Intermediate shock substructures within a slow-mode shock occurring in partially ionised plasma
AAstronomy & Astrophysics manuscript no. intermediate_shocks c (cid:13)
ESO 2019April 30, 2019
Intermediate shock substructures within a slow-mode shockoccurring in partially ionised plasma
B. Snow and A. Hillier University of Exeter, Exeter, EX4 4QF, UK e-mail: [email protected]
April 30, 2019
ABSTRACT
Context.
Slow-mode shocks are important in understanding fast magnetic reconnection, jet formation and heating in thesolar atmosphere, and other astrophysical systems. The atmospheric conditions in the solar chromosphere allow bothionised and neutral particles to exist and interact. Under such conditions, fine substructures exist within slow-modeshocks due to the decoupling and recoupling of the plasma and neutral species.
Aims.
We study numerically the fine substructure within slow-mode shocks in a partially ionised plasma, in particular,analysing the formation of an intermediate transition within the slow-mode shock.
Methods.
High-resolution 1D numerical simulations are performed using the (PIP) code using a two-fluid approach.
Results.
We discover that long-lived intermediate (Alfvén) shocks can form within the slow-mode shock, where there isa shock transition from above to below the Alfvén speed and a reversal of the magnetic field across the shock front.The collisional coupling provides frictional heating to the neutral fluid, resulting in a Sedov-Taylor-like expansion withovershoots in the neutral velocity and neutral density. The increase in density results in a decrease of the Alfvén speedand with this the plasma inflow is accelerated to above the Alfvén speed within the finite width of the shock leading tothe intermediate transition. This process occurs for a wide range of physical parameters and an intermediate shock ispresent for all investigated values of plasma- β , neutral fraction, and magnetic angle. As time advances the magnitudeof the magnetic field reversal decreases since the neutral pressure cannot balance the Lorentz force. The intermediateshock is long-lived enough to be considered a physical structure, independent of the initial conditions. Conclusions.
Intermediate shocks are a physical feature that can exist as shock substructure for long periods of time inpartially ionised plasma due to collisional coupling between species.
Key words. magnetohydrodynamics (MHD), shock waves, Sun:chromosphere
1. Introduction
Shocks occur readily in the lower solar atmosphere, drivenby wave steepening, e.g., umbral flashes (Beckers & Tal-lant 1969; Houston et al. 2018), or magnetic reconnectiondriven events, such as Ellerman bombs (Ellerman 1917; Nel-son et al. 2013). The listed phenomena occur in regions ofthe sun where partially ionised effects are thought to playa key role in the underlying physics. As such, to under-stand phenomena occurring in the lower solar atmosphere,we must also understand the role of partial ionisation inshocks.In magnetohydrodynamics (MHD), there are three char-acteristic wave speeds (slow, Alfvén and fast) which leadsto a multitude of potential shock transitions (see for ex-ample Delmont & Keppens 2011). The slow-mode shock(transition from super-slow to sub-slow flow speeds) is ofparticular importance due to its role in magnetic reconnec-tion, where a change in connectivity of the magnetic fieldresults in a release of stored magnetic energy. The energycan be released instantaneously via Joule heating, or post-reconnection by the influence of the magnetic field on theplasma. The classical 2D schematic for fast reconnectionfeatures slow-mode shocks (Petschek 1964). Recent worksuggests more complicated reconnection configurations oc-cur in the solar atmosphere, e.g., plasmoid instability, 3D topology, extended MHD (see reviews by Yamada et al.2010; Pontin 2011; Loureiro & Uzdensky 2016; Cassak et al.2017). In the Petschek model, the presence of slow-modeshocks in the outflow region allow efficient transport of en-ergy away from the reconnection region, increasing the re-connection rate. Slow-mode shocks are also found to formin alternative models for fast reconnection (Liu et al. 2012;Innocenti et al. 2015; Shibayama et al. 2015).The intermediate shock (transition from above to belowthe Alfvén velocity) has been shown to exist in single-fluidresistive MHD representative of the Earth’s magnetopause(Karimabadi 1995). The physical mechanism being that thenon-ideal region around the shock allows a separation of themagnetic field and fluid, and the formation of an interme-diate shock between the two ideal regions. The stabilityof intermediate shocks was proved by Wu & Hada (1991)who studied numerically the formation of these shocks dueto wave steepening. The relationship between intermediateshocks and slow-mode shocks was shown analytically byHau & Sonnerup (1989) for resistive MHD shocks, findinga relation between the downstream slow-mode speed andthe strength of the intermediate transition.In partially ionised plasma, such as in the solar chro-mosphere or prominances, shocks become more complex.Hillier et al. (2016) studied the formation of slow-modeshocks in partially ionised plasmas. Around the shock front,
Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . S R ] A p r &A proofs: manuscript no. intermediate_shocks the plasma and neutral species decouple and recouple re-sulting in a finite width slow-mode shock. The Lorentz forceindirectly affects neutrals through collisions and hence thedrift between ion and neutral species becomes an importantparameter in partially-ionised plasma. However, observingthe ion-neutral drift has been found to be difficult due toline-of-sight effects (Anan et al. 2017). A review of partially-ionised modelling can be found in Khomenko (2017).In this paper, we utilise high-resolution two-fluid nu-merical simulations to investigate the substructure withinslow-mode shocks. We discover that long-lived intermedi-ate shocks can form within the finite width of the shockas a result of the fluid coupling and decoupling around theshock front. This result has applications to the solar chro-mosphere, as well as interplanetary and interstellar par-tially ionised plasma. We study the formation of compoundintermediate shocks in partially ionised plasma, and theconditions in which such shock structures can form.The outline of this paper is as follows. First we definethe shock transitions and classifications to be used in thispaper, and the analytical solution to the magnetohydro-dynamic (MHD) and partially ionised plasma (PIP) equa-tions in the Hoffman-Teller shock frame. Next, the numer-ical methods and initial conditions under consideration areintroduced. Following this, a reference case is presentedwhere we compare an MHD and a PIP simulation, and iden-tify the key differences, investigating the physical evolutionthat produces the intermediate shock. Finally, we considera parameter study and investigate the effect of the plasmaproperties on the lifetime and magnitude of the shock. Wefind that for our initial conditions, an intermediate shockwill always form in a partially-ionised plasma. However thelifetime and magnitude depend heavily on the plasma prop-erties.
