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Two competing mechanisms have been proposed to explain the generation of auroral roar emissions in the ionosphere: direct excitation of X-mode electromagnetic waves by the auroral electrons via the cyclotron maser instability [Weatherwax et al., 1995; Yoon et al., 1996] and generation of electrostatic upper hybrid waves at altitudes where $f_{uh}=n$ $f_{ce}$, followed by conversion of these waves to electromagnetic waves via one of several linear and nonlinear mechanisms [e.g., Gough and Urban, 1983; Weatherwax et al., 1995]. Recently, Yoon et al., [1998] present a unified model of these mechanisms, showing that for auroral parameters the growth rate of the electrostatic upper hybrid waves is 2-3 orders of magnitude greater than that of the cyclotron maser stimulated X-mode waves; however, the efficiency with which the upper hybrid waves convert to electromagnetic radiation remains an open question.

Both of the candidate generation mechanisms predict that the frequency of the excited waves nearly equals 2 or 3 times the local electron gyrofrequency. This condition implies that the observed frequency of the emission is related to the source altitude, and the frequency drift of the emissions is related to the motion of the source. For a dipole magnetic field the component of the source velocity along the field is given by [e.g., Gurnett and Anderson, 1981; LaBelle et al., 1995]:

\begin{displaymath}{dr \over dt}~=~-{R_E \over 3} ({n f_0 \over f^4})^{1/3}{df \over dt}

where $f_0$ is the electron gyrofrequency at ground level, $f = n$$f_{ce}$ the emission frequency, and $R_E$ is the radius of the Earth. For parameters typical of auroral roar observations ($f_0$ = 1.6 MHz and $f=2.8$ MHz) [Weatherwax et al., 1995], (1) can be simplified to $dr/dt \cong -0.79 df/dt$, where the spatial variation is given in kilometers and frequency is given in kilohertz.

Using this relation, the maximum observed frequency drift (-790 kHz s$^{-1}$) corresponds to a source moving upward at ~620 km s$^{-1}$, and typical observed frequency drifts ($\pm$0-10 kHz s$^{-1}$) correspond to source motions upward and downward of magnitude 0-800 m s$^{-1}$. For comparison, at F-region altitudes the ion sound speed is typically 1.4-2.5 km s$^{-1}$ assuming that O$^+$ is the dominant ion, and the Alfvén speed is about 900 km s$^{-1}$. The drift velocity estimated from the maximum observed frequency drift measured approaches that of the local Alfvén speed, but typical drift velocities < 10 km s$^{-1}$ are closer to the ion sound speed. The thermal velocities of 10 keV oxygen ions, 100 eV protons, and 0.1 eV electrons also fall in the range of observed source velocities, as shown in Figure 6.

Assuming generation of the auroral roar fine structures at locations where $f=2$$f_{ce}$ in a dipole field, the separation of fine structure features by 100-1000 Hz implies source regions vertically separated by a few hundred meters. The upper bound on the minimum bandwidth of 6 Hz restricts the vertical spatial extent of the source to as small as a few meters, 2 orders of magnitude less than the free space wavelength at $f$ = 2.9 MHz. The effective $Q \equiv f/\delta f$ of the emission process is $5 \times 10^5$.

Two other observed emissions, auroral kilometric radiation (AKR) and Jovian decametric S-bursts, also exhibit fine structure. Table 1
\tablecaption{Comparision of Auroral...
...andwidth &
$\le$\ 6 & 5 & $<2$x$10^3$\tablenotemark{f} & hertz
summarizes some charateristics of these emissions for comparison to the less well known auroral roar.

AKR exhibits fine structure strikingly similar to that of auroral roar. As in auroral roar, both upward and downward drifting features are observed, and the velocities inferred from these drifts assuming generation at $f=$$f_{ce}$ generally comparable to the sound speed and are much less than the Alfvén speed [Gurnett and Anderson, 1981]. Nonharmonic multiplet features such as those seen in Figure 5 are also present in AKR fine structure [Calvert 1982]. As in auroral roar, the minimum bandwidths of the AKR fine structure correspond to vertical source dimensions smaller than the free space wavelength [Baumback and Calvert, 1987]. Of course, AKR is generated at much higher altitudes than auroral roar, and the intensity of AKR averaged over its bandwidth (of the order of $10^{-11}$ V$^2$ m$^{-2}$ Hz$^{-1}$ at 25 R$_E$) far exceeds that of auroral roar
( $\sim\! 10^{-15}$ V$^2$ m$^{-2}$ Hz$^{-1}$ at a few hundred kilometers).

