PFC/JA-87-50
Terminated Whistler Emissions in the
Magnetosphere
Miller, R.H.; Molvig, K.
January 1988
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA
This work was supported by the Office of Naval Research No. N00014-83-K-0319.
Reproduction, translation, publication, use and disposal, in whole of part, by or for the
United States government is permitted.
Paper presented in part at:1987 Cambridge Workshop in Theoretical Geoplasma Physics,
July 28-August 1, 1987.
Paper published in part: 1987 Cambridge Workshop in Theoretical Geoplasma Physics,
volume 7, SPI Conference Proceedings and Reprint Series, Scientific Publishers, INC.
TERMINATED WHISTLER EMISSIONS IN THE
MAGNETOSPHERE
Ronald H. Miller
and
Kim Molvig
Plasma Fusion Center, M.I.T., Cambridge, MA 02139
Abstract
Terminated emissions are obtained using a fully self-consistent nonlinear whistler wave theory. The construction of a variable perpendicular
distribution function dependent upon the inhomogeneous resonant velocity
results in terminating the wave-particle interactions. This turning-off of
the interaction is known as a terminated emission.
I
Introduction
There has been a great deal of work by Nunn et at. [1,2,3], Vomvordis et
al. [4,5] and Molvig et al. [6,7] in understanding the theoretical nonlinear
dynamics. The theory by Molvig et al. [6,7], has a relatively simple analytical solution which can be used to illuminate the nonlinear dynamics
and will form the basis of the work performed in this paper. The theory
presented here will use an asymptotic results for the nonlinear currents expressed in terms of r, the time the particle is trapped. What necessitates
an asymptotic analysis is the complexity of the nonlinear dynamics. In
this asymptotic limit, the current integrals are simplified to a sum over the
trapped perpendicular velocities where the integrand becomes an explicit
function of the trapping time, r. The nonlinear dynamics are reduced to
solving a simple convection equation for r,
a
+ V.
a)r=1()
which is integrated subject to the trapping condition:the trapping condition
comes from a detailed balance between field aligned gradients and longitudinal Lorentz force. The emission mechanism is greatly simplified but
accurately models the nonlinear dynamics; provided, the resonant particles
remain trapped for several trapping oscillations.
The theory by Molvig et al. [6,7] used a simplified perpendicular distribution function which was independent of the spatial resonant velocity.
In the actual magnetosphere, the resonant velocity has a spatial variation
along the geomagnetic field line with a minimum occurring at the geomagnetic equator. This variation must be included in the perpendicular
1
distribution which is used in calculating the nonlinear currents. Without
the variable distribution, the emission remains attached to the equator.
The emission form then amplifies without disconnecting from the equator
and becomes numerically unstable. The inclusion of the new distribution
results in disconnection and termination of the whistler pulse. The resulting emission. forms produced by the variable distribution function will be
discussed in this paper.
II
Theory
A ducted [8], coherent, right hand circularly polarized wave propagates
from the south pole where it interacts near the geomagnetic equator with
electrons traversing in the opposite direction. The wave-particle interactions occur near the geomagnetic equator where the magnetic and density
inhomogeneities are small to allow for trapping of the resonant particle.
Through an axial bunching in phase a nonlinear current is produced which
drives the slowly varying amplitude and phase resulting in an emission.
II.1
Nonlinear Equations of Motion
The equations of motion for the electron in a monochromatic whistler wave
propagating along an inhomogeneous geomagnetic field are [1,6,7,9]
ds
=so(2)
dvs
-
=(v1 1 -
w
B'"
-)wce- cos?,+
1 v||vjeB
s(3)
2
dvjj =v -ce Bw vi cos 1 - 1 vi2 B
di
B
2Bas
(4)
db
d ==
dt
(5)
(v- 1 - wik )
BW
Wce -- sin
Vi
B
+ we,
where Ampere's Law has been used to relate the electric and magnetic
fields, i.e.,
ck
ck
Bw= -E" = -ETA
(6)
is the triggering input wave electric field and A is a slowly varying
amplitude, initially of order unity. The last terms in Eq.(3) and Eq.(4)
ET
2
are from the inhomogeneity parallel to the geomagnetic field. The inhomogeneity perpendicular to the geomagnetic field has been neglected since
this variation scales as 1/R,. The triggered emissions originate at 4 earth
radii from the earth, near the plasmapause, therefore R, ~ 4RE. The relative phase between the perpendicular velocity vector and the electric field
vector can be written as
0 = 0 - (0. + 0)
(7)
where 4 is the gyrophase.
phases:
The electric field vector is described by two
1. p, a slow phase that describes the amplitude evolution and the resulting frequency variation,
bw -
(8)
2. 0,,, a fast phase that describes the input electric field frequency,
19.t
WO =
(9)
The total frequency, at any given time, is the sum of Eq.(8) and Eq.(9),
= wO + Sw
(10)
The equation that describes the gyrophase evolution, Eq.(5), can be transformed into,
dk
dit
_
=
(v1
- _VII
-
V
e)
Wce
BW
sin4o + kvj - w + w,,
B
where Eq.(7) was used to define -0. Using the resonant velocity, which
is given by matching the doppler-shifted wave frequency and the gyrofrequency of the electron,
VR
=
(12)
k
Eq.(11) becomes
d
vp
(v - a )
= k(I - VR) -w
Vs
dt
Bw
B-e sin
3
(13)
The second term on the right hand side of Eq.(13) comes from the wave electric and transverse Lorentz forces. The first term comes from a longitudinal
Lorentz force which is larger than the second by ????. It is this term, that
is responsible for the trapping that generates the nonlinear currents. The
exact equations for the orbits that we will be using are Eqs.(2),(3),(4),and
(13).
