Magnetohydrodynamic Spectrum Research Articles

A spectroscopic investigation of thermal instability for cylindrical equilibria with background flow

Context. Flows are omnipresent and govern the dynamics of plasma. Solar tornadoes are a class of apparently rotating prominences that might be formed by thermal instability. In spectroscopic studies on thermal instability, background flow is commonly neglected. Aims. We here determine the effect of background flow on thermal instability in cylindrical magnetic field configurations. How various parameters affect the distribution of eigenmodes in the magnetohydrodynamic (MHD) spectrum is also explored. We investigate whether discrete thermal modes exist. Methods. In an analytical study, we extended upon the literature by including a generic background flow in a cylindrical coordinate system. The non-adiabatic MHD equations are linearised, Fourier-analysed, and examined to understand how a background flow changes the continua. An approximate expression for discrete thermal modes is derived using a Wentzel-Kramers-Brillouin (WKB) analysis. The analytical results are then verified for a benchmark equilibrium using the eigenvalue code Legolas. The eigenfunctions of discrete thermal modes are visualised in 2D and 3D. Results. The thermal continuum is Doppler-shifted due to the background flow, just like the slow and Alfvén continua. Discrete modes are altered because the governing equations contain flow-related terms. An approximate expression to predict the appearance of discrete thermal modes based on the equilibrium parameters is derived. All analytical expressions match the numerical results. The distribution of the density perturbations of the discrete thermal modes is not a uniform or singular condensation, due to the shape of the eigenfunctions and the dependence of the assumed waveform on the coordinates and wavenumbers. A 3D visualisation of the total velocity field shows that the helical field is heavily influenced by the radial velocity perturbation. Conclusions. We derived analytic expressions for non-adiabatic MHD modes of a cylindrical equilibrium with background flow and verified them using a coronal equilibrium. However, the equations are valid for and can be applied in other astrophysical environments.

Magnetohydrodynamic Spectroscopy of a Non-adiabatic Solar Atmosphere

The quantification of all possible waves and instabilities in a given system is of paramount importance, and knowledge of the full magnetohydrodynamic (MHD) spectrum allows one to predict the (in)stability of a given equilibrium state. This is highly relevant in many (astro)physical disciplines, and when applied to the solar atmosphere it may yield various new insights in processes like prominence formation and coronal loop oscillations. In this work we present a detailed, high-resolution spectroscopic study of the solar atmosphere, where we use our newly developed Legolas code to calculate the full spectrum with corresponding eigenfunctions of equilibrium configurations that are based on fully realistic solar atmospheric models, including gravity, optically thin radiative losses and thermal conduction. Special attention is given to thermal instabilities, known to be responsible for the formation of prominences, together with a new outlook on the thermal and slow continua and how they behave in different chromospheric and coronal regions. We show that thermal instabilities are unavoidable in our solar atmospheric models and that there exist certain regions where both the thermal, slow and fast modes all have unstable wave mode solutions. We also encounter regions where the slow and thermal continua become purely imaginary and merge on the imaginary axis. The spectra discussed in this work illustrate clearly that thermal instabilities (both discrete and continuum modes) and magneto-thermal overstable propagating modes are ubiquitous throughout the solar atmosphere, and may well be responsible for much of the observed fine-structuring and multi-thermal dynamics.

Low-frequency oscillations while magnetic storms as a tool to determine the types of solar plasma flows

Comparison of wavelet spectrum (skeletons) local maxima for disturbed components of solar plasma flow parameters and geomagnetic field disturbances recorded along the meridional station chain during geomagnetic storm intervals is performed in the range of magnetohydrodynamic (MHD) waves. An algorithm for quantitative evaluation of analyzed skeletons consistency has been developed. It has been used to demonstrate the possibility of the type of solar wind plasma flow elaboration on unique spectral signs of Solar wind speed disturbances, density and interplanetary magnetic field. It is shown that the energy spectrum of oscillations for these parameters reflects the internal structure of the corresponding plasma formation. The skeletons application to the analysis of the interplanetary space main parameters made it possible to estimate the magnetosphere reaction time in geomagnetic field horizontal component oscillations at different latitudes on the disturbance. As a result, the distributed magnetosphere reaction over latitude was determined in the form of geomagnetic field oscillations on the disturbed solar flow parameters. It is shown that the dynamics of the components of the solar wind parameters disturbance spectra corresponding to plasma flows manifest themselves in the MHD spectra of high-latitude stations magnetograms and can be used as a diagnostic tool.

