The mirror mode (Chandrasekhar, 1961; Vedenov et al., 1961; Hasegawa, 1969; Davidson, 1972; Hasegawa, 1975; Gary, 1993; Southwood and Kivelson, 1993; Kivelson and Southwood, 1996; Pokhotelov et al., 2000; Pokhotelov et al., 2001; Constantinescu, 2002; Pokhotelov et al., 2002; Constantinescu et al., 2003; Pokhotelov et al., 2004; Sulem, 2011; Rincon et al., 2015; Noreen et al., 2017; Yoon, 2017) which, in high temperature plasma, evolves under anisotropic A i = P i ⊥ / P i ∥ − 1 > 0 pressure conditions, can be interpreted as a phase transition from (unstable) normal to a (stationary) second-kind quasi-superconducting state (Treumann and Baumjohann, 2019; Treumann and Baumjohann, 2020). It causes the plasma to become magnetically perforated. This raises the question, investigated in this letter, in what way closely spaced mirror bubbles may interact, possibly producing identifiable effects other than localized diamagnetic field depletions.

This second-kind superconducting phase transition (Ginzburg and Landau, 1950) is known from low temperature solid state physics (Bardeen et al., 1957; Callaway, 1990), evolving Meissner diamagnetism based on electron pairing and condensation that pushes the magnetic field locally out. In mirror modes the possibility of similar condensations has recently been demonstrated (Treumann and Baumjohann, 2019). The transition is initiated by the mirror instability, which starts under the necessary condition of positive ion pressure anisotropy A i > 0 . In addition, the sufficient condition for instability requires the magnetic field strength B to drop below a threshold B_c B < B c ≈ 2 μ 0 N T i ⊥ A i | sin θ m | (1) (with temperature in energy units) which follows from the linear ion-mirror growth rate γ ω c i ≈ k ∥ λ i 1 + A i β p ∥ π [ A i + T e ⊥ T i ⊥ A e − k 2 k ⊥ 2 β p ⊥ ] (2) The electron anisotropy A e ≈ 0 is assumed negligible initially, k ∥ ≪ k ⊥ , λ i = c / ω i ion inertial length, plasma β p = 2 μ 0 N ( T i + T e ) / B 0 2 . Sometimes this is inverted into a condition on the (arbitrary) angle of wave propagation (Southwood and Kivelson, 1993) θ m = tan − 1 ( k ⊥ / k ∥ ) but the relevant physics is contained in the field threshold. The two conditions together yield that the critical (ion) temperature is T c i ⊥ = T i ∥ which means that T c i ⊥ − T i ⊥ ≥ 0   stability ,   “ normal ”  state T c i ⊥ − T i ⊥ < 0   instability ,   “ quasi − superconducting ” (3) which indeed reminds at the solid state superconducting phase transition.

The mirror instability readily saturates quasilinearly on the expense of the ion anisotropy (cf., e.g., Davidson, 1972; Treumann and Baumjohann, 1997) forming elongated k ∥ ≪ k ⊥ magnetic bottles. Landau diamagnetic theory (cf., e.g., Huang, 1973) suggests that any finite temperature diamagnetism is macroscopically very small, which is confirmed by simulations (Noreen et al., 2017) which show the saturation amplitude to remain minuscule. The observation of large-amplitude localized quasi-stationary magnetic depletions of ≲ 50 % (see Treumann and Baumjohann, 2018a, for examples, in high resolution) must be enforced by conditions which are not included in linear or quasilinear theory (Treumann et al., 2004). We do not go into discussing this problem here as it has been the subject of previous publications (cf., Treumann and Baumjohann, 2018b; Treumann and Baumjohann, 2019). We just note that a number of simulations (Rincon et al., 2015; Yao et al., 2019) and theory (Sulem, 2011) claim that nonlinear interactions between ions and waves provide large magnetic amplitudes (see the discussion in Treumann and Baumjohann, 2019). In simple words, nonlinear scattering off waves, as sometimes assumed, increases diffusivity and entropy which primarily thermodynamically inhibits structure formation. The argument of increased internal pressure fails because pressure is compensated by the large elastic volume of the environment. It dilutes the magnetic field only infinitesimally. Bubbles can deepen only on the expense of their neighbours along the same flux tube by sucking in plasma, which contradicts the observation of long mirror chains while supporting observation of isolated bubbles (Luehr and Kloecker, 1987; Treumann et al., 1990). In thermodynamic terms, entropic structure formation requires chaotic single particle interactions. These are provided by condensate formation.

