Low Azimuthal Wavenumber Research Articles

Constraining Saturn’s Interior with Ring Seismology: Effects of Differential Rotation and Stable Stratification

Abstract Normal mode oscillations in Saturn excite density and bending waves in the C ring, providing a valuable window into the planet’s interior. Saturn’s fundamental modes (f-modes) excite the majority of the observed waves, while gravito-inertial modes (rotationally modified g-modes) associated with stable stratification in the deep interior provide a compelling explanation for additional density waves with low azimuthal wavenumbers m. However, multiplets of density waves with nearly degenerate frequencies, including an m = 3 triplet, still lack a definitive explanation. We investigate the effects of rapid and differential rotation on Saturn’s oscillations, calculating normal modes for independently constrained interior models. We use a non-perturbative treatment of rotation that captures the full effects of the Coriolis and centrifugal forces, and consequently the mixing of sectoral f-modes with g-modes characterized by very different spherical harmonic degrees. Realistic profiles for differential rotation associated with Saturn’s zonal winds can enhance these mode interactions, producing detectable oscillations with frequencies separated by less than 1%. Our calculations demonstrate that a three-mode interaction involving an f-mode and two g-modes can feasibly explain the finely split m = 3 triplet, although the fine-tuning required to produce such an interaction generally worsens agreement with seismological constraints provided by m = 2 density waves. Our calculations additionally demonstrate that sectoral f-mode frequencies are measurably sensitive to differential rotation in Saturn’s convective envelope. Finally, we find that including realistic equatorial antisymmetry in Saturn’s differential rotation profile couples modes with even and odd equatorial parity, producing oscillations that could in principle excite both density and bending waves simultaneously.

Polygonal Eyewall Asymmetries During the Rapid Intensification of Hurricane Michael (2018)

AbstractPolygonal eyewall asymmetries of Hurricane Michael (2018) during rapid intensification (RI) are analyzed from ground‐based single Doppler radar. Here, we present the first observational evidence of the evolving wind field of a polygonal eyewall during RI to Category 5 intensity by deducing the axisymmetric and asymmetric winds at 5‐min intervals. Spectral time decomposition of the retrieved tangential wind structure shows quantitative evidence of low (1–4) azimuthal wavenumbers with propagation speeds that are consistent with linear wave theory on a radial vorticity gradient, suggesting the presence of rapidly evolving vortex Rossby waves. Dual‐Doppler winds from the NOAA P‐3 Hurricane Hunter airborne radar provide further evidence of the three‐dimensional vortex structure that supports growth of asymmetries during RI. Both reflectivity and tangential wind fields show polygonal structure and propagate at similar speeds, suggesting a close coupling of the dynamics and the convective organization during the intensification.

Global‐Scale ULF Waves Associated With SSC Accelerate Magnetospheric Ultrarelativistic Electrons

AbstractWe study electron behavior in the outer radiation belts during the 16 July 2017 storm sudden commencement (SSC), in which prompt intensification of ultrarelativistic electron fluxes was observed at around L = 4.8 by Van Allen Probe B immediately after an interplanetary shock. The electron fluxes in multiple energy channels show clear oscillations in the Pc5 frequency range, although the oscillation characteristics are quite different in different energy channels. At energies above ∼1 MeV, the oscillation periods were very close to the electron drift period, which resembles an energy spectrogram evolution expected for an energetic particle injection event and its drift echoes. At lower energies, however, the oscillation periods hardly depended on the energy: They were very close to the ultralow frequency (ULF) wave period derived from electric field measurements (about 250 s according to wavelet analysis). These complex signatures are consistent with the picture of drift resonance between electrons and short‐lived ULF waves with low azimuthal wave numbers. Good agreement between the observations and numerical simulations confirms that shock‐induced global‐scale ULF waves can efficiently accelerate outer belt ultrarelativistic electrons up to 3.4 MeV over a time scale shorter than 1 hr.

Optimal Perturbations of an Oceanic Vortex Lens

The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed.

