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^{1}, V. M. Canuto

^{2}and J. R. Chasnov

^{2}

### Abstract

If a random and statistically homogeneous density distribution is created in a fluid, a random motion of the fluid is subsequently generated by buoyancy forces. This motion is resisted by viscous stresses, while the density variation is smoothed by molecular diffusion of the relevant scalar property of the fluid. Furthermore, advective mixing generates smaller‐scale components of the scalar quantity, increasing the rate of smoothing. If the Reynolds number of the motion is sufficiently large, the motion becomes turbulent in the ordinary sense. Such turbulence may be said to be generated by an ‘‘active’’ conserved scalar quantity. To elucidate the nature of this buoyancy‐driven turbulence, we consider an infinite fluid that is initially at rest everywhere, with a given distribution of the conserved scalar quantity. In order to have a well‐defined initial state specified by a manageably small number of parameters, the initial distribution of the scalar is assumed to be statistically homogeneous and isotropic, and to be characterized by a single length *L*, and a measure of the magnitude of the scalar variations, say, the initial root‐mean‐square scalar fluctuation θ_{0}. The other parameters on which the field of buoyancy‐driven turbulence depends are the kinematic viscosity ν, the diffusivity of the conserved scalar quantity *D*, and the gravitational constant *g*.

Dimensionless variables are constructed by choosing *L* as the unit of length and [*L*/*g*θ_{0}]^{1/2} as the unit of time. The choice of these units introduces two dimensionless parameters into the problem, a Reynolds number *R*=[*g*θ_{0} *L* ^{3}]^{1/2}/ν and a Schmidt number σ=ν/*D*. Since analytical treatment of the problem is limited to the special cases where the nonlinear interactions may be neglected (e.g., *R* and σ*R*≪1), the primary means of inquiry is a numerical simulation of the flow field. We have performed numerical simulations (using a subgrid model) which sample the entire parameter space of the flow. Of particular interest is the asymptotic behavior of the flow as *R* and σ*R* increase. One observes that a competition arises between the increase in magnitude of the fluid velocity according to the linear equations and the nonlinear generation of small‐scale density and velocity fluctuations. The linear analysis predicts that, in dimensionless units, the time *t* _{ m } at which the mean‐square velocity fluctuations reach a maximum, as well as the value of this maximum 〈*u* _{ i } *u* _{ i }〉_{ m }, increases without limit as *R* and σ*R* increase. In contrast, the results of the numerical simulation show that *t* _{ m } and 〈*u* _{ i } *u* _{ i }〉_{ m } become independent of *R* and σ*R* for sufficiently large values of these parameters. The linear amplification of the flow is thus checked by nonlinear effects. Buoyancy‐generated turbulence at high Reynolds number has a period of growth from an initial state of rest, lives for a time in which there is a burst of activity, and then dies in consequence of this activity. This example of self‐induced mixing can be given a quantitative description which may be useful in other contexts.

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