One way out of the above mentioned basic physical dilemma between observation and theory may be related to the resonance of bouncing particles in the mirror bubble and the persistent thermal ion-acoustic background noise which is independent of the presence of mirror modes (Rodriguez and Gurnett, 1975; Treumann and Baumjohann, 2018a; Treumann and Baumjohann, 2019). These resonant bouncing particles (we here restrict to electrons, but ions if bouncing could contribute in a similar way as well) form the required condensate for phase transition.

The problem of interaction of two superconductors (in our case two quasi-superconducting partially magnetic field-depleted mirror bubbles separated by a non-superconducting magnetized sheet) is the celebrated Josephson problem (Josephson, 1962; Josephson, 1964). It makes use of the Landau-Ginzburg mesoscopic theory of superconductivity (Ginzburg and Landau, 1950) which is applicable in this case. The order parameter is the expectation value of the wave function ψ given by 〈 ψ ψ * 〉 = N , which in our case is the above introduced fractional density N of the bouncing electrons in resonance with the ion-acoustic thermal background fluctuations that form the condensate.

The interaction includes of course the boundaries of the two bubbles and hence takes into account the current while, in the mutual interaction, the interior is of little interest. It just responses by the exponential partial Meissner screening of the magnetic field B = B 0 exp ( − x / λ m ) with λ m the actual penetration depth which for N < 1 is larger than the inertial length λ i = c / ω i ≈ 43 λ e and the London length in a proton plasma. Since mirror bubbles are ion modes even though driven unstable by electrons, one here must use the ion inertial length, with ω i the ion plasma frequency. Observations suggest that the mirror penetration length is roughly λ m ≈ ( 10 − 20 ) λ L which on its own suggests that N≲ 10 − 2 .

The wave function of superconduction which in the above spirit we apply to the case of mirror modes obeys the above used first Ginzburg-Landau equation (as was proposed in Treumann and Baumjohann, 2018b; Treumann and Baumjohann, 2019). The current, being purely electronic, is given by the well known quantum mechanical expression (Ginzburg and Landau, 1950; Bardeen et al., 1957; Huang, 1973), the second Ginzburg-Landau equation j ( x , t ) = − i e ℏ N 0 2 m e ( ψ * ∇ ψ − ψ ∇ ψ * ) − | ψ | 2 N 0 κ G A μ 0 λ e 2 (9) = e ℏ N 0 m e | ψ | 2 ( ∇ ϕ − e ℏ κ G A ) (10) with A the magnetic vector potential, ϕ the phase of the complex wave function, and boundary condition that the normal current must be continuous n ⋅ [ ( ℏ ∇ − i e A κ G ) ψ ] 2 − 1 = 0 (11) where the brackets mean the difference between the quantities to both sides of the boundary, as indicated by the subscript 2 − 1 , and n is the normal to the boundary. Clearly, in a purely classical treatment only the last term in the current expression survives when putting ℏ = 0 . In our semiclassical approach we retain the quantum part of the current, which is the Josephson approximation. Classically the quantum part is neglected, and one has j ( x ) = − | ψ | 2 N 0 κ G A μ 0 λ e 2 (12) and from Mawell’s equations trivially □ A = − | ψ | 2 N 0 κ G A μ 0 λ e 2 (13) whose stationary solution in one dimension only is clearly B = B 0 exp ( − x / λ m ) , the explicit partial Meissner skin effect caused by the condensate | ψ | 2 . (Note again that the current does not explicitly depend on mass, which implies that a hypothetical ion condensate would as well contribute to the phase transition.) Thus the important physics is contained in the generation of the condensate as described in the previous section. If assuming B / B 0 ≈ 0.5 as is typical in the magnetosheath, one has that λ m ≈ 1.5 x , where x is the measured penetration length. This, in the magnetosheath, is about x ≈ 100 km. Hence, with λ i ∼ 10 km one confirms the order of magnitude of λ m as given above.

Now, when in contrast to the above semi-classical use of the first Ginzburg-Landau equation considering the interaction of the mirror bubbles, the quantum property of the phase has to be retained because it is just the phase which contains the microscopic information. Moreover, space plasmas are ideal conductors and no resistors. Hence the normal current will naturally be different from zero and will reflect the microscopic effect of the interaction. For this reason the quantum part of the current must be retained. We will see that this is important in the case under consideration.