The particles of interest will have their parallel velocities on the order
of the resonant velocity so a new velocity variable will be defined,
V= v11 - VR
(14)
which describes the perturbation of the parallel velocity about the resonant
velocity. Inserting Eq.(14) into the equations of motion, they become
= Vi + V
dv
=
(15)
Ew2
-v.-
w"cos
(1+")
dvr
= -(
dV)
k
- =kvI
we
-
[
- (VR + v)
c
(3 -2
L2
(1
W
-
)
k "(16)
-v)we,
(vj -
1f)
B"
ee
1
18B
cos 4 + 1vi(VR+ v) B 8B
Bw
-- sin
(17)
(18)
The explicit spatial dependence of the electron density and magnetic field
were assumed parabolic, i.e.
B = Bo(1+
)
(19)
n
)
(20)
= no(1+
where BO and no are the values of the density and magnetic field at the
geomagnetic equator and their corresponding scale lengths, LN and L.
4
11.2
Asymptotic Theory
The orbit equations have a common factor B'/B which is a small quantity
in the magnetosphere. For triggered emissions, the wave magnetic field,
B"',
is on the order of 10 pT or less. This is five orders of magnitude smaller
than the geomagnetic field at the plasmapause. This provides a natural
expansion parameter for the equations of motion.
If we define a smallness parameter, e, as
Wt= < 1
(21)
Wce
where wt,. is the trapping frequency, an asymptotic theory can be developed
[6,71. For trapping to occur, the inhomogeneity factors and wave terms must
be comparable in Eq.(16). Using this and the requirement that there be
many trapping oscillations as the particle moves thought the interaction
region, an expansion parameter el/" is obtained [6,7](For further information on this derivation, the reader is suggested to consider the papers by
Molvig et al.)
The rescaling of the equations of motion is done by choosing an appropriate time and velocity scale. The velocity scale is set by the resonant
condition,
V, = IVR(w ~ we,/2)
(22)
By assuming that Lo
w/2, due to ducting [8], one obtains VR - We,/k.
For wave frequencies near the gyrofrequency, the cold electron dispersion
relation
N2
L0
V
VRI
= 1 - W(W - U.e,)
(23)
yields a wavevector on the order of the collisionless skin depth, k - W,/c.
Using this relation for the wavevector, the velocity scale can be written as
V|
k
W-
(24)
C-
U;,
An important relation that will be need later in performing the asymptotic
expansion is the ratio,
V
VR
Wc
U~t
(25)
kVR uje,
The time scale is set by the trapping period which will be discussed
later in the paper but can be seen from the first term on the right hand
side of Eq.(16). The trapping is a longitudinal trapping and will cause
an oscillation about the perturbed velocity, v. For this reason, the scale
frequency, which is proportional to the inverse of the time scale, will be on
the order of the trapping frequency which scales as,
5
w,2 ~ kv- Bw
B c
(26)
therefore, the scaling frequency will be defined as,
(Wcewpe
W,
)1/2
(27)
Using the scaling arguments w, and V,, Eq.(15)-(18) can be rescaled.
Rescaling the equations of motion and performing the asymptotic expansion correct to order el/ 3 , we obtain for the approximate nonlinear equations of motion for the electron [6,7],
ds
dt
=-=
W
We,
du
--
v1 A
W
(28)
-e
cos $ - S
(29)
S(30)
dt
where u
kv and S has the form
S = S',+ T(
(31)
and S' can be written as
(32)
S'=CvI +D
The factor S is the effective inhomogeneity and it will play an important
role in energy exchange and trapping. The coefficients D, C and T are
complicated functions of the frequency, scale lengths and magnetospheric
parameter,
C=
2
(
e
)1/2
L,2L2 ETW,,
D
W
W
C
Wce
W
("
1(33)
- 1)1/2
L2
)(3
- 270
L,2
W))
-
We
)(34)
(35)
T = 1 + W"
2w
The equation for the change in perpendicular velocity is of higher order.
Once rescaled , we obtain
dvj
A
=T
w Cos T
dt
(36)
For the trapping dynamics, the perpendicular velocity is a constant,
v 1 ~ const .
6
II.2.1
Nonlinear Trapping
The electron undergo an axial bunching due to trapping by the wave. This
trapping can be seen by combining Eq.(28) and Eq.(29),
cos = -S
dt2 + 1 - W-oi=-
(37)
where
W=
W
(38)
is the trapping frequency. Eq.(36) is a nonlinear oscillator equations whose
solutions are elliptic integrals. There exist bounded oscillatory solutions to
this equation as long as the inhomogeneity parameter , S, satisfies
ISI(1 - -)
j "A - 1Q1 < 1
v1 A
(39)
For this reason, one can see that trapping occurs near the geomagnetic
equator where the inhomogeneity becomes small. This doesn't mean that
the inhomogeneity is not important. It is the inhomogeneity that yields
the energy transfer between the electrons and the wave.
Suppose we expand 0 about a trapping vortex,
0 = cos~ 1 (-Q) + 6b
(40)
The insertion of Eq.(39) into Eq.(36) yields an oscillator equation for small
perturbations about the trapping vortex, cos- 1 (-Q),
d0(41)
2b
+
2
2&0
dt
with an average phase
<,O >= cos 1 (-Q)
(42)
Recalling Eq.(35), we obtain for the energy transfer
dvo
~.const cos(cos(-Q))
-
Q
The energy transfer is proportional to the rescaled inhomogeneity,
a homogeneous case, Q = 0, there is no energy transfer.
7
(43)
Q. For
11.3
Wave Equation and Nonlinear Current
Since the amplitude and phase are slowly varying quantities relative to the
frequency and wavelength, the WKBJ wave equations can be used [1,6,7].
