the map is not the territory: challenges in extracting science from global simulations john lyon and...
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The Map is not the Territory:
Challenges in Extracting Science fromGlobal 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
Major Global Codes
Code TVD Order Grid div B Ion.
Elliptic clean
?
Y
Y
Y (strong
?
Y (strong)
Y (strong)
YSpherical ?
2YTanaka
YAMR1?Jahunnen
NCartesian1NWinglee
NCartesian2NOgino
Y Mixed2YMRC
YAMR2YMichigan
Y Stretched4YUCLA (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
1 1
2 2
( ( ) ( ) /i i
dUf U f U x
dt
V SUdV Fds
t
Discretization of Linear Advection
• Spatial domain is broken up into cells • Only a few time levels are carried
0// xuvtu
Donor Cell
• A simple first order algorithm
• Maintains monotonic solution
• Linear advection problem clearly shows diffusive character
1
1/ 2 1/ 2
1
n
i
ni i i
n n ni i i
v tu u F F
xv t
u u ux
Second Order
• A simple second order algorithm
• Does not maintain monotonic solution
• Introduces dispersion errors as seen in linear advection example
1
1/ 2 1/ 2
1 1
1 1
2 2
n
i
ni i i
n n n n ni i i i i
v tu u F F
xv t
u u u u ux
• Combines low order and high order fluxes
• Limiter to keep solution monotonic
• Provides nonlinear numeric resistivity and viscosity
1
1/ 2 1/ 2
1/ 2 1 1
1 1
1 1
1 1
2 2
max 0,
1
2
n
i
ni i i
n n n ni i i i i
n n n ni i i i i
n n n ni i i i i
v tu u F F
x
F u u sign u u
u u Bs u u
s sign u u sign u u
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.– 4th order means that the solution accurately
represents the Taylor series through the 4th 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)
2
'
B
B B
( )0
d
dt
B
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
1 1 1 1 1, , , , , ,
2 2 2 2 2
1 1 1 1, , , ,
2 2 2 2
/
/
x y yi j k i j k i j k
z yi j k i j k
B E E zt
E E y
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 RE 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
2
( )
B
Bv
LFMIonospheric 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/B2
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)
2 1/ 2o o s oc
1/ 2oRJ
78 7 0
0
o
o
o
o
e
e
o
3/ 2 1/ 2
0.85H2
5 0.45
1 0.0625p p
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
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
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
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
Comparison between Flow channels and BBFs
• Flow channels have properties similar to BBF results reported by Angelopolous
• FWHM of VX 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
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
Huh?
• what are those fingers?
• where's the growth
phase?
• magnetic connectivity
• Why is velocity vortex just
in the magnetosphere?
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