The mirror mode: a "superconducting" space plasma analogue
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Ann. Geophys., 36, 1015–1026, 2018
https://doi.org/10.5194/angeo-36-1015-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
The mirror mode: a “superconducting” space plasma analogue
Rudolf A. Treumann1 and Wolfgang Baumjohann2
1 International Space Science Institute, Bern, Switzerland
2 Space Research Institute, Austrian Academy of Sciences, Graz, Austria
Correspondence: Wolfgang Baumjohann (wolfgang.baumjohann@oeaw.ac.at)
Received: 1 May 2018 – Discussion started: 8 May 2018
Accepted: 17 July 2018 – Published: 26 July 2018
Abstract. We examine the physics of the magnetic mirror 1 Introduction
mode in its final state of saturation, the thermodynamic equi-
librium, to demonstrate that the mirror mode is the ana-
logue of a superconducting effect in a classical anisotropic- Under special conditions high-temperature collisionless plas-
pressure space plasma. Two different spatial scales are iden- mas may develop properties which resemble those of super-
tified which control the behaviour of its evolution. These conductors. This is the case with the mirror mode when the
are the ion inertial scale λim (τ ) based on the excess den- anisotropic pressure gives rise to local depletions of the mag-
sity Nm (τ ) generated in the mirror mode, and the Debye netic field similar to the Meissner effect in metals where it
scale λD (τ ). The Debye length plays the role of the corre- signals the onset of superconductivity (Kittel, 1963; Fetter
lation length in superconductivity. Their dependence on the and Walecka, 1971; Huang, 1987; Lifshitz and Pitaevskii,
temperature ratio τ = Tk /T⊥ < 1 is given, with T⊥ the refer- 1998), i.e. the suppression of friction between the current
ence temperature at the critical magnetic field. The mirror- and the lattice. In collisionless plasmas there is no lattice,
mode equilibrium structure under saturation is determined the plasma is frictionless, and thus it already is ideally con-
by the Landau–Ginzburg ratio κD = λim /λD , or κρ = λim /ρ, ducting which, however, does not mean that it is supercon-
depending on whether the Debye length or the thermal-ion ducting. To be superconducting, additional properties are re-
gyroradius ρ – or possibly also an undefined turbulent corre- quired. These, as we show below, are given in the saturation
lation length `turb – serve as correlation lengths. Since in all state of the mirror mode.
space plasmas κD 1, plasmas with λD as the relevant cor- The mirror mode is a non-oscillatory plasma instability
relation length always behave like type II superconductors, (Chandrasekhar, 1961; Hasegawa, 1969; Gary, 1993; South-
naturally giving rise to chains of local depletions of the mag- wood and Kivelson, 1993; Kivelson and Southwood, 1996)
netic field of the kind observed in the mirror mode. In this which evolves in anisotropic plasmas (for a recent review,
way they would provide the plasma with a short-scale mag- see Sulem, 2011, and references therein). It has been argued
netic bubble texture. The problem becomes more subtle when that it should readily saturate by quasilinear depletion of the
ρ is taken as correlation length. In this case the evolution of temperature anisotropy (cf. e.g. Noreen et al., 2017, and ref-
mirror modes is more restricted. Their existence as chains erences therein). Observations do not support this conclu-
or trains of larger-scale mirror bubbles implies that another sion. In fact, we recently argued (Treumann and Baumjo-
threshold, VA > υ⊥th , is exceeded. Finally, in case the corre- hann, 2018a) that the large amplitudes of mirror-mode oscil-
lation length `turb instead results from low-frequency mag- lations observed in the Earth’s magnetosheath, magnetotail,
netic/magnetohydrodynamic turbulence, the observation of and elsewhere, like other planetary magnetosheaths, in the
mirror bubbles and the measurement of their spatial scales solar wind and generally in the heliosphere (see e.g. Tsuru-
sets an upper limit on the turbulent correlation length. This tani et al., 1982, 2011; Czaykowska et al., 1998; Constan-
might be important in the study of magnetic turbulence in tinescu et al., 2003; Zhang et al., 1998, 2008, 2009; Lucek
plasmas. et al., 1999a, b; Volwerk et al., 2008, and many others), are
a sign of the impotence of quasilinear theory in limiting the
growth of the mirror instability. Instead, mirror modes should
Published by Copernicus Publications on behalf of the European Geosciences Union.1016 R. A. Treumann and W. Baumjohann: Mirror instability
be subject to weak kinetic turbulence theory (Sagdeev and spacecraft observations, the reader may consult Constanti-
Galeev, 1969; Davidson, 1972; Tsytovich, 1977; Yoon, 2007, nescu (2002) and Constantinescu et al. (2003). Such an ap-
2018; Yoon and Fang, 2007), which allows them to evolve proach is rather alien to space physics. It follows the path
until they become comparable in amplitude to the ambient prescribed in solid-state physics but restricts itself to the do-
magnetic field long before any dissipation can set in. main of classical thermodynamics only.
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 2 Properties of the mirror instability
turbulence theory of the mirror mode is available yet as
The mirror instability evolves whence the magnetic field B in
it is difficult to identify the various modes which interact
a collisionless magnetized plasma with an internal pressure–
to destroy quasilinear quenching. The frequent claim that
temperature anisotropy T⊥ > Tk , where the subscripts refer
whistlers (lion roars) excited in the trapped electron compo-
to the directions perpendicular and parallel to the ambient
nent would destroy the bulk (global) temperature anisotropy
magnetic field, drops below a critical value
is erroneous, because whistlers (Thorne and Tsurutani, 1981; s 1
Baumjohann et al., 1999; Maksimovic et al., 2001; Zhang
p Te⊥ 2
et al., 1998) grow at the expense of a small component of B < Bcrit ≈ 2µ0 N Ti⊥ 2i + 2e sin θ , (1)
Ti⊥
anisotropic resonant particles only (Kennel and Petschek,
1966). Depletion of the resonant anisotropy by no means af- where 2j = T⊥ /Tk − 1 j > 0 is the temperature anisotropy
fects the bulk temperature anisotropy that is responsible for of species j = e, i (for ions and electrons) and θ is the angle
the evolution of the mirror instability. On the other hand, of propagation of the wave with respect to the ambient mag-
construction of a weak turbulence theory of the mirror mode netic field (cf. e.g. Treumann and Baumjohann, 2018a). Here
poses serious problems. One therefore needs to refer to other any possible temperature anisotropy in the electron popula-
means of description of its final saturation state. tion has been included, but will be dropped below as it seems
Since measurements suggest that the observed mirror (Yoon and López, 2017) that it does not provide any further
modes are about stationary phenomena which are swept over insight into the physics of the final state of the mirror mode.
