= 2^QnkBTi±/B2 is the perpendicular ion ¡, and the electrons do not play any role in the instability. The growth rate of this mode (Hasegawa, 1969) is proportional to the ratio (k||/ki)2 where (k^/k±)2 ^ 1. The k-vector is thus nearly perpendicular to the magnetic field and the mode has a small growth rate. However, because it is practically non-propagating and is therefore convected with the plasma flow, it has plenty of time to grow and so can reach large amplitudes which ultimately cannot be described by simple linear theory. In the limit of Te — 0, theory predicts that the cold electrons will wipe out any parallel electric field and therefore that k| should be zero and the mode cannot exist. However, a small but finite temperature of the electrons will allow for the mode to exist in slightly oblique direction (Pantellini and Schwartz, 1995; Pokhotelov et al., 2001, 2003, 2004).

The ordinary ion-mirror mode grows fastest Pokhotelov et al. (2004) at perpendicular wavelengths comparable to the ion gyroradius, k±pi — 1. The above threshold for the short wavelength mirror mode is higher by a factor of 2 than in the very long wavelength case k±pi ^ 1. Thus, depending on the anisotropy, the fastest growing waves will be those which have a wavelength just long enough for the anisotropy to exceed the instability threshold. The inclination of the mode with respect to the magnetic field implies that the bottles are no longer symmetric around the field direction. Field aligned currents should flow within the structure, generating a non-coplanar magnetic field component, which twists the mirror mode magnetic field around the bottle.

The mirror mode is never observed in the state of linear small magnetic field compressions. Magnetic field compression ratios of 30-80% are observed, deep in the nonlinear regime. Since the mode is non-oscillatory, it is unsurprising that a quasilinear approach (Treumann and Baumjohann, 1997) does not explain the observations. That particle trapping occurs has been suggested by Kivelson and Southwood (1996). Such trapping is inferred from lion roar excitation (Treumann et al., 2000) and observation within mirror modes (Baumjohann et al., 1999), as well as by direct electron observation (Chisham et al., 1998). It has been recognised recently (Treumann et al., 2004) that in the nonlinear marginally stable state the mirror modes should evolve into three-dimensional cylindrical structures with zero parallel wave number extended along the ambient magnetic field. Any remaining inclination with respect to the field then indicates that the mode is still in evolution.

3.4.2 Nonlinear static bottle model of mirror modes

Constantinescu (2002) used the marginal mirror equilibrium condition to consider the stationary equilibrium state of a mirror bottle. Pressure equilibrium in the plasma reference frame is written v(+£) + v

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