On the instabilities of tropical cyclones generated by cloud resolving models

An approximate method is developed for finding and analysing the main instability modes of a tropical cyclone whose basic state is obtained from a cloud resolving numerical simulation. The method is based on a linearised model of the perturbation dynamics that distinctly incorporates the overturning secondary circulation of the vortex, spatially inhomogeneous eddy diffusivities, and diabatic forcing associated with disturbances of moist convection. Although a general formula is provided for the latter, only parameterisations of diabatic forcing proportional to the local vertical velocity perturbation and modulated by local cloudiness of the basic state are implemented herein. The instability analysis is primarily illustrated for a mature tropical cyclone representative of a category 4 hurricane. For eddy diffusivities consistent with the fairly conventional configuration of the simulation that generates the basic state, perturbation growth is dominated by a low azimuthal wavenumber instability having greatest asymmetric kinetic energy density in the lower tropospheric region of the inner core of the vortex. The characteristics of the instability mode are inadequately explained by nondivergent 2D dynamics. Moreover, the growth rate and modal structure are sensitive to reasonable variations of the diabatic forcing. A second instability analysis is conducted for a mature tropical cyclone generated under conditions of much weaker horizontal diffusion. In this case, the linear model predicts a relatively fast high-wavenumber instability that is insensitive to the parameterisation of diabatic forcing. The prediction is in very good quantitative agreement with a previously published analysis of how the instability develops in a cloud resolving model on the way to creating mesovortices slightly inward of the central part of the eyewall.

Auto-oscillations in supersonic boundary layer

The production of periodic oscillations in a supersonic boundary layer at the moderate and high Mach numbers (M = 2 and 5.35) is investigated within the framework of the weakly nonlinear stability theory of the second order in nonlinearity. The model includes the effects of self-action, such as the generation of stationary secondary harmonics and the disturbances of double frequencies. It is shown that for two-dimensional vortex disturbances, the character of the excitation of vortex disturbances changes from the mild one to the stiff one with the increasing Mach number, which leads to a reduction of the critical Reynolds number Rec. For three-dimensional disturbances of low azimuthal wave numbers, a supercritical auto-oscillatory regime sets in. A complex regime realizes for two-dimensional acoustic disturbances at M = 5.35 with a stiff excitation in the region of Rec.

Large-eddy simulations of the near field of a turbulent annular jet

Using high-order spectral element method and large eddy simulations we study a turbulent annular jet at the Reynolds number of 8900, based on the bulk velocity in the annular channel and its half-width. The region of high turbulent kinetic energy is located inside the recirculation zone behind the bluff-body where the low-frequency motion is known to occur which is detected with energy spectra. Further analysis includes the azimuthal Fourier and Proper orthogonal decomposition (POD). It is shown that the main eigenmodes with low azimuthal wavenumbers m rotate around the symmetry axis in both directions providing the explanation for the precessing motion of the recirculating zone.

Three-dimensional instabilities and inertial waves in a rapidly rotating split-cylinder flow

The nonlinear dynamics of the flow in a differentially rotating split cylinder is investigated numerically. The differential rotation, with the top half of the cylinder rotating faster than the bottom half, establishes a basic state consisting of a bulk flow that is essentially in solid-body rotation at the mean rotation rate of the cylinder and boundary layers where the bulk flow adjusts to the differential rotation of the cylinder halves, which drives a strong meridional flow. There are Ekman-like layers on the top and bottom end walls, and a Stewartson-like side wall layer with a strong downward axial flow component. The complicated bottom corner region, where the downward flow in the side wall layer decelerates and negotiates the corner, is the epicentre of a variety of instabilities associated with the local shear and curvature of the flow, both of which are very non-uniform. Families of both high and low azimuthal wavenumber rotating waves bifurcate from the basic state in Eckhaus bands, but the most prominent states found near onset are quasiperiodic states corresponding to mixed modes of the high and low azimuthal wavenumber rotating waves. The frequencies associated with most of these unsteady three-dimensional states are such that spiral inertial wave beams are emitted from the bottom corner region into the bulk, along cones at angles that are well predicted by the inertial wave dispersion relation, driving the bulk flow away from solid-body rotation.

