Magnetohydrodynamic Modes Research Articles

LONG-LIVED MAGNETIC-TENSION-DRIVEN MODES IN A MOLECULAR CLOUD

We calculate and analyze the longevity of magnetohydrodynamic (MHD) wave modes that occur in the plane of a magnetic thin sheet. Initial turbulent conditions applied to a magnetically subcritical cloud are shown to lead to relatively rapid energy decay if ambipolar diffusion is introduced at a level corresponding to partial ionization primarily by cosmic rays. However, in the flux-freezing limit, as may be applicable to photoionized molecular cloud envelopes, the turbulence persists at "nonlinear" levels in comparison with the isothermal sound speed $\cs$, with one-dimensional rms material motions in the range of $\approx 2\,\cs -5\,\cs$ for cloud sizes in the range of $\approx 2\,\pc - 16\,\pc$. These fluctuations persist indefinitely, maintaining a significant portion of the initial turbulent kinetic energy. We find the analytic explanation for these persistent fluctuations. They are magnetic-tension-driven modes associated with the interaction of the sheet with the external magnetic field. The phase speed of such modes is quite large, allowing residual motions to persist without dissipation in the flux-freezing limit, even as they are nonlinear with respect to the sound speed. We speculate that long-lived large-scale MHD modes such as these may provide the key to understanding observed supersonic motions in molecular clouds.

A numerical matching technique for linear resistive magnetohydrodynamics modes

A new numerical matching technique for linear stability analysis of resistive magnetohydrodynamics (MHD) modes is developed. The solution to the resistive reduced MHD equations in an inner layer with a finite width is matched onto the solution to the inertialess ideal MHD or the Newcomb equation by imposing smooth disappearance of parallel electric field in addition to the continuity of perturbed magnetic field and its spatial gradient. The boundary condition for the parallel electric field is expressed as a boundary condition of the third kind for the stream function of the perturbed velocity field. This technique can be applied for the reversed magnetic shear plasmas of their minimum safety factors being rational numbers, for which the conventional asymptotic matching technique fails. In addition, this technique resolves practical difficulties in applying the conventional asymptotic matching technique, i.e., the sensitivity of the outer-region solution on the accuracy of the local equilibrium as well as the grid arrangements, even in normal magnetic shear plasmas. Successful applications are presented not only for the eigenvalue problem but also for the initial-value problem.

Magnetohydrodynamic Stabilization of Axisymmetric Mirror Systems Using Pulsed ECRH

This paper, part of a continuing study of means for the stabilization of magnetohydrodynamic interchange modes in axisymmetric mirror-based plasma confinement systems, represents a preliminary look at a technique that would employ a train of plasma pressure pulses produced by electron cyclotron resonance heating (ECRH) to accomplish the stabilization. The use of sequentially pulsed ECRH rather than continuous-wave ECRH facilitates the localization of the heated-electron plasma pulses in regions of the magnetic field with positive field-line curvature, e.g., in the “expander” region of the mirror magnetic field, outside the outermost mirror. The technique proposed relies on the time-averaged effect of plasma pressure pulses generated in regions of positive field-line curvature to overcome the destabilizing effect of plasma pressure in regions of negative field-line curvature within the confinement region. The plasma pulses, when produced in regions of the confining field having a negative gradient, create transient ambipolar electric potentials, an effect studied in 1964 in the PLEIADE experiment in France. These electric fields preserve the localization of the hot-electron plasma pulse for times determined by ion inertia. It may be possible to use this aspect of pulsed ECRH not only to stabilize the plasma but also to plug mirror losses in a manner similar to that employed in the tandem mirror.

