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- About
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**Regular paper**
26 Jul 2018

**Regular paper** | 26 Jul 2018

Mirror instability

^{1}International Space Science Institute, Bern, Switzerland^{2}Space Research Institute, Austrian Academy of Sciences, Graz, Austria

Abstract

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We examine the physics of the magnetic mirror mode in its final state of
saturation, the thermodynamic equilibrium, to demonstrate that the mirror
mode is the analogue of a superconducting effect in a classical
anisotropic-pressure space plasma. Two different spatial scales are
identified which control the behaviour of its evolution. These are the ion
inertial scale *λ*_{im}(*τ*) based on the excess density
*N*_{m}(*τ*) generated in the mirror mode, and the Debye scale
*λ*_{D}(*τ*). The Debye length plays the role of the correlation length
in superconductivity. Their dependence on the temperature ratio
$\mathit{\tau}={T}_{\Vert}/{T}_{\u27c2}<\mathrm{1}$ is given, with *T*_{⟂} the reference temperature
at the critical magnetic field. The mirror-mode equilibrium structure under
saturation is determined by the Landau–Ginzburg ratio
${\mathit{\kappa}}_{D}={\mathit{\lambda}}_{\mathrm{im}}/{\mathit{\lambda}}_{D}$, or
${\mathit{\kappa}}_{\mathit{\rho}}={\mathit{\lambda}}_{\mathrm{im}}/\mathit{\rho}$, depending on whether the Debye length
or the thermal-ion gyroradius *ρ* – or possibly also an undefined
turbulent correlation length ℓ_{turb} – serve as correlation
lengths. Since in all space plasmas *κ*_{D}≫1, plasmas with *λ*_{D}
as the relevant correlation length always behave like type II
superconductors, naturally giving rise to chains of local depletions of the
magnetic field of the kind observed in the mirror mode. In this way they
would provide the plasma with a short-scale magnetic bubble texture. The
problem becomes more subtle when *ρ* is taken as correlation length. In
this case the evolution of mirror modes is more restricted. Their existence
as chains or trains of larger-scale mirror bubbles implies that another
threshold, ${V}_{\mathrm{A}}>{\mathit{\upsilon}}_{\u27c2\mathrm{th}}$, is exceeded. Finally,
in case the correlation length ℓ_{turb} instead results from
low-frequency magnetic/magnetohydrodynamic turbulence, the observation of
mirror bubbles and the measurement of their spatial scales sets an upper
limit on the turbulent correlation length. This might be important in the
study of magnetic turbulence in plasmas.

How to cite

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top
How to cite.

Treumann, R. A. and Baumjohann, W.: The mirror mode: a “superconducting” space plasma analogue, Ann. Geophys., 36, 1015-1026, https://doi.org/10.5194/angeo-36-1015-2018, 2018.

1 Introduction

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Under special conditions high-temperature collisionless plasmas may develop properties which resemble those of superconductors. This is the case with the mirror mode when the anisotropic pressure gives rise to local depletions of the magnetic field similar to the Meissner effect in metals where it signals the onset of superconductivity (Fetter and Walecka, 1971; Huang, 1987; Kittel, 1963; Lifshitz and Pitaevskii, 1998), i.e. the suppression of friction between the current and the lattice. In collisionless plasmas there is no lattice, the plasma is frictionless, and thus it already is ideally conducting which, however, does not mean that it is superconducting. To be superconducting, additional properties are required. These, as we show below, are given in the saturation state of the mirror mode.

The mirror mode is a non-oscillatory plasma instability (Chandrasekhar, 1961; Gary, 1993; Hasegawa, 1969; Kivelson and Southwood, 1996; Southwood and Kivelson, 1993) which evolves in anisotropic plasmas (Sulem, 2011). It has been argued that it should readily saturate by quasilinear depletion of the temperature anisotropy (Noreen et al., 2017). Observations do not support this conclusion. In fact, we recently argued (Treumann and Baumjohann, 2018a) that the large amplitudes of mirror-mode oscillations observed in the Earth's magnetosheath, magnetotail, and elsewhere, like other planetary magnetosheaths, in the solar wind and generally in the heliosphere (Constantinescu et al., 2003; Czaykowska et al., 1998; Lucek et al., 1999a, b; Tsurutani et al., 1982, 2011; Volwerk et al., 2008; Zhang et al., 2008, 2009, 1998), are a sign of the impotence of quasilinear theory in limiting the growth of the mirror instability. Instead, mirror modes should be subject to weak kinetic turbulence theory (Davidson, 1972; Sagdeev and Galeev, 1969; Tsytovich, 1977; Yoon, 2007, 2018; Yoon and Fang, 2007), which allows them to evolve until they become comparable in amplitude to the ambient magnetic field long before any dissipation can set in.

This is not unreasonable, because all those plasmas where the mirror instability evolves are ideal conductors on the scales of the plasma flow. On the other hand, no weak turbulence theory of the mirror mode is available yet as it is difficult to identify the various modes which interact to destroy quasilinear quenching. The frequent claim that whistlers (lion roars) excited in the trapped electron component would destroy the bulk (global) temperature anisotropy is erroneous, because whistlers (Baumjohann et al., 1999; Maksimovic et al., 2001; Thorne and Tsurutani, 1981; Zhang et al., 1998) grow at the expense of a small component of anisotropic resonant particles only (Kennel and Petschek, 1966). Depletion of the resonant anisotropy by no means affects the bulk temperature anisotropy that is responsible for the evolution of the mirror instability. On the other hand, construction of a weak turbulence theory of the mirror mode poses serious problems. One therefore needs to refer to other means of description of its final saturation state.

Since measurements suggest that the observed mirror modes are about
stationary phenomena which are swept over the spacecraft at high flow speeds
(called Taylor's hypothesis, though, in principle, it just refers to the
Galilei transformation), it seems reasonable to tackle them within a
*thermodynamic* approach, i.e. assuming that in the observed
large-amplitude saturation state they can be described as the
*stationary* state of interaction between the ideally conducting
plasma and magnetic field. This can be most efficiently done when the free
energy of the plasma is known, which, unfortunately, is not the case.
Magnetohydrodynamics does not apply, and the formulation of a free energy in
the kinetic state is not available. For this reason we refer to some
phenomenological approach which is guided by the phenomenological theory of
superconductivity. There we have the similar phenomenon that the magnetic
field is expelled from the medium due to internal quantum interactions, known
as the Meissner effect. This resembles the evolution of the mirror mode,
though in our case the interactions are not in the quantum domain. This is
easily understood when considering the thermal length
${\mathit{\lambda}}_{\mathrm{\hslash}}=\sqrt{\mathrm{2}\mathit{\pi}{\mathrm{\hslash}}^{\mathrm{2}}/{m}_{e}T}$ and comparing it to the shortest
plasma scale, viz. the inter-particle distance ${d}_{N}\sim {N}^{-\frac{\mathrm{1}}{\mathrm{3}}}$.
The former is, for all plasma temperatures *T*, in the atomic range, while
the latter in space plasmas for all densities *N* is at least several orders
of magnitude larger. Plasmas are classical. In their equilibrium state
classical thermodynamics applies to them.

In the following we boldly ask for the *thermodynamic* equilibrium
state of a mirror unstable plasma. For other non-thermodynamical attempts at
modelling the equilibrium configuration of magnetic mirror modes and
application to multi-spacecraft observations, the reader may consult
Constantinescu (2002) and Constantinescu et al. (2003). Such an approach
is rather alien to space physics. It follows the path prescribed in
solid-state physics but restricts itself to the domain of classical
thermodynamics only.

2 Properties of the mirror instability

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The mirror instability evolves whence the magnetic field *B* in a
collisionless magnetized plasma with an internal pressure–temperature
anisotropy ${T}_{\u27c2}>{T}_{\Vert}$, where the subscripts refer to the directions
perpendicular and parallel to the ambient magnetic field, drops below a
critical value

$$\begin{array}{}\text{(1)}& B<{B}_{\mathrm{crit}}\approx \sqrt{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}N{T}_{i\u27c2}}({\mathrm{\Theta}}_{i}+\sqrt{{\displaystyle \frac{{T}_{e\u27c2}}{{T}_{i\u27c2}}}}{\mathrm{\Theta}}_{e}{)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}|\mathrm{sin}\phantom{\rule{0.33em}{0ex}}\mathit{\theta}|,\end{array}$$

where ${\mathrm{\Theta}}_{j}=({T}_{\u27c2}/{T}_{\Vert}-\mathrm{1}{)}_{j}>\mathrm{0}$ is the temperature anisotropy
of species $j=e,i$ (for ions and electrons) and *θ* is the angle of
propagation of the wave with respect to the ambient magnetic field (Treumann and Baumjohann, 2018a). Here any possible temperature anisotropy in the
electron population has been included, but will be dropped below as it seems
(Yoon and López, 2017) that it does not provide any further insight into the
physics of the final state of the mirror mode.

