A Novel Technique for
Rapid L* Calculation
Kyungguk Min1, Jacob Bortnik2 and Jeongwoo Lee1
1 CSTR / NJIT, 2 Department of Atmospheric and Oceanic Sciences / UCLA
The Scientific Magnetic Mapping and Techniques Focus Group
Jun 18th
Motivation
• L* calculation is essential in radiation belt
modeling.
• It is currently computationally expensive to
compute L*.
• We propose a new rapid method that
calculates L* from first principles.
• Calculating L* at 181x181 (32761) grid is
done in about 2 min.
L* is...
• Roederer's generalized L value:
• L* = 2πk /ΦR , Φ =∫B⋅dA =∮A⋅dl
• Commonly used for quantifying radial
0
E
transport in the radiation belts.
• Flux integration should be performed along
a closed curve in guiding drift shell.
Closed Curve in Energy Space
Bm(x, y, K)
Total energy:
W = PE + KE = qU + μmBm
– Bm(K): magnetic field magnitude at mirror
point as function of K.
– K: modified longitudinal invariant
K = ∮√(Bm-B(s)) ds
W0
(xi, yi)
Ki
Under conservation of energy and first two
invariants, particle trajectory is iso-
energy contours on a mirror point.
References:
"(U, B, K) Coordinates: A natural system for studying magnetospheric convection," Whipple, 1978
"Particle tracing in the magnetosphere: new algorithms and results," Sheldon and Gaffey, 1993
K
er and S. Zaharia: LANL⇤ V2.0: global modeling and validation
Comparison with LANL* and IRBEM: Input Setup
Table 1. Input parameters for the neural network LANLstar.
Table from Koller and Zaharia [2010]
Number
Parameter
Description
1
2
3
4
5
6
7
8–13
14
15
16
17
18
19
tY
tDOY
tUT
Dst
psw
By
Bz
W1 6
Lm
Bmirr
↵loc
rGSM
✓GSM
'GSM
Integer number representing the year = 1996
Day of the year (int) = 6
Universal Time in units of hours (float) = 1.24 (01h 14m 34s)
Disturbance storm time index (nT) = 7.78
Solar wind dynamic pressure (nPa) = 4.10
Y component of the IMF field (nT) = 3.72
Z component of the IMF field (nT) = -.13
See Tsyganenko and Sitnov (2005) = [.12, .25, .09, .05, .23, 1.05]
McIllwain value; Roederer (1970) = From IRBEM and UBK
Magnetic field strength at mirror point (nT) = From IRBEM and UBK
Local pitch angle (deg) = [90 60 30 10]˚
Radial coordinate in GSM system (RE ) = [9 8 7 6 5 4 3]
Latitudinal coordinate in GSM (deg) = 0˚
Longitudinal coordinate in GSM (deg) = 180˚
20
21
Vsw
Φ
Solar wind velocity (km/s) = 400.10
Electric potential (kV/RE) = constant (i.e. W' = μmBm)
Tile≈-30˚
er and S. Zaharia: LANL⇤ V2.0: global modeling and validation
Comparison with IRBEM: Input Setup
Table 1. Input parameters for the neural network LANLstar.
Table from Koller and Zaharia [2010]
Number
Parameter
Description
1
2
3
4
5
6
7
8–13
14
15
16
17
18
19
tY
tDOY
tUT
Dst
psw
By
Bz
W1 6
Lm
Bmirr
↵loc
rGSM
✓GSM
'GSM
Integer number representing the year = 1996
Day of the year (int) = 6
Universal Time in units of hours (float) = 1.24 (01h 14m 34s)
Disturbance storm time index (nT) = 7.78
Solar wind dynamic pressure (nPa) = 4.10
Y component of the IMF field (nT) = 3.72
Z component of the IMF field (nT) = -.13
See Tsyganenko and Sitnov (2005) = [.12, .25, .09, .05, .23, 1.05]
McIllwain value; Roederer (1970) = From IRBEM and UBK
Magnetic field strength at mirror point (nT) = From IRBEM and UBK
