Mathematics and the cloud_fit program

     This is a summary of the program created by Brian Lundberg (REU student during the summer of 2002 under Dr. Robert Leamon) and the underlying mathematical model. For a more thorough explanation, please take a look at Brian's website.



Mathematical Model


Assumptions and SimplificationsResulting Equations
  • 'Force-free' plasma
  • Cylindrically-symmetric flux tube
  • Globally electrically neutral plasma

  • Magnetic-force-free plasma
  • Neglect pressure gradients and/or gravitational and viscous terms
  • Maxwell's Equations (or what's left of them)
    • Ñ•(B) = 0
    • Ñ×(B) = μ0J
  • J×B = 0 or J = αB

Ñ×(Ñ×B) = Ñ( ÑB) - Ñ 2(B) → Ñ×(αB) = - Ñ 2(B) → Ñ 2(B) = - α2B


This vector equations is solved for the magnetic field components in cylindrical coordinates with zeroth and first order Bessel functions of the first kind.
  • BA = B0J0(αR)
  • BT = B0H J1(αR)
  • BR = 0
When R = R0, BA = 0. The first zero of the zeroth order Bessel function occurs at 2.4048; therefore, α = 2.4048/R0.

Integrating the equations for current density, J = αB, and magnetic field, BA, over the cross sectional area gives us equations for current and flux.

I = (0.5192)*(2π/μ0)*B0R0

Φ = (0.21588)*2π*B0R02




cloud_fit


     The program models the magnetic field of the cloud by fitting six parameters (and handedness = +/-1) and minimizing the error between the cloud model and WIND magnetic field data.

These six parameters are as follows:

θ0
initial latitude (deg)
φ0
initial longitude (deg)
t0
'center time,' time from initial cloud contact to the
spacecraft's closest approach to the center of the
cloud (hrs)
B0
magnetic field magnitude along the cloud axis (nT)
R0
cloud radius (km)
Y0
distance of closest approach as a fraction of the cloud radius
[Magnetic Cloud Model]

Feb 17-18, 1998

     IDL's AMOEBA function is used to minimize the error calculating function by adjusting the six input parameters. As AMOEBA often converges on local minima, rather than the global minimum, it is sometimes necessary to try several sets of initial parameters in order to obtain the anticipated solution. With this in mind, we set about auditting the program with several sets of initial parameters hoping that although some of these sets would converge on undesirable answers (local minima) others would converge on the correct answer (global minimum). Since values of φ0, B0, R0, and t0 are always positive and errors in these parameters seem less likely to mislead AMOEBA, we took the average value of each from our cloud_fit results to use in the sets of parameters. But since the parameters θ0, Y0, and the handedness of the cloud which take both positive and negative values tend to cause problems when the sign of the initial value is incorrect, we needed to create sets of initial parameters for both positive and negative values and so we ended up with 8 sets of initial parameters. Unfortunately, these sets of average values could not always provide desirable solutions and trying additional combinations was still sometimes necessary.




Magnetic cloud events


YearStart TimeStop Timeθ0 φ0B0 (nT)R0 (km) (χ/B0)2IMC (A)ΦMC (Wb)αMC(m-1)
1995Feb 08 03:00Feb 08 22:00-18.181.212.71.66E+070.0165.47E+084.74E+121.45E-10
Mar 04 11:00Mar 05 04:00-54.8154.411.81.42E+070.0354.38E+083.26E+121.69E-10
Dec 16 05:00Dec 16 22:0039.049.611.63.42E+070.0231.03E+091.85E+137.01E-11
1997Aug 03 14:00Aug 04 04:00-2.990.016.21.27E+070.0325.36E+083.56E+121.89E-10
1998Feb 17 10:00Feb 18 18:00-4.731017.6 1.90E+070.0328.73E+088.69E+121.26E-10
May 02 12:00May 03 17:00-8.535710.53.36E+070.0359.14E+081.60E+137.16E-11
1999Feb 18 14:00Feb 19 12:0040.958.316.83.54E+070.0111.54E+092.86E+136.79E-11
Apr 16 20:00Apr 17 21:0035.211822.31.44E+070.0188.34E+086.27E+121.67E-10
Aug 09 10:00Aug 10 16:0059.027012.82.35E+070.0237.81E+089.59E+121.02E-10
*Sept 21 21:00Sept 22 05:005.690.014.77.08E+060.0112.70E+089.99E+113.40E-10
2000*Jul 15 19:00Jul 16 09:000.962.348.12.64E+070.0153.29E+094.55E+139.11E-11
Jul 28 21:00Jul 29 11:004.931914.61.64E+070.0266.22E+085.33E+121.47E-10
Aug 01 00:00Aug 01 16:0018.211812.34.09E+070.0121.31E+092.79E+135.88E-11
Aug 12 06:00Aug 13 05:00-10.981.829.12.51E+070.0241.90E+092.49E+139.58E-11
Oct 03 17:00Oct 04 15:0015.833826.32.69E+070.0151.84E+092.58E+138.94E-11
Nov 06 23:00Nov 07 18:0023.010023.82.26E+070.0071.40E+091.65E+131.06E-10
2001Mar 19 22:00Mar 22 02:00-85.669.223.32.10E+070.0051.27E+091.39E+131.15E-10
Apr 22 00:00Apr 23 02:00-56.427013.21.85E+070.0276.34E+086.13E+121.30E-10
May 28 11:00May 29 11:00-7.631.512.42.80E+070.0159.01E+081.32E+138.59E-11
Jul 10 17:00Jul 12 08:00-12.92029.02.01E+070.0124.70E+084.93E+121.20E-10
Nov 24 15:00Nov 25 14:0031.110.418.55.13E+070.0242.46E+096.60E+134.69E-11

*Colors correspond with best fit, good fit, and worst fit given by the cloud_fit program.


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