the spacecraft at high flow speeds (called Taylor’s hypothe- The important observation is that the existence of the mir-
sis, though, in principle, it just refers to the Galilei transfor- ror mode depends on the temperature difference T⊥ − Tk and
mation), it seems reasonable to tackle them within a thermo- the critical magnetic field. Commonly only the temperature
dynamic approach, i.e. assuming that in the observed large- anisotropy is reclaimed as being responsible for the growth
amplitude saturation state they can be described as the sta- of the mirror mode. Though this is true, it also implies the
tionary state of interaction between the ideally conducting above condition on the magnetic field. To some degree this
plasma and magnetic field. This can be most efficiently done resembles the behaviour of magnetic fields under supercon-
when the free energy of the plasma is known, which, unfortu- ducting conditions. To demonstrate this, we take T⊥ as the
nately, is not the case. Magnetohydrodynamics does not ap- reference – or critical – temperature. The critical magnetic
ply, and the formulation of a free energy in the kinetic state is field becomes a function of the temperature ratio τ = Tk /T⊥ .
not available. For this reason we refer to some phenomeno- Once τ < 1 and B < Bcrit the magnetic field will be pushed
logical approach which is guided by the phenomenological out of the plasma to give space to an accumulated plasma
theory of superconductivity. There we have the similar phe- density and also weak diamagnetic surface currents on the
nomenon that the magnetic field is expelled from the medium boundaries of the (partially) field-evacuated domain.
due to internal quantum interactions, known as the Meiss- The τ dependence of the critical magnetic field can be cast
ner effect. This resembles the evolution of the mirror mode, into the form
though in our case the interactions are not in the quantum do- i 12
1 1
Bcrit (Tk ) h −1 T⊥ 2 Tk 2
main. This is peasily understood when considering the thermal 0
= τ 1 − τ = 1 − , (2)
length λ} = 2π }2 /me T and comparing it to the shortest Bcrit Tk T⊥
1
plasma scale, viz. the inter-particle distance dN ∼ N − 3 . The which indeed resembles that in the phenomenological theory
former is, for all plasma temperatures T , in the atomic range, of superconductivity. Here
while the latter in space plasmas for all densities N is at least 0
Bcrit
p
= 2µ0 N Ti⊥ sin θ (3)
several orders of magnitude larger. Plasmas are classical. In
their equilibrium state classical thermodynamics applies to and the critical threshold vanishes for τ = 1 where the range
them. of possible unstable magnetic field values shrinks to zero; the
In the following we boldly ask for the thermodynamic limits Tk = 0 or T⊥ = ∞ make no physical sense.
equilibrium state of a mirror unstable plasma. For other non- Though the effects are similar to superconductivity, the
thermodynamical attempts at modelling the equilibrium con- temperature dependence is different from that of the Meiss-
figuration of magnetic mirror modes and application to multi- ner effect in metals in their isotropic low-temperature super-
conducting phase. In contrast, in an anisotropic plasma the
Ann. Geophys., 36, 1015–1026, 2018 www.ann-geophys.net/36/1015/2018/R. A. Treumann and W. Baumjohann: Mirror instability 1017
effect occurs in the high-temperature phase only while being attributing the dimension of energy density to the expansion
absent at low temperatures. Nevertheless, the condition τ < 1 coefficients a and b. An expansion like this one is always
indicates that the mirror mode, concerning the ratio of paral- possible in the spirit of a perturbation approach as long as the
lel to perpendicular temperatures, is a low-temperature effect total density N/N0 = 1 + Nm with Nm < 1. It is thus clear
in the high-temperature plasma phase. This may suggest that that Nm is not the total ambient plasma density N0 , which is
even in metals high-temperature superconductivity might be itself in pressure equilibrium with the ambient field B0 under
achieved more easily for anisotropic temperatures, a point we static conditions expressed by N0 T = B02 /2µ0 under the as-
will follow elsewhere (Treumann and Baumjohann, 2018b). sumption that no static current J 0 flows in the medium. Oth-
Since the plasma is ideally conducting, any quasi- erwise its Lorentz force J 0 × B 0 = −T ∇N0 is compensated
stationary magnetic field is subject to the penetration for by the pressure gradient force already in the absence of
depth, which is the inertial scale λim = c/ωim , with ωim 2 = the mirror mode and includes the magnetic stresses generated
2
e Nm /0 mi based on the density Nm of the plasma compo- by the current. This case includes a stationary contribution of
nent involved in the mirror effect. The mirror instability is a the free energy F0 around which the mirror state has evolved.
