On the influence of a CW whistler on magnetospheric noise

Phw.

Space Sci.. Vol.

25. pp. 879 to 885.

ON THE

Pergamon

Pms,

1977.

Printed

in Northern

Ireland

INFLUENCE OF A CW WHISTLER MAGNETOSPHERIC NOISE ARMN’IDO

ON

I,. BRINCA

Centro de Electrodinfimica, Instituto Superior Tknico, Lisbon 1, Portugal (Receioed 19 November 1976) Ah&net-Analysis of the modifications introduced in a turbulent whistler noise spectrum with the injection of a coherent whistler leads to a nonlinear dispersion equation for the stochastic modes. These modes are submitted to real frequency shifts and corrections to their growth rates which are in qualitative agreement with observations made in the Siple Station VLF wave injection experiment showing the creation of noise-free bands when CW whistler modes are transmitted. 1. INTRODUCllON

coexistence of energetic particles and waves in the Earth’s magnetosphere leads to a dynamic equilibrium where the continuously generated waves interact with the particle distribution. Quasilinear (QL) analysis of the interaction (Davidson, 1972) shows a general tendency toward stabilization: the modifications brought about by the growing waves on the velocity distribution reduce the source of instability, eventually originating a macroscopic state of turbulent noise immersed in an average distribution of particles. This process is described by a dispersion relation depending on the average velocity distribution (which defines the growth rates of the waves), and an evolution equation for the average distribution involving the waves’ spectral energy density and the precipitation and injection of particles. Although plagued with time-secularities arising in the iterative solution of the Vlasov equation, quasi-linear theory has been remarkably successful in explaining several magnetospheric phenomena. In the whistler domain that concerns us here, the works of Kennel and Petschek (1966) and Etcheto et al. (1973) are examples of application of quasilinear theory to the magnetosphere. Information on both the turbulent noise spectrum and the average particle distribution can thus be obtained. Given this wave-particle interdependence, artificial alteration of the magnetospheric wave and particle contents might be achieved by the injection of particles or waves. The concept adopted by the Siple Station VLF wave injection experiment (Helliwell and Katsufrakis, 1974) falls into the latter category. The transmitted coherent signals produce monochromatic whistlers which may follow (geomagnetic) field-aligned ducts and be observed at the Siple conjugate point near Roberval, The

879 6

Quebec. The injected continuous wave (CW) modifies both the magnetospheric modes and the distribution of particles, albeit observations might emphasize only one of these two complementary aspects. For example, Helliwell(l975) has reported that the switching on of the monochromatic mode at Siple creates a “quiet” band in the background magnetospheric noise, about 6 dB in depth and about SO-200 Hz wide, located just below the transmitter signal (-6 kHz) at both conjugate points. Bearing in mind the interdependence of the dispersion relation for the stochastic modes and the energetic particle distribution the analysis of this phenomenon should consider the influence of the coherent mode on both the evolution equation of the average distribution function and the dispersion equation of the turbulent modes. These two aspects constitute the aim of the study to be presented. the difficulties associated with the However, analysis of turbulence with a monochromatic signal in its midst requires the adoption of several simplifying hypotheses. We assume (i) the interaction between the injected wave and the whistler noise to take place in a homogeneous magnetoplasma (equatorial region), (ii) the waves to propagate parallel to the local geomagnetic field, and (iii) the coherent (moderate amplitude) whistler to be known throughout the interaction time. The study will not be strictly self consistent because of (iii), though this approximation has been used in several nonlinear problems with success (O’Neil, 1965). The hypothesis of parallel propagation, (ii), is reasonable for ducted whistler propagation in the equatorial region, at frequencies close to half the local electron gyrofrequency, as happens to be the case here. The homogeneity assumed in (i) is more controversial. Indeed, pitch angle scattering will modify the energetic particle distribution and

A. L. BRINCA

880

its consequences depend on the actual inhomogeneity of the magnetosphere. Hence, the evolution equation for the average velocity distribution formally derived below, being associated with a homogeneous model, will not be discussed. The mechanism of pitch angle scattering by the coherent whistler might play an important role in the creation of the quiet bands (Helliwell, 1976; private communication), and we shall wme back to this point in the concluding discussion. Because the analogous electrostatic problem (modification of a turbulent spectrum through the injection of a monochromatic wave in its midst) has been analyzed in detail elsewhere (Brinca, 1976), we shall invoke the parallelism between the two similar situations and omit most of the mathematical manipulations. The solution of the relevant equations (Section 2) relies on the operational formalism introduced by Misguich and Balescu (1975a) in the investigation of electrostatic turbulence. It leads to an evolution equation for the average distribution function of the particles which is different from the classical QL diffusion equation, and (Section 3) a nonlinear dispersion equation for the turbulent modes which, when wntrasted with the QL one, shows (Section 4) the existence of real frequency shifts and corrected growth rates in the noise spectrum. By analogy with the electrostatic case, we expect the noise spectrum just below the injected whistler frequency to be attenuated, and the upper turbulence band to be enhanced.

