1

Polarization of elliptic E region plasma irregularities and implications for

coherent radar backscatter from Farley Buneman waves

D. L. Hysell and J. Drexler

Department of Earth and Atmospheric Science, Cornell University, Ithaca, New York

Short title: POLARIZED E REGION PLASMA IRREGULARITIES

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Abstract.

The problem of two-dimensional, homogeneous, elliptical irregularities in an otherwise

homogeneous plasma with anisotropic conductivity is considered. We find an analytic

solution for the potential inside and outside the irregularities. In the special case of circular

irregularities, the internal electric field is reduced from the background field in both depletions

and enhancements. The internal field is rotated in a different directions for depletion and

enhancements, however. When the irregularity is elongated, the electric field inside can be

larger or smaller than the background field in both depletions and enhancements, depending

on the attack angle of the background field. The effects of ion inertia can further suppress

the internal electric field in small-scale circular irregularities. These electrodynamics

considerations may help explain some aspects of radar observations of irregularities excited

by Farley Buneman waves and instabilities in the electrojets, in particular their tendency

to exhibit Doppler shifts significantly smaller than the line-of-sight background electron

convection speed and proportional to the cosine of the flow angle. The analysis generalizes

that of St.-Maurice and Hamza [2001] who introduced this avenue of investigation.

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Introduction

It is well established that Farley Buneman waves in the equatorial and auroral E region

ionospheres propagate at phase speeds significantly less than the background electron

convection speed and roughly equal to the ion acoustic speed (e.g. Farley [1985], Haldoupis

[1989], Sahr and Fejer [1996]). This is inconsistent with the prediction of linear theory based

on plane-wave analysis, although linear theory does suggest that waves propagating at this

speed will be marginally stable (e.g. Fejer et al. [1984]). Evidence that nonlinear effects

are responsible for limiting the propagation speed of the waves to the ion acoustic speed is

compelling [Otani and Oppenheim, 1998, 2006]. The main nonlinear process at work seems

to be the generation of secondary Farley Buneman waves that propagate at oblique angles to

the primary waves [Oppenheim, 1997]. The superposition of the primary and secondary waves

produces a checkerboard pattern of discrete enhancements and depletions that propagate en

masse in (nearly) the direction of the background flow but at significantly slower speeds

and with opposing transverse drifts in the direction normal to it. This behavior is seen

clearly in numerical simulations of Farley Buneman instabilities driven well above threshold

[Oppenheim et al., 1996; Otani and Oppenheim, 1998]. Figure 3 of the latter manuscript along

with figures 7–9 of Otani and Oppenheim [2006] in particular show the configurations and

flow patterns to which we refer.

Recent experimental evidence indicates that the Doppler shifts of radar echoes from

Farley Buneman waves can obey a cosine dependence on flow angle [Woodman and Chau,

2002; Bahcivan et al., 2005]. The same cosine dependence has also been found in the

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numerical simulations using advanced diagnostics [Oppenheim et al., 2005]. This suggests

that the Doppler shifts may be indicative of the line-of-sight projection of the proper motion

of the irregularity patches in the simulations, each one behaving like a hard target (see also

Drexler and Maurice [2006] for an interpretation of the radar backscatter). We are therefore

interested in investigating the dynamics of these irregularities, which can be modeled as

elliptical regions of enhanced or depleted plasma. The purpose of this paper is to examine

how such irregularities drift apart from explicit consideration of any instability physics.

Note that there is no contradiction between the cosine dependence alluded to above and the

preponderance of echoes with Doppler shifts close to the ion acoustic speed observed with

the STARE and Millstone Hill radars [Haldoupis, 1989]. So called “type 1” echoes with this

property dominate coherent scatter from Farley Buneman waves at frequencies above 50 MHz

[Balsley and Farley, 1971] and are associated with scatter from small flow angles. At longer

wavelengths, echoes from all flow angles can be more readily observed and exhibit Doppler

shifts bounded by the ion acoustic speed.