slow purely growing instability with real frequency ω ≈ 0. At Regarding the presence of the mirror mode, we know that
these low frequencies the plasma is quasi-neutral. In metal- it must also be in balance between the local plasma gradi-
lic superconductivity this length is the London penetration ent ∇Nm of the fluctuating pressure and the induced mag-
depth which refers to electrons as the ions are fixed to the netic pressure (δB)2 /2µ0 . Note that all quantities are station-
lattice. Here, in the space plasma, it is rather the ion scale be- ary; the prefix δ refers to deviations from “normal” thermo-
cause the dominant mirror effect is caused by mobile ions in dynamic equilibrium, not to variations. Moreover, we have
the absence of any crystal lattice. Such a “magnetic lattice” Maxwell’s equations which in the stationary state reduce to
structure is ultimately provided under conditions investigated
below by the saturated state of the mirror mode, where it col- ∇ × δB = µ0 δJ , and δB = ∇ × A, (7)
lectively affects the trapped ion component on scales of an
internal correlation length. accounting for the vanishing divergence by introducing the
fluctuating vector potential A (where we drop the δ prefix on
the vector potential). This enables us to write the kinetic part
3 Free energy of the free energy of the particles involved in the canonical
operator form
In the thermodynamic equilibrium state the quantity which
describes the matter in the presence of a magnetic field B is p2 1 2
the Landau–Gibbs free energy density = − iα∇ − qA , (8)
2m 2m
1
GL = FL − δB · B, (4) referring to ions of positive charge q > 0, and the constant
2µ0 α naturally has the dimension of a classical action. (There is
where FL is the Landau free energy density (Kittel and Kroe- a little problem as to what is meant by the mass m in this
mer, 1980) which, unfortunately, is not known. In magneto- expression, to which we will briefly return below.) In this
hydrodynamics it can be formulated but becomes a messy form the momentum acts on a complex dimensionless “wave
expression which contains all stationary, i.e. time-averaged, function” ψ(x) whose square
nonlinear contributions of low-frequency electromagnetic
2
plasma waves and thermal fluctuations. The total Landau– ψ(x) = ψ ∗ (x)ψ(x) = Nm (9)
Gibbs free energy is the volume integral of this quantity over
all space. In thermodynamic equilibrium this is stationary, we below identify with the above-used normalized excess
and one has in plasma density known to be present locally in any of the
d
Z mirror-mode bubbles.
d 3 x GL = 0. (5) Unlike quantum theory, ψ(x) is not a single-particle wave
dt
function: it rather applies to a larger compound of trapped
In order to restrict to our case we assume that FL in the above
particles (ions) in the mirror modes which behave similarly
expression, which contains the full dynamics of the plasma
and are bound together by some correlation length (a very
matter, can be expanded with respect to the normalized den-
important parameter, which is to be discussed later). It en-
sity Nm < 1 of the plasma component which participates in
ters the expression for the free energy density, thus provid-
the mirror instability:
ing the units of energy density to the expansion coefficients
1 a, b. In the quantum case (as for instance in the theory of su-
FL = F0 + aNm + bNm2 + . . ., (6)
2 perconductivity), we would have α = }; in the classical case
with F0 the Helmholtz free energy density, which is inde- considered here, α remains undetermined until a connection
pendent of Nm corresponding to the normal (or mirror sta- to the mirror mode is obtained. Clearly, α } cannot be
ble) state. Normalization is to the ambient density N0 , thus very small because the gradient and the corresponding wave
www.ann-geophys.net/36/1015/2018/ Ann. Geophys., 36, 1015–1026, 20181018 R. A. Treumann and W. Baumjohann: Mirror instability
vector k involved in the operation ∇ are of the scale of the prescription and varying the Gibbs free energy with respect
inverse ion gyroradius in the mirror mode. Hence, we sus- to A and ψ, ψ ∗ yields (for arbitrary variations) an equation
pect that α ∝ T /ωp , where T is a typical plasma temperature for the “wave function” ψ(x):
(in energy units), and ωp is a typical frequency of collective
ion oscillations in the plasma. Any such oscillations naturally 1 2 2
− iα∇ − qA + a + b ψ ψ = 0, (13)
imply the existence of correlations which bind the particles 2m
(ions) to exert a collective motion and which give rise to the
field A and density fluctuations
√ δN . Such frequencies can be which is recognized as a nonlinear complex Schrödinger
either plasma ωp = ωi = e N/0 m, cyclotron ωc = eB/m equation. Such equations appear in plasma physics whence
frequencies, or some unknown average turbulent frequency waves undergo modulation instability and evolve towards the
ωturb on turbulence scales shorter than the typical average general family of solitary structures.
mirror-mode scale. For the ion mirror mode the choice is that It is known that the nonlinear Schrödinger equation can be
q ∝ +e, and m ∝ mi . solved by inverse scattering methods and, under certain con-
Inspecting Eq. (8) we will run into difficulties when as- ditions, yields either single solitons or trains of solitary solu-
suming q = e and m = mi because with a large number tions. To our knowledge, the nonlinear Schrödinger equation
of particles collectively participating, each contributing a has not yet been derived for the mirror instability because
charge e and mass m, the ratio q 2 /m will be proportional no slow wave is known which would modulate its amplitude.
to the number of particles. In superconductivity this provides Whether this is possible is an open question which we will
no problem because pairing of electrons tells that mass and not follow up here. Hence the quantity α remains undeter-
charge just double, which is compensated for in Eq. (8) by mined for the mirror mode. Instead, we chose a phenomeno-
m → 2m. Similarly, in the case of the mirror mode we have logical approach which is suggested by the similarity of both
for the normalized density excess Nm = δN /N ≡ ζ < 1, the mirror-mode effect in ideally conducting plasma and the
where N is the total particle number, and δN its excess. We above-obtained nonlinear Schrödinger equation to the phe-
thus identify an effective mass m∗ ≡ 1mi , where 1 = 1 + ζ . nomenological Landau–Ginzburg theory of metallic super-
Because of the restriction on ζ < 1 this yields for the effec- conductivity.
tive mass in mirror modes the preliminary range In the thermodynamic equilibrium state the above equa-
mi < m∗ < 2mi , (10) tion does not describe the mirror-mode amplitude itself.