The similarities between AKR fine structure and auroral roar fine structure suggest similar mechanisms. Calvert [1982] explains the extremely narrow bandwidth of AKR fine structures with a laser-feedback model: the boundaries of field-aligned density depletions provide the mirrors, and the unstable electron distribution provides the energy to the waves. Excitation occurs where an integer number of wavelengths fit across the density cavity. Provided that the wave growth across the cavity is sufficient to make up for the loss upon reflection, the wave grows to saturation. The excited wave is, in principle, perfectly monochromatic with frequency determined by the cavity dimensions. As the cavity dimensions change, the frequency drifts. Both upward and downward frequency drifts are thus naturally explained. Multiplet structures arise when m, m+1, m+2, etc. wavelengths fit into the density cavity at various locations along the field line. In the case of AKR the cyclotron maser growth rate is so large that an effective reflection coefficient of 1% or less suffices to produce the overall unity gain required for laser action. In the Calvert [1982] model the bandwidths and frequency spacings of multiplet structures composing AKR can be explained based on electron density cavities that are expected to exist in the AKR source region.

Figure 8. A model of an ionospheric density cavity in which waves reflect at the cavity walls such that an integer number of wavelengths fit in the cavity. Two adjacent feedback paths are shown with waves containing two and three wavelengths. The walls of the cavity make an angle $\alpha$ with the vertical. The wave along the lower altitude feedback path has a shorter wavelength because of the stronger magnetic field. Adjacent feedback paths are possible with vertical walls ($\alpha$ = 0), but the vertical separation between paths, and hence frequency separation, is reduced by vertically converging walls ($\alpha$ > 0).
\begin{figure}\figbox*{\hsize}{}{\epsfig{file=figs/cavity.eps, width=6cm}}

Unlike the case of AKR, the cyclotron maser mechanism at ionospheric altitudes is characterized by a low growth rate [Weatherwax et al., 1995; Yoon et al., 1996, 1998]. Hence, if the laser-feedback mechanism is to explain the fine structure of auroral roar, the reflection coefficient at the density cavity boundaries must be of the order of 99%, which seems unlikely, especially if the cavity walls are not perpendicular to the wave path. Nevertheless, it is instructive to consider the plausibility of the laser-feedback mechanism from a purely geometric perspective. Figure 8 illustrates a density cavity with walls making an angle $\alpha$ with vertical. A density cavity with this geometry is realized by considering a horizontal density cavity and an upward density gradient. The result is a field-aligned cavity with vertically converging walls similar to the geometry used by Calvert [1982] and not uncommon in the auroral zone. Two adjacent feedback paths are illustrated: one correpsonding to two wavelengths fitting across the cavity and the other corresponding to three wavelengths. The wavelength of the waves is shorter at lower altitude paths, where the frequency is higher, and therefore it is possible to have multiplet structures even in a cavity with vertical walls ($\alpha$ = 0). Assuming the X mode dispersion relation and generation of the waves in a dipole field geometry where $f=2$ $f_{ce}$ , it is possible to evaluate the vertical separation of adjacent feedback paths and hence the frequency spacing of the multiplet structures as a function of $\alpha$. Figure 9 illustrates the result. For vertical walls ($\alpha$ = 0) the observed frequency spacings of 100-500 Hz would require a density cavity with horizontal dimension $10^2$ - $10^3$ km, larger than typically observed in the ionosphere. F region density structures exist on a wide range of scales, but well-defined, field-aligned cavities most commonly have scales of tens of kilometers [Doe et al., 1993]. A cavity dimension of 50 km requires a cavity wall angle ($\alpha$) of over 13$^\circ$ to explain the observed 200-Hz frequency spacings. This geometry is perhaps not out of line with observations; however, it is hard to understand how > 99% of the wave energy can be reflected back along the feedback path under this condition. Of course, the cyclotron maser instability might still be responsible for the unstructured roar, which exhibits no multiplet features, or even structured roar if an approach other than the laser-feedback one is used.

