MHD Modes Research Articles

Coupling among neoclassical tearing modes, edge localized modes and Alfvén eigenmodes in HL-2A high β H-mode plasmas

In this work, the coupling among several MHD modes across different spatial regions, including the neoclassical tearing mode (NTM) and two branches of Alfvén eigenmode (AE) in the core and the edge localized mode (ELM), has been investigated in the HL-2A high beta H-mode plasmas. The NTMs induce a saturated m/n = 1/1 helical core (m and n are the poloidal and toroidal mode numbers, respectively) through the ‘magnetic-flux pumping’ effect. The ELM crash results in a rapid (<1 ms) decrease of the NTM island width followed by a much slower recovery. The degree of the island-width drop is proportional to the normalized beta as well as the ELM size, and can be up to 60%. In addition, two branches of AEs, in the toroidal Alfvén eigenmode (TAE) and beta-induced Alfvén eigenmode (BAE) bands, become evident after the 2/1 NTM onset and their magnitudes are modulated by the 2/1 NTM rotation. Besides, the changes of the TAE and BAE amplitudes are closely related to the temporal evolution of the ELM crash event, implying the strong interaction between AEs and the ELM. It is found that the coupling among these MHD modes in the core region during the NTM phase regulates the edge transport, i.e., relaxation of the pressure profile, mitigation of the peeling-ballooning instability, reduction of the radial electric field shear and enhancement of the turbulent transport in the pedestal region.

Multi-chamber GEM-based concept of radiated power/SXR measurement system for use in high radiation environment of DEMO

This work describes the progress of development of total core radiation power and soft X-ray (SXR) diagnostic system designed in accordance with the DEMO control requirements. Monitoring of energy loss through the separatrix is necessary for reliable plasma control. Data gathered by this system could also be helpful for studying spatial distribution of heavy impurities, MHD modes and their localization, plasma shape and position.Gas electron multiplier (GEM) technology is assessed as a base for a new radiated SXR power measurement system. The main advantages of this technology are the compactness of GEM detector, good temporal and spatial resolution, the ability to discriminate energy of incident photons and better neutron resilience than existing systems. All of these make it potentially a good candidate for SXR diagnostic system in ITER and DEMO.The scope of the study includes analysis of the feasibility of selected approaches, the design of the diagnostic system, necessary simulations and engineering considerations, and the proposal of an integration scenario. Preliminary design considerations for the GEM-sensor-based system under development have been done. Plasma radiation intensities and spectra have been simulated using data from the appropriate DEMO scenario. A concept of the photosensitive chamber has been assessed. Photoabsorption simulations with the proposed structure and parameters have been conducted and the results have been discussed.

Dynamics of ultralow-q plasmas in the RFX-mod device

The results are presented of an experimental activity performed in the RFX-mod device aimed at characterizing plasma dynamics in the so-called Ultralow-q (ULq) magnetic configuration, which corresponds to edge safety factor values below 1. The role of the edge safety factor in determining plasma dynamics is studied. In particular, a characterization of MHD activity is performed. The results of dedicated non-linear 3D visco-resistive MHD simulations are in good quantitative agreement with the experimental observations. In particular, the predicted tendency for ULq plasmas to be characterized by magnetic spectra dominated by a single mode (either a kink or a double resonant internal mode) is confirmed by experiment. Magnetic reconnection plays a relevant role in determining the dynamics of the magnetic topology. Both almost quiescent and largely fluctuating plasmas are observed with a strong sensitivity on the edge safety factor. The main MHD properties of the ULq are compared to those of RFP and tokamak discharges, also produced in the RFX-mod device. MHD modes exhibit toroidal rotation at a frequency depending on mode amplitude. Differently from what encountered in RFP plasmas at comparable current levels, no wall locking is detected.

On the role of preexisting MHD activity for the plasma response to massive deuterium injection

As part of a reliable disruption mitigation system (SPI) for ITER, pure deuterium shattered pellet injection (SPI) has been proposed as a way of avoiding hot tail runaway electron generation. It offers the possibility of diluting the plasma and, thereby, cooling it down by a large factor without immediately triggering a thermal quench (TQ). However, the reliability of this and similar SPI approaches could be reduced by preexisting MHD modes, which are usually present during the pre-TQ phase, when the disruption mitigation scheme is being triggered. To address this question, this theoretical study investigates massive deuterium injection into an MHD active ASDEX Upgrade plasma using the non-linear MHD code JOREK. Cases with and without preexisting 2/1 islands are studied. Scans are performed in the preexisting island size, the number of atoms injected, and the relative phase of the injection location with respect to the island. Realistic values of resistivity and heat diffusion anisotropy are considered. This provides insights into the physical mechanisms at play and the relevant time scales involved. Results largely indicate that plasma dilution by deuterium also seems to work reliably in the presence of preexisting MHD activity. Nevertheless, when injecting in phase with the X-point of a large preexisting island, the TQ can occur earlier than without. Altogether, simulations increase confidence in the reliability of plasma dilution by deuterium injection and its applicability to ITER.

