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*Last Update January 2014*

If you have any questions regarding this benchmark effort please contact Hiro Matsui.

Snapshot of the radial magnetic field and radial velocity. Radial magnetic field B_r at CMB r= r_o = 20/13 (left) and radial velocity u_r (right) at mid-depth of the shell in quasi-steady state for insulated magnetic boundary case. Step of contour lines for B_r and u_r are 0.25 and 2.0, respectively. | Time evolution of magnetic energy E_mag (top) and kinetic energy E_kin (bottom) for insulated magnetic boundary case to t = 3.0. |

Snapshot of the radial magnetic field and radial velocity. Radial magnetic field B_r at CMB r= r_o = 20/13 (left) and radial velocity u_r (right) at mid-depth of the shell in quasi-steady state for pseudo vacuum boundary case. Step of contour lines for B_r and u_r are 0.25 and 2.0, respectively. | Time evolution of magnetic energy E_mag (top) and kinetic energy E_kin (bottom) for pseudo vacuum boundary case to t = 3.0. |

Code/Group Name | Authors | Description | Accuracy Results |
---|---|---|---|

ASH Mob | Nick Featherstone, Mark Miesch, Sacha Brun | ||

Calypso | Hiroaki Matsui | Spherical harmonics expansion on sphere and 2nd-order FDM in the radial direction. Crank-Nicolson Scheme is used for the diffusion terms, and 2nd-order Adams-Bashforth scheme is used for the other terms. Vorticity equation and Poisson equation for the toroidal vorticity is used for the time integration of the fluid motion. | Received |

ETH Code | Andrey Sheyko | Spectral simulation using spherical harmonics for the angular component and finite differences in radius. The incompressibility condition is guaranteed by the use of a toroidal/poloidal decomposition of the vector fields. A second order predictor-corrector scheme is used for the time integration | Received |

GeoFEM-MHD | Hiroaki Matsui | Spatial discretization: Finite-element methods with tri-linear hexahedral elements. Time integration: Fractional step scheme is applied. Crank-Nicolson scheme is used for the diffusion terms, and the other terms are solved by Adams-Bashforth scheme. Pressure and electric potential are solved to satisfy mass conservation and Coulomb gauge for magnetic vector potential. | Received |

GFD-dennou group | Shin-ichi Takehiro, Youhei Sasaki, Yoshi-Yuki Hayashi | Fully spectral method. Spherical harmonics and Chebyshev polinomials are used to expand the variables in the horizontal and radial directions, respectively. The nonlinear terms and the Coriolis terms are evaluated in the physical space and convert back to the spectral space (so called transform method). Diffusion terms are integrated with Crank-Nicolson scheme while other terms are done with Adams-Bashforth scheme. | Received |

Goddard Code | Weijia Kuang, Weiyuan Jiang | For the momentum equation, we solve the radial component of the velocity, the radial component of the vorticity, and the modified pressure simultaneously. For the induction equation, we solve the radial components of the magnetic field and current simultaneously. In the model, the velocity and the magnetic field are described by the poloidal and toroidal scalars. On the spherical surfaces, spherical harmonic expansions are used. In the radial direction, both compact finite difference algorithms and the Chebyshev collocation method are employed. For this benchmark effort, the Chebyshev collocation method is used. | Received |

H2000 | Rainer Hollerbach | Poloidal-toroidal decomposition, then expanded in spherical harmonics in angle (fully de-aliased). In r quantities are expanded in Chebyshev polynomials. The time-stepping is 2nd order Runge-Kutta, with the diffusive terms treated implicitly. | Received |

Helsinki Pencil Code | Petri Käpylä, Joern Warnecke | ||

IPGP / JHU team | Maylis Landeau, Julien Aubert | The code uses a spherical harmonic expansion in lateral directions, and finite differences in the radial direction (Dormy et al., 1998). The radial mesh interval decreases in geometrical progression towards the boundaries. Three-point stencil are used for second-order derivatives and five-point stencil for biharmonic operators (second-order accurate). Time integration involves a Crank-Nicolson scheme for diffusion terms and a second-order Adams-Bashforth scheme for other terms. To ensure numerical stability the time step is chosen as the minimum between the characteristic times of advection and Alfven wave propagation in one grid. | Received |

Leeds Spherical Dynamo (LSD) code | David Gubbins, Ashley Willis, Chris Davies, Maggie Avery, Chris Jones | The Leeds Spherical Dynamo (LSD) code solves the Boussinesq dynamo equations by representing velocity, v, and magnetic field, B, as poloidal and toroidal scalars. It is pseudospectral;the theta and phi variations are expanded in spherical harmonics, and r variations by variable order, variable mesh, finite differences with non-equidistant grid using Chebyshev zeros as grid points. The nonlinear terms are evaluated by the transform method. Time stepping is by a predictor-corrector method and the time step controlled first by the CFL (Δt < Δx/v) condition, and then modified using information from the predictor step. | Received |