Rather it describes the evolution of the “wave function” of
which is similar to the mass in metallic superconductivity. the compound of particles trapped in the mirror-mode mag-
However, each mirror bubble contains a different number δN netic potential A which it modulates. This differs from su-
of trapped particles. Hence ζ (x) becomes a function of space perconductivity where we have pairing of particles and es-
x which varies along the mirror chain, and 1(x) then be- cape from collisions with the lattice and superfluidity of the
comes a function of space. The restriction on ζ < 1 makes paired particle population at low temperatures. In the ideally
this variation weak. For an observed chain of mirror modes conducting plasma we have no collisions, but, under normal
one defines some mean effective mass meff by conditions, also no pairing and no superconductivity, though
meff ≡ m∗ (x) = 1(x) mi . (11) in the presence of some particular plasma waves, attractive
forces between neighbouring electrons can sometimes evolve
Averaging reduces 1, making the effective mass closer to (Treumann and Baumjohann, 2014). In superconductivity the
the lower bound mi , which is to be used below for m → meff pairing implies that the particles become correlated, an effect
wherever the mass appears. which in plasma must also happen whence the superconduct-
Retaining the quantum action and dividing by the charge ing mirror-mode Meissner effect occurs, but it happens in
q, the factor of the Nabla operator becomes }/q = 80 e/2π q. a completely different way by correlating large numbers of
Hence, α is proportional to the number ν = 8/80 of elemen- particles, as we will exemplify further below.
tary flux elements in the ion-gyro cross section, which in a The wave function ψ(x) describes only the trapped par-
plasma is a large number due to the high temperature T⊥ . ticle component which is responsible for the maintenance
This makes α }. of the pressure equilibrium between the magnetic field and
With these assumptions in mind we can write for the free plasma. In a bounded region one must add boundary condi-
energy density up to second order in Nm : tions to the above equation which imply that the tangential
1 1 2 component of the magnetic field is continuous at the bound-
2 4
F = F0 + a ψ + b ψ + − iα∇ − qA ψ ary and the normal components of the electric currents vanish
2 2m
δB · B 0 at the boundary because the current has no divergence. The
+ . (12) current, normalized to N0 , is then given by
2µ0
Inserted into the Gibbs free energy density, the last term is iqα ∗ q2
2
absorbed by the Gibbs potential. Applying the Hamiltonian δJ = ψ ∇ψ − ψ∇ψ ∗ − ψ A, (14)
2m m
Ann. Geophys., 36, 1015–1026, 2018 www.ann-geophys.net/36/1015/2018/R. A. Treumann and W. Baumjohann: Mirror instability 1019
which shows that the main modulated contribution to the cur- which implies that either a or b is negative. In addition there
rent is provided by the last term, the product of the mirror is the trivial solution ψ = 0 which describes the initial stable
2 state when no instability evolves. The Helmholtz free energy
particle density ψ = Nm times the vector potential fluc-
tuation A, which is the mutual interaction term between the density in this state is F = F0 . Equation (12) shows that the
density and magnetic fields. One may note that the vector thermodynamic equilibrium has free energy density
potential from Maxwell’s equations is directly related to the 2
magnetic flux 8 in the wave flux tube of radius R through its q 2 aN0 2 a 2 q 2 aN0 2 Bcrit
F = F0 − A − = F0 − A − , (19)
circumference A = 8/2π R. 2mb 2b 2mb 2µ0
One also observes that under certain conditions in the
last expression for the current density the two gradient where the last term is provided by the critical magnetic field,
terms of the ψ function partially cancel. Assuming ψ = which is the external magnetic field. Thus b > 0 and a < 0,
ψ(x) e−ik·x , the current term becomes and the dependence on temperature τ can be freely attributed
to a. Comparison with Eq. (2) then yields
qα 2 q2 2
kψ ψ A. (15)
s
δJ = − b −1 1
m m 0
a(τ ) = −Bcrit τ 2 1−τ 2. (20)
The first term is small in the long-wavelength domain kα µ0
1. Assuming that this is the case for the mirror mode, which
At critical field one still has A = 0. Hence the density at crit-
implies that the first term is important only at the bound-
ical field is
aries of the mirror bubbles where it comes up for the diamag-
netic effect of the surface currents, the current is determined |a(τ )| B0 1 1
mainly by the last term, which can be written as Nm (τ ) = = √ crit τ − 2 1 − τ 2 , (21)
b bµ0
q 2 N0 2
δJ ≈ − Nm A = −0 ωim A. (16) which shows that the distortion of the density vanishes for
m τ = 1, as it should be. This expression can be used in the
This is to be compared to µ0 δJ = −∇ 2 A, thus yielding the magnetic penetration depth to obtain its critical temperature
penetration depth dependence
λim (τ ) = c/ωim (τ ), (17) 1
m2 b
τ 4
λim (τ ) = 2 m, (22)
which is the ion inertial length based on the relevant tempera- 0
µ0 q 4 N0 Bcrit 1−τ
ture dependence of the particle density Nm (τ ) for the mirror
mode, where we should keep in mind that the latter is nor- which suggests that the critical penetration depth vanishes
malized to N0 . Thus, identifying the reference temperature for τ = 0. However, τ = 0 is excluded by the argumenta-
as T⊥ , we recover the connection between the mirror-mode tion following Eqs. (2) and (21), because it would imply infi-
penetration depth and its dependence on temperature ratio τ nite trapped densities. In principle, τ ≥ τmin cannot become
from thermodynamic equilibrium theory in the long wave- smaller than a minimum value which must be determined by
length limit with main density N0 constant on scales larger other methods referring to measurements of the maximum
than the mirror-mode wavelength. density in thermodynamic equilibrium. One should, however,
0 (θ ) ∝ sin θ still depends on the angle
keep in mind that Bcrit
θ which enters the above expressions.
