© Crown copyright Met Office Turbulent dispersion: Key insights of G.I.Taylor and L.F.Richardson and developments stemming from them Dave Thomson, 17 th November 2010 Royal Meteorological Society Meeting: Turbulence - a 'resolved' problem?

© Crown copyright Met Office Key papers Two papers by Taylor and Richardson from the 1920’s form the foundation for much of what we know about turbulent dispersion: Taylor (1921) “Diffusion by continuous movements” Richardson (1926) “Atmospheric diffusion shown on a distance-neighbour graph” Contrasting styles!

© Crown copyright Met Office Contents Why focus on turbulent dispersion? Setting the scene for Taylor’s and Richardson’s work Taylor’s (1921) paper and ideas arising Richardson’s (1926) paper and ideas arising

© Crown copyright Met Office Why focus on turbulent dispersion? “The only reason we are interested in turbulence is because of its dispersive properties” – Philip Chatwin A (deliberatively provocative) exaggeration of course but one with some underlying truth, especially if include dispersion of momentum as well as heat & material Dispersion and mixing is fundamental to turbulence Whenever one tries to characterise what turbulence is, the ability to disperse and mix material is always a key characteristic

© Crown copyright Met Office Setting the scene Reynolds (1883, 1894) Statistical description, Reynolds stresses/fluxes Turbulence requires viscosity << velocity x length scale Boussinesq (1877, 1899) Turbulent flux ≈ “eddy-diffusivity” × gradient Einstein (1905) – Brownian motion Independent random jumps leads to flux proportional to concentration gradient Langevin (1908) Refined view of Brownian motion with continuously changing velocity

© Crown copyright Met Office Taylor (1921) Instead of assuming “flux = eddy-diffusivity × gradient”, express dispersion in terms of how fluid elements move: Hence Note is the Lagrangian correlation function of the velocity t t

© Crown copyright Met Office Taylor (1921) Expect correlation to decay in time, on time-scale say This leads to Effective eddy diffusivity is different for material of different ages t << >> t

© Crown copyright Met Office Importance of allowing for finite time scale of velocity changes Two examples: Dispersion in convective boundary layer: Upwind diffusion in light winds: Both cases very hard to simulate with an Eulerian “fixed frame of reference” approach, but can be treated easily and naturally with a Lagrangian approach.

© Crown copyright Met Office Convective boundary layer See: Deardorff & Willis – water tank results de Baas, Nieuwstadt & van Dop – Lagrangian model simulations

© Crown copyright Met Office Upwind diffusion in light winds Material of different ages at same location makes Eulerian treatment difficult Wind Source Puffs growing following Taylor (1921) Puffs growing following a diffusion equation – too much upwind spread for same plume growth downwind

© Crown copyright Met Office A Lagrangian model (“NAME”) ‘Particles’ tracked through the flow following resolved flow and modelled turbulence Turbulence represented by adding a random component of motion following ideas derived from Taylor (1921) (Chernobyl simulation)

© Crown copyright Met Office “Taylor” diffusion (Taylor ) Along-flow dispersion in a pipe (channel/river/boundary- layer) due to the variation in the mean flow: time scale for mixing across pipe Correlation time for along-flow velocity = time to mix across pipe Hence along-flow diffusivity = Slow mixing across pipe implies fast along pipe diffusion In meteorology can be important in stable boundary layers

© Crown copyright Met Office Richardson (1926) “Does the wind possess a velocity? This question, at first sight foolish, improves on acquaintance … it is not obvious that Δx/Δt attains a limit as Δt → 0” He gives an example: In fact Richardson is wrong here – Δx/Δt can’t fluctuate wildly or there would be infinite turbulent energy However “Does the wind possess an acceleration?” is a useful question In the limit of small viscosity Δu/Δt blows up: position velocity acceleration

© Crown copyright Met Office Richardson (1926) “The so-called constant K (the eddy-diffusivity) varies in a ratio from 2 to a billion (cm 2 s -1 )” t Wind tunnel Atmosphere Time- (or ensemble-) average picture Growth relative to puff centroid As cloud gets bigger ever larger eddies come into play What was the “mean flow” becomes an “eddy”

© Crown copyright Met Office Richardson (1926) Consider all pairs of “particles” in the dispersing cloud: Distribution of pair separations seems to obey diffusion equation with separation dependent diffusivity K ~ ar 4/3 r.m.s. pair separation ~ (at) 3/2 for t >> r 0 2/3 /a (r 0 = initial r) distribution of pair separation r : – strongly peaked a ~ ε 1/3 Pair separation r Mean square separation of all pairs = 2 × mean square spread of cluster r

© Crown copyright Met Office Statistical structure of concentration fluctuations Fluctuations important for toxic/flammable/reactive materials and odours Depends on peaked nature of pair separation distribution Also depends on the way triads, tetrads etc of particles separate

© Crown copyright Met Office Richardson’s law: r.m.s. pair separation ~ ε 1/2 t 3/2 for t >> r 0 2/3 /ε 1/3 In complex flow with chaotic trajectories: expect r ~ r 0 exp(t/T) – can make r small if r 0 small enough For Richardson’s law, can’t reduce r by making r 0 small – trajectories cease to be deterministic (in the limit of small viscosity) This is the analogue for trajectories of butterfly effect for whole weather system Understanding this non-determinism properly requires understanding of the zero viscosity limit – hence turbulence is not a ‘resolved’ (or resolvable) problem