Centrifugal, barotropic and baroclinic instabilities of isolated ageostrophic anticyclones in the two-layer rotating shallow water model and their nonlinear saturation

AbstractInstabilities of isolated anticyclonic vortices in the two-layer rotating shallow water model are studied at Rossby numbers up to two, with the main goal to understand the interplay between the classical centrifugal instability and other ageostrophic instabilities. We find that different types of instabilities with low azimuthal wavenumbers exist, and may compete. In a wide range of parameters, an asymmetric version of the standard centrifugal instability has larger growth rate than the latter. The dependence of the instabilities on the parameters of the flow, i.e. Rossby and Burger numbers, vertical shear and the ratios of the layers’ thicknesses and densities, is investigated. The zones of dominance of each instability are determined in the parameter space. Nonlinear saturation of these instabilities is then studied with the help of a high-resolution finite-volume numerical scheme, by using the unstable modes identified from the linear stability analysis as initial conditions. Differences in nonlinear development of the competing centrifugal and ageostrophic barotropic instabilities are evidenced. A nonlinear mechanism of axial symmetry breaking during the saturation of the centrifugal instability is displayed.

Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

The core of a terrestrial-type planet consists of a spherical shell of rapidly rotating, electrically conducting, fluid. Such a body supports two distinct classes of quasi-geostrophic (QG) eigenmodes: fast, primarily hydrodynamic, inertial modes with period related to the rotation time scale and slow, primarily magnetic, magnetostrophic modes with much longer periods. Here, we investigate the properties of these hydromagnetic quasi-geostrophic modes as a function of non-dimensional parameters controlling the strength of the background magnetic field, the planetary rotation rate, and the amount of magnetic dissipation.We first present analytic solutions that illustrate the essential parameter dependences of the modes and provide a convenient benchmark for our numerical scheme. A comparison between known three-dimensional inertial modes in a sphere and our axially invariant QG modes shows encouraging agreement at low azimuthal wavenumbers, particularly for the slowest modes. The container geometry and background magnetic field structure are found to influence the radial structure of the modes, but not the scaling of their frequency with the control parameters. When the background magnetic field decreases toward the outer boundary in a spherical shell, QG modes tend to be compressed towards the outer boundary. Including magnetic dissipation, we find a continuous transition from diffusionless slow magnetic modes into quasi-free decay magnetic modes. During that transition (which is controlled by the magnitude of the Elsasser number), we find that slow magnetic modes weakly modified by diffusion exhibit a distinctive spiralling planform. When magnetic diffusion is significant (Elsasser number much smaller than unity), we find quasi-free decay slow magnetic modes whose decay time scale is comparable to, or shorter than, their oscillation time scale.Based on our analysis, we expect Mercury to be in a regime where the slow magnetic modes are of quasi-free decay type. Earth and possibly Ganymede, with their larger Elsasser numbers, may possess slow modes that are in the transition regime of weak diffusion, depending on the details of their poorly known internal magnetic fields. Fast QG modes, that are almost unaffected by the background magnetic field, are expected in the cores of all three bodies.

Linear stability analysis of supersonic axisymmetric jets

Stabilities of supersonic jets are examined with different velocities, momentum thicknesses, and core temperatures. Amplification rates of instability waves at inlet are evaluated by linear stability theory (LST). It is found that increased velocity and core temperature would increase amplification rates substantially and such influence varies for different azimuthal wavenumbers. The most unstable modes in thin momentum thickness cases usually have higher frequencies and azimuthal wavenumbers. Mode switching is observed for low azimuthal wavenumbers, but it appears merely in high velocity cases. In addition, the results provided by linear parabolized stability equations show that the mean-flow divergence affects the spatial evolution of instability waves greatly. The most amplified instability waves globally are sometimes found to be different from that given by LST.