8
9
8
+v 9( )A =-IR
S
8
I
+ Vy)$
(
(44)
A -
(45)
Eq.(42) and Eq.(43) are simple advection equations for the amplitude and
phase, respectively. The nonlinear currents have a very complex form which
can be written as follows
INL = IR + iII =
4 7r exp(-i(O, + Wp))e* - JNL
(w8D/8w)wET
(46)
where e* is the complex unit vector describing a RHCP wave and D is the
dispersion relation satisfying the following equation,
c2 k
D = 1 -
w
2
2
.2
Pe
(47)
W(W - W')
The nonlinear current density is calculated by taking a velocity moment
which yields,
INL = -(
exp -dvF
"1/2
We
Wce
(48)
where F is the distribution function normalized to unity. Using Liouville's
theorem and expanding F about v1 = VR and v 1 = v±c, we can push time
back along the characteristics of the particles to where the distribution
function is independent of 0 [6,7]. The distribution becomes,
F(v,,
F~~/'v±t
(V~±)
vi-, t) = F(VR, v)
+vC
8F
8F
(Vv±e) + AVv± 9 ,, (VR,v±e ) (49)
where the subscript, c, denotes characteristics. Using Eq.(49) and keeping
to
61/3,
the nonlinear current becomes
INL
~(-
Wee
1 -- -- )/2Wce
)r
--5J
I
WCe
.
(50)
The change in the perpendicular velocity can be calculated by integrating
Eq.(36) backwards along the particle characteristics. The current can be
expressed as
IR = -OA1/2j
dv vI 2 a,(Q)Q(Sr - VS'r2) ftr
2
8
BO-V
(51)
Ij= OA' 1
av(Q) = 27r(1
= (-)1/2(l
Wee
1, 2 )
/2/2
dvv 'Lav(Q) (1 - Q2 )(Sr - 2 V.S T
-
Q2 )1/4 j
_'
U/WC
dxx J2(x)
)Wce
F
(52)
(53)
(54)
wee
where IR and II are the real current and out of phase current, i.e. imaginary current and av(Q) is the trapping volume. (The reader is referred to
the papers by Molvig et al. for a detailed derivation of the nonlinear currents). The integrals for the imaginary currents are over a range of trapped
perpendicular velocities. The range of velocities is given by the assumption
that at least one trapping oscillation occurs during a trapped state which
is given by
wt,.r > 27r
(55)
where wt,. is the trapping frequency. The above equation along with r being
defined as
r = 6./VY
(56)
which is the length of the interaction region divided by the resonant velocity,
yields a cubic equation for the amplitude in terms of v 1 ,
A 3 > 7r2 V 2 C 2 (1 _ W)(VI +
(57)
)2
Solution of this equation determines the upper and lower limits of the perpendicular velocity integral. We refer the reader to the papers by Molvig
et al. [6,7) which further illuminates the asymptotic derivation of the nonlinear currents. The nonlinear asymptotic currents still have a complex
functional form, but the important results are that they depend upon:
1. The time the particle is trapped, r
2. The effective inhomogeneity
Q and
S 11to Bo
3. The trapping volume, a,(Q)
4. The velocity derivative of the distribution function, F
9
Trapping Time
II.3.1
The fundamental quantity in the nonlinear currents is the time the particle is trapped by the wave or the trapping time. This quantity, r, can be
calculated by propagating the trapping condition back in time along the
particle's orbit to when the particle is first entrapped. The trapping time
can also be calculated by noticing that the trapping condition is invariant under particular transformations. The latter method is the simplest
numerically and analytically; therefore, we shall adopt that methodology.
The trapping condition, Eq.(39), is invariant under the following transformations,
t-
(58)
t +At
T -+
+ Ar
(59)
s
- IVRJAt
(60)
therefore the root, r = r(s, t), satisfies
ir(a,t) = r(s + IVRIAt,t - At) + At
(61)
One can form a differential equation from Eq.(61) by Taylor expanding the
R.H.S. and taking the limit At, as -* 0, i.e.,
(9 + Va )r = 1
(62)
Note that the resonant velocity is not the absolute value, VR = - VRI.
Eq.(62) is true provided the particles satisfy the trapping condition. In
that case, Eq.(62) simply advances r. If the trapping condition is not met,
then r = 0.
II.4
Distribution Function
The emphasis of this paper is on the functional form of the resonant perpendicular distribution function. Fig.(3) shows the approximate form of
the distribution function used in our theory. The distribution is the solid
contour that are representative of a loss-cone distribution. For Fig.(3), the
parallel part of the distribution function was assumed a maxwellian.
F(vj1 , v±) = const exp(-v /vth) exp( -v2 /vth) (1 - exp(-v. /vb))
(63)
This was done so that contours could be drawn along with the illumination
of a perpendicular distribution as a function of the resonant velocity. In
our theory, all the parallel dependence at a particular resonant velocity has
been incorporated into an input constant A,
10
A ~ const f(oj = VR)
(64)
where the full distribution has the following form
F(v±, V,.)
conat f (vj = VR) G(v±)
= A exp(-v'/v') {1 - exp(-±/v,)I
(65)
The dashed vertical line marks the resonant velocity and the dashed contours represent the single-wave characteristics. The wave characteristics
are calculated by using Quasi-Linear theory where they represent the diffusion paths of the particles under the influence of a multi-frequency wave.
The application of Q.L. theory on a monochromatic wave isn't really useful
but it does illuminate the nature of instabilities. The characteristics are
obtained by the coexistance of an asymptotic steady state of nonzero wave
excitation and a steady particle distribution. Under these conditions, one
can show from an H-theorem
dH
dt =-0
(66)
which implies that the zero order distribution function satisfies
G(vi,vl1 )k Fo = [(1 - ka)
Wk
a +
Ovj
Wk
-]Fo
8
= 0.0
(67)
VII
The result from Eq.(67) is that the distribution function is a constant
along the characteristics of the velocity space operator, G(v±,v1 1 ) [10]. Let
s stand for the distance along the diffusion path or characteristic, then
dFO
dvj OF0
dvOF
OF(
da de
0v+ dvj Mo
da
das
da Ov~l
(68)
dv-(
da
(69)
where
1
-
)
w
dvl = a(70)
da
Wk
Integrating Eqs.(69) and (70), one obtains the single-wave characteristics
(vil - w)
kl
1
(71)
+ v' = conat
The wave characteristics are concentric circles displaced from the origin by
the amount, wk/k 1 , in the voj direction. Eq.(70) also represents constant
energy contours in the rest frame of the wave. By looking at the diffusion
paths of the particles in reference to the constant energy curves,
11
v2
Vj1 +V2
+1.
