6.4 Optimization approach

The optimization approach as proposed in Wheatland et al. (2000Jump To The Next Citation Point) is closely related to the MHD relaxation approach. It shares with this method that a similar initial non-equilibrium state is iterated towards a NLFFF equilibrium. It solves a similar iterative equation as Equation (65View Equation)
∂B-- ∂t = μF, (67 )
but F has additional terms, as explained below. The force-free and solenoidal conditions are solved by minimizing the functional
∫ [ ] L = B −2|(∇ × B) × B |2 + |∇ ⋅ B |2 dV. (68 ) V
If the minimum of this functional at L = 0 is attained then the NLFFF equations (2View Equation) – (4View Equation) are fulfilled. The functional is minimized by taking the functional derivative of Equation (68View Equation) with respect to an iteration parameter t:
1dL ∫ ∂B ∫ ∂B ---- = − ---⋅ FdV − ----⋅ GdS, (69 ) 2 dt V ∂t S ∂t
( ) [(∇-×--B-) ×-B-] ×-B F = ∇ × B2 { ( ) + − ∇ × ((∇-⋅ B-)B-) ×-B B2 − Ω × (∇ × B ) − ∇ (Ω ⋅ B ) } + Ω (∇ ⋅ B) + Ω2B , (70 )
−2 Ω = B [(∇ × B ) × B − (∇ ⋅ B )B]. (71 )
For vanishing surface terms the functional L decreases monotonically if the magnetic field is iterated by
∂B ----= μF. (72 ) ∂t
The first term in Equation (70View Equation) is identical with FMHS as defined in Equation (66View Equation).

A principal problem with the optimization and the MHD-relaxation approaches is that using the full magnetic field vector on the lower boundary does not guarantee the existence of a force-free configuration (see the consistency criteria in Section 4.6. Consequently, if fed with inconsistent boundary data, the codes cannot find a force-free configuration, but a finite residual Lorentz force and/or a finite divergence of the field remains in the 3D equilibrium. A way around this problem is to preprocess the measured photospheric data, as explained in Section 4.7. An alternative approach is that one allows deviations of the measured horizontal field vector and the corresponding field vector on the lower boundary of the computational box during the minimization of the functional (68View Equation). Wiegelmann and Inhester (2010Jump To The Next Citation Point) extended this functional by another term

∫ ν (B − Bobs) ⋅ W ⋅ (B − Bobs)dS, (73 ) S
where ν is a free parameter and the matrix W contains information how reliable the data (mainly measurements of the horizontal photospheric field) are. With this approach inconsistencies in the measurement lead to a solution compatible with physical requirements (vanishing Lorentz force and divergence), leaving differences between Bobs and the bottom boundary field B in regions where W is low (and the measurement error high). Consequently, this approach takes measurement errors, missing data, and data inconsistencies into account. Further tests are necessary to investigate whether this approach or preprocessing, or a combination of both, is the most effective way to deal with noisy and inconsistent photospheric field measurements. This approach, as well as a variant of the Grad–Rubin method, have been developed in response to a joint study by DeRosa et al. (2009Jump To The Next Citation Point), where one of the main findings w as that force-free extrapolation codes should be able to incorporate measurement inconsistencies (see also Section 6.6).
  Go to previous page Go up Go to next page