Confinement improvement during detached phase with RMP application in deuterium plasmas of LHD

In order to explore the compatibility of good core plasma performance with divertor heat load mitigation, the interaction between cold edge plasma and core plasma transport, including the edge transport barrier (ETB), has been analysed in the divertor detachment discharges of deuterium plasmas in LHD with resonant magnetic perturbation (RMP) field application. The RMP application introduces a widened edge stochastic layer and sharp boundary in the magnetic field structure between the confinement region and the edge stochastic layer. The widened edge stochastic layer enhances impurity radiation and provides stable detachment operation as compared with the case without RMP. It is found that ETB is formed at the confinement boundary at the onset of detachment transition. However, as the detachment deepens, the resistive pressure gradient-driven MHD mode is excited, which degrades the ETB. At the same time, however, the core transport decreases to keep global plasma stored energy (W p) unchanged, showing clear core-edge coupling. After a gradual increase of density fluctuation during the MHD activity, a spontaneous increase of W p and the recovery of ETB are observed while the detachment is maintained. Then, the coherent MHD mode ceases and ELM-like bursts appear. In the improved mode, impurity decontamination occurs, and the divertor heat load increases slightly. Key controlling physical processes in the interplay between core and cold edge plasma are discussed. A comparison between deuterium and hydrogen plasmas shows that hydrogen plasmas exhibit similar features to the deuterium ones in terms of density and magnetic fluctuations, impurity decontamination towards higher confinement, etc. But most of the features are modest in the hydrogen plasmas and thus no clear confinement mode transition with clear ETB formation is defined. Better global confinement is obtained in the deuterium plasmas than the hydrogen ones at a higher radiation level.

Resistive MHD modes in hollow cathodes external plasma

A significant number of plasma instabilities occur in the region just outside of hollow cathodes, depending on the injected gas flow, the current level and the application of an external magnetic field. In particular, the presence of an axial magnetic field induces a helical mode, affecting all the plasma parameters and the total current transported by the plasma. To explore the onset and behavior of this helical mode, the fluctuations in the plasma parameters in the current-carrying plume outside of a hollow cathode discharge have been investigated. The hollow cathode was operated at a current of 25 A, and at variable levels of propellant flow rate and applied magnetic fields. Electromagnetic probes were used to measure the electromagnetic fluctuations, and correlation analysis between each of the probe signals provided spatial-temporal characterization of the generated waves. Time-averaged plasma parameters, such as plasma potential and ion energy distribution function, were also collected in the near-cathode plume region by means of scanning emissive probe and retarding potential analyzer. The results show that the helical mode exists in the cathode plume at sufficiently high applied magnetic field, and is characterized by the presence of a finite electromagnetic component in the axial direction, detectable at discharge currents ⩾25 A. A theoretical analysis of this mode reveals that one possible explanation is consistent with the hypotheses of resistive magnetohydrodynamics, which predicts the presence of helical modes in the forms of resistive kink. The analysis has been carried out by linear perturbation of the resistive MHD equations, from which it is possible to obtain the dispersion relation of the mode and find the k–ω unstable branch associated with the instability. These findings provided the basis for more detailed investigation of resistive MHD modes and their effect in the plume of hollow cathodes developed for electric propulsion application.

Propagating Oscillations in the Lower Atmosphere Under Coronal Holes

The subject of this study is oscillations in the lower atmosphere in coronal-hole regions, where the conditions are favorable for propagation between the atmospheric layers. Based on spectroscopic observations in photospheric and chromospheric lines, we analysed the features of the oscillations that show signs of propagation between layers of the solar atmosphere. Using the cross-spectrum wavelet algorithm, we found that both chromospheric and photospheric signals under coronal holes share a range of significant oscillations of periods around five minutes, while the signals outside of coronal holes show no mutual oscillations in the photosphere and chromosphere. The phase shift between the layers indicates a predominantly upward propagation with partial presence of standing waves. We also tested the assumption that torsional Alfv\'en waves propagating in the corona originate in the lower atmosphere. However, the observed line-width oscillations, although similar in period to the Alfv\'en waves observed earlier in the corona of open-field regions, seem to be associated with other MHD modes. If we assume that the oscillations that we observed are related to Alfv\'en waves, then perhaps this is only through the mechanisms of the slow MHD wave transformation.

