Pergamon Adv. Space Res. Vol. 18. No. 8, pp. (8)‘285-@)289. 1996

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SCALING PARAMETERS AND PARAMETRIC COORDINATES IN STATIC AND TIME DEPENDENT MAGNETIC REVERSALS

S. C. Chapman and N. W. Watkins

Space Science Centre (MAPS), Universiry of Sussex, Brighton BN1 9QH, U.K.

ABSTRACT

We examine single charged particle dynamics in simple magnetic reversals first where the field is

static and then where it has time dependence, including the associated induction electric field. The

nature of the different regimes of particle dynamics in specific static and time dependent models

may be investigated both analytically by reference to the limiting regular behaviour of the system, and numerically by analysis of integrated trajectories. Here we examine the delay coordinate plot

technique as an alternative to Poincare plots to reveal the phase space portrait of systems; necessary in the time dependent system in which Poincare plots cannot generally be constructed directly.

INTRODUCTION

A considerable body of work exists (e.g. see references in [7], also [l], [2], [lo]) on single charged

particle dynamics in simple static models of the magnetic field reversal in the earth’s magnetotail.

More recently dynamics in smoothly varying time dependent reversals (intended to model the pre

substorm plasmasheet) have been investigated (131, [4)). In the static parabolic model regular and

stochastic regimes of particle behaviour have been identified, the particle remaining within a given regime for all t [l], [2]. In a parabolic field model with simple time dependence (and corresponding

induction electric field) [3], [4] th e same regimes of behaviour existed as in the static parabolic

model, but a single trajectory may exhibit the properties of the regular and stochastic regimes at

different times. A standard technique for static systems is to numerically integrate the particle

equations of motion to construct a Poincare surface of section, which represents a ‘cut’ through the phase space torus of a given particle trajectory. The nature of this torus as revealed by the

Poincare plot then essentially describes the nature of the dynamics. As the system is static we can

in principle integrate for arbitrarily large t to allow the trajectory to explore all available regions

of the Poincare plot. In a time dependent system this is no longer the case; in a smoothly time

varying system (the fields do not oscillate) the nature of the torus changes with time. Here we

give preliminary examples of use of an alternative technique; delay coordinate plots (eg [9] and references therein) to examine the dynamics. Examples will be given for both the static parabolic

model with the corresponding Poincare plots and for a time dependent model.

THE STATIC PARABOLIC MODEL

Normalizing the magnetic field to the linking field B,, temporal scales to the inverse of the gy-

rofrequency in this field at, = s and spatial scales to the particle gyroradius pr = & the static parabolic field model is:

B = (croz, 0,l) &o ~z --

a’ = B, ho

with E = 0 in the de Hoffman Teller frame. The parameter cue scales the degree of nonintegrability

in the sense that cro + 0 is an integrable limit [5]. We can deduce this from the Hamiltonian

Ho = j2 22

2+5- + @o(~o, 272) go = s2 i2 1 t=

T+ 1 2 T + $YO~ - z)~ = 5A (2)

(8)285

(81286 S. C. fl.lqmm and N. W. Warkins

stalilsphlo-o.0795 -.-

69.4.

99.3. a \

-69.2.

(I: \ \

st*icmmoam5

ol------ 0.1 . 0.05. f -\\

5 0. :

-0.05. \ -0.1 . /

:;I I 99.9 69 99.1 99.2 99.3 99.4 1.5 0.05 0 0.05 0.1 0.15 0.2

dww x

1 Figure 1: Static model (oe = 0.0795): delay coordinate (left) and Poincare (right) plots.

which in this limit, corresponding to a zero reversing field B, = 0 (or an infinite sheet thickness

compared to the gyroradius 2 ---t oo), yields gyromotion about the constant linking field B,.