=
(72)
const
which are centered at the origin, a stable or unstable system can be determined.
Figs.(1) and (2) are plots of anisotropic distribution functions: Fig.(1)
has an anisotropy in the parallel direction, T11 > Ti; Fig.(2) has an anisotropy
in the perpendicular direction, Ti > T11. The direction in which the particles move along the characteristics will determine whether energy is gained
or lost. By examining the point at which the resonant velocity and characteristic meet, this determination can be made. The motion of the particle is
confined to the diffusion paths so given the direction of motion the particles
will gain of loss energy to the wave. The particle will always move in the
direction of decreased density, i.e., towards a distribution contour of lower
magnitude. Summarizing the analysis,
1. Find the point at which the resonant velocity meets the single-wave
characteristic
2. Determine in which direction the particles will move along the characteristics by noting that motion is towards a decreasing density contour
3. Given the direction of motion, determine if the particles move to lower
or higher energy. Constant energy curves are concentric circles centered about the origin where motion towards larger contours implies
a gain in energy and vice versa.
If the particle moves in the direction of a lower constant energy contour
then the particle losses energy which is an indication of instability, where
motion on the contour to higher energies implies stability. For T11 > TL , the
particles move to higher energies and the system is stable. For T1 > T1, the
particles move to lower energies and the system is unstable. The nonlinear
instability that we are considering is independent of 8F/Oj which is not
the case in linear and quasi-linear theory. Linear theory may be important
only in raising the wave amplitude toward the required threshold value but
it doesn't apply to the driving mechanism responsible for our emissions.
By moving the resonant velocity in Fig.(3), the slice of the distribution
function that is used in the current calculation varies. Figs.(4)-(7) show the
distribution function for various resonant velocities. With this particular
choice of distribution(excluding energy inversion), the number of resonant
particles decreases as the resonant velocity increases, i.e., there are less
particles as one moves to higher energies. The decrease in particles is
solely controlled by the parallel part of the distribution function. To model
this variation, we will have to include a functional dependence in the input
12
parameter A. We can imagine a similar dependence in the input parameter
Vth and vb of the distribution.
Eq.(65) will be the distribution function used in the current calculations
where a resonant velocity dependence will be inserted. The functional form
assumed to model this variation in A, Vth and Vb is an exponential whose
argument is proportional to the difference between the equatorial resonant
velocity and the resonant velocity at the arclength value, s:
A(s)
Vth(S)
Vb(s)
A(0) exp(-e1 [VR(s)
VR(0)])
-
vth(0) exp(-e6 2[V(s)
-
VR(0)])
vb(O) exp(-e 3 [1iq(s) - 1R(0)])
where vth(0), vb(O)
(73)
(74)
(75)
and A(0) are evaluated at the geomagnetic equator
and e1 2 3 are input constants. The resonant velocity has a minimum at
the equator and correspondingly the functional form is maximized at the
equator and falls off exponentially from the geomagnetic equator.
III
Results
The inclusion of the resonant velocity dependence yields a variable distribution which solved a numerical instability. Prior to this theoretical development, the amplitude would not disconnect from the origin. Figs.(8)-(18)
show the time development of the emission without a variable distribution
function. Fig.(8) has the initial triggering wave that undergoes the nonlinear amplification and frequency variation. Figs.(9)-(18) are plots at every
200 time steps in the computer simulation. At each 200 time steps or every
74 msec, two plots are displayed: (1) the amplitude and real current; (2)
the frequency, w = wo + bw, and the resonant velocity at each grid point in
space.
The run with no resonant velocity stops at 800 time steps because of a
numerical instability. This instability comes about because of the inherent
size of the real current that develops from the nonlinear wave-particle interactions. Once the pulse moves to the downstream side of the equator,
i.e., the R.H.S. of the equator, the pulse remains fixed at the equator. The
continual trapping at the equator at zero amplitude and the production of
a large current causes the amplitude to remain fixed to the equator and
amplify. This large current production, just off the equator, off-sets convection of the pulse. The reason for trapping at the equator comes from the
ability to trap particles with zero amplitude. This issue has been explain
in great detail in the previous papers by Molvig et al.. Summarizing the
13
explanation, the parameter that determines whether or not trapping occurs
is given by Eq.(39), Q. This parameter remains finite at the origin since,
A - const - s
(76)
const- s
(77)
S
-
which yields,
Q
-
const
(78)
The continued trapping allows for the currents to become very large.
Figs.(19)-(26) are the 3-Dimensional plots of the trapping time and the real
current. Fig.(19)-(22) show the trapping time for varied time step; 600, 800,
1000, 1200 time steps. The trapping time is centered near the geomagnetic
equator and slopes downward towards higher v 1 . The reason for the sloping
is due to the trapping windows which are calculated by solving Eq.(57).
Initially, the amplitude is small and correspondingly the window in velocity
space is also small. As the nonlinear interaction continues, the amplitude
increases. This increase in the amplitude causes the trapping windows to
expand where the whole distribution can be sampled. The sloping of r
will flatten off as the interactions continues which is an indication that
the velocity windows are becoming larger. This can be seen by viewing
Figs.(19)-(22).
Figs.(23)-(26) are the corresponding 3-Dimensional real current plots.
The current is small initially but as the time increases it become very large.
