Turbulent diffusion in stably stratified non-decaying turbulence

Abstract

We develop a Lagrangian model of both one-particleIn this paper ‘particle’ and ‘fluid element’ are synonymous. and two-particle turbulent diffusion in high Reynolds number and low Froude number stably stratified non- decaying turbulence. This model is a kinematic simulation (KS) that obeys both the linearized Boussinesq equations and incompressibility. Hence, turbulent diffusion is anisotropic and is studied in all three directions concurrently with incompressibility satisfied at the level of each and every trajectory.

Horizontal one-particle and two-particle diffusions are found to be independent of the buoyancy (Brünt–Väissälä) frequency N. For one-particle diffusion we find that

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and

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where i = 1,2 and u′ and L are a r.m.s. velocity and a length-scale of the energy-containing motions respectively, and

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This capping of one-particle vertical diffusion requires the consideration of the entire three-dimensional flow, and we show that each and every trajectory is vertically bounded for all times if the Lagrangian vertical pressure acceleration a₃ is bounded for all times. Such an upper bound for a₃ can be derived from the linearized Boussinesq equations as a consequence of the coupling between vertical pressure acceleration and the horizontal and vertical velocities.

Two-particle vertical diffusion exhibits two plateaux. The first plateau's scaling is different according to whether the initial separation Δ₀ between the two particles is larger or smaller than η, the smallest length-scale of the turbulence:

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The second plateau is reached when the two particles become statistically independent, and therefore

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The transition between the two plateaux coincides with the time when the two particles start moving significantly apart in the horizontal plane.