Experimental and numerical investigation of a variable density swirling-jet stability

Linear stability of a variable density, and incompressible, swirling jet is studied at moderate Reynolds numbers, both experimentally and numerically. Mean flow characteristics are obtained experimentally. They show a correlation between axial velocity and density profiles. For low density ratios, s, the presence of self-sustained oscillations of the flow (global modes) is highlighted. Using direct numerical simulations (at low azimuthal wavenumbers m = 0, −1, −2) to study the local convective/absolute transition at the jet exit-nozzle, it is shown that frequencies of the local absolute-instability are in good agreement with those of the experimentally obtained global modes. Furthermore, it is found that increasing the swirl number q enhances the absolute instability for helical modes m = −1 and −2. Finally, it is also shown that the unstable selected mode, experimentally obtained, has the spatio-temporal properties of the absolutely unstable one obtained numerically.

Stability of steady flow through an axially corrugated pipe

The linear stability of steady flow in pipes with circular cross-section and sinusoidal axial variation in diameter is studied by finding global eigenmodes with axial wavelength commensurate with that of the wall corrugation, chosen to be equal to one pipe mean radius. The maximum peak-to-peak height of corrugation considered is approximately 8% of the mean diameter. At low corrugation amplitude and at low Reynolds numbers, the base flow remains attached to the wall, while at larger amplitudes and Reynolds numbers, an axisymmetric separation bubble forms within the corrugation. For all Reynolds numbers considered, flows remain stable to axisymmetric perturbations, but become unstable to standing-wave modes of low azimuthal wavenumber, with critical Reynolds number first falling, then increasing with increasing corrugation height. Both attached and separated flows exhibit similar types of instability modes, which in the case of separated flow are most energetic near the reattachment line of the base flow. The leading instability modes consist of counter-rotating vortices situated near the pipe wall.

Numerical study of eigenmode forcing effects on jet flow development and noise generation mechanisms

The effect of nonlinear interaction of instability eigenmodes on jet flow transition and its near acoustic field for a high-subsonic round jet at a Reynolds number of Re=4.5×105 and a Mach number of Ma=0.9 is investigated using large-eddy simulations. At the inflow, helical perturbations of azimuthal wavenumbers |n|=4,…,8 determined from linear stability theory are superimposed on a laminar base flow in order to trigger transition to turbulence. The disturbance amplitude is varied parametrically in the range from 1.5% to 4.5% of the jet exit velocity Uj. Thereby we aim to characterize sources of noise generation and, in particular, underlying mode interactions. With increasing forcing amplitude, the transitional behavior of the jet changes which affects the mean flow and also the acoustic near-field, which are both analyzed in detail. As the forcing amplitude is increased, the axial root-mean-square peak levels along the jet centerline are reduced by approximately 7%. Simultaneously, pronounced dual-peak distributions are generated along the jet lip line which are related to the localization of vortex pairings of the jet column mode. For low-amplitude excitation the azimuthal turbulent kinetic energy spectra show that the unexcited, naturally least stable axisymmetric mode n=0 and the helical mode n=1 dominate the early nonlinear regimes between z≈6r0 and 9r0 where r0 is the jet radius. An analysis of the Fourier mode amplitude clarifies that this energy rise is linked to the helical mode n=1. For higher forcing amplitudes, in addition to the varicose mode n=0 interactions between the excited even mode n=4 and higher azimuthal harmonics thereof dominate the azimuthal energy spectra. These differences in the early nonlinear development of the eigenmodes are found to alter the acoustic near-field. At small angles from the downstream jet axis, the peak acoustic frequency occurs at a Strouhal number based on the angular frequency ω and the jet diameter Dj of St=ωDj/(2πUj)≈0.4. For low-amplitude forcing sound pressure levels are slightly enhanced which can be linked to the dominant low azimuthal wavenumbers identified in the transitional region. In the sideline direction, regardless of the excitation level, broadbanded spectra with maxima in the band 0.7≤St≤0.8 are found which is maintained at intermediate observer angles. For high forcing amplitude, however, a tonal component outside the initially excited frequency range is observed. This peak at St≈0.88 can be explained by weakly nonlinear interactions of initially forced eigenmodes n=4 and n=8 together with the jet column mode.

