Magnetic Rossby Waves Research Articles

Two-dimensional ideal magnetohydrodynamic waves on a rotating sphere under a non-Malkus field: I. Continuous spectrum and its ray-theoretical interpretation

Two-dimensional ideal incompressible magnetohydrodynamic (MHD) linear waves at the surface of a rotating sphere are studied as a model to imitate the outermost layer of the Earth's core or the solar tachocline. This thin conducting layer is permeated by a toroidal magnetic field the magnitude of which depends only on the latitude. The Malkus background field, which is proportional to the sine of the colatitude, provides two well-known groups of branches; on one branch, retrograde Alfvén waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfvén waves undergo a gradual transition into slow MR waves. In the case of non-Malkus fields, we demonstrate that the associated eigenvalue problems can yield a continuous spectrum instead of Alfvén and slow MR discrete modes. The critical latitudes attributed to the Alfvén resonance eliminate these discrete eigenvalues and produce an infinite number of singular eigenmodes. The theory of slowly varying wave trains in an inhomogeneous magnetic field shows that a wave packet related to this continuous spectrum propagates toward a critical latitude corresponding to the wave and is eventually absorbed there. The expected behaviour whereby the retrograde propagating packets pertaining to the continuous spectrum approach the latitudes from the equatorial side and the prograde ones approach from the polar side is consistent with the profiles of their eigenfunctions derived using our numerical calculations. Further in-depth discussions of the Alfvén continuum would develop the theory of the “wave–mean field interaction” in the MHD system and the understanding of the dynamics in such thin layers.

The Dynamics of Magnetic Rossby Waves in the Quasigeostrophic Shallow Water Magnetohydrodynamic Theory

The dynamics of magnetic Rossby waves are investigated by applying a quasigeostrophic shallow water magnetohydrodynamic system, which is linearized with respect to both uniform background flow and uniform magnetic field. Due to the influence of the free surface divergence, the phase speed for magnetic Rossby waves can be either a monotonically increasing or a monotonically decreasing function, and the resulting difference between the group velocity and the phase speed can be either positive or negative. This is determined by whether the corresponding Alfvén wave speed is the upper limit or not. Differently, the phase speed is always a monotonically increasing function and the difference between the group velocity and the phase speed is always positive for incompressible magnetic Rossby waves. Multiplying a factor, the wavenumber vector shares the same endpoint with the group velocity vector. The endpoint moves on a cycle that has a center at the k-axis and is tangent to the l-axis in the wavenumber space. The circle is quite similar to the Longuet-Higgins circle for Rossby waves on Earth’s atmosphere and ocean. The fundamental dynamics is the theoretical basis for deeply understanding the meridional energy transport by waves and the interaction between waves and the background states.

The energy dispersion of magnetic Rossby waves in zonally nonuniform basic states

The energy dispersion of magnetic Rossby waves was investigated by applying two-dimensional incompressible magnetohydrodynamic equations in both zonally varying basic flow and basic magnetic field. A derived cubic dispersion relation suggests that there are at most three types of magnetic Rossby wave. Two of them represent waves that gradually tend to Alfvén waves during the energy dispersion process. The energy dispersion trajectories (wave rays) finally move with the zonal group velocity that tends to be equal to the zonal phase speed after being reflected by at least one turning location at which the meridional group velocity equal to zero. Along the marching rays, both the wave action density and wave energy tend to be constant values while the wave amplitude will decrease with increasing total wavenumber. The third one represents a wave that gradually have the constant meridional wavenumber, wave action density, wave energy, and wave amplitude. However, the difference in the zonal group velocity and the zonal phase speed suggests that the wave is still dispersive. This type of wave will disappear if specifying uniform basic magnetic field. The cubic dispersion relation is then reduced to a quadratic one. Correspondingly, the remaining two dispersion relations feature a fast- and a slow-propagating magnetic Rossby wave, respectively. They finally tend to be Alfvén waves with no energy dispersion when the energy dispersion process completes.

Information-theoretic Analysis of Longitude Distribution of Photospheric Magnetic Fields from MDI/HMI Synoptic Maps: Evidence for Rossby Waves

Much of the research on the magnetic activity of the Sun has been focused on its axisymmetric component. However, the longitudinal complexity plays a fundamental role in the solar magnetic activity. Rossby waves have recently been proposed as a fundamental mechanism regarding the nonaxisymmetric nature of the solar magnetic fields. Here, we use HMI and MDI magnetic field synoptic maps to evaluate the magnetic field structures’ (mainly active regions) organization and propagation as a function of time and latitude. We demonstrate, using information theory, that the organization of longitudinal structures observed on synoptic maps is proportional to the level of activity at a given latitude. We further show that this organization on the longitudinal structures is persistent and due to long-lived features. The drift velocity of these long-lived photospheric features is inferred and is shown to significantly vary with latitude, and is compatible with the phase speed of tachocline magnetic Rossby waves with a toroidal field in the range of 5–10 kG. Our results suggest that Rossby waves contribute to the organization and propagation of photospheric magnetic features on the timescale of several months and beyond.

