Cavity Mode Oscillations Research Articles

Ultra-Wide Bandwidth Brillouin-Raman Random Fiber Laser With Augmented Synergistic Nonlinearity

We propose a Brillouin-Raman random fiber laser with enhanced synergistic nonlinearity for ultra-wide bandwidth Brillouin comb generation. The augmented synergistic nonlinearity, including enhanced Rayleigh scattering (RS) effect, stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS), is achieved in a hybrid fiber (a fiber combination of a 7.2-km dispersion compensation fiber and a 10-km single-mode fiber) under a strong Raman pumping condition. Under the strong Raman pumping condition, more Stokes lines are generated due to the enhanced RS-induced oscillations of Brillouin Stokes lines at short wavelength band, thus suppressing self-lasing cavity mode oscillations at long wavelength band. In the experiments, the multi-wavelength Brillouin comb with an ultra-wide bandwidth of 63.5 nm (1513 nm-1576.5 nm) with frequency spacing of double Brillouin frequency shift, which is the largest bandwidth from Brillouin fiber lasers, to the best of our knowledge, has been achieved. The proposed ultra-wide bandwidth Brillouin-Raman fiber laser has potential applications in optical communication, optical wavelength division multiplexing, and fiber sensing.

Statistical Analysis of Pi2 Pulsations Observed by Van Allen Probes

AbstractThe relative importance of propagating and cavity mode waves remains an important question regarding the generation of Pi2 pulsations detected on the ground. To determine the wave mode, we statistically generate spatial maps of magnetospheric oscillations that are coherent with ground Pi2 pulsations. The magnetospheric observations were made by the two Van Allen Probes spacecraft over a 7‐year period. The amount and quality of the spacecraft data allow us to investigate the mode structure of Pi2 pulsation in ways that were not possible in previous studies. We use the H component of low‐latitude ground Pi2 pulsations detected in the 22–02 magnetic local time (MLT) sector as the reference signal to generate L‐MLT and meridional maps of the coherence, amplitude, and phase of the magnetospheric electric and magnetic field components defined in magnetic field aligned coordinates. We identify low‐frequency and high‐frequency components in Pi2 power spectra, and we are able to determine the mode structure of the high‐frequency events for the first time. The maps demonstrate that the poloidal components have higher coherence than the toroidal components. For each frequency component, the maps of the poloidal components agree with those of cavity mode oscillations obtained in a numerical simulation using realistic models for the magnetospheric mass density and magnetic field. This result is conclusive evidence of the cavity mode nature of Pi2 pulsations detected in the inner magnetosphere.

Observational study of ionosphere TEC fluctuations triggered by interplanetary shock

<p indent=0mm>Interplanetary shocks can cause violent disturbances in near-earth space. In this paper, we use high time-resolution <sc>(1 s)</sc> data from ground-based global position system (GPS) receivers to analyze the perturbations of ionosphere total electron content (TEC) when an interplanetary shock impacted the Earth’s magnetosphere on March 17, 2015. We found that after the shock arrived at the Earth, TEC perturbations demonstrated the characteristics of ULF waves, and the fluctuation characteristics were different in different latitude regions. The amplitude of TEC fluctuations observed by the receivers in the polar region was approximately 0.4 TECU, which lasted for more than <sc>10 min,</sc> and the fluctuation period was 2 to <sc>3 min,</sc> which increased with the increasing latitude of the pierce point. The amplitude of the TEC fluctuations observed by the receivers in middle and low latitude areas was approximately 0.2 TECU, which lasted for 5 to <sc>10 min,</sc> and the fluctuation period was approximately 2 min. We suggest that the existence of plasmasphere may be the reason for such difference in TEC fluctuations, and that the TEC fluctuations in polar regions may be related to field line resonances. The TEC fluctuations in the middle and low latitude areas may be related to cavity mode oscillations. We used GPS TEC technology to detect ionosphere TEC fluctuations triggered by interplanetary shock, which can help us further understand the nature of ULF waves in the Earth’s magnetospheric-ionospheric system and can further demonstrate the important role of GPS TEC technology in the study of the near-earth space environment.