The positive and negative maximas of the current integrand are given by the
slope of the distribution function. The current integrand would continue to
grow if it were not for the numerical instability that destroys the computer
simulation. At the equator, the current must be exactly zero since the origin
is the detrapping point for the particles, i.e., r(s = 0) = 0. The current is
changing very rapidly throughout the calculation and at each instant the
current must be zero at the equator. The summation of large positive and
negative contribution from different velocities at the equator yields some
uncertainty in the sum being exactly equal to zero at each time step. This
uncertainty manifest itself in Fig.(15) where IR has an oscillating value at
the equator. This oscillation continues until 'R is oscillatory throughout
the whole spatial grid. This current variation is felt by the amplitude
through Eq.(44) so that the amplitude eventually becomes meaningless.
The corresponding 3-D plot of the current in Fig.(15) is Fig.(24).
Fig.(17) is the distribution function used in the calculation of the currents which is held constant throughout this run. The frequency and resonant velocity, however, have a variation which can be seen from time step
14
to time step. The nonlinear wave-particle interaction generates a change
in the frequency which is also seen in the resonant velocity, Eq.(28). The
resonant velocity has a near parabolic form since it depends on Wc, and w,.
As the interaction continues, the change in frequency becomes large causing a larger depression in the resonant velocity. This depression would also
continue without bound since the pulse is trapped at the equator. Fig.(18)
is the non-terminated emission frequency which is out of the graphical windows. The frequency has a fairly constant slope and would also continue
without bound. The constant depression of the resonant velocity indicates
that the particles which are resonant with the wave must have an increase
energy. The theory only sampled particles at one parallel energy and therefore it became necessary to include a variable distribution function into
the calculation of the nonlinear current.
Figs.(27)-(44) are the plots with the variable distribution. The beginning of the run is very similar to the old case. The run, however, continues
for 1200 time steps more than the non-terminated case. Every 200 time
steps the same plots are outputted. The emission becomes significantly
different around the 600 time step or 223 msec into the run. Fig.(31) has
a smaller amplitude and current than in Fig.(13). The current is no longer
constant through the linear slope region of the pulse: this is due to the inclusion of the new distribution function. The 600 time step emissions have
roughly the same frequency and resonant velocity plots, but the distribution functions are different. The distribution function at the entrapping
point has not been included because of the numerous plots already being
displayed. The distribution undergoes a maximum change which occurs
near the end of the run and can be seen in Fig.(43) for comparison with
Fig.(17).
One finds that the maximum of Fig.(43) is smaller than that in Fig.(17).
This decrease causes a smaller current to be produced and therefore a
smaller emission amplitude. As the new run continues, the current becomes
smaller through out the interaction region which is a direct result of the
variable distribution and the increased depression of the resonant velocity.
The resonant velocity becomes larger as the change in frequency continues. This depression in the resonant velocity causes a different sampling
of the distribution. The depression in the resonant velocity becomes large
enough such that the current near the origin decreases sufficiently that the
convection of the pulse cannot be overcome. The pulse disconnects from
the origin and propagates downstream. Examining Fig.(37), one can see
that this did in fact happen. The emission has disconnected from the origin
and has started to propagate towards the receiver. The current form has
also changed where it has positive and negative values. This is due to the
15
change in the distribution function and the other parameters in the current
such as the trapping volume and the trapping parameter, c,(Q) and Q,
respectively. Figs.(45)-(52) are the 3-D plots of the trapping time and the
current. Figs.(45)-(48) are the trapping time plots which have a very different form than Figs.(19)-(22). The trapping time in Figs.(45)-(48) doesn't
have as large a maximum and the limit in velocity space which is trapped
is smaller than in Figs.(19)-(22). The range of trapped velocities decreases
as the interaction continues which is indication of a shutting-off of the interaction or termination. This can also be seen in the 3-D current plots
in Figs.(49)-(52). The currents don't reach the same magnitude as can be
seen in Figs.(23)-(26) where the variable distribution is not included. Not
only can one see a decrease in the magnitude of the 3-D current but the
range of velocity contributions is decrease which is a direct result from the
decrease trapped velocity space seen in Figs.(45)-(48). The current shutsoff as time continues along with moving spatially to higher s values. By
the 2000 time step, the emission has terminated. There is no longer any
current being produced and the pulse is free from the geomagnetic equator
to propagate to a receiver. The terminated emission frequency can be seen
in Fig.(44). The frequency doesn't continue to grow but stops once the
nonlinear wave-particle interactions terminate. This is called a terminated
emission.
IV
Conclusion
Terminated emissions were obtained by using a variable perpendicular distribution function. This distribution, dependent upon the spatial value of
the resonant velocity, shuts-off the wave-particle interaction which yielded
a terminated emission. Future work will continue in applying the theory
to various magnetospheric conditions along with various magnetosphere.
A new frequency theory is being developed and along with the variable
distribution should be able to predict risers and hooks.
16
I
',
t-
I
-
- 1
,
x
1
f
iV
\
*
-
-r'
Figure 1: Anisotropic Distribution Function, T11 > TL.
17
l
'
,
0 ' t-
r'.
-N
-
E
I
c
na
-L
-
..
96
I--
9'-
C/O
0' 1-
E x
-- ----- - - - -
z
4!
CN
a
e-,
-1-~IU'1IAA
Figure 2: Anisotropic Distribution Function, T±. > T .
18
-Ie
I
0' i
I.
S-
-
;L
I-
1'-
.1
-
N-
rr
V)
rzn
it
/0
Itt
if
S
Ni
Figure 3: Loss-cone Distribution.
19
0'
e
-
I
--
cc.
-0
- -
-
-
-
-
c~-
-
-
--
-
1
-
C4.
-
-
-
---
-
0
(JA dsadA)t
=
Figure 4: Loss-cone Distribution VR= -0.8.
20
1
0' 1
-
,
a
9-
ro
S-
r
S
.0
4'
(JA' dj.dA))
Figure 5: Loss-cone Distribution VR= -2.0 .
21
-
T
-4
-.
4
Cm.
- ~ ~~
~
~
~
~
~
~
-- g
*0.
-44
(JAUdjadA?~j
Figure 6: Loss-cone Distribution VR= -3.2 .
