Transcript Wheatland method for NLFFF MHD simulation

Optimization NLFFF Model
J.McTiernan
SSL/UCB
HMI/SDO 27-Jan-2005
Optimization method (cont):
• The first step is to set up the initial conditions. The
method requires that the B field on all boundaries be
unchanged. For the test case of the Low-Lou field, the
B field on all boundaries is known.
• For real data, only the lower boundary is known; the
bottom boundary is the magnetogram, the upper and
side boundaries are the initial field.
• The initial field can be a potential field or a linear
force-free field that is extrapolated from the
magnetogram. In the test case with all known
boundaries, this results initially in discontinuities at the
outer boundaries. These smooth out during iterations.
Optimization method (cont):
• Iterative process, starts with initial B field. Calculate F,
set new B = B + F*dt (typical dt =1.0e-5)
• “Objective function”, L, is guaranteed to decrease,
because dL/dt is guaranteed to be negative since dB/dt
is proportional to F.
• The change in L becomes smaller as the number of
iterations increases, L asymptotically approaches zero,
but never quite gets there.
• Keep iterating until L stops decreasing, or until some
preset iteration limit (currently set to 500000).
Optimization method (cont):
• For the test case, the quantity F, that drives the
minimization, is non zero only on the boundaries.
Differences between the calculated and initial field
propagate into the volume as we iterate.
• For the case with only the lower boundary, F is only
non-zero at the lower boundary, the difference between
the calculated and initial fields propagates upwards
from the lower boundary.
• The final extrapolation is dependent on the boundary
conditions. If only the lower boundary condition is
specified the solution also depends on the initial field.
Tests of NLFFF model:
• 4 different tests, based on the Low-Lou model discussed
in previous presentation:
• 1st test, 64x64x64 cube, all boundary fields are specified
by LL model.
• On 2.4 GHz Linux machine, this took 4300 minutes,
50000 (max) iterations, and L was still noticeably
decreasing at the end of the calculation. (Note that the
code is written in IDL. Run time scales with the total
number of grid points, e.g., a 32x32x32 cube will take
1/8th the time.) A typical value for real data, for a
64x64x64 cube, is 200 minutes. This is because real
data has noise.
Tests of NLFFF model:
• 2nd Test, added random noise to the initial field; the
maximum noise level on the lower boundary was set to
be 2% of the max total field strength on that boundary,
approximately 6 Gauss. (How realistic is this?)
• This “converged”-- the objective function, L, stopped
decreasing after about 4140 iterations, or about 370
minutes. If the noise level is doubled, the iterations
stop after 211 minutes.
Tests of NLFFF model:
• 3rd Test, ignored the outer boundaries of the LL field,
and set the outer boundaries to the initial potential
field. This is still running, (as of 26-Jan-2005 2pm)
after 33000 iterations, and about 1790 minutes…
• 4th test, same as 3rd test, but with noise added to lower
boundary as in 2nd test. Converged after 8600 iterations
and 500 minutes. (For some reason, the tests with only
the lower BC’s are about 60% faster, per iteration,
about 3 sec per iteration, instead of 5 sec per iteration
for the tests with all BC’s specified.)
Tests of NLFFF model:
• We can conclude that noise matters, and the level of
noise will restrict the eventual use of the field
extrapolation, in the sense that the field will not
converge to as force-free a state.
• Also the field lines can be different. This could be an
issue when interpreting model results.
• Future work will include more realistic noise estimates
and Monte Carlo tests for simulated and real
magnetogram data, spherical coordinates version is
being debugged, will have to redo in Fortran…