1. Ash3d: A new USGS tephra fall model
Hans Schwaiger
Larry Mastin
Roger Denlinger

2. Why do we need a tephra fall model?
Forecasting ash distribution during unrest
Constraining eruption parameters through observation & modeling
Research into the physics & hazards of ash eruptions

3. What we were using
Ashfall: A 2-D model Developed by Tony Hurst

4. For Redoubt, Evan Thoms & Rob Wardwell wrote a Python script to automatically run Ashfall & plot these maps.
Main disadvantages:
We don’t have the source code
It’s limited to 2-D runs in a 1-D wind field.

5. What the model does
Calculated advection-diffusion of multiple grain sizes for 4-D wind field
Calculates deposit thickness and its variation with time.
Calculates time of arrival of ash at airports.
Writes out 3-D ash-cloud migration at time steps, for animation.

6. Model overview
The equation for advection of ash by wind and diffusion of ash by turbulent eddies is solved by method of fractional steps, treating advection and diffusion independently.
Advection step: Solve advection equation to get concentration at intermediate time step (q*):
Diffusion step: Solve diffusion equation, integrating the remaining fractional step:
Where q is ash concentration in kg/km3, u is velocity in km/hr, and K is an eddy diffusivity in units of km2/hr

7. Methods of Solution
A domain of cells is constructed either in a spherical (lat./lon.) or Cartesian coordinates
Wind fields must be provided and are used to passively transport ash as it settles.
The numerical schemes used are:
Finite volume methods with Riemann solvers, in which ash flux occurs at cell boundaries.
Semi-Lagrangian methods that backtrack ash transport along wind streamlines in a fixed Eulerian framework.
Turbulent diffusion is treated either explicitly (Forward Euler) or implicitly (Crank-Nicolson)

8. Illustration of advection schemes
Donor Cell
Upwind with
Dimension
Splitting
Corner
Transport
Upwind
t+Dt t
Semi-
Lagrangian

9. Illustration of advection schemes
Donor Cell
Upwind with
Dimension
Splitting
Conserves mass
Moderately fast
Dt = Dx/v
Increased numerical diffusion
Conserves mass
Slow
Dt = Dx/v
Low numerical diffusion
Corner
Transport
Upwind
Conserves mass only approximately
Fast
Dt = c Dx/v
Low numerical diffusion
Accuracy depends on order of interpolation
Semi-
Lagrangian

10. Illustration of diffusion schemes
Explicit Forward Euler
Easily implemented
First-order accurate
Dt = (Dx)2/K
t+Dt t
Implicit Crank-Nicolson
Assumes linearity
Requires solving Ax=b
Second-order accurate
Dt limited only by accuracy
t+Dt t

11. Why so many options?
Fast, but non-conservative, calculations can be automated for ensemble forecast runs
Fully conservative calculations (slower) might be necessary for greater confidence in particular results
Full mass conservation might also be required when including additional physics (aggregation)

12. Verification and Validation
Verification – Is your code solving the equations correctly?
Construct suite of test cases to check behavior in idealized conditions
Linear advection in x,y
Linear advection in z
Diffusion in x,y,z
Circular advection
Method of manufactured
solutions
Validation – Are you even solving the right equations?
Comparison with experimental data
Comparison with field data

13. Convergence for different schemes

14. Circular advection test case
Smooth boundaries are modeled well
Sharp boundaries are smoothed by numerical diffusion

15. Types of calculation that Ash3d can do
<12-50 km
deg.
( km)
Calculation on a sphere
Calculation on a plane
Uses lower-resolution Global Forecast System winds
Doesn’t conserve mass as well
But, it can model an eruption from any volcano on Earth.
Uses high-resolution winds from projected models (e.g. NAM, WRF)
Conserves mass very well
But, it can only model eruptions in certain geographic locations.

16. Projected Meteorological models used
NAM AK 45 km
NAM 11 km

17. Model inputs Wind files (1-D, 3-D, 4-D) Grid parameters: dx,dy,dz
Grain size distribution, fall velocities
Eruption source parameters:
Number of eruptions
Time, duration, plume height, erupted volume, Suzuki constant

18. Model outputs ESRI ASCII (For import to Arc products)
Deposit thickness (final, at specified times)
Ash cloud elevation & concentration
Ash arrival times & thickness at airports & other points of interest
3-D data in various formats:
ESRI ASCII (For import to Arc products)
Kml/kmz (Google Earth)
NetCDF
Raw binary

19. Example: Iceland (4-14-2010) 24-hours after start of simulation
Resolution = 0.33 degrees

20. Example: Iceland (4-14-2010) 24-hours after start of simulation
Resolution = 0.20 degrees

21. Example: Iceland (4-14-2010) 24-hours after start of simulation
Resolution = 0.10 degrees

22. Example: Iceland (4-14-2010) 42-hours after start of simulation
Resolution = 0.10 degrees

23. Example: Ensemble simulations
Probability of 1mm ash deposit
Contours at 5%, 50%, 95%
Redoubt event 6
dx,dy=5 km
dz = 1 km
K = 0 km2/hr
Grain sizes (1,2,4 m/s)
ESP:
Duration = 0.25 hr
Erup. Vol = km3
Plume H = random
uniform (6-20 km)
50 realizations
Approximately 90 minutes
to run 50 realizations

24. Example: Ensemble simulations
Probability of 1mm ash deposit
Contours at 5%, 50%, 95%
Redoubt event 6
dx,dy=5 km
dz = 1 km
K = 0 km2/hr
Grain sizes (1,2,4 m/s)
ESP:
Duration = uniform hr
Erup. Vol = uniform
km3
Plume H = normal
m= 12 km, s=3 km
50 realizations

25. Next steps Finish verification
Start validation: compare with field data
Automate operational runs for volcanoes in unrest
Aggregation

26. Further work Consider plume dynamics as input conditions
More realistic initial ash distribution
Weak plumes vs strong plumes in presence of ambient wind

27. Dusty gas software
Built on clawpack (LeVeque) and authored by Marica Pelanti
2-D axi-symmetric
3-D Cartesian

28. Thanks!