STREAMER WAVES DRIVEN BY CORONAL MASS EJECTIONS

Between July 5th and July 7th 2004, two intriguing fast coronal mass ejection(CME)-streamer interaction events were recorded by the Large Angle and Spectrometric Coronagraph (LASCO). At the beginning of the events, the streamer was pushed aside from their equilibrium position upon the impact of the rapidly outgoing and expanding ejecta; then, the streamer structure, mainly the bright streamer belt, exhibited elegant large scale sinusoidal wavelike motions. The motions were apparently driven by the restoring magnetic forces resulting from the CME impingement, suggestive of magnetohydrodynamic kink mode propagating outwards along the plasma sheet of the streamer. The mode is supported collectively by the streamer-plasma sheet structure and is therefore named "streamer wave" in the present study. With the white light coronagraph data, we show that the streamer wave has a period of about 1 hour, a wavelength varying from 2 to 4 solar radii, an amplitude of about a few tens of solar radii, and a propagating phase speed in the range 300 to 500 km s$^{-1}$. We also find that there is a tendancy for the phase speed to decline with increasing heliocentric distance. These observations provide good examples of large scale wave phenomena carried by coronal structures, and have significance in developing seismological techniques for diagnosing plasma and magnetic parameters in the outer corona.

THREE-DIMENSIONAL PROPAGATION OF MAGNETOHYDRODYNAMIC WAVES IN SOLAR CORONAL ARCADES

We numerically investigate the excitation and temporal evolution of oscillations in a two-dimensional coronal arcade by including the three-dimensional propagation of perturbations. The time evolution of impulsively generated perturbations is studied by solving the linear, ideal magnetohydrodynamic (MHD) equations in the zero-beta approximation. As we neglect gas pressure the slow mode is absent and therefore only coupled MHD fast and Alfven modes remain. Two types of numerical experiments are performed. First, the resonant wave energy transfer between a fast normal mode of the system and local Alfven waves is analyzed. It is seen how, because of resonant coupling, the fast wave with global character transfers its energy to Alfvenic oscillations localized around a particular magnetic surface within the arcade, thus producing the damping of the initial fast MHD mode. Second, the time evolution of a localized impulsive excitation, trying to mimic a nearby coronal disturbance, is considered. In this case, the generated fast wavefront leaves its energy on several magnetic surfaces within the arcade. The system is therefore able to trap energy in the form of Alfvenic oscillations, even in the absence of a density enhancement such as that of a coronal loop. These local oscillations are subsequently phase-mixed to smaller spatial scales. The amount of wave energy trapped by the system via wave energy conversion strongly depends on the wavelength of perturbations in the perpendicular direction, but is almost independent from the ratio of the magnetic to density scale heights.

KELVIN–HELMHOLTZ INSTABILITY IN CORONAL MAGNETIC FLUX TUBES DUE TO AZIMUTHAL SHEAR FLOWS

Transverse oscillations of coronal loops are often observed and have been theoretically interpreted as kink magnetohydrodynamic (MHD) modes. Numerical simulations by Terradas et al. (2008, ApJ 687, L115) suggest that shear flows generated at the loop boundary during kink oscillations could give rise to a Kelvin-Helmholtz instability (KHI). Here, we investigate the linear stage of the KHI in a cylindrical magnetic flux tube in the presence of azimuthal shear motions. We consider the basic, linearized MHD equations in the beta = 0 approximation, and apply them to a straight and homogeneous cylindrical flux tube model embedded in a coronal environment. Azimuthal shear flows with a sharp jump of the velocity at the cylinder boundary are included in the model. We obtain an analytical expression for the dispersion relation of the unstable MHD modes supported by the configuration, and compute analytical approximations of the critical velocity shear and the KHI growth rate in the thin tube limit. A parametric study of the KHI growth rates is performed by numerically solving the full dispersion relation. We find that fluting-like modes can develop a KHI in time-scales comparable to the period of kink oscillations of the flux tube. The KHI growth rates increase with the value of the azimuthal wavenumber and decrease with the longitudinal wavenumber. However, the presence of a small azimuthal component of the magnetic field can suppress the KHI. Azimuthal motions related to kink oscillations of untwisted coronal loops may trigger a KHI, but this phenomenon has not been observed to date. We propose that the azimuthal component of the magnetic field is responsible for suppressing the KHI in a stable coronal loop. The required twist is small enough to prevent the development of the pinch instability.