22
011
6
4
ao
9-
46
i
.
- .
-.
of
.0
CDi
.. 1
'0
4-
0
(JAFdjadA)g
Figure 7: Loss-cone Distribution VR= -4.4 .
23
I I I I I II
I
I
I
I
I
I
I
T I
I
I
C
01
(-)
E
0
II
I I
I
I
I
C
*0
C
0
0
C*4
*
CL
E
0
III
I
. I
-)
I I I I II
I II
I
Ul)
(u a IoJ q g
i
u aJ Jn
I
-
L-
U)
C0
x )
I I
I
D
a IJ
(
p !I
o
) )pn Q p duJD
Figure 8: Initial amplitude wave form w/o VR dependence.
24
+3
0
I
I
I - I
I
I
I
I
I
~TI
II
II
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
I
I
I
II
I
I
II
I
I
(0W W
Eu>
C
01
E
2
W
CL
E
a
I
II
I
.
.
.
I
.
I
I
I
C14
J
I
I
I
II
U->
C0
(u
. I
X
o
i x ) 1u n
o
(a
Figure 9: A and IR at t=200 w/o dependence.
25
) pn
LUJ
I
dwD
II
)O~
L111
11'i11
1
r11v
[rTU1l11
IIj
II
II
I'
III
II
II
11111
I
II
II
I
I
I
II
II
I
II
I
/
E
E
C
C
4)>
0
f+ 3
IIIIIIIIIII
IIIIII I
II
LI
CD
CO
I I II
I I I I I II
I
( ug j q) s~ ai A P UD
I
II I
I
x.* i
I
*
I I I I I I I I I II I I
mm
II I , I
IO
00
0S
->
D(p ~6 9W 0
Figure 10: w and VR at t=200 w/o dependence.
26
a a
II
-
II
ii
.
II I V
I
I
C-1).
. iI
I I I I
I
I
I
F
a
.
I
I
I
I
I
I
I
E
CL
E
I
I
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I
C14
(u g Io i q
I
I
~
I
I
4-L
..-
S
I
1 D .1J
x ) 1u a i jn 3
( p ! Io s ) Q p n I ! I d w D
Figure 11: A and IR at t=400 w/o dependence.
27
I
1
T T
I
I
1
I
I
II I
I
I
I
I
I
I
I
I
I
I
I I
I
TTrr1Fr~~F~~
Ev>
0'
I
I
2'
E
c
0
E
0
.1...
I
.1
.1....
c'~J
lemma' a,,
I
*
C'4
I
111111111
*j
I
I
I
LO
00
I
I
I
(C>
A
PU D ( P!1
0
s ) o 6a
Figure 12: w and VR at t=400 w/o dependence.
28
I
J
-
--
-I
( u Q Io jq )&
~
I
WQ
II
I
Y
I
1~
C
V
E
9
0
-
V
)9
.---.-----
C.
E
0
I
I
I
I
I
I
I
U-)
C)
(u a
o g
.
I
.
.I
0
X
)
I
L
IUa JD
IDa
iJ
(p
! Io s
Figure 13: A and IR at t=600 w/o dependence.
29
) apn
I
-
0
.-
I
dw D
I I
Ti
I
II
I
I
I
I
I
/
('3
OOm1
E
9'
-
C
03
E
0
,, I I II II I I I II IA II
I
0
00
I
.~..
m m m li.
a I I I I I I I I I I
to ~.o
a
I
*~-
I
II
.
Ii
.
I
S~. I
i I I I I I I
Ni
L~
A
P UD
I
I
( P!0S
6 9W 0
)0D
Figure 14: w and VR at t=600 w/o dependence.
30
III
to
I
( ugIo jq )S J
I I I
.
I
I
L
II II II vI Ij I,
I
I
I
T 1
I
r
*
I
I
I
I
QC
0' L
E
9'
a.
-
I7
-- - - - - - - - - -
------ .---.--- ---
0
0
E
0
I
I I
I I I I
u)
(u
goJq
I I I1 1 1
.1
.
1_1_1_1_,_IIIIIIIAI
-
gx)
I
i
---
ugJ
jn:D
ioJ
(pio
Figure 15: A and IR at t=800 w/o dependence.
31
i
i
I
L.
) pn i
iduwo
*1 1
1 1 1
j
I
I
I
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
I
I
I
I
I
I
I
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A
0' L.
9.
E
9.
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C
C)
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E
0
0
I
II I'
I Ii Ii ~I I I I
I I
i
I
I I I I
I 1 11 1 1 1 1 1 1 1
I
1 1 1
1
1
1 1
I
I
i
i
1 1 -L I I I I I I
I
I
I I
I
I I.11I
a I I ...I . . I . I
0
1"
00
IO
OI
--
C=)
( u~joiq)&si
A
P UD
( P!1 s) DE aWQ
Figure 16: w and VR at t=800 w/o dependence.
32
d
V>
Eca_
ca
o
0>
--- --- --- --- -----
-M -O
- --- --- -- -- --- --- --- ---
(.
--
d a dA
u o ~nq
i
Figure 17: Distribution function.
33
J0
.4
-
sp
I
I I
I
I
I
I
I
I
~
I
II
I
I I
I
I I
I I I
I
I
I
I
I I
I
I
I I
I
I
I
,1L
01
6'
9,'
W
E
E
E
£
(.)
£
II
U-,
I
I
I
I
P")r
. I
I
CN
, , I.,.
i i
-
00
-
( zq I
U!I)
L
A ou Qn b i Q)
Figure 18: Emission frequency at t=800 w/o termination.
34
I.
.1...
I
CN
-
0
+3
I
I
I
III
0.
<;
..
4.
Figure 19: 3-D trapping time at t=600 w/o dependence.
35
.L.
Figure 20: 3-D trapping time at t=800 w/o dependence.
36
,4
10,00,
.
5.4
4.
ta=00
Figure 21: 3-D trapping time at t=1000 w/o dependence.
