Nonlinear (thermo)hydrodynamical waves

Mathieu Jenny & Emmanuel Plaut


      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 leading-order balance in the momentum equation is between the Coriolis force and the pressure gradient, and this imposes the Proudman-Taylor 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):

On the right are shown, in a meridional cut, the levels of the mean or zonal flow (in the azimuthal direction) generated by this wave; these zonal flows would play a major role in the dynamo mechanisms, see e.g. Kageyama & Sato, Phys. Rev. E 55 (1997). Since the 3D computations are very demanding, and in order to better understand the mechanics of these waves, Friedrich Busse, with whom we also cooperated, suggested to develop quasi geostrophic 2D models. Friedrich Busse also suggested to build analogous laboratory experiments, where the centrifugal force plays the role of the gravity field ; the applied thermal gradient should be inversed to set up an unstable stratification. Motivated by such experiments by Markus Jaletzky, using liquid metals, which gave unexpected results, we have developed a systematic study of a simplified model where the caps are cones and not spherical shells (like in the experiments by Markus Jaletzky) [1].
We have then studied the effects of a change of slope of the caps in the radial direction. This leads notably to multicellular quasi inertial modes at low Prandtl number [3].
More recently we have developped a model in cylindrical coordinates, which gives good results at Prandtl numbers of order unity [4]. We thus recover `spiralling waves', and observe that the properties of 2D waves approach quantitatively those of 3D waves when the Ekman number is decreased. The following comparison, which concerns also the mean flows, is a `visual validation' of our model:

The curves on the right show the mean or zonal flows. The mechanisms that control their precise form are not very well understood. To clarify them, we have reformulated the Reynolds stresses generated by a pure wave [4]. The formula that we have obtained, which highlights the importance of both the kinetic energy and the geometrical form of the waves, is of general interest. Thus we study also in [4] the case of open shear flows, and of Tollmien-Schlichting and Kelvin-Helmholtz waves.


      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, 4391-4394, 1997 and Phys. Rev. E 59, 4091-4105, 1999) :



We have developped a new cartesian model of these instabilities [2], which gives rather good results when compared to the experiments of Liu & Ecke:



Moreover this model has been extended to the case of rotating convection in an annulus. In this case we have shown that the multiple connexity of the flow domain generates a topological constraint that drives nonlocal effects [2].


      This motivated us to build a new isothermal setup to produce hydrodynamical waves in possibly rotating flows. This experiment is made with two plates rotating independently, an inferior plate where an annular channel is digged, and a superior plate which shears the fluid contained in this channel:

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] Low-Prandtl-number convection in a rotating cylindrical annulus, E. Plaut and F. H. Busse, J. Fluid Mech. 464, 345-363 (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 mean-flow develops at quadratic order. The Poiseuille part of this mean-flow is determined by an equation obtained by averaging of the Navier-Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling 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 short-wavelength 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 Rayleigh-Bénard convection: effects of the boundary conditions and of the topology, E. Plaut, Phys. Rev. E 67, 046303-1-11 (2003).

Motivated by the experimental results of Liu & Ecke (1997, 1999), different models are developed to analyze the weakly nonlinear dynamics of the traveling-wave sidewall modes appearing in rotating Rayleigh-Bénard 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 multiply-connected geometry is shown to lead generally to the existence of a global mean-flow 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 Ginzburg-Landau envelope equation contains a nonlocal term. This new term, however, vanishes in the large-gap 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 small-gap annular geometry if a small-Prandtl-number fluid is used, despite the fact that no regime of Benjamin-Feir instability is predicted to occur.

[3] Multicellular convection in rotating annuli, E. Plaut and F. H. Busse, J. Fluid Mech. 528, 119-133 (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 quasi-inertial modes and analyze the physical mechanisms leading to multicells. This is done for both stress-free and no-slip 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, 302-326 (2008).

A general reformulation of the Reynolds stresses created by two-dimensional 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 wave-flow. 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 shear-flow waves, a new expression of the Reynolds-Orr 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 three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic 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, 35-46 (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 inter-radii 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, co-rotation and counter-rotation cases. The basic flows obtained at small shear are characterized. In the absence of co-rotation, the centrifugal force linked with the curvature of the system plays an important role, whereas, when co-rotation 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 co-rotation cases, but to weak turbulence in the counter-rotation case. The Reynolds number for the onset of instabilities, constructed with the velocity difference between the lid and bottom at mid-radius, and the height of the channel, increases from 1000 in the counter-rotation case to 1260 in the pure shear case and higher and higher values when co-rotation increases, i.e., when the Coriolis effect increases. The relevance of uni-dimensional Ginzburg-Landau 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