A theory of mixing in a stably stratified fluid

Abstract

A theory is developed for turbulence in a stably stratified fluid, for example in the experiments of Rouse & Dodu and of Turner where there is no shear and the turbulence is induced by a source of energy near the lower boundary of the fluid. A growing mixed layer of thickness D appears in the lower portion of the fluid and is separated from the non-turbulent fluid above, in which the buoyancy gradient is given, by an interfacial layer of thickness h. The lower mixed layer has a very weak buoyancy gradient and the large buoyancy difference across the interfacial layer is Δb.

As indicated by the experiments of Thompson & Turner and Hopfinger & Toly, and derived by the author in a recent paper, if u is the root-mean-square horizontal velocity and l is the integral length scale, the eddy viscosity ul is a constant in a homogeneous fluid agitated by a grid. When there is stratification, the theory indicates that the fluid motion is unaffected by buoyancy forces in the mixed layer, so that ul should again be constant in the lower portions of the mixed layer. Since l is proportional to distance, we may conveniently suppose that the source of the disturbances is at a level z = 0 where u is infinite in accordance with uz = K. Thus we may take K to be a fundamental parameter characterizing the turbulent energy source. Then z is distance above the plane of the virtual energy source. If the non-turbulent fluid has uniform buoyancy, DΔb = U² may be shown to be constant. In general, whether constant or not, U may be taken to be a fundamental parameter expressing the stability. The quantity $\hat{R}i = U^{2}D^{2}/K^2$ is the most fundamental of the several Richardson numbers that have been introduced in this problem because, with its use, ‘constants’ of proportionality do not depend on the molecular coefficients of viscosity or diffusion (for high Reynolds number turbulence) or on the geometry of the grid.

The theory contains a number of results:

1. $u_e D/K \sim \hat{R}i^{-\frac{7}{4}},$ where $u_e = dD/dt$ is the entrainment velocity. Integration yields $D \infty t^{\frac{2}{11}}$ for a homogeneous upper fluid and $D \infty t^{\frac{2}{9}}$ for a linear upper density field. This $-\frac{7}{4}$ entrainment law compares with a $-\frac{3}{2}$ law suggested by several experimenters.

2. Turbulence in the interfacial layer is intermittent with intermittency factor $I_3 \sim \hat{R}i^{-\frac{3}{4}}$. The turbulent patches have dimension $\sigma_3 \sim D\hat{R}i^{-\frac{3}{4}}$.

3. If the (equal) root-mean-square velocities in an infinite homogeneous fluid at a distance D from the grid are denoted by u₁ ∼ v₁ ∼ w₁, we find that the r.m.s. velocities near the interface are u₂ ∼ u₁, v₂ ∼ u₁ and $w_2 \sim u_1 \hat{R}i^{-\frac{1}{4}}$.

4. The buoyancy flux q₂ near the interface may be expressed as $q_2 \sim w^3_2/D$.

5. h ∼ D, as observed in several experiments.