Quantum frequency locking and downconversion in a driven qubit-cavity system

We study a periodically driven qubit coupled to a quantized cavity mode. Despite its apparent simplicity, this system supports a rich variety of exotic phenomena, such as topological frequency conversion as recently discovered in [Martin et al, PRX 7, 041008 (2017)]. Here we report on a qualitatively different phenomenon that occurs in this platform, where the cavity mode's oscillations lock their frequency to a rational fraction $r/q$ of the driving frequency $\Omega$. This phenomenon, which we term quantum frequency locking, is characterized by the emergence of $q$-tuplets of stationary (Floquet) states whose quasienergies are separated by $\Omega/q$, up to exponentially small corrections. The Wigner functions of these states are nearly identical, and exhibit highly-regular and symmetric structure in phase space. Similarly to Floquet time crystals, these states underlie discrete time-translation symmetry breaking in the model. We develop a semiclassical approach for analyzing and predicting quantum frequency locking in the model, and use it to identify the conditions under which it occurs.

L Versus Time Structures of Dayside Magnetic Pulsations Detected by the European Quasi‐Meridional Magnetometer Array

AbstractWe studied the spatiotemporal structure of ground magnetic pulsations on the dayside by displaying magnetic field perturbations detected by the European quasi‐Meridional Magnetometer Array (EMMA) as 2‐D images in the magnetic L value versus time space, called EMMAgrams. We generated EMMAgrams from observations made on 15 August 2015, including a previously studied pulsation event associated with an interplanetary shock. In addition to signatures of field line resonance driven by a cavity mode oscillation, we found poleward propagating structures with L‐independent periods in the Pc2 band. The Pc2 structures are attributed to periodic magnetohydrodynamic pulses (upstream waves) originating from the ion foreshock and propagating in the magnetosphere along the path proposed by Tamao. Ringing of local field lines at L‐dependent periods (transient pulsations) is also clearly detected as dispersive poleward propagating structures not only immediately after the shock impact but also during time periods of less obvious external disturbances. A transient pulsation decays after a few wave periods, and cross‐spectral analysis of transient pulsations detected at two stations with a small latitudinal separation indicates elevation of the cross phase in a band delimited by the field line resonance frequencies at the stations. Successive excitation of transient pulsations by variations of the solar wind dynamic pressure appears to contribute significantly to formation of similar cross‐phase peaks that are widely used in magnetoseismic studies.

Observation and Numerical Simulation of Cavity Mode Oscillations Excited by an Interplanetary Shock

AbstractCavity mode oscillations (CMOs) are basic magnetohydrodynamic eigenmodes in the magnetosphere predicted by theory and are expected to occur following the arrival of an interplanetary shock. However, observational studies of shock‐induced CMOs have been sparse. We present a case study of a dayside ultralow‐frequency wave event that exhibited CMO properties. The event occurred immediately following the arrival of an interplanetary shock at 0829 UT on 15 August 2015. The shock was observed in the solar wind by the Time History of Events and Macroscale Interactions during Substorms‐B and ‐C spacecraft, and magnetospheric ultralow‐frequency waves were observed by multiple spacecraft including the Van Allen Probe‐A and Van Allen Probe‐B spacecraft, which were located in the dayside plasmasphere at L ∼1.4 and L ∼ 2.4, respectively. Both Van Allen Probes spacecraft detected compressional poloidal mode oscillations at ∼13 mHz (fundamental) and ∼26 mHz (second harmonic). At both frequencies, the azimuthal component of the electric field (Eϕ) lagged behind the compressional component of the magnetic field (Bμ) by ∼90°. The frequencies and the Eϕ‐Bμ relative phase are in good agreement with the CMOs generated in a dipole magnetohydrodynamic simulation that incorporates a realistic plasma mass density distribution and ionospheric boundary condition. The oscillations were also detected on the ground by the European quasi‐Meridional Magnetometer Array, which was located near the magnetic field footprints of the Van Allen Probes spacecraft.

Q-switched fiber laser based on an acousto-optic modulator with injection seeding technique.

The operation mechanism and the pulse property of an actively Q-switched erbium-doped fiber laser based on an acousto-optic modulator (AOM) switch with the injection seeding technique are investigated. Our results show that the Q-switched pulses can be locked to oscillate near a fixed frequency higher than that of the seed laser, though the frequency-shift effect of the AOM impedes stable cavity mode oscillations. The operation mechanism of such Q-switch fiber lasers can be explained by the mutual locking-in among the shifted frequency components originated from the injected coherence seed with the help of the gain dynamics of the Q-switch cavity. Moreover, narrow-linewidth Q-switched pulses with different repetition rates can be obtained with different cavity lengths for incredibly stable output pulses without any use of cavity-stabilized techniques.