Study of Resonance Helical Magnetic Field (RHF) Effect In IR-T1 Tokamak Plasma Using Hilbert-Huang Transform

In the IR-T1 Tokamak the effect of a resonance helical magnetic field (RHF) on plasma confienment are presented using Hilbert-Huang Transform method.Mirnov coil data analyzed by Hilbert-Huang Transform (HHT) and decomposed to the main intrinsic mode functions(imfs) and their instantanous frequencies (IFs). We find that, HHT method can extract MHD activities with different amplitudes in the different times. Then, the IFs of these modes can be calculated by Hilbert Transform (HT). As a comparison of WT and HHT analysis the Hilbert spectrum has much better frequency definition. Amplitude modulation of Mironov oscillations in IR-T1 tokamak plasma generate intra-wave frequency modulation, In the Hilbert spectrum analysis this effect is small compared to the STFT and WT spectrums. In our study, Magnetic islands with frequency around 40 kHz can be seen in wavlet Transform(WT) and HHT results in ohmic and RHF discharges. The HHT results after application of l=2/n=1 resonant field show that, the amplitude of m=2 poloidal MHD mode are amplified and then suppressed for a few milliseconds after amplification. Similarly, the amplitude of the m=3 MHD mode are amplified and then strongly suppressed by the start of the l=3/n=1 resonant field.

Abel inversion of soft x-ray fluctuations associated with fast particle-driven fishbone instabilities in MAST plasmas

A set of soft x-ray cameras provided measurements of high frequency instabilities as well as steady-state emission in the Mega Amp Spherical Tokamak (MAST). It is shown that Abel inversion can be readily applied to fluctuating soft x-ray emission from the MAST midplane associated with fast particle-driven ‘fishbone’ instabilities, characterised by toroidal mode number n = 1. Each fishbone burst had an early phase in which high amplitude fluctuating soft x-ray signals from the plasma core were close to being in phase with each other, and there was a region close to the outboard plasma edge in which the fluctuations were relatively weak and in antiphase with those in the core. The major radius of the ‘phase axis’ at which the mode amplitude changed sign R p was initially outboard of the tokamak magnetic axis at R 0, but moved inboard during the burst, eventually becoming close to R 0, at which time the oscillations were of similar amplitude inboard and outboard of R p . The fishbone radial structure early in the burst can be understood in part by recognising that the mode is supported by energetic ions with a high average toroidal rotation rate: in a co-rotating frame, the effective magnetic axis is shifted outboard by a distance that is comparable to the difference between the major radii of the phase axis early in the burst and the laboratory frame magnetic axis. It is conjectured that the transition to a mode with R p ≃ R 0 occurred because most of the energetic ions were expelled from the plasma core region where the mode amplitude peaked, so that the instability could no longer be characterised as an energetic particle mode. Abel inversion of fishbone soft x-ray emission thus provides useful insights into the nature of energetic particle modes in tokamak plasmas and their relationship with MHD modes.

Suppression of MHD modes with active phase-control of probe-injected currents

Active phase-control of probe-injected current is shown to both suppress and amplify long-wavelength rotating magnetohydrodynamic instabilities in the HBT-EP tokamak. Four probes are connected in quadrature and energized to drive non-axisymmetric currents through the edge of the tokamak, creating magnetic perturbations comparable to previously-studied saturated kink modes or resonant magnetic perturbations that are generated by an external control coil array. Measurements of the magnetic perturbations from the probe-injected currents determine a set of current-carrying helical filaments used to model active feedback control of resistive wall modes. These experiments suggest current-injection feedback may be an effective alternative to external control coils for control of RWMs and other long-wavelength kink-like modes at the edge of tokamaks.

