1. WRF plume modelling update NCAS, Leeds
Ralph Burton, Stephen Mobbs, Alan Gadian

2. WRF plume model: simple tests
Weather Research and Forecasting model
WRF configuration: 100m resolution (25km x 25km),
141 vertical levels, 30km top
Resting atmosphere – U.S. Standard atmosphere; dry; no ambient wind
Different thermal perturbations at “vent”
“Circular” vent
Results look like plumes, but -
Do they follow various theoretical models of plume behaviour
(cf. Mark Woodhouse & Co. work)?

3. Aspect ratio = 1:1
~15km
25km

4. Section through umbrella cloud
Very complex! – do simple tests on gross features.

5. Tests 1. Radial lengthscale, eruption / jet column.4 3
Height (km)
Turner, Buoyancy effects in Fluids 2 2
gradient ~1/8
b = (6/5)αz,
b = radial lengthscale
α = entrainment constant
α = 0.10 (Turner again)
gives b ~ z/8 1

6. Tests II. Radial spread of umbrella cloud
H = 13.5km; ΔT = 86K
H = 14.5km; ΔT = 111K
H = 15.3km; ΔT = 136K
WRF
From Sparks et al. Volcanic Plumes
p283

7. Tests III. Height of plume.
Theory
WRF
Courtesy Mark Woodhouse
gradient = 0.25
1/4
H = k (ΔT)

8. Summary. Results seem to agree with theoretical work
This is a simple visual inspection –
More rigorous tests to do
But no evidence of fundamental differences
Compare with Mark Woodhouse & Bristol team’s work.

9. Further Work. Similar approach adopted by e.g.
Neri and Macedonio, “Numerical simulation of collapsing
volcanic columns with particles of two sizes”
J. Geophy Res. B4,

10. Further work: full multiphase WRF
N + 1 phases: 1 air (gas + liquid and solid water) phase, N particulate phases (size bins)
Fundamentally N + 1 momentum equations, one for each phase, with interaction forces (drag) between them
Integrate N particulate momentum equations plus the combined (summed) momentum equation
There is only one shared pressure field and so the combined momentum equation is simply the usual one in the model, taking account of the contribution of the particles to the density.
All interaction forces between phases are equal and opposite (Newton's 3rd law) so cancel in the combined momentum equation
Drag terms in each particulate momentum equation
Modified equation of state taking account of the compressible fraction (air).

11. From Elghobashi (1994)
“On predicting
turbulent-laden flows”,
Applied Scientific Research, 52