The Map is not the Territory: Challenges in Extracting Science from Global Simulations John Lyon and a host of others A map is not the territory it represents, but if correct, it has a similar structure to the territory, which accounts for its usefulness." - Alfred Korzybski

Outline Basic LFM Science from a code? –Verification Is it working properly? –Validation How does it compare to observations? –Scientific Understanding? flow channels Kelvin-Helmholtz Future directions

MHD equations in LFM

Major Global Codes Code TVD Order Grid div B Ion. Elliptic clean ? Y Y Y (strong ? Y (strong) YSpherical ? 2YTanaka YAMR1?Jahunnen NCartesian1NWinglee NCartesian2NOgino YMixed2YMRC YAMR2YMichigan YStretched4YUCLA (Raeder) YAdapted8YCISM (LFM)

Design Considerations for LFM High resolving power transport Adapted grid Div B = 0 Operation in low β, high Alfven speed regime Integral ionosphere

Spatial Discretization (finite volume) Conservative Finite Difference Scheme State variables are cell centered quantities and we discretize our model equation with numerical fluxes through the cell interfaces Scheme is conservative

Discretization of Linear Advection Spatial domain is broken up into cells Only a few time levels are carried

Donor Cell A simple first order algorithm Maintains monotonic solution Linear advection problem clearly shows diffusive character

Second Order A simple second order algorithm Does not maintain monotonic solution Introduces dispersion errors as seen in linear advection example

Combines low order and high order fluxes Limiter to keep solution monotonic Provides nonlinear numeric resistivity and viscosity Partial Interface Method

TVD switches Used to keep numerical solutions from overshoots in regions where they would normally happen Idea is to use a dissipative scheme where needed, and elsewhere go with a more accurate ( in sense of Taylor series, say ) scheme Leads to a non-linear technique because the solution algorithm itself depends on the local solution Addition of diffusion is generally triggered by sharp gradients, just how sharp can be a parameter of the switch

Numerical Order Usually describes the accuracy of an interpolation scheme in terms of a Taylor series. –4 th order means that the solution accurately represents the Taylor series through the 4 th order terms –also can represent a formal convergence rate for non-discontinuous problems Not always relevant for problems with shocks and other discontinuities

Treatment of the Magnetic Field Various approaches can be used to satisify the constraint that  B=0 –Projection method  B convection Modify the MHD equations so that  B convects through the system, allows jumps in Bװ –Use a magnetic flux conservative scheme that keeps  B=0 –Marder Scheme (diffuse divB)

Magnetic Flux Conservative Scheme Magnetic field placed on center of cell faces Electric field is placed at center of cell edges so that Cancellation occurs when field components of all six faces are summed up

Computational Grid of the LFM Distorted spherical mesh – Places optimal resolution in regions of a priori interest – Grid is not orthogonal – Logically rectangular nature allows for easy code development Finite Volume Calculation – Requires the calculation metric quantities Surface normal direction Surface Area Cell Volume

Magnetosphere-Ionosphere Coupling Inner boundary of MHD domain is placed between 2-4 R E from the Earth – High Alfven speeds in this region would impose strong limitations on global step size – Physical reasonable since MHD not the correct description of the physics occuring within this region – Covers the high latitude region of the ionosphere (45  -90  ) Parameters in MHD region are mapped along static dipole field lines into the ionosphere Field aligned currents (FACs) and precipitation parameters are used to solve for ionospheric potential which is mapped back to inner boundary as boundary condition for flow

LFM Ionospheric Simulation 2D Electrostatic Model      J || –J || determined at magnetospheric BC Conductivity Models – Solar EUV ionization Creates day/night and winter/summer asymmetries – Auroral Precipitation Empirical determination of energetic electron precipitation Electric field used for flow at magnetosphere – v =  B/B 2

Auroral Precipitation Model Empirical relationships are used to convert MHD parameters into a characteristic energy and flux of the precipitating electrons –Initial flux and energy –Parallel Potential drops (Knight relationship) –Effects of geomagnetic field –Hall and Pederson Conductance from electron precip. (Hardy)