Classification of Solar Plasma Flows by Analyzing Out- and Intra-Magnetospheric Magnetohydrodynamic Oscillations During Magnetic Storms

Comparison of wavelet spectrum (skeletons) local maxima for disturbed components of solar plasma flow parameters and geomagnetic field disturbances recorded along the meridional station chain during geomagnetic storm intervals is performed in the range of magnetohydrodynamic (MHD) waves. An algorithm for quantitative evaluation of analyzed skeletons consistency has been developed. It has been used to demonstrate the possibility of the type of solar wind plasma flow elaboration on unique spectral signs of Solar wind speed disturbances, density and interplanetary magnetic field. It is shown that the energy spectrum of oscillations for these parameters reflects the internal structure of the corresponding plasma formation. The skeletons application to the analysis of the interplanetary space main parameters made it possible to estimate the magnetosphere reaction time in geomagnetic field horizontal component oscillations at different latitudes on the disturbance. As a result, the distributed magnetosphere reaction over latitude was determined in the form of geomagnetic field oscillations on the disturbed solar flow parameters. It is shown that the dynamics of the components of the solar wind parameters disturbance spectra corresponding to plasma flows manifest themselves in the MHD spectra of high-latitude stations magnetograms and can be used as a diagnostic tool.

Low-frequency magnetohydrodynamic spectra of plasmas with anisotropic pressure in axisymmetric toroidal systems

The low-frequency ideal magnetohydrodynamic (MHD) spectra of anisotropic pressure plasmas in axisymmetric toroidal systems are analyzed. In the framework of the Chew-Goldberger-Low model, the continuum spectrum equations for the coupled localized Alfvén and slow magnetosonic modes are derived. The equations are applied to study the continuous spectra for low-pressure plasmas in large aspect ratio tokamaks. It is shown that for some classes of plasma equilibria the pressure anisotropy results in instability of continuous modes. Such an instability is due to anisotropic plasma stratification over poloidal angle on the magnetic surfaces. It takes place for both the zonal flow modes with (m, n) = 0 and the general electromagnetic modes with (m, n) ≠ 0. The latter are unstable only if they are localized near the rational magnetic surface. A stabilization of this instability due to the Alfvén effect is shown for the modes localized away from the rational magnetic surface.

Тороидальные альфвеновские моды в плазме токамака Глобус-М

AbstractConditions of the excitation of toroidal Alfvén eigenmodes (TAEs) and their inf luence on the confinement of fast particles into the Globus-M spherical tokamak have been studied by KINX code calculations of the magnetohydrodynamic spectra of reconstructed divertor equilibrium configurations with stability margin q _0 > 1 on the magnetic axis. The sensitivity of the frequencies of TAE modes with toroidal wavenumber n = 1 to the type of boundary conditions and choice of boundary surface has been studied. It has been established that the frequencies of modes with dominating poloidal harmonics m = 1 and 2 in the gap of continuum are significantly higher than those observed in the spectra of signals measured in the Mirnov coil probes, especially under the assumption of free plasma boundary with allowance for its compressibility. The TAE modes with lower frequencies and higher poloidal wavenumbers localized near the plasma boundary may be responsible for the oscillations observed in the experiment. However, these modes are characterized by the interaction with continuum and, probably, exhibit related damping.

