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

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

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

Plume models: e.g., conservation of mass [πr2 ρ W]z+dz = [πr2 ρ W]z + 2πr ρ0 UE dz assume ρ ≈ ρ0 d (r2 W) = 2r UE dz or d (r2 W) = 2r UE dz 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

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

Surface T – T0 ~ 350° C 500 m 750 m 1000 m 15 minutes 20 minutes 45 minutes 60 minutes 1500 m 1750 m 1250 m U / Wmax 2000 m 2250 m 2500 m distance (m) [ RHS = approx. edge of plume]

Surface T – T0 ~ 650° C 500 m 750 m 1000 m 15 minutes 20 minutes 45 minutes 60 minutes 1500 m 1750 m 1250 m U / Wmax 2000 m 2250 m 2500 m distance (m) [ RHS = approx. edge of plume]

All times, all levels T – T0 ~ 350° C T – T0 ~ 650° C U / Wmax distance (m) [ RHS = approx. edge of plume]

All times, all levels T – T0 ~ 350° C T – T0 ~ 650° C a0 + a1x +a2x2+a3x3 (r =0.9999) U / Wmax distance (m) [ RHS = approx. edge of plume]

Additionally …theory predicts that r ≈ α z

Look at spread of plume with height: Gaussian half-widths T – T0 ~ 350° C T – T0 ~ 650° C Height (m) r ≈ α z gives α = 0.08, 0.09 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)

Courtesy Mark Woodhouse Latest work has been a collaboration between Leeds and Bristol Courtesy Mark Woodhouse This side to show that the latest work has been a direct collaboration between Leeds and Bristol groups. Leeds have been running a series of WRF runs specified by Bristol. The work is motivated by the problem of plume response to wind shear.

WRF results with theory   WRf results superimposed on Mark’s theory graph. There is some agreement (described on next slide), WRF results fit within the envelope of the theoretical approaches. N.B. last statement not true for shear/N > 0.7 but this represents very high wind speeds. (see next slide). N.B. plume height varies in WRF model, there are wavelike features, etc; so the standard deviation of plume height is shown also (error bars). N fixed = 0.01 s-1

Summary WRF modelling can be meaningfully compared with simple plume theory and suggests an entrainment coefficient of the order of 0.08 WRF wind shear results being investigated at Bristol for comparison with integral plume models. - Example of close collaboration between modelling and theory: mutually beneficent.

gradient ~1/8 Ejection column Lateral spreading

r W

α = entrainment coefficient usually taken to be ~ 0.1 but… Peter Tate, PhD thesis, Univ. of NSW

Cf. Fox 1970 “Forced Plume in a stratified fluid”