2. Methodology
MHD waves have three characteristic speeds: Alfvén ( V A ),slow ( V s ), and fast ( V f ). As such, multiple shock transitionsare possible, depending on the magnitude of the velocity,relative to the characteristic speeds in the pre- and post-shock regions.Following the approach of Delmont & Keppens (2011),we classify shock transitions using the relationship betweenthe normal flow velocity v ⊥ and the characteristic speeds: – (1) superfast: V f < | v ⊥ | , – (2) subfast: V A < | v ⊥ | < V f , – (3) superslow: V s < | v ⊥ | < V A , – (4) subslow: < | v ⊥ | < V s , – ( ∞ ) static: v ⊥ = 0 .Defining the upstream condition u and downstream condi-tion d, several shocks of the form u → d are possible: – → fast shocks, – → slow shocks, – → switch-on, – → switch-off, – → , → , → , → intermediate shocks. Fig. 1.
Hau-Sonnerup shock solution for plasma- β u = 0 . and θ u = 0 . . Shaded region shows impossible transitions. Possibletransitions include: slow-mode shock (blue), intermediate shock(red), rotational discontinuity (circle), and fast-mode shocks(green). The Hoffman-Teller frame allows for jump relations to bederived from the MHD equations. In this choice of restframe, the velocity and magnetic field vectors are in thesame plane either side of the shock, i.e., the electric fieldacross the shock is zero. This reduces the MHD equationsto a 2-dimensional problem for variables perpendicular ( ⊥ )and parallel ( (cid:107) ) to the shock front. In the Hoffmann-Teller frame, the MHD equations can beintegrated and the upstream ( u ) and downstream ( d ) con-ditions can be equated: ρ u v u ⊥ = ρ d v d ⊥ , (1) ρ u v u ⊥ v u (cid:107) − µ B u ⊥ B u (cid:107) = ρ d v d ⊥ v d (cid:107) − µ B d ⊥ B d (cid:107) , (2) ρ u v u ⊥ v u ⊥ + P u + B u µ = ρ d v d ⊥ v d ⊥ + P d + B d µ , (3) γγ − P u ρ u + v u γγ − P d ρ d + v d , (4) v u ⊥ B u (cid:107) − v u (cid:107) B u ⊥ = v d ⊥ B d (cid:107) − v d (cid:107) B d ⊥ (5) B u ⊥ = B d ⊥ . (6)We can also define the following: Alfvén Mach number as A ⊥ = v ⊥ µ ρB x , plasma- β as β = µ PB , and the angle betweenthe velocity and magnetic field θ .A solution to Equations (1-6) can be found in Hau& Sonnerup (1989) relating the upstream Alfvén velocityto the upstream plasma- β , upstream θ , and downstreamAlfvén velocity, i.e., A u ⊥ = (cid:20) A d ⊥ (cid:18) γ − γ (cid:18) γ + 1 γ − − tan θ u (cid:19) (cid:0) A d ⊥ − (cid:1) + tan θ u (cid:18) γ − γ A d ⊥ − (cid:19) (cid:0) A d ⊥ − (cid:1)(cid:19) − β u cos θ u (cid:0) A d ⊥ − (cid:1) (cid:21) / (cid:34) γ − γ (cid:0) A d ⊥ − (cid:1) cos θ u − A d ⊥ tan θ u (cid:18) γ − γ A d ⊥ − (cid:19)(cid:35) . (7) Article number, page 2 of 10. Snow and A. Hillier: Intermediate shock substructure
This is shown graphically in Figure 1 for a choice ofplasma- β u = 0 . and θ u = π/ . The trivial solution isthat the upstream and downstream velocities are identi-cal, i.e., no shock transition. The shaded region shows im-possible solutions, where the velocity is higher downstreamthan upstream. The non-trivial solutions show the possi-ble shock transitions for given plasma- β and θ values. Thiscurve intersects the A u = A d line at three points. The in-tersect labelled s u denotes the upstream slow speed. At thepoint A u ⊥ = A d ⊥ = 1 , a rotational discontinuity is possible,where there is a change in the angle of the magnetic fieldbut plasma properties remain the same. Slow-mode transi-tions (blue line) are bounded by A u ⊥ = 1 . The downstreamslow speed (point labelled on Figure 1 as s d ) correspondsto the critical value that separates strong ( → ) andweak ( → ) intermediate transitions. The strong tran-sition ( → ) is closely linked to the slow-mode transition( → ) through its relation to the