Parameters

The AshDisperse package contains the setup() function that launches the command-line interface to guide the user through the setup of the model, producing a parameters object that contains the parameter settings needed to run a model simulation and a Met object that contains meteorological data (see meteorology for more details of the meteorological data inputs).

The parameter setup involves setting a number of model inputs split into categories that collect their purpose:

1. Source settings

2. Grain parameters

3. Solver settings

4. Physical parameters

5. Output settings

6. Model parameters

7. Meteorological data parameters

Source settings

The source settings characterise the volcanic eruption column and umbrella cloud from which ash is delivered into the atmosphere and dispersed.

We begin by setting the volcano source location, either as a named volcano (with location taken from the GVP Volcanoes of the World Database) or as a WGS84 latitude and longitude in decimal degrees. The location provides for georeferenced outputs and is used to extract meteorological data (see meteorology).

The emission model adopted here is a Gaussian distribution in the horizontal (x & y) and a Suzuki-style distribution in the vertical (z). These produce a three-dimensional spatial distribution over which the ash emission occurs.

The parameters set in the source settings are contained in parameters.source and are listed below, with their query in the CLI:

• Enter Volcano name or give latitude and longitude of source (as decimal degrees in format lat, lon)

  • The location of the volcanic source, either as named location from GVP dataset, or as a latitude and longitude.

  • Tnput either non-numeric text with name from GVP dataset (e.g. Etna), or comma-separated numeric values (e.g. 30.0, 15.5).

  • The latitude and longitude can be accessed as parameters.source.latitude and parameters.source.longitude, respectively. If the name is given, it can be accessed as parameters.source.name.

• Mass eruption rate

  • The intensity of the eruption, which characterized using a Mass Eruption Rate (MER, in kg/s). Note, here we required the MER of tephra (so excluding gases).

  • Enter as numeric value, including scientific notation (e.g. as 1000 or 1e3)

  • Accessed as parameters.source.MER.

• Eruption duration

  • The duration of the eruption (in seconds), which is used to determine the mass loading.

  • Enter as numeric value, including scientific notation.

  • Accessed as parameters.source.duration.

• Plume height

  • A plume height (in metres) is input to set the vertical height over which the emission occurs, denoted by \(H\).

  • Enter as numeric value, including scientific notation.

  • Accessed as parameters.source.PlumeHeight.

• Gaussian source radius

  • The radius (in metres) of the Gaussian source distribution in the horizontal plane, denoted by \(R\).

  • Enter as numeric value, including scientific notation.

  • Accessed as parameters.source.radius.

• Select Suzuki emission profile parameter

  • Choice of parameterization for the Suzuki-style distribution, either specifying the Suzuki k-parameter is input, or the altitude of the peak emission rate is specified

  • Enter either k or peak

• Suzuki emission profile k-parameter

  • If Suzuki emission profile parameter is specified as k, the Suzuki k-parameter is required and is denoted by \(k\).

  • Enter a positive numeric value.

  • Accessed as parameters.source.Suzuki_k.

• Suzuki emission profile peak-parameter

  • If Suzuki emission profile parameter is specified as peak, the emission profile is specified using the altitude of the peak of the emission rate, input here as a proportion of the plume height (e.g. a value of 0.9 corresponds to a peak in the emission rate at 90% of the plume height).

  • Enter numeric value in range [0, 1].

  • Accessed as parameters.source.Suzuki_peak.

Note, there are two methods in parameters.source to compute the Suzuki_peak value from Suzuki_k (parameters.source.peak_from_k()) and vice versa (parameters.source.k_from_peak()).

Grain parameters

The grain parameters set characteristics of the tephra particles emitted during the eruption and are contained in the parameters.grains object.

We can set multiple grain classes, and in the CLI we must add grain classes until the sum of their proportions is equal to unity.

For each grain class, we specify:

• Grain diameter

  • The diameter of the grain (in metres), denoted by \(d\), which is a major factor in determining the settling speed.

  • Enter as a positive numeric value, including scientific notation.

  • Accessed as parameters.grains.diameter[j] for grain class j.

• Grain density

  • The density of the grain (in kg/m³), denoted by \(\rho\), which also contributes to the settling speed.

  • Enter as a positive numeric value, including scientific notation

  • Accessed as parameters.grains.density[j] for grain class j.