37
742rpA
.
Ln
ra
0!
P.
F-LA
Figure 22: 3-D trapping time at t=1200 w/o dependence.
38
0
0
00
Figure 23: 3-D real current at t=600 w/o dependence.
39
Ite
Figure 24: 3-D real current at t=800 w/o dependence.
40
U.,
Figure 25: 3-D real current at t=1000 w/o dependence.
41
0,
0
0
8(4)0-07
-i'erp
Figure 26: 3-D real current at t=1200 w/o dependence.
42
I
I
I
I
I
I
I
I
I
I
I
I
I
I
S
I
I
I
I
I
I
I
C
01
4)
9.
E
9'
0
-
4)
~0
C
Q>)
0
"
.0
C-
CL
., -
.E
a
-I I I I I
I
I
I I I I
i
i
i
i
I
i
i
0
I
I II I ]I I I
I
U-)
.,
-
0
C0
(
u93
Jq S
x
) I u9
D J (p !Ios) Qpn
J jn: D
Figure 27: A and IR at t=200 with VR dependence.
43
I ;dw o
.
I I I
I
I
I
I
I
v C.
I
I
I
I
I
I
I I
I
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E
0*'
I
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S
9 '
-
V
-L
E
0
z ,
f+
I
I
I
I I i I
I i,
1
1
1I
1
1
1
I
IIIIIIIIIIIIIIIII.IIIIIItIIII
1,1
I
I I
I
f ,i
0
I
I
I
I
( uP
Io jq )S9 A P U
(P!I os) obgwo
Figure 28: w and VR at t=200 with VR dependence.
44
I
-
I
coo
I
I
I
I
0
9E
E
9~
0
*0
C
0
z
,
C.
0
E
I
.
0
.
.
I
.
.
I
.
I
a
I.
aI
a
I
I
.a
0
LO
J
oiq
g
x
) I u aJ
II
0
*
(u
a
I
n
D
J
(p! I o &)
Figure 29: A and IR at t=400 with VR dependence.
45
+
apdn I doD
-
I
~-T
I
lIllIllI,,
I~
ii
I
ii
~I
ii
it
ii
Ii]!
I
I
-
/
01
-
I'
I
-
II
E
i-
I'
e>
E
C~)
0
. 0
I
I aI
II
II
ii
I A
I
I
II
I
Ii
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i
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I i
I
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i
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i
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I
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0>
'1
0
C::>
(
U; oJq) sa
JA
I
I
I
pu D
I I
(P
I
1Ios)oawo
Figure 30: w and VR at t=400 with VR dependence.
46
i
I
I
I
Ii
I I
C-
I
I
0
CNO
I
I
I
I
I
I
I
I
Eu
Z
0 '
9.
Em
C.)
0
10
----
-
z,
C.
E
a
I
C0
I I I I aI I
I
I
a
I
I
I
I
x
) I uQ JJi D I
Figure 31: A and
I
0
0
U-)
(u a 0 J o g
I
IR
De J
(p. ! o&) Q pn I
at t=600 with VR dependence.
47
d wd D
I IT
I
Z '1L
0*
I
p
I
-
/
I
E
I-
I
C)
E
0
0
i I
o
I I I II II I II
I II I I .
Do
I .II .II I . . I
-D
N
, , I I I . ..
,l.,\.
II
II
o
*
I
(u 91oiq
I~%I
. I. .II. . . .. .
I
a I
II I hI~IIa
o
I
I
-
(p!1os)o6awo
pUD
) S .i A
Figure 32: w and VR at t=600 with VR dependence.
48
0
I
I
I
I
I
I
P
I
I
I
P
C4
Et
E
9'
W
C.
E
a
- I I I I I I
I I I I I
i
i
i
I
.. . . .
I
i
0
0
*
01
I
Figure 33: A and IR at t=800 with VR dependence.
49
4.
L~.J
-
~.L'
I
I
I
111
11
I
E
* ,
IIIIIIvII II I III
III IIIIIIIIIvIII
z
9
I.
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C
C,)
0
K.
CD
I
0
C.
Lo
i
LO
I
i
I I I I I I I I I
j
q) S
9 J A
p UD ( P! oS )0 W 0
Figure 34: w and VR at t=800 with VR dependence.
50
I ill.
0
1
( u@ I
I
V-
I
I
I
I
I
I
I
I
I
I
I
I
I
C
0*
z
V
E
9~
0
Q>
C
0
--
-
E
-1 1 1 1
.1
1
I
I
1
LO
0
1
1
1
1
1
1
1
I
1
1
1
1q
g x )Iu aj n D
I
I
0
0
*
( U a1
I
I
1
Da
( p!I o s ) Q p n I
Figure 35: A and In at t=1000 with VR dependence.
51
+
d WD
'
I'
' I
I '
'
' I '
' '
I
'
I
'
'
I
I II
' ' ' 1 1 1 Ii
I'
1 1
1I '
' '
'
I I
A
V
0*' L
/
/
E
V.
-*
5n
"0
-
I
C
II
. ,i. I..
co
ii
,
III1 II
1
I
LO
e
11
1
-
C1
1
a
( u@ Io jq )S a
1
1
1,
lid
cl-
0-
A
e
o
II
CI
I
I
e
i I i, ,1 I 11
1 11 7
0
) D 6 a wo
U D ( p!o 0s
Figure 36: w and VR at t=1000 with VR dependence.
52
1
1
,
1,,
1 1
r j
I
I
I
Ir
II I I
I
I
I
I
I
Z .I~
E
9.
CL
E
*0
I
I
~
I
i
i
I
0
I
I
I
I
I
U->)
.,
I
....
0
0
u
o i
F x )
d In aD I 2
(p d !I o ) ;p n
Figure 37: A and IR at t=1200 with VR dependence.
53
ddw
-o
.L I I I I I I I I I I I I I III I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I=
Z
-
I/i
I.