downstream slow speed. In the PIP equations, collisional terms are taken into ac-count and therefore, the equations have a non-zero right-hand side, see Equations (8-17). As such, when the equa-tions are integrated in the Hoffmann-Teller frame, thereare integral terms on the right hand side, see Equations(A.1-A.13), as opposed to constants in the MHD Equations.However, by adding the neutral and plasma species to-gether, the integral terms can be eliminated, and by choos-ing a point upstream and downstream such that the driftvelocity equals zero, one recovers the MHD equations in theHoffman-Teller frame, where the Alfvén speed and plasma- β values depend on the total plasma density ( ρ n + ρ p ), seeAppendix A. These equations are also independent fromthe neutral fraction and hence, in the steady state solu-tion, the values either side of the shock are governed by theMHD solution, when points are chosen such that the driftvelocity is zero. However, within the shock, the species de-couple and recouple hence in partially-ionised plasmas, itis of interest to study the substructure that occurs withinthe finite-width of the shock. Two-fluid numerical simulations are performed using the(PIP) code (Hillier et al. 2016) which solves the interactionsof a neutral fluid, and a coupled ion electron plasma. Thesimulations are 1D and use high resolution to resolve thesubstructure present within the shock. A first order HLLD(Harten-Lax-van Leer-Discontiunities) scheme is used toprevent spurious oscillations from occurring around theshock interface. The normalised PIP equations are given below: ∂ρ n ∂t + ∇ · ( ρ n v n ) = 0 , (8) ∂∂t ( ρ n v n ) + ∇ · ( ρ n v n v n + P n I )= − α c ρ n ρ p ( v n − v p ) , (9) ∂e n ∂t + ∇ · [ v n ( e n + P n )]= − α c ρ n ρ p (cid:20)
12 ( v n − v p ) + 3 (cid:18) P n ρ n − P p ρ p (cid:19)(cid:21) , (10) e n = P n γ − ρ n v n , (11) ∂ρ p ∂t + ∇ · ( ρ p v p ) = 0 (12) ∂∂t ( ρ p v p ) + ∇ · (cid:18) ρ p v p v p + P p I − B B + B I (cid:19) = α c ρ n ρ p ( v n − v p ) , (13) ∂∂t (cid:18) e p + B (cid:19) + ∇ · [ v p ( e p + P p ) − ( v p × B ) × B ]= α c ρ n ρ p (cid:20)
12 ( v n − v p ) + 3 (cid:18) P n ρ n − P p ρ p (cid:19)(cid:21) , (14) ∂ B ∂t − ∇ × ( v p × B ) = 0 , (15) e p = P p γ − ρ p v p , (16) ∇ · B = 0 , (17)for neutral (subscript n) and plasma (subscript p) species.The neutral equations (8-11) are independent of the mag-netic field. The plasma equations (12-17) are similar to theMHD equations however include collisional terms that cou-ple the plasma to the neutral fluid. The collisional coeffi-cient α c is defined as α c = α (cid:114) T n + T p T (18)where T is the normalisation temperature such that thesound speed c s = V A and, in the normalised form, α = 1 .The magnetic field is normalised using B = B / √ µ . Detailsof the equations and their implementation in the (PIP) codecan be found in Hillier et al. (2016).Our simulation data is translated into the shock frameby calculating the shock propagation speed in the MHDsolution ( v s ). The position and velocity variables areremapped to the shock frame, i.e., v x n s,x p s = v x n ,x p − v s , x s = x/t − v s . The initial conditions in this paper are an extension of themodel for slow-mode shock created in reconnection pro-posed by Petschek (1964). Initially, a discontinuous mag-netic field is specified across the boundary. The plasma andneutral fluids are assumed to be in thermal equilibrium atthe start of the simulation.
Article number, page 3 of 10 &A proofs: manuscript no. intermediate_shocks v x B y Magnetic fieldreversal v y Slow-modeshock front0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x/t0.00.20.40.60.81.0 P r Rarefaction waveInflowOutflow (a) (b)(c) (d)
Fig. 2.
