Marco Conceptual

Small-scale compressive (the high-mAlfv´en waves in a hot plasma with curved mag- netic field lines) ULF waves are subject to intensive interaction with plasma particles, and hence demand a special treatment beyond standard MHD.

12.4.1 Slow modes

Some compressive ULF pulsations could be associated with slow magnetoacoustic waves (Leonovich et al. 2006; Du et al. 2011), similar to those that are abundant in the solar corona (Sec. 6.3). However, the diamagnetic property of these waves cannot be considered as confirmative observational evidence. While the slow mode is a natural solution to the MHD equations, the very applicability of the MHD approximation to a collisionless plasma is questionable. Indeed, the condition for the slow mode is the transfer of momentum along field lines. In dense plasmas (like the main part of the low solar corona), momentum transfer is provided by particle collisions. Since the magnetospheric plasma is almost collisionless, this mechanism is ineffective.

Let us consider how the slow mode is derived in a collisionless plasma. To obtain the slow mode dispersion relation, ions with concentrationnimust be treated as particles

other hand, electrons must be treated as particles with thermal velocities much higher than the wave phase speed. Additionally, let us assume that the plasma also contains cold electronsnc. If the cold electron number density is much smaller than the hot

electron number density then we have thatncnh. The electromagnetic field of the

compressive wave under such conditions can then be described by two variables: the longitudinal magnetic field of the wavebk and the “parallel potential”ϕk related to

the parallel electric field of the wave. In these variables, the quasi-neutrality condition and the perpendicular Ampere’s law can be written as

ϕ_ke k 2 k ω2 − 1 u2 ! +bk B∆= 0, Bb_k 4π +eϕknh= 0, (83) with 1 u2 = nc me + ni mi −1 neh meV_k2_h , ∆= nc me + ni mi −1 (ni−nh),

whereV_khis the longitudinal velocity of hot electrons. Since Landau damping effects

are not relevant here, all imaginary terms are assumed to be small, and can be omitted in Eq. (83).

Further, let us consider the particular case when the cold electron number is much smaller than the hot electron number

ncnh

me

mi

. (84)

Then, from Eq. (83) we have after some algebra:

ω2=k_k2 V

2

sVA2

Vs2+V_A2

, (85)

Eq. (85) coincides with the MHD slow magnetoacoustic dispersion relation.

It is evident from these calculations that contrary to the basic assumptions of MHD theory the parallel electric field of the wave plays a key role in formation of the slow mode in a collisionless plasma, providing the key momentum transfer mechanism. In fact, the slow mode in kinetics is an electrostatic mode associated withekand modified

by the compression of the magnetic field. Magnetic field non-uniformity complicates the situation due to bounce effects (Le Contel et al. 2000). In this case, the electrostatic mode dispersion properties, known as the Kadomtsev–Pogutse mode, have nothing in common with given by Eq. (85) (Kadomtsev and Pogutse 1967).

The condition given by Eq. (84) is not likely to be satisfied in space plasmas, so let us consider the opposite case:

ncnh

me

mi

, ncnh. (86)

In this case, the expressions for∆andu2become∆=nime/nc andu2= (nc/nh)Vk2h

and the set of equations (83) can be combined into a single equation

k_k2 ω2 − 1 u2 ! −4πme B2 ninh nc = 0. (87)

The solution to this equation is a mode with a dispersion relation similar to Eq. (85), but with a different phase speed,u:

ω2=k_k2u2, (88)

where u2/Vs2 =ncmi/nhme 1. Consequently, its frequency is very high and can

overstep the limits of the ULF range. When it is said that the cold electron fraction shorts out the parallel electric field it is usually meant that the frequency of the elec- trostatic mode with the finiteE_kis too high for the ULF range in the magnetospheric context.

12.4.2 Drift compressive modes.

A nontrivial solution to Eq. (83) appears when the plasma inhomogeneity is taken into account. This oscillation branch is called thedrift compressive mode (Ng et al. 1984; Crabtree and Chen 2004; Porazik and Lin 2011). Its characteristic frequency is of the same order as theplasma diamagnetic drift frequency,

ω∗= ky ωc T n0 n − 3 2 T0 T , (89)

where n is the plasma concentration, and T is the temperature. Probably, the drift compressive mode is the most common high-m compressive mode, since its existence requires only finite β and plasma inhomogeneity. The generation of the drift com- pressive mode can occur due to wave–particle interaction in the presence of a sharp spatial gradient of particle energy (Klimushkin and Mager 2011) and/or an inverted (bump-on-tail) distribution.

The spatial structure of drift compressive modes in a 2D inhomogeneous plasma with parallel and radial non-uniformities was studied by Mager et al. (2013). It was shown that when the wave frequencyωis much less than the proton bounce frequency

ωb, the mode is narrowly localised along the field line near the equator, being concen-

trated in the region with highβ. As long as the wave frequencyωis approaching the drift compressive mode eigenfrequencyω∗, the compressive resonance appears, when

the field-aligned magnetic field component of the wave field becomes very large. Fur- thermore, at the compressive resonance the radial component of the wave vector tends to infinity,kx(x) → ∞, which means that the wave has a large azimuthal magnetic

component. This resonance can be considered as a compressive analogue of the MHD Alfv´en resonance. However, the radial structure of the drift compressive modes has not been studied in sufficient detail to date.

The drift compressive modes are coupled with the Alfv´en modes because of field line curvature and equilibrium current (Klimushkin and Mager 2011). As was shown by Klimushkin et al. (2012), these coupled modes comprise three oscillation branches. The first of them propagates in the azimuthal direction opposite to the diamagnetic drift velocity of ions. It is simply an Alfv´en mode weakly modified by the coupling. The two other branches are the Alfv´en and drift compressive modes with the azimuthal phase velocity directed parallel to the ion diamagnetic drift velocity. These modes converge to each other with increase in the wave vector azimuthal componentkyand the azimuthal

current J, and merge at some critical values of ky and J. If ky and J exceed these

critical values, these two branches form two other modes, one of which is unstable, and the other is damped. The unstable mode can be called the drift ballooning coupling

mode. Opposite to the ordinary ballooning instability in MHD, the drift ballooning instability is not aperiodic, since there is a real part of the oscillation frequency of the order of the drift frequency. Then, only the mode with the same direction of the azimuthal phase speed as the ion diamagnetic drift velocity can be unstable, and the instability threshold depends on the value ofky.

12.4.3 Drift mirror modes.

Compressive ULF waves were also identified asdrift–mirror modes(Woch et al. 1990; Rae et al. 2007; Sibeck et al. 2012). These modes can be considered as the drift com- pressive modes in an anisotropic plasma where the particle transverse thermal speed is higher than the parallel thermal speed. This drift-mirror mode mode is unstable when the parameterτ is negative (Hasegawa 1969),

τ= 1−β⊥ β⊥ β_k −1 <0. (90)

However, the theory of drift-mirror modes is far from complete. For example, the pioneering works of Cheng and Lin (1987); Cheng and Qian (1994) on the effect of parallel inhomogeneity need to be further developed. Even if near the equator the mirror instability condition is satisfied, τ < 0, it becomes close to τ ' 1 near the ionosphere because theβparameter quickly grows towards the ionosphere due to field line convergence. Thus, somewhere on field lines theτ value must pass through zero. The mode structure and the stability in this case have not been studied to date. Other open questions include

– A realistic kinetic theory of the spatial structure of compressive modes, such as drift–compressive and drift–mirror modes, taking into account their coupling with Alfv´en modes must be developed.

– Probably, these waves emerge in the magnetosphere owing to some impulsive pro- cesses in space plasmas. Any theory of such process is still lacking.