The purpose of this paper is to give an overview of the characteristics of MHD **waves**, fire-hose and mirror instabil- ities in the double-polytropic MHD model. The gyrotropic MHD formulation used in this study is different from the pre- vious one in that both ions and electrons are treated as sep- arate fluids and may possess different pressure anisotropy, plasma β values and energy equations. The two-fluid based MHD formulation is more suitable for modeling the space plasma where ions and electrons may have distinct temper- ature and pressure anisotropy (e.g., Hau et al., 1993; Phan et al., 1996). Indeed, it has been shown that for the earth’s magnetosheath the double-polytropic laws not only provide a viable energy closure for protons but also for electrons with different polytropic exponents though the temperature of electrons is usually much lower than that of ions (Hau et al., 1993). In this study separate pairs of polytropic laws are thus used as energy equations for both ion and electron flu- ids. The properties of hydromagnetic **waves**, the conditions for anomalous slow **waves** and the criteria for fire-hose and mirror **instabilities** are examined based on the linear double- polytropic MHD model. The nonlinear evolutions of anoma- lous slow **waves** and **instabilities** are studied based on the gyrotropic MHD simulations; especially, the conditions for nonlinear saturation of fire-hose and mirror **instabilities** are examined.

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Abstract. We review the basic approximations underlying magnetohydrodynamic (MHD) theory, with special empha- sis on the closure approximations, i.e. the approximations used in any fluid approach to close the hierarchy of moment equations. We then present the main closure models that have been constructed for collisionless plasmas in the large-scale regime, and we describe our own mixed MHD-kinetic model, which is designed to study low-frequency linear **waves** and **instabilities** in collisionless plasmas. We write down the full dispersion relation in a new, general form, which gathers all the specific features of our MHD-kinetic model into four polytropic indices, and which can be applied to standard adi- abatic MHD and to double-adiabatic MHD through a simple change in the expressions of the polytropic indices. We study the mode solutions and the stability properties of the full dis- persion relation in each of these three theories, first in the case of a uniform plasma, and then in the case of a stratified plasma. In both cases, we show how the results are affected by the collisionless nature of the plasma.

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A numerical method that employs a combination of contour advection and pseudo- spectral techniques is used to simulate shear-induced **instabilities** in an internal solitary wave. A three-layer configuration for the background stratification in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones is consid- ered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck- Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady state, internal solitary **waves**. The results of the simulations show that the in- stability takes place in the pycnocline and manifests itself as Kelvin-Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ri c ) and the critical amplitude required for insta-

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Among recent publications, it is worth noting the following. Del Zanna et al. (2001) and Del Zanna and Velli (2002) studied the stability and nonlinear evolution of pump **waves** in three dimensions numerically. Hertzberg et al. (2003, 2004a,b) and Cramer et al. (2003) investigated the parametric **instabilities** of pump **waves** in multi-component and dusty plasmas. Matsukiyo and Hada (2003) considered the **instabilities** of pump **waves** in a relativistic electron–positron plasma. Ruderman and Simpson (2004b, 2005), Simpson and Ruderman (2005) and Simpson et al. (2006) investigated absolute and convective **instabilities** of pump **waves** in the ideal MHD approximation.

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The plan of the paper is as follows: In Sect. 2, we re- view and compare the parametric **instabilities** in the Hall- MHD system and in the DNLS equation. In Sect. 3, we dis- cuss the relationship between generation of solitary **waves** and nonlinear wave-wave interaction in detail, by examining numerically produced Alfv´enic turbulence using the DNLS equation. Our main discussions on the phase coherence is presented in this section. In Sect. 4, we discuss phase and frequency features in the Hall-MHD system. From the rela- tion between the plasma density and the envelope of the mag- netic field, we examine validity of the so-called quasi-static approximation used in the DNLS. When the longitudinal and transverse wave modes have similar phase velocities, they can couple strongly, and as a result the phase coherence can be generated. We summarize the results and discuss some of the fundamental issues presented in the paper in Sect. 5.

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The concept of absolute and convective **instabilities** was first developed in plasma physics (Briggs 1964; Bers 1973). Later it was applied to hydrodynamic stability problems, in particular, to stability of geophysical and astrophysical flows (Kulikovskii & Shikina 1977; Huerre & Monkewitz 1985; Brevdo 1988; Ruderman 2000; Wright et al. 2000, 2002; Mills et al. 2000; Terra-Homem & Erdélyi 2003, 2004; Ruderman et al. 2004). Recently Ruderman & Simpson (2004b) and Simpson & Ruderman (2005) (Papers I and II in what follows) studied the absolute and convective **instabilities** of circularly po- larized Alfvén **waves** analytically using the dimensionless pump wave amplitude a as a small parameter. In this paper we use numerical technics to extend their analysis for arbitrary a. The paper is organized as follows. In the next section we give a very brief description of Briggs’ method for studying the absolute and convective **instabilities**, and present the numerical results of our analysis of the absolute and convective **instabilities**. In Sect. 3 we investigate spatially amplifying **waves**. In Sect. 4 we discuss possible implications of our results on the interpretation of ob- servational data obtained onboard of space missions. Section 5 contains the summary of results and our conclusions.