MHD (black) and PIP (red neutral, blue plasma) reference solution for β = 1 . , B x = 0 . , ξ n = 0 . . (a) v x velocity. (b) B y magnetic field. (c) v y velocity. (d) Pressure. Green line indicates total pressure ( P n + P p ). The PIP solution is plotted after 2500collisional times. B x = 0 . (19) B y = − . x > , . x < (20) ρ n = ξ n ρ t (21) ρ p = ξ i ρ t = (1 − ξ n ) ρ t (22) P n = ξ n ξ n + 2 ξ i P t = ξ n ξ n + 2 ξ i β B (23) P p = 2 ξ i ξ n + 2 ξ i P t = 2 ξ i ξ n + 2 ξ i β B (24)where ξ n and ξ i are the neutral and ion fractions respec-tively.The x = 0 boundary is treated as reflective, but suchthat magnetic field can penetrate the boundary. The sys-tem is normalised to have an Alfvén velocity of unity. Thecollisional time as determined by the bulk fluid density iscalculated as τ = 1 / ( α ( T ) ρ t ) = 1 .These initial and boundary conditions were used in thework of Hillier et al. (2016) to investigate the slow-modeshock. Here we investigate the formation and lifetimes ofa feature that was present in Figures 4a and 5a of Hillieret al. (2016) but not discussed, namely the intermediateshock transition, which features a reversal of magnetic fieldacross the shock interface.
3. Results
In this section, we present an MHD simulation and a PIPsimulation and highlight the key differences. A snapshotof the results is shown in Figure 2. Both simulations use256,000 grid cells to resolve the spatial dimension. Thesetwo simulations use the following parameters: β = 1 . , B x = 0 . , ξ n = 0 . . Note that the neutral fraction isonly used in the PIP case and the MHD simulation is fullyionised. The effect of different parameters is investigated inSection 4.2. In the MHD case (Figure 2, black lines), the initial con-ditions produce a rarefaction wave that drives fluid at thelocal Alfvén speed towards a slow-mode shock. The solutionis expanding in time, however, is steady-state when plottedon a time-normalised axis ( x/t ), hence can be described aspseudo-steady.It is important to note that in the MHD solution, theslow-mode shock possesses no complex substructure, has adiscontinuous jump (no finite width), and is purely a → transition. There is no reversal in magnetic field across theshock (see Figure 2). These features do not hold when themodel is extended to include additional effects (e.g. par-tial ionisation) where substructure can form in the shockfront affecting the shock dynamics and resultant heatingand energy transport. Article number, page 4 of 10. Snow and A. Hillier: Intermediate shock substructure
The red and blue lines in Figure 2 show the neutral and ionfluid respectively for the PIP simulation, using the same pa-rameters as the MHD case. The snapshot is taken after 2500collisional times when the system can be assumed to be in aquasi-self-similar state. In the PIP case, the ion and neutralspecies decouple and recouple around the shock front, re-sulting in a finite width slow-mode shock where shock sub-structure can occur (compared to the discontinuous jumpin the MHD case).The initial conditions are a discontinuity in the mag-netic field only, therefore directly affect the plasma only.The neutral fluid can only be influenced indirectly by col-lisions with the ionised plasma. As such, the system tendstowards a pseudo-steady-state as time tends towards infin-ity. Hillier et al. (2016) presented the dynamic evolution fora similar case: – Initialisation
The initial conditions are a discontinu-ity in the magnetic field and hence directly affect theplasma only, and the neutral fluid is affected via colli-sions. At τ = 1 the system is highly decoupled. A Sedov-Taylor-like expansion occurs in the neutral fluid due tofrictional heating between the two species. – Weak coupling A fast-mode rarefaction wave formsand drives fluid towards a slow-mode shock. At thistime, the rarefaction wave and slow-mode shock frontare coupled and interact. – Strong coupling
In this phase, the rarefaction waveand slow-mode shock decouple. The slow-mode shockcan be considered to be independently evolving. – Quasi-self-similar state
System tends towards a self-similar state as time tend towards infinity. This is plot-ted in Figure 2. The rarefaction wave is separated fromthe slow-mode shock, Figure 2a. The two fluids are rea-sonably well coupled except within the slow-mode shockwhere large drift velocities are present, Figure 2a. Theneutral overshoot is present at late times indicating thatit is a stable feature of the system (Figure 2a).A feature present but not discussed in the work of Hillieret al. (2016) is the reversal of the magnetic field across theshock front (Figure 2b). This is a long-lived (but transient)feature of the system that exists long after the rarefactionwave and slow-mode shock have separated. The magneticfield reversal is a signature that an intermediate transitionexists within the slow-mode shock.
