Energy spectrum and mode selection in thermally driven turbulence: the nonlinear core unresolved by Chandrasekhar's 1952 statistical theory

Chandrasekhar's 1952 statistical theory of turbulence caused by thermal instability treats a homogeneous, axisymmetric region of Boussinesq fluid with the nonlinear terms of the equations of motion and heat conduction neglected. Its stationary correlation solutions admit a generalized Fourier analysis exhibiting a minimum eddy wavelength and two mode families, with kinetic energy principally vertical or principally horizontal. Because the theory is linear, each Fourier component is treated independently: the nonlinear energy transfer between components, the resulting distribution of modal energy over wavenumbers (the energy spectrum), the dynamical circumstances selecting one or the other mode family, and the nonlinear modification of the modal shapes and of the minimum-wavelength bound are all left undetermined. The problem is to build the nonlinear statistical description of thermally driven turbulence that determines these quantities. The later literature - Malkus's marginal-stability theory of the heat transport and spectrum (1954), Kraichnan's mixing-length treatment (1962), and modern direct numerical simulations, precision experiments, and superstructure studies - characterizes aspects of the spectrum and of large-scale mode selection phenomenologically, but no closed predictive determination exists, so the nonlinear core remains open.

Background

Rayleigh–Bénard convection - a horizontal fluid layer heated from below and cooled from above - is the canonical setting for turbulence driven by thermal instability. Two nondimensional parameters govern it: the Rayleigh number $Ra$, which measures the ratio of buoyant driving to viscous and thermal damping, and the Prandtl number $Pr$, the ratio of the fluid's momentum diffusivity to its thermal diffusivity. In the Boussinesq approximation, density variations are retained only where they multiply gravity, so temperature acts as an active scalar: buoyancy converts thermal energy into kinetic energy while the velocity field advects and stirs the temperature field.

In a 1952 paper in Philosophical Transactions of the Royal Society A, S. Chandrasekhar developed a statistical theory of the turbulence that results from this instability. Restricting attention to a region far from the horizontal boundaries, so that the turbulence is approximately homogeneous (statistics invariant under translations) and axisymmetric (invariant under rotations about the vertical direction, using his theory of axisymmetric tensors), he defined two-point correlation functions of the velocity components and temperature fluctuations - for example, the correlation of vertical velocities at two points directly above one another - and derived a closed system of equations for the scalar functions defining these correlations for the case in which the nonlinear terms of the equations of motion and heat conduction are neglected and a constant mean adverse temperature gradient is maintained. In the statistically stationary state, the theory yields an exact balance between the viscous dissipation of kinetic energy and the liberation of potential energy by gravity. The stationary solutions vary periodically in the vertical direction and permit a generalized Fourier analysis of the correlations, with two structural consequences: the Fourier analysis of the vertical-velocity correlation cannot contain wavelengths below a certain minimum that depends on the physical parameters and the maintained gradient - a smallest eddy size - and the turbulence field can be decomposed into two mode families, one with kinetic energy principally in the vertical direction and one with it principally in the horizontal direction.

The limitation is inherent to the construction. By discarding the nonlinear terms, the theory treats each Fourier component independently: no energy is transferred between components, so the nonlinear cascade that defines turbulence is absent, and the amplitudes of the admissible modal solutions - the distribution of energy over wavenumbers, that is, the energy spectrum - are not determined by the theory. Nor does it say under what dynamical circumstances a realized turbulent state favors the vertical-energy or the horizontal-energy mode family, or how nonlinear mode coupling would reshape the modal structures and the minimum-wavelength bound.

The question was taken up almost immediately in a different idealization. W. V. R. Malkus (1954) measured discrete transitions in the heat transport of turbulent convection between horizontal conducting plates and found an apparent lack of uniqueness of the flow patterns at given heat transport - direct evidence that mode selection is a real question - and, in a companion paper, constructed a theory of the heat transport and spectrum of thermal turbulence in which the spectrum and mean gradients are determined once a minimum eddy size is given, that size being fixed by marginal-stability arguments (each successively excited mode is taken to sit exactly at its onset of instability) for the successively unstable modes. That framework bounds and parameterizes the spectrum rather than deriving it from the governing equations. R. H. Kraichnan (1962) extended the mixing-length program - modeling turbulent transport as diffusion by eddies of an assumed characteristic size - to arbitrary Prandtl number, obtaining heat-transport predictions including a high-$Ra$ asymptotic regime, again phenomenologically. In astrophysical convection the closure difficulty persisted: a 1967 review of the solar convection zone records aerodynamicists' severe reservations about mixing-length approaches to convective velocity fields.

The modern era treats turbulent Rayleigh–Bénard convection through direct numerical simulation (DNS), precision experiments, and scaling theories for the heat transport. The large-scale circulation, the boundary layers, and most recently the turbulent superstructures - very large, long-lived roll-and-cell patterns with horizontal extents of several layer heights - are characterized empirically, and the kinetic-energy spectrum in the bulk is found to be approximately Kolmogorov-like, $E(k)\sim k^{-5/3}$ with an approximately constant energy flux, for $Pr\sim1$ fluids, in contrast to the Bolgiano–Obukhov spectrum $E(k)\sim k^{-11/5}$ of stably stratified turbulence. None of these results constitutes a closed nonlinear statistical theory that predicts the energy spectrum and the selected mode from the governing equations and parameters; that is the unresolved core posed by the 1952 theory.

Problem Statement

For a Boussinesq fluid sustaining statistically stationary turbulence driven by thermal instability in the boundary-free, homogeneous and axisymmetric idealization (a horizontal layer with a maintained adverse, that is destabilizing, mean temperature gradient, characterized by its Rayleigh number $Ra$ and Prandtl number $Pr$), incorporate the nonlinear terms of the equations of motion and heat conduction into the statistical description of the turbulence so as to determine what the linearized 1952 theory necessarily leaves free:

(1) the inter-modal (nonlinear) energy transfer among Fourier components and the resulting spectral distribution of kinetic and thermal energy over wavenumbers - the energy spectrum of the stationary turbulent state, as a function of $Ra$ and $Pr$;

(2) the dynamical circumstances under which the realized turbulent state is dominated by modes whose kinetic energy is principally vertical versus principally horizontal (the two mode families exhibited by the 1952 theory); and

(3) how nonlinear mode coupling modifies the modal shapes and the minimum-wavelength (smallest-eddy) bound that the linearized theory exhibits.

A determination must not assume what is to be found: fixing a minimum eddy size by hypothesis, or postulating marginal stability as an input, does not by itself constitute a solution, since the linear theory's construction leaves precisely these quantities free. The three items form one scientific objective - the nonlinear stationary statistics of thermally driven turbulence - obtained as the stationary solutions of a closed finite system of equations for the two-point correlations that retains the nonlinear terms and determines items (1)-(3). If a closure relation for the higher-order (third-order) correlations is used, the energy spectrum, the mode selection, and the minimum-wavelength bound must still be obtained from the stationary solutions of that system rather than postulated as inputs.

References

1. Subrahmanyan Chandrasekhar, On turbulence caused by thermal instability, Philosophical Transactions of the Royal Society of London. Series A 244(884), 357–384 (1952), DOI 10.1098/rsta.1952.0009, https://doi.org/10.1098/rsta.1952.0009
2. W. V. R. Malkus, Discrete transitions in turbulent convection, Proceedings of the Royal Society of London. Series A 225(1161), 185–195 (1954), DOI 10.1098/rspa.1954.0196, https://doi.org/10.1098/rspa.1954.0196
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