MHD Simulations Run a number of cases with Northward IMF –n=5/cc, v = 400 km/s, Bz = 5 nT Chose Northward to avoid complications with activity –still interesting though for NBz current position and strength Relative strength of convection in the four cells Nature of reconnection above cusp All cases have run with N IMF for at least a two hours

Simulations continued Have varied the non-linear, TVD, switch sharpness, the spatial order of differencing, and have done one high resolution run for comparison standard grid is 50x48x96 => 0.5 Re sorts of resolution at 10 Re high resolution is 100x96x128

Simulation Results Show three panels for each run –Ionosphere FAC is false color surface (lower color bar) Potential is shown in contours (upper color bar) –Magntitude of B Noon-midnight plane shown vectors are unit vectors in direction of B –Vx same plane and vectors

Moderate TVD/ High Spatial Order (8)

High TVD/ 2 nd Order

High TVD/4 th Order

No TVD/ 8 th Order

Low TVD/ 8 th Order

High TVD/ 8 th Order

Doubled Resolution: High TVD/ 8 th Order

Simulation Results Show three panels for each run –Ionosphere FAC is false color surface (lower color bar) Potential is shown in contours (upper color bar) –Magntitude of B Noon-midnight plane shown vectors are unit vectors in direction of B –Vx same plane and vectors

Moderate TVD/ High Spatial Order (8)

High TVD/ 2 nd order spatial

High TVD/ 4 th Order

No TVD/ 8 th Order

Low TVD/ 8 th Order

High TVD/ 8 th Order

Doubled Resolution: High TVD/ 8 th Order

Conclusions Once you go to at least a moderately aggressive TVD scheme, the results start to look very similar qualitatively Your conclusion as to how similar given simulations are depends on what you look at –|B| shows little difference across all runs –Vx and the ionospheric structure are more sensitive to the numerics The differences, however, are sufficiently large that quantitative comparisons of many quantities will be quite different

Verification? Pressure for N IMF

Validation Major CISM Activity Both event and statistical

Comparison with geostationary observations ?? agreement for all three components of B Resolution improves the results for Bz, in particular Dayside works better than nightside –Most likely a ring current effect

Velocity histogram for tail plasma sheet

For Substorms

Flow Channels

Comparison between Flow channels and BBFs Flow channels have properties similar to BBF results reported by Angelopolous FWHM of V X profile and magnitude comparable BBF properties Use code to determine if they result from localized reconnection or interchange instability

It is also a good rule not to put overmuch confidence in the observational results that are put forward until they are confirmed by theory. -- A. Eddington

simulation

Flow channel

Specific entropy –Consistent with “bubbles”/interchange –But still don’t know why and how form

In search of the elusive K-H Here it is!

And we see it in the electric field

and in the density

the magnetic connectivity doesn't show rolling up of magnetic boundary

streamlines in and around density curl

Huh? what are those fingers? where's the growth phase? magnetic connectivity Why is velocity vortex just in the magnetosphere?

Southward as well –not as well organized

density doesn't show as clear roll-ups less internal structure associated with boundary

flank reconnection in vortex?

preliminary conclusions seem to have K-H depends on IMF direction magnetic reconnection/diffusion seems to maintain a cleaner magnetic boundary –higher resolution needed? sub-solar velocity acceleration in N case seems to give high growth rate, but where does it come from? vortices seem to lie on magnetosphere side

to do diagnostics –K-H growth rate as finction of position –careful study of Lagrangian behavior of fluid and magnetic field do they remain coupled? if not, why not? higher resolution

Future Code Directions Parallelization –“standard” CISM LFM shortly –MHD code for magnetosphere decoupled from solution of ionosphere/thermosphere General purpose capability ( Ogen ) Multi-fluid Code Hall Term Adaptive Grid Capability –Single grid adaptivity –Overture multiple grid + AMR

Multi-fluid currently in OpenMP, moving soon to MPI

Adaptivity Lapenta algorithm for Brackbill-Salzmann type adaptivity

Overture framework for solving PDE's on overset grids will use mostly grid facilities –overset, moving, AMR