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

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

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

4. 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

5. 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

6. 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

7. 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]

8. 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]

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

10. 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]

11. Additionally …theory predicts that
r ≈ α z

12. 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)

13. 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)

14. 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.

16. 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

17. 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.

18. gradient ~1/8
Ejection column
Lateral spreading

19. r W

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

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