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, ν, and thermal diffusivity, κ, are lowered to zero, with σ ≡ ν/κ fixed, then the energy dissipation per unit mass, ε, 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 σ 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 ε → 0 in the inviscid limit. 1. Introduction A key feature of the ocean and the stratosphere is that the solar irradiance forces the fluid differentially in latitude and sets up strongly stable density stratification. This situation is very different from the Rayleigh–Bénard paradigm, in which a fluid is heated from below and cooled at the top. A more appropriate geophysical idealization, suggested by Stommel (1962), is the experiment of Rossby (1965) illustrated in figure 1. We follow Stern (1975) in referring to this flow configuration as 'horizontal convection'. Because the surface temperature is uneven, there are horizontal density and pressure gradients and the fluid is in motion. Indeed, the critical Rayleigh number for the onset of horizontal convection is zero: the smallest ∆T sets the fluid into motion. But in the geophysical situation the Rayleigh and Reynolds numbers (denoted generically by R) are enormous. In many situations, this statement is the prelude to a discussion of turbulent flow. Yet we show below that Rossby's experiment cannot become 'truly turbulent', even if R → ∞. By 'true turbulence' we do not mean merely sensitive dependence on initial conditions, or even the dynamics of many coupled modes. We mean that the two experimental laws of fully developed turbulence apply (Frisch 1995). The first law requires a spectral cascade, as envisaged by Richardson. Our main focus is however: The law of finite energy dissipation: If, in an experiment on turbulent flow, all the control parameters are kept the same, except for the viscosity, ν, which is lowered as much as possible, the energy dissipation per unit mass behaves in a way consistent with a finite positive limit. The main implication of the law of …