The important observation is that the existence of the mirror mode depends on
the temperature difference ${T}_{\u27c2}-{T}_{\Vert}$ and the critical magnetic field.
Commonly only the temperature anisotropy is reclaimed as being responsible
for the growth of the mirror mode. Though this is true, it also implies the
above condition on the magnetic field. To some degree this resembles the
behaviour of magnetic fields under superconducting conditions. To demonstrate
this, we take *T*_{⟂} as the reference – or critical – temperature. The
critical magnetic field becomes a function of the temperature ratio
$\mathit{\tau}={T}_{\Vert}/{T}_{\perp}$. Once *τ*<1 and *B*<*B*_{crit} the magnetic field
will be pushed out of the plasma to give space to an accumulated plasma
density and also weak diamagnetic surface currents on the boundaries of the
(partially) field-evacuated domain.

The *τ* dependence of the critical magnetic field can be cast into the
form

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{{B}_{\mathrm{crit}}\left({T}_{\Vert}\right)}{{B}_{\mathrm{crit}}^{\mathrm{0}}}}=\left[{\mathit{\tau}}^{-\mathrm{1}}\right(\mathrm{1}-\mathit{\tau}){]}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}=({\displaystyle \frac{{T}_{\u27c2}}{{T}_{\Vert}}}{)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}(\mathrm{1}-{\displaystyle \frac{{T}_{\Vert}}{{T}_{\u27c2}}}{)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}},\end{array}$$

which indeed resembles that in the phenomenological theory of superconductivity. Here

$$\begin{array}{}\text{(3)}& {B}_{\mathrm{crit}}^{\mathrm{0}}=\sqrt{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}N{T}_{i\u27c2}}\left|\mathrm{sin}\phantom{\rule{0.125em}{0ex}}\mathit{\theta}\right|\end{array}$$

and the critical threshold vanishes for *τ*=1 where the range of possible
unstable magnetic field values shrinks to zero; the limits ${T}_{\Vert}=\mathrm{0}$ or
${T}_{\u27c2}=\mathrm{\infty}$ make no physical sense.

Though the effects are similar to superconductivity, the temperature
dependence is different from that of the Meissner effect in metals in their
isotropic low-temperature super-conducting phase. In contrast, in an
anisotropic plasma the effect occurs in the high-temperature phase only while
being absent at low temperatures. Nevertheless, the condition *τ*<1
indicates that the mirror mode, concerning the ratio of parallel to
perpendicular temperatures, is a *low-temperature* effect in the
high-temperature plasma phase. This may suggest that even in metals
high-temperature superconductivity might be achieved more easily for
anisotropic temperatures, a point we will follow elsewhere
(Treumann and Baumjohann, 2018b).

Since the plasma is ideally conducting, any quasi-stationary magnetic field
is subject to the penetration depth, which is the inertial scale
${\mathit{\lambda}}_{\mathrm{im}}=c/{\mathit{\omega}}_{\mathrm{im}}$, with
${\mathit{\omega}}_{\mathrm{im}}^{\mathrm{2}}={e}^{\mathrm{2}}{N}_{\mathrm{m}}/{\mathit{\u03f5}}_{\mathrm{0}}{m}_{i}$ based on the density
*N*_{m} of the plasma component involved in the mirror effect. The
mirror instability is a slow purely growing instability with real frequency
*ω*≈0. At these low frequencies the plasma is quasi-neutral. In
metallic superconductivity this length is the London penetration depth which
refers to electrons as the ions are fixed to the lattice. Here, in the space
plasma, it is rather the ion scale because the dominant mirror effect is
caused by mobile ions in the absence of any crystal lattice. Such a
“magnetic lattice” structure is ultimately provided under conditions
investigated below by the saturated state of the mirror mode, where it
collectively affects the trapped ion component on scales of an internal
correlation length.

3 Free energy

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In the thermodynamic equilibrium state the quantity which describes the
matter in the presence of a magnetic field ** B** is the Landau–Gibbs
free energy density

$$\begin{array}{}\text{(4)}& {G}_{\mathrm{L}}={F}_{\mathrm{L}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}}}\mathit{\delta}\mathit{B}\mathbf{\cdot}\mathit{B},\end{array}$$

where *F*_{L} is the Landau free energy density (Kittel and Kroemer, 1980)
which, unfortunately, is not known. In magnetohydrodynamics it can be
formulated but becomes a messy expression which contains all stationary, i.e.
time-averaged, nonlinear contributions of low-frequency electromagnetic
plasma waves and thermal fluctuations. The total Landau–Gibbs free energy is
the volume integral of this quantity over all space. In thermodynamic
equilibrium this is stationary, and one has

$$\begin{array}{}\text{(5)}& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int {d}^{\mathrm{3}}x\phantom{\rule{0.33em}{0ex}}{G}_{\mathrm{L}}=\mathrm{0}.\end{array}$$

In order to restrict to our case we assume that *F*_{L} in the above
expression, which contains the full dynamics of the plasma matter, can be
expanded with respect to the normalized density *N*_{m}<1 of the
plasma component which participates in the mirror instability:

$$\begin{array}{}\text{(6)}& {F}_{\mathrm{L}}={F}_{\mathrm{0}}+a{N}_{\mathrm{m}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}b{N}_{\mathrm{m}}^{\mathrm{2}}+\mathrm{\dots},\end{array}$$

with *F*_{0} the Helmholtz free energy density, which is independent of
*N*_{m} corresponding to the normal (or mirror stable) state.
Normalization is to the ambient density *N*_{0}, thus attributing the dimension
of energy density to the expansion coefficients *a* and *b*. An expansion
like this one is always possible in the spirit of a perturbation approach as
long as the total density $N/{N}_{\mathrm{0}}=\mathrm{1}+{N}_{\mathrm{m}}$ with
$\left|{N}_{\mathrm{m}}\right|<\mathrm{1}$. It is thus clear that *N*_{m} is not the
total ambient plasma density *N*_{0}, which is itself in pressure equilibrium
with the ambient field *B*_{0} under static conditions expressed by
${N}_{\mathrm{0}}T={B}_{\mathrm{0}}^{\mathrm{2}}/\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}$ under the assumption that no static current *J*_{0}
flows in the medium. Otherwise its Lorentz force
${\mathit{J}}_{\mathrm{0}}\times {\mathit{B}}_{\mathrm{0}}=-T\mathrm{\nabla}{N}_{\mathrm{0}}$ is compensated for by the pressure
gradient force already in the absence of the mirror mode and includes the
magnetic stresses generated by the current. This case includes a stationary
contribution of the free energy *F*_{0} around which the mirror state has
evolved.

Regarding the presence of the mirror mode, we know that it must also be in
balance between the local plasma gradient ∇*N*_{m} of the
fluctuating pressure and the induced magnetic pressure
(*δ*** B**)

$$\begin{array}{}\text{(7)}& \mathrm{\nabla}\times \mathit{\delta}\mathit{B}={\mathit{\mu}}_{\mathrm{0}}\mathit{\delta}\mathit{J},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{and}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathit{\delta}\mathit{B}=\mathrm{\nabla}\times \mathit{A},\end{array}$$

accounting for the vanishing divergence by introducing the fluctuating vector
potential ** A** (where we drop the

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{{\mathit{p}}^{\mathrm{2}}}{\mathrm{2}m}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}m}}|-i\mathit{\alpha}\mathrm{\nabla}-q\mathit{A}{|}^{\mathrm{2}},\end{array}$$

referring to ions of positive charge *q*>0, and the constant *α*
naturally has the dimension of a classical action. (There is a little problem
as to what is meant by the mass *m* in this expression, to which we will
briefly return below.) In this form the momentum acts on a complex
dimensionless “wave function” *ψ*(** x**) whose square

$$\begin{array}{}\text{(9)}& \left|\mathit{\psi}\right(\mathit{x}){|}^{\mathrm{2}}={\mathit{\psi}}^{*}(\mathit{x}\left)\mathit{\psi}\right(\mathit{x})={N}_{\mathrm{m}}\end{array}$$

we below identify with the above-used normalized excess in plasma density known to be present locally in any of the mirror-mode bubbles.