Magnetically Arrested Disks and the Origin of Poynting Jets: A Numerical Study

The dynamics and structure of accretion disks, which accumulate a vertical magnetic field in their centers, are investigated using two- and three-dimensional MHD simulations. The central field can be built up to the equipartition level, where it disrupts a nearly axisymmetric outer accretion disk inside a magnetospheric radius, forming a magnetically arrested disk (MAD). In the MAD, the mass accretes in the form of irregular dense spiral streams, and the vertical field, split into separate bundles, penetrates through the disk plane in low-density magnetic islands. The accreting mass, when spiraling inward, drags the field and twists it around the axis of rotation, resulting in collimated Poynting jets in the polar directions. These jets are powered by the accretion flow with an efficiency of up to ~1.5% (in units of inc2). The spiral flow pattern in the MAD is dominated by modes with low azimuthal wavenumbers m ~ 1–5 and can be a source of quasi-periodic oscillations in the outgoing radiation. The formation of the MAD and the Poynting jets can naturally explain the observed changes of spectral states in galactic black hole binaries. Our study is focused on black hole accretion flows; however, the results can also be applied to accretion disks around nonrelativistic objects, such as young stellar objects and stars in binary systems.

Nonaxisymmetric Instabilities of Convection around Magnetic Flux Tubes

On the surface of the Sun, magnetic flux elements collect in regions of converging flow, grow in field strength, and become pores, which have been observed to exhibit nonaxisymmetric structure over a range of scales. Around a fully developed sunspot, as well as the fine scale of the penumbra, the moat sometimes shows a clearly observable spokelike structure at low azimuthal wavenumbers. We investigate the formation of azimuthal structure by computing the linear stability properties of fully nonlinear axisymmetric magnetoconvection, which takes the form of a central flux tube surrounded by a convecting field-free region. We find steady and oscillatory instabilities with a preferred azimuthal wavenumber. The unstable modes are concentrated in the convecting region close to the outer edge of the flux tube. The instability is driven by convection and is not a magnetic fluting instability.

Propagation and stability of vorticity–entropy waves in a non-uniform flow

The evolution of disturbances in an annular duct with a non-isentropic radially varying mean flow is studied. Linear and nonlinear analyses are carried out to examine how the mean velocity and density gradients affect the stability and coupling between the disturbances. To isolate the effect of the mean-velocity gradients from that of the mean-density gradients two mean flows are considered, one with a Gaussian density profile and a uniform axial velocity and the other with Gaussian density and Gaussian axial-velocity distributions. For small-amplitude disturbances with the former mean flow profile, the vortical disturbances convect with the mean flow and density fluctuations grow linearly in space as a result of the interaction of the mean-density gradient with the disturbance radial velocity. Eigenmode analysis of the latter profile shows that unstable modes with exponential growth occur owing to the inflection point in the mean-velocity profile. These modes are almost independent of the mean-density profile and are most unstable for low azimuthal wavenumbers. Nonlinear solutions support the linear results and show an algebraic growth of the density for a range of azimuthal wavenumbers and both uniform and non-uniform mean-velocity profiles. The growth of the velocity fluctuations, however, is strongly dependent on the azimuthal wavenumber of the incident disturbance and the mean-velocity profile. The largest growth in the disturbance is observed at radial locations where the largest mean-flow gradients exist. Owing to the growth of the density fluctuations, coupled vorticity–entropy waves are observed downstream of a forced harmonic excitation in a non-isentropic flow. The forcing amplitudes of the incident waves were varied to see how the solutions change with amplitude. As the amplitude is increased, the waves continue to grow and a steepening of the gradients is observed as they propagate downstream until eventually very sharp density and velocity fronts form. These results show that the mean-flow and density profiles play an important role in the evolution of low-azimuthal-wavenumber disturbances which can couple strongly to the duct acoustic modes during combustion instabilities.