4. Analysis
The reversal in magnetic field in Figure 2b is a key indicatorof an intermediate shock, where the plasma transitions fromabove the local Alfvén speed, to below it. An intermediateshock is present in all partially ionised simulations as atransient feature of the system, however is not present in theMHD cases. The feature arises due to the interactions of ionand neutral particles. To analyse this feature, a new frameof reference is chosen such that the shock is stationary, i.e., x s = x/t − v s where v s is the propagation speed of the MHDshock. In this frame, the velocity is also adjusted to accountfor the shock propagation speed, i.e., v x − v s . The shockframe allows us to correctly compute the transitions across the shock front and compare with the analytical results.The evolution of this structure is as follows: (1) Sedov-Taylor-like expansion of the neutral fluid: The ini-tial conditions are a discontinuity in the magnetic field thataffects the plasma only, resulting in a separation of neu-tral and plasma species in the shock front (Figure 3a). Thelarge drift velocity creates a rapid heating of the neutralfluid. Subsequently, a hydrodynamic, Sedov-Taylor-like ex-pansion occurs in the neutral fluid, resulting in an overshootof neutral velocity (Figure 3a), and a sudden increase inneutral density (Figure 3b). (2) Acceleration of plasma:
The rarefaction wave drives ve-locity towards the slow-mode shock at the Alfvén speed(Figure 3c). This is also true in the MHD case and is a fea-ture of the shock problem being studied. Within the slow-mode shock front, the bulk Alfvén speed decreases due tothe increase in neutral density (Figure 3d) and the reversalin the magnetic field (Figure 2b). The plasma inside theshock is accelerated to above the Alfvén speed (Figure 3c),resulting in an intermediate transition (Figure 3e). (3) Decay with time:
The intermediate shock is sustainedby the neutral pressure (Figure 3f), which, because of thefinite coupling, cannot balance the Lorentz force. As timetends to infinity, the neutral pressure equalises and the mag-netic field reversal becomes decreases in magnitude, see Fig-ure 4. This is an indicator that the intermediate shock dis-appears with time. For these parameters, the magnetic fieldreversal is still prominent after 10000 collisional times, indi-cating that whilst this is a transient feature of the system,it is sufficiently long-lived to be physical and independentof the initial conditions.The frictional heating ( α c ( T n , T p ) ρ n ρ p ( v n − v p ) ) andthermal damping ( α c ( T n , T p ) ρ n ρ p ( P n /ρ n − P p / (2 ρ p )) ) be-tween the two species are shown in Figure 5 through thefinite-width of the shock. Both the frictional heating andthermal damping reach their maximum values at the inter-mediate transition. Here, there is the largest drift velocity(Figure 3a). Either side of the shock, the species are rea-sonably well coupled and hence there is minimal frictionalheating. The simulations in this section use β = 0 . , B x = 0 . withthe ionisation fraction ξ i varying. The results can be scaledby a rough estimate of the time scale changes by multi-plying by the ionisation fraction, i.e., tξ i . As the neutralfraction increases, the influence of the neutral pressure be-comes more important. The initial evolution stages generatean overshoot in the neutral pressure due to the collisionalcoupling. There is then an interplay between the neutralpressure and the magnetic tension. When ξ n ≤ . , theneutral pressure increases with time to balance the mag-netic tension. When ξ n > . the neutral pressure over-shoot decreases with time. This is shown in Figure 6 withthe ξ n = 0 . (black dashed) and the ξ n = 0 . (magentadashed) lines. Whilst the two results look very different, Article number, page 5 of 10 &A proofs: manuscript no. intermediate_shocks
Neutral overshootIntermediateshockSlow-mode shock-0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s -0.12-0.11-0.10-0.09-0.08-0.07-0.06 v x - v s Neutral overshoot-0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s D e n s i t y Alfvenic inflow-0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s M v A -0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s v A -0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s P r e ss u r e Slow-mode transitionIntermediate transition-0.020.00 0.02 0.04 0.06 0.08 0.10 0.12x s SuperfastSubfastSuperslowSubslowStatic (a) (b)(c) (d)(e) (f)
Fig. 3.
MHD (black dashed) and PIP (red neutral, blue plasma) solutions within the shock. The figures are in the shock frame( x s = x/t − v s , where v s is the propagation speed of the shock in the MHD solution). (a) perpendicular velocity in the shock frame v x − v s . (b) Density. Green shows the total ( ρ n + ρ p ) density. (c) Alfvén Mach number using the plasma density (magenta) andtotal density (green). (d) Alfvén speed using the total density. (e) Shock transitions. (f) Pressure. Green line shows total pressure( P n + P p ). the same mechanism is occurring whereby the system seeksan equilibrium; the main difference is the magnitude of theneutral pressure overshoot due to the initial heating. Allcases tend towards a constant maximum neutral pressure.All neutral fractions investigated feature the intermedi-ate transition as a substructure within the slow mode shock.The normalised time scale does not map the behaviour per-fectly and there are differences in magnitude and gradientsof the magnetic field across the different neutral fractions.The neutral pressure is the key variable due to its role in theequalising the Lorentz force. As the neutral fraction tendsto unity, the maximum neutral pressure is reasonably sim-ilar across simulations. Figure 7 shows the neutral pressure and magnetic field re-versal for the low-beta regime. All three cases plotted con-tain roughly the same gradient in magnetic field over timebut have different equilibrium neutral pressures. As theplasma- β increases, the maximum equilibrium neutral pres-sure increases drastically. This results in a larger restoringforce and the magnetic field decreases more rapidly.For large plasma- β values, a sonic shock can occur inthe neutral fluid, as seen in Hillier et al. (2016). In thesesimulations, there can be two shocks occurring within thefinite-width shock region: intermediate shock in the plasma,and sonic shock in the neutrals. Article number, page 6 of 10. Snow and A. Hillier: Intermediate shock substructure
100 1000 10000t0.010.10 M a g n e t i c f i e l d r e v e r s a l M a x i m u m n e u t r a l p r e ss u r e Fig. 4.
