1. WRF plume modelling update, WP1 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
Different thermal perturbations at “vent”
“Circular” vent
This is just a summary slide showing the basic WRF configuration.

3. 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
Recap #1. This was shown at the last Vanaheim meeting: showing how the ejection column has the slope expected by theory. This is for a typical run.

4. 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
Recap #2. Showing the radial spread with time, similar to plots in Sparks “Volcanic Plumes” book.

5. Tests III. Height of plume.
Theory
WRF
Courtesy Mark Woodhouse
gradient = 0.25
Recap #3. How the height of the plume varies withsurface thermal perturbation: exact agreement with theory here.
1/4
H = k (ΔT)

6. Addition of more complexity. I. Background wind speed
10 m/s
15 m/s
20 m/s
30 m/s
40 m/s
New stuff starts here. This shows the effect of changing the background wind field, which is constant with height, no friction at surface. As expected, the plume shows a visual decrease in height with increasing speed. But is it the expected decrease?...colours show concentration, logarithmic scale. Inset text boxes show the background wind speed. Plots are x-sections through the plume.

7. Results. Bent-over plume. Ambient wind speeds in red
U=10m/s
U=15m/s
U=20m/s
U=30m/s
U=40m/s
Theory: maximum plume
height scales as H ~ U-0.33
gradient of line of best fit =
Divergence from theoretical results
may be caused by difficulty in defining “rise height” for modelled cases.
E.g. if only the three rightmost points are used, the gradient is , very close to the
theoretical value.
This shows the agreement with the theoretical results for a bent-over plume. There is pretty good agreement. It is not obvious how to define the rise height for modelled results, several approaches are available. This one uses the rise height of an isosurface of constant value. Other methods such as looking at the location of maximum perturbation (theta-theta_background) produce similar results. Other modelling studies have found departure from the power, e.g. “Buoyancy-driven plumes in rotating, stratified cross-flows: Plume dependence on rotation, turbulent mixing, and cross-flow strength” Lavelle, J. W. Jour. Gephys. Res (1997) who found the gradient to be -0.40, similar to this case. Also, it can be seen that the flow is possibly in several different regimes here, with higher background winds producing less turbulent flow. (Regimes being presumably the transition from ‘strong plume’ to ‘weak plume’) N.B. if the three rightmost points only are used, the slope is much closer to the theoretical value. This shows the sensitivity of this type of analysis to the definition of plume height. DISCUSSION POINT? Is this type of work suitable for comparison with the Bristol folks’ approaches? Are these results sensitive to the plume radius?

8. Whirlwind -like structures
Under certain circumstances (depending upon upstream wind speed, for fixed heat source),
ash is transported downwards along relative vorticity “tubes”, similar to observations
of certain cases.
WIND
Ash
Sparks et al. Volcanic Plumes
Here is a case when the background wind speed is 10 m/s. Vortices can be seen in the pollutant concentration (see also next few slides). This is shown schematiccally in a diagram from Sparks “Volcanic Plumes”. These “whirlwinds” (the term used by the authrors in the report on the Surtsey eruption shown here) have also been seen in man-made events e.g. oil fires.
Surtsey volcano BAMS
Oil tank fire MWR

9. Animation of ash concentration, this shows (visually) the presence of vortex tubes which appear to be shed from the bent-over plume. The frequency of shedding (Strouhal frequency) agrees well with the frequency shown in lab studies, e.g. Fric and Roshko, “Vortical structure in the wake of a transverse jet” JFM 279, 1-47 (1994)

10. Relative vorticity 6 5 4 3 2 1
The next 3 slides show relative vorticity (coloured) with contours of concentration (solid lines.) The vorticity tubes appear to “drag” ash downwards as can be seen by flicking through the various 3 slides. The plots are cross sections through the centre of the plume, wind is from left to right.

11. Relative vorticity 6 5 4
height (km) 3 2 1

12. Relative vorticity 6 5 4
height (km) 3 2 1

13. Surface fields, 1 min after eruption Density Pressure “Whirlwind”
U=10m/s
Surface fields,
1 min after eruption
Density
Pressure
No “Whirlwind”
U=15m/s
This shows the solenoidal term in the vorticity equation (z-compt anway.) When the wind speed is 15m/s, the flow is more barotropic and there is no obvious source of vorticity generated by this term. When the background wind speed decreases to zero, the pressure and density surfaces are colocated. So there is a region of parameter space (the relevant parameters being updraught velocity and background wind speed) where whirlwind-type structures appear. DISCUSSION POINT: is this important or merely interesting? Lab studies e.g. Fric and Roshko (ibid.) found these structures by using a jet of air. However the jet (to best of my knowledge) was not heated. Hence no solenoidal term? If these whirlwinds effectively transport ash downwards, it may be important. How often do they occur in reality? I have seen no high-res modelling of this.

14. Addition of more complexity. II. Orography.
100m horizontal resolution, initialised with wind profile derived from averaged
JJA ERA40 reanalyses (courtesy S. Sparks et al.) and N = 0.012, using 90m topography data
Images show time-averaged concentration at surface (4-hour simulation)
Plume height = 0.8km
Plume height = 1.8km
Plume height = 2.1km
These show the effect (or rather possibility) of incorporating orography into the high-res runs. The background wind comes from Steve Sparks and postdoc (sorry, forgotten name, not Vanaheim I don’t think). Briefly, flow is southerly (westerly) below (above) 1km. These are equivalent to long time-exposure photographs. if the plume height is below ~1km, the whole southern half of the island may be severely affected.
Plume height increasing; plume interaction with orography decreasing

15. Addition of more complexity. III. Addition of moisture.
Include moisture at lower levels; no ambient wind
WRF has sophisticated microphysics schemes
Results compare well with other models e.g. ATHAM
Herzog et al. Journal of Volcanology and
Geothermal Research 1998
WRF
This one just shows that including microphysics is possible and gives similar results to other studies, e.g. results from the ATHAM model. No attempt has been made to reproduce the ATHAM results shown here, but it is interesting that the same structure is found in both cases. DISCUSSION POINT: would this be a good way to test the theoretical advances of the Bristol group? Can they include moisture in their bent-over plume model? If so, it would be difficult to verify, presumably? This might be a good alternative…
Ice mixing ratio in plume. Same colour scale in all plots

16. Summary Work with Bristol Group to find
Model has been extended to include background wind
– some issues need resolving. (e.g. exponent in power law)
Interesting cases e.g. whirlwinds
Can include orography – results might be useful
Can include moisture – needs further investigation
Work with Bristol Group to find
most useful applications of these model runs.
Summary slide. Main point: work more closely with Bristol. Perhaps I (RRB) should go down to Bristol for a day/couple of days.

17. 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,

18. 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).
extra slides if needed

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