On the Connection between Rieger-type and Magneto-Rossby Waves Driving the Frequency of the Large Solar Eruptions during Solar Cycles 19–25

Global solar activity variation mainly occurs over about an 11 yr cycle. However, both longer and shorter periodicities than the solar cycle are also present in many different solar activity indices. The longer timescales may be up to hundreds of years, while the shorter timescales for global solar variability could be within 0.5–2 yr, which include, e.g., from the Rieger-type periods (150–160 days) to quasi-biennial oscillations of 2 yr. The most likely origin of this short-timescale quasi-periodicity is attributed to magnetic Rossby waves, which have periods of 0.8–2.4 yr. In this work, we present findings of a unique evolution of identified shorter periodicities, like the Rieger-type, arising from magnetic Rossby waves, throughout Solar Cycles 19–25. We report further observational evidence of the strong relationship between the Rieger-type periodicity, magneto-Rossby waves, and major solar flare activity. Moreover, this study also reveals that the global solar magnetic field has a continuous periodic longitudinal conveyor belt motion along the solar equator, together with an up-and-down movement in the latitudinal directions. We found that when these longitudinal and latitudinal movements have Rieger-type periodicity and magneto-Rossby waves during the same period of a solar cycle, major flare activity is present.

On the Energy Dispersion of Magnetic Rossby Waves

The energy dispersion of magnetic Rossby waves has been investigated by applying the two-dimensional incompressible magnetohydrodynamic (MHD) equations in both uniform basic flow and basic magnetic field. The dispersion relation suggests that the magnetic Rossby waves can be divided into fast- and slow-propagating modes, respectively. The fast-propagating mode propagates eastward and is similar to the fast Alfvén waves. The energy dispersion speed is faster than the phase speed, which means the perturbation energy can lead the perturbations themselves to arrive downstream. The slow-propagating waves with smaller (larger) horizontal scales are similar to those of the slow Alfvén waves (Rossby waves). The zonal group velocity is slower than the zonal phase speed for the slow-propagating magnetic Rossby waves. For the slow-propagating waves that propagate eastward, this means that the perturbation energy may trigger new perturbations that are located upstream of the perturbations themselves. The group velocity vector is basically same as (opposite of) the wavevector for the fast-propagating (slow-propagating) magnetic Rossby waves that propagate eastward. The endpoints of the group velocity vectors and the wavevector multiplying a factor are located on a cycle in the wavenumber space. Due to the uniform basic flow and the uniform basic magnetic field, the energy dispersion paths (called rays) are straight lines. Along the straight-line rays, the wave action, wave energy, and amplitude keep their initial values, and the wave neither develops nor decays.

Dynamo and the Adiabatic Invariant

Abstract The paper considers a dynamo generated by a shallow fluid layer in a celestial body (planet or star). This dynamo is based on the extra invariant for interacting magnetic Rossby waves.

Signatures of High‐Latitude Waves in Observations of Geomagnetic Acceleration

AbstractModels for the second time‐derivative of the geomagnetic field reveal prominent activity at high latitudes. Alternating patches of positive and negative geomagnetic acceleration propagate to the west at speeds that exceed nominal fluid velocities in the core. We show that waves are a viable interpretation of these observations. Magnetic Rossby waves produce a high‐latitude response with suitable phase velocities. However, the spatial complexity of the prediction is not compatible with the observations. Our preferred interpretation involves zonal MAC waves. These waves can account for the observed geomagnetic field when a stratified layer exists at the top of the core. The required layer has a thickness in excess of 100 km and a buoyancy frequency comparable to the rotation frequency. We anticipate a gradual reduction in the phase velocity over time, leading to a future change in the propagation direction.