frame become ;+

+3,i+B.~)=pJ+~$ -$-E$+E&=

L(z,v, t)=L(t)=

(oOt - koz)$]

B, = B,[ws

(wet - kOz)j]

En -=-

00

B,

ko’

(o,,t- k,z)i+sin

is applied to the system at time t = 1,. The ensuing behaviour is determined by the Vlasov and Maxwell equations. The assumption that all the waves propagate parallel to B. makes the problem homogeneous in the transverse directions (x, y). The governing equations in the laboratory

-$

“&;

[E+vx(B+B,)]-i, * (9

E=E,+E',

(6)

B=B,+B',

(7)

J= -e

dvvf,

(8)

I

where e and m. are the charge (magnitude) and mass of the electron, c is the vacuum speed of light, p. is the vacuum magnetic permeability, L(r) is a Vlasov operator and J is the current density. In this stochastic problem we deal with fluctuating components and ensemble average values. We shall resolve the dependent variables Y into their two parts with the application of the ensemble averaging operator A introduced by Weinstock (1969): ?=AY=(Y),

y=T+y’,

E, = E,[sin (oOt - kOz)i -ws

(3)

with

2.TREORY

Our idealized system comprises an infinite homogeneous magnetoplasma of stationary ions with the d.c. magnetic field along the z axis, B. = B&, and whistler turbulent fields (E’, B')created by an unstable velocity distribution of the particles, f. A monochromatic coherent whistler mode (E,, B,) of frequency w0 and wavenumber k,,,

(2)

1

Uz, v, 1) f(z, v, t) = 0,

Y’=(l-A)Y.

The fluctuating part of the wave fields (E',B') is associated with the ensemble average component of the fields @ = E,, fi= B, + B,).Previous works on whistler turbulence did not take into consideration the imposed coherent field, making E, =B, = 0: the average Vlasov operator was thus time independent. In order to restore this simplifying characteristic, we move from the laboratory frame to the monochromatic CW whistler frame. Since the whistler phase velocities are much smaller than c, we use a Galilean transformation, R=r-+

V=+,

WZ

T)=B(z,

1),

-f-=t, 0

0 E,(Z

7-l

=

I

E(z,r)+FixB(z, 0

t)

1I’

where the subscript I denotes the transverse (with respect to f) components.

881

On the influence of a CW whistler on magnetospheric noise

In the CW frame the cob-sent whistler electric field is cancelled, and the splitting of the basic equations into their stochastic parts leads to

through

= iJT

A(T, To)=X+e[T,

T,I-L+AL’(l-A)]

X, exp

(

]

$+

;+(I

E F(T) = - (L’(T)P(T)), > -

A)U+‘(T) = - L’(T)flT),

(10)

dT,[-L(T,)+AL’(T,)

l-0

(9)

f(T) = NT, ‘I’,)f(T,)-

(I -&)$

(11)

, (18)

JTo

(1%

T dT,A(T, T,)L’(T,)fiT,),

(&I, I&‘)

+J?b dTA~‘UM(T, TI)L’U-I))~T (20)

=~(-r,.,J,‘)-~~(B,‘,B,‘) 0 -- ;* & (Ex’, -JG? -F

5

0

(I%‘, J%‘) = $

5

(12)

(I%‘, -EY’)].

(Bx’, --BY’),

(13)

1

i,=V,-&+ et<-a;ms(k,Z+i#z) E i

To proceed, we need to introduce simplifying assumptions in these exact equations. The type of approximation adopted at this stage classifies the ensuing theory. Quasi-linear theory utilizes a timeevolution propagator, V,(T, To), which does not take into consideration the effect of the turbulent fields in the particle’s orbits. The propagator h(T, To) is then approximated by NT, To)--+ UT,

x$-c

sin(k,Z+4)I,

To)= X+47;

To 1-El

(14) = exp [-(T-

L’= -;[E’(Z,

Tf+VxB’(Z,

(Ti-J&p)=

V, = V,,

T)]-&,

(15)

dVV(f;f’),

cylindrical

coordinates

in

(Vx + iv,) = V,e”,

we define the electron cyclotron frequencies e& u, =-, %

c& a=----, 4

Jr% is the ion current density in the CW frame, and I-= v&-v& I a time-evolution

I propagator

(17) A(T, T,)