The Kelvin–Helmholtz instability has been proposed as a mechanism to extract energy from magnetohydrodynamic (MHD) kink **waves** in flux tubes, and to drive dissipation of this wave energy through turbulence. It is therefore a potentially important process in heating the solar corona. However, it is unclear how the instability is influenced by the oscillatory shear flow associated with an MHD wave. We investigate the linear stability of a discontinuous oscillatory shear flow in the presence of a horizontal magnetic field within a Cartesian framework that captures the essential features of MHD oscillations in flux tubes. We derive a Mathieu equation for the Lagrangian displacement of the interface and analyse its properties, identifying two different **instabilities**: a Kelvin–Helmholtz instability and a parametric instability involving resonance between the oscillatory shear flow and two surface Alfv´en **waves**. The latter occurs when the system is Kelvin–Helmholtz stable, thus favouring modes that vary along the flux tube, and as a consequence provides an important and additional mechanism to extract energy. When applied to flows with the characteristic properties of kink **waves** in the solar corona, both **instabilities** can grow, with the parametric instability capable of generating smaller scale disturbances along the magnetic field than possible via the Kelvin–Helmholtz instability. The characteristic time-scale for these **instabilities** is ∼ 100 s, for wavelengths of 200 km. The parametric instability is more likely to occur for smaller density contrasts and larger velocity shears, making its development more likely on coronal loops than on prominence threads.

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viously supported a number of rocket campaigns, such as MaCWAVE/MIDAS in summer 2002 (Goldberg et al., 2004), winter MaCWAVE in Jan 2003 (Goldberg et al., 2006), DELTA in Dec 2004, and ROMA in Jan 2005. In the summer MaCWAVE/MIDAS campaign, we observed very large positive and negative temperature gradients (Fritts et al., 2004) and large gravity wave variability (Williams et al., 2004). During the winter MaCWAVE campaign, a very large semidiurnal tide caused a large temperature inversion and areas of instability that propagated down with the tidal phase (Williams et al., 2006). The formation of tempera- ture inversions, the creation and evolution of **instabilities**, and their relationship with atmospheric gravity **waves** are not fully understood yet and continue to be an active area of research (Li et al., 2005; Hecht et al., 2004; She et al., 2004) with several recent reviews (Meriwether and Gerrard, 2004; Hecht, 2004; Fritts and Alexander, 2003).

the nonlinear evolution of circularly polarized Alfvén **waves** both in homogeneous 6 and stratified 7 plasmas using a one- dimensional numerical code, and applied their results to the acceleration of the solar wind. Del Zanna et al. 8,9 and Del Zanna and Velli 10 developed a three-dimensional MHD code to study the stability and nonlinear evolution of Alfvén **waves**. They applied their numerical results to the evolution of Alfvén wave spectra in the solar wind, and to plasma heating in coronal holes. Shevchenko et al. 11 used the deriva- tive nonlinear Schrödinger sDNLSd equation to study the parametric decay instability of Alfvén packets propagating in the opposide directions. Hertzberg et al. 12–14 and Cramer et al. 15 extended the linear theory of parametric **instabilities** of Alfvén **waves** to multicomponent and dusty plasmas. Mat- sukiyo and Hada 16 studied the parametric **instabilities** of cir- cularly polarized Alfvén **waves** in a relativistic electron- positron plasma.

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In summary, we have developed a uniﬁed methodology for the treatment of low-frequency beam-driven **instabilities** in relativistic quantum plasmas. A detailed comparison was made considering different parameter regimes (formally, ¯ h = 0 and 1/c = 0; ¯ h = 0 and 1/c = 0; both ¯ h = 0 and 1/c = 0). The corresponding Buneman **instabilities** were shown to be associated to certain beam-plasma negative energy modes. Both quantum and relativistic effects cause a stabilizing effect, with the remark that the quantum effects produce a splitting of the original Buneman mode into two modes, due to the quantum recoil term as discussed in Sec. IV. In addition, observe that the model equations have certain limitations in particular because of the semiclassic nature (no quantized ﬁelds); see [17] for more details. Moreover, our collective Klein-Gordon plasma model neglects collisions, thermal and degeneracy pressure, and kinetic (Landau damping) effects, besides exchange interactions [26], which should be included in more sophisticated models for relativistic beams in quantum plasmas. Nevertheless, the dispersion relation (7) can be equally found using more complete, ﬁeld-theoretic methods [24]. In this context, the positive-negative energy **waves** found here have a more broad applicability, as long as the low-frequency assumption is valid.