Toroidal Alfvén Eigenmodes in Globus-M Spherical Tokamak Plasma

Conditions of the excitation of toroidal Alfven eigenmodes (TAEs) and their inf luence on the confinement of fast particles into the Globus-M spherical tokamak have been studied by KINX code calculations of the magnetohydrodynamic spectra of reconstructed divertor equilibrium configurations with stability margin q0 > 1 on the magnetic axis. The sensitivity of the frequencies of TAE modes with toroidal wavenumber n = 1 to the type of boundary conditions and choice of boundary surface has been studied. It has been established that the frequencies of modes with dominating poloidal harmonics m = 1 and 2 in the gap of continuum are significantly higher than those observed in the spectra of signals measured in the Mirnov coil probes, especially under the assumption of free plasma boundary with allowance for its compressibility. The TAE modes with lower frequencies and higher poloidal wavenumbers localized near the plasma boundary may be responsible for the oscillations observed in the experiment. However, these modes are characterized by the interaction with continuum and, probably, exhibit related damping.

MHD normal mode analysis with equilibrium pressure anisotropy

In this work, we generalise linear magnetohydrodynamic (MHD) stability theory to include equilibrium pressure anisotropy in the fluid part of the analysis. A novel ‘single-adiabatic’ (SA) fluid closure is presented which is complementary to the usual ‘double-adiabatic’ (CGL) model and has the advantage of naturally reproducing exactly the MHD spectrum in the isotropic limit. As with MHD and CGL, the SA model neglects the anisotropic perturbed pressure and thus loses non-local fast-particle stabilisation present in the kinetic approach. Another interesting aspect of this new approach is that the stabilising terms appear naturally as separate viscous corrections leaving the isotropic SA closure unchanged. After verifying the self-consistency of the SA model, we re-derive the projected linear MHD set of equations required for stability analysis of tokamaks in the MISHKA code. The cylindrical wave equation is derived analytically as done previously in the spectral theory of MHD and clear predictions are made for the modification to fast-magnetosonic and slow ion sound speeds due to equilibrium anisotropy.

Low-frequency Alfvén gap modes in rotating tokamak plasmas

As a result of toroidal rotation, sequences of new global modes are predicted to arise in magnetically confined plasmas. The frequencies of these Alfvén modes lie inside gaps of the continuous magnetohydrodynamic (MHD) spectrum that are created or enlarged by toroidal flow. The numerically obtained results are compared with an analytical investigation, yielding a useful criterion for mode existence. Because of their low frequencies, these modes may be easily destabilized by energetic particles. Because of their sensitivity to the Mach number however, these modes can provide a valuable extension to MHD spectroscopy by giving information on the rotational velocity.

New construction of the magnetohydrodynamic spectrum of stationary plasma flows. II. Rayleigh–Taylor and Kelvin–Helmholtz instability

In a preceding paper [J. P. Goedbloed, Phys. Plasmas 16, 122110 (2009)] a new method was developed to compute the magnetohydrodynamic spectrum of waves and instabilities of stationary plasma flows by means of the construction of the solution paths, Ps and Pu, of stable waves and instabilities in the complex ω plane of an extended boundary value problem and the monotonicity of the alternator R≡ξ/Π (ratio of displacement and total pressure perturbation) along those paths to find the eigenvalues. This method is applied in this paper to explicitly construct the solution paths, and the eigenvalues on it, for a plane gravitating plasma slab with shear flow. The topology of the solution paths for Rayleigh–Taylor and Kelvin–Helmholtz instabilities that are constructed here for the first time exhibits a surprising complexity. This is not a mathematical artifact, but characteristic for the structure of the magnetohydrodynamic spectrum of plasmas with background flow. Hence, its construction is crucial for the success of magnetohydrodynamic spectroscopy, aimed at determining the internal equilibrium characteristics of astrophysical and laboratory fusion plasmas.