The other limit, of vanishing linking field B, = 0 so that (~0 + co, cannot legitimately be taken

under this normalization; under a different normalization the integrable limit of serpentine motion

in a neutral sheet may be recovered [5]. For small values of (~0 the system might therefore be

expected to be close to integrable, and if chaotic, small changes in cue will yield substantial changes

in the dynamics. We can illustrate this by means of delay coordinate plots (e.g. 191). These are constructed from individual numerically integrated trajectories by recording the times t,, at which

the particle crosses the z = 0 plane (in the parabolic model all particles are trapped, ie they

will mirror at some finite 2). We then plot sucessive intervals between centre plane crossings, ie

delt(n + 1) = tn+2 - tn+l versus delt(n) = tn+l - t,. Such a plot essentially reveals structure

in the frequency information in the (in this case, z) oscillator. Examples for different values of

cre are shown in Figures 1-4, in each case the particle was started at the origin and at 45’ pitch

angle and normalized speed of 1. The integration was performed using a variable order, variable

stepsize technique (as in [3]). The method can be seen to reveal markedly different characteristic

structures in frequency space for small changes in as. The Poincare plots, constructed from the 2,

z coordinates of the trajectories as they cross the z = 0 plane are also shown, their morphology

can be seen to be reflected in the delay coordinate plots. In particular, the delay coordinate plots

can be extremely sensetive indicators of the topological properties of the system (see figures 2 and

3).

SIMPLE TIME DEPENDENT MODEL

We will consider a magnetic field of the form:

(using the same normalization as in the static model) with corresponding induction electric field E = -$&. Here CYO is as defined for the static model and the frame dependent convectional

E,(T) = -A&‘), which in general cannot be removed by a de Hoffman Teller frame transformation. The time dependent Hamiltonian equation of motion is

fi = ;1”1. (;(& + L2) + Q) = v.E = g !I’ = ;A; = ff; (3 - 2~~; + $)2 (4)

so that time dependence results in a pseudopotential which, as well as having the same spatial dependence as in the static model is a function of time dependent parameters Al(t) = czty fi/fi and As(t) = fi and the frame dependent At(t). The Hamiltonian equation of motion can be shown [S]

87.2.

67.

Y&B e7 * 67.2 67.4 67.8 87.0 68 88.2 80.4 WN

St5lbQlph5O-O.Oel 0.15

0.1 .

0.05.

?i o-

-0.05~

-0.1 .

-0.05 0 0.05 0.1 0.15 0.2 x

i

1

1 Figure 2: Static model (a~ = 0.081): delay coordinate (left) and Poincare (right) plots .

static alphaoEo.081 0.025

0.02.

0.015.

0.01 .

0.005~

3! 0.

-0.005~

-0.01

-0.015-

-0.02-

-“TQJ5 -0.04 -0.03 -0.02 -0.01 0 x

static a@hao=o.oel 0.025

0.02

0.015-

0.01.

i 0.005.

/

5! o- i

-o.ocs-

Figure 3: Static model (~0 = 0.081): a zoom of the Poincare plot, which shows two distinct resonant surfaces (right) which overlap (left).

slew: al@ac=o.oB5 7wo

6ooo-

5ooo-

-4cQo-

I

LO-

2ooo-

Figure 4: Static model ((~0 = 0.0835): delay coordinate (left) and Poincare (right) plots .

(8)288 S. C. (3hqmm and N. W. Watkins

%L----J 65 40.66 40.67 40.l38 40.60 40.7 40.71 dam

A-----J id 10' 10' IO"

d*l(n)

Figure 5: Time Dependent Model: the lhs plot shows a fast transition, the rhs plot, a slow transition.

to yield regular behaviour in the limit fi t 0, fi finite ie Xi -+ 0 or in the limit fi + 0, fi finite ie Ar --t 00; hence a single trajectory can at different times be in one or other of these limits. These limits will correspond to the regular oscillatory solutions of gyromotion about the z component of the field, and serpentine motion in the neutral sheet (ie the x component of the field) if the rate of change of Q is sufficiently small. The rate of change of !P has been quantified with two parametric coordinates, dXr(t) =I x’r 1 /A 2 and d&(t) =I x’z 1 /Af (31, (41, [51 which which appear since there are two timescales on which the pseudopotential changes. These timescales are just the characteristic particle Larmor period given by h2 = w, = fi, and the timescale for a transition in dynamics given by the time taken for Xr < 1 to /\I 4 1 as measured by this Larmor period.

