Nonlinear (thermo)hydrodynamical waves 
Nonlinear waves are idealized, very coherent structures that can be used to approximate the coherent structures that appear in many natural or industrial flows. They can often be used to model the first step of the transition to turbulence. It is therefore important to develop methods to compute these nonlinear waves efficiently, analyze their structures and the relevant mechanisms that sustain or modify them. We focussed on systems with cylindrical symmetry, even if the article [4] below also considers systems with cartesian symmetry.
A first class of nonlinear waves is given by the thermal Rossby waves appearing through thermoconvection instabilities of a liquid planetary core, between the solid hot inner core and the solid outer mantle, which is relatively cool. Such turbulent waves in the core of the Earth are the `dynamo' that sustain the geomagnetic field, see e.g. Busse, Annu. Rev. Fluid Mech. 32 (2000). In smaller planets where no such magnetohydrodynamical effects exist, or where they are less important, thermal Rossby waves still pose challenging questions, because of the many competing effects that come into play: hydrodynamical and thermal effects couple, and in addition the inertial forces linked with the diurnal rotation of the planet should play an important role, because, at the global scale, this rotation is typically `fast'. Thus the leadingorder balance in the momentum equation is between the Coriolis force and the pressure gradient, and this imposes the ProudmanTaylor constraint: the flow should be approximately 2D in the plane perpendicular to the rotation axis. Thus the thermal Rossby waves are `columnar' (numerical simulation by Radostin Simitev, with whom we cooperated):
A second class of nonlinear waves is given by the sidewall waves which appear for instance near the coasts in the oceans. Similar sidewall waves have been observed in a rotating disk convection experiment by Liu & Ecke (cf Phys. Rev. Lett 78, 43914394, 1997 and Phys. Rev. E 59, 40914105, 1999) :
The photography on the right shows a wave obtained experimentally
by shearing the channel with the lid;
the contrast has been enhanced digitally.
The wave is visible in the gray zone thanks to a `tracer',
see the regularly spaced `commas'
[5].
We have also performed a numerical study of this system with
Éric Serre
[5].
[1] 
LowPrandtlnumber convection in a
rotating cylindrical annulus,
E. Plaut and F. H. Busse,
J. Fluid Mech. 464, 345363 (2002).
Motivated by recent experimental results obtained in a low Prandtl number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde meanflow develops at quadratic order. The Poiseuille part of this meanflow is determined by an equation obtained by averaging of the NavierStokes equation. It thus gives rise to a new globalcoupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this globalcoupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a shortwavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.

[2] 
Nonlinear dynamics of traveling waves
in rotating RayleighBénard convection:
effects of the boundary conditions and of the topology,
E. Plaut,
Phys. Rev. E 67, 046303111 (2003).
Motivated by the experimental results of Liu & Ecke (1997, 1999), different models are developed to analyze the weakly nonlinear dynamics of the travelingwave sidewall modes appearing in rotating RayleighBénard convection. These models assume fully rigid boundary conditions for the velocity field. At the linear level, this influences most strongly the critical frequencies: they appear to be proportional to the logarithm of the Coriolis number, which is twice the inverse of the Ekman number. An annular flow domain is considered. This multiplyconnected geometry is shown to lead generally to the existence of a global meanflow mode proportional to the average, over the azimuthal coordinate, of the square of the modulus of the envelope of the waves. Because this mode feeds back on the active wave modes at cubic order, the resulting GinzburgLandau envelope equation contains a nonlocal term. This new term, however, vanishes in the largegap limit relevant to the experiments of Liu & Ecke. As compared with previous theoretical work, the present models lead to reduced discrepancies with the results of these experiments concerning the coefficients of the envelope equation. It is also shown that the new nonlocal effects may be realized experimentally in a smallgap annular geometry if a smallPrandtlnumber fluid is used, despite the fact that no regime of BenjaminFeir instability is predicted to occur.

[3] 
Multicellular convection
in rotating annuli,
E. Plaut and F. H. Busse,
J. Fluid Mech. 528, 119133 (2005).
The onset of convection in a rotating cylindrical annulus with sloping conical boundaries is studied in the case where this slope increases with the radius. The critical modes assume the form of drifting spiralling columns attached to the inner cylindrical wall at moderate and large Prandtl numbers, but they become attached to the outer wall at low Prandtl numbers. These latter `equatorially attached' modes are multicellular at intermediate rotation rates. Through a perturbation analysis which is validated by a numerical code, we show that all equatorially attached modes are quasiinertial modes and analyze the physical mechanisms leading to multicells. This is done for both stressfree and noslip boundary conditions. At finite amplitudes the convection generates a Reynolds stress which leads to the development of a mean zonal flow, and a geometrical analysis of the mechanisms leading to this zonal flow is presented. The influence of Ekman friction on the zonal flow is also studied.

[4] 
Reynolds stresses
and mean fields generated by pure waves:
applications to shear flows and convection in a rotating shell,
E. Plaut, Y. Lebranchu, R. Simitev and F. H. Busse,
J. Fluid Mech. 602, 302326 (2008).
A general reformulation of the Reynolds stresses created by twodimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the waveflow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows and thermal Rossby waves in spherical shell convection with rotation. In the case of shearflow waves, a new expression of the ReynoldsOrr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a threedimensional code using noslip boundary conditions are presented in the nonlinear regime, and compared with those of a twodimensional quasigeostrophic model. A semiquantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasigeostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found. 
Erratum : in the second part of equation (2.16c), the coefficient in front of the inverse Reynolds number must be 3 instead of 3/2. 
[5] 
Structure and stability of annular sheared channel flows:
effects of confinement, curvature and inertial forces  waves,
E. Plaut, Y. Lebranchu, M. Jenny and E. Serre,
Eur. Phys. J. B 79, 3546 (2011).
The structure and stability of the flows in an annular channel sheared by a rotating lid are investigated experimentally, theoretically and numerically. The channel has a square section, and a small curvature parameter: the ratio of the interradii to the mean radius is 9.5%. The sidewalls and the bottom of the channel are integral and can rotate independently of the lid, permitting pure shear, corotation and counterrotation cases. The basic flows obtained at small shear are characterized. In the absence of corotation, the centrifugal force linked with the curvature of the system plays an important role, whereas, when corotation is fast, the Coriolis force dominates. These basic flows undergo some instabilities when the shear is increased. These instabilities lead to supercritical traveling waves in the pure shear and corotation cases, but to weak turbulence in the counterrotation case. The Reynolds number for the onset of instabilities, constructed with the velocity difference between the lid and bottom at midradius, and the height of the channel, increases from 1000 in the counterrotation case to 1260 in the pure shear case and higher and higher values when corotation increases, i.e., when the Coriolis effect increases. The relevance of unidimensional GinzburgLandau models to describe the dynamics of the waves is studied. The domain of validity of these models turns out to be quite narrow. 
Emmanuel Plaut 
Last modified: Thu Feb 18 11:43:37 CET 2016 