The problem of elliptical irregularities in two dimensional, homogeneous, anisotropic

conductors was solved formally by Jones [1945] but not in the space physics context of

interest here. If it is found that the irregularities move much more slowly than and turn from

the background plasma because of purely electrodynamic considerations, it may be possible to

explain the failure of linear theory based on plane-wave analysis to predict the irregularity drift

speeds and the associated radar Doppler shifts. This was the reasoning that led St.-Maurice

and Hamza [2001] to undertake the first study of the electrodynamics of discrete, small-scale

E region plasma enhancements and depletions (blobs and holes) undergoing Farley Buneman

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turbulence. They showed that discrete, small-scale density irregularities both slow and turn

from the background flow in a predictable way. The present study aims to generalize their

findings somewhat utilizing a different mathematical approach (conformal mapping) that

permits the explicit assessment of the effects of elliptical irregularity geometry and orientation.

We limit our analysis to two-dimensional irregularities and therefore neglect the effects

of finite aspect sensitivity, which are outside the scope of this study. The analysis considers

irregularities with jump discontinuity boundaries and therefore must also exclude the effects of

diffusion and diamagnetic drift. We consider irregularities of sufficiently small scale to avoid

issues pertaining to electrostatic potential mapping along geomagnetic field lines, although

the results could be generalized by replacing local conductivities with flux tube-integrated

conductivities. Although we focus on Farley-Buneman waves, many of the results could be

applicable to other waves and flows.

This paper is organized as follows. We begin by finding an analytic solution for the

electrostatic potential in the vicinity of two-dimensional elliptical irregularities in an E region

plasma in a background electric field. We highlight the special case of circular irregularities

but consider also how elongation affects their dynamics. The effects of ion inertia are also

considered. Finally, we compare our results to those from other related studies and assess the

implications for Farley Buneman waves.

Electric field solutions

This will be a two-dimensional treatment of plasma density irregularities and the

associated electric fields in the plane perpendicular to the background magnetic field. The

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plasma has uniform Pedersen and Hall conductivities, and the irregularity is a uniform,

elliptical enhancement or depletion with a step boundary. We seek solutions for steady flow

around the irregularity boundary at the moment its center coincides with the origin of a polar

coordinate system. As the electrons are magnetized, equipotential contours are streamlines of

the electron flow around and through the irregularity. Speeding, slowing, and deflection of the

irregularity will be signaled by an internal electric field differing from the background field. Figure 1.

We regard the plasma as a dielectric and assert that the electrostatic potential obeys

Laplace’s equation inside and outside the irregularity boundary. The boundary conditions

are that the potential and the normal component of the current density be continuous across

the irregularity wall. In addition, the electric field outside the irregularity should transition

to a uniform background field at large distances. In an isotropic dielectric, the electric field

inside is shielded by induced dipole moments and polarization surface charge at the wall that

emerges to maintain the boundary conditions. In an anisotropic conductor (i.e. with second

rank conductivity), the field inside is also rotated.

Following Smythe [1939], we solve this problem by solving two related ones. The

construction involved is counterintuitive but mathematically expedient and accurate. Consider

a disk of radius r1 surrounded by a concentric ring of enhanced or depleted plasma of outer

radius r2, itself surrounded by homogeneous background plasma. This situation is depicted in

the left panel of Figure 1. A uniform background electric field is applied. We seek solutions

to Laplace’s equation everywhere outside the disk for two distinct configurations where

Dirichlet and Neumann boundary conditions are applied, respectively, at the surface of the

disk. Results from the two configurations will contribute to the complete solution of the

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elliptical irregularity problem.

Given the configuration with Dirichlet boundary conditions, the solutions for the potential

in the regions within and outside the ring of plasma must have the forms:

φr1<r<r2 =

(

r − r21r

)

(a cos θ + b sin θ) (1)

φr>r2 =1

r(A cos θ + B sin θ)− Er cos θ (2)

where E is the amplitude of the background electric field and θ is the polar angle measured

counterclockwise from the background field direction. The coefficients are set by applying

boundary conditions. One of these is that the potential should be continuous across the r = r2

boundary. The other is that the component of the current density J normal to the boundary

should be continuous across it. (This is the integral form of the plasma quasineutrality

condition.) For the moment, we regard the current density to be comprised of Pedersen and

Hall terms, with radial components given by

J · r = σPEr + σHEθ

= −σP∂φ/∂r − (σH/r)∂φ/∂θ

where σP and σH are the Pedersen and Hall conductivities, respectively, which we take as:

σP ≈ neΩi

νiB

σH ≈ ne/B

The equations to be solved are consequently:

(a cos θ + b sin θ)r−

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= (A cos θ + B sin θ)r−1

2 − Er2 cos θ (3)

−(aσpir+ + bσhir−) cos θ − (bσpir+ − aσhir−) sin θ

= [(Aσpo −Bσho) cos θ + (Bσpo + Aσho) sin θ] r−1

2

+(σpoE cos θ − σhoE sin θ) r2 (4)

with (3) imposed by the continuity of the potential and (4) by the continuity of the current

density across the irregularity boundary. Here, the i and o subscripts denote quantities inside

and outside the irregularity, respectively. We also introduce the notation r± = r2 ± r21/r2. Sine

and cosine terms must equate separately. Solving the system gives:

a =−2σpoEr2 (σpir+ + σpor−)

(σpir+ + σpor−)2 + (σhor− − σhir−)

2

b = aσhir− − σhor−σpir+ + σpor−

A = ar2r− + Er2

2

B = br2r−

Substituting these expressions into (1) and (2) yields the potential everywhere outside the

disk, which behaves like a conductor in this configuration. Equipotential lines for this solution

are shown in the left panel of Figure 1 as solid lines. Note that the solution has been rotated

so that the zero equipotential inside the plasma ring is coincident with the horizontal axis. In

view of (1), the necessary rotation angle is given by θ = tan−1(b/a).

Given the configuration with Neumann boundary conditions at r = r1, the correct form of

the potential becomes:

φ′

r1<r<r2=

(

r +r21r

)

(a′ sin θ − b′ cos θ) (5)

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φ′

r>r2=

1

r(A′ sin θ −B′ cos θ)− Er sin θ (6)

The primed coefficients can be solved by applying the same boundary conditions at r = r2.

Doing so produces expressions for a′, b′, A′ and B′ of precisely the same form as listed above

except with the r+ and r− terms exchanging roles. (That is, wherever r+ appears, replace

it with r−, and vice versa.) It can be shown that (5) is conjugate to (1), meaning that the

equipotential curves they give rise to fall at mutual right angles. The equipotentials for one

configuration are the lines of force for the other. The same relationship does not hold for (6)

and (2), however.

Equipotential contours of φ′ are plotted in the left panel of Figure 1 using dashed lines.

This time, the solution has been rotated so that the zero equipotential inside the plasma ring is

coincident with the vertical axis. The angle of rotation this time is given by θ′

= tan−1(b′/a′),

where the primed coefficients are calculated according to the prescription in the preceding

paragraph. While both the primed and unprimed potential solutions denote uniform electric

fields at large distances from the origin, those electric fields are not orthogonal to one another

when the solutions inside the plasma ring are orthogonal, as θ 6= θ′

in general.

The horizontal and vertical axes of the left panel of Figure 1 may be regarded as the real

and imaginary axes of the complex plane z = x + iy. The next objective is to map this plane

into another one, w = u + iv, where the boundaries transform into the desired ellipse but where

the form of Laplace’s equation and the boundary conditions are maintained. The necessary

conformal transformation is given by

z = w ±√

w2 − r21 (7)

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which maps the circle of radius r1 in the z plane to a line segment on the real axis between

u = ±r1 in the w plane (see middle and right panels of Figure 1). Concentric circles with radii

r2 > r1 in the z plane map to ellipses with foci at the endpoints of the line segment in the w

plane. The major and minor axes of the ellipse are given by r± ≡ r2 ± r21/r2, respectively.

At large radial distances, the z and w planes become the same except for a constant factor

of two. This means that the boundary conditions at infinity are also the same and that the

background electric field that dominates at large distances in both is the same field (within a

constant). Note that the w plane panels in Figure 1 have been scaled by this factor of two for

easier viewing; correct proportions are otherwise maintained.

The potential solutions for the two configurations illustrated in the left panel of Figure 1

have been transformed according to (7) and plotted as solid and dashed lines, respectively, in

the middle panel. In the configuration with Dirichlet boundary conditions, the circle of radius

r1 in the z plane was an equipotential, making the line segment in the w plane between the foci

of the ellipse an equipotential. Lines parallel to that inside the ellipse are also equipotentials.