Multicomponent decompositions for a sample of S0 galaxies

We have estimated the bulge-to-total (B/T) light ratios in the Ks band for a sample of 24 SO, S0/a and Sa galaxies by applying a two-dimensional multicomponent decomposition method. For the disc an exponential function is used, the bulges are fitted by a Sersic R 1/n function and the bars and ovals are described either by a Sersic or a Ferrers function. In order to avoid non-physical solutions, preliminary characterization of the structural components is made by inspecting the radial profiles of the orientation parameters and the low azimuthal wavenumber Fourier amplitudes and phases. In order to identify also the inner structures, unsharp masks were created: previously undetected inner spiral arms were found in NGC 1415 and marginally in NGC 3941. Most importantly, we found that S0s have a mean (B/T) K ratio of 0.24 ± 0.11, which is significantly smaller than the mean (B/T) R = 0.6 generally reported in the literature. Also, the surface brightness profiles of the bulges in S0s were found to be more exponential-like than generally assumed, the mean shape parameter of the bulge being = 2.1 ± 0.7. We did not find examples of barred S0s lacking the disc component, but we found some galaxies (NGC 718, 1452 and 4608) having a non-exponential disc in the bar region. To our knowledge, our study is the first attempt to apply a multicomponent decomposition method for a moderately sized sample of early-type disc galaxies.

Latitudinal Shear Instabilities during Type I X‐Ray Bursts

Coherent oscillations have been observed during thermonuclear (type I) X-ray bursts from 14 accreting neutron stars in low-mass X-ray binaries, providing important information about their spin frequencies. However, the origin of the brightness asymmetry on the neutron star surface producing these oscillations is still not understood. We study the stability of a zonal shearing flow on the neutron star surface using a shallow-water model. We show that differential rotation of 2% between the pole and equator, with the equator spinning faster than the poles, is unstable to hydrodynamic shear instabilities. The unstable eigenmodes have a low azimuthal wavenumber m, wave speeds 1% or 2% below the equatorial spin rate, and e-folding times 1 s. Instability is related to low-frequency, buoyantly driven r-modes that have a mode frequency within the range of rotation frequencies in the differentially rotating shell. In addition, a modified Rayleigh's criterion based on potential vorticity rather than vorticity must be satisfied. The eigenfunctions of the unstable modes have the largest amplitudes nearer to the rotational pole than the equator. We discuss the implications for burst oscillations. Growth of shear instabilities may explain the brightness asymmetry in the tails of X-ray bursts. However, some fine-tuning of the level of differential rotation and a spin frequency near 300 Hz are required in order for the fastest growing mode to have m = 1. If shear instabilities are to operate during a burst, temperature contrasts of 30% across the star must be created during ignition and spreading of the flash. The frequency drift may be related to the nonlinear development of the instability as well as the cooling of the burning layer. The properties and observability of the unstable modes vary strongly with neutron star rotation frequency.

Ground observations of high‐latitude Pc3‐4 ULF waves

A detailed study has been undertaken of Pc3‐4 waves recorded on the ground with the IMAGE magnetometer array (56° < Λ < 76°) during January and March 1998. We focus only on daytime events exhibiting high coherence (>0.6) across the entire station array. Most of these had well‐defined wave packet appearance in time series records and a clear peak in power spectra. Their occurrence and frequency suggest the waves are generated by the upstream ion‐cyclotron resonance mechanism, with no evidence of generation by the Kelvin‐Helmholtz instability. For each event the amplitude, phase, coherence, ellipticity, azimuth angle, and degree of polarization across the ground array were examined. The coherence length, azimuthal wave number, and hence the apparent wave propagation velocity were thus determined, with emphasis on the precision and significance of these measurements. It was found that these daytime Pc3‐4 pulsations usually have maximum amplitude near the magnetopause projection, meridional coherence lengths of order 1.5–2.0 × 103 km, and low azimuthal wave numbers during morning hours, averaging around −4.0 (indicating westward propagation). Over 80% of events propagated poleward and westward, with average equivalent ground velocity of 41 km/s N43°W for the H component. About 24–30% of the events are higher harmonics of field line resonances. There is no evidence that the remaining events arise from cavity modes or localized modulated electron precipitation. The observations instead suggest a mechanism involving mode coupling and field‐guided propagation. In this model, fast mode waves in the Pc3‐4 range entering near the subsolar point propagate earthward and due to the inhomogeneity of the magnetosphere couple to the field‐guided Alfvén mode. At certain latitudes, standing oscillations are established at harmonics of the local resonant frequency, while at other latitudes traveling waves convey energy to low altitudes. The expected L dependence of wave power and travel time agree well with observed amplitude and phase profiles.