Benchmarks

An Introduction and Tutorial to the "McKenzie Equations" for Magma Migration

A new formulation for the equations of magma migration in viscous materials as originally derived by McKenzie is presented, as well as a set of well-understood special case problems that form a useful benchmark-suite for developing and testing new codes.

Running stgMADDs Benchmarks

The Magma Development team has finished the alpha release of the Magma Dynamics Demonstration Suite (MADDs). The initial code implements the zero porosity/no melting magma benchmark for mid-ocean ridge solid flows in 2D and 3D built on the Underworld framework. The purpose of this code is principally to validate accurate pressure solvers for Stokes flow in current CIG supported software. The stgMADDs source code is available in CIG's Mercurial Repository (geodynamics.org/hg).

Milestone 1 Results and Analysis

Details how to run the first milestone of the MADDs project in 2D and 3D and provides some results of these simulations. It also gives the rates of convergence of the pressure gradient solutions as the resolution is increased.

2D Ridge Model

Velocity, pressure and pressure gradients solutions and L2 errors for a 2D ridge model with 120 x 60 elements.

3D Ridge Model

Velocity, pressure and pressure gradient solutions and L2 error fields for 3D ridge model.

Global Pressure Gradient Errors for 2D Ridge Model

Normalised global L2 errors.

Global Pressure Gradient Errors for 3D Ridge Model

Global normalised L2 pressure gradient errors at varying resolutions.

Milestone 2 Results and Analysis

Details the results of the Milestone 2 simulations and analyzes the accuracy of the advection scheme.

Gaussian Porosity Field Advection

Advection of Gaussian porosity field as a Stokes equation force term. The lower density porosity region rises due to gravity.

Ridge Model with Gaussian Porosity Field

Stokes flow with 2D ridge model boundary conditions and Gaussian porosity initial distribution, driven by a porosity dependent force term.

Semi Lagrangian Advection Scheme Test - Step Function

Diagonal step function initial distribution subjected to a shearing velocity field.

Semi Lagrangian Advection Scheme Test - Gaussian Distribution

Gaussian initial distribution subjected to a shearing velocity field.

Error Convergence for Advection Scheme - Step Function IC

Normalised global L2 errors for semi Lagrangian advection scheme with a diagonal step function initial condition as a function of resolution.

Error Convergence for Advection Scheme - Gaussian IC

Normalised global L2 errors for semi Lagrangian advection scheme with Gaussian initial distribution as a function of resolution.

Milestone 3 Results

Details the results for the third milestone, in which the melt velocity was determined given the existing solid velocity and pressure fields.

Melt Model - 2D Ridge with Constant Porosity

Solid and melt velocity, pressure and pressure gradient fields for 2D ridge model with constant porosity. Melt velocity magnitudes are significantly larger near the point of discontinuity due to their proportionality to the pressure gradients, which are largest at these points.

Melt Model - Gaussian Porosity Driven Flow

Solid and melt velocity, pressure and pressure gradient fields for Stokes flow driven by a Gaussian initial porosity distribution.

Milestone 4 Results and Analysis

Discussion of the system being modeled, and details of how to run the model with different initial conditions in 2 and 3D.

2D Solitary Wave

A 2D solitary wave with a wave speed of 7 rising through a solid with a constant speed of -2. The wave shows no visible diffusive behavior.

Noisy 1D Solitary Wave Initial Condition

Initial condition of a vertically changing 1D solitary wave with a certain amount of introduced noise, which allows 2D solitary waves to emerge over time.

Emerging 2D Solitary Waves

Solitary waves emerging from a noisy 1D solitary wave initial condition.

Emergent 2D Solitary Waves

Solitary waves having emerged from a noisy 1D solitary wave initial distribution.

Emergent 3D Solitary Waves.

3D Solitary Waves emerging from a noisy 1D Solitary Wave initial distribution

Milestone 5 Results and Analysis

Results and analysis for the isoviscous McKenzie equations (with melting) driven by a corner flow velocity BC.

Isoviscous McKenzie System with Corner Flow BC - 1

After 1 time step

Isoviscous McKenzie System with Corner Flow BC - 50

After 50 time steps

Isoviscous McKenzie System with Corner Flow BC - 3200

After 3200 time steps.

Velocity - x component

x component of the velocity field for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Velocity - y component

y component of the velocity field for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Dynamic (Stokes) pressure

Dynamic pressure due to viscous shear for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Porosity

Porosity field for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Compaction pressure

Compaction pressure due to compressibility of the solid phase for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Melt fraction

Melt fraction field representing the melt to solid phase of the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Melt velocity - x component

x component of the melt velocity field for the 3D isoviscous McKenzie model with ridge BCs at time step 150.

Melt velocity - y component

y component of the Melt velocity field for the 3D isoviscous McKenzie model with ridge BCs at time step 150.