Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near $20^\circ$-$30^\circ$, and decrease at higher latitudes. We search for an explanation that takes into account solar latitudinal differential rotation. In the equatorial $β$ plane, we study the propagation of linear Rossby waves (phase speed $c <0$) in a parabolic zonal shear flow, $U = - \overline{U}\ ξ^2<0$, where $\overline{U} = 244$ m/s and $ξ$ is the sine of latitude. In the inviscid case, the eigenvalue spectrum is real and continuous and the velocity stream functions are singular at the critical latitudes where $U = c$. We add eddy viscosity in the problem to account for wave attenuation. In the viscous case, the stream functions are solution of a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number $100 \le Re \le 700$), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode with a real frequency close to that of the classical Rossby mode, which we call the R mode. For $Re \approx 300$, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model. Conclusion: Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers towards the equator.
Rossby waves arise due to the conservation of total vorticity in rotating fluids and may govern the large-scale dynamics of stellar interiors. Recent space missions collected huge information about the light curves and activity of many stars, which triggered observations of Rossby waves in stellar surface and interiors. We aim to study the theoretical properties of Rossby waves in stratified interiors of uniformly rotating radiative stars with sub-adiabatic vertical temperature gradient. We use the equatorial beta-plane approximation and linear vertical gradient of temperature to study the lin…
The influence of a toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. A sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to trapping of the waves near the equator and to an increase of the Rossby wave period up to the timescale of solar cycles. Dispersion relations of all equatorial magnetohydrodynamic (MHD) shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal mag…
Annual oscillations have been detected in many indices of solar activity during many cycles. Recent multi spacecraft observations of coronal bright points revealed slow retrograde toroidal phase drift (with the speed of 3 m/s of 1 yr oscillations, which naturally suggested their connection with Rossby-type waves in the interior. We have studied from a theoretical point of view the dynamics of global magneto-Kelvin and magneto-Rossby waves in the solar tachocline with toroidal magnetic field. Using spherical coordinates, the dispersion relations of the waves and latitudinal structure of solutio…
The Sun's complex dynamics is controlled by buoyancy and rotation in the convection zone and by magnetic forces in the atmosphere and corona. While small-scale solar convection is well understood, the dynamics of large-scale flows in the solar convection zone is not explained by theory or simulations. Waves of vorticity due to the Coriolis force, known as Rossby waves, are expected to remove energy out of convection at the largest scales. Here we unambiguously detect and characterize retrograde-propagating vorticity waves in the shallow subsurface layers of the Sun at angular wavenumbers below…
In a previous paper we studied the effect of latitudinal rotation on solar equatorial Rossby modes in the beta-plane approximation. Since then, a rich spectrum of inertial modes has been observed on the Sun, which is not limited to the equatorial Rossby modes and includes high-latitude modes. Here we extend the computation of toroidal modes in 2D to spherical geometry, using realistic solar differential rotation and including viscous damping. The aim is to compare the computed mode spectra with the observations and to study mode stability. At fixed radius, we solve the eigenvalue problem numer…
In the context of secular evolution, we describe the dynamics of the radiative core of low-mass stars to understand the internal transport of angular momentum in such stars which results in a solid rotation in the Sun from 0.7R_sun to 0.2R_sun and a weak radial core-envelope differential rotation in solar-type stars. This study requires at least a 2D description to capture the latitudinal variations of the differential rotation. We build 2D numerical models of a radiative core on the top of which we impose a latitudinal shear so as to reproduce a cylindrical differential rotation in a convecti…
Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged quantities that erase latitudinal differences in the physics. Here we quantify those latitudinal differences by analyzing spherical shell simulations in terms of their regionalized convective heat transfer properties. This is done by measuring local Nusselt numbers in two specific, …
