Pis'ma v ZhETF, vol.86, iss. 10, pp.725-730
© 2007 November 25
Nonlinear theory of mirror instability near threshold
E. A. Kuznetsov+*, T.Passotv, P.L.Sulemv + P.N. Lebedev Physical Institute RAS, 119991 Moscow, Russia *L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia vCNRS, Observatoire de la Cote d'Azur, 06304 Nice Cedex 4, France Submitted 18 October 2007
An asymptotic model based on a reductive perturbative expansion of the drift kinetic and the Maxwell equations is used to demonstrate that, near the instability threshold, the nonlinear dynamics of mirror modes in a magnetized plasma with anisotropic ion temperatures involves a subcritical bifurcation, leading to the formation of small-scale structures with amplitudes comparable with the ambient magnetic field.
PACS: 52.25.Xz, 52.35.Py, 94.05.^a, 94.30.Cj
1. In regions of planetary magnetosheaths close to the magnetopause and in the solar wind as well, magnetic structures with a cigar form elongated along the direction of the ambient magnetic field are commonly observed (see e.g. [1, 2]). According to recent observations [3], more than 60% of such structures are magnetic depressions (holes) associated with maxima of the density and pressure fluctuations. A typical depth of magnetic holes is about 20% of the mean magnetic field value and can sometimes achieve 50%. The characteristic width of such structures is of the order of a few ion Larmor radii, and they display an aspect ratio of about 7-10. The origin of these structures is not fully understood but they are often viewed as associated with the nonlinear development of the mirror instability, a kinetic instability first predicted by Vedenov and Sagdeev [4] in 1957.
The linear mirror instability has been extensively studied both analytically (see, e.g. [5, 6]), and by means of particle-in-cell (PIC) simulations [7]. This instability develops in a collisionless plasma, when the anisotropy of the ion temperature exceeds the threshold,
Ti/Ty -1 = 021. (l)
Here j3± = 8irp±/B2 (similarly, /3y = 8irp^/B2), where p_l and p|| are perpendicular and parallel plasma pressures respectively. Such conditions can be met under the effect of the plasma compression in front of the magnetopause [8]. As shown in [9, 10, 5], the instability is arrested at large k due to finite ion Larmor radius (FLR) effects.
Mirror structures are also observed when the plasma is linearly stable [11, 12], which may be viewed as the signature of a bistability regime. This property was also established in the framework of anisotropic magnetohy-
drodynamics, using an energetic argument [13]. The aim of the present paper is to demonstrate that the bistability of mirror structures results from a subcritical bifurcation. As well known, for such a bifurcation, non trivial stationary states below threshold are linearly unstable, while above threshold, initially small-amplitude solutions undergo a sharp transition to a large-amplitude state, associated with a blowup behavior within an asymptotic theory. After reviewing the nonlinear theory of the mirror instability, briefly announced in [14], we demonstrate the subcritical character of the bifurcation in three steps: absence of small-amplitude stationary solution above threshold, existence of an unstable branch of solutions below threshold and blowup behavior for the initial value problem above threshold.
The approach is based on a mixed hydrodynamic-kinetic description, assuming a weak nonlinear regime near threshold. Close to threshold, the unstable modes have wavevectors almost perpendicular to the ambient magnetic field B (kz/k± -C 1) with k±pi -C 1, so that the perturbations can be described using a long-wave approximation. The latter allows one to apply the drift kinetic equation (see, e.g., [15, 16]) to estimate the main nonlinear effects that correspond to a local shift of the instability threshold (1). All other nonlinearities connected, for example, with ion inertia are smaller. As the result, we obtain an asymptotic equation for the parallel magnetic field fluctuation, Bz that displays a quadratic nonlinearity. We show that this equation belongs to the generalized gradient type with a free energy that decreases in time, associated with the development of magnetic holes. This process has a self-similar blowup behavior. This means that possible stabilization of the instability can only take place for amplitudes of order one, a regime that is beyond the framework of the present
asymptotics. The present approach contrasts with the quasi-linear theory [17] that also assumes vicinity of the instability threshold but, being based on a random phase approximation, cannot predict the appearance of coherent structures. Phenomenological models based on the cooling of trapped particles were proposed to interpret the existence of deep magnetic holes [18, 19]. These models do not however address the initial value problem in the mirror unstable regime.
2. Consider for the sake of simplicity, a plasma with cold electrons. To describe the mirror instability in the long-wave limit it is enough to use the drift kinetic equation for ions ignoring parallel electric field and transverse electric drift:
0/ dt
+ «lib • V/ - ßb
= 0.
OVII
(2)
In this approximation ions move along the magnetic field (b = B¡B) due to the magnetic force jih ■ VB where fi = v\/2B is the adiabatic invariant which plays the role of a parameter in equation (2). Both pressures py and p_l are given by
'I
PH = vi 13 / vfifdßdvndip = to / v?,fd3v
7
p_l = mB / ßfdßdvndip
-to
/ vlfd3 V.
