Corso di Laurea Magistrale in Fisica della Materia
Curriculum Plasmi
Kelvin-Helmholtz mediated magnetic reconnection
at the Earth’s magnetospheric flanks
Candidato:
Manuela Sisti
Relatori:
Prof. Francesco Califano Prof. Matteo Faganello
1 Introduction 5
2 Fluid Equations and MHD 9
2.1 Validity regimes of non-relativistic fluid equations . . . 9
2.2 Two-fluid model . . . 10
2.3 Two fluid model in a one-fluid framework and Magnetohydrohynamics . . . 12
2.3.1 Magnetohydrodynamics (MHD) . . . 13
2.3.2 MHD normal modes . . . 13
2.3.3 Mach numbers . . . 15
3 Magnetic Reconnection 17 3.1 Introduction . . . 17
3.1.1 Conservation theorems and magnetic reconnection . . . 18
3.1.2 Conditions for magnetic reconnection . . . 20
3.2 Resistive reconnection . . . 21
3.2.1 Linearized reconnection equations . . . 21
3.2.2 Boundary layer approach . . . 24
3.2.3 Physical interpretation of magnetic reconnection . . . 29
3.2.4 Magnetic insland: X and O-points . . . 30
3.3 Inertial reconnection . . . 30
3.3.1 Inertial reconnection in MHD . . . 30
3.3.2 Inertial Hall-reconnection in a two-fluid framework . . . 33
4 Kelvin-Helmholtz instability 37 4.1 Introduction to Kelvin-Helmholtz instability . . . 37
4.2 Linear theory for Kelvin-Helmholtz instability . . . 37
4.2.1 Linear MHD theory . . . 38
4.3 Effects of finite thickness shear layer . . . 41
4.4 Effects of compressibility . . . 45
4.5 Effects of density inhomogeneity . . . 45
5 Plasmas in the Earth magnetosphere context 47 5.1 The Earth magnetic context. . . 47
5.2 Interaction between solar wind and Earth’s magnetosphere . . . 47
5.2.1 On the 2D dynamics of the interaction between KHI and magnetic recon-nection . . . 48
5.2.2 On the 3D dynamics of the interaction between KHI and magnetic recon-nection: double mid-latitude reconnection . . . 50
5.2.3 On satellites in situ evidences of KHI and magnetic reconnection . . . 52 3
6 The model and numerical results 55
6.1 The code . . . 55
6.1.1 Box size and boundary conditions . . . 56
6.1.2 Plasma model implemented . . . 57
6.1.3 Physical values . . . 58
6.1.4 Equilibrium equations and normalized physical values . . . 58
6.1.5 The passive tracer . . . 60
6.2 Analysis of the linear phase of KHI . . . 63
6.2.1 Shape in x of the mode amplitude . . . 64
6.2.2 Most excited modes and angle for the maximal growth rate . . . 65
6.2.3 The non-linear evolution of the vortices until time t = 550. . . 69
6.3 Latitudinal shift of the vortices . . . 70
6.4 Nonlinear evolution and secondary processes . . . 72
6.5 Magnetic reconnection processes . . . 74
6.5.1 On the reconnection events . . . 77
6.5.2 On the latitudinal distribution of proxies and reconnection sites . . . 78
6.5.3 A statistical analysis of field lines connectivity. . . 79
7 Conclusions and future work 91
A A method to integrate magnetic field lines 93
Bibliography 95
Introduction
The plasma state is usually referred as the fourth state of matter. In practice it is a collection of a very large number of charged particles, globally neutral, dominated by the electromagnetic forces while binary interactions are very weak. Thus, the behaviour of a plasma is very different from that of a gas. Much of the matter in the Universe exists in a plasma state (near 99.9% of the ordinary matter): the Sun and stars, the interstellar medium, the upper athmosphere (ionosphere and magnetosphere),. . . Instead the lower atmosphere of the Earth being a neutral gas is an exception (except for lightning and auroras) but plasma can be created artificially in the laboratory for energetic (controlled thermonuclear fusion), medical or engineering research. The study of astrophysical and space plasmas, in particular of solar violent events (solar flares), coronal mass ejections, solar wind interaction with magnetosphere or turbulence, are very im-portant for the whole field of research. Indeed space plasmas with direct in-situ measurments by space-craft constitute a natural laboratory for plasma physics. Only from space we can obtain detailed microscopic measures underlining the physics of important phenomena which occur also in laboratories on Earth (for example, we can obtain informations about the distribution function of