Unlike quantum theory, *ψ*(** x**) is not a single-particle wave
function: it rather applies to a larger compound of trapped particles (ions)
in the mirror modes which behave similarly and are bound together by some
correlation length (a very important parameter, which is to be discussed
later). It enters the expression for the free energy density, thus providing
the units of energy density to the expansion coefficients

Inspecting Eq. (8) we will run into difficulties when assuming *q*=*e*
and *m*=*m*_{i} because with a large number of particles collectively
participating, each contributing a charge *e* and mass *m*, the ratio *q*^{2}∕*m*
will be proportional to the number of particles. In superconductivity this
provides no problem because pairing of electrons tells that mass and charge
just double, which is compensated for in Eq. (8) by *m*→2*m*.
Similarly, in the case of the mirror mode we have for the normalized density
excess ${N}_{\mathrm{m}}=\mathit{\delta}\mathcal{N}/\mathcal{N}\equiv \mathit{\zeta}<\mathrm{1}$, where
𝒩 is the total particle number, and *δ*𝒩 its
excess. We thus identify an effective mass ${m}^{*}\equiv \mathrm{\Delta}{m}_{i}$, where
$\mathrm{\Delta}=\mathrm{1}+\mathit{\zeta}$. Because of the restriction on *ζ*<1 this yields for the
effective mass in mirror modes the preliminary range

$$\begin{array}{}\text{(10)}& {m}_{i}<{m}^{*}<\mathrm{2}{m}_{i},\end{array}$$

which is similar to the mass in metallic superconductivity. However, each
mirror bubble contains a different number *δ*𝒩 of trapped
particles. Hence *ζ*(** x**) becomes a function of space

$$\begin{array}{}\text{(11)}& {m}_{\mathrm{eff}}\equiv \langle {m}^{*}\left(\mathit{x}\right)\rangle =\langle \mathrm{\Delta}\left(\mathit{x}\right)\rangle {m}_{i}.\end{array}$$

Averaging reduces Δ, making the effective mass closer to the lower
bound *m*_{i}, which is to be used below for *m*→*m*_{eff} wherever the mass
appears.

Retaining the quantum action and dividing by the charge *q*, the factor of
the Nabla operator becomes $\mathrm{\hslash}/q={\mathrm{\Phi}}_{\mathrm{0}}e/\mathrm{2}\mathit{\pi}q$. Hence, *α* is
proportional to the number $\mathit{\nu}=\mathrm{\Phi}/{\mathrm{\Phi}}_{\mathrm{0}}$ of elementary flux elements in
the ion-gyro cross section, which in a plasma is a large number due to the
high temperature *T*_{⟂}. This makes *α*≫ℏ.

With these assumptions in mind we can write for the free energy density up to
second order in *N*_{m}:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}F={F}_{\mathrm{0}}+a\phantom{\rule{0.33em}{0ex}}|\mathit{\psi}{|}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}b\phantom{\rule{0.33em}{0ex}}|\mathit{\psi}{|}^{\mathrm{4}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}m}}\left|\right(-i\mathit{\alpha}\mathrm{\nabla}-q\mathit{A})\phantom{\rule{0.33em}{0ex}}\mathit{\psi}{|}^{\mathrm{2}}\\ \text{(12)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+{\displaystyle \frac{\mathit{\delta}\mathit{B}\mathbf{\cdot}{\mathit{B}}_{\mathrm{0}}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}}}.\end{array}$$

Inserted into the Gibbs free energy density, the last term is absorbed by the
Gibbs potential. Applying the Hamiltonian prescription and varying the Gibbs
free energy with respect to ** A** and $\mathit{\psi},{\mathit{\psi}}^{*}$ yields (for arbitrary
variations) an equation for the “wave function”

$$\begin{array}{}\text{(13)}& \left[{\displaystyle \frac{\mathrm{1}}{\mathrm{2}m}}\right(-i\mathit{\alpha}\mathrm{\nabla}-q\mathit{A}{)}^{\mathrm{2}}+\phantom{\rule{0.33em}{0ex}}a\phantom{\rule{0.33em}{0ex}}+\phantom{\rule{0.33em}{0ex}}b\phantom{\rule{0.33em}{0ex}}\left|\mathit{\psi}{|}^{\mathrm{2}}\right]\phantom{\rule{0.33em}{0ex}}\mathit{\psi}=\mathrm{0},\end{array}$$

which is recognized as a nonlinear complex Schrödinger equation. Such equations appear in plasma physics whence waves undergo modulation instability and evolve towards the general family of solitary structures.

It is known that the nonlinear Schrödinger equation can be solved by
inverse scattering methods and, under certain conditions, yields either
single solitons or trains of solitary solutions. To our knowledge, the
nonlinear Schrödinger equation has not yet been derived for the mirror
instability because no slow wave is known which would modulate its amplitude.
Whether this is possible is an open question which we will not follow up
here. Hence the quantity *α* remains undetermined for the mirror mode.
Instead, we chose a phenomenological approach which is suggested by the
similarity of both the mirror-mode effect in ideally conducting plasma and
the above-obtained nonlinear Schrödinger equation to the phenomenological
Landau–Ginzburg theory of metallic superconductivity.

In the thermodynamic equilibrium state the above equation does not describe
the mirror-mode amplitude itself. Rather it describes the evolution of the
“wave function” of the compound of particles trapped in the mirror-mode
magnetic potential ** A** which it modulates. This differs from
superconductivity where we have pairing of particles and escape from
collisions with the lattice and superfluidity of the paired particle
population at low temperatures. In the ideally conducting plasma we have no
collisions, but, under normal conditions, also no pairing and no
superconductivity, though in the presence of some particular plasma waves,
attractive forces between neighbouring electrons can sometimes evolve
(Treumann and Baumjohann, 2014). In superconductivity the pairing implies that the
particles become correlated, an effect which in plasma must also happen
whence the superconducting mirror-mode Meissner effect occurs, but it happens
in a completely different way by correlating large numbers of particles, as
we will exemplify further below.

The wave function *ψ*(** x**) describes only the trapped particle
component which is responsible for the maintenance of the pressure
equilibrium between the magnetic field and plasma. In a bounded region one
must add boundary conditions to the above equation which imply that the
tangential component of the magnetic field is continuous at the boundary and
the normal components of the electric currents vanish at the boundary because
the current has no divergence. The current, normalized to

$$\begin{array}{}\text{(14)}& \mathit{\delta}\mathit{J}={\displaystyle \frac{iq\mathit{\alpha}}{\mathrm{2}m}}({\mathit{\psi}}^{*}\mathrm{\nabla}\mathit{\psi}-\mathit{\psi}\mathrm{\nabla}{\mathit{\psi}}^{*})-{\displaystyle \frac{{q}^{\mathrm{2}}}{m}}|\mathit{\psi}{|}^{\mathrm{2}}\mathit{A},\end{array}$$

which shows that the main modulated contribution to the current is provided
by the last term, the product of the mirror particle density
$|\mathit{\psi}{|}^{\mathrm{2}}={N}_{\mathrm{m}}$ times the vector potential fluctuation ** A**,
which is the mutual interaction term between the density and magnetic fields.
One may note that the vector potential from Maxwell's equations is directly
related to the magnetic flux Φ in the wave flux tube of radius

One also observes that under certain conditions in the last expression for
the current density the two gradient terms of the *ψ* function partially
cancel. Assuming $\mathit{\psi}=\left|\mathit{\psi}\right(\mathit{x}\left)\right|{e}^{-i\mathit{k}\mathbf{\cdot}\mathit{x}}$, the
current term becomes

$$\begin{array}{}\text{(15)}& \mathit{\delta}\mathit{J}={\displaystyle \frac{q\mathit{\alpha}}{m}}\mathit{k}|\mathit{\psi}{|}^{\mathrm{2}}-{\displaystyle \frac{{q}^{\mathrm{2}}}{m}}|\mathit{\psi}{|}^{\mathrm{2}}\mathit{A}.\end{array}$$