4 Magnetic penetration scale The last two expressions still contain the undetermined co-
efficient b. This can be expressed through the minimum value
So far we considered only the current. Now we have to re- of the anisotropy τmin at maximum critical density Nm .1 as
late the above penetration depth to the plasma, the mirror
mode. What we need is the connection of the mirror mode 2
B0
b = crit −1
to the nonlinear Schrödinger equation. Because treating the τmin 1 − τmin . (23)
nonlinear Schrödinger equation is very difficult even in two µ0
dimensions, this is done in one dimension, assuming for in- (Note that for Nm > 1 the above expansion of the free en-
stance that the cross section of the mirror structures is circu- ergy F becomes invalid. It is not expected, however, that the
lar with the relevant dimension the radius. In the presence of mirror mode will allow the evolution of sharp density peaks
a magnetic wave field A 6 = 0, Eq. (13) under homogeneous or which locally double the density.) With this expression the
nearly homogeneous conditions, with the canonical gradient inertial length becomes
term neglected, has the thermodynamic equilibrium solution
1
2 a q 2 N0 2 λim (τ ) τ 1 − τmin 4
Nm = ψ =− − A > 0, (18) = . (24)
b 2mb λi0 τmin 1−τ
www.ann-geophys.net/36/1015/2018/ Ann. Geophys., 36, 1015–1026, 20181020 R. A. Treumann and W. Baumjohann: Mirror instability
When the mirror mode saturates away from the critical that α is not an elementary constant like }. It depends on the
field, the magnetic fluctuation grows until it saturates as well, critical reference temperature T⊥ , and it depends on τ . Its
and one has A 6 = 0. The increased fractional density Nm is in constancy is understood in a thermodynamic sense.
perpendicular pressure equilibrium with the magnetic field Our argument applies when A 6 = 0. In this case Eq. (13)
distortion δB through reads as
1 B02 α 2 d 2 f (x)
− f (1 − f 2 ) = 0,
2
Nsat T⊥ = B 0 − δB sat − −
2µ0 N0 2µ0 N0 2ma dx 2
1 h 2 2 2 i B0 · ∇ × A ψ(x)
= ∇ A − ∇A − where f (x) =R. A. Treumann and W. Baumjohann: Mirror instability 1021
1.0 Landau-Ginzburg parameter κρ
1.0
N m /N max
τ = T|| /T
T
κρ /κρ sat
0.0 τsat = 0.25
0.0 0.5 1.0 x/λ√2
0.0
√ τsat 1.0 τ
Figure 1. Shape of excess density in dependence on x/ 2λα . The
shape of the magnetic field depression can be obtained directly from
Figure 2. The Landau–Ginzburg parameter κρ /κρ,sat as a func-
pressure balance. It mirrors the excess density.
tion of the anisotropy ratio τ = Tk /T⊥ < 1 for the particular choice
τsat = 41 . The parameter κρ refers to the thermal gyroradius as the
short-scale correlation length, as explained in the text. It maximizes
Identification of α is an important step. With its knowl-
at saturation anisotropy τ = τsat and vanishes for τ = 1 when no
edge in mind the nonlinear Schrödinger equation for the hy- instability sets in. For any given ratio τ the value of κρ lies on a
pothetical saturation state of the mirror mode is (up to the curve like the one shown. There is a threshold for the mirror mode
coefficient b, which, however, is defined in Eq. (23) and can to evolve into bubbles which it must overcome. It is given by the ra-
be obtained from measurement) completely determined and tio κρ,sat > 1 of the critical Alfvén speed to perpendicular thermal
thus ready for application of the inverse scattering proce- velocity.
dure 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. Execut- not yet been reported. On the other hand, in no case known
ing this programme should, under various conditions, pro- to us has a global reduction of the gross magnetic field in an
vide the different forms of the mirror mode in its final ther- anisotropic plasma been identified yet.
modynamic equilibrium state. This is left as a formally suf- It is clear that in any real collisionless high-temperature
ficiently complicated exercise which will not be treated in plasmas neither can Nm become infinite nor can τ drop to
the present communication. Instead, we ask for the condi- zero. Since it is not known how and in which way, i.e. by
tions under which the mirror mode evolves into a chain of which exactly known process mirror modes saturate in their
separated mirror bubbles, which requires the existence of a final thermodynamic equilibrium state, their growth must ul-
microscopic though classical correlation length. timately 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
6 The problem of the correlation length types of candidates for an effective correlation length, the
Debye scale λD , the ion gyroradius ρ, and some turbulent
The present phenomenological theory of the final thermo-
correlation length `turb .
dynamic equilibrium state of the mirror mode is modelled
In a plasma the shortest natural correlation length is the
after the phenomenological Landau–Ginzburg theory of su-
Debye length λD which under all conditions is much shorter
perconductivity as presented in the cited textbook literature.
