison of double precision and
extended precision computations. Through the analysis we give practical recipes
to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure
Compressible MHD in spherical geometry.
none2L. Valdettaro;M. MeneguzziValdettaro, Lorenzo; M., Meneguzz
Consequences of Rotation In Energetics of Accretion Disks
A synthesis on recent results about rotating turbulence is done. They prove that rotation tends to inhibit energy transfer from large scales to small scales because of the generation of strong inertial waves aligned with the rotation axis. Implications on the theory of turbulent transport in accretion disks are discussed. It is argued that the concept of turbulent viscosity should be revisited to take into account the possibility of non-local transport by large scale wave-dominated turbulence. An "energetic puzzle" resulting from a possible inverse cascade of energy is brought out. Various solutions to solve it are suggested and are linked to currently available observations. New prescriptions for turbulent dissipation in the presence of rotation or large scale waves are given. Special emphasis is put on compressibility effects, for which a detailed analysis of dissipation processes is provided and translated in term of turbulent viscosities. When the turbulence is strongly influenced by the rotation, most of the energy is transported by sonic waves. The corresponding turbulent viscosity is computed numerically for several types of disks. A universal model for thin accretion disks is suggested in relation with the results
Large Eddy Simulation of gravity currents with a high order DG method
This work deals with Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of a turbulent gravity current in a gas, performed by means of a Discontinuous Galerkin (DG) Finite Elements method employing, in the LES case, LES-DG turbulence models previously introduced by the authors. Numerical simulations of non-Boussinesq lock-exchange benchmark problems show that, in the DNS case, the proposed method allows to correctly reproduce relevant features of variable density gas ows with gravity. Moreover, the LES results highlight, also in this context, the excessively high dissipation of the Smagorinsky model with respect to the Germano dynamic procedure