Comparison of MHD-induced rotation damping with NTV predictions on MAST

Plasma rotation in tokamaks is of special interest for its potential stabilizing effect on micro- and macro-instabilities, leading to increased confinement. In MAST, the torque from neutral beam injection can spin the plasma to a core velocity ∼300 km s−1 (Alfvén Mach number ∼0.3). Low density plasmas often exhibit a weakly non-monotonic safety factor profile just above unity. Theory predicts that such equilibria are prone to magneto-hydro-dynamic (MHD) instabilities, which was confirmed by recent observations. The appearance of the mode is accompanied by strong damping of core rotation on a timescale much faster than the momentum confinement time.The mode's saturated structure is estimated using the CASTOR code together with soft x-ray measurements, enabling the calculation of the plasma braking by the MHD mode according to neoclassical toroidal viscosity (NTV) theory. The latter exhibits strong similarities with the torque measured experimentally.

Drift-resistive-inertial ballooning modes in quasihelical stellarators

A linear stability theory of nonideal magnetohydrodynamic (MHD) ballooning modes is investigated using a two fluid model for electron-ion plasmas. Drift-resistive-inertial ballooning mode eigenvalues and eigenfunctions are calculated for a variety of equilibria including axisymmetric shifted circular geometry (ŝ−α model) as well as for three dimensional configurations relevant for the Helically Symmetric Stellarator (HSX) [F. S. B. Anderson, A. F. Almagri, D. T. Anderson, et al., Fusion Technology 27, 273 (1995)]. For typical HSX parameters, characteristic ballooning mode growth rates exceed the electron collision frequency. In this regime, electron inertial effects dominate plasma resistivity and produce an instability whose growth rate scales with the electromagnetic skin depth. However, as plasma β is increased, the resistive and inertial effects become unimportant. Under these conditions, the mode is completely stabilized by drift frequency effects, which dominate resistivity and inertia. Numerical results indicate that in the absence of drift effects, the resistive-inertial MHD modes are purely growing and persist in regimes where ideal MHD ballooning modes are stable. It is found that the magnitudes of the linear growth rates are not sensitive to the addition of a mirror term to the magnetic spectrum that spoils the quasihelical symmetry of the configuration. The eigenvalues and eigenvectors in the strong ballooning approximation are used together with a quasilinear mixing length estimate to determine particle flux and particle diffusivity. The particle diffusivity increases with rising density gradient and collisionality in a plasma with a low electron temperature. This increase in transport is consistent with the increase observed in the edge region of HSX plasmas. The magnitude of the particle diffusivity is computed to be in the range from 5 to 10 m2/s, which is consistent with the experimental measured particle diffusivity at the edge of HSX plasmas.

Stabilization of pressure-driven magnetohydrodynamic modes by separatrix in dipole plasma confinement

The eigenvalue problem is solved for the short-wavelength pressure-driven magnetohydrodynamic modes in configuration with closed magnetic field lines in the poloidal direction. Here we show that the magnetic separatrix (which determines the boundary of the confinement region) provides a substantial stabilizing effect by which the total volume inside the separatrix becomes stable even for very high beta values. The plasma inertia and compressibility are properly formulated to give the correct growth rate of the mode.

Equilibrium and global MHD stability study of KSTAR high beta plasmas under passive and active mode control

The Korea Superconducting Tokamak Advanced Research, KSTAR, is designed to operate a steady-state, high beta plasma while retaining global magnetohydrodynamic (MHD) stability to establish the scientific and technological basis of an economically attractive fusion reactor. An equilibrium model is established for stability analysis of KSTAR. Reconstructions were performed for the experimental start-up scenario and experimental first plasma operation using the EFIT code. The VALEN code was used to determine the vacuum vessel current distribution. Theoretical high beta equilibria spanning the expected operational range are computed for various profiles including generic L-mode and DIII-D experimental H-mode pressure profiles. Ideal MHD stability calculations of toroidal mode number of unity using the DCON code shows a factor of 2 improvement in the wall-stabilized plasma beta limit at moderate to low plasma internal inductance. The planned stabilization system in KSTAR comprises passive stabilizing plates and actively cooled in-vessel control coils (IVCCs) designed for non-axisymmetric field error correction and stabilization of slow timescale MHD modes including resistive wall modes (RWMs). VALEN analysis using standard proportional gain shows that active stabilization near the ideal wall limit can be reached with feedback using the midplane segment of the IVCC. The RMS power required for control using both white noise and noise taken from NSTX active stabilization experiments is computed for beta near the ideal wall limit. Advanced state-space control algorithms yield a factor of 2 power reduction assuming white noise while remaining robust with respect to variations in plasma beta.