Rabi splitting induced by a metamaterial plasmon cavity

We studied the surface plasmon resonance properties of transverse electric (TE) wave in a μ-negative (MNG) material/dielectric/μ-negative (MNG) material waveguide with a finite length which works as a subwavelength cavity. The wavelength of the surface plasmon becomes shorter when decrease the thickness of the dielectric core and decrease the plasma frequency of MNG material. The resonance in this cavity can be understood as a Fabry-Perot resonance caused by the reflection of the TE guided mode at the entrance and the exit surfaces. The electromagnetic fields and power flow are concentrated around the dielectric core at the resonant frequency, the magnetic field is maximized at the dielectric core entrance and exit. When a subwavelength magnetic resonator is put at the core entrance and the resonance frequency is tuned to the plasmon cavity mode, Rabi splitting and Rabi oscillation can appear because of the strong coupling between this resonator and the cavity mode.

Phase relation between Pi2‐associated ionospheric Doppler velocity and magnetic pulsations observed at a midlatitude MAGDAS station

We examined the correlation between nighttime Pi2 pulsations detected simultaneously by a frequency modulated continuous wave (FM‐CW) (HF) radar and by a ground magnetometer, both located at a midlatitude (L = 2.05) Magnetic Data Acquisition System station. Eighty‐three Pi2 events were observed during the 43 day period from 23 September 2006 to 4 November 2006. The variations of the ground magnetic H component and ionospheric Doppler velocity (V*) exhibited high coherence for 80% of the 83 Pi2 events, for about a half of which the H and V* variations have the same dominant frequency. For such events, V* led H by 90° in phase, in the midnight sector of 2230–0300 LT. The average Ey (east‐west electric field) amplitude derived from V* is 0.27 mV/m. The 90° phase delay was not found for the five events that were observed near dusk and dawn. The phase relation of H and V* for Pi2s in the midnight sector may be explained in terms of the radial standing structure of compressional waves, i.e., cavity mode oscillation.

Dominant mode in closed photonic crystal microcavity filled with high scattering irregular gain medium

The experimental results demonstrate a dominant vertical cavity mode oscillation in a closed microcavity filled with high scattering and irregular structure gain medium. The mechanisms are (A) ultralow loss in a closed microcavity, (B) strong feedback from the microcavity causes light to be dominant in the vertical cavity mode, and (C) the dominant light in the vertical cavity mode synchronizes light in random directions and localization state.

A case study testing the cavity mode model of the magnetosphere

Abstract. Based on a case study we test the cavity mode model of the magnetosphere, looking for eigenfrequencies via multi-satellite and multi-instrument measurements. Geotail and ACE provide information on the interplanetary medium that dictates the input parameters of the system; the four Cluster satellites monitor the magnetopause surface waves; the POLAR (L=9.4) and LANL 97A (L=6.6) satellites reveal two in-situ monochromatic field line resonances (FLRs) with T=6 and 2.5 min, respectively; and the IMAGE ground magnetometers demonstrate latitude dependent delays in signature arrival times, as inferred by Sarafopoulos (2004b). Similar dispersive structures showing systematic delays are also extensively scrutinized by Sarafopoulos (2005) and interpreted as tightly associated with the so-called pseudo-FLRs, which show almost the same observational characteristics with an authentic FLR. In particular for this episode, successive solar wind pressure pulses produce recurring ionosphere twin vortex Hall currents which are identified on the ground as pseudo-FLRs. The BJN ground magnetometer records the pseudo-FLR (alike with the other IMAGE station responses) associated with an intense power spectral density ranging from 8 to 12 min and, in addition, two discrete resonant lines with T=3.5 and 7 min. In this case study, even though the magnetosphere is evidently affected by a broad-band compressional wave originated upstream of the bow shock, nevertheless, we do not identify any cavity mode oscillation within the magnetosphere. We fail, also, to identify any of the cavity mode frequencies proposed by Samson (1992). Keywords. Magnetospheric physics (Magnetosphereionosphere interactions; Solar wind-magnetosphere interactions; MHD waves and instabilities)

