Errores psicológicos de los juicios:

The solar wind is a turbulent flow, however, even the almost fluctuation-free steady solar wind flow would be able to drive waves at the magnetopause through shear flow instabilities. Among these, the Kelvin–Helmholtz instability (KHI) is the most widely studied (e.g. Pu and Kivelson 1983; Walker 2005) and commonly accepted mechanism (see, e.g. Mazur and Chuiko 2015, and references therein, for the recent status of this research). The KHI can develop at a boundary between two counter-streaming MHD media, such as the magnetopause, by amplifying small scale fluctuations into large am- plitude waves and vortices, as schematically illustrated in Figure 34. Speed fluctuations at the perturbed surface cause pressure fluctuations according to Bernoulli’s theorem. Where the pressure has a minimum, fluid elements experience a net force opposite to the pressure gradient. In an incompressible fluid this happens where the flow speed is higher than the bulk speed of the medium, i.e. at the largest displacements. The waves grow due to the instability. In the compressible case, density variations can reduce, or in extreme cases even quench the instability (Walker 2005). In a magnetised plasma, the magnetic field component parallel to the relative shear velocity of the media also has an important stabilising effect. This is because perturbations due to the instability that acts towards bending field lines are counteracted by the force of magnetic tension. At the magnetospheric boundary KH waves manifest themselves in the fluctuations of the plasma (density, velocity, temperature) and magnetic field, typically with a pe- riod of the order of minutes. Theory of KHI has developed through several stages since it was first proposed. In the first models of the KHI, the instability was considered

Fig. 34 Schematic illustration of Kelvin–Helmholtz instability onset at the magnetosphere flanks. The excited surface modes are coupled with Alfv´en field line oscillations.

an infinitely thin tangential discontinuity separating two incompressible fluids. Pu and Kivelson (1983) were the first to study the KHI in a compressible MHD plasma and model the phenomenon at the terrestrial magnetopause. They identified two compress- ible modes: the quasi-slow mode and the quasi-fast mode, both coupled to evanescent waves on both sides of the magnetopause. The most important effect introduced by the compressibility is the appearance of a second critical flow speed, above which the instability quenched.

A boundary layer of finite width was first investigated by Lee et al. (1981), and the idea was further developed by Walker (1981) and Miura and Pritchett (1982) by including compressibility. In all cases two surface modes were identified, one at each side of the boundary layer. The highest growing rate was found to be achieved atkd≈1, wherekis the wave number anddis the thickness of the boundary layer. AsVph=ω/k,

it also means that for a given thicknessdand phase speedVpha quasi-monochromatic

KH wave grows.

All of the above theoretical results were achieved based on models valid only in the linear MHD regime. These models can explain how and where the instability may appear, but cannot describe the fully developed, nonlinear phase of the instability. In particular, development of KHI can increase the thickness of the shear flow interface via generation of the effective viscosity (e.g. Mishin 2005). Global MHD models including the nonlinear development of the KHI, based upon full-MHD numerical modelling, appeared almost a decade ago. Claudepierre et al. (2008) was successful in reproducing two KH modes propagating tailward along the outer and inner edges of the low latitude boundary layer (LLBL) for a southward IMF. These two modes were found to occur forkd= 0.5–1.0 and to have different phase velocities and wavelengths, but oscillating at the same frequency. Larger shear flow velocities were found to excite KH waves of higher frequencies. Merkin et al. (2013) found that the KHI has a 3D nature: the magnetopause is perturbed not only in the equatorial plane, but also in the noon- midnight meridional plane. Surface mode perturbations were found to couple to body modes past the terminator plane. They also found field-aligned currents along closed field lines connecting the shear layer and the ionosphere. Their finding that magnetic

and plasma pressures are spatially decorrelated is attributed to nonlinear effects. They identified two regions where the waves grow: one closer to the subsolar point (the point on the Earth where the Sun is perceived to be directly in zenith), and another at the flanks. According to this model, the KHI is initiated around 30◦ off the noon meridian (Guo et al. 2010; Li et al. 2012). The second unstable region starts prior to the terminator and lasts to about −5RE (meaning that this location is in the anti-

sunward direction). The plasma closer the subsolar point is less compressible and hence here the growth rate is higher.

Early observations supporting the KHI origin of high latitude Pc4–5 waves were the reversal of wave polarity at noon and amplitude rapidly decaying with increasing distance from the magnetopause. It was found that these waves propagate away from noon with a phase speed that is independent of frequency. These observations are in agreement with theoretical predictions, i.e. with a tailward propagating surface wave coupled to an evanescent wave in the magnetosphere. In the magnetosphere compressive waves of KH origin are coupled to local Alfv´en field line resonances. Rolled up KH vortices were identified more frequently on the post noon, dusk-flank (Taylor et al. 2012) indicating that the KHI develops rapidly into the non-linear phase. The observed KH wavelengths were found to be in general larger than the wavelength predicted by linear theory (Kivelson and Chen 1995).

Recent multi-spacecraft missions Cluster, THEMIS have offered unique opportu- nities to observe the KHI in action. Using the observations of the Cluster satellites, Hasegawa et al. (2004)) demonstrated the nonlinear growth of KHI and associated vor- tices along the dusk-flank during a time period when the IMF was northward. Nishino et al. (2007) observed KHI waves on both flanks simultaneously with Cluster and Geo- tail spacecraft.

Beside the KHI, the Miles–Phillips instability has also been proposed as a candidate source of surface wave activity on the magnetopause (e.g. Kurazhkovskaya and Klain 2012). The Miles–Phillips instability is also a shear flow instability that is thought to grow waves at any flow speed. Thus for the shear flow speeds lower or higher than the KH critical flow speeds the Miles–Phillips instability could be dominant (Lushnikov 1998). The Miles–Phillips instability requires a boundary layer with a continuously changing flow speed. The growth rate is proportional to the curvature of the flow speed profile at a distanced from the boundary, where the flow speed equals to the phase velocity of the surface wave. The prerequisite of both the KH and Miles–Phillips instabilities is the existence of some initial seed fluctuation. Perturbations of any origin can serve as seed fluctuations for these shear flow instabilities. However, the Miles– Phillips mechanism has not yet been investigated in depth for conditions (compressive MHD plasma, finite boundary layer thickness, etc.) typical at the magnetopause.

In the solar corona, development of KHI-driven plasma vortices was observed in the interface between the dimming region and the surrounding corona (Ofman and Thompson 2011), and off-limb at a flank of a rising CME plasmoid (Foullon et al. 2011). In the former study, the vortices were found to range in size from a few to several Mm, and traveled along the interface at the apparent speed of 6–14 km s−1.

2.5D numerical MHD modelingsupported the interpretation of the observations in terms of KHI. In the latter work, a street of four Alfv´enic vortices was seen to develop in about 30 s. It moved along the flank at the speed of about 400 km s−1 that was approximately two times lower than the projected speed of the plasmoid. The typical size of a vortex was about 10 Mm. The increment was estimated as 0.05–0.03 s−1.

The magnetospheric KHI model by Miura and Pritchett (1982) was employed for the interpretation of the observed results.

In the corona, the attention has also been paid to shear-flow instabilities caused by the effect of negative energy waves (Joarder et al. 1997; Holzwarth et al. 2007). In this regime, the presence of a mechanism for draining the energy from magnetoacoustic waves, e.g. dissipation by finite viscosity, thermal conductivity or resistivity, transfor- mation to another mode, or leakage from the system, leads to their amplification. For these instabilities, the threshold value of the flow shear is lower than the KHI thresh- old. A more detailed study of this phenomenon and its manifestation in the corona and magnetosphere is required.