Temporal variations of solar inertial mode parameters from GONG (2002-2024) and HMI (2010-2024): Rossby modes ($3 \leq m \leq 16 $) and $m=1$ high-latitude mode
Max-Planck-Institut für Sonnensystemforschung（马克斯·普朗克太阳系统研究所）；National Solar Observatory（美国国家太阳天文台）；Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen（哥廷根大学天体物理与地球物理研究所）；Center for Astrophysics and Space Science, NYUAD Institute, New York University Abu Dhabi（纽约大学阿布扎比分校天体物理学与空间科学中心）；LIRA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CY Cergy Paris Université, CNRS（巴黎天文台拉伊尔实验室（隶属于巴黎文理研究大学、索邦大学、巴黎西岱大学、塞吉-蓬图瓦兹大学及法国国家科学研究中心））
Several types of inertial modes have been detected on the Sun. Properties of these inertial modes have been studied in the linear regime but have not been studied in nonlinear simulations of solar rotating convection. Comparing the nonlinear simulations, the linear theory, and the solar observations is important to better understand the differences between the models and the real Sun. We wish to detect and characterize the modes present in a nonlinear numerical simulation of solar convection, in particular to understand the amplitudes and lifetimes of the modes. We developed a code with a Yin-…
Context. Large-scale equatorial Rossby modes have been observed on the Sun over the last two solar cycles. Aims. We investigate the impact of the time-varying zonal flows on the frequencies of Rossby modes. Methods. A first-order perturbation theory approach is used to obtain an expression for the expected shift in the mode frequencies due to perturbations in the internal rotation rate. Results. Using the time-varying rotation from helioseismic inversions we predict the changes in Rossby mode frequencies with azimuthal orders from m = 1 to m = 15 over the last two solar cycles. The peak-to-pea…
Long-term records of sunspot number and concentrations of cosmogenic radionuclides (10Be and 14C) on the Earth reveal the variation of the Sun's magnetic activity over hundreds and thousands of years. We identify several clear periods in sunspot, 10Be, and 14C data as 1000, 500, 350, 200 and 100 years. We found that the periods of the first five spherical harmonics of the slow magnetic Rossby mode in the presence of a steady toroidal magnetic field of 1200-1300 G in the lower tachocline are in perfect agreement with the time scales of observed variations. The steady toroidal magnetic field can…
On the one hand, several types of global-scale inertial modes of oscillation have been observed on the Sun. They include the equatorial Rossby modes, critical-latitude modes, and high-latitude modes. On the other hand, the columnar convective modes (predicted by simulations; also known as banana cells or thermal Rossby waves) remain elusive. We aim to investigate the influence of turbulent diffusivities, non-adiabatic stratification, differential rotation, and a latitudinal entropy gradient on the linear global modes of the rotating solar convection zone. We solve numerically for the eigenmode…
Photospheric magnetic fields are studied using synoptic maps from 1976 to 2003 produced at the National Solar Observatory, Kitt Peak (NSO/KP). Synoptic maps were averaged over the time interval of nearly three solar cycles (Solar Cycles 21-23). The change in the latitudinal distribution was considered for the following groups of magnetic field values: B = 0-5 G, B = 5-15 G, B = 15-50 G, and B>50 G. Magnetic fields in each of the above groups have common latitudinal distribution features, while for different field groups these features are significantly different. Each of the groups is closely …
In this paper, we have studied the convectively coupled equatorially trapped waves in rotating stars, with and without magnetic field. The equatorial trapped HD and MHD Poincaré, Rossby, mixed Rossby-Poincaré, and Kelvin waves were identified. The effects of stratification and non-traditional Coriolis force terms have been investigated. When the flow is strongly stratified, the wave frequencies of the convectively coupled model are almost the same as those of shallow water model. However, when the flow is weakly stratified, the wave frequencies are constrained by the buoyancy frequency. The no…
Asteroseismology has undergone a profound transformation as a scientific field following the CoRoT and Kepler space missions. The latter is now yielding the first measurements of latitudinal differential rotation obtained directly from oscillation frequencies. Differential rotation is a fundamental mechanism of the stellar dynamo effect. Our goal is to measure the amount of differential rotation in the solar analogues 16 Cyg A and B, which are the components of a binary system. These stars are the brightest observed by Kepler and have therefore been extensively observed, with exquisite precisi…
Systematic Investigation of Solar Modulation of Galactic Protons for Solar Cycle 23 using a Monte Carlo Approach with Particle Drift Effects and Latitudinal Dependence