The first term is small in the long-wavelength domain *k**α*≪1.
Assuming that this is the case for the mirror mode, which implies that the
first term is important only at the boundaries of the mirror bubbles where it
comes up for the diamagnetic effect of the surface currents, the current is
determined mainly by the last term, which can be written as

$$\begin{array}{}\text{(16)}& \mathit{\delta}\mathit{J}\approx -{\displaystyle \frac{{q}^{\mathrm{2}}{N}_{\mathrm{0}}}{m}}{N}_{\mathrm{m}}\mathit{A}=-{\mathit{\u03f5}}_{\mathrm{0}}{\mathit{\omega}}_{\mathrm{im}}^{\mathrm{2}}\mathit{A}.\end{array}$$

This is to be compared to ${\mathit{\mu}}_{\mathrm{0}}\mathit{\delta}\mathit{J}=-{\mathrm{\nabla}}^{\mathrm{2}}\mathit{A}$, thus yielding the penetration depth

$$\begin{array}{}\text{(17)}& {\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)=c/{\mathit{\omega}}_{\mathrm{im}}\left(\mathit{\tau}\right),\end{array}$$

which is the ion inertial length based on the relevant temperature dependence
of the particle density *N*_{m}(*τ*) for the mirror mode, where we
should keep in mind that the latter is normalized to *N*_{0}. Thus, identifying
the reference temperature as *T*_{⟂}, we recover the connection between
the mirror-mode penetration depth and its dependence on temperature ratio
*τ* from thermodynamic equilibrium theory in the long wavelength limit
with main density *N*_{0} constant on scales larger than the mirror-mode
wavelength.

4 Magnetic penetration scale

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So far we considered only the current. Now we have to relate the above
penetration depth to the plasma, the mirror mode. What we need is the
connection of the mirror mode to the nonlinear Schrödinger equation.
Because treating the nonlinear Schrödinger equation is very difficult even
in two dimensions, this is done in one dimension, assuming for instance that
the cross section of the mirror structures is circular with the relevant
dimension the radius. In the presence of a magnetic wave field ** A**≠0, Eq. (13) under homogeneous or nearly homogeneous conditions,
with the canonical gradient term neglected, has the thermodynamic equilibrium
solution

$$\begin{array}{}\text{(18)}& {N}_{\mathrm{m}}=|\mathit{\psi}{|}^{\mathrm{2}}=-{\displaystyle \frac{a}{b}}-{\displaystyle \frac{{q}^{\mathrm{2}}{N}_{\mathrm{0}}}{\mathrm{2}mb}}{A}^{\mathrm{2}}>\mathrm{0},\end{array}$$

which implies that either *a* or *b* is negative. In addition there is the
trivial solution *ψ*=0 which describes the initial stable state when no
instability evolves. The Helmholtz free energy density in this state is
*F*=*F*_{0}. Equation (12) shows that the thermodynamic equilibrium has
free energy density

$$\begin{array}{}\text{(19)}& F={F}_{\mathrm{0}}-{\displaystyle \frac{{q}^{\mathrm{2}}a{N}_{\mathrm{0}}}{\mathrm{2}mb}}{A}^{\mathrm{2}}-{\displaystyle \frac{{a}^{\mathrm{2}}}{\mathrm{2}b}}={F}_{\mathrm{0}}-{\displaystyle \frac{{q}^{\mathrm{2}}a{N}_{\mathrm{0}}}{\mathrm{2}mb}}{A}^{\mathrm{2}}-{\displaystyle \frac{{B}_{\mathrm{crit}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}}},\end{array}$$

where the last term is provided by the critical magnetic field, which is the
external magnetic field. Thus *b*>0 and *a*<0, and the dependence on
temperature *τ* can be freely attributed to *a*. Comparison with Eq.
(2) then yields

$$\begin{array}{}\text{(20)}& a\left(\mathit{\tau}\right)=-{B}_{\mathrm{crit}}^{\mathrm{0}}\sqrt{{\displaystyle \frac{b}{{\mathit{\mu}}_{\mathrm{0}}}}}{\mathit{\tau}}^{-{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}(\mathrm{1}-\mathit{\tau}{)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}.\end{array}$$

At critical field one still has ** A**=0. Hence the density at critical
field is

$$\begin{array}{}\text{(21)}& {N}_{\mathrm{m}}\left(\mathit{\tau}\right)={\displaystyle \frac{\left|a\right(\mathit{\tau}\left)\right|}{b}}={\displaystyle \frac{{B}_{\mathrm{crit}}^{\mathrm{0}}}{\sqrt{b{\mathit{\mu}}_{\mathrm{0}}}}}{\mathit{\tau}}^{-{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}(\mathrm{1}-\mathit{\tau}{)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}},\end{array}$$

which shows that the distortion of the density vanishes for *τ*=1, as it
should be. This expression can be used in the magnetic penetration depth to
obtain its critical temperature dependence

$$\begin{array}{}\text{(22)}& {\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)=[{\displaystyle \frac{{m}^{\mathrm{2}}b}{{\mathit{\mu}}_{\mathrm{0}}{q}^{\mathrm{4}}({N}_{\mathrm{0}}{B}_{\mathrm{crit}}^{\mathrm{0}}{)}^{\mathrm{2}}}}{\displaystyle \frac{\mathit{\tau}}{(\mathrm{1}-\mathit{\tau})}}{]}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{4}}}}\phantom{\rule{1em}{0ex}}\mathrm{m},\end{array}$$

which suggests that the critical penetration depth vanishes for *τ*=0.
However, *τ*=0 is excluded by the argumentation following Eqs. (2)
and (21), because it would imply infinite trapped densities. In
principle, *τ*≥*τ*_{min} cannot become smaller than a minimum value
which must be determined by other methods referring to measurements of the
maximum density in thermodynamic equilibrium. One should, however, keep in
mind that ${B}_{\mathrm{crit}}^{\mathrm{0}}\left(\mathit{\theta}\right)\propto \left|\mathrm{sin}\mathit{\theta}\right|$ still depends on
the angle *θ* which enters the above expressions.

The last two expressions still contain the undetermined coefficient *b*. This
can be expressed through the minimum value of the anisotropy *τ*_{min} at
maximum critical density *N*_{m}*≲*1 as

$$\begin{array}{}\text{(23)}& b={\displaystyle \frac{({B}_{\mathrm{crit}}^{\mathrm{0}}{)}^{\mathrm{2}}}{{\mathit{\mu}}_{\mathrm{0}}}}{\mathit{\tau}}_{\mathrm{min}}^{-\mathrm{1}}(\mathrm{1}-{\mathit{\tau}}_{\mathrm{min}}).\end{array}$$

(Note that for *N*_{m}>1 the above expansion of the free energy *F*
becomes invalid. It is not expected, however, that the mirror mode will allow
the evolution of sharp density peaks which locally double the density.) With
this expression the inertial length becomes

$$\begin{array}{}\text{(24)}& {\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)}{{\mathit{\lambda}}_{i\mathrm{0}}}}=\left[{\displaystyle \frac{\mathit{\tau}}{{\mathit{\tau}}_{\mathrm{min}}}}\right({\displaystyle \frac{\mathrm{1}-{\mathit{\tau}}_{\mathrm{min}}}{\mathrm{1}-\mathit{\tau}}}){]}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{4}}}}.\end{array}$$

When the mirror mode saturates away from the critical field, the magnetic
fluctuation grows until it saturates as well, and one has ** A**≠0. The
increased fractional density

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{N}_{\mathrm{sat}}{T}_{\u27c2}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}({\mathit{B}}_{\mathrm{0}}-\mathit{\delta}{\mathit{B}}_{\mathrm{sat}}{)}^{\mathrm{2}}-{\displaystyle \frac{{B}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}\left[{\mathrm{\nabla}}^{\mathrm{2}}\right({A}^{\mathrm{2}})-(\mathrm{\nabla}\mathit{A}{)}^{\mathrm{2}}{]}_{\mathrm{sat}}-{\displaystyle \frac{{\mathit{B}}_{\mathrm{0}}\cdot \mathrm{\nabla}\times \mathit{A}}{{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}}}\\ \text{(25)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\approx -{\displaystyle \frac{{q}^{\mathrm{2}}a\left({\mathit{\tau}}_{\mathrm{sat}}\right)}{\mathrm{2}mb}}{A}_{\mathrm{sat}}^{\mathrm{2}}.\end{array}$$

There is also a small local contribution from the magnetic stresses which results from the surface currents at the mirror boundaries in which only a minor part of the trapped particles is involved. This is indicated by the approximate sign.

