1. © 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?

2. © 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!

3. © 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

4. © 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

5. © 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

6. © 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

7. © 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

8. © 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.

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

10. © 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

11. © 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)

12. © Crown copyright Met Office “Taylor” diffusion (Taylor 1953-4) 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

13. © 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

14. © 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”

15. © 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

16. © 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

17. © 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