Figure 9. The angle ($\alpha$) a cavity wall (as modeled in Figure 8) must make with vertical in order that adjacent feedback paths are separated by a frequency $\Delta f$ is shown for cavity widths ranging from 12 to 1200 km. For reference, the dotted line indicates that for a 50-km-wide cavity the observed frequency separation between multiplets of ~200 Hz (Figure 5) requires $\alpha >

Jovian decametric radiation contains substructure
such as ``S bursts'' which superficially resemble the fine structure in auroral roar reported here. Ellis, [1974] proposed a mechanism for the Jovian S bursts involving radiation from electrons moving adiabatically within the Jovian magnetic field and radiating at the local cyclotron frequency. To consider whether this idea has relevance to the auroral roar fine structure emissions, following Zarka et al., [1996], we write an expression for the time derivative of the frequency generated by an electron moving adiabatically in a dipole magnetic field and radiating at twice the local electron cyclotron frequency:

\begin{displaymath}{df \over dt}~=~{-3 \over LR_{\rm E}} g( \theta )f v \left[ 1...
...\Phi_{eq}){\pi m_e f L^3 \over B_{eq}e} \right] ^{1/2} \eqno(2)\end{displaymath}

where $f$ is the wave frequency, L the McIlwain L parameter of the field line, $v$ is the electron's velocity, $\Phi_{eq}$ is its equatorial pitch angle, $m_e$ is its mass, $B_{eq}$ is the magnetic field strength at the equator at L = 1, $\theta $ is the magnetic colatitude, and $g(\theta)$ is a factor nearly constant and of the order of unity at auroral latitudes:

\begin{displaymath}g(\theta )={ {\rm cos} \theta \over {\rm sin}^2 \theta }
...os}^2 \theta ) \over (1 + 3{\rm cos}^2 \theta )^{3/2}} \eqno(3)\end{displaymath}

The observed fine structures often extend over all or most of the 10-kHz bandwidth of the receiver, which corresponds to an altitude range of about 10 km if the radiation is at the electron cyclotron harmonic in a dipole field. Even within ~10 km of the electron's reflection height, $df/dt$ greatly exceeds the observed values for all except thermal electrons. For example, within 1-10 km above its reflection point a 1-keV electron with an equatorial pitch angle of 10.5$^\circ$ acquires a parallel velocity sufficient to make $df/dt$ = -200 to -2000 kHz s$^{-1}$, overlapping with the upper range of the observations. To produce $df/dt$ of the order of 10 kHz s$^{-1}$ over a 10-kHz band requires $\le 0.1$ eV electrons. Hence only for thermal electrons is the adiabatic motion of the electrons consistent with the drifting patterns of the fine structures.

However, another problem with this hypothesis is that the time durations of the fine structures (Figure 4) generally exceed the inverse electron-neutral collision frequency, which is of the order of 10$^{-3}$-10$^{-1}$ s at F region altitudes, depending on the phase of the solar cycle [Kelley, 1989]. If the emissions are nonthermal, they must be related to a nonthermal feature of the electron velocity distribution such as a loss cone, but for a given population of electrons such a feature will be isotropized on the timescale of the collision frequency. Since the fine structure features last much longer than the collision time, it is unlikely that they result from the adiabatic motion of an individual batch of electrons, even if for thermal electrons this motion can explain the frequency variation. The long timescale of the fine structure features relative to the collision time suggests that the fine structure frequency is selected by a condition on the wave conversion or excitation process rather than by the motion of individual electrons or batches of electrons. The frequency selection could be linked directly or indirectly to ion motion, however, as the ion collision frequency is much lower.

next up previous
Next: Conclusion Up: Further investigation of auroral Previous: Observations

Simon Shepherd 2002-05-02