a magnetocentrifugal mechanism accelerates plasma along poloidal field lines threading the accretion disk; the plasma is then collimated by a toroidal dominant magnetic field at larger distance. Lynden-Bell (1996,2003) [74, 75] constructed an analytical magnetostatic MHD model where the upward flux of a dipole magnetic field is twisted relative to the downward flux. The height of the magnetically dominant cylindrical plasma grows in this configuration. The toroidal component of the twisted field is responsible for both collimation and propagation. The Lynden-Bell (1996,2003) [74, 75] model and various following models, (typically numerical simulations with topologically similar magnetic field configurations; e.g., [68, 69, 73, 84, 122]), are called “magnetic tower” models. In these models, the large scale magnetic fields are often assumed to possess “closed” field lines with both footprints residing in the disk. Because plasma at different radii on the accretion disk and in the corona have different angular velocity, the poloidal magnetic field lines threading the disk will become twisted up [14, 69, 73, 74, 75, 100], giving rise to the twist/helicity or the toroidal component of the magnetic fields in the jet. Faraday rotation measurements to 3C 273 show a helical magnetic field structure and an increasing pitch angle between toroidal and poloidal component along the jet [128]. These results favor a magnetic structure suggested by magnetic tower models. Furthermore, it is (often implicitly) assumed that the mass loading onto these magnetic fields is small, so the communication by Alfv´ en **waves** is often fast compared to plasma flows.

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We have investigated the observational signatures of transverse MHD **waves** by performing 3D MHD numerical simulations. We have focused on the case of standing (global) kink **waves** in coronal flux tubes (with radial density structuring) and considered the signatures for both imaging and spectroscopic devices. We have further taken two different coronal models, with and without a temperature variation in the perpendicular cross-section, and considered 2 spectral lines, the ‘core’ and ‘boundary’ lines, catching more the dynamics of the loop core and boundary, respectively. Modulation of intensity, Doppler velocities, line width and loop width are obtained, mainly due to the combination of the Kelvin-Helmholtz instability and resonant absorption, and to a minor degree to the inertia and fluting modes produced by the initial kink mode. It is important to realise therefore that non-linear effects such as those obtained in our model can lead to line widths and loop widths modulation that would be usually associated with sausage modes (Antolin & Van Doorsselaere 2013). Long, non-periodic trends include dimming/enhancement and loop width thinning/broadening in the core/boundary line, and pre- to post KHI positive jumps of intensity, Doppler and line widths. In Table 1 we summarise the phase relations and main features of the observable quantities, combined with previous findings (Antolin et al. 2015, 2016). Despite the relatively simple model we can see that several factors combine to create a very complex set of observable features. This illustrates how hard it must be to disentangle all physical mechanisms in real observations. Our main results are the following:

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The results obtained in this paper can have serious im- plication on the interpretation of observational data obtained in space missions. Alfvén **waves** propagating in the solar wind plasma have been observed during space missions. Since the Alfvén speed is much smaller than the solar wind speed at distances of the order of or larger than 1 a.u. (astro- nomical unit), the phase speed of Alfvén **waves** in the solar reference frame is approximately equal to the solar wind speed. The speeds of space stations with respect to the solar reference frame are much smaller than the solar wind speed. This means that the phase speed of Alfvén **waves** is approxi- mately equal to the solar wind speed in a space station ref- erence frame, i.e., it is much larger than the Alfvén speed. In accordance with the results obtained in this paper this im- plies that circularly polarized Alfvén **waves** propagating in the solar wind plasma are always convectively unstable in a space station reference frame. Hence, we cannot observe temporal growth of the Alfvén wave instability in space mis- sions.

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Localized LFEW emissions near the crossing of the Phobos- 2 spacecraft with orbit of the Martian moon Phobos (so- called Phobos events). It is suggested that these emissions give evidence of a heavy ion source (gas torus) along the Phobos orbit. 2.2 Strong deceleration of the solar wind upstream of the Martian bow shock. Simultaneous obser- vation of proton cyclotron **waves** are seen as hints of their role in momentum coupling between the solar wind and the exospheric plasma. 2.3 Proton cyclotron **waves** during the late stages of the AMPTE Ba and Li releases. The observed spectral peaks near the proton cyclotron frequency seem to be a convincing observation of LFEW excitation by minor heavy ion beam-plasma interaction. In Section 3 we derive the dispersion relation of low-frequency electromagnetic **waves** in a cold plasma composed of protons and heavy, unmagnetized ions moving relative to the protons regardless of the magnetic field orientation. Results of the dispersion analysis are shown. Section 4 contains a discussion with a outlook to future work which requires a kinetic treatment of the considered instability.