In the Neumann boundary condition configuration, the circle of radius r1 was a line of force.

Consequently, the associated equipotentials in the w plane are normal to the line segment

between the foci (vertical).

The potential in the vicinity of the ellipse arising from an arbitrary background field

can be formed from the superposition of the results from the two configuration. This is

illustrated in the right panel of Figure 1, which gives the result of adding equal parts of the

solutions in the middle panel. Far from the irregularity center, the potential is that of a uniform

background electric field. Inside the irregularity, the electric field is uniform, but its direction

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and magnitude differ from the background field. The rotation angle is between θ and θ′,

depending on the weighting. Fringing fields distort the flow just outside the irregularity

boundary.

Circular irregularities

The formulas derived above become considerably less cumbersome in the illustrative

case of a circular irregularity, for which r1 = 0 and r± = r2 ≡ r. The potential is given by (1)

and (2) without further transformation or construction. Furthermore, the coefficients on the

harmonics assume relatively simple forms:

A =∆Σ+R2∆2

Σ2 +R2∆2Er

2

B =R∆(∆− Σ)

Σ2 +R2∆2Er

2

a = A/r2− E

b = B/r2

where use has been made of the following abbreviations:

∆ = σPi − σPo

Σ = σPi + σPo

R = σHi/σPi = σHo/σPo ∼ 10− 20

and where the subscripts i and o continue to imply quantities inside (r < r) and outside

(r > r) the circular irregularity. Here, ∆ > 0 (∆ < 0) denotes and enhancement (depletion).

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The solution describes electron flow around the circular irregularity that is nearly

tangential to the boundary except in the ram and wake. Inside the irregularity, the electric field

is uniform. The magnitude of the interior field is

|Er<r| = E

|Σ−∆|√Σ2 +R2∆2

(8)

While it is possible for (8) to predict electric fields stronger than E in a depleted irregularity

given small enough values of R, values of R greater than√3 imply attenuation of the field

inside the irregularity for either sign of ∆.

The angle the interior field makes with the background field is:

tan θ =b

a=

R∆

Σ(9)

which can have either sign depending on the sign of ∆. The effects of anisotropic conductivity

(R > 0) are made clear by (8) and (9). Figure 2.

Figure 3.Potential solutions for a circular density depletion and an enhancement are shown in

Figure 2 and Figure 3, respectively. The interior number density is 20% below and above

background in the two cases. We take R = 15 in both examples. Equipotential contours are

plotted. The background electric field points toward the right, and the angle θ is measured

counterclockwise from the 3 o’clock position. Taking the magnetic field direction to be into

the paper, the streamlines of the flow go from bottom to top.

Eq. (8) predicts internal electric fields 57% and 54% of the background field for the

depletion and enhancement, respectively. Eq. (9) predicts rotations of -59 and 54 for

these two cases. The rotation has opposite senses for depletions and enhancements, but the

internal electric field is reduced (shielded) in both cases (as first pointed out by St.-Maurice

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and Hamza [2001]). Note also that depletions and enhancements behave asymmetrically, the

former giving rise to stronger electric field perturbations than the latter. The asymmetry arises

from the fact that it is the difference between the inner and outer conductivities that causes

polarization but the sum of the inner and outer conductivities that arrests it and maintains

quasineutrality. Symmetry is approximately restored when the density perturbations are small.

Effect of irregularity elongationFigure 4.

Figure 5.The effects of irregularity elongation can be assessed from two extreme examples.

Figure 4 shows the equipotentials in the vicinity of an elliptical irregularity oriented such that

the background electric field is parallel to its minor axis. The irregularity is a 20% depletion

with an aspect ratio of 6:1. The equipotentials outside the irregularity are nearly parallel to its

surface, and the flow around it is almost uninterrupted. The interior electric field is slightly

smaller than what (8) predicts, but the rotation angle is much less than (9). This is a case

where elongation counteracts rotation.

In contrast, Figure 5 shows the same irregularity only oriented with its major axis

parallel to the background electric field. This time, flow around the irregularity is drastically

interrupted. The rotation angle of the electric field is somewhat greater this time than what (9)

predicts. Notably, the electric field inside the irregularity is larger than the background field.