Toroidal plasma acceleration due to NBI fast ion losses in LTX-β

The recent Lithium Tokamak Experiment-Beta (LTX-β) upgrade includes the addition of neutral beam injection (NBI) in the same direction as the plasma current (co-I P ) and a new toroidal Mirnov array for MHD characterization. In initial NBI experiments, a spontaneously rotating n = 1 MHD mode is seen to accelerate during NBI in the counter-beam direction, accompanied by a rise in electron density consistent with the beam-injected inventory but without a clear increase in plasma pressure. Together with analytic and numerical modeling of beam optics and fast ion confinement, these observations indicate the prompt loss of all or nearly all beam ions. However, the same modeling also suggests that planned upgrades to the Ohmic heating system should provide the fast ion confinement necessary for beam heating and core fueling. A simple analytic model relates the momentum confinement time to the observed evolution of mode rotation due to the combination of NBI momentum coupling, fast ion loss , and anomalous viscous torques, yielding values consistent with past measurements of electron energy confinement time τ E,e .

Effect of m/n = 2/1 neoclassical tearing mode on sawtooth collapse in JT-60U

We have investigated the role of the m/n = 2/1 neoclassical tearing mode (NTM) for suppression of the sawtooth collapse in JT-60U from the viewpoint of the anomalous transport of the current diffusion, namely flux pumping. In the stabilization experiments of m/n = 2/1 NTMs by electron cyclotron current drive of JT-60U, it has been clarified that the sawtooth collapses occur during or after the stabilization of m/n = 2/1 NTMs. It is also confirmed that the minimum safety factor, q min, is nearly unity before and after the stabilization of an m/n = 2/1 NTM. While the flux pumping by ELM-NTM coupling was reported in DIII-D, the suppression of the sawtooth collapse is observed without ELMs in this study. On the other hand, it is observed that the sawtooth precursor appears during the stabilization of the m/n = 2/1 NTM accompanying the disappearance of the fluctuation of the n = 1 helical core (HC), which is induced by the m/n = 2/1 NTM. In addition, the modulation of the toroidal rotation velocity having the frequency of the n = 1 HC is observed in the core region. Because the helical flow with HCs is a possible source of the dynamo loop voltage in tokamaks, these observations suggest that the suppression of the sawtooth collapse in JT-60U is realized by the dynamo loop voltage due to the n = 1 HC induced by the m/n = 2/1 NTM. Our result indicates that n = 1 HCs induced by other MHD modes also may induce anomalous current transport in tokamaks.

A study of the heating and current drive options and confinement requirements to access steady-state plasmas at Q ∼ 5 in ITER and associated operational scenario development

Access to steady-state (SS) plasma scenarios that can be potentially achieved in ITER by the exploitation of the neutral beam (NB) and electron cyclotron (EC) heating and current drive capabilities, including their foreseen upgrades, has been assessed and development of the Q ∼ 5 ITER SS operation (SSO) scenario has been carried out by combining all the necessary physics and engineering components. A set of modelling codes including 0.5D/1.5D transport and source modelling suites (METIS, ASTRA and CORSICA) and ideal MHD stability analysis codes (KINX and MISHKA) are applied to combined analysis and scenario development. Operating the plasma with a moderate density peaking factor (n e0/<n e> ∼ 1.3) for sufficient fusion gain (Q up to 5) at a low density (f GW = 0.6–0.8) and the required current drive efficiency produced an internal inductance (l i(3) ∼ 0.8–0 9) which improves the stability margin for low-n external ideal MHD modes. Based on this first observation, a combined operational space and ideal MHD stability analysis has recently identified Q ∼ 5 ITER SS target plasmas (Polevoi et al 2020 Nucl. Fusion 60 096024), and then the CORSICA scenario modelling suite has been applied to study access to the target plasmas as well as to develop a candidate operation scenario. Ideal MHD stability analysis on the SS plasmas has been performed using DCON embedded in the CORSICA scenario modelling suite. The use of off-axis electron cyclotron current drive (ECCD) with a power level of 20–30 MW from the equatorial and upper launchers was essential for the tailoring of the current profile to maintain ideal MHD stability of SS plasmas in ITER. Upgrading the NBI power from the baseline 33 MW to 49.5 MW provided the necessary current drive capability to allow fully non-inductive Q ∼ 5 operation with an energy confinement enhancement requirement (H 98 ≲ 1.6) similar to that achieved in some experiments in present tokamaks (Petty et al 2020 62nd Annual Meeting of the APS Division of Plasma Physics, Virtual Meeting (USA, 9–13 November 2020); Snyder et al 2019 Nucl. Fusion 59 086017; Solomon et al 2013 Nucl. Fusion 53 093033). The CORSICA simulations of the access to the Q ∼ 5 ITER SS target plasmas show that, provided that the enhancement of the energy confinement (H 98 ∼ 1.5–1.6) is achieved, it is possible for ITER to demonstrate the Q ∼ 5 SSO with NB and EC heating and current drive within the limits of the ITER Central Solenoid (CS)/Poloidal Field (PF) coil systems.

