1

Two Modeling Problems in Lightning Physics(both Laplacian)

Richard SonnenfeldDepartment of Physics and Langmuir Laboratory, New Mexico Institute of Mining and Technology

2Physics Question 1“the leader problem”

If you want to mount a conducting cylindrical instrument on a conducting (or insulating) airframe and interpret the induced electric field in terms of what lightning in the distance is doing, what is the field/charge configuration on the instrument?

If know potential phi, then know surfacecharge density sigma.

σϵ0

=∇ Φ⋅n̂

3Instrument problem is easierthan leader problem

There are SOME closed-form solutions.The aspect ratio is not strange.

For the leader problem, you have a channel that is kilometers long and centimeters across … a real scale problem.

4Poor-man's numerical solution to leader problem

Most charge is on ends of leader.

5Low res numerical 2D solution to leader problem

6Physics Question 2“the instrument problem”

Given that a lightning leader “shorts out” a cloud (creates an equipotential connecting different charge regions)

What should the charge distribution on the lightning channel look like?

If know potential phi, then know surfacecharge density sigma.

σϵ0

=∇ Φ⋅n̂

710,000 Samples/s16-bits/Sample,Measure 8 channels

E-field (Channels 0-3)Timing (Channel 4)B-field (Channels 5-7)

A vector field-change sonde

8It worked fine on a balloon …

So let's put it on a small drone.

MUCH HARDER

A vector field-change sonde

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12LMA Plot for IC flash “C”

Planview

Altitude vs. time

x

Planview

Altitude vs. time

13Distributed Charge Analysis for IC flash “C”

Planview

Distance vs. time

‘Expected’ field

14Distributed Charge Analysis for IC flash “C”

Planview

Distance vs. time

‘Expected’ field

15Comparing Expectation and Experiment for flash “C”

q= 23 C

16

Instrument problem closed form solutions

Solve laplace's equation for a conducting disk in a plane for a constant electric field.

Field enhancement of 2X at poles.

3D problem (sphere in free space) also has a closed form solution. Field enhancement of 3X at poles.

17Cylindrical airframe problem

18Cylindrical airframe problem

19Numerical Method

Relaxation method – Every cell is set to the mean of its nearest neighbors. Trivial to do on a square, equi-spaced grid.

Minor programming challenges to efficiently set and reset boundary conditions and to exclude areas in interior of conductors from calculations.

20Cylindrical airframe problem

Do not know how to set an “equipotential” boundary condition. (Instead am using a fixed potential BC that is set to potential at middle of object).

Do not know WHAT potential to set object to.

21Motivation

22Motivation

23Progress

Using COMSOL 3D solutions More complex shapes Fake equipot (solves for dielectrics

and set dielectric to large #) Needed to write code to regrid

Comsol's polygons to rectangular grid (AFTER solution)

New Matlab function:

24Progress

Comsol calculates “V” and gridpoints “x”, “y”, “z”

aa=linspace(xmin,xmax,resolution);bb=linspace(ymin,ymax,resolution);[XX, YY]=meshgrid(aa,bb);F=scatteredInterpolant(x(j),y(j),V(j),'linear','none');imgV=F(XX,YY);

25Large E-field changes can occur during “RF quiet” periods. They are consistent with growing +

charges near the flash initiation point.

Lu model

Hager model

LMA

26Conclusions

Solving Laplace's equation for constant potential boundary conditions is useful both for designing instruments to study lightning and to understand lightning itself.

Developing mathematical tools that physicists can understand and does not require supercomputers would be useful both in pedagogy and research.

AE43A-04

27AE43A-04

Bill Winn, John Battles, Gaopeng Lu, William Hager

G. Aulich, K. Eack, H. Edens, S. Hunyady, S. Kieft, P. Krehbiel, B. Mong, W.Rison, M. Stock, and R. Thomas for technical advice and assistance.

National Science Foundation for primary support (ATM)

NASA Space Grant Program for some undergraduate support

National Forest Service for permitting field work in Cibola National Forest

Acknowledgements