The last two lines yield for the macroscopic penetration depth the expression
Eq. (22). We thus conclude that Eq. (22) is also valid
at saturation with *τ*=*τ*_{sat}. Measuring the saturation wavelength
*λ*_{sat} and saturation temperature anisotropy *τ*_{sat} determines
the unknown constant *b* through Eq. (23) with *τ*_{min} replaced with
*τ*_{sat}. Clearly

$$\begin{array}{}\text{(26)}& {\mathit{\tau}}_{\mathrm{min}}\le {\mathit{\tau}}_{\mathrm{sat}}<\mathrm{1}\end{array}$$

as the mirror mode might saturate at temperature anisotropies larger than the
permitted lowest anisotropy. Moreover, measurement of *τ*_{sat} at
saturation, the state in which the mirror mode is actually observed,
immediately yields the normalized saturation density excess
*N*_{m}(*τ*_{sat}) from Eq. (21) which then from
pressure balance yields the magnetic decrease, i.e. the mirror amplitude. To
some extent this completes the theory of the mirror mode in as far as it
relates the density at saturation to the saturated normalized temperature
anisotropy at given *T*_{⟂} and determines the scale
*λ*_{im} and *δ**B*(*τ*_{sat}).

5 The equivalent action *α*

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Since observations always refer to the final thermodynamic state, when the
mirror mode is saturated, the anisotropy at saturation can be measured, and
the value of the unknown constant *α* in the Schrödinger equation can
also be determined. Expressed through *b* and *λ*_{im} at
*τ*_{sat}, it becomes

$$\begin{array}{}\text{(27)}& \mathit{\alpha}=\sqrt{\mathrm{2}m}{\mathit{\lambda}}_{\mathrm{sat}}={\displaystyle \frac{m}{q}}\sqrt{{\displaystyle \frac{b}{{\mathit{\mu}}_{\mathrm{0}}{N}_{\mathrm{0}}\left|a\right({\mathit{\tau}}_{\mathrm{sat}}\left)\right|}}}.\end{array}$$

What is interesting about this number is that it is much larger than the
quantum of action ℏ but at the same time is sufficiently small, which
in retrospect justifies the neglect of the gradient term in the former
section. It represents the elementary action in a mirror unstable plasma,
where the characteristic length is given by the inertial scale
$\mathit{\alpha}/\sqrt{\phantom{\rule{0.125em}{0ex}}\mathrm{2}m}={\mathit{\lambda}}_{\mathrm{sat}}$ or the maximum of the normalized
density *N*_{m}. One may note that *α* is not an elementary
constant like ℏ. It depends on the critical reference temperature
*T*_{⟂}, and it depends on *τ*. Its constancy is understood in a
thermodynamic sense.

Our argument applies when ** A**≠0. In this case Eq. (13)
reads as

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}-{\displaystyle \frac{{\mathit{\alpha}}^{\mathrm{2}}}{\mathrm{2}ma}}{\displaystyle \frac{{d}^{\mathrm{2}}f\left(x\right)}{\mathrm{d}{x}^{\mathrm{2}}}}-f(\mathrm{1}-{f}^{\mathrm{2}})=\mathrm{0},\\ \text{(28)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\mathrm{where}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}f\left(x\right)={\displaystyle \frac{\mathit{\psi}\left(x\right)}{\left|{\mathit{\psi}}_{\mathrm{\infty}}\right|}}<\mathrm{1}\end{array}$$

and $|\mathit{\psi}{|}_{\mathrm{\infty}}=\sqrt{{N}_{\mathrm{max}}\left({x}_{\mathrm{\infty}}\right)}$ is given by the maximum
density excess in the centre *x*_{∞} of the magnetic field decrease.
Clearly this equation defines a natural scale length which is given by

$$\begin{array}{}\text{(29)}& {\mathit{\lambda}}_{\mathit{\alpha}}=\mathit{\alpha}/\sqrt{\mathrm{2}m\left|a\right({T}_{\u27c2},\mathit{\tau}\left)\right|},\end{array}$$

which, identifying it with *λ*_{sat}, yields the above
expression for *α*. For *x*_{∞} large the equation for *f* can be
solved asymptotically when $\mathrm{d}f/\mathrm{d}x=\mathrm{0}$ for *f*^{2}=1
corresponding to a maximum in *N*_{m}. It is then easy to show by
multiplication by d*f*∕d*x* that

$$\begin{array}{}\text{(30)}& {\displaystyle \frac{\mathrm{d}f\left(x\right)}{\sqrt{\mathrm{1}-{f}^{\mathrm{2}}}}}=\sqrt{\mathrm{2}}{\mathit{\lambda}}_{\mathit{\alpha}}\mathrm{d}x,\end{array}$$

which has the Landau–Ginzburg solution

$$\begin{array}{}\text{(31)}& f\left(x\right)=\mathrm{tanh}\left[{\displaystyle \frac{x}{\sqrt{\mathrm{2}}{\mathit{\lambda}}_{\mathit{\alpha}}}}\right].\end{array}$$

This implies that the excess density assumes the shape

$$\begin{array}{}\text{(32)}& {N}_{\mathrm{m}}={N}_{\mathrm{max}}{\mathrm{tanh}}^{\mathrm{2}}\left[{\displaystyle \frac{x}{\sqrt{\mathrm{2}}{\mathit{\lambda}}_{\mathit{\alpha}}}}\right].\end{array}$$

It approaches *N*_{max} for *x*→*x*_{∞}. The above condition on
the vanishing gradient of *f* at *x*_{∞} warrants the flat shape of the
excess density at maximum (*x*_{∞}) and the equally flat shape of the
magnetic field in its minimum. At *x*=0 the amplitude *f*(*x*) starts
increasing with finite slope ${f}^{\prime}\left(\mathrm{0}\right)=\sqrt{\mathrm{2}}{\mathit{\lambda}}_{\mathit{\alpha}}$. On the other
hand, the initial slope of *N*_{m} is ${N}_{\mathrm{m}}^{\prime}\left(\mathrm{0}\right)=\mathrm{0}$. The
normalized excess density has a turning point at
*x*_{t}≈0.48*λ*_{α} with value *N*_{m}(*x*_{t})≈0.11*N*_{max}. This behaviour is schematically shown in
Fig. 1. Of course, these considerations apply strictly
only to the one-dimensional case. It is, however, not difficult to generalize
them to the cylindrical problem with radius *r* in place of *x*. The main
qualitative properties are thereby retained. In the next section we will turn
to the question of generation of chains of mirror-mode bubbles, as this is
the case which is usually observed in space plasmas.

Since the quantum of action enters the magnetic quantum flux element ${\mathrm{\Phi}}_{\mathrm{0}}=\mathrm{2}\mathit{\pi}\mathrm{\hslash}/e$, we may also conclude that in a mirror-unstable plasma the relevant magnetic flux element is given by ${\mathrm{\Phi}}_{\mathrm{m}}=\mathit{\alpha}/q$.

Identification of *α* is an important step. With its knowledge in mind
the nonlinear Schrödinger equation for the hypothetical saturation state of
the mirror mode is (up to the coefficient *b*, which, however, is defined in
Eq. (23) and can be obtained from measurement) completely determined
and thus ready for application of the inverse scattering procedure which
solves it under any given initial conditions for the mirror mode. It thus
opens up the possibility of further investigating the final evolution of the
mirror mode. Executing this programme should, under various conditions,
provide the different forms of the mirror mode in its final thermodynamic
equilibrium state. This is left as a formally sufficiently complicated
exercise which will not be treated in the present communication. Instead, we
ask for the conditions under which the mirror mode evolves into a chain of
separated mirror bubbles, which requires the existence of a microscopic
though classical correlation length.

6 The problem of the correlation length

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The present phenomenological theory of the final thermodynamic equilibrium
state of the mirror mode is modelled after the phenomenological
Landau–Ginzburg theory of superconductivity as presented in the cited
textbook literature. From the existence of *λ*_{im} we would
conclude that, under mirror instability, the magnetic field inside the plasma
volume should decay to a minimum value determined by the achievable minimum
*τ*_{sat} of the temperature ratio. This conclusion would, however,
be premature and contradicts observation where chains or trains (Zhang et al., 2009) of mirror-mode fluctuations are usually
observed (Luehr and Kloecker, 1987; Treumann et al., 1990), which presumably are in their saturated
state having had sufficient time to evolve beyond quasilinear saturation
times and reached saturation amplitudes much in excess of any predicted
quasilinear level. In fact, observations of mirror modes in their growth
phase have to our knowledge not yet been reported. On the other hand, in no
case known to us has a global reduction of the gross magnetic field in an
anisotropic plasma been identified yet.

