Bose condensation gives new insight into turbulent advection (Vol. 41, No. 6)

Passive scalar turbulence describes the advection and diffusion of a scalar quantity (such as temperature or pollutant concentration) in a turbulent flow. It was rigorously proven in the 1990’s that the Kraichnan model, now widely hailed as the “Ising model of turbulence”, leads to statistical intermittency and anomalous scaling of the advected field, which means power law behaviour of the structure functions of the scalar with scaling exponents ζ_N depending in a nonlinear way on their order N. The emergence of anomalous scaling was traced to the existence of statistical integrals of motion showing up in the evolution of Lagrangian fluid particles and exponents ζ_N identified with the highest scaling dimension or degree of the corresponding so-called “zero modes” (homogeneous functions of interparticle distances, whose average in the N-particle configurational space is left invariant by the dynamics).

Analytical computations of zero modes and their degrees were, up to now, mostly done using perturbative methods around limiting values of parameters for which anomalous scaling disappears. It is shown in this paper that scaling dimensions of zero modes can be recast as eigenvalues of a many-body pseudo-Hamiltonian describing the dynamics of the Lagrangian particles in an appropriate comoving frame. A variational estimate for ζ_N is then obtained by using techniques borrowed from condensed matter physics and assuming Bose condensation of particles. A connection of zero modes with the extremal events leading to the formation of fronts of the scalar, as caught by the instanton formalism, is also established.

By bridging up the gap between the two most powerful tools (zero modes and instantons) introduced these last years in the study of the inertial range intermittency in turbulent systems, this works revives the hope that they might help us to unravel, once, the whole complexity of incompressible 3D Navier-Stokes turbulence.