The effect of an instability on the spectral structure of homogeneous turbulence

Abstract

The hydrodynamic turbulence, i.e. a chaotic flow dominated by inertia (nonlinearity) is a common phenomenon both in natural systems and in engineering laboratory flows. The standard description of turbulence involves determination of the spectral energy profile, in other words the so-called turbulent energy cascade. An important, unanswered question concerns the effect of instabilities on the features of fully developed turbulence, since instabilities lead to chaotic, nonlinear behaviour in the first place and the final structure of turbulence is not independent of its original source. However, hydrodynamic instabilities are strongly anisotropic and often inhomogeneous, therefore analytic results are not achievable. Here, to study this problem a simpler case of an active scalar transport is considered, that is the evolution of temperature through advection, diffusion and heat sources; the advecting flow is influenced and slaved by the temperature through buoyancy and assumption of rapid background rotation. Homogeneous instability is induced by the heat source. This allows to apply the renormalization group technique to extract the final simple recursion differential equations for the turbulent diffusivities from the fully nonlinear dynamical equation, which are solved numerically. The effect of the presence of linear instability on the final energy spectrum is studied.