It is clear that in any real collisionless high-temperature plasmas neither
can *N*_{m} become infinite nor can *τ* drop to zero. Since it is
not known how and in which way, i.e. by which exactly known process mirror
modes saturate in their final thermodynamic equilibrium state, their growth
must ultimately become stopped when the particle correlation length comes
into play. The nature of such a correlation length is unknown, nor is it
precisely defined. There are at least three types of candidates for an
effective correlation length, the Debye scale *λ*_{D}, the ion gyroradius
*ρ*, and some *turbulent* correlation length ℓ_{turb}.

In a plasma the shortest *natural* correlation length is the Debye
length *λ*_{D} which under all conditions is much shorter than the
above-estimated penetration length *λ*_{im}. Referring to the
Debye length, the Landau–Ginzburg parameter, i.e. the ratio of penetration
to correlation lengths, in a plasma as a function of *τ* becomes

$$\begin{array}{}\text{(33)}& {\mathit{\kappa}}_{D}\equiv {\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)}{{\mathit{\lambda}}_{D}\left(\mathit{\tau}\right)}}\approx {\displaystyle \frac{c}{{\mathit{\upsilon}}_{\u27c2\mathrm{th}}\left(\mathit{\tau}\right)}}\gg \mathrm{1},\end{array}$$

a quantity that is large. Writing for the Debye length

$$\begin{array}{}\text{(34)}& {\mathit{\lambda}}_{D}^{\mathrm{2}}\left(\mathit{\tau}\right)={\mathit{\lambda}}_{D\u27c2}^{\mathrm{2}}{\displaystyle \frac{\mathrm{1}+\mathit{\tau}/\mathrm{2}}{{N}_{\mathrm{m}}\left(\mathit{\tau}\right)}},\phantom{\rule{1em}{0ex}}{\mathit{\lambda}}_{D\u27c2}^{\mathrm{2}}={\displaystyle \frac{\mathrm{4}}{\mathrm{3}}}{\displaystyle \frac{{T}_{\u27c2}/{m}_{i}}{{\mathit{\omega}}_{i\mathrm{0}}^{\mathrm{2}}}},\end{array}$$

the Landau–Ginzburg parameter can be expressed in terms of *τ*,
exhibiting only a weak dependence on the temperature ratio *τ*<1:

$$\begin{array}{}\text{(35)}& {\mathit{\kappa}}_{D}\left(\mathit{\tau}\right)={\displaystyle \frac{{\mathit{\lambda}}_{i\mathrm{0}}}{{\mathit{\lambda}}_{D\perp}}}\sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{1}+\mathit{\tau}/\mathrm{2}}}}\gg \mathrm{1}.\end{array}$$

Thus, *κ*_{D} is practically constant and about independent on the
temperature anisotropy. Its value ${\mathit{\kappa}}_{D\mathrm{0}}={\mathit{\lambda}}_{i\mathrm{0}}/{\mathit{\lambda}}_{D}$ at
*τ*=1, ${T}_{\Vert}={T}_{\u27c2}$ refers to the isotropic case when no mirror
instability evolves.

This is an important finding because it implies that in a plasma the case
that the magnetic field would be completely expelled from the volume of the
plasma cannot be realized. Different regions of extension substantially
larger than *λ*_{D} are (electrostatically) uncorrelated. They therefore
behave separately, lacking knowledge about their (electrostatically)
uncorrelated neighbours separated from them at distances substantially
exceeding *λ*_{D}. Each of them experiences the penetration scale and
adjusts itself to it. This is in complete analogy to Landau–Ginzburg theory.
Thus, once the main magnetic field in an anisotropic plasma drops below
threshold, the plasma will necessarily evolve into a chain of nearly
unrelated mirror bubbles which interact with each other because each occupies
space. In superconductivity this corresponds to a type II superconductor.
Mirror unstable plasmas in this sense behave like type II superconductors.
They decay into regions of normal magnetic field strength and embedded
domains of spatial-scale *λ*_{m}(*τ*) with a reduced magnetic
field. These regions contain an excess plasma population which is in pressure
and stress balance with the magnetic field. Its diamagnetism (perpendicular
pressure) keeps the magnetic field partially out and causes weak diamagnetic
currents to flow along the boundaries of each of the partially
field-evacuated domains. This trapped plasma behaves analogously to the pair
plasma in metallic superconductivity, this time however at the high plasma
temperature being bound together not by pairing potentials, but – in the
case of the Debye length playing the role of the correlation length – by the
Debye potential over the Debye correlation length.

However, the Debye length is a *very short* scale, in fact the
shortest collective scale in the plasma, and though it must have an effect on
the collective evolution of particles in plasmas, it should be doubted that,
on the mirror-mode saturation scale, it would have a substantial or even
decisive effect. Instead, there could also be larger scales on which the
particles are correlated.

Such a scale is, for instance, the thermal-ion gyroradius *ρ*(*τ*). For
the low frequencies of the mirror mode, the magnetic moment
$\mathit{\mu}\left(\mathit{\tau}\right)={T}_{\u27c2}/B\left(\mathit{\tau}\right)=\mathrm{const}$ of the particles is conserved in
their dynamics, which implies that all particles with the same magnetic
moment *μ*(*τ*) behave about collectively, at least in the sense of a
gyro-kinetic theory.

However, though *μ*(*τ*) is a constant of motion, it still is a function
of the anisotropy through the dependence of the magnetic field on *τ*.
Expressing the thermal gyroradius through the magnetic moment

$$\begin{array}{}\text{(36)}& \mathit{\rho}\left(\mathit{\tau}\right)=\sqrt{{\displaystyle \frac{\mathrm{2}\mathit{\mu}\left(\mathit{\tau}\right)}{e{\mathit{\omega}}_{ci}\left(\mathit{\tau}\right)}}}={\mathit{\rho}}_{\mathrm{0}}\sqrt{{\displaystyle \frac{\mathit{\tau}}{\mathrm{1}-\mathit{\tau}}}},\phantom{\rule{1em}{0ex}}{\mathit{\rho}}_{\mathrm{0}}=\sqrt{{\displaystyle \frac{\mathrm{2}m{T}_{\u27c2}}{{e}^{\mathrm{2}}({B}_{\mathrm{crit}}^{\mathrm{0}}{)}^{\mathrm{2}}}}},\end{array}$$

it can be taken as another kind of collective correlation scale as on scales
*larger* than *ρ* it collectively binds particles of the same
magnetic moment which, in particular, are magnetically trapped like those
which are active in the mirror instability. Below the gyroradius charged
particles are magnetically free. *ρ* is the scale where the particles
magnetize, start feeling the magnetic field effect, and collectively enter
another phase in their dynamics. This scale is much larger than the Debye
length and may be more appropriate for describing the saturated behaviour of
the mirror mode. Thus one may argue that, as long as the penetration depth
(inertial sale) exceeds *ρ*, the thermal gyroradius is the relevant
correlation length. Only when it drops below the gyroradius does the Debye
length take over. The Landau–Ginzburg parameter then becomes

$$\begin{array}{}\text{(37)}& {\mathit{\kappa}}_{\mathit{\rho}}\left(\mathit{\tau}\right)={\displaystyle \frac{{\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)}{\mathit{\rho}\left(\mathit{\tau}\right)}}={\displaystyle \frac{{\mathit{\lambda}}_{i\mathrm{0}}}{{\mathit{\rho}}_{\mathrm{0}}}}[{\displaystyle \frac{{\mathit{\tau}}_{\mathrm{sat}}}{\mathit{\tau}}}{\displaystyle \frac{\mathrm{1}-\mathit{\tau}}{\mathrm{1}-{\mathit{\tau}}_{\mathrm{sat}}}}{]}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{4}}}}.\end{array}$$