RESONANTLY DAMPED KINK MAGNETOHYDRODYNAMIC WAVES IN A PARTIALLY IONIZED FILAMENT THREAD

Transverse oscillations of solar filament and prominence threads have been frequently reported. These oscillations have the common features of being of short period (2–10 minutes) and being damped after a few periods. The observations are interpreted as kink magnetohydrodynamic (MHD) wave modes, whereas resonant absorption in the Alfvén continuum and ion-neutral collisions are candidates to be the damping mechanisms. Here, we study both analytically and numerically the time damping of kink MHD waves in a cylindrical, partially ionized filament thread embedded in a coronal environment. The thread model is composed of a straight and thin, homogeneous filament plasma, with a transverse inhomogeneous transitional layer where the plasma physical properties vary continuously from filament to coronal conditions. The magnetic field is homogeneous and parallel to the thread axis. We find that the kink mode is efficiently damped by resonant absorption for typical wavelengths of filament oscillations, the damping times being compatible with the observations. Partial ionization does not affect the process of resonant absorption, and the filament plasma ionization degree is only important for the damping for wavelengths much shorter than those observed. To our knowledge, this is the first time that the phenomenon of resonant absorption is studied in a partially ionized plasma.

Numerical calculations demonstrating complete stabilization of the ideal magnetohydrodynamic resistive wall mode by longitudinal flow

The cylindrical ideal magnetohydrodynamic (MHD) stability problem, including flow and a resistive wall, is cast in the standard mathematical form, ωA⋅x=B⋅x, without discretizing the vacuum regions surrounding the plasma. This is accomplished by means of a finite element expansion for the plasma perturbations, by coupling the plasma surface perturbations to the resistive wall using a Green’s function approach, and by expanding the unknown vector, x, to include the perturbed current in the resistive wall as an additional degree of freedom. The ideal MHD resistive wall mode (RWM) can be stabilized when the plasma has a uniform equilibrium flow such that the RWM frequency resonates with the plasma’s Doppler-shifted sound continuum modes. The resonance induces a singularity in the parallel component of the plasma perturbations, which must be adequately resolved. Complete stabilization within the ideal MHD model (i.e., without parallel damping being added) is achieved as the grid spacing in the region of the resonance is extrapolated to 0 step size.

Low-frequency magnetohydrodynamics and geodesic acoustic modes in toroidally rotating tokamak plasmas

This paper analyses low-frequency magnetohydrodynamic (MHD) modes, especially the geodesic acoustic modes (GAMs), in toroidal plasmas with large aspect ratio and circular cross section, including the effects of toroidal plasma rotation. A system of equations describing MHD modes with frequency of the order of the sound frequency in such plasmas is derived from the Frieman–Rotenberg equation, using a technique where the plasma perturbation \U0001d6cf and the perturbed magnetic field Q are expanded separately in the inverse aspect ratio ε = r/R, where r and R denote the minor and major radii of the plasma torus, respectively. The large-scale, ideal MHD properties of the GAM induced by toroidal rotation (Wahlberg 2008 Phys. Rev. Lett. 101 115003) are thereafter analysed in more detail employing this system of equations. It is shown that both the axisymmetric GAMs existing in rotating plasmas are localized on a specific magnetic surface only to leading order in ε, and that a ‘halo’ consisting of finite components of both \U0001d6cf and Q with dominant poloidal mode numbers m = ±2 appears outside this magnetic surface to higher orders in ε.