than the above-estimated penetration length λim . Referring to
From the existence of λim we would conclude that, under
the Debye length, the Landau–Ginzburg parameter, i.e. the
mirror instability, the magnetic field inside the plasma vol-
ratio of penetration to correlation lengths, in a plasma as a
ume should decay to a minimum value determined by the
function of τ becomes
achievable minimum τsat of the temperature ratio. This con-
clusion would, however, be premature and contradicts obser- λim (τ ) c
vation where chains or trains (cf. e.g. Zhang et al., 2009, κD ≡ ≈ 1, (33)
λD (τ ) υ⊥th (τ )
for examples) of mirror-mode fluctuations are usually ob-
served (though isolated “solitary mirror” modes have also oc-
a quantity that is large. Writing for the Debye length
casionally been reported; see e.g. Luehr and Kloecker, 1987;
Treumann et al., 1990, where they were dubbed “magnetic
1 + τ/2 4 T⊥ /mi
cavities”), which presumably are in their saturated state hav- λ2D (τ ) = λ2D⊥ , λ2D⊥ = 2
, (34)
ing had sufficient time to evolve beyond quasilinear satura- Nm (τ ) 3 ωi0
tion times and reached saturation amplitudes much in excess
of any predicted quasilinear level. In fact, observations of the Landau–Ginzburg parameter can be expressed in terms
mirror modes in their growth phase have to our knowledge of τ , exhibiting only a weak dependence on the temperature
www.ann-geophys.net/36/1015/2018/ Ann. Geophys., 36, 1015–1026, 20181022 R. A. Treumann and W. Baumjohann: Mirror instability
ratio τ < 1: through the magnetic moment
s s r s
λi0 2 2µ(τ ) τ 2mT⊥
κD (τ ) = 1. (35) ρ(τ ) = = ρ0 , ρ0 =
0 2
, (36)
λD⊥ 1 + τ/2 eωci (τ ) 1−τ e2 Bcrit
it can be taken as another kind of collective correlation scale
Thus, κD is practically constant and about independent on as on scales larger than ρ it collectively binds particles of
the temperature anisotropy. Its value κD0 = λi0 /λD at τ = 1, the same magnetic moment which, in particular, are mag-
Tk = T⊥ refers to the isotropic case when no mirror instabil- netically trapped like those which are active in the mirror
ity evolves. instability. Below the gyroradius charged particles are mag-
This is an important finding because it implies that in a netically free. ρ is the scale where the particles magnetize,
plasma the case that the magnetic field would be completely start feeling the magnetic field effect, and collectively enter
expelled from the volume of the plasma cannot be realized. another phase in their dynamics. This scale is much larger
Different regions of extension substantially larger than λD than the Debye length and may be more appropriate for de-
are (electrostatically) uncorrelated. They therefore behave scribing the saturated behaviour of the mirror mode. Thus
separately, lacking knowledge about their (electrostatically) one may argue that, as long as the penetration depth (inertial
uncorrelated neighbours separated from them at distances sale) exceeds ρ, the thermal gyroradius is the relevant corre-
substantially exceeding λD . Each of them experiences the lation length. Only when it drops below the gyroradius does
penetration scale and adjusts itself to it. This is in complete the Debye length take over. The Landau–Ginzburg parameter
analogy to Landau–Ginzburg theory. Thus, once the main then becomes
magnetic field in an anisotropic plasma drops below thresh-
old, the plasma will necessarily evolve into a chain of nearly 1
λim (τ ) λi0 τsat 1 − τ 4
unrelated mirror bubbles which interact with each other be- κρ (τ ) = = . (37)
ρ(τ ) ρ0 τ 1 − τsat
cause each occupies space. In superconductivity this corre-
sponds to a type II superconductor. Mirror unstable plasmas This ratio depends on the temperature anisotropy τ = Tk /T⊥ ,
in this sense behave like type II superconductors. They de- which is a measurable quantity and the important parame-
cay into regions of normal magnetic field strength and em- ter, while it saturates at κρ,sat = λi0 /ρ0 , the ratio of inertial
bedded domains of spatial-scale λm (τ ) with a reduced mag- length to gyroradius at the critical field. This ratio is not nec-
netic field. These regions contain an excess plasma popula- essarily large. It can be expressed by the ratio of Alfvén ve-
tion which is in pressure and stress balance with the mag- locity VA to perpendicular ion-thermal velocity υ⊥th :
netic field. Its diamagnetism (perpendicular pressure) keeps
0
the magnetic field partially out and causes weak diamagnetic λi0 VA Bcrit
currents to flow along the boundaries of each of the partially κρ,sat = = > 1. (38)
ρ0 υ⊥th
field-evacuated domains. This trapped plasma behaves anal-
ogously to the pair plasma in metallic superconductivity, this Hence, when referring to the thermal-ion gyroradius as the
time however at the high plasma temperature being bound correlation length, the mirror mode would evolve and satu-
together not by pairing potentials, but – in the case of the rate into a chain of mirror bubbles only, when the Alfvén
Debye length playing the role of the correlation length – by speed VA > υ⊥th exceeds the perpendicular thermal veloc-
0 ∝ sin θ , highly oblique angles
ity of the ions. (Since Bcrit
the Debye potential over the Debye correlation length.
However, the Debye length is a very short scale, in fact are favoured. The range of optimum angles has recently been
the shortest collective scale in the plasma, and though it must estimated in Treumann and Baumjohann, 2018a.) This is to
have an effect on the collective evolution of particles in plas- be multiplied by the τ dependence, of which Fig. 2 gives an
mas, it should be doubted that, on the mirror-mode saturation example. The value of this function is always smaller than
scale, it would have a substantial or even decisive effect. In- one. For a chain of mirror bubbles to evolve in a plasma, the
stead, there could also be larger scales on which the particles requirement κρ > 1 can then be written as
are correlated.
4
κρ,sat
Such a scale is, for instance, the thermal-ion gyroradius τ
1≤ < , (39)
ρ(τ ). For the low frequencies of the mirror mode, the mag- τsat 4
1 + κρ,sat − 1 τsat
netic moment µ(τ ) = T⊥ /B(τ ) = const of the particles is
conserved in their dynamics, which implies that all particles which is always satisfied for τsat < 1 and κρ,sat > 1, i.e. the
with the same magnetic moment µ(τ ) behave about collec- Alfvén speed exceeding the perpendicular thermal speed,
tively, at least in the sense of a gyro-kinetic theory. which indeed is the crucial condition for mirror modes to
However, though µ(τ ) is a constant of motion, it still evolve into chains and become observable, with the gyrora-
is a function of the anisotropy through the dependence of dius playing the role of a correlation length. Mirror-mode
the magnetic field on τ . Expressing the thermal gyroradius chains in the present case are restricted to comparably cool
Ann. Geophys., 36, 1015–1026, 2018 www.ann-geophys.net/36/1015/2018/R. A. Treumann and W. Baumjohann: Mirror instability 1023
anisotropic plasma conditions, a prediction which can be its nature. The above theory should open a way of relating
checked experimentally to decide whether or not the gyro- a turbulent correlation length to the properties of a mirror
radius serves as a correlation length. unstable plasma. The condition is simply that the turbulent
Otherwise, when the above condition is not satisfied and Landau–Ginzburg parameter
τ < 1 is below threshold, a very small and thus probably
λim (τ )
not susceptible reduction in the overall magnetic field is κturb (τ ) = >1 (40)
produced in the anisotropic pressure region over distances `turb (τ )
L ρ, much larger than the ion gyroradius. Observation is large, depending on the anisotropy parameter τ and the av-
of such domains of reduced magnetic field strengths under erage transverse scales of the mirror bubbles. This expression
anisotropic pressure/temperature conditions would indicate yields an upper limit for the turbulent correlation length
the presence of a large-scale type I classical Meissner effect
in the plasma. Such a reduction of the magnetic field would `turb (τ ) < λim (τ ) , (41)
be difficult to explain otherwise and could only be under- where λim (τ ) is known as a function of τ and the plasma
stood as confinement of plasma by discontinuous boundaries parameters. Investigating this in further detail both observa-
of the kind of tangential discontinuities. tionally and theoretically should throw additional light on the
The relative rarity of observations of mirror-mode chains nature of magnetic turbulence in high-temperature plasmas
or trains seems to support the case that the gyroradius, not like those of the solar wind and magnetosheath. It would even
the Debye length, plays the role of the correlation length in a contribute to a more profound understanding of magnetic tur-
magnetized plasma under conservation of the magnetic mo- bulence in general as well as in view of its application to as-
ments of the particles. From basic theory it cannot be decided trophysical problems.