New construction of the magnetohydrodynamic spectrum of stationary plasma flows. I. Solution path and alternator

A new method of systematically constructing the full structure of the complex magnetohydrodynamic spectra of stationary flows is presented. It is based on the self-adjointness of the generalized force operator G and the Doppler–Coriolis shift operator U, and the associated quadratic forms for the normalized energy W¯ and the normalized Doppler–Coriolis shift V¯, which may be constructed for all complex values of ω if the original eigenvalue problem is converted into a one-sided boundary value problem. This turns W¯ into a complex expression, while V¯ remains real. Whereas the solution path Ps of stable modes is just the real axis, the solution path Pu of unstable modes in the complex ω plane is found by requiring that the solution-averaged Doppler–Coriolis shifted real part of the frequency vanishes, σ−V¯[ξ(ω)]=0, or that the energy is real, Im W¯[ξ(ω)]=0. The location of the eigenvalues on these solution paths is determined by two quadratic forms, which may straightforwardly be evaluated in any of the finite element spectral codes in existence. A new oscillation theorem is proved about the monotonicity of complex eigenvalues for one-dimensional systems. Instead of counting internal nodes of the real displacement vector ξ (as in static plasmas), it is based on counting the zeros of the alternating ratio, or alternator, R≡ξ/Π of the boundary values of the complex functions ξ and the total pressure perturbation Π, which is real on the solution path. This finally provides the generalization of the basic structural properties of the magnetohydrodynamic spectrum of static plasmas, which has been known for a long time, to stationary plasmas.

Modeling Solar Wind Turbulence: The Kolmogorov-like Way

The spectral energy distributions of the magnetohydrodynamic (MHD) fluctuations of the solar wind turbulence are derived using the dimensional arguments a la Kolmogorov within the framework of the Hall magnetohydrodynamics. While the velocity and the magnetic field fluctuations are dynamically related, the density fluctuations could behave as a passive scalar and be simply convected by the velocity or the magnetic field fluctuations. The Hall effect removes the degeneracy of the ideal Alfvenic spectra of the velocity and the magnetic fluctuations, at spatial scales shorter than or equal to the ion- inertial scale, adding steeper branches to the ideal MHD spectra. Which spectrum would the density fluctuations, behaving as a passive scalar, follow in such a case? The answer leads to the interesting consequence that the electron density fluctuations should follow the magnetic spectra since the electrons are frozen to the magnetic field and the ion density fluctuations should follow the kinetic energy spectra as ions carry the inertia. Thus the electron and the ion density would have different spectra at spatial scales equal to and smaller than the ion-inertial scale. However this raises the issue of the quasineutrality that must be maintained at each scale within the Hall-MHD. One way to accommodate both the quasineutrality as well as the electron- magnetic freezing in the Hall MHD is to discard the passive nature of the density fluctuations; they must be dynamically active in the turbulence. The quasineutrality could also be restored by a third species of particles providing a stationary background.

Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics

Action-angle variables corresponding to singular (or improper) eigenmodes are rigorously formulated for the Alfvén and slow (or cusp) continuous spectra of ideal magnetohydrodynamics. The perturbation energy is then transformed into the normal form, namely, the eigenfrequency multiplied by the action variable. It is shown that the Laplace transform approach expedites this action-angle formulation more efficiently than the existing ones devoted to other kinds of continuous spectra. The presence of flow that is either nonparallel to the magnetic field or supersonic at some places brings about singular eigenmodes with negative energy. The Alfvén and slow singular eigenmodes are neutrally stable even in the presence of any external potential fields, but may cause instability when coupled with another singular or nonsingular eigenmode with the opposite sign of energy.

PHOENIX: MHD spectral code for rotating laboratory and gravitating astrophysical plasmas