T&],

(21)

where the last step follows in our problem because the average Vlasov operator is time-independent: henceforth we do not need to deal with the direct time-ordering operator X,. To avoid the time-secularities arising in the QL theory, Dupree (1966) and Weinstock (1969) introduced a renormalized propagator that, in the weakcoupling approximatjon, includes the influence of the stochastic fields up to second order in their ampiitude. The approach is known as the renormalized quasi-linear (RQL) theory, and introduces several modifications in the results of the QL theory. Among these we emphasize the appearance of real frequency shifts and corrected growth rates in the turbulent spectrum (Weinstock, 1972). The results are obtained for a zero ensemble average field, which is not the case here. An application of the RQL theory to the whistler mode (again with a zero average field) was recently done by Yamamoto (1975).

(16)

J

-e

where we have adopted velocity space,

Defining

I

where X, is the direct time-ordering operator @Iisguich and Balescu, 1975a, b), we can obtain a formal solution for the fluctuating distribution f’(T), and an evolution equation for RT):

(l_~)~‘“‘.B”,“(_j,,~~, = & (&, 8,) = k&,, (cos k,Z, -sin koZ),

x(l-A)l

882

A. L. Bru~cr,

We adopt the quasi-linear viewpoint implicit in (21), and negiect the effect of the initial fluctuations involving f(T) in (19) and (20). Our basic equations for the distribution function become

lace transforming &K

in time according to dZ g(Z , T)ef(-nr+Kz)f

=

_

1 g(z,

drU,(T, T-T) x@‘(Z, T-T)+VXB’(Z,

T-T))

where obtain

+T(T-r),

7’) = m

B

J

B represents

dfl

I

__

j(nT-KZ)

dK&-iKe

the Bromwich

9

Contour, we

(22) with

(27) <[E’(Z, T)+VxB’(Z,

Tf]Uo(T, T-4

x[E’(Z, T-T)+VXB’(Z,

and

T-T)]

( )‘= ( )X’+i( )Y’, t )‘=(

)x’-i(

)Y’.

>-&T-r),

Equation (26) leads to the dispersion relation in the CW whistler wave frame when the fluctuating curwhere, recognizing that the static magnetic field is rent densities are expressed in terms of the ampmuch larger than the coherent magnetic field (w, >> litudes of the stochastic modes. a), we find for the particle propagator Analysis of equations (11) and (16) shows that the injection of a coherent whistler with constant amplitude makes the ensemble average dis~hution U&T, T-r) = exp function periodic in Z, with period ho= 2dko. Since f must also be periodic in 4, the average distribution function is doubly periodic in 2 and 4. -D sin(k,Z+4)T , (24) in particular, equation (11) is satisfied when we use and (Brinca, 1976)

11

f(T-r)=

&(T-T,

T)fiT).

(25)

The upper limit of integration in (22) and (23) becomes +m because, having neglected the contribution of the initial fluctuations, we take T,-, --m. The Appendix outlines the main steps of the computation, up to second order in the CW whistier amphtude, of the particle propagator U&T, T7). Once this propagator is known, the dispersion relation for the stochastic modes and the evolution equation for the average distribution function are obtained through combination of the basic equations established above. 3. DISPERSION RELATION

The dispersion relation for the stochastic modes is obtained from the combination of the fluctuating component of the distribution function, (22), with Maxwell’s equations for the fluctuating fields, (12) and (13). By Fourier analyzing in space, and Lap-

&‘z, V,, 4, Z, T)= nF(V,, V,, T)+anfi,

(28)

where 2

-gFws(k.Z+z#). P 1 -1” _ dV,V12F, dVz v.L=2 I __ f0 fi=

n denotes the average electron number density, and vP is the plasma frequency. Utilization in (22) and (25) of this average distribution function and the results obtained in the Appendix show that, to second order in the CW whistler amplitude, the fluctuating component of the velocity distribution becomes

On the influence of a CW whistler on magnetospheric noise + S,D’ . ;

(S,‘F)

1

(31) and (32) as [K’-f(n+zK)1-b.(il,K)]=a’P(QK).