Determination of poloidal mode numbers of MHD modes and their radial location using a soft x-ray camera array in the Wendelstein 7-X stellarator

A forward modeling technique is developed for determining the characteristic features of observed MHD modes from the line-of-sight data of the soft x-ray (SXR) tomography diagnostics in the Wendelstein 7-X (W7-X) stellarator. In particular, forward modeling is used to evaluate the poloidal mode numbers m, radial location, poloidal rotation direction and ballooning character of the MHD modes. The poloidal mode structures have been modeled by the radially localized Gaussian-shaped emission regions rotating along the magnetic surfaces. In the present study the cases of rigid-shape emission regions and flexible emission regions are modeled. Various mode phase velocity dependences on the magnetic surface position are simulated. The modeled phase dynamics of line-integrated oscillations and the distribution of oscillation amplitudes are compared with the experimental signals of the SXR cameras which observe the plasma at various viewing angles in the poloidal cross-section. Application of this technique enables describing of the 1–50 kHz modes. In particular, in the discharge W7X-PID 20180918.045 three identified branches with the poloidal mode numbers m= 8, m= 10 and m= 11 localized at ρ ≈ 0.3 are rotating in the clockwise poloidal direction. The present paper reports the first application of the forward modeling technique to the data from the SXR diagnostics in W7-X. The high m-modes are identified by forward modeling in W7-X.

Intermittency of Fast MHD Modes and Regions of Anomalous Gradient Orientation in Low-β Plasmas

Abstract The strong alignment of small-scale turbulent Alfvénic motions with the direction of magnetic field that percolates the small-scale eddies and imprints the direction of the magnetic field is a property that follows from the MHD theory and the theory of turbulent reconnection. The Alfvénic eddies mix magnetic fields perpendicular to the direction of the local magnetic field, and this type of motion is used to trace magnetic fields with the velocity gradient technique (VGT). The other type of turbulent motion, fast modes, induces anisotropies orthogonal to Alfvénic eddies and interferes with the tracing of the magnetic field with the VGT. We report a new effect, i.e., in a magnetically dominated low-β subsonic medium, fast modes are very intermittent, and in a volume with a small filling factor the fast modes dominate other turbulent motions. We identify these localized regions as the cause of the occasional change of direction of gradients in our synthetic observations. We show that the new technique of measuring the gradients of gradient amplitudes suppresses the contribution from the fast-mode-dominated regions, improving the magnetic field tracing. In addition, we show that the distortion of the gradient measurements by fast modes is also applicable to the synchrotron intensity gradients, but the effect is reduced compared to the VGT.

MHD mode identification by higher order singular value decomposition of C-2W Mirnov probe data.

The C-2W device (also known as "Norman") at TAE Technologies has proven successful at generating stable, long-lived field-reversed configuration (FRC) plasmas with record temperatures. The largest Mirnov probe array in C-2W measures three components of the magnetic field just inside the vessel wall at 64 locations distributed approximately evenly in the cylindrical vessel's azimuthal and axial dimensions. This nearly rectangular array of probes creates a unique opportunity to apply higher order singular value decomposition (HOSVD) to efficiently analyze the external magnetic field data for the purposes of reconstructing the magnetohydrodynamic mode structures in the FRC. In the first application of this method for this purpose, HOSVD is shown to quickly and effectively detect and separate toroidal modes while indicating longitudinal dependence of mode phases and amplitudes, enhancing the coherence and utility of the vast quantity of data produced by this array. Analysis of the data from the entire array at once via HOSVD proves not only computationally more efficient than methods that separately analyze groups of probes at different axial locations but also leads to improved mode resolution at axial locations where these modes are weaker.