There is, however, a difference in the region between the two bubbles. It is void of any condensate and thus that narrow domain is void of the Meissner effect. The magnetic field and density it contains are spatially constant. Hence the difference between the two regions is just in the quantum mechanical term in the boundary condition and thus cannot be neglected while the conditions in the two adjacent bubbles may be different. Moreover, the tangential currents (which we do not consider here as they contribute to the partial Meissner effect but are not involved into the normal current which must by itself be continuous) flowing in the adjacent bubble boundaries are in opposite directions. This implies that the two bubbles do not merge. They do not attract each other because of the repulsive Lorentz forces such that they remain separated. Nevertheless one may assume that the separation is narrow with non-compensating currents.

Since all regions are conducting a normal current will necessarily flow. In real superconductors separated by insulators electron tunnelling takes care of normal currents. Here, in the classical case, these currents are real. Nevertheless because of the retained quantum mechanical part of the current, Josephson conditions of current continuity apply to both its sides are given as ∂ n ψ 1 − i e ℏ κ G A n ψ 1 = b ψ 2 (14) ∂ n ψ 2 − i e ℏ κ G A n ψ 2 = − b ψ 1 (15) with b = const some real constant whose value is only of secondary importance here. These boundary conditions follow directly from the general condition of continuity of the normal current (11) writing it in the normal coordinates not as a difference but as the finite mismatch between the two wave functions. This is easily seen when subtracting them. As the mismatch is not known a priori because it depends on the properties of the transition region, it can be accounted for simply by the constant b. One may note that for a perfect insulator with no classical current flow b = 0 . However, the medium between two mirror bubbles clearly permits current flow. So b ≠ 0 is an appropriate assumption ² . This means that seen from each bubble’s side the effect of the other on the transition is constant and opposite. It is thus assumed that the transition layer between the bubbles is thin enough to consider a constant current when crossing it, thereby for simplicity neglecting any spatial fluctuation or divergence of the current. This may hold as long as the transition distance is short compared with the bubble diameter, a condition satisfied in the magnetosheath, for instance. Moreover, the layer is neither an ideal conductor nor an ideal insulator such that current flow across it is permitted. In the case when it is an ideal conductor it should be narrower than the skin depth outside the mirror bubbles, but even if this does not apply current flow is permitted anyway.

Inserting these boundary conditions into the current Eq. 9 and cancelling some terms yields for the perpendicular current crossing the thin layer that j n = − i e b ℏ N 0 2 m e ( ψ 1 * ψ 2 − ψ 1 ψ 2 * ) = j J sin ( ϕ 2 − ϕ 1 ) (16) where j J = ( e b ℏ / m e ) N 0 | ψ | 2 is the Josephson current, and we for simplicity assumed condensate symmetry | ψ 1 | = | ψ 2 | up to the phases. (One may note that this current is a pure electron current; any ion-condensate contribution can be neglected because of the inverse proportionality to the mass. Clearly this is an effect of the large electron mobility.) This is most easily seen when replacing the functions ψ 1,2 by the sum of their real and imaginary parts ψ = ψ r + i ψ i . The difference in the round brackets is the phase difference between the two wave functions ψ = | ψ | e i ϕ in the two bubbles which distinguishes them.

This transverse current is a retained quantum effect even in the macroscopic case. One might argue that the two phases might be the same and thus cancel the current. There is, however, no reason for this to happen even when the condensates are identical. Continuity of the current does not require equal phases if only it can be achieved otherwise. Its importance comes into account when remembering that the gauge potentials are defined just up to additional functions which leave the fields unaffected. The vector potential A → A + ∇ U is defined up to the gradient of a potential U, and consequently the electric field E = − ∇ V − ∂ t A up to its time derivative ∂ t U . This implies from Eq. 9 that the phase changes as ϕ → ϕ + e U / ℏ κ G and the electric potential as V → V − ∂ t U or after comparison and elimination of U ∂ t ϕ = e V / ℏ κ G (17) showing that the phase is affected by the gauge potential thereby exhibiting a real change in the phase.

Before continuing, it is most interesting to reflect about what has happened. In principle the electrodynamic equations are gauge invariant which means that the vector and scalar potentials can be changed by adding particular gauge functions while leaving the fields unchanged. This it true also here. However, by applying an external potential V to the two mirror bubbles one fixes one particular gauge. This still does not change anything on the fields, it however breaks the gauge symmetry locally. By providing the mirror modes with a particular electric potential field V ( x ) they shift to one special particular gauge, and no other gauge can be chosen anymore. In the following we will see which consequences this produces.