S,D’~--&(S,‘j,+S,‘F)

Going back to the laboratory relation takes the form

+ SID’ . & (S,‘f; + S,‘F) + S2D’ * -& (S,‘F)

with

II

(29)

K2-$(R+f-f K)2]Box’ =

h,%.,+aUO%,,+,+

b,Bb.,-,.

a*[ P(f-4 K)Bh.K

+ CI&,,K+~+c,B&_+

+ Wb.K+tko+ b.B;).m~g+ c&,.~+~lro (30) + ~,Bb.~-~lro 1 where the coefficients bi, ci and P are functions of R, K and F. Concentrating on the turbulent modes in the spectral vicinity of the coherent whistler (assumed to be a r-wave propagating along i), that is n-0 and IKI - ko, we recognize that the only modes entering (30) which correspond to whistler waves propagating along i are Bb,K and B&-2ko. Thus (30) becomes 9

= a2 c,B,&-~~+P(R,

[

K)B;,,

1 .

(31)

(n-2w,+~

= a2 c,‘B~,,+P’(fl, [

0

where Dar is the classical quasi-linear dispersion function,

whistler

4. DISCUSSION

The coherent whistler (o,,, k,) interacts with whistler noise modes (0, k) and gives rise to beat waves at (o foO, k* k,). These new waves bring about nonlinear wave-particle interactions at the resonant velocities (o + o0 - 2o,)/( k + k,) and (w o,)/(k - k,) which modify the classical QL whistler dispersion relation into the form given in (33). The interpretation of the alterations introduced in the stochastic (whistler noise) spectrum by the injection of the CW whistler is facilitated if we think of an ideal unperturbed QL turbulent system whose average distribution function coincides with F(v,, vl, t). We can then contrast the solutions of our perturbed real system, as defined by the nonlinear dispersion relation (33), with the behaviour of the unperturbed idealized medium. For an initial-value problem (imposed real k), the solutions of the ideal system should satisfy DQL(ou, k)=O,

co, =w,‘+iwui.

(35)

Since the injection of the monochromatic whistler creates a perturbation in the QL system, we expect the new solution to be of the form o, = w, + 60, and + SW, k) = a’P(w, + So, k).

(36)

Moderate amplitude coherent modes should give rise to perturbations where (801<<10~1. Taylor expansion of both sides of (36) yields thus

K)2-bo)]B;.,_,,,

K -2k,)B&_,,

with v, = (o - w,)/k.

D,(o,

Similarly, we could obtain [(K-2k,)2-$

(33)

T-7).

The explicit evaluation of (29) and calculation of its Fourier-Laplace transform fnk with the weak time dependence of F being neglected, allows the stochastic current density (27) to be obtained which, in turn, enters into (26). This procedure yields

[

frame, the dispersion

Dor(o, k) = 02P(o, k),

T-T)+VXB’(Z,

D’=E’(Z,

883

1 .

(32)

Neglecting terms of order higher than a2, the dispersion relation in the moving frame is defined by

sO=tiw’+i&b’

a2P =ao,, am

-az-

P(G apaa

a2

k)

aDo, (a,, k) ’ aa

(37)

A. L. BRIXA

884

The monochromatic wave impressed in the midst of a whistler noise spectrum alters the dispersion

characteristics of the turbulent modes, introducing a real frequency shift So’ and a correction SW’ to their growth rates. Relying on the analogy with the electrostatic problem analyzed elsewhere (Brinca, 1976), we expect to find in the neighborhood of k. a negligible real frequency shift, 80,-o, and correction to the growth rates, 60,, proportional to a*(l-k/k,). Hence, in the immediate vicinity of the injected CW whistler (k-k,), we witness an enhancement of the spectrum (6wi CO) for whistler noise modes with k > kO, and an abatement of the noise band at k < ko. In terms of a boundary-value problem where the injection is performed at a fixed location (z -10) with an imposed real frequency, we conclude that the noise spectrum above the CW whistler frequency is reinforced whereas the noise band lying immediately below the coherent mode is attenuated. This interaction mechanism leads to a qualitative agreement with respect to the location of the noisefree bands in the Siple experiments. One should bear in mind, however, that the concurring pitch angle scattering of energetic electrons in the actual, inhomogeneous geomagnetic field might be a dominant process in the creation of the quiet bands. Helliwell (private communication, 1976) suggests that the reduction in electron flux caused by pitch angle scattering by the coherent whistler in a region off the equator is responsible for the creation of the quiet bands: the scattered electrons are those that, prior to the CW injection, resonate with the (suppressed) noise at the equator. REFERENCES