CRRES electric field study of the radial mode structure of Pi2 pulsations

The radial mode structure of Pi2 pulsations in the inner magnetosphere (L &lt; 7) and its relation to the plasmapause are studied using data acquired by the Combined Release and Radiation Effects Satellite (CRRES) between August 1990 and September 1991. Low‐latitude Pi2 pulsations detected on the ground at Kakioka (L = 1.25) are used as the reference signal to determine the relative amplitude and phase of the electric field oscillations detected at CRRES. The plasmapause is identified using electron density inferred from the plasma wave spectra observed on CRRES. Pi2 events at CRRES are defined to be 10‐min intervals of high coherence between oscillations in the Kakioka horizontal northward magnetic field (H) and CRRES dusk‐to‐dawn electric field (Eφ) components within the Pi2 band (6–25 mHz). The Eφ component represents the poloidal oscillation of the geomagnetic field lines for satellite local times near midnight. Fifty‐five high‐coherence Eφ‐H Pi2 events occurred when both CRRES and Kakioka were within 3 hours of magnetic midnight. For these events CRRES was on L shells ranging from 2 to 6.5 and was either in the plasmasphere or in the close vicinity of the plasmapause, providing evidence for the plasmaspheric origin of low‐latitude Pi2 pulsations. The amplitude of Eφ varied significantly but there is an indication of a maximum near L = 4. The phase of Eφ (relative to Kakioka H) remained near −90° at all distances. These properties are consistent with the radial structure of the fundamental cavity mode oscillations confined in the plasmasphere. For some events observed at L &gt; 3.5 it was also possible to determine the amplitude and phase of the compressional component Bz at CRRES. In contrast to Eφ, the phase of Bz (relative to H) was clustered both at ∼180° and ∼0 for events occurring near the plasmapause. This observation still is consistent with the cavity mode according to a numerical simulation using a dipole magnetic field and a realistic plasmapause plasma density structure, which indicates that the node of Bz is located near the plasmapause. Depending on the satellite position relative to the node, the phase can be either –180° or 0. A negative correlation is found between the Pi2 frequency and the distance of the plasmapause, which is additional support for the cavity mode origin of low‐latitude Pi2 pulsations.

Microwave emission from two-stacked arrays of long Josephson junctions

We have recorded the radiation generated by magnetic flux quanta oscillations and by cavity mode oscillations in two-stacked arrays each consisting of five long Josephson junctions biased in parallel. The two arrays have in common the middle electrode, which is thinner than the London penetration depth, and can be biased independently. Basically, two kind of fluxon states can be generated, one consisting of a row of five fluxons oscillating in one of the arrays, the other consisting of a row of five fluxons in one array and five antifluxons in the other array, bound in a coherent state. It is found from the microwave emission that the latter configuration is less sensitive to low-frequency noise.

Effect of magnetopause leakage on the lifetime of magnetospheric cavity modes

Magnetospheric cavity modes can decay through various loss processes. Energy leakage across the magnetopause is one such process that is commonly neglected. Here we consider what the actual effect of the neglected magnetopause leakage might be on cavity mode lifetime and hence on coupled transverse mode wave growth. First, we review the linear theory of fast mode wave reflection at a tangential discontinuity to show that magnetopause leakage is significant. For realistic subsolar magnetopause conditions we estimate that 20–60% of magnetospheric fast mode wave Poynting flux incident on the magnetopause is transmitted through the magnetopause and out of the magnetospheric cavity. Second, we demonstrate the effect of magnetopause leakage on magnetospheric cavity modes using an MHD simulation. The simulation is of a box‐like cavity with a dipolar variation of the magnetospheric magnetic field and realistic conditions at the subsolar magnetopause, based upon in situ spacecraft observations. The simulation results show that a purely fast mode cavity oscillation decays due to leakage of fast mode wave energy across the magnetopause. The decay rate that arises from magnetopause leakage is comparable to that found in previous simulations from coupling of fast mode energy into transverse mode field line resonances. Thus we conclude that magnetopause leakage is likely to enhance the decay rate of fast and transverse cavity mode oscillations and should be taken into account in field line resonance theory and especially its application to observations.