Maximum neutral pressure (dashed) and maximum mag-netic field reversal (solid) through time. Both quantities aremaximum near the intermediate transition. -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12x s -6 -5 -4 -3 -2 P o w e r Fig. 5.
Frictional heating (black line) and thermal damping(red and blue lines). The red line indicates ions losing heat toneutrals, and vice versa for the blue line. Dashed lines indicatethe finite width of the shock.
10 100t x i M a g n e t i c f i e l d r e v e r s a l M a x i m u m n e u t r a l p r e ss u r e Fig. 6. (dashed) Maximum neutral pressure and (solid)Magnetic field reversal for different neutral fractions ξ n = 0 . (black), . (red), . (blue), . (green), . (magenta). The initial B x component of magnetic field is modified toalter the angle of the magnetic field. Note that the initial B y component remains the same and hence the total B strength varies across simulations. Interestingly, the pre-shock region has | B | ≈ in all simulations as a consequenceof the investigated system.
10 100 1000t0.00010.00100.01000.1000 M a g n e t i c f i e l d r e v e r s a l M a x i m u m n e u t r a l p r e ss u r e Fig. 7. (dashed) Neutral pressure and (solid) Magnetic fieldreversal for different beta values β = 0 . (black), . (red), and . (blue).
100 1000 10000t10 -5 -4 -3 -2 -1 M a g n e t i c f i e l d r e v e r s a l M a x i m u m n e u t r a l p r e ss u r e Fig. 8. (dashed) Neutral pressure and (solid) Magnetic fieldreversal for B x = 0 . (black), . (red), . (blue), . (green). Changing the B x value changes the propagation speedsof waves and hence phenomena occurs on different timescales. From Figure 8 it is clear that a larger B x resultsin faster equalisation of the neutral pressure and hence themagnetic field reversal and the intermediate shock decayfaster, compare to low B x values. Magnetic tension is theequalising force for the intermediate shock. The magnetictension increases for larger B x values and hence the inter-mediate shock decays faster.
5. Comments on potential observations
In the MHD case, the fluid and the magnetic field are frozentogether, so in the rest frame of the fluid the electric fieldbecomes E ∝ v × B = 0 . However, for the two fluid case,there is a separation of the species within the finite-widthof the shock. Within the finite-width, even though there isno electric field felt by the plasma fluid there is therefore anelectric field felt by the neutrals. This can be calculated bylooking at how fast the neutrals move across the magneticfield, which is the the drift velocity, i.e., E n ∝ ( v n − v p ) × B . This metric is zero either side of the shock since thespecies are fully coupled. Inside the shock, the ions andneutrals decouple and hence there is a localised increase inthe neutral electric field inside the shock substructure. Theelectric field felt by the neutrals is a potential observablefor measuring the neutral-ion drift (e.g., Anan et al. 2014).The maximum electric field felt by the neutrals withinthe shock is plotted through time in Figure 9 for the refer- Article number, page 7 of 10 &A proofs: manuscript no. intermediate_shocks ( v n - v p ) ´ B Fig. 9.
Peak neutral electric field increase within the shockthrough time. Note that the electric field outside the shock inthis frame is zero. ence PIP simulation ( β = 1 , B x = 0 . , ξ = 0 . ). As timeadvances, this tends towards a constant value indicatingthat there will always be a localised increase in neutral elec-tric filed within the shock. The presence of this enhancedneutral electric field could be a potential observable for theeffects of partial ionisation in shock waves.
6. Summary
This paper has demonstrated that intermediate shocks canoccur as substructure inside slow-mode shocks due to par-tial ionisation. High-resolution 1D numerical simulationswere performed to fully resolve the substructure that oc-curs within the finite width of the slow-mode shock. Thephysical process involved in forming and dissipating the in-termediate shock substructure is as follows:1.
Collisional coupling results in overshoots in neu-tral velocity and density
The initial conditions drivethe plasma only. As such, there is a large drift velocitybetween the two species. The neutral fluids response isto create a Sedov-Taylor-like expansion which createslocalised increases in the neutral density and velocity atthe interface.2.
Acceleration of the plasma
Our system has velocitydriven towards the slow-mode shock at the Alfvén speed.Within the finite-width of the shock, the Alfvén veloc-ity decreases due to the increased neutral pressure andreversal of the magnetic field, and hence the plasma ve-locity is accelerated inside the shock to above the Alfvénspeed.3.
