Ash3d:A new USGS tephra fall model

Hans SchwaigerLarry Mastin

Roger Denlinger

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

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What we were using

Ashfall: A 2-D model Developed by Tony Hurst

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.

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.

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

( ) 0q

u qt x

* 2 *

20

q qK

t x

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)

Illustration of advection schemes

Donor CellUpwind withDimensionSplitting

CornerTransportUpwind

Semi-Lagrangian

t t+t

Illustration of advection schemesDonor CellUpwind withDimensionSplitting

CornerTransportUpwind

Semi-Lagrangian

•Conserves mass•Moderately fast•t = x/v•Increased numerical diffusion

•Conserves mass•Slow•t = x/v•Low numerical diffusion

•Conserves mass only approximately•Fast•t = c x/v•Low numerical diffusion•Accuracy depends on order of interpolation

Illustration of diffusion schemesExplicit Forward Euler

•Easily implemented•First-order accurate •t = (x)2/K

tt+t

Implicit Crank-Nicolson

•Assumes linearity•Requires solving Ax=b•Second-order accurate •t limited only by accuracyt

t+t

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)

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,yo Linear advection in zo Diffusion in x,y,zo Circular advection

• Method of manufacturedsolutions

• Validation – Are you even solving the right equations?• Comparison with experimental data• Comparison with field data

Convergence for different schemes

Circular advection test case

• Smooth boundaries are modeled well

•Sharp boundaries are smoothed by numerical diffusion

Types of calculation thatAsh3d can do

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.

<12-50 km

0.5-2.5 deg.(50-250 km)

Projected Meteorological models used

NAM 11 km

NAM AK 45 km

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

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

Example: Iceland (4-14-2010)24-hours after start of simulation

Resolution = 0.33 degrees

Example: Iceland (4-14-2010)24-hours after start of simulation

Resolution = 0.20 degrees

Example: Iceland (4-14-2010)24-hours after start of simulation

Resolution = 0.10 degrees

Example: Iceland (4-14-2010)42-hours after start of simulation

Resolution = 0.10 degrees

Example: Ensemble simulations

Redoubt event 6

dx,dy=5 kmdz = 1 kmK = 0 km2/hrGrain sizes (1,2,4 m/s)

ESP:Duration = 0.25 hrErup. Vol = 0.007 km3

Plume H = random uniform (6-20 km)

50 realizations

Approximately 90 minutesto run 50 realizations

Probability of 1mm ash deposit Contours at 5%, 50%, 95%

Example: Ensemble simulationsRedoubt event 6

dx,dy=5 kmdz = 1 kmK = 0 km2/hrGrain sizes (1,2,4 m/s)

ESP:Duration = uniform

0.25-0.50 hrErup. Vol = uniform

0.005-0.007 km3

Plume H = normal= 12 km, =3 km

50 realizations

Probability of 1mm ash deposit Contours at 5%, 50%, 95%

Next steps

Finish verification Start validation: compare with field data Automate operational runs for volcanoes in

unrest Aggregation

Further work

Consider plume dynamics as input conditions More realistic initial ash distribution Weak plumes vs strong plumes in presence of

ambient wind

Dusty gas software

Built on clawpack (LeVeque) and authored by Marica Pelanti 2-D axi-symmetric 3-D Cartesian

Thanks!