Possible evidence of damped cavity mode oscillations stimulated by the January, 1997 magnetic cloud event

A sudden drastic change in the solar wind dynamic pressure is one proposed mechanism for the excitation of cavity mode resonances in the plasmasphere. To study the possibility of cavity mode occurrence following a 180 cm−3 solar wind density pulse on January 11, 1997, a numerical MHD simulation was run on a dipole grid. The simulated spectra recorded by a virtual satellite are compared with those produced from the Polar magnetic time signals during the interval 0228–0304 UT. In both sets of spectra, power is concentrated between 2 and 5 mHz; observed FLR‐like spectral peaks in the range 6–20 mHz are in rough agreement with simulated plasmaspheric FLRs between 7–21 mHz. In the simulation, the fundamental plasmaspheric cavity mode is at 3.5 mHz; phase analysis of observed 3.5 mHz electric and magnetic field signals from the interval 0230–0237 UT supports the possibility of a heavily damped standing wave at this frequency.

Pi2 pulsations observed from the Akebono satellite in the plasmasphere

Magnetic field, electric field, and electron density measurements from the Akebono satellite are used to study the properties of two Pi2 pulsations that occurred in succession on February 13, 1990, when the satellite was in the plasmasphere at L = 2.4–3.8, 24°–40° magnetic latitude, and 22.5 hours magnetic local time. Magnetic pulsations with a nearly identical waveform were observed in the same time interval at three ground stations (Aedey, L ≈ 6.84; York, L ≈ 2.55; and Hermanus, L ≈ 1.83), which were located near midnight, confirming that the pulsations propagated to the ground. At the satellite the pulsations had comparable radial and azimuthal components in both the magnetic and electric fields. In contrast to the observations near the magnetic equator by the Active Magnetospheric Particle Tracer Explorers Charge Composition Explorer spacecraft [Takahashi et al., 1995], no compressional component was detected in the magnetic field. The orthogonal components of the electric and magnetic fields oscillated either in phase or 180° out of phase, a property of a propagating (rather than standing) wave. The Poynting flux of the Pi2 pulsations was parallel to the ambient magnetic field and directed toward the nearer ionosphere, with little indication of ionospheric reflection. This unidirectional flow of electromagnetic energy is consistent with the strong ionospheric damping of Alfvén waves estimated from a numerical calculation. It is significant that the measured Poynting flux could damp a cavity mode oscillation in ∼10 s, assuming that the previously reported equatorial compressional Pi2 pulsations represent the cavity mode. Since the transverse Pi2 pulsations at Akebono lasted ∼400 s, they cannot be due to a gradual energy leakage from a simple cavity mode oscillation. Consequently, if the observed energy flow is a general property of plasmaspheric Pi2 pulsations, a simple cavity mode oscillation excited by an impulsive source is not an appropriate model for Pi2 pulsations.

Ground‐satellite coherence analysis of Pc3 pulsations

Although Pc3 magnetic pulsations are commonly observed in the dayside magnetosphere and are believed to propagate into the inner magnetosphere and to the ground, how and where they establish themselves as a regular oscillation are not completely understood. In particular, it is not clear whether the cavity mode plays a significant role in determining the spectral properties of Pc3 pulsations. The mode, in principle, can be identified by multipoint observations of the spatial variation of the amplitude and phase of magnetic field perturbations, with at least one satellite providing evidence of compressional magnetic field oscillation in the magnetosphere. Motivated by these requirements for detection of the cavity mode, we surveyed combined magnetic field data from the Active Magnetospheric Particle Tracer Explorers Charge Composition Explorer (AMPTE CCE) satellite and the Kakioka ground station. We identified an interval on November 4–5, 1984, during which both the satellite (dipole L = 3–6) and ground station (L = 1.25) were on the morningside (near 8‐hour local time) with a small local time separation (within 0.9 hour) and observed Pc3 magnetic pulsations exhibiting a compressional component in space. Unlike previous studies, which used satellites located at L &gt; 6, we observed high coherence in the Pc3 band between the compressional component (bz)in space and the horizontal component (H) on the ground. A cross‐phase analysis of the merged bz and H time series indicates that these components oscillated in antiphase over a range of satellite radial distance. We discuss whether the observations can be taken as evidence of a cavity mode oscillation in the magnetosphere.

One-Dimensional Transient Response of the Inner Magnetosphere at the Magnetic Equator, 1. Transfer Function and Poles.