which of the two correlation lengths, the Debye length λD or
the ion gyroradius ρ, dominates the dynamics and saturation
of the mirror mode. A decision can only be established by 7 Conclusions
observations.
The mirror mode is a particular zero-frequency mesoscale
However, the thermal-ion gyroradius, though the statisti-
plasma instability which provides some mesoscopic struc-
cal average of the distribution of gyroscales, is itself just a
ture to an anisotropic plasma. It has been observed surpris-
plasma parameter which officially lacks the notion of a gen-
ingly frequently under various conditions in space, in the so-
uine correlation length. For this reason one would rather re-
lar wind, cometary environments, near other planets and, in
fer to the third possibility, a turbulent correlation length `turb
particular, behind the bow shock (Czaykowska et al., 1998),
which evolves as the result of either high-frequency plasma
such that one also believes that they occur in shocked plas-
or – in the case of mirror modes probably better suited – mag-
mas if the shock causes a temperature anisotropy τ < 1 (cf.
netic turbulence in the plasma.
e.g. Balogh and Treumann, 2013, chap. 4). Since mirror
It is well known that, for instance, the solar wind or
modes are long scale, they provide the plasma with a very
the magnetosheath carries a substantial level of turbulence
particular spatial texture. Mirror unstable plasmas are appar-
which mixes plasmas of various properties and obeys a par-
ently built of a large number of magnetic bottles which con-
ticular spectrum. In the solar wind such spectra have been
tain a trapped particle population. This makes mirror modes
shown to exhibit approximate Kolmogorov-type properties,
most interesting even in magnetohydrodynamic terms as a
at least in certain domains of frequencies or wave numbers,
kind of long-wavelength source of turbulence. In addition,
and similarly in the magnetosheath, where the conditions
their boundaries are surfaces which separate the bottles and
are more complicated because of the boundedness of the
thus have the character or tangential discontinuities or sur-
magnetosheath and the resulting spatial confinement of the
faces of diamagnetic currents which are produced by the in-
plasma and its streaming. Such spectra imply that particles
ternal interaction between the plasma and magnetic field. We
and waves are not independent but contain some information
have shown above that such an interaction resembles super-
about their behaviour in different spatial and frequency do-
conductivity, i.e. a classical Meissner effect.
mains; in other words, they are correlated.
Mirror modes in the anisotropic collisionless space plasma
Unfortunately, the turbulent correlation length is impre-
apparently represent a classical thermodynamic analogue to
cisely defined. No analytical expressions have been provided
a “superconducting” equilibrium state. One should, how-
yet which would allow us to refer to it in the above deter-
ever, not exaggerate this analogy. This equilibrium state is
mination of the Landau–Ginzburg parameter. This inhibits
no macroscopic quantum state. It is a classical effect. The
prediction of the range and parameter dependences of the
analogy is just formal, even though it allows us to conclude
turbulent Landau–Ginzburg ratio. Nonetheless, turbulent cor-
about the final mirror equilibrium. Sometimes such an ana-
relation scales might dominate the development of the mir-
logue helps in understanding the underlying physics1 like
ror mode. The observation of a spectrum of mirror modes
that is highly peaked around a certain wavelength not very 1 In a recent paper (Treumann and Baumjohann, 2018c), we have
much larger than the ion gyroradius may tell something about shown that a classical Higgs mechanism is responsible for bend-
www.ann-geophys.net/36/1015/2018/ Ann. Geophys., 36, 1015–1026, 20181024 R. A. Treumann and W. Baumjohann: Mirror instability
here, where it paves the way to a global understanding of low. The parallel temperature cannot drop below a minimum
the final saturation state of the mirror mode even though this value. This value is open to determination by observations.
does not release us from understanding in which way this The observation of chains of mirror bubbles, for instance
final state is dynamically achieved. in the magnetosheath, which provide the mirror-unstable
In contrast to metallic superconductivity which is de- plasma with a particular intriguing magnetic texture, sug-
scribed by the Landau–Ginzburg theory to which we refer gests that the plasma, in addition to being mirror unstable,
here or, on the microscopic quantum level, by BCS-pairing is subject to some correlation length which determines the
theory, the problem of circumventing friction and resistance spatial structure of the mirror texture in the saturated ther-
is of no interest in ideally conducting space plasmas which modynamic quasi-equilibrium state. This correlation length
evolve towards mirror modes. High temperature plasmas are can be either taken as the Debye scale λD , which then nat-
classical systems in which no pairing occurs and BCS theory urally makes it plausible that many such mirror bubbles
is not applicable. Those plasmas are already ideally conduct- evolve, because in all magnetized plasmas the magnetic pen-
ing. In contrast, there is a vital interest in the opposite prob- etration depth by far exceeds the Debye length and makes
lem, how a finite sufficiently large resistance can develop un- the Landau–Ginzburg parameter based on the Debye length
der conditions when collisions and friction among the parti- κD 1. This, however, should lead to rather short-scale mir-
cles are negligible. This is the problem of generating anoma- ror bubbles. Otherwise, the role of a correlation length could
lous resistivity which may develop from high-frequency ki- also be played by the thermal-ion gyroradius ρ. In this case
netic instabilities or turbulence and is believed to be urgently the conditions for the evolution of the mirror mode with
needed, for instance causing dissipation in reconnection. In the many observed bubbles become more subtle, because
the zero-frequency mirror mode it is of little importance even then κρ &1 occurs under additional restrictions, implying that
asymptotically, in the long-term thermodynamic limit, where the Alfvén speed exceeds the perpendicular thermal speed.