This ratio depends on the temperature anisotropy $\mathit{\tau}={T}_{\Vert}/{T}_{\perp}$, which is
a measurable quantity and the important parameter, while it saturates at
${\mathit{\kappa}}_{\mathit{\rho},\mathrm{sat}}={\mathit{\lambda}}_{i\mathrm{0}}/{\mathit{\rho}}_{\mathrm{0}}$, the ratio of inertial
length to gyroradius at the critical field. This ratio is not necessarily
large. It can be expressed by the ratio of Alfvén velocity *V*_{A}
to perpendicular ion-thermal velocity *υ*_{⟂th}:

$$\begin{array}{}\text{(38)}& {\mathit{\kappa}}_{\mathit{\rho},\mathrm{sat}}={\displaystyle \frac{{\mathit{\lambda}}_{i\mathrm{0}}}{{\mathit{\rho}}_{\mathrm{0}}}}={\displaystyle \frac{{V}_{\mathrm{A}}\left({B}_{\mathrm{crit}}^{\mathrm{0}}\right)}{{\mathit{\upsilon}}_{\u27c2\mathrm{th}}}}>\mathrm{1}.\end{array}$$

Hence, when referring to the thermal-ion gyroradius as the correlation
length, the mirror mode would evolve and saturate into a chain of mirror
bubbles only, when the Alfvén speed
${V}_{\mathrm{A}}>{\mathit{\upsilon}}_{\u27c2\mathrm{th}}$ exceeds the perpendicular
thermal velocity of the ions. (Since
${B}_{\mathrm{crit}}^{\mathrm{0}}\propto \left|\mathrm{sin}\mathit{\theta}\right|$, highly oblique angles are
favoured. The range of optimum angles has recently been estimated in
Treumann and Baumjohann, 2018a.) This is to be multiplied by the *τ* dependence,
of which Fig. 2 gives an example. The value of this function is
always smaller than one. For a chain of mirror bubbles to evolve in a plasma,
the requirement *κ*_{ρ}>1 can then be written as

$$\begin{array}{}\text{(39)}& \mathrm{1}\le {\displaystyle \frac{\mathit{\tau}}{{\mathit{\tau}}_{\mathrm{sat}}}}<{\displaystyle \frac{{\mathit{\kappa}}_{\mathit{\rho},\mathrm{sat}}^{\mathrm{4}}}{\mathrm{1}+({\mathit{\kappa}}_{\mathit{\rho},\mathrm{sat}}^{\mathrm{4}}-\mathrm{1}){\mathit{\tau}}_{\mathrm{sat}}}},\end{array}$$

which is always satisfied for *τ*_{sat}<1 and
${\mathit{\kappa}}_{\mathit{\rho},\mathrm{sat}}>\mathrm{1}$, i.e. the Alfvén speed exceeding the
perpendicular thermal speed, which indeed is the crucial condition for mirror
modes to evolve into chains and become observable, with the gyroradius
playing the role of a correlation length. Mirror-mode chains in the present
case are restricted to comparably cool anisotropic plasma conditions, a
prediction which can be checked experimentally to decide whether or not the
gyroradius serves as a correlation length.

Otherwise, when the above condition is not satisfied and *τ*<1 is below
threshold, a very small and thus probably not susceptible reduction in the
overall magnetic field is produced in the anisotropic pressure region over
distances *L*≫*ρ*, much larger than the ion gyroradius. Observation of
such domains of reduced magnetic field strengths under anisotropic
pressure/temperature conditions would indicate the presence of a large-scale
type I classical Meissner effect in the plasma. Such a reduction of the
magnetic field would be difficult to explain otherwise and could only be
understood as confinement of plasma by discontinuous boundaries of the kind
of tangential discontinuities.

The relative rarity of observations of mirror-mode chains or trains seems to
support the case that the gyroradius, not the Debye length, plays the role of
the correlation length in a magnetized plasma under conservation of the
magnetic moments of the particles. From basic theory it cannot be decided
which of the two correlation lengths, the Debye length *λ*_{D} or the ion
gyroradius *ρ*, dominates the dynamics and saturation of the mirror mode.
A decision can only be established by observations.

However, the thermal-ion gyroradius, though the statistical average of the
*distribution* of gyroscales, is itself just a plasma parameter which
officially lacks the notion of a *genuine* correlation length. For
this reason one would rather refer to the third possibility, a
*turbulent* correlation length ℓ_{turb} which evolves as
the result of either high-frequency plasma or – in the case of mirror modes
probably better suited – magnetic turbulence in the plasma.

It is well known that, for instance, the solar wind or the magnetosheath carries a substantial level of turbulence which mixes plasmas of various properties and obeys a particular spectrum. In the solar wind such spectra have been shown to exhibit approximate Kolmogorov-type properties, at least in certain domains of frequencies or wave numbers, and similarly in the magnetosheath, where the conditions are more complicated because of the boundedness of the magnetosheath and the resulting spatial confinement of the plasma and its streaming. Such spectra imply that particles and waves are not independent but contain some information about their behaviour in different spatial and frequency domains; in other words, they are correlated.

Unfortunately, the turbulent correlation length is imprecisely defined. No
analytical expressions have been provided yet which would allow us to refer
to it in the above determination of the Landau–Ginzburg parameter. This
inhibits prediction of the range and parameter dependences of the turbulent
Landau–Ginzburg ratio. Nonetheless, turbulent correlation scales might
dominate the development of the mirror mode. The observation of a spectrum of
mirror modes that is highly peaked around a certain wavelength not very much
larger than the ion gyroradius may tell something about its nature. The above
theory should open a way of relating a turbulent correlation length to the
properties of a mirror unstable plasma. The condition is simply that the
*turbulent* Landau–Ginzburg parameter

$$\begin{array}{}\text{(40)}& {\mathit{\kappa}}_{\mathrm{turb}}\left(\mathit{\tau}\right)={\displaystyle \frac{\langle {\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)\rangle}{{\mathrm{\ell}}_{\mathrm{turb}}\left(\mathit{\tau}\right)}}>\mathrm{1}\end{array}$$

is large, depending on the anisotropy parameter *τ* and the average
transverse scales of the mirror bubbles. This expression yields an upper
limit for the turbulent correlation length

$$\begin{array}{}\text{(41)}& {\mathrm{\ell}}_{\mathrm{turb}}\left(\mathit{\tau}\right)<\langle {\mathit{\lambda}}_{\mathrm{im}}\left(\mathit{\tau}\right)\rangle ,\end{array}$$

where 〈*λ*_{im}(*τ*)〉 is known as a
function of *τ* and the plasma parameters. Investigating this in further
detail both observationally and theoretically should throw additional light
on the nature of magnetic turbulence in high-temperature plasmas like those
of the solar wind and magnetosheath. It would even contribute to a more
profound understanding of magnetic turbulence in general as well as in view
of its application to astrophysical problems.

7 Conclusions

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The mirror mode is a particular zero-frequency mesoscale plasma instability
which provides some mesoscopic structure to an anisotropic plasma. It has
been observed surprisingly frequently under various conditions in space, in
the solar wind, cometary environments, near other planets and, in particular,
behind the bow shock (Czaykowska et al., 1998), such that one also believes that
they occur in shocked plasmas if the shock causes a temperature anisotropy
*τ*<1 (Balogh and Treumann, 2013). Since mirror modes are long
scale, they provide the plasma with a very particular spatial texture. Mirror
unstable plasmas are apparently built of a large number of magnetic bottles
which contain a trapped particle population. This makes mirror modes most
interesting even in magnetohydrodynamic terms as a kind of long-wavelength
source of turbulence. In addition, their boundaries are surfaces which
separate the bottles and thus have the character or tangential
discontinuities or surfaces of diamagnetic currents which are produced by the
internal interaction between the plasma and magnetic field. We have shown
above that such an interaction resembles superconductivity, i.e. a classical
Meissner effect.

Mirror modes in the anisotropic collisionless space plasma apparently
represent a classical *thermodynamic* analogue to a
“superconducting” equilibrium state. One should, however, not exaggerate
this analogy. This equilibrium state is *no macroscopic quantum state*. It is a classical effect. The analogy is just formal, even though it
allows us to conclude about the *final mirror equilibrium*. Sometimes
such an analogue helps in understanding the underlying physics^{1} like here, where it paves the way to a global understanding of the
final saturation state of the mirror mode even though this does not release
us from understanding in which way this final state is dynamically achieved.