The cause of the appearance or disappearance of the Rieger-type periodicity in the northern and southern hemispheres of the sun

Abstract We use a continuous wavelet transform to analyse the daily hemispheric sunspot area data from the Greenwich Royal Observatory during cycles 12–24 and then study the cause of the appearance or disappearance of the Rieger-type periodicity in the northern and southern hemispheres during a certain cycle. The Rieger-type periodicity in the northern and southern hemispheres should be developed independently in the two hemispheres. This periodicity in the northern hemisphere is generally anti-correlated with the long-term variations in the mean solar cycle strength of hemispheric activity, but the correlation of the two parameters in the southern hemisphere shows a weak correlation. The appearance or disappearance of Rieger-type periodicity in the northern and southern hemispheres during a certain solar cycle is not directly correlated with their corresponding hemispheric mean activity strength but should be related to the strength of the hemispheric activity during sunspot maximum times, which hints the Rieger-type periodicity is more related to temporal evolution of toroidal magnetic field. The Rieger-type periodicity in the two hemispheres disappears in those solar cycles with relatively weak hemispheric activity during sunspot maximum times. The reason for the disappearance of this periodicity may be due to the combined influence of relatively weak toroidal magnetic fields and torsional oscillations, the differential rotation parameters vary through the solar cycle and may not remain more or less unchanged during some time, which does not permit the strong growth of magnetic Rossby waves.

Nonlinear MHD Rossby wave interactions and persistent geomagnetic field structures

The geomagnetic field presents several stationary features that are thought to be linked to inhomogeneities at the core-mantle boundary. Particularly important stationary structures of the geomagnetic field are the flux lobes, which appear in pairs in mid- to high mid- to high latitudes. A recently discovered stratified layer at the top of the Earth's core poses important constraints on the dynamics at this layer and on the interaction of the core dynamics and the base of the mantle. In this article, we introduce the linear and nonlinear theories of magnetic Rossby waves in a thin shell at the top of the Earth's core. We study the nonlinear interaction of these waves in the presence of prescribed forcings at the base of the mantle of both a thermal and a topographic nature. We show that the combined effects of forcing and nonlinear interaction can lead the wave phases to be locked around a particular geographical longitude, generating a quasi- stationary flow pattern with a significant meridional component. The solutions of the system are shown to be analogous to atmospheric blocking phenomena. Therefore, we argue that persistent and long-lived structures of the geomagnetic field, such as the geomagnetic lobes, might be associated with a blocking at the top of the Earth's core due to nonlinear stationary waves.

Wave Processes in Three-Dimensional Stratified Flows of a Rotating Plasma in the Boussinesq Approximation

Magnetohydrodynamic (MHD) waves in a stratified rotating plasma in a gravitational field are investigated in the Boussinesq approximation. A theory of flows on an f-plane, on a nontraditional f-plane (with regard to the horizontal component of the Coriolis force), on a β-plane, and on a nontraditional β‑plane is developed. In each case, linear solutions to systems of three-dimensional MHD equations in the Boussinesq approximation are obtained that describe magnetic gravito-inertial waves, magnetostrophic waves, and magnetic Rossby waves. All existing types of three-wave interactions are found with the use of dispersion equations. In the case of magnetic Rossby waves in the β-plane approximation, it is shown that the low-frequency mode of a magnetic Rossby wave in the Boussinesq approximation is equivalent to that in the shallow-water MHD approximation. By the multiscale expansion method, a system of amplitude equations for interacting waves and the increments of two types of instability occurring in the system, decay and amplification, are obtained. For each type of three-wave interactions, it is shown that there is a difference in the coefficients and differential operators in the system of three-wave interactions.

Manifestation of Rossby Waves in the Global Magnetic Field of the Sun during Cycles 21–24

Abstract The evolution of the solar global magnetic field (GMF) was investigated and compared with oscillations of different periods during cycles 21–24. The data from the Wilcox Solar Observatory were used. The results indicate that GMF structures were associated with certain oscillations. Oscillations of 81.83–163.65 days were associated with structures lasting 10–20 CRs. They occurred as some pulses during solar maxima. Oscillations of 163.652–463.680 days were associated with GMF structures lasting ≈20–50 CRs. They match two peaks in the magnetic-field cycle. Oscillations of 1.270–5.231 yr were formed during the dominance of the sectorial GMF structure. Oscillations of 5.231–13.451 yr were the most intense. From cycle 22 to 24, their intensity decreased, and the range of periods narrowed and shifted to longer periods, determining the corresponding changes in the oscillations of 81.83 days–5.231 yr and associated GMF structures. They match well the evolution of the zonal GMF structure. They are assumed to be a descending part of a period of ≈≥60 yr. A decrease in the magnetic field from cycle 22 to 24 may be due to a decrease in the intensity of these oscillations. Some periodicities of the revealed oscillations are in good agreement with the estimates of the periods for fast magnetic Rossby waves. The GMF structure connection with Rossby waves is discussed.