This is a case where elongation counteracts the suppression of the internal field. Enhanced

irregularities behave similarly, producing internal fields both weaker and stronger than the

background field depending on the attack angle of the latter. The angle of rotation is reversed

from that of depletions. Figure 6.

Figure 7.

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The effects of elongation are summarized by Figure 6, which plots the magnitude and

direction of the electric field inside a 20% depleted irregularity relative to the external field.

The abscissa is the angle that the background field makes with the major axis of the ellipse.

Three ellipses, with aspect ratios of 5:3, 3:1, and 6:1, are considered. The figure illustrates

that elongation invalidates the results from the circular irregularity analysis in the manner

exemplified above. The rotation and attenuation of the internal electric field is counteracted

when the irregularity is elongated perpendicular to and parallel to the background electric

field, respectively. Sufficiently elongated depletions can have internal electric fields stronger

than the background field and rotation angles greater than 90. Similar remarks apply to

Figure 7, which shows the results for 20% enhanced elliptical irregularities.

The tendency for the interior field in circular E region irregularities to be reduced for

either sign of ∆ arises from the dissimilar role of the Hall conductivity inside and outside

the irregularity. Outside, the equipotential lines are nearly tangent to the irregularity, the

electric field is nearly normal to the boundary, and the Hall current does not contribute

significantly to the radial current flowing through the boundary. This is not true in the interior,

where the equipotentials are oblique to the irregularity wall and the Hall current contributes

substantially to the radial current balance. In that regard and given sufficiently large ratios R,

density depletions and enhancements tend to behave like conductivity enhancements alike.

Only for small values of R can the interior electric field in circular depletions exceed the

background field. When the irregularity is elongated and oriented as it is in Figure 5, however,

the interruption in the flow causes the equipotentials outside the irregularity to approach it

normally instead of tangentially. In that event, Hall currents flowing outside the irregularity

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can contribute to the boundary current, and the electric field inside the depletion can exceed

the background electric field.

Effects of ion inertia

Thusfar, only Pedersen and Hall currents have been included in the boundary conditions

of the potential problem. For small-scale irregularities, it is also necessary to include

polarization currents associated with ion inertia. The “irregularity” in question here is the

boundary between the background and the depleted or enhanced plasma. Since the electrons

are incompressible, the boundary must move with the velocity of the electrons inside. The

component of the electron drift velocity normal to the boundary is continuous across the

boundary, which is just another statement of incompressibility. The tangential component is

not continuous, but tangential flow is immaterial to the boundary motion.

Ions generally move much more slowly than the electrons. Quasineutrality is nevertheless

maintained as ions converge or diverge in front of and behind the irregularity so as to preserve

the integrity of the boundary. To the extent they are unable to change configuration rapidly,

the electric field inside the irregularity and the irregularity velocity are also limited. Such

limiting depends on the shape of the irregularity and on the inner and outer conductivities, in

the manner shown above.

Ion inertia causes an ion current to flow when the electric field structure associated with

the irregularity boundary drifts past. The smaller the scale of the structure r and the faster

the drift speed v, the stronger the current. Polarization currents should compete with Pedersen

currents when the time it takes for an irregularity to drift past is comparable to a collision

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time, or when r/v ∼ ν−1

i . Extending the analysis to include polarization currents introduces

a scale-size dependence to the problem.

We consider the effects of polarization currents on the dynamics of circular irregularities

only. The polarization current density is carried by the ions and has the form (in the collisional

limit):

J = −neΩi

ν2i B

dE

dt= −σP

νi

dE

dt

which is obtained by solving the ion momentum equation, retaining the Lorentz force,

ion-neutral collision, and inertia terms. The strategy for evaluating the time derivative is to

consider steady flow around and through the circular irregularity. The ions have negligible

velocity in the lab frame of reference considered to this point in the analysis. In the frame

of reference fixed to the irregularity and moving with velocity v in the lab frame, the total

time derivative above can be replaced with the convective derivative, −(v · ∇), where the

irregularity velocity is again the E × B drift velocity associated with the internal electric

field. Note that ∇E is invariant and can be evaluated in any inertial frame of reference; the

remainder of the calculations take place in the lab frame.