In contrast to metallic superconductivity which is described by the
Landau–Ginzburg theory to which we refer here or, on the microscopic quantum
level, by BCS-pairing theory, the problem of circumventing friction and
resistance is of no interest in ideally conducting space plasmas which evolve
towards mirror modes. High temperature plasmas are classical systems in which
no pairing occurs and BCS theory is not applicable. Those plasmas are
*already* ideally conducting. In contrast, there is a vital interest
in the opposite problem, how a finite sufficiently large resistance can
develop under conditions when collisions and friction among the particles are
negligible. This is the problem of generating *anomalous* resistivity
which may develop from high-frequency kinetic instabilities or turbulence and
is believed to be urgently needed, for instance causing dissipation in
reconnection. In the zero-frequency mirror mode it is of little importance
even asymptotically, in the long-term thermodynamic limit, where such an
anomalous resistance may contribute to decay of the mirror-surface currents
which develop and flow along the boundaries of the mirror bubbles. The times
when this happens are very long compared with the saturation time of the
mirror instability and transition to the thermodynamic quasi-equilibrium
which has been considered here.

The more interesting finding concerns the explanation *why at all*, in
an ideally conducting plasma, mirror *bubbles* can evolve. Fluid and
simple kinetic theories demonstrate that mirror modes occur in the presence
of temperature anisotropies, thereby identifying the *linear growth rate* of the instability. Trapping of large numbers of charged particles
(ions, electrons) in accidentally forming magnetic bottles/traps causes the
mirror instability to grow. The present theory contributes to clarification
of this mechanism and its *final thermodynamic* equilibrium state as a
*nonlinear* effect which is made possible by the available free energy
which leads to a particular nonlinear Schrödinger equation. The
*perpendicular* temperature in this theory plays the role of a
critical temperature. When the parallel temperature drops below it, which
means that $\mathrm{1}>\mathit{\tau}>{\mathit{\tau}}_{\mathrm{min}}$, mirror modes can evolve. Interestingly
the anisotropy is restricted from below. The parallel temperature cannot drop
below a minimum value. This value is open to determination by observations.

The observation of chains of mirror bubbles, for instance in the
magnetosheath, which provide the mirror-unstable plasma with a particular
intriguing magnetic texture, suggests that the plasma, in addition to being
mirror unstable, is subject to some correlation length which determines the
spatial structure of the mirror texture in the saturated thermodynamic
quasi-equilibrium state. This correlation length can be either taken as the
Debye scale *λ*_{D}, which then naturally makes it plausible that many
such mirror bubbles evolve, because in all magnetized plasmas the magnetic
penetration depth by far exceeds the Debye length and makes the
Landau–Ginzburg parameter based on the Debye length *κ*_{D}≫1. This,
however, should lead to rather short-scale mirror bubbles. Otherwise, the
role of a correlation length could also be played by the thermal-ion
gyroradius *ρ*. In this case the conditions for the evolution of the
mirror mode with the many observed bubbles become more subtle, because then
*κ*_{ρ}*≳*1 occurs under additional restrictions, implying that
the Alfvén speed exceeds the perpendicular thermal speed. This prediction
has to be checked and possibly verified experimentally. A particular case of
the dependence of the gyroradius-based Landau–Ginzburg parameter
*κ*_{ρ} is shown graphically in Fig. 2.

It may be noted that the Debye length and the ion gyroradius are fundamental
plasma scales. Correlations can of course also be provided by other means, in
particular by any form of turbulence. In that case a *turbulent*
correlation length would play a similar role in the Landau–Ginzburg
parameter, whether shorter or larger than the above-identified penetration
scale. Regarding mirror modes in the magnetosheath to which we referred
(Treumann and Baumjohann, 2018a), it is well known that the magnetosheath hosts a broad
turbulence spectrum in the magnetic field as well as in the dynamics of the
plasma (fluctuations in the velocity and density).

Though this makes it highly probable that turbulence intervenes and affects the evolution of mirror modes, any “turbulent correlation length” is, unfortunately, rather imprecisely defined as some average quantity. To our knowledge, though referring to multi-spacecraft missions is not impossible, it has even not yet been precisely identified in any observations of turbulence in space plasmas. Even when identified, its functional dependence on temperature and density is required for application in our theory. If these functional dependencies are not available, it becomes difficult to include any turbulent correlation length. In addition, one expects that its turbulent nature would make the theory nonlocal. Attempts in that direction must, at this stage of the investigation, be relegated to future efforts.

Finally, it should be noted that the magnetic penetration depth
*λ*_{m} which lies at the centre of our investigation is rather
different from the ordinary inertial length scale of the plasma. It is based
on the excess density *N*_{m}<1 less than the bulk plasma density
*N*_{0}. It thus gives rise to an enhanced (excess) plasma frequency
${\mathit{\omega}}_{\mathrm{m}}={\mathit{\omega}}_{i}\sqrt{{N}_{\mathrm{m}}+\mathrm{1}}={\mathit{\omega}}_{i}\sqrt{\mathrm{1}+\mathit{\zeta}}\mathit{\lesssim}\sqrt{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathit{\omega}}_{i}$,
which implies that $L>c/{\mathit{\omega}}_{i}>{\mathit{\lambda}}_{\mathrm{m}}$ is shorter than the
typical scale of the volume *L* and (slightly) shorter than the bulk inertial
length *c*∕*ω*_{i}. This becomes clear when recognizing that the mirror mode
evolves inside the plasma from some thermal fluctuation (Yoon and López, 2017), which causes the magnetic field locally to
drop below its critical value – Eq. (2). Then
*λ*_{m} identifies the local perpendicular scale of a mirror
bubble after it has saturated and is in thermodynamic equilibrium. One
expects that the transverse diameter of a single mirror bubble in the ideal
case would be roughly 2*λ*_{m}. However, since each bubble
occupies *real* space, in a mirror-saturated plasma state the bubbles
compete for space and distort each other (Treumann and Baumjohann, 1997), thereby providing the plasma with an *irregular magnetic texture* of some, probably narrow, spectrum of transverse scales
which peaks around some typical transverse wavelength and resembles a
strongly distorted crystal lattice that is elongated along the ambient
magnetic field.

It also relates the measurable saturated magnetic amplitudes of mirror modes
to the saturated anisotropy *τ*_{sat} and the Landau–Ginzburg
parameter *κ*, transforming both into experimentally accessible
quantities. These should be of use in the development of a weak-kinetic
turbulence theory of magnetic mirror modes as the result of which mirror
modes can grow to the observed large amplitudes which are known to far exceed
the simple quasilinear saturation limits. It also paves the way to the
determination of a (possibly turbulent) correlation length in mirror unstable
plasmas of which so far no measurements have been provided.

To the space plasma physicist the present investigation may look a bit academic. However, it provides some physical understanding of how mirror modes do really saturate, why they assume such large amplitudes and evolve into chains of many bubbles or magnetic holes, and what the conditions are when this happens. Moreover, since the mirror mode in some sense resembles superconductivity, which also implies that some population of particles involved behaves like a superfluid, it would be of interest to infer whether such a population exhibits properties of a superfluid. One suggestion is that the untrapped ions and electrons which escape from the magnetic bottles along the magnetic field resemble such a superfluid population. This also suggests that other high-temperature plasma effects like the formation of purely electrostatic electron holes in beam–plasma interaction may exhibit superfluid properties. In conclusion, the unexpected working of the thermodynamic treatment in the special case of the magnetic mirror mode shows once more the enormous explanatory power of thermodynamics.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgement

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Acknowledgement.

This work was part of a Visiting Scientist Programme at the International
Space Science Institute (ISSI) Bern. We acknowledge the interest of the ISSI
directorate and the hospitality of the ISSI staff. We thank the referees
Karl-Heinz Glassmeier (TU Braunschweig, DE), Ovidiu Dragos Constantinescu (U
Bukarest, RO), and Hans-Reinhard Mueller (Dartmouth College, Hanover, NH,
USA) for their critical remarks on the manuscript, in particular the
non-trivial questions on the effective mass and the role of turbulence.

The topical editor, Anna
Milillo, thanks Karl-Heinz Glassmeier, Dragos Constantinescu, and
Hans-Reinhard Müller for help in evaluating this paper.

References

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In a recent paper (Treumann and Baumjohann, 2018c), we have shown that a classical Higgs mechanism is responsible for bending the free space O-L and X-R electromagnetic modes in their long-wavelength range away from their straight vacuum shape when passing a plasma. The plasma in that case acts like a Higgs field and attributes a tiny mass to the photons, making them heavy. This is interesting because it shows that any bosons become heavy only in permanent interaction with a Higgs field and only in a certain energy–momentum–wavelength range. It also shows that earlier attempts at measuring a permanent photon mass by observing scintillations of radiation (and also by other means) have just measured this effect. Their interpretations as upper limits for a real permanent photon mass are incorrect because they missed the action of the plasma as a classical Higgs field.