The gauge is in fact a (Weyl) gauge like in field theory. Time integration, with applied constant external potential V, yields the well known form of the Josephson phase ϕ 2 − ϕ 1 = ( ϕ 2 − ϕ 1 ) 0 − ω J t ,     ω J = e ℏ κ G ( V 2 − V 1 ) (18) which enters into the exponent of the wave function ψ. In the presence of a potential difference V 2 − V 1 the current (16) in the junction consisting of the two mirror bubbles with their common boundary of finite thickness will thus oscillate at the Josephson circular frequency ω J . Correspondingly the normal current becomes j n = j J sin ( Δ ϕ 0 − ω J t ) , Δ ϕ 0 = ( ϕ 2 − ϕ 1 ) 0 (19) This current is a real oscillating classical normal current flowing in the boundary region of the two adjacent mirror bubbles. It is a current that varies with time, oscillating back and forth between the bubbles (For instance, for Δ ϕ 0 = 0 one has j n = − j J sin ω J t (20) showing that the normal current in the boundary oscillates spatially back and forth between the two bubbles, which is a real classical spatially localized effect.) Physically it is not difficult to understand the origin of this oscillation. The two oppositely directed tangential currents at the boundaries of the mirror bubbles could indeed be closed by a normal current across the highly conducting gap which separates them. In the presence of an electric potential this happens on the small scale when the transverse current temporarily for a very short time breaks through and connects the two tangential currents. This break through happens however on the microscopic scale and transports very few magnetic flux elements only. Therefore its high frequency. Interestingly, there should as well be a spatial dependence of this process along the bubbles which we have not considered here. It causes a tangential variation of the oscillation frequency.

Denoting the potential difference as V 2 − V 1 = Δ V and introducing the magnetic flux quantum Φ 0 = π ℏ / e , the observable Josephson frequency becomes ν J = Δ V Φ 0 κ G , Φ 0 = 2 × 10 − 15   Vs . (21) This frequency corresponds to an energy κ G ℏ ω J ≈ 1   e v multiplied by the applied potential difference Δ V . For Δ V ≈ 1   V the oscillation frequency is κ G ν J ≈ 5 × 10 14   Hz , which is in the near-optical infrared. The applied potential is measured in units of the elementary magnetic flux. It thus reflects the oscillations or transport of elementary flux tubes at high frequency, perfectly suited to measure very small potential differences as used in Josephson SQUIDs.

This result discovered by Josephson (Josephson, 1962) is remarkable as, according to the above discussion, it also occurs under semi-classical conditions if only an electric potential V ( t ) is applied to the two adjacent mirror bubbles. This is the case in a streaming plasma with plasma flow across the magnetically depleted region or where an externally applied cross potential exists.

Examples are the magnetosheath (Lucek et al., 2005) or other regions like, for instance, mirror mode chains in the solar wind (Winterhalter et al., 1994; Zhang et al., 2008). Other examples are collisionless shocks (Balogh and Treumann, 2013) which have comparably narrow transition scales, develop current sheet overshoots between magnetic depletions resembling a similar kind of junctions. Ion-inertial scale plasma turbulence or flow-driven reconnection are further examples.

The Josephson frequency of oscillation is comparably high. Its large value is due to retaining the quantum effect which implies normalization of the potential difference Δ V to the small flux quantum Φ 0 . Thus the Josephson current oscillates at high frequency. For any classical macroscopic process it averages out as neither the flow nor the density can follow the fluctuation. It is just the current whose real phase oscillates between the two adjacent mirror bubbles. The quantum effect on the macroscopic plasma behaviour of the fields thus disappears. This is, of course, what is expected if considering the flow or evolution of the magnetic field.

However, there is one effect that, in addition to the fluctuating electric potential, is retained even in classical physics. This is radiation which can, in principle, be observed even though its cause is to be found in quantum physics. In this sense the Josephson effect and the frequency resemble the generation of electromagnetic radiation by atomic processes, which are pure quantum effects with macroscopically measurable consequences: emission of radiation. In close similarity the Josephson radiation can, in principle, be observed from remote by monitoring its intensity.