Polarization current flows only outside the irregularity where the electric field is

nonuniform. Only the radial component of the current density enters into the boundary

conditions for the problem. In polar coordinates, we evaluate this current density at the

boundary using (1) and (2):

v · ∇Er|r = −vr2

r3

(A cos θ + B sin θ) (10)

+vθr3

[(

−(A+ ar2) sin θ

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+(B + br2) cos θ

)

− Er2

sin θ

]

where vr and vθ are the radial and azimuthal components of the irregularity drift velocity

at the irregularity boundary. A general solution to the potential problem incorporating the

current density implied by (10) in the boundary conditions does not exist, and it appears that

the flow is not steady but rather evolves when the effects of ion inertia are included in the

analysis. The boundary will consequently distort in a manner this analysis cannot predict.

However, we can arrive at an approximate solution by assuming steady flow at the leading

edge of the irregularity, near the ram, and enforcing the boundary conditions there. Near the

ram, vr ∼ v and vθ ∼ 0 by definition, and the surviving terms in (10) can be added to the

right side of (4) and combined there with the terms representing the Pedersen current flowing

outside the irregularity, which have the same form. In effect, ion inertia reduces or counteracts

the Pedersen current driven by polarization charge accumulated at the irregularity boundary.

These Pedersen currents tend to diminish the polarization charge and its inherent shielding

effect, but inertia helps to restore the shielding. Since the background electric field ultimately

responsible for the charge accumulation is uniform, it gives rise to no polarization current to

offset the Pedersen current it drives.

The prescription for including the effects of ion inertia is then to define an effective

Pedersen conductivity for the region outside the irregularity:

σpo = σpo

(

1− 2v

rνi

)

(11)

and to substitute this term into (4), which now reads:

−(aσpi + bσhi) cos θ − (bσpi − aσhi) sin θ

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= [(Aσpo −Bσho) cos θ + (Bσpo + Aσho) sin θ] r−2

+σpoE cos θ − σhoE sin θ (12)

Notice that σpo still modifies the E term, which is unaffected. Propagating the changes

through the preceding formalism leads to the following results:

Σ = σPi + σPo

|Er<r| = E

|Σ−∆|√

Σ2 +R2∆2

(13)

tan θ =R∆

Σ(14)

Finally, (13) must be solved self consistently since the v in (11) is E/B. The result depends

on the irregularity radius r through (11) as well as on the collision frequency and irregularity

amplitude.

Comparison to other studies

There have been a number of studies evaluating the electric field inside F region plasma

density irregularities associated with barium cloud releases and equatorial spread F “bubbles”.

Ott [1978] calculated the ascent rate of circular bubbles in the collision- and inertia-dominated

flow regimes. His collisional regime calculation was general, but his inertial regime calculation

was limited to the case of 100% plasma depletions. Ossakow and Chaturvedi [1978] undertook

a related study, considering partial enhancements and depletions and allowing for ellipsoidal

geometries but neglecting inertial effects. Most recently, [McDaniel and Hysell, 1997]

considered the case of partial, circular enhancements and depletions, including inertial effects.

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Taken together, the studies found that F region depletions can develop interior electric fields

much larger than the background field but that inertia limits the field amplitude, particularly

when the irregularity is small. Enhancements, meanwhile have reduced interior electric

fields. The interior field does not rotate. Depletions give rise to relatively larger electric field

perturbations than enhancements for the same reason discussed above. Our results (taking

R=0) agree with the aforementioned studies in the collisional regime. As the effects of ion

inertia are different in E and F region plasmas, we do not expect agreement in the inertial

regime.

The present results can also be compared to those obtained computationally by Hysell

et al. [2002]. In their simulations, which followed on the analysis of Shalimov et al. [1998],

large electric fields many times background were produced inside some E region plasma

density enhancements. However, those results were obtained with a three-dimensional model

that included electrical coupling to the F region. Coupling was scale-size dependent, and

electric field enhancements were only obtained for elongated E region structures with major

and minor axes perpendicular to B longer and shorter than about 1 km, respectively. The

results had a narrow range of applicability and do not apply to the sub meter-scale structures

of primary interest here.

Finally, this study is obviously most closely related to the pioneering one by St.-Maurice

and Hamza [2001], who calculated the electric field inside elongated E region density

irregularities using a limiting approach and assuming a balance between inertia and diffusion.