The new PHOENIX code is discussed together with a sample of many new results that are obtained concerning magnetohydrodynamic (MHD) spectra of axisymmetric plasmas where flow and gravity are consistently taken into account. PHOENIX, developed from the CASTOR code [W. Kerner, J.P. Goedbloed, G.T.A. Huysmans, S. Poedts, E. Schwarz, J. Comput. Phys. 142 (1998) 271], incorporates purely toroidal, or both toroidal and poloidal flow and external gravitational fields to compute the entire ideal or resistive MHD spectrum for general tokamak or accretion disk configurations. These equilibria are computed by means of FINESSE [A.J.C. Beliën, M.A. Botchev, J.P. Goedbloed, B. van der Holst, R. Keppens, J. Comp. Physics 182 (2002) 91], which discriminates between the different elliptic flow regimes that may occur. PHOENIX makes use of a finite element method in combination with a spectral method for the discretization. This leads to a large generalized eigenvalue problem, which is solved by means of Jacobi–Davidson algorithm [G.L.G. Sleijpen, H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17 (1996) 401]. PHOENIX is compared with CASTOR, PEST-1 and ERATO for an internal mode of Soloviev equilibria. Furthermore, the resistive internal kink mode has been computed to demonstrate that the code can accurately handle small values for the resistivity. A new reference test case for a Soloviev-like equilibrium with toroidal flow shows that, on a particular unstable mode, the flow has a quantifiable stabilizing effect regardless of the direction of the flow. PHOENIX reproduces the Toroidal Flow induced Alfvén Eigenmode (TFAE, [B. van der Holst, A.J.C. Beliën, J.P. Goedbloed, Phys. Rev. Lett. 84 (2000) 2865]) where finite resistivity in combination with equilibrium flow effects causes resonant damping. Localized ideal gap modes are presented for tokamak plasmas with toroidal and poloidal flow. Finally, we demonstrate the ability to spectrally diagnose magnetized accretion disk equilibria where gravity acts together with either purely toroidal flow or both toroidal and poloidal flow. These cases show that the MHD continua can be unstable or overstable due to the presence of a gravitational field together with equilibrium flow-driven dynamics [J.P. Goedbloed, A.J.C. Beliën, B. van der Holst, R. Keppens, Phys. Plasmas 11 (2004) 28].

On the continuous spectrum of leaky magnetohydrodynamic modes and the associated quasimodes

We aim to clarify the mathematical status of the leaky modes in an unbounded magnetohydrodynamic (MHD) plasma. The initial value problem of a one-dimensional unbounded MHD plasma is solved by means of a Laplace transform in time. It is shown that the MHD operator remains self-adjoint in the sense that a self-adjoint extension of the minimal MHD operator exists and that the eigenfrequencies should therefore be real. However, the classical picture of the MHD spectrum is to be extended with a slow and a fast leaky continuous spectrum when unbounded spatial domains are considered. The inversion integral for the Laplace transform is evaluated in such a way as to recast the continuum contribution in terms of a complete spectral representation. This involves the construction of the spectral measure associated with the continuous spectra. In the case of a slab structure, the spectral measure is not a monotonic function of the frequency, but peaks appear in the spectral measure around specific frequencies. These frequencies correspond to the discrete damped leaky modes described in the context of solar physics. The spectral measure can be interpreted physically in terms of the energy carried in and out by the incoming and outgoing waves.

Local analysis of MHD spectra for cylindrical plasmas with flows

An analytical and numerical study of the ideal magnetohydrodynamic (MHD) spectrum of waves and instabilities of a cylindrical plasma column with flows is presented. Our analytical results are relevant for thermally stratified, rotating, magnetized cylindrical equilibria. The presence of azimuthal flow makes the general analysis of the MHD spectrum difficult, except in cases where the continuous parts of the spectrum are absent. In the presence of Doppler shifted Alfvén and slow continua, a local analysis at resonant surfaces or internal extrema can provide a simple and reliable way to access information on MHD spectroscopy. In this paper, local cluster conditions, which govern the occurrence of sequences of discrete global modes, have been generalized for rotating equilibria. The generalized Suydam criterion for instability is revisited. A numerical study confirms our analytical results and clearly demonstrates how the local criteria govern the existence of the accumulating eigenmodes.