If both dX1 << 1 and dA2 < 1 throughout the transition from the AI * 0 to the Ai --$ 00 limits (‘slow passage’), the limits can yield regular motion and for finite XI segments of stochastic behaviour can occur. If either dX1 >> 1 or dX2 > 1 through the transition (‘fast passage’) the motion may no longer be stochastic. We can illustrate this using numerical results obtained for a specific simple model. A model has been selected which has the property that Ai = Al(t), dX1 = dAl(t), and dX2 is constant; trajectories in this model then exhibit transitions with respect to both A1 and dX1 and are ordered with respect to dA2, Using the above normalization the magnetic field is

B= oosVoY

1

(& + &) >

which represents a “thinning” field; that is, the linking field decreases with t. The corresponding electric field has E,(T) = 0, this corresponds to a frame of reference in which the field line passing through X = 0, 2 = 0 is at rest. The parametric coordinates for this system are

For the delay plots shown in Figure 5, the field model has (~0 = 0.01 (a weak reversal) and for the left hand plot S&r = 0.02 (a fast changing reversal) and for the right hand plot S&T = 25 (a slowly changing reversal). In both csses to/(i&) = 0.01, and the particle has initial position r = (5 x 10m2, 0, l), such that initially the particle lies on the rest field line defined by the choice of E,(T) = 0. The initial v = (0,0.5, -0.01) at t = 0, with the velocity components chosen to ensure that the particle crosses the center plane z = 0 on a shorter timescale than the transition timescales of the system. The delay coordinate plots clearly distinguish between the fast and slow transition; the left hand plot reveals a system with a single frequency component that slowly changes with

Static and Time Dependent Magnetic Reversals (8)289

time ( dX2 > 1 for all t) whereas the right hand plot shows a period during which there is a wide

range of frequencies, corresponding to the period when X1 N 1 and dX1 < 1 (dX2 < 1 for all t).

CONCLUSIONS

We have shown examples of how delay coordinate plots can be used to establish the nature of

the dynamics of a single particle in a magnetic reversal. In the case of a static reversal, we have

produced delay coordinate plots for small values of QO (close to a known integrable limit). These

plots reveal dramatic changes in the dynamics for small incremental changes in cyo and detailed

structure in the frequency information of the trajectory, consistent with being in the parameter range for the onset of the destruction of KAM surfaces. In the case of a time dependent reversal,

the plots clearly show the distinction between systems which have the same CYO but distinct (ie fast

or slow) rates of change, as determined by adiabaticity parameters for the system. It is anticipated that the delay coordinate plot technique will be useful in exploring dynamics in time dependent

models and establishing in what sense the motion of trapped particles in the parabolic field is

chaotic.

REFERENCES

1. Buchner, J., and L. M. Zelenyi, Phys. Lett. A, 118, 395, 1986.

2. Buchner, J., and L.M. Zelenyi, J. Geophys. Res., 94, 11821, 1989.

3. Chapman, S. C., and N. W. Watkins, J. Geophys. Res., 98, 165, 1993. 4. Chapman, S. C., J. Geophys. Res., 99, 5977, 1994.

5. Chapman, S. C., submitted to J. Geophys. Res., 1994.

6. Chen, J., and P. Palmadesso, J. Geophys. Res., 91, 1499, 1986.

7. Chen, J., J. Geophys. Res, 97, 15011, 1992.

8. Lichtenberg, A. J. and M. A. Lieberman, Regular and Chaotic Dynamics, second edition,

Springer-Verlag, NY, 1992.

9. Shaw R., The dripping faucet as a model chaotic system, Science Frontier Express Series, Aerial

Press, Santa Cruz, 1984.

10. Wang, Z, J. Geophys. Res, 99, 5949, 1994