0* L
9.
E
9'
.t7
0
E
0
0
----
K II I
C
i
Co
I
Lo
i
ii
I
i
I
I
i
i
I
i
I
I
I
CN
I
I
I
I
I
I I
I I I
>
CN
-
-
I
(u
Do jq)SaJ A puo
(p !os
) o 6 wo
Figure 38: w and VR at t=1200 with VR dependence.
54
I I I I
I
II
I"-
0
I
I
i
I
I
I
I
I
0 t
CN
-0
9'
-
99
Q
0
E
I
I
I
I
i
I
I
0)
0
+
(u a Io iq
x ) 1u a i jn D
ID a i
( p!I o s ) ;;p fl
Figure 39: A and IR at t=1400 with VR dependence.
55
IdW
D
0
V-0o
(NO
II I
II
11
I
1
I
I
1
I
1
1
1
EO>
S
-
E
9'
-~
0
E
0
0
11 1 III
00
to
II
o
111111111
*
(uajoJq)SaJA
III
N
pUO
II
II
1
4
I
II
II
r.0
(P
! 1 s ) 0 au
Figure 40: w and VR at t=1400 with VR dependence.
56
I
I.
C4
I
I
I I I
I
I
f ,I
F
I
I
I
C
OIL
V
2'
91
E
-~
0
V
"0
V
C
0
0.
+ 3~
E
0
- 0
I
I
I
I
I
I
i
I
u~)
i
i
i
I
I
i
0
I
I
J
(u a Io i q g x ) Iu a j jn D
Io a i
( p !I o s ) ; p n
Figure 41: A and IR at t=1600 with VR dependence.
57
i
-
I
I
-
0
!j d Lu D
.1. I
I
I
I
I
I
I
I
I
II
1
1
I.
I.
p
-
p
9,
/
E
.0
0
+3
I I
0
Co
..
I
. I .
Lo
11 . .1111111111
I
VIII
111111111
r.o
~
I
I
( ua Io jq )saiA puD
N
I
I
~
I
I
I
I
I
0
e-.J
~
-
-
-
(p P !o
s ) o5 6 awo
Figure 42: w and VR at t=1600 with VR dependence.
58
I I
I' I
I
I
0
I
I
I
I
I
zL
C
0*
9.
(-)
E
9~
CL
a.
E
. .
I
I
I
I
~
~
I
I
I
I
I
0
I
0
C0-
(u
a o.i q g
x)
u a.ij n
Figure 43: A and
IR
D IDa
( p
!Io s)
4pn I
at t=1800 with VR dependence.
59
dwLuo
I
II
11 1I 1
1'
III
111 11 1 11
1
1
II
1 1 II
I
OL
8'
/
E
9'
-~
-C
V
C
()
0
0
+Z
I I
o
I, ,
co
I ,. . . . .I II I
~-
(NJ
-
i
I
i
I
(u @ Io j q )
S@ J A
pUD
, ,
,I
L
*-
I
I I I I
cc
I 0S
I
C>
D
( p Ios ) D6oJ
Figure 44: w and VR at t=1800 with V dependence.
60
I
II
II
NC
II
I
"-
0
-a
_I
-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
r
r i
0\
C
E
V
z ,
E
a
I I I I I I I I I
C
i
I
I
I
I
I
I
I
C)
C
4-
( U a I o i q g x ) I u a j j n 3 1o a i
( p!I o & ) Q p n
Figure 45: A and IR at t=2000 with VR dependence.
61
I Uw o
*
I
I
I
II
11-
Eu
z
L.
,
'~
E
9-~
W
0
.
.
0
+
i
I
I
I
I
II
I.
, , ,
,,iI
o
0
0
C4
*
I
(
U91 Jq)Sa
JA
puo
(Pi
! os)o6awo
Figure 46: w and VR at t=2000 with VR dependence.
62
I
3
*2
.9
-
g
- -.. .
-
-,- - - - - - -,---
(d j ad
A)
u
--
o In qi
- - - - - - - - - - - ---
I s ip
Figure 47: Distribution function with VR dependence,
63
I IiilIIIIIIiiilI
111111
IlillillI
II
I 11111111111111
II
111111111
I
Ij
III
0'1
6
8'
E
L
C',
E
cr,
*
I
a a a
I
'"j-
U,
U-)
a a a
I a a * *l**** I
r"n
C14
U-)
U-
*
>
-
t0
£+ 3
.1...
.11.11111
01)
11.1.1.11
11111 liii
r-
00
''l IV
W-
( z qI
u)
A~ o u Q n
b~
Qj j
Figure 48: Terminated Emission frequency
64
till
liii
Ii
-ll
<0
E'
U,0
Elka)
46.4
rp
Fa0
Figure 49: 3-D trapping time at t=600 with VR dependence.
65
Figure 50: 3-D trapping time at t=800 with VR dependence.
66
Ui
~LP
U,
~~(
F:
SNIka
1,0
74
I
6,44
I
4tw.4
vpr
Figure 51: 3-D trapping time at t=1000 with VR dependence.
67
U,)
-Ia
S
7,4 6,.4
0,0
vperp
Fiur
Figure 52: 3-D trapping time at t=1200 with VR dependence.
68
0
C0
r2.
7.
Figure 53: 3-D real current at t=600 with Va dependence.
69
p rP
U)
0:
W,
3,j 2-4 ''t
S.O
P
Figure 54: 3-D real current at t=800 with VR dependence.
70
P
Ul
8
Fu.0
4w.4
0-0
%IN)
7.4
.
Figure 55: 3-D real current at t=1000 with VR dependence.
71
'.4
Fige
Figure 56: 3-D real current at t=1200 with VR dependence.
72
Acknowledgement
We would like to thank C.W. Roberson for initiating this project. I am also
grateful for the stimulating conversation with Jacek Myczkowski and my
fellow graduate students. This research has been sponsored by the Office
of Naval Research.
REFERENCES
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73