Effects of nonresonant hot ions with large orbits on Alfvén cascades and on magnetohydrodynamic instabilities in tokamaks

The effects of nonresonating hot ions on the spectrum of magnetohydrodynamic (MHD) waves and instabilities in tokamaks are studied in the limit when the width of the hot ion drift orbits is much larger than the radial scale length of the MHD perturbations. Due to the large magnetic drift velocities the hot ions cannot contribute to the MHD perturbations directly, but two main effects of the hot ions, the hot-ion density-dependent effect and the hot-ion pressure-dependent effect, influence the MHD perturbations indirectly. The physics of both effects is elucidated and it is shown that both these effects can be described in MHD approach. A new code, MISHKA-H (MISHKA including the hot-ion indirect effects), is developed as an extension of the ideal MHD code MISHKA-D [Huysmans et al., Phys. Plasmas 8, 4292 (2002)]. Analytical benchmarks for this code are given. Results of the MISHKA-H code on Alfvén spectrum in a shear-reversed discharges with ion-cyclotron resonance frequency (ICRF) heating are presented. Modeling of Alfvén cascades and their transition into toroidal Alfvén eigenmodes in shear-reversed tokamak equilibrium is considered. The hot-ion effect on the unstable branch of the MHD spectrum is studied for the test case of an n=1 ideal MHD internal kink mode, which is relevant to short-period sawteeth in low-density plasmas observed in Joint European Torus (JET) [Rebut et al., Proceedings of the 10th International Conference, Plasma Physics and Controlled Nuclear Fusion, London (International Atomic Energy Agency, Vienna, 1985), Vol. I, p. 11] experiments with high-power ICRF heating.

Transsonic Instabilities in Laboratory and Astrophysical Plasmas

Waves and instabilities of transsonically rotating axisymmetric plasmas present a highly complex problem that is of interest for two unrelated fields of research, viz. laboratory tokamak confinement for the eventual thermonuclear energy production and the dynamics of a vast number of astrophysical plasmas rotating about compact objects, broadly indicated as accretion disks. The complexity originates from the transsonic transitions of the poloidal flow which causes the character of the rotating equilibrium states to change dramatically, from elliptic to hyperbolic or vice versa, when the poloidal velocity surpasses certain critical speeds. Associated with these transitions the different types of magnetohydrodynamic (MHD) shocks may appear. Obviously, at such transitions the possible waves and instabilities of the system also change dramatically. We investigate these changes for the two mentioned classes of physical systems, starting from the point of view that the continuous spectrum of magnetohydrodynamics presents the best organizing principle for the structure of the complete spectrum of waves and instabilities since it is the most robust part of it, for that reason called the “essential spectrum” by mathematicians. The physical importance of the problem is that it provides the simplest approach to local waves and instabilities of the system and, possibly, to the onset of MHD turbulence in accretion disks.

Electron inertial effects on the resistive magnetohydrodynamic spectrum of a magnetic X-point

The mode spectrum associated with a magnetic X-point is determined, taking into account resistivity and electron inertia. The assumed equilibrium configuration has zero current. The eigenmodes are expressed in terms of solutions of the hypergeometric equation, and are also computed numerically. The spectrum is shown to exhibit a rich structure of continuum and discrete modes. When both resistivity and electron inertia are present, there are two distinct continua associated with singular current sheets and infinite kinetic energy, corresponding to zero and finite real mode frequencies. A finite number of discrete modes can also exist, depending on the Lundquist number and the collisionless skin depth.

Waves and Instabilities in Accretion Disks: Magnetohydrodynamic Spectroscopic Analysis

A complete analytical and numerical treatment of all magnetohydrodynamic waves and instabilities for radially stratified, magnetized accretion disks is presented. The instabilities are a possible source of anomalous transport. While recovering results on known hydrodynamic and both weak- and strong-field magnetohydrodynamic perturbations, the full magnetohydrodynamic spectra for a realistic accretion disk model demonstrate a much richer variety of instabilities accessible to the plasma than previously realized. We show that both weakly and strongly magnetized accretion disks are prone to strong nonaxisymmetric instabilities. The ability to characterize all waves arising in accretion disks holds great promise for magnetohydrodynamic spectroscopic analysis.