Brinca, A. L. (1976). Modification of a turbulent spectrum through the injection of a monochromatic wave. J. Plasma Phys. 16, 353. Davidson, R. C. (1972) Methods in Nonlinear Plasma Physics, Chapters 9, 10 and 12. Academic Press, New York. Dupree, T. H. (1966). A perturbation theory for strong plasma turbulence. Phys. Fluids 9, 1773. Etcheto, J., Gendrin, R., Solomon, J. and Roux, A. (1973). A self-consistent theory of magnetospheric ELF hiss. j. geophys. Res. 78, 8150. . Helliwell, R. A. (1975). Siple Station VLF wave injection experiments. EOS Trans. Am. geophys. Union 56,426. Helliwell, R. A. and Katsufrakis, J. P. (19’74). VLF wave injection into rhe magnetosphere from Siple Station, Antarctica. .I. geophys. Res. 79, 25 11. Kennel, C. F. and Petschek, H. E. (1966). Limit on stably trapped particle fluxes. J, geophys. Res. 71, 1. Misauich. J. H. and Balescu, R. (1975a). Re-normalized q&i-linear approximation of plasma turbulence. J. phma Pkys. 13, 385.

Misguich, J. H. and Balescu, R. (1975b). Direct and inverse time-evolution propagators. General properties and application to plasma turbulence. Bull. Acad. R. Setg. CL Sci. S-LXI, 210. O’Neil, T. (1965). CoUisionIess damping of nonlinear plasma oscillaGons. Pkys. Fluids 8, 525% Weinstock, J. (1969). Formulation of a statistical theory of strong plasma turbulence. Phys. Fluids 12, 1045. Weinstock, J. (1972). Nonlinear theory of frequency shifts and broadening of plasma waves. Phys. Fluids 15,454. Yamamoto, T. (1975). Nonlinear theory of a whisder wave. J. Plasma Phys. 14, 543. APPENDIX The particle propagator Equation (21) defines the particle propagator in the CW frame. It represents the solution of the Vlasov equation when the self-consistent fields are neglected: Uo(TO, T,,)= 1,

z= v&

at,-+ws(koZ+4) I

1

$ - a sin (koZ + +)f.

Because we assume the stationary magnetic field to be much larger than the CW whistler field (o, >>a>,and particles with small V, fall into the geomagnetic losscone, we substitute w, for [* . .] in the expression of the average Vlasov operator. Following Misguich and Balescu (1975b), we resolve - 7; into two operators ii and B.

E= a sin (k,Z+ 4)I’, and define

UACCTO)=X+e(l; TO1A) =exp[ -(T-T~I(Vz~+_~)] =exp

f

-(T-

TO)Vz$

-(T-To~$

I

.

The desired propagator U&T, To) can be written as a formal power expansion of the E operator. Up to second order in E (and a), and making To= T -7, we have L&(X T-r)=

L’,(T, T-T) F + dT, U,O-, T,l%(T,, T- 7) IT-r ‘I ‘1 + dT1 ’ dT,UAU’, T,) I, -I IT-r x Bu,vl, T#U,(T,, T-T).

Noting that V,(Ti, T,) is a simple finite displacement

On the influence of a CW whistler on magnetospheric

noise

885

operator in the coordinates Z and 4,

4-o-i-~)o,l,

fJ,(T,, ‘I;)g(Z,6)=g[Z-Pi-Tj)V*,

+*]&~+VL’[-$cos2X6-&sin2X, 2

and performing the time integrations, yields

--&cos2Xd--&cos2X+-&-1sin(ro,)

1

U,(T, T-T)=SO+OS,+&,

,Z 2013

1

with

+*sin(Xd+X)-&cos(w,)

SO=eeA

+&os(x,+x)

I

1

1

+- 1 20,~

-a2 I az2I ’

where

, X= k,Z+&

X,=k,(Z+rV,)+&+rW,, o,=o,+k,V,.

-&

1

1

sin (70~) +&sin

(X, +X)

3 zcos2X,-~sin2X,-~sin2X+-%j 40,* 8~1 -$sin(7W1)+& 1

+$ 1

1

sin(X,+X)

The inverse time-evolution particle propagator can be obtained from this direct propagator. Since the average Vlasov operator is independent of time in the monochromatic wave frame, we find

1 $

U~(T-T,T)=S~i+aS*‘+o~S,‘,

1

1

20,

1

where the S,’ are obtained from the S, by changing 7 into -7. The perturbation parameter of these expansions is

Vz$

In low-temperature plasmas we have (o12)- oo2, so that Q <<1, because ak,V, represents the square of the bounce frequency of an electron with perpendicular velocity V,, phase-trapped in the bottom of the effective potential well of the coherent whistler, and for magnetospheric whistlers,

r:cos2X,-fcos2X-cos(7~,)

-cos(x,+x)+1+cos2x]rr

Thus, these orders of magnitude justify the adopted per-

-~c0S(X,+X)+~Sin(&+X)-isin2X

1

WI

2Wl

turbation whistlers.

scheme

when

dealing

with magnetospheric