Persistent westward drift of the geomagnetic field at the core–mantle boundary linked to recurrent high-latitude weak/reverse flux patches

SUMMARY Observations of changes in the geomagnetic field provide unique information about processes in the outer core where the field is generated. Recent geomagnetic field reconstructions based on palaeomagnetic data show persistent westward drift at high northern latitudes at the core–mantle boundary (CMB) over the past 4000 yr, as well as intermittent occurrence of high-latitude weak or reverse flux patches. To further investigate these features, we analysed time-longitude plots of a processed version of the geomagnetic field model pfm9k.1a, filtered to remove quasi-stationary features of the field. Our results suggest that westward drift at both high northern and southern latitudes of the CMB have been a persistent feature of the field over the past 9000 yr. In the Northern Hemisphere we detect two distinct signals with drift rates of 0.09° and 0.25° yr−1 and dominant zonal wavenumbers of m = 2 and 1, respectively. Comparisons with other geomagnetic field models support these observations but also highlight the importance of sedimentary data that provide crucial information on high-latitude geomagnetic field variations. The two distinct drift signals detected in the Northern Hemisphere can largely be decomposed into two westward propagating waveforms. We show that constructive interference between these two waveforms accurately predicts both the location and timing of previously observed high-latitude weak/reverse flux patches over the past 3–4 millennia. In addition, we also show that the 1125-yr periodicity signal inferred from the waveform interference correlates positively with variations in the dipole tilt over the same time period. The two identified drift signals may partially be explained by the westward motion of high-latitude convection rolls. However, the dispersion relation might also imply that part of the drift signal could be caused by magnetic Rossby waves riding on the mean background flow.

A New Mechanism for Maunder-like Solar Minima: Phase Synchronization Dynamics in a Simple Nonlinear Oscillator of Magnetohydrodynamic Rossby Waves

Abstract The long-term solar cycle variability and Grand solar minima remain open questions from a theoretical point of view. Recently, a growing basis of evidence points out to the role of the magnetic Rossby waves in the solar cycle. Here we present a simple deterministic model, based on a low-order spectral representation of the barotropic quasi-geostrophic-magnetohydrodynamic equations for the Solar tachocline. This model supports the idea of the long-term behavior of the solar activity as a result of nonlinear interaction of magnetic Rossby modes. Solutions show that Rossby waves undergo irregular switches between periods of high activity and periods of suppressed activity, resembling the Maunder minimum. Low-energy states in the model are associated with the synchronization of the dynamical phases of the waves. These irregular transitions in the amplitudes of the waves are reminiscent of the observed time series of the solar activity. This suggests that Maunder-like states arise from chaotic transitions between regimes with different degrees of organization in the system.

The Shallow-Water Magnetohydrodynamic Theory of Stratified Rotating Astrophysical Plasma Flows: Beta-Plane Approximation and Magnetic Rossby Waves

Rotating magnetohydrodynamic flows of a thin stratified plasma layer in a gravitational field with a free boundary in an external vertical magnetic field were investigated. Magnetohydrodynamic equations were obtained in the two-layer shallow-water approximation in an external magnetic field when plasma is divided into two layers of different densities. A system of shallow-water equations for rotating stratified plasma in an external magnetic field was obtained in the beta-plane approximation. A linear theory was developed, and solutions in the form of magnetic Rossby waves, along with corrections that describe the effects of stratification, were obtained for stationary solutions in the form of vertical or horizontal magnetic field. Qualitative analysis of dispersion curves reveals the presence of three-wave nonlinear interactions of magnetic Rossby waves for each of the stationary states. The existence of parametric instabilities was demonstrated and their increments were found.

Role of Interaction between Magnetic Rossby Waves and Tachocline Differential Rotation in Producing Solar Seasons

Abstract We present a nonlinear magnetohydrodynamic shallow-water model for the solar tachocline (MHD-SWT) that generates quasi-periodic tachocline nonlinear oscillations (TNOs) that can be identified with the recently discovered solar “seasons.” We discuss the properties of the hydrodynamic and magnetohydrodynamic Rossby waves that interact with the differential rotation and toroidal fields to sustain these oscillations, which occur due to back-and-forth energy exchanges among potential, kinetic, and magnetic energies. We perform model simulations for a few years, for selected example cases, in both hydrodynamic and magnetohydrodynamic regimes and show that the TNOs are robust features of the MHD-SWT model, occurring with periods of 2–20 months. We find that in certain cases multiple unstable shallow-water modes govern the dynamics, and TNO periods vary with time. In hydrodynamically governed TNOs, the energy exchange mechanism is simple, occurring between the Rossby waves and differential rotation. But in MHD cases, energy exchange becomes much more complex, involving energy flow among six energy reservoirs by means of eight different energy conversion processes. For toroidal magnetic bands of 5 and 35 kG peak amplitudes, both placed at 45° latitude and oppositely directed in north and south hemispheres, we show that the energy transfers responsible for TNO, as well as westward phase propagation, are evident in synoptic maps of the flow, magnetic field, and tachocline top-surface deformations. Nonlinear mode–mode interaction is particularly dramatic in the strong-field case. We also find that the TNO period increases with a decrease in rotation rate, implying that the younger Sun had more frequent seasons.