such an anomalous resistance may contribute to decay of This prediction has to be checked and possibly verified ex-
the mirror-surface currents which develop and flow along the perimentally. A particular case of the dependence of the
boundaries of the mirror bubbles. The times when this hap- gyroradius-based Landau–Ginzburg parameter κρ is shown
pens are very long compared with the saturation time of the graphically in Fig. 2.
mirror instability and transition to the thermodynamic quasi- It may be noted that the Debye length and the ion gyro-
equilibrium which has been considered here. radius are fundamental plasma scales. Correlations can of
The more interesting finding concerns the explanation course also be provided by other means, in particular by any
why at all, in an ideally conducting plasma, mirror bub- form of turbulence. In that case a turbulent correlation length
bles can evolve. Fluid and simple kinetic theories demon- would play a similar role in the Landau–Ginzburg parameter,
strate that mirror modes occur in the presence of tempera- whether shorter or larger than the above-identified penetra-
ture anisotropies, thereby identifying the linear growth rate tion scale. Regarding mirror modes in the magnetosheath to
of the instability. Trapping of large numbers of charged par- which we referred (Treumann and Baumjohann, 2018a), it is
ticles (ions, electrons) in accidentally forming magnetic bot- well known that the magnetosheath hosts a broad turbulence
tles/traps causes the mirror instability to grow. The present spectrum in the magnetic field as well as in the dynamics of
theory contributes to clarification of this mechanism and its the plasma (fluctuations in the velocity and density).
final thermodynamic equilibrium state as a nonlinear effect Though this makes it highly probable that turbulence inter-
which is made possible by the available free energy which venes and affects the evolution of mirror modes, any “turbu-
leads to a particular nonlinear Schrödinger equation. The lent correlation length” is, unfortunately, rather imprecisely
perpendicular temperature in this theory plays the role of defined as some average quantity. To our knowledge, though
a critical temperature. When the parallel temperature drops referring to multi-spacecraft missions is not impossible, it
below it, which means that 1 > τ > τmin , mirror modes can has even not yet been precisely identified in any observa-
evolve. Interestingly the anisotropy is restricted from be- tions of turbulence in space plasmas. Even when identified,
its functional dependence on temperature and density is re-
ing the free space O-L and X-R electromagnetic modes in their quired for application in our theory. If these functional de-
long-wavelength range away from their straight vacuum shape when pendencies are not available, it becomes difficult to include
passing a plasma. The plasma in that case acts like a Higgs field any turbulent correlation length. In addition, one expects that
and attributes a tiny mass to the photons, making them heavy. This its turbulent nature would make the theory nonlocal. At-
is interesting because it shows that any bosons become heavy only
tempts in that direction must, at this stage of the investiga-
in permanent interaction with a Higgs field and only in a certain
energy–momentum–wavelength range. It also shows that earlier at-
tion, be relegated to future efforts.
tempts at measuring a permanent photon mass by observing scin- Finally, it should be noted that the magnetic penetration
tillations of radiation (and also by other means) have just measured depth λm which lies at the centre of our investigation is
this effect. Their interpretations as upper limits for a real perma- rather different from the ordinary inertial length scale of
nent photon mass are incorrect because they missed the action of the plasma. It is based on the excess density Nm < 1 less
the plasma as a classical Higgs field. than the bulk plasma density N0 . It thus gives rise to an
Ann. Geophys., 36, 1015–1026, 2018 www.ann-geophys.net/36/1015/2018/R. A. Treumann and W. Baumjohann: Mirror instability 1025
√
enhanced
√ (excess)
√ plasma frequency ωm = ωi Nm + 1 = Data availability. No data sets were used in this article.
ωi 1 + ζ . 2 ωi , which implies that L > c/ωi > λm is
shorter than the typical scale of the volume L and (slightly)
shorter than the bulk inertial length c/ωi . This becomes clear Competing interests. The authors declare that they have no conflict
when recognizing that the mirror mode evolves inside the of interest.
plasma from some thermal fluctuation (cf. Yoon and López,
2017, for the calculation of low-frequency thermal magnetic
fluctuation levels in a stable isotropic plasma; similar calcu- Acknowledgement. This work was part of a Visiting Scientist Pro-
lations in stable anisotropic plasmas have not yet been per- gramme at the International Space Science Institute (ISSI) Bern. We
acknowledge the interest of the ISSI directorate and the hospitality
formed), which causes the magnetic field locally to drop be-
of the ISSI staff. We thank the referees Karl-Heinz Glassmeier (TU
low its critical value – Eq. (2). Then λm identifies the local Braunschweig, DE), Ovidiu Dragos Constantinescu (U Bukarest,
perpendicular scale of a mirror bubble after it has saturated RO), and Hans-Reinhard Mueller (Dartmouth College, Hanover,
and is in thermodynamic equilibrium. One expects that the NH, USA) for their critical remarks on the manuscript, in particular
transverse diameter of a single mirror bubble in the ideal case the non-trivial questions on the effective mass and the role of
would be roughly 2λm . However, since each bubble occu- turbulence.
pies real space, in a mirror-saturated plasma state the bubbles
compete for space and distort each other (cf. e.g. Treumann The topical editor, Anna Milillo, thanks Karl-Heinz Glassmeier,
and Baumjohann, 1997, for a sketch), thereby providing the Dragos Constantinescu, and Hans-Reinhard Müller for help in
plasma with an irregular magnetic texture of some, probably evaluating this paper.
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. References
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