Oscillating currents represent sources of electromagnetic radiation, as prescribed by the wave equation □   A r a d ( x , t ) = − μ 0   j ( x , t ) (22) where A r a d is the radiated vector potential, and j = j n n , the sinusoidal real current Eq. 19, where it should once more be noted that this is a real classical current. Any natural system which acts, even semi-classically, like a Josephson junction should therefore emit electromagnetic radiation at frequency around ν J = ω J / 2 π . The spectral width of the radiation depends on the spectral width Δ ω of the applied time dependent external potential V ( t ) , whose Fourier transform is V ( ω ) = ∫ ​ d t   V ( t ) e i ω t (23) whose width can be quite large, compared with the theoretical sharpness of the Josephson frequency, in particular when the flow is highly turbulent. It leads to a time dependent Josephson phase ϕ J ( t ) = Δ ϕ 0 − e ℏ κ G ∫ ​ 0 t d t ′   V ( t ′ ) (24) and, consequently, to a radiation spectrum of some typical width Δ ν in frequency. An electric oscillation spectrum V ( ω ) yields for the fluctuating part of the phase ϕ J ( t ) − Δ ϕ 0 = − e 2 π i ℏ κ G ∫ ​ d ω ω   V ( ω ) ( e − i ω t − 1 ) (25) which, for the realistic case of comparably low frequency oscillations ω t ≪ 1 yields that the Josephson frequency becomes ν J ≈ 1 4 π κ G Φ 0 ∫ ​ d ω   V ( ω ) ≈ V ( ω m a x ) 4 π κ G Φ 0 Δ ω (26) The emission spectrum is only as broad as Δ ω . Since the electric potential arises from external motions in plasma its spectrum is limited from above by the plasma frequency Δ ω ≲ ω e ≪ ω J . As the plasma frequency is low, any radiation of Josephson frequency ν J will therefore remain to be of very narrow bandwidth Δ ν ≪ ν J and thus, when observed, will form a narrow emission line. Any spectral broadening would then have other reasons having to be retraced to the angle or the tininess of the applied potential. One does of course not expect that this kind of radiation would be intense. Its intensity per unit volume and frequency is well known (Jackson, 1975) to be proportional to the average spectral square of the radiated vector potential, d 2 I d Ω d ω ∝ | A r a d ( ω ) | 2 ∝ μ 0 2 | j J ( ω ) | 2 ∝ ( e b ℏ m ) 2 μ 0 2 N 0 2 N 2 (27) where we have just noted the proportionalities without solving the above wave equation as this would go beyond the purpose of the present letter.

Because of the weakness of the maximum Josephson current j J , any susceptibly high enough radiation intensity requires a large number of emitters closely distributed in the volume, i.e., a large number of interacting mirror bubbles and hence large volumes, a condition that is probably not realistic in near-Earth space but may sometimes be realized under astrophysical conditions. Hence there should be little doubt about the presence of such a radiation effect, its detectability might however be questioned.

Following earlier work on condensate formation in magnetic mirror modes we have provided the conditions for a quasi-superconducting phase transition in high temperature plasma, following the linear mirror instability. In this process bouncing charged particles in discrete particle resonance with the thermal ion acoustic background noise lock to the ion sound wave and temporarily escape from bounce motion while generating a large anisotropy. These particles form a condensate in the mirror bubble. Since bouncing particles are abundant, the condensate is quite dense and permanently present, longer than the life time of the discrete resonance of each single particle. It continuously reforms. Through production of a weak attracting electric potential in their wakes the condensate particles give rise to a correlation length ξ. The phase transition is governed by the semi-classical GL theory and results in a second-kind quasi-superconducting state exhibiting a partial Meissner effect. Since the Ginzburg ratio κ G > 1 is large, the phase transition perforates the plasma causing a magnetic texture which consists of chains of mirror bubbles.

We then investigated the interaction of two closely spaced bubbles finding that it can be described as a Josephson junction which produces a classical signature in weak high frequency electromagnetic radiation at frequency depending on the equivalent electric field and direction of the plasma flow. Its frequency is sufficiently far above the plasma frequency cut-off such that it would be observable from remote. Though weak and if observable it maps the mirror mode region into frequency space. Similar effects are expected in reconnection and shocks and could be of interest in application to astrophysical objects. On the other hand, putting SQUIDS onto spacecraft in order to measure potential differences produced in mirror bubbles, reconnection, and shocks with extremely high accuracy in situ might be advantageous.