Entrainment in volcanic plumes Ralph Burton Stephen Mobbs

Entrainment in volcanic plumes Ralph Burton, Stephen Mobbs, Alan Gadian, Mark Woodhouse (University of Bristol)

Why? The near-field influences the far-field. Courtesy Mark Woodhouse, Bristol University From UKMO Website

Back to basics! Plume models: Morton, Taylor, Turner … 1940 s onward

Plume models: e. g. , conservation of mass [πr 2 ρ W]z+dz = [πr 2 ρ W]z + 2πr ρ0 UE dz assume ρ ≈ ρ0 d (r 2 W) = 2 r UE dz or d (r 2 W) = 2 r U dz E dz

Do similar thing for momentum, buoyancy, …leads to set of equations. But what is UE? (closure!) Assume! UE = αW α = entrainment coefficient, usually taken to be ~ 0. 1

WRF model in LES mode U Wmax Take U, Wmax along several height levels… … calculate U / Wmax

Surface T – T 0 ~ 350° C U / Wmax 500 m 1250 m 2000 m 750 m 1000 m 1500 m 1750 m 2250 m 2500 m distance (m) [ RHS = approx. edge of plume] 15 minutes 20 minutes 45 minutes 60 minutes

Surface T – T 0 ~ 650° C U / Wmax 500 m 1250 m 2000 m 750 m 1000 m 1500 m 1750 m 2250 m 2500 m distance (m) [ RHS = approx. edge of plume] 15 minutes 20 minutes 45 minutes 60 minutes

All times, all levels U / Wmax T – T 0 ~ 350° C T – T 0 ~ 650° C distance (m) [ RHS = approx. edge of plume]

All times, all levels U / Wmax T – T 0 ~ 350° C T – T 0 ~ 650° C a 0 + a 1 x +a 2 x 2+a 3 x 3 (r =0. 9999) distance (m) [ RHS = approx. edge of plume]

Additionally …theory predicts that r ≈ α z

r ≈ α z gives α = 0. 08, 0. 09 Height (m) Look at spread of plume with height: Gaussian half-widths T – T 0 ~ 350° C T – T 0 ~ 650° C Half-width (m)

α appears to be “constant” with height “constant” in time “constant” over the two runs studied has a consistent variation with distance from source of the order of ~ 0. 05 – 0. 09 (agrees very well with commonly used values)

Latest work has been a collaboration between Leeds and Bristol Courtesy Mark Woodhouse

WRF results with theory WRF N fixed = 0. 01 s-1

Summary Ø WRF modelling can be meaningfully compared with simple plume theory and suggests an entrainment coefficient of the order of 0. 08 Ø WRF wind shear results being investigated at Bristol for comparison wi integral plume models. - Ø Example of close collaboration between modelling and theory: mutually beneficent.

gradient ~1/8 Ejection column Lateral spreading

W r

α = entrainment coefficient usually taken to be ~ 0. 1 but… Peter Tate, Ph. D thesis, Univ. of NSW

Cf. Fox 1970 “Forced Plume in a stratifie d fluid”