MagIC | Johannes Wicht | ||

SFEMaNS | Jean-Luc Guermond, Francky Luddens, and Caroline Nore | SFEMaNS uses quadratic finite elements in the meridian section and Fourier expansions in the azimuthal direction. The time stepping is done with second-order backward Euler. The diffusive terms are treated implicitly. The divergence of the magnetic induction is controlled through a magnetic pressure. Parallelism is done in the meridian section using Petsc and with respect to the Fourier modes using FFTW. | Received |

Simitev-Busse Dynamo Code | Radostin Simitev, Friedrich Busse | This is a pseudospectral numerical code for the solution of the governing equations using a poloidal-toroidal representation for the velocity and magnetic fields. Unknown scalar fields are expanded in spherical harmonics in the angular variables and Chebyshev polynomials in radius. Linear and diffusion terms are computed in spectral space, while Coriolis and non-linear terms are computed in physical space and transformed to spectral space at every time step. Time stepping is implemented by a combination of an implicit Crank–Nicolson scheme for the diffusion terms and an explicit Adams–Bashforth scheme for the Coriolis and the non-linear terms; both schemes are second order accurate. Adaptive time-stepping is possible through a modification of this scheme. | Received |

UCSC Code | Gary Glatzmaier | Spectral: spherical harmonics and Chebyshev polynomials. Semi-implicit time integration: Adams-Bashforth and Crank-Nicolson. | Received |

xshells | Nathanael Schaeffer | xshells uses finite differences in radius and spherical harmonic expansion. The diffusive terms are treated using Cranck-Nicholson scheme, while the other terms are treated explicitely with a second order Adams-Bashford scheme. Non-linear terms are computed in physical space using the SHTns spherical harmonic transform library. | Received |

- Clarification of initial condition, boundary value [10-29-2013]
- Added an expression of horizontal perturbation of initial temperature by using the spherical harmonics (Section 2.1.4). Added an insulated boundary solution by H. Harder in Wicht et.al. (2009) (Appendix B.1). Thanks to A. Jackson for the suggestion.
- Time evolution data fix, addition to documentation [09-27-2013]
- The initial temperature was computed incorrectly in the example shown above. The time history data for the both insulated and psedo vacuum cases are now updated to the correct values (the old graph for the insulated case and pseudo vacuum case are available). Thanks for Jean-Luc Guermond for finding this.

The benchmark cases have azimuthally symmetrical solutions (m=4) and have symmetry (or anti-symmetry) with respect to the equator. Please report solutions from calculations made with no imposed symmetries.

Jackson et. al. revises his manuscript of the paper for the pseudo-vacuum benchmark, and proposes suggested solutions for the benchmark. We follow their suggested solution as the reference solution in the pseudo vacuum case. Thanks for Andrew Jackson for informing this.

The drift frequency has the largest difference between suggested solution and the old reference solution, but the difference is only 0.02%.Reference solution for Pseudo vacuum case Old value New value Kinetic enegy 40.679 40.678 Magnetic energy 219.40 219.39 Local Temperature 0.4259 0.42589 Local velocity -11.6357 -11.636 Local magnetic field 1.404 1.4043 Drift frequency 0.74976 0.74990 - Documentation is updated by using the suggested solution by Jackson et. al. and results using the correct initial temperature.
- Pseudo-vacuum value fix, addition to documentation [09-17-2013]
- The initial condition for the pseudo-vacuum case was computed incorrectly in the example shown above. The above pseudo-vacuum time history and data are now updated to the correct values (the old graph is available here). Thanks for Jean-Luc Guermond for finding this. Documentation for direction of drift and dependence on the direction of initial magnetic field is also added.
- Kinetic energy value correction [09-10-2013]
- The suggested solution for the kinetic energy in the insulated magnetic boundary case should be 30.773 rather than 30.733 (Tables 2, 4, 5 in the benchmark document). Further details are at http://www.sciencedirect.com/science/article/pii/S0031920108002707

- Benchmark Invitation Letter
- Letter to the scientific community describing the benchmark effort.
- Benchmark Overview Document [10-29-2013]
- An overview of the geodynamo benchmark effort, including the motivation and science behind it and details of the problems and measurements.
- Benchmark Results Form
- User fillable PDF form to report results from the insulated and pseudo-vacuum geodynamo code benchmarks.
- Evolution insulated and evolution pseudo-vacuum [updated 09-27-2013]
- Evolution of several energy values in benchmark runs using Calypso for the insulated and pseudo-vacuum cases.