(3)
(4)
Equation (2) with relations (3) and (4) are supplemented with the equation expressing the balance of forces in a plane transverse to the local magnetic field
r in ( \iB-VBi
= 0.
(5)
Here, consistently with the long-wave approximation, we neglect both the plasma inertia and the non-gyrotropic contributions to the pressure tensor. Furthermore, nik = Sik — bibk denotes the projection operator in the plane transverse to the local magnetic field. In this equation, the first term describes the action of the magnetic and perpendicular pressures, the second term being responsible for magnetic lines elasticity.
The equation governing the mirror dynamics is then obtained perturbatively by expanding Eqs. (2) and (5). In this approach, the ion pressure tensor elements are computed from the system (2), (5), near a bi-Maxwellian equilibrium state characterized by temperatures T± and T|| and a constant ambient magnetic field B0 taken along the z-direction.
From Eq. (5) linearized about the background field B0 by writing B = B0 + B (B0 > B) with B ~ ~ c ' ,kr, we have
PÏ>
BqBz in
ß± - ß\\ \ B0BZ in
(6)
Here kz and k± are the projections of the wave vector k, and p^ is calculated from the linearized drift kinetic equation (2):
d/M d/W dBz
1 v\\—--n—---— = 0.
dt
dz
dz dvu
In Fourier space, this equation has the solution
ßBz
df( 0)
w — kzv\\ Or
(7)
The mirror instability is such that w/kz -C vth^ = = ^/2T||/to. This means that the ions contributing to the resonance w — kzv\\ = 0, correspond to the maximum of the ion distribution function.
After substituting (7) into the first order term for perpendicular pressure (4) and performing integration, we get
PÏ> = ß±
ß\\
BpBz _ iyfiw ß\B0Bz in \kz\vm ßn in
(8)
The first term in (8) is due to the difference between perpendicular and parallel pressures, while the second one accounts for the Landau pole.
Equation (8) together with (6) yield the growth rate for the mirror instability in the drift approximation where FLR corrections are neglected [4]
7 = \kz\vthW
ß\\
V^ßi
ß\
1
ßl
k2,ß±
X
(9)
where % = 1 + (0± — /3||)/2. The instability takes place when /3j_//3|| — 1 > (321 and, near threshold, develops in quasi-perpendicular directions, making the parallel magnetic perturbation dominant.
As shown in Refs. [9, 10, 5], when the FLR corrections are relevant, the growth rate is modified into
7 = \kz\vthl\
ß\\X
V^ßl
k2 3 * " £ " ^
fc
(10)
where e = ~ 1 — /^J1) and the ion Lar-
mor radius pi = vth±/uCi is defined with the transverse thermal velocity vth± = \/2Tj_/to and the ion gyrofre-quency wCi = eB0/mc. This growth rate can be recovered by expanding the general expression given in
Nonlinear theory of mirror instability.
727
[5], in the limit of small transverse wavenumbers. It can also be obtained directly from the Vlasov-Maxwell (VM) equations in a long-wave limit which retains non gyrotropic contributions [20]. It is important to note that the expression (10) for 7 is consistent with the applicability condition w/kz -C i.e. when the supercritical parameter |e| -C 1. In this case the instability saturation happens at small k± oc y/e due to FLR and for almost perpendicular direction in a small cone of angles, kz/k± oc y/e. As a result, the growth rate 7 oc e2, so that, when defining new stretched variables by
kz=eKzpi1(2/V3)X1/2,
k± = (2/V3 )y/iK±pr1x1/\
7 = r(2/V3)e20 (Xß\\/ß±) 3/2,
it takes the form
T=\Kz\(l^K2/K2±^K2±).
(11)
(12)
Hence it is seen that, in the (K± — 0) plane (0 = = Kz/K1), the instability takes place inside the unit circle: 02 - K2 < 1. The maximum of F is obtained for K_l = 1/2, 0 = ± 1/2 and is equal to Fmax = 1/8. Outside the circle the growth rate becomes negative (in agreement with [10]).
3. As it follows from (6), in the linear regime, near the instability threshold, the fluctuations of perpendicular and magnetic pressures almost compensate each other (compare with (9)). Therefore, in the nonlinear stage of this instability, we can expect that the main nonlinear contributions come from the second order corrections to the total (perpendicular plus magnetic) pres-
sure, i.e.
PÏ>
BnB
0 -»z
in
(2) ■P±
B2 d2 B0BZ ^
8tt XA
in
This result can be obtained rigorously by means of a multi-scale expansion based on the linear theory scal-ings (11). For this purpose, we introduce a slow time T a
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