1. Ash3d: A new USGS tephra fall model Hans Schwaiger 1 Larry Mastin 2 Roger Denlinger 2 1 Alaska Volcano Observatory 2 Cascade Volcano Observatory

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/km 3, u is velocity in km/hr, and K is an eddy diffusivity in units of km 2 /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 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 Semi- Lagrangian t t+t

9. Illustration of advection schemes Donor Cell Upwind with Dimension Splitting Corner Transport Upwind Semi- Lagrangian Conserves mass Moderately fast Uses most 2 nd order terms t = x/u Increased numerical diffusion Conserves mass Slow Uses all 2 nd order terms t = x/u Low numerical diffusion Conserves mass only approximately Fast t = c x/u Low numerical diffusion Accuracy depends on order of interpolation

10. 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)

11. Types of calculation that Ash3d can do Calculation on a sphereCalculation on a plane Uses lower-resolution Global Forecast System winds But, it can model an eruption from any volcano on Earth. Uses high-resolution winds from projected models (e.g. NAM, WRF) But, it can only model eruptions in certain geographic locations. <12-50 km 0.5-2.5 deg. (50-250 km))

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

13. 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

14. Model outputs  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

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

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

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

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

19. Example: Ensemble simulations Redoubt: March 24, 2009 (event 6) dx,dy=5 km dz = 1 km nx,ny,nz = 140,140,22 nt = ~600 K = 0 km 2 /hr grain sizes (1,2,4 m/s) ESP: Duration = 15 min Erup. Vol = 0.007 km 3 Plume H = random uniform (6-20 km) 50 realizations (~90 min)

20. Example: Ensemble simulations Redoubt 2009 event 6 ESP: Plume H uniform 6-20 km Erup. Vol. 7x10 -4 km 3 Duration 15 min Probability of ash deposit > 1mm Contours at 5%, 50%, 95%

21. Example: Ensemble simulations Redoubt: March 24, 2009 (event 6) dx,dy=5 km dz = 1 km nx,ny,nz = 140,140,22 nt = ~600 K = 0 km 2 /hr grain sizes (1,2,4 m/s) ESP: Duration = uniform 15-30 min Erup. Vol = uniform 0.005-0.009 km 3 Plume H = normal = 14 km, =3 km 50 realizations (~90 min)

22. Example: Ensemble simulations Redoubt 2009 event 6 ESP: Plume H normal = 14 km =3 km Erup. Vol. uniform 5-9x10 -4 km 3 Duration uniform 15-30 min Probability of ash deposit > 1mm Contours at 5%, 50%, 95%

23. Next steps  Finish verification  Start validation: compare with field data  Automate operational runs for volcanoes in unrest  Sensitivity analysis  Adaptive mesh refinement  Include Aggregation  Topography

24. Summary  New USGS tephra fall model nearly complete  Automated simulations for volcanoes in unrest coming soon  Feedback appreciated on output formats  Simulation output  Ensemble output

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

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

27. Convergence for different schemes

28. Execution time for different schemes

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