3D multi-fluid model Erika Harnett University of Washington

Equations solved For each species  }

Numerical scheme Second-order Runge-Kutta with flux correction smoothing on the plasma parameters only (B not smoothed), no time averaging. Solved on a Cartesian grid. The spacing on each grid is equal in all three dimensions Nested grid system is used for higher resolution (see next section) Assume continuous boundary conditions at the outer boundaries Assume a constant density and temperature (thus pressure) for all species at inner (planetary) boundary, and plasma assumed to have no bulk speed (but has a thermal vel.) No chemistry used in this version of the model

Numerical scheme - continued Variable time step used, determined from stability conditions i.e.  t < (smallest  x) / (fasted speed) divB correction can be turned on/off. Only necessary when simulating strong shocks at unmagnetized planets after initialization

Nested Grid System All grids are assumed to have shape (nx,ny,nz) All formulas are solved on each grid independently x os Remaining boundary values (S) are determined from interpolation Values at outer boundary of high resolution grid (O) are taken from overlapping points in lower res. grid Values from inner portion of higher resolution grids (X) reset overlapping points in lower resolution grids when the boundary conditions are determined in each time step. x x x ooo o o ooo o o o ss ss s s sss s s

Grid 1 - resolution:  x = 168 km outer edge of grid along x: R M & 2.77 R M y,z: R M & < 2.18 R M Grid 2 - resolution:  x = 336 km outer edge of grid along x: R M & 6.53 R M y,z: R M & 4.36 R M Grid 3 - resolution:  x = 673 km outer edge of grid along x: R M &15.8 R M y,z: R M & 8.71 R M Grid 4 - resolution:  x = 1345 km outer edge of grid along x: R M & 73.3 R M y,z: R M & 17.4 R M 4 grids, each with the shape (nx,ny,nz) = 111 x 89 x 89 Gridding System

Model Input Parameters Solar Wind (set statically or driven with changing conditions read in from an input file): Density = 3 cm -3 Velocity = 400 km s -1 in x direction T e = 10 eV, T i = 4.3 eV B x = nT, B y = 2.52 nT, B z = 0 nT Three species: H + (solar wind), H + (ionosphere), O + (ionosphere) Note: A small amount of ionospheric species must be present in solar wind and a small amount of solar wind species at inner boundary to prevent divide by zero errors.

Model Input Parameters - continued Planetary (inner) boundary (single grid point width): Altitude = 302 km O + Density = 875 cm equator (decreases to 70% less at poles) H + Density = 100 cm equator (decreases to 70% less at poles) No day/night asymmetry in density T e = 0.3 eV, T i = 0.07 eV Rotation period = 24.6 earth hours

Odds and Ends Code written in Fortran, fastest code with Intel compiler. Parallelized to run on multi-core processors using OpenMP. Currently run on a high-end desktop (dual quad-core Intel chips ~ 3 GHz, ~ 4-8G of RAM) Output necessary to restart simulation (density, momentum, pressure, magnetic field for each species at all grid points) written in binary during course of simulation and at the end of a run. Time between outputs varies, set as in input parameter. Wave reflection only an issue for sub-sonic/sub-Alfvenic incident winds. Prevent reflections at outflow boundary by having large outer simulation grid such that bow shock contacts outer boundaries down-stream of planet

Validation Comparison to hybrid results: Harnett, E., R. Winglee, and P. Delamere (2005), "3D multi-fluid simulations of Pluto's magnetosphere - a comparison with 3D hybrid simulation results", Geophys. Res. Lett., 32, L19104, doi: /2005GL Standard reconnection (Harris current sheet) problem: Winglee R. M., E. Harnett, A. Stickle, J. Porter (2008), "Multiscale/multifluid simulations of flux ropes at the magnetopause within a global magnetospheric model", J. Geophys. Res., 113, A02209, doi: /2007JA Comparison to satellite observations: Paty, C., W. Paterson, and R. Winglee (2008), Ion energization in Ganymede’s magnetosphere: Using multifluid simulations to interpret ion energy spectrograms, J. Geophys. Res., 113, A06211, doi: /2007JA

Validation - continued Comparison to satellite observations: Snowden, D., R. Winglee, C. Bertucci, and M. Dougherty (2007), Three-dimensional multifluid simulation of the plasma interaction at Titan, J. Geophys. Res., 112, A12221, doi: /2007JA Convergence Spatial tested with results from: Harnett E. M., R. M. Winglee, C. Paty (2006), Multi-scale/multi-fluid simulations of the post plasmoid current sheet in the terrestrial magnetosphere, Geophys. Res. Lett., 33, L21110, doi: /2006GL Temporal not really an issue.