Their approach did not permit the precise specification of the irregularity shape and neglected

the fringing fields that form outside the irregularity. Given the differences in approaches, it is

20

interesting to note that their eq. (19) predicts internal field rotations in agreement with our (9).

The amplitude of the internal field they predicted is consistent with (8) except with the ∆ term

in the numerator in our expression deleted. Their results evidently describe the behavior of

circular irregularities with small density perturbations (but large values of the ratio R).

In a plane-wave treatment of Farley Buneman waves, currents driven by ion inertia and

diffusion balance in the case of marginal stability. While the effects of diffusion have not been

addressed in the present analysis, we might expect inertia and diffusion to balance in a global

sense but not in a local sense, the dependence of the two mechanisms on angle being different.

Our treatment of inertia should therefore hold, at least approximately, in the ram and the wake

of the irregularity where polarization currents are largest and, presumably, uncompensated by

diffusion. This hypothesis can be verified through numerical simulation.

Discussion

For illustrative purposes, we have considered irregularities with 20% density depletions

and enhancements in the examples shown throughout the paper. Density fluctuations in

Farley Buneman turbulence are meanwhile limited to about 5% RMS [Pfaff , 1991], implying

much weaker polarization effects. In such cases, (8) and (9) predict electric field rotations of

about ∼ ±20 and modest field reductions proportional to the amplitude perturbation. When

the irregularity is elongated to a 6:1 aspect ratio, however, the internal electric field can be

reduced or enhanced over background by as much as 20%, depending on the attack angle of

the background field. Larger aspect ratios imply even larger perturbations and rotations. We

can model a primary Farley Buneman wave packets as series of elongated, parallel plasma

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irregularities, oriented to the background field as shown in Figure 5 and alternating between

enhancements and depletions, the crests and troughs of the primary waves. The rotation

angles within the irregularities would alternate signs accordingly. This picture is consistent

with the wave turning described by Oppenheim et al. [1996], who pointed out the asymmetry

between enhancements and depletions that we have accounted for on purely electrodynamic

considerations.

Because of the rotations, convection electric fields with components transverse to the

background field would be available to drive secondary Farley Buneman waves at right angles

to the primary wave. The superposition of the secondary waves and the primary would then

result in a patchwork of roughly circular enhancements and depletions. In simulation, these

irregularities predominate and have sub-meter scales [Oppenheim et al., 1996].

Ion inertia and the resulting polarization currents would tend to arrest the motion of

the small-scale irregularities. For example, setting the parameter 2v/rνi in (11) to unity,

considering a circular irregularity with a 5% density perturbation, and propagating the results

through to (13) predicts an internal electric field reduced to 4/5 the value of the background

field. Taking E/B to be 1200 m/s then implies an internal drift velocity of E/B = 960 m/s.

The rotation angles predicted by (14) in this case are about ∼ ±40 and is slightly different

for enhancements and depletions. The component of the irregularity drift velocity in the

direction of propagation of the primary wave is therefore about 740 m/s, which is something

like the propagation speed of the waves observed in simulation. Given an ion-neutral collision

frequency of 1000 s−1 then requires, in consideration of (11), an irregularity radius of 0.8

m for self consistency. This is even larger than the size of the irregularities emerging in the

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numerical simulations of Oppenheim et al. [1996] and consistent with wave speed saturation

well below the background electron convection speed. In view of their opposing and slightly

asymmetric rotations, irregularities in opposing phases of the primary wave would tend to

drift almost at right angles with respect to each other, with the average direction being slightly

different than the background convection direction.

Secondary wave pumping and the subsequent amplification of the secondary waves

continues until the primary wave speed is reduced to the ion acoustic speed, at which time the

primary wave becomes marginally stable. Assuming the Doppler shift of radar backscatter

from the irregularities is indicative of their proper motion, we would expect a mean Doppler

shift of ∼ Cs cos θ and a Doppler width of ∼ Cs sin θ, where θ is the angle between the

background electron convection and the radar line of sight. Experiments conducted recently

support this picture [Woodman and Chau, 2002; Bahcivan et al., 2005].