Analysis of the structure and dynamics of the magnetic field of the Sun is fundamental for understanding of the origin of solar activity and variability as well as for the study of solar-terrestrial relations. Observations of the large scale magnetic field in the photosphere taken at the Wilcox Solar Observatory from 1976 up to 2007 have been analysed to deduce its latitudinal and longitudinal structures, its differential rotation, and their variability in time. This paper is dedicated to the analysis and dynamics of the latitudinal structure of the solar magnetic field over three solar cycles…
In this dissertation, several components of large-scale solar flows are studied observationally: solar equatorial Rossby waves (waves of radial vorticity), large-scale convection, and surface flows around active regions. Maps of horizontal flows are derived from photospheric observations by the Helioseismic and Magnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO) using two different techniques: granulation tracking and local helioseismology. First, the eigenfunctions of solar Rossby waves are measured from helioseismic ring-diagram flow maps with a correlation method and a spectra…
The observation of global acoustic waves (p modes) in the Sun has been key to unveiling its internal structure and dynamics. A different kind of wave, known as sectoral Rossby modes, have been observed and identified, which potentially opens the door to probing internal processes that are inaccessible through p mode helioseismology. Yet another set of waves, appearing as retrograde-propagating, equatorially antisymmetric vorticity waves, have also been observed but their identification remained elusive. Here, through a numerical model implemented as an eigenvalue problem, we provide evidence s…
In a previous study on thermonuclear (type I) nursts on accreting neutron stars we addressed and demonstrated the importance of the effects of rotation, through the Coriolis force, on the propagation of the burning flame. However, that study only analysed cases of longitudinal propagation, where the Coriolis force coefficient $2Ω\cosθ$ was constant. In this paper, we study the effects of rotation on propagation in the meridional (latitudinal) direction, where the Coriolis force changes from its maximum at the poles to zero at the equator. We find that the zero Coriolis force at the equator, wh…
Simulations of the onset and dynamical evolution of inertial waves in solar/stellar interior
Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Observatório Nacional（国家天文台）；Royal Observatory of Belgium（比利时皇家天文台）；High Altitude Observatory, NSF National Center for Atmospheric Research（美国国家大气研究中心高空观测站）；National Center for Atmospheric Research（美国国家大气研究中心）
Rapid rotation in massive stars imposes a latitudinal variation in the mass loss from radiatively driven winds that can lead to enhanced mass loss at the poles (with little angular momentum loss) and/or equator (with maximal angular momentum loss). Here we present an examination of the stellar wind lines of the two O-type stars with the fastest known equatorial velocities, VFTS 102 ($V\sin i = 610 \pm 30$ km/s; O9:Vnnne+) and VFTS 285 ($V\sin i = 609 \pm 29$ km/s; O7.5 Vnnn) in the Large Magellanic Cloud. Ultraviolet spectra of both stars were obtained with the Hubble Space Telescope Cosmic Or…
Effects of Magnetic Fields on Solar Inertial Modes in Global Nonlinear Simulations
Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Royal Observatory of Belgium（比利时皇家天文台）；High Altitude Observatory, NSF National Center for Atmospheric Research（美国国家科学基金会国家大气研究中心高海拔天文台）；National Center for Atmospheric Research（国家大气研究中心）
The oscillations of a slowly rotating star have long been classified into spheroidal and toroidal modes. The spheroidal modes include the well-known 5-min acoustic modes used in helioseismology. Here we report observations of the Sun's toroidal modes, for which the restoring force is the Coriolis force and whose periods are on the order of the solar rotation period. By comparing the observations with the normal modes of a differentially rotating spherical shell, we are able to identify many of the observed modes. These are the high-latitude inertial modes, the critical-latitude inertial modes,…
The Rossiter-McLaughlin (RM) effect has been widely used to estimate the sky-projected spin-orbit angle, $λ$, of transiting planetary systems. Most of the previous analysis assume that the host stars are rigid rotators in which the amplitude of the RM velocity anomaly is proportional to $v_\star \sin i_\star$. When their latitudinal differential rotation is taken into account, one can break the degeneracy, and determine separately the equatorial rotation velocity $v_\star$ and the inclination $i_{\star}$ of the host star. We derive a fully analytic approximate formula for the RM effect adoptin…