Local pitch angle (deg) = [90 60 30 10]˚
Radial coordinate in GSM system (RE ) = [9 8 7 6 5 4 3]
Latitudinal coordinate in GSM (deg) = 0˚
Longitudinal coordinate in GSM (deg) = 180˚
20
21
Vsw
Φ
Solar wind velocity (km/s) = 400.10
Electric potential (kV/RE) = constant (i.e. W' = μmBm)
Tile≈-30˚
Dipole vs. IGRF
Φ(L=1) = 1.90e5
Φ(L=8) = 2.37e4
L* = 7.9991
Ldip=8 Footprints (20.7˚)
Φ(L=1) = 1.95e5
Φ(L=8) = 2.35e4
L* = 8.2972
L* (U=0): Comparison with IRBEM: Dipole Only
L* comparison for dipole
0.08
UBK
IRBEM
RMS(L*true − L*est)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−9
2⋅10-3
3⋅10-4
−8
−7
−6
x (y =0) [R ]
i
err = √(∑PA(dL*)2/nPA)
i
E
−5
−4
−3
L* (U=0): Comparison with IRBEM: IGRF Only
L* comparison for IGRF
0.35
L*
−UBK
UBK
dip – IRBEM
L*
L*dip−IRBEM
– UBK
0.3
dip
UBK−IRBEM
L*dip – IRBEM
RMS(dL*)
0.25
0.2
0.15
0.1
0.05
0
−9
−8
−7
−6
xi (yi=0) [RE]
−5
−4
L*dip =∮surfAdip⋅dl /∮orbAdip⋅dl
−3
Comparison with IRBEM: Dipole+TS89 [SM]
L* comparison for Dipole+TS89
RMS(dL*)
60˚
8
L*(90°) []
7
Large deviation at
tail-like structure
90˚
6
0.16
0.14
0.12
Same order of
error predicted
in Dipole only 0.1
5
4
0.08
3
2
−9
UBK
IRBEM
−8
−7 −6 −5
xi (yi=0) [RE]
−4
−3
−9
−8
−7 −6 −5
xi (yi=0) [RE]
−4
0.06
−3
dL*
9
Comparison with IRBEM: Dipole+TS05 [SM]
L* comparison for Dipole+TS05
10
60˚
9
RMS(dL*)
0.16
8
0.14
7
0.12
6
90˚
5
0.08
4
3
2
−9
0.1
0.06
UBK
IRBEM
−8
−7 −6 −5
xi (yi=0) [RE]
−4
−3
−9
−8
−7 −6 −5
xi (yi=0) [RE]
−4
0.04
−3
dL*
L*(90°) []
0.18
Comparison with IRBEM: Dipole+TS89 [GSM]
L* comparison for Dipole+TS89
7
RMS(dL*)
6.5
0.12
0.1
5.5
0.08
5
0.06
4.5
0.04
4
0.02
3.5
−9
UBK
IRBEM
−8
−7 −6 −5
xi (yi=0) [RE]
−4
−3
−9
−8
−7 −6 −5
xi (yi=0) [RE]
−4
0
−3
dL*
6
L*(90°) []
0.14
Comparison with IRBEM: Dipole+TS05 [GSM]
L* comparison for Dipole+TS05
7
RMS(dL*)
0.02
6.5
L*(90°) []
6
5.5
0.01
5
4.5
0.005
4
3.5
−9
UBK
IRBEM
−8
−7 −6 −5
x (y =0) [R ]
i
i
E
−4
−3
−9
−8
−7 −6 −5
x (y =0) [R ]
i
i
E
−4
0
−3
dL*
0.015
Speed & L* Map
MATLAB
UBK
IRBEM (DIP only)
LANL*
181x181
~ 2 min
> 24+ min
< .5 sec
Dip + TS05
8
8
6
6
4
4
9
2
0
0
5
−2
L* map: none+geodip sm
−4
8
L*
2
y [RE]
q=e
μm = 1 keV/nT
K = 0.1 √nT RE
9
Φ = const.
L* map: ts05+geodip sm
5
−4
8
−6
6
1
1
−8
4
0
x [RE]
5
0
−2
−5
9
2
y [RE]
−5
2
0
5
−2
0
x [RE]
5
9
5
−4
−4
−6
−6
1
1
−8
−8
−5
0
x [RE]
5
−5
0
x [RE]
5
L*
−8
4
−6
6
L*
y [RE]
L* map: ts05+geodip sm
L*
Dipole only
y [RE]
Fortran
(x, y)
L* map: none+geodip sm
−2
Fortran
Φ = – A/r – B r sin(φ)
, where A=92 kV/RE,
B=10 kV.
Summary
1. Trajectory calculation in energy space is preferable because
– No error accumulation due to numerical integration (only errors are
from numerical interpolation),
– Strict conservation of energy and 2nd invariant (accurate contour and
perfectly closed if closed),
– Fast computation: contour calculation is rapid (ex. contour plot) and
independent of initial value of a particle, and
– Based on first principles only.
2. Preparation step is required to compute Bm(K) and the time depends on
the field model (≈10 minutes for TS05).
3. The difference (dL*) is generally less than 0.1 and this method
outperforms, at least in a dipole-like magnetic field, the previous method
based on Lagrangian approach.
End of slide