• Grain class proportion

  • The proportion (mass fraction) of the grain class in the total grain size distribution.

  • Enter as a positive numeric value, including scientific notation.

  • Accessed as parameters.grains.proportion[j] for grain class j.

Solver settings

There are settings for the numerical solver that are specified in the solver parameters object, parameters.solver.

Each of the solver parameters has a default value.

The solver parameters are:

• Dimensionless domain size in x

  • Denoted by domX and accessed as parameters.solver.domX.

  • The advective distance in the x-direction (given by \(UH/W_s\), where \(U\) is the maximum wind speed in x, \(H\) is the column height and \(W_s\) is the particle settling speed) is used to non-dimensionalize the x-coordinate.

  • Diffusion can carry particles further than the advective distance, so the domain is taken to be larger than the advective distance by a factor domX.

  • Default value domX = 1.5.

  • Require domX > 1.

• Dimensionless domain size in y

  • Denoted by domY and accessed as parameters.solver.domY.

  • The advective distance in the y-direction (given by \(VH/W_s\), where \(V\) is the maximum wind speed in y, \(H\) is the column height and \(W_s\) is the particle settling speed) is used to non-dimensionalize the y-coordinate.

  • Diffusion can carry particles further than the advective distance, so the domain is taken to be larger than the advective distance by a factor domY.

  • Default value domY = 1.5.

  • Require domY > 1.

• Minimum resolution in z

  • Denoted by minN and accessed as parameters.solver.minN.

  • Input as \(log_2\) of minN, denoted by minN_log2 and accessed as parameters.solver.minN_log2.

  • Default value is minN_log2 = 4 (minN = 16).

• Maximum resolution in z

  • Denoted by maxN and accessed as parameters.solver.maxN.

  • Input as \(log_2\) of maxN, denoted by maxN_log2 and accessed as parameters.solver.minN_log2.

  • Default value is maxN_log2 = 8 (maxN = 256).

• Tolerance for Chebyshev series

  • Denoted by \(\epsilon\) and accessed as parameters.solver.epsilon.

  • The Chebyshev spectral series (with coefficients \(a_n\)) is taken of degree \(N\) so that \(|a_n|<\) epsilon for \(n > N\).

  • Default value is \(\epsilon = 10^{-8}\).

• Resolution in x

  • Denoted by Nx and accessed as parameters.solver.Nx.

  • Input as \(log_2\) of Nx, denoted by Nx_log2 and accessed as parameters.solver.Nx_log2.

  • Number of Fourier modes in the x-direction.

  • Default value is Nx_log2 = 8 (Nx = 256).

• Resolution in y

  • Denoted by Ny and accessed as parameters.solver.Ny.

  • Input as \(log_2\) of Ny, denoted by Ny_log2 and accessed as parameters.solver.Ny_log2.

  • Number of Fourier modes in the y-direction.

  • Default value is Ny_log2 = 8 (Ny = 256).

Note

The default resolutions for the horizontal coordinates, Nx_log2 = 8 and Ny_log2 = 8, are recommended as the minimum for usable output. Lower values will compute solutions with fast runtimes but that the outputs of the calculation are potentially under-resolved. Lower values could be used as an initial check of parameter settings, followed by runs with higher resolution. See Copy and update parameters for details of how to update parameters.

Physical parameters

It is possible to change the values of the physical parameters in the model, which are contained in the parameters.physical object.

The physical parameters, with their default values, are:

• Horizontal diffusion coefficient

  • Denoted by \(\kappa_{h}\) and accessed as parameters.physical.Kappa_h.

  • The turbulent diffusivity (m²/s) of the atmosphere in the horizontal directions (x and y).

  • Default value is \(\kappa_{h} = 100\) m²/s.

• Vertical diffusion coefficient

  • Denoted by \(\kappa_{v}\) and accessed as parameters.physical.kappa_v.

  • The turbulent diffusivity (m²/s) of the atmosphere in the vertical direction (z).

  • Default value is \(\kappa_v = 10\) m²/s.

• Gravitational acceleration

  • Denoted by \(g\) and accessed as parameters.physical.g.

  • The gravitation acceleration (m/s²), taken to be constant with altitude.

  • Default value is \(g = 9.81\) m/s².

• Viscosity of air

  • Denoted by \(\mu\) and accessed as parameters.physical.mu.