Decay with time
The neutral pressure is not sufficientto balance the Lorentz force and hence there is a grad-ual evolution whereby the magnetic field reversal tendsto zero as time tends towards infinity. The intermediateshock is however sufficiently long-lived to be indepen-dent of the initial conditions and considered a physicalfeature of partially ionised shocks.The intermediate shock was present as substructure forall tested parameter regimes but the magnitude of the mag-netic field reversal was parameter dependent. The larger theequilibrium neutral pressure, the larger the magnetic fieldreversal. As such, the intermediate shock is strongest forhigh plasma- β values. We would therefore expect this fea-ture to be most significant in the lower atmosphere, e.g.,Ellerman bombs. The work in this paper is analogous to the magneto-spheric intermediate shocks discussed in Karimabadi (1995)where the resistivity creates a dispersive region accelerat-ing the plasma towards the shock region and resulting inan intermediate transition. Here we have a similar effectexcept it is driven by two-fluid interaction in the absenceof resistivity.We have shown that an intermediate shock exists for awide range of parameters, with varying degrees of magni-tude and lifetime. Theoretically, this has the potential tobe present across a wide range of phenomena in the solaratmosphere where partial-ionisation effects are important,from wave-steepening events (e.g., umbral flashes), to mag-netic reconnection (Ellerman bombs, spicules). Future workwill be to analyse the implications and observability of in-termediate shocks in the lower solar atmosphere.In summary, there are four main conclusions in this pa-per:1. Ideal MHD heating across the shock
We haveshown analytically that the two-fluid equations reduceto the ideal MHD shock equations when the species arecoupled either side of the shock (see Appendix A). Theconsequence of this is that one would expect to obtainideal MHD-like heating across the shock, with no addi-tional heating from the collisions.2.
Shocks as substructure in PIP case
Within thefinite width of a partially-ionied shock, interactions be-tween the two species can lead to the formation of in-termediate transitions within the larger shock structure.Intermediate shocks are a transition from above to be-low the Alfvén speed (here super-Alfvén to sub-slow)and feature a reversal in the magnetic field (see Figure2). This feature forms due to the collisional effects be-tween the two species inside the finite-width shock. Anintermediate transition was present for all tested param-eter regimes.3.
Potential for large currents
The formation of an in-termediate shock features a sharp reversal in the mag-netic field across the intermediate shock front, hencethere is the potential for large currents to form insidethe large-scale slow-mode transition. Large currents areknown to play a role in particle acceleration. Hencethe formation of intermediate shocks in partially ionisedplasma may lead to an additional particle accelerationmechanism.4.
Localised electric field experienced by the neu-trals
Within the finite width of the shock, there is alocalised increase in neutral electric field. This tends to-wards a constant value as time tends to infinity, hencemay be a potential observable of ion-neutral interactionswithin shock fronts.
Acknowledgements
BS and AH are supported by STFC research grantST/R000891/1. AH is also supported by STFC ErnestRutherford Fellowship grant number ST/L00397X/2.
References
Anan, T., Casini, R., & Ichimoto, K. 2014, ApJ, 786, 94Anan, T., Ichimoto, K., & Hillier, A. 2017, A&A, 601, A103Beckers, J. M., & Tallant, P. E. 1969, Sol. Phys., 7, 351
Article number, page 8 of 10. Snow and A. Hillier: Intermediate shock substructure