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This distribution describes a homogeneous, isotropic plasma species in thermal equilibrium. Using the Maxwellian as his f^ , Landau found that Yj < 0, and so E(k,t) decays as t tends to infinity, since a negative exponential factor is included in it. Any other plasma parameter that can be derived from f^ decays in a similar manner. Now Yj > 0 for any j would imply unlimited wave growth, or instability. Thus **waves** propagating

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smallest possible periodic cell, we have found that such indeterminacy can be avoided. The method we have described in this paper for analysing certain types of symmetry- breaking **instabilities** bifurcating from a non-trivial basic state is based entirely on the observed spatial symmetries of these patterns. However, information on spatial symme- tries of the new pattern alone may not be su!cient for our approach to be applicable in some problems. For example, consider a bifurcation problem dened on a spherical domain. Suppose a basic state with O(3) symmetry loses stability and the observed bifurcating solutions are axisymmetric, then the isotropy subgroup of the bifurcating modes is given by O(2). If the eigenfunctions are expanded in spherical harmonics, it is known that O(2) is a maximal isotropy subgroup of O(3) for all even values of the

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linear eigenvalue formulation, solving for the frequency ω (the imaginary part of the eigenvalue) as a function of wavenumber k, and then plotting ω/k and dω/dk. Both quantities vary considerably with k, but in the range k ⇡ 2 − 6, where fig. 4 indicates that the power is concentrated, the phase velocity is around 0.04, and the group velocity around 0.02. That these numbers are close to the slopes extracted from fig. 2 suggests that even in the fully nonlinear regime in figs. 2 and 3, there are still aspects of the solutions that are not so different from a simple linear analysis of these **waves**. See also R¨ udiger et al (2006), where linear wave speeds like these are compared with the experimental results.

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consequences. First, near integer multiples of h/2, where sin(κz) ≈ 0, the HMRI cannot exist at all, since it relies crucially on both axial and azimuthal components of the imposed field being non-zero. Second, in between these nodes, where the axial component has opposite signs, the **waves** will necessarily also travel in opposite directions, because it is the relative sign of the axial and azimuthal components that determines the up/down symmetry breaking (Knobloch 1996). We see therefore that even though the cylinders are infinite in extent, any **waves** that develop cannot arise from infinity, but inevitably have to emerge at a particular location, travel a certain distance, and then fade away again. That is, these **waves** correspond to an absolute instability, at least as far as global eigenmodes are concerned.

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Multiple free energy sources for **instabilities** coexist in magnetized plasmas with density gradient and velocity shear. Density and temperature gradients destabilize drift wave type **instabilities** such as resistive drift wave, and ion temperature gradient mode [1]. Inhomogeneous flows such as poloidal mean sheared flow, zonal flow and toroidal rotations are driven intrinsically [2–5] and/or externally [6–8]. Strong inhomogeneity of the perpendicular flow can be a free energy source of the Kelvin-Helmholtz (KH) in- stability [9–12] and interchange mode [13, 14]. When the parallel flow shear becomes strong, the parallel compres- sion for ion acoustic **waves** can become negative to drive KH-type instability. This instability is called the D’Angelo mode [15, 16], which has been observed in basic plasma experiments [17]. Turbulence simulation and theory sug- gest the importance of the D’Angelo mode at scrape o ﬀ layers and spherical tokamaks [18, 19]. The coexistence of the D’Angelo mode and drift wave has been observed [17]. On one hand, the drift wave induces the particle transport that reduces the density gradient, and simultaneously drive the momentum transport to form the parallel flow [20]. On the other hand, the D’Angelo mode relaxes the parallel flow gradient by momentum transport, and steepens the density gradient via the particle pinch eﬀect [16]. Thus, a direct cross-interference between the particle and momen- tum transport occurs in a system where **instabilities** due to

We now present a summary of formulas for the growth rates of stimulated Raman (SR) and stimulated Brillouin (SB) scattering **instabilities**, as well as of modulational **instabilities** of a constant amplitude pump that is scattered off a quantum electron plasma wave, a quantum ion mode, and a spectrum of nonresonant electron and ion density perturbations. For three- wave decay interactions, one assumes that D − = 0 and D + = 0. Thus, one ignores D + from Eqs. (12) and (13). Letting = K · V g − δ + iγ R,B , and = L + iγ R and = r ( I ) +