Horizontal convection is non-turbulent

Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, v, and thermal diffusivity, kappa, are lowered to zero, with sigma equivalent to nu/kappa fixed, then the energy dissipation per unit mass, epsilon, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number sigma sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not 'truly turbulent' because epsilon --> 0 in the inviscid limit.