STABILITY OF THE MAGNETOPAUSE OF DISK-ACCRETING ROTATING STARS

We discuss three modes of oscillation of accretion disks around rotating magnetized neutron stars which may explain the separations of the kilo-Hertz quasi periodic oscillations (QPO) seen in low mass X-ray binaries. The existence of these compressible, non-barotropic magnetohydrodynamic (MHD) modes requires that there be a maximum in the angular velocity $\Omega_\phi(r)$ of the accreting material larger than the angular velocity of the star $\Omega_*$, and that the fluid is in approximately circular motion near this maximum rather than moving rapidly towards the star or out of the disk plane into funnel flows. Our MHD simulations show this type of flow and $\Omega_\phi(r)$ profile. The first mode is a Rossby wave instability (RWI) mode which is radially trapped in the vicinity of the maximum of a key function $g(r){\cal F}(r)$ at $r_{R}$. The real part of the angular frequency of the mode is $\omega_r=m\Omega_\phi(r_{R})$, where $m=1,2...$ is the azimuthal mode number. The second mode, is a mode driven by the rotating, non-axisymmetric component of the star's magnetic field. It has an angular frequency equal to the star's angular rotation rate $\Omega_*$. This mode is strongly excited near the radius of the Lindblad resonance which is slightly outside of $r_R$. The third mode arises naturally from the interaction of flow perturbation with the rotating non-axisymmetric component of the star's magnetic field. It has an angular frequency $\Omega_*/2$. We suggest that the first mode with $m=1$ is associated with the upper QPO frequency, $\nu_u$; that the nonlinear interaction of the first and second modes gives the lower QPO frequency, $\nu_\ell =\nu_u-\nu_*$; and that the nonlinear interaction of the first and third modes gives the lower QPO frequency $\nu_\ell=\nu_u-\nu_*/2$, where $\nu_*=\Omega_*/2\pi$.

On the merits of heating and current drive for tearing mode stabilization

Neoclassical tearing modes (NTMs) are magnetohydrodynamic modes that can limit the performance of high β discharges in a tokamak, leading eventually to a plasma disruption. A NTM is sustained by the perturbation of the ‘bootstrap’ current, which is a consequence of the pressure flattening across a magnetic island. Control and suppression of this mode can be achieved by means of electron cyclotron waves (ECWs) which allow the deposition of highly localized power at the island location. The ECW power replenishes the missing bootstrap current by generating a current perturbation either inductively, through a temperature perturbation (electron cyclotron resonance heating), or non-inductively by direct current drive (electron cyclotron current drive). Although both methods have been applied successfully to experiments showing a predominance of ECRH for medium-sized limiter tokamaks (TEXTOR, T-10) and of ECCD for mid-to-large-sized divertor tokamaks (AUG, DIII-D, JT-60), conditions determining their relative importance are still unclear. We address this problem with a numerical study focused on the contributions of heating and current drive to the temporal evolution of NTMs as described by the modified Rutherford equation. For the effects of both heating as well as current drive, simple analytical expressions have been found in terms of an efficiency fore-factor times a ‘geometrical’ term depending on the power deposition width wdep, location and modulation. When the magnetic island width w equals the width of the deposition profile, w ≈ wdep, both geometric terms are practically identical. Whereas for current drive the geometric term approaches a constant for small island widths and is inversely proportional to (w/wdep)2 for large island widths, the heating term approaches a constant for large island widths and is proportional to (w/wdep) for small island widths. For medium-sized tokamaks (TEXTOR, AUG) the heating and current drive efficiencies are of the same order of magnitude, whereas in a future, large reactor like ITER the current drive efficiency is expected to be significantly larger.

Disruption control on FTU and ASDEX upgrade with ECRH

The use of ECRH has been investigated as a promising technique to avoid or postpone disruptions in dedicated experiments in FTU and ASDEX Upgrade. Disruptions have been produced by injecting Mo through laser blow-off (FTU) or by puffing deuterium gas above the Greenwald limit (FTU and ASDEX Upgrade). The toroidal magnetic field is kept fixed and the ECRH launching mirrors have been steered before every discharge in order to change the deposition radius. The loop voltage signal is used as disruption precursor to trigger the ECRH power before the plasma current quench. In the FTU experiments (Ip = 0.35–0.5 MA, Bt = 5.3 T, PECRH = 0.4–1.2 MW) it is found that the application of ECRH modifies the current quench starting time depending on the power deposition location. A scan in deposition location has shown that the direct heating of one of the magnetic islands produced by magnetohydrodynamic (MHD) resistive instabilities (either m/n = 3/2, 2/1 or 3/1) prevents its further growth and also produces the stabilization of the other coupled modes and the delay of the current quench or its full avoidance. Disruption avoidance and complete discharge recovery are obtained when the ECRH power is applied on rational surfaces. The modes involved in the disruption are found to be tearing modes stabilized by a strong local ECRH heating. The Rutherford equation has been used to reproduce the evolution of the MHD modes. In the ASDEX Upgrade experiments L-mode plasmas (Ip = 0.6 MA, Bt = 2.5 T, PECRH = 0.6 MW ∼ POHM) the injection of ECRH close to q = 2 significantly delays the 2/1 onset and prolongs the duration of the discharge: during this phase the density continues to increase. No delay in the onset of the 2/1 mode is observed when the injected power is reduced to 0.35 MW.

