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

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

The mirror mode is a non-oscillatory plasma instability

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

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

In the following we boldly ask for the

The mirror instability evolves whence the magnetic field

The important observation is that the existence of the mirror mode depends on
the temperature difference

The

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

Since the plasma is ideally conducting, any quasi-stationary magnetic field
is subject to the penetration depth, which is the inertial scale

In the thermodynamic equilibrium state the quantity which describes the
matter in the presence of a magnetic field

Regarding the presence of the mirror mode, we know that it must also be in
balance between the local plasma gradient

Unlike quantum theory,

Inspecting Eq. (

Retaining the quantum action and dividing by the charge

With these assumptions in mind we can write for the free energy density up to
second order in

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

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

The wave function

One also observes that under certain conditions in the last expression for
the current density the two gradient terms of the

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

The last two expressions still contain the undetermined coefficient

When the mirror mode saturates away from the critical field, the magnetic
fluctuation grows until it saturates as well, and one has

The last two lines yield for the macroscopic penetration depth the expression
Eq. (

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

Our argument applies when

Shape of excess density in dependence on

Since the quantum of action enters the magnetic quantum flux element

Identification of

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

The Landau–Ginzburg parameter

It is clear that in any real collisionless high-temperature plasmas neither
can

In a plasma the shortest

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

However, the Debye length is a

Such a scale is, for instance, the thermal-ion gyroradius

However, though

Otherwise, when the above condition is not satisfied and

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

However, the thermal-ion gyroradius, though the statistical average of the

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

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

Mirror modes in the anisotropic collisionless space plasma apparently
represent a classical

In a
recent paper

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

The more interesting finding concerns the explanation

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

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

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

It also relates the measurable saturated magnetic amplitudes of mirror modes
to the saturated anisotropy

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.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

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.