Acknowledgments. DLH is thankful for helpful discussions with Prof. J. P. St. Maurice.

This work was supported by the National Science Foundation through NSF grant ATM-0436114 to

Cornell University. The Jicamarca Radio Observatory is a facility of the Instituto Geofısico del Peru

and is operated with support from the NSF Cooperative Agreement ATM-0432565 through Cornell

University. The help of the staff was much appreciated.

23

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Received X; revised X; accepted X.

To appear in Radio Science, 2006

This manuscript was prepared with AGU’s LATEX macros v5, with the extension package

‘AGU++’ by P. W. Daly, version 1.6b from 1999/08/19.

26

Figure Captions

Figure 1. Figures illustrating the electrostatic potential in the vicinity of an elliptical, anisotropic con-

ductor in a background electric field. The problem is scale invariant, so the scaling and labeling is

arbitrary. We consider a 20% elliptical depletion with an aspect ratio of 5:3. The ratio of the Hall

to Pedersen conductivity is 15. Left panel: equipotentials outside a disk of radius r1, surrounded by

a concentric ring-shaped irregularity with outer radius r2, itself surrounded by homogeneous plasma.

Solid (dashed) lines result from applying Dirichlet (Neumann) boundary conditions at the surface of the

disk. Center panel: same as left panel, only transformed into the w plane. Here, the solid and dashed

lines represent equipotentials aligned with the major and minor axes of the ellipse with foci at u = ±r1.

Right panel: superposition of equipotentials from the center panel. Equipotentials in the vicinity of an

elliptical irregularity in an arbitrary background electric field are thus constructed (see text).

27

Figure 2. Potential contours for a 20% circular depletion.

Figure 3. Potential contours for a 20% circular enhancement.

28

Figure 4. Elliptical depletion irregularity with an aspect ratio of 6:1. The minor axis aligned with the

background electric field.

Figure 5. Elliptical depletion irregularity with an aspect ratio of 6:1. The major axis is aligned with the

background electric field.

29

Figure 6. Amplitude and rotation angle of electric field within elliptical depletion irregularities as a

function of the angle between the major axis and the background electric field. Solid lines depict the

modulus of the electric field as a fraction of the background field. Dashed lines give the angle between

the internal and background field. Three sets of curves represent irregularities with aspect ratios of 5:3,

3:1, and 6:1. Larger aspect ratios give more eccentric curves.

Figure 7. Same as previous figure, only for 20% elliptical enhancement irregularities.

30

Figures

Figure 1. Figures illustrating the electrostatic potential in the vicinity of an elliptical, anisotropic con-

ductor in a background electric field. The problem is scale invariant, so the scaling and labeling is

arbitrary. We consider a 20% elliptical depletion with an aspect ratio of 5:3. The ratio of the Hall

to Pedersen conductivity is 15. Left panel: equipotentials outside a disk of radius r1, surrounded by

a concentric ring-shaped irregularity with outer radius r2, itself surrounded by homogeneous plasma.

Solid (dashed) lines result from applying Dirichlet (Neumann) boundary conditions at the surface of the

disk. Center panel: same as left panel, only transformed into the w plane. Here, the solid and dashed

lines represent equipotentials aligned with the major and minor axes of the ellipse with foci at u = ±r1.

Right panel: superposition of equipotentials from the center panel. Equipotentials in the vicinity of an

elliptical irregularity in an arbitrary background electric field are thus constructed (see text).

31

Figure 2. Potential contours for a 20% circular depletion.

Figure 3. Potential contours for a 20% circular enhancement.

32

Figure 4. Elliptical depletion irregularity with an aspect ratio of 6:1. The minor axis aligned with the

background electric field.

Figure 5. Elliptical depletion irregularity with an aspect ratio of 6:1. The major axis is aligned with the

background electric field.

33

Figure 6. Amplitude and rotation angle of electric field within elliptical depletion irregularities as a

function of the angle between the major axis and the background electric field. Solid lines depict the

modulus of the electric field as a fraction of the background field. Dashed lines give the angle between

the internal and background field. Three sets of curves represent irregularities with aspect ratios of 5:3,

3:1, and 6:1. Larger aspect ratios give more eccentric curves.

Figure 7. Same as previous figure, only for 20% elliptical enhancement irregularities.