MHD memes

The celebration of Allan Kaufman's 80th birthday was an occasion to reflect on a career that has stimulated the mutual exchange of ideas (or memes in the terminology of Richard Dawkins) between many researchers. This paper will revisit a meme Allan encountered in his early career in magnetohydrodynamics, the continuation of a magnetohydrodynamic mode through a singularity, and will also mention other problems where Allan's work has had a powerful cross-fertilizing effect in plasma physics and other areas of physics and mathematics. To resolve the continuation problem we regularize the Newcomb equation, solve it in terms of Legendre functions of imaginary argument, and define the small weak solutions of the Newcomb equation as generalized functions in the manner of Lighthill, i.e. via a limiting sequence of analytic functions that connect smoothly across the singularity.

Fast imaging of transients and coherent magnetohydrodynamic modes in DIII-D

A fast framing camera is used to image plasma waves and instabilities in the DIII-D tokamak [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] in unprecedented detail including tearing modes (TMs) and sawtooth crashes. To image core magnetohydrodynamic (MHD) activity, the fast camera detects visible bremsstrahlung emission εB in moderate to high density plasmas. For coherent MHD activity such as TMs, high-resolution two-dimensional images of mode amplitude and phase are obtained by Fourier filtering each pixel’s time series at the mode frequency. Images of m/n=2/1 TMs show that inside the q=2 surface, the camera measurements are in excellent agreement with an analytic model of a 2/1 island superimposed on the equilibrium εB profile. Direct comparison of the measurements to a NIMROD simulation shows significant discrepancies, most likely due to artificially high-density diffusion used in the code for numerical stability. The first visible-light images of transient sawtooth crashes show the structure and location of the perturbed emission from an m=1 precursor oscillation and show that during the nonlinear crash phase the instability extends to more than half of the plasma minor radius.

Propagating transverse waves in soft X-ray coronal jets

Aims. The theoretical model for magnetohydrodynamic (MHD) modes guided by a field-aligned plasma cylinder with a steady flow is adapted to interpret transverse waves observed in solar coronal hot jets, discovered with Hinode/XRT in terms of fast magnetoacoustic kink modes. Methods. Dispersion relations for linear magnetoacoustic perturbations of a plasma jet of constant cross-section surrounded by static magnetised plasma are used to determine the phase and group speeds of guided transverse waves and their relationship with the physical parameters of the jet and the background plasma. The structure of the perturbations in the macroscopic parameters of the plasma inside and outside the jet, and the phase relations between them are also established. Results. We obtained a convenient expansion for the long wave-length limit of the phase and group speeds and have shown that transverse waves observed in soft-X-ray solar coronal jets are adequately described in terms of fast magnetoacoustic kink modes by a magnetic cylinder model, which includes the effect of a steady flow. In the observationally determined range of parameters, the waves are not found to be subject to either the Kelvin-Helmholtz instability or the negative energy wave instability, and hence they are likely to be excited at the source of the jet.

A digital control system for external magnetohydrodynamic modes in tokamak plasmas

A feedback system for controlling external, long-wavelength magnetohydrodynamic activity is described. The system is comprised of a network of localized magnetic pickup and control coils driven by four independent, low-latency field-programable gate array controllers. The control algorithm incorporates digital spatial filtering to resolve low mode number activity, temporal filtering to correct for frequency-dependent amplitude and phase transfer effects in the control hardware, and a Kalman filter to distinguish the unstable plasma mode from noise.