Overshooting of turbulent motions from convective regions into adjacent stably stratified zones plays a significant role in stellar interior dynamics as this process may lead to mixing of chemical species, and contribute to the transport of angular momentum and magnetic fields. We present a series of fully non-linear, three-dimensional (3D) anelastic simulations of overshooting convection in a spherical shell which are focused on the dependence of the overshooting dynamics on the density stratification and the rotation, both key ingredients in stars which however have not been studied systemat…
A major outstanding problem in solar physics is the confinement of the solar tachocline, the thin shear layer that separates nearly solid-body rotation in the radiative interior from strong differential rotation in the convection zone. Here, we present the first 3-D, global solar simulation in which a tachocline is confined by a self-excited dynamo. The non-axisymmetric magnetism is initially built in the convection zone and then diffusively imprints downward. Additionally, the field is locally amplified throughout the radiative interior by vigorous horizontal motions that arise from equatoria…
A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The …
For the period of 1997-2006, coronal holes detected in the SOHO/EIT 195 $Å$ full disk calibrated images are used to compute the rotation rates of high latitude and near polar coronal holes and, their latitudinal variation is investigated. We find that, for different latitude zones between $80^{o}$ north and $75^{o}$ south, for all their area, the number of days observed on the solar disk, and their latitudes, coronal holes rotate rigidly. Estimated magnitudes of sidereal rotation rate of the coronal holes are: $13.051 \pm 0.206$ deg/day for the equator, $12.993 \pm 0.064$ deg/day in the region…
Can atmospheric waves in planet-hosting solar-like stars substantially resonate to tidal forcing? Substantially at a level of impacting the space weather or even of being dynamo-relevant? In particular, low-frequency Rossby waves, which have been detected in the solar near-surface layers, are predestined at responding to sunspot cycle-scale perturbations. In this paper, we seek to address these questions as we formulate a forced wave model for the tachocline layer, which is widely considered as the birthplace of several magnetohydrodynamic planetary waves, i.e., Rossby, inertia-gravity (Poinca…
Rotating stellar convection transports angular momentum towards the equator, generating the characteristic equatorial acceleration of the solar rotation while the radial flux of angular momentum is always inwards. New numerical box simulations for the meridional cross-correlation $\langle u_θu_ϕ\rangle $, however, reveal the angular momentum transport towards the poles for slow rotation and towards the equator for fast rotation. The explanation is that for slow rotation a negative radial gradient of the angular velocity always appears, which in combination with a so-far neglected rotation-indu…
The rotational profile of the Sun is considered to be one of the key inputs in a solar dynamo model. Hence, precise and long-term measurements of this quantity is important for our understanding of solar magnetism and its variability. In this study, we use the newly digitized, white light sunspot data (1923 -- 2011) from Kodaikanal Solar Observatory (KoSO) to derive the solar rotation profile. An automated correlation based sunspot tracking algorithm is implemented to measure the rotation parameters, $A$, the equatorial rotation rate and $B$, the latitudinal gradient. Our measurements of $A=14…
The solar convection zone rotates differentially, with its equatorial region rotating more rapidly than the polar regions. This form of differential rotation, also observed in many other low-mass stars, is understood to arise when Coriolis effects are stronger than those associated with buoyant driving of the convection. When buoyancy dominates, a so-called antisolar state of differential rotation results, characterized by rapidly-rotating poles and a slow equator. The transition between these two states has been shown to occur when the intensity of these two forces is roughly equal or, equiva…
We analyze the simulation result shown in Hotta & Kusano, 2021 in which the solar-like differential rotation is reproduced. The Sun is rotating differentially with the fast equator and the slow pole. It is widely thought that the thermal convection maintains the differential rotation, but recent high-resolution simulations tend to fail to reproduce the fast equator. This fact is an aspect of one of the biggest problems in solar physics called the convective conundrum. Hotta & Kusano, 2021 succeed in reproducing the solar-like differential rotation without using any manipulation with unpreceden…