Observation of TAE mode driven by off-axis ECRH induced barely trapped energetic electrons in EAST tokamak

A toroidal Alfvén eigenmode (TAE) is excited by electron cyclotron resonance heating (ECRH) induced barely trapped energetic electrons in experimental advanced superconducting tokamak . This TAE appears in the low density EAST discharges under pure off-axis ECRH heating. After analysing the ECRH power modulation induced local density and temperature oscillations, the location of this TAE mode and the power deposition of ECRH are determined. This edge localized TAE mode drifts in ion-diamagnetic direction may be driven by barely trapped energetic electrons considering the contribution of poloidal bounce effect in the general wave-particle resonance condition. The experimental observations also demonstrate that ECRH power modulation with fixed frequency could be used as an effective diagnostic tool to study the internal properties of MHD modes as well as particle and heat transports.

Non-Resonant n = 1 Helical Core Induced by m/n = 2/1 Neoclassical Tearing Mode in JT-60U

In JT-60U, simultaneous excitation of n = 1 helical cores (HCs) and m/n = 2/1 Tearing Modes (TMs) was observed [T. Bando et al., Plasma Phys. Control. Fusion 61 115014 (2019)]. In this paper, we have investigated the excitation mechanism of n = 1 HCs with m/n = 2/1 TMs based on the experimental observations and a simple quasi-linear MHD model. In the previous study, it was reported that a "coupling" on the phase of the MHD mode is observed between n = 1 HCs and m/n = 2/1 TMs. In this study, it is found that the coupling is observed with the mode frequency from several Hz to 6 kHz. This indicates that the resistive wall and the plasma control system do not induce the coupling because the both time scales are different from the mode frequency. In addition, n = 1 HCs appear to be the non-resonant mode from the two observations: n = 1 HCs do not rotate with the plasma around the q = 1 surface in the core and the coupling is also observed even when qmin > 1. It is also observed that the electron fluctuation due to an n = 1 HC in the core region disappears with the stabilization of an m/n = 2/1 neoclassical tearing mode by electron cyclotron current drive, implying that n = 1 HCs are driven by m/n = 2/1 TMs. This perspective, n = 1 HCs are driven by m/n = 2/1 TMs, is supported by the observation that the saturated amplitude of the m/n = 1/1 component of the radial displacement in the core is smaller than that of the m/n = 2/1 component. Finally, we revisit a quasi-linear MHD model where the m/n = 1/1 HC is induced directly by the sideband of the current for the m/n = 2/1 TM, which allows to excite the non-resonant m/n = 1/1 mode. The model also describes the characteristic of the coupling, fm/n=1/1(HC) = 2fm/n=2/1(TM).

The theory of cosmic ray scattering on pre-existing MHD modes meets data

ABSTRACT We present a comprehensive study about the phenomenological implications of the theory describing Galactic cosmic ray scattering on to magnetosonic and Alfvénic fluctuations in the GeV−PeV domain. We compute a set of diffusion coefficients from first principles, for different values of the Alfvénic Mach number and other relevant parameters associated with both the Galactic halo and the extended disc, taking into account the different damping mechanisms of turbulent fluctuations acting in these environments. We confirm that the scattering rate associated with Alfvénic turbulence is highly suppressed if the anisotropy of the cascade is taken into account. On the other hand, we highlight that magnetosonic modes play a dominant role in Galactic confinement of cosmic rays up to PeV energies. We implement the diffusion coefficients in the numerical framework of the dragon code, and simulate the equilibrium spectrum of different primary and secondary cosmic ray species. We show that, for reasonable choices of the parameters under consideration, all primary and secondary fluxes at high energy (above a rigidity of $\simeq 200 \, \mathrm{GV}$) are correctly reproduced within our framework, in both normalization and slope.

МГД-волны в столкновительной плазме солнечной короны и земной ионосферы

We have studied MHD waves (Alfvén and fast compressional modes) in a homogeneous collisional three-component low-β plasma. The three-component plasma consists of electrons, ions, and neutrals with arbitrary ratio between collision frequencies and wave time scales. We have derived a general dispersion equation and relationships for phase velocity and collisional damping rates for MHD modes for various limiting cases: from weak collisions to a strong collisional coupling, and for longitudinal and oblique propagation. In a weak collision limit, the MHD eigen-modes are reduced to ordinary low-damping Alfvén and fast magnetosonic waves. For a partially ionized plasma with a strong collisional coupling of neutrals and ions, velocities of magnetosonic and Alfvén waves are substantially reduced, as compared to the Alfvén velocity in the ideal MHD theory. At a very low frequency, when neutrals and ions are strongly coupled, a possibility arises of weakly damping MHD modes, called “decelerated” MHD modes. These modes can be observed in the solar corona/chromosphere and in the F layer of the terrestrial ionosphere.