One-dimensional transient response of the inner magnetosphere at the magnetic equator is investigated using two models of altitude distribution of the Alfven speed VA. The present paper concentrates on the transfer function of the system under consideration and its poles, which govern the transient response of the system. The poles, which are mathematical counterparts of the cavity resonances, appear owing to the inhomogeneity of VA and their locations depend on the altitude distribution of VA as well as the position of external source (or outer boundary of the inner magnetosphere). Even if there exists no strong Alfven velocity gradient at the outer boundary, an observable cavity-mode oscillation in the Pi2 range can be excited because of the existence of a strong gradient of the plasmapause within the inner magnetosphere. However, the existence of a strong gradient at the outer boundary brings about a long-lived nature of the cavity-mode oscillation as well as calls some new poles into existence. While the surface of the solid earth forms the inner boundary at which the almost perfect reflection of wave takes place, the ionosphere is of secondary importance as a reflector of wave. The existence of the solid earth plays an essential role in the observability of the compressional oscillation arising from the cavity resonance all over the inner magnetosphere. The real part of each pole has a negative value, meaning that the cavity-mode oscillation decays with a damping factor of absolute value of the real part of the pole. Such a damping is primarily due to the leakage of energy through the outer boundary of the inner magnetosphere.

One-Dimensional Transient Response of the Inner Magnetosphere at the Magnetic Equator, 2. Analysis of Waveforms.

Under a model of altitude distribution of the Alfven speed V A , one-dimensional transient response of the inner magnetosphere at the magnetic equator to earthward propagating impulse- and step-like MHD disturbances is considered. The waveforms of transient compressional oscillations due to these disturbances at some L shells are directly simulated by a numerical inversion of the Laplace transform with orthonormal Laguerre functions. The present paper concentrates on the analysis of waveforms. Then, it is verified that the compressional oscillations are due to the poles of the system under consideration. The oscillation arising from the cavity resonance all over the inner magnetosphere is most dominant. However, its amplitude becomes smaller as the characteristic time scale T of an incident disturbance grows large, and it is negligibly small for T greater than several times of eigenperiod of the resonance. On the other hand, when T is relatively small (e.g., T ≤ 10 s), the oscillations due to the cavity resonances trapped around the trough in V A are outstanding. It is also found that the relative phase between the cavity-mode oscillations all over the inner magnetosphere at the earth's surface and another L shell increases monotonically with L when the inner magnetosphere has no strong gradient or a strong positive gradient of V A at its outer boundary. However, the relative phase is nearly zero and nearly 180° inside and outside a specific L shell, respectively, when the inner magnetosphere has a strong negative gradient at its outer boundary. The one-dimensional cavity-mode type resonance of the inner magnetosphere is certainly a cause of equatorial Pi2 pulsations. However, some constituents of the Pi2's may be not cavity-mode oscillations but quasi-steady-state oscillations forced by some damped sinusoidal waves incident on the outer boundary of the inner magnetosphere.

A Possible Signature of Magnetic Cavity Mode Oscillations in ISEE Spacecraft Observations.

The structure of magnetohydrodynamic waves in the terrestrial magnetosphere is controlled by the dispersion relations of the wave modes, the inhomogeneities of the system and the boundary conditions. If the waves are confined within the magnetospheric cavity with proper boundary conditions, two types of spectra are predicted by MHD theory. One is a discrete compressional spectrum and the other is a continuous shear Alfvén spectrum. The discrete spectrum refers to compressional eigenmodes with spatially constant eigenfrequencies. The continuous shear Alfvén spectrum corresponds to field line resonances with spatially varying resonant frequencies. There have been many reports of observations consistent with the theoretical predictions for shear Alfvén waves. However, only a few spacecraft observations of the global cavity eigenmodes or the coupling between the two modes have been reported. Here we report an observation using ISEE 1 and 2 spacecraft magnetometer data. On a quiet day with low solar wind dynamic pressure and low geomagnetic activity, the ISEE 1 and 2 spacecraft identified a compressional wave with constant frequency throughout most of its inbound orbit in the outer magnetosphere. Multiple harmonics of shear Alfvén waves with spatially varying frequencies were observed in the azimuthal component. Evidence of the coupling between these two waves in found. Comparing our observations with the results of a computer simulation in a dipole field, we find qualitative (not quantitative) agreement. We also consider why so few examples of radially-extended monochromatic compressional oscillations have been found. We conclude that the exceptional circumstances required for development and identification of these wave structures occur very rarely.