Cassak, P. A., Liu, Y.-H., & Shay, M. A. 2017, Journal of PlasmaPhysics, 83, 715830501Delmont, P., & Keppens, R. 2011, Journal of Plasma Physics, 77, 207Ellerman, F. 1917, ApJ, 46, 298Hau, L.-N., & Sonnerup, B. U. O. 1989, J. Geophys. Res., 94, 6539Hillier, A., Takasao, S., & Nakamura, N. 2016, A&A, 591, A112Houston, S. J., Jess, D. B., Asensio Ramos, A., et al. 2018, ApJ, 860,28Innocenti, M. E., Goldman, M., Newman, D., Markidis, S., & Lapenta,G. 2015, ApJ, 810, L19Karimabadi, H. 1995, Advances in Space Research, 15, 507Khomenko, E. 2017, Plasma Physics and Controlled Fusion, 59,014038Liu, Y.-H., Drake, J. F., & Swisdak, M. 2012, Physics of Plasmas, 19,022110Loureiro, N. F., & Uzdensky, D. A. 2016, Plasma Physics and Con-trolled Fusion, 58, 014021Nelson, C. J., Shelyag, S., Mathioudakis, M., et al. 2013, ApJ, 779,125Petschek, H. E. 1964, NASA Special Publication, 50, 425Pontin, D. I. 2011, Advances in Space Research, 47, 1508Shibayama, T., Kusano, K., Miyoshi, T., Nakabou, T., & Vekstein,G. 2015, Physics of Plasmas, 22, 100706Wu, C. C., & Hada, T. 1991, J. Geophys. Res., 96, 3769Yamada, M., Kulsrud, R., & Ji, H. 2010, Reviews of Modern Physics,82, 603
Article number, page 9 of 10 &A proofs: manuscript no. intermediate_shocks
Appendix A: Hoffman-Teller PIP Equations
In the Hoffman-Teller frame, the 2-fluid PIP equations are: ρ n v ⊥ n = const , (A.1) ρ n v ⊥ n v (cid:107) n = − I + const , (A.2) ρ n v ⊥ n v ⊥ n + P n = − I + const , (A.3) v ⊥ n (cid:18) γγ − P n + 12 ρ n v n (cid:19) = − I + const , (A.4) ρ p v ⊥ p = const , (A.5) ρ p v ⊥ p v (cid:107) p − µ B ⊥ B (cid:107) = I + const , (A.6) ρ p v ⊥ p v ⊥ p + P p + B µ = I + const , (A.7) v ⊥ p (cid:18) γγ − P p + 12 ρ p v p (cid:19) = I + const , (A.8) v ⊥ p B (cid:107) − v (cid:107) p B ⊥ = 0 , (A.9) B ⊥ = const , (A.10) I = (cid:90) α c ( T n , T p ) ρ n ρ p ( v (cid:107) n − v (cid:107) p )d ⊥ , (A.11) I = (cid:90) α c ( T n , T p ) ρ n ρ p ( v ⊥ n − v ⊥ p )d ⊥ , (A.12) I = (cid:90) α c ( T n , T p ) ρ n ρ p × (cid:20)
12 ( v n − v p ) + 3 R g ( T n − T p ) (cid:21) d ⊥ . (A.13)The integral terms ( I , I , I ) can be removed by addingthe neutral and ion equations together, i.e., ρ n v ⊥ n + ρ p v ⊥ p = const , (A.14) ρ n v ⊥ n v (cid:107) n + ρ p v ⊥ p v (cid:107) p − µ B ⊥ B (cid:107) = const , (A.15) ρ n v ⊥ n v ⊥ n + P n + ρ p v ⊥ p v ⊥ p + P p + B µ = const , (A.16) v ⊥ n (cid:18) γγ − P n + 12 ρ n v n (cid:19) + v ⊥ p (cid:18) γγ − P p + 12 ρ p v p (cid:19) = const , (A.17) v ⊥ p B (cid:107) − v (cid:107) p B ⊥ = 0 , (A.18) B ⊥ = const . (A.19)The partial pressure and density can be expressed interms of a total value using the neutral fraction ξ n : ρ n = ξ n ρ t , (A.20) ρ p = (1 − ξ n ) ρ t , (A.21) P n = ξ n ξ n + 2(1 − ξ n ) P t , (A.22) P p = 2(1 − ξ n ) ξ n + 2(1 − ξ n ) P t . (A.23)Substituting these into Equations (A.14-A.19) gives: ξ n ρ t v ⊥ n + (1 − ξ n ) ρ t v ⊥ p = const , (A.24) ξ n ρ t v ⊥ n v (cid:107) n + (1 − ξ n ) ρ t v ⊥ p v (cid:107) p − µ B ⊥ B (cid:107) = const , (A.25) ξ n ρ t v ⊥ n v ⊥ n + ξ n ξ n + 2(1 − ξ n ) P t + (1 − ξ n ) ρ t v ⊥ p v ⊥ p + 2(1 − ξ n ) ξ n + 2(1 − ξ n ) P t + B µ = const , (A.26) v ⊥ n (cid:18) γγ − ξ n ξ n + 2(1 − ξ n ) P t + 12 ξ n ρ t v n (cid:19) + v ⊥ p (cid:18) γγ − − ξ n ) ξ n + 2(1 − ξ n ) P t + 12 (1 − ξ n ) ρ t v p (cid:19) = const , (A.27) v ⊥ p B (cid:107) − v (cid:107) p B ⊥ = 0 , (A.28) B ⊥ = const . (A.29)Furthermore, we can impose an additional constraintsuch that either side of the shock the drift velocity equalszero ( v (cid:107) p = v (cid:107) n = v (cid:107) and v ⊥ p = v ⊥ n = v ⊥ ): ρ t v ⊥ = const , (A.30) ρ t v ⊥ v (cid:107) − µ B ⊥ B (cid:107) = const , (A.31) ρ t v ⊥ v ⊥ + P t + B µ = const , (A.32) v ⊥ (cid:18) γγ − P t + 12 ρ t v (cid:19) = const , (A.33) v ⊥ B (cid:107) − v (cid:107) B ⊥ = 0 , (A.34) B ⊥ = const . (A.35)Equations (A.30-A.35) are identical to the MHD equa-tions except here the density is the total density ρ t andpressure P t . Therefore, the solution to these equations isidentical to the Hau & Sonnerup (1989) solution for MHD(Equation 7), independent of the neutral fraction. It shouldbe noted that this is only true over the larger shock struc-ture and inside the shock there is substructure that is highlydependent on the collisional effects.. Therefore, the solution to these equations isidentical to the Hau & Sonnerup (1989) solution for MHD(Equation 7), independent of the neutral fraction. It shouldbe noted that this is only true over the larger shock struc-ture and inside the shock there is substructure that is highlydependent on the collisional effects.