The Origin of the \u201cSeasons\u201d in Space Weather

Powerful ‘space weather’ events caused by solar activity pose serious risks to human health, safety, economic activity and national security. Spikes in deaths due to heart attacks, strokes and other diseases occurred during prolonged power outages. Currently it is hard to prepare for and mitigate the impact of space weather because it is impossible to forecast the solar eruptions that can cause these terrestrial events until they are seen on the Sun. However, as recently reported in Nature, eruptive events like coronal mass ejections and solar flares, are organized into quasi-periodic “seasons”, which include enhanced bursts of eruptions for several months, followed by quiet periods. We explored the dynamics of sunspot-producing magnetic fields and discovered for the first time that bursty and quiet seasons, manifested in surface magnetic structures, can be caused by quasi-periodic energy-exchange among magnetic fields, Rossby waves and differential rotation of the solar interior shear-layer (called tachocline). Our results for the first time provide a quantitative physical mechanism for forecasting the strength and duration of bursty seasons several months in advance, which can greatly enhance our ability to warn humans about dangerous solar bursts and prevent damage to satellites and power stations from space weather events.

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.

North–South Asymmetry in Rieger-type Periodicity during Solar Cycles 19–23

Abstract Rieger-type periodicity has been detected in different activity indices over many solar cycles. It was recently shown that the periodicity correlates with solar activity having a shorter period during stronger cycles. Solar activity level is generally asymmetric between northern and southern hemispheres, which could suggest the presence of a similar behavior in the Rieger-type periodicity. We analyze the sunspot area/number and the total magnetic flux data for northern and southern hemispheres during solar cycles 19–23, which had remarkable north–south asymmetry. Using wavelet analysis of sunspot area and number during the north-dominated cycles (19–20), we obtained the periodicity of 160–165 days in the stronger northern hemisphere and 180–190 days in the weaker southern hemisphere. On the other hand, south-dominated cycles (21–23) display the periodicity of 155–160 days in the stronger southern hemisphere and 175–188 days in the weaker northern hemisphere. Therefore, the Rieger-type periodicity has the north–south asymmetry in sunspot area/number data during solar cycles with strong hemispheric asymmetry. We suggest that the periodicity is caused by magnetic Rossby waves in the internal dynamo layer. Using the dispersion relation of magnetic Rossby waves and observed Rieger periodicity, we estimated the magnetic field strength in the layer as 45–49 kG in more active hemispheres (north during cycles 19–20 and south during cycles 21–23) and 33–40 kG in weaker hemispheres. The estimated difference in the hemispheric field strength is around 10 kG, which provides a challenge for dynamo models. Total magnetic flux data during cycles 20–23 reveals no clear north–south asymmetry, which needs to be explained in the future.

The dynamics of magnetic Rossby waves in spherical dynamo simulations: A signature of strong-field dynamos?

We investigate slow magnetic Rossby waves in convection-driven dynamos in rotating spherical shells. Quasi-geostrophic waves riding on a mean zonal flow may account for some of the geomagnetic westward drifts and have the potential to allow the toroidal field strength within the planetary fluid core to be estimated. We extend the work of Hori et al. (2015) to include a wider range of models, and perform a detailed analysis of the results. We find that a predicted dispersion relation matches well with the longitudinal drifts observed in our strong-field dynamos. We discuss the validity of our linear theory, since we also find that the nonlinear Lorentz terms influence the observed waveforms. These wave motions are excited by convective instability, which determines the preferred azimuthal wavenumbers. Studies of linear rotating magnetoconvection have suggested that slow magnetic Rossby modes emerge in the magnetostrophic regime, in which the Lorentz and Coriolis forces are in balance in the vorticity equation. We confirm this to be predominant balance for the slow waves we have detected in nonlinear dynamo systems. We also show that a completely different wave regime emerges if the magnetic field is not present. Finally we report the corresponding radial magnetic field variations observed at the surface of the shell in our simulations and discuss the detectability of these waves in the geomagnetic secular variation.