  • The dynamic viscosity of air (kg/m/s), taken to be constant with altitude.

  • The viscosity can affect the settling speed of grains, particularly for small sizes.

  • Default value is \(\mu = 1.85\times 10^{-5}\) kg/m/s.

Output settings

One of the outputs from AshDisperse is ash concentration in three-dimensions. The horizontal resolution is determined by the solver settings and grain parameters, but the vertical levels for output can be specified in the output settings, contained in the parameters.output object. A set of linearly spaced levels are specified using:

• Lower altitude

  • The lowest altitude level (in metres) to compute.

  • Zero is taken to be ground level.

  • Default is 0.0.

• Upper altitude

  • The highest altitude level (in metres) to compute.

  • Default is \(2H\) where \(H\) is the plume height specified in source parameters.

• Altitude step

  • The increment in altitude (in metres) for output levels between the lower and upper altitudes.

  • Default is for 11 elevation levels including the lower and upper altitudes.

Model parameters

AshDisperse computes several model parameters (typically dimensionless numbers constructed from parameters specified above) which are stored in the parameters.model object. There parameters cannot be set directly by the user. The model parameters are:

• Settling speed scale

  • An array of dimensional scales for the settling speed (m/s), with one value for each grain class.

  • Accessed as parameters.model.SettlingScale.

• Velocity ratio

  • An array of dimensionless numbers giving the ratio of the settling speed scale to the horizontal wind speed (taken at the plume top height), with one value for each grain class.

  • Accessed as parameters.model.Velocity_ratio.

• Concentration scale

  • An array of dimensional values (in kg/m³) that are used to scale dimensionless ash concentrations computed in the model, with one value for each grain class.

  • Accessed as parameters.model.cScale.

• Horizontal length scales

  • An array of dimensional values used to scale the dimensionless horizontal distances in the model, with one value for each grain class. Note, the values are much larger than the dispersion distances, since the model requires negligable ash concentration at the edge of the grid.

  • Accessed as parameters.model.xyScale.

• Dimensionless domain size in x

  • An array of dimensionless values for the domain size in the x-direction, with one value for each grain class.

  • This is an adjustment of domX to account for the radius of the source. If the source radius is larger than the horizontal length scale (parameters.model.xyScale), then the domain must be made larger so that the source is not too close to the boundary. If the source radius is smaller than parameters.model.xyScale, the value is equal to domX.

  • Accessed as parameters.model.Lx.

• Dimensionless domain size in y

  • An array of dimensionless values for the domain size in the y-direction, with one value for each grain class.

  • This is an adjustment of domY to account for the radius of the source. If the source radius is larger than the horizontal length scale (parameters.model.xyScale), then the domain must be made larger so that the source is not too close to the boundary. If the source radius is smaller than parameters.model.xyScale, the value is equal to domY.

  • Accessed as parameters.model.Ly.

• Source flux scale

  • An array of dimensional source fluxes, with one value for each grain class.

  • The total mass eruption rate is divided according to the proportion specified for each grain class.

  • Accessed as parameters.model.QScale.

• Peclet number

  • Defined as \(\mathrm{Pe} = \kappa_{h} / (UH)\) where \(\kappa_{h}\) is the lateral diffusivity, \(U\) is a scale for the wind speed, and \(H\) is the eruption plume height.

  • Characterizes the relative importance of diffusion to advection. If \(\mathrm{Pe}\gg 1\) then diffusive transport dominates over advective transport, while advection dominates over diffusion if \(\mathrm{Pe}\ll 1\).

  • Accessed as parameters.model.Peclet_number.

• Diffusion ratio

  • Defined as \(\kappa_{h}/\kappa_{v}\), i.e. the ratio of the lateral to the vertical diffusivities of the atmosphere.

  • Accessed as parameters.model.Diffusion_ratio.

Meteorological parameters

The model parameters require characteristic scales from the meteorological data. These are stored in the parameters.met object and are:

• Wind speed scale

  • Denoted by \(U\) and accessed as parameters.met.U_scale.

  • Taken to be the maximum value of the wind speed over altitudes from ground level to the plume height \(H\).

• Settling speed scale

  • Denoted by \(W_{s}\) and accessed as parameters.met.Ws_scale.

  • Each grain class has its own settling speed scale, so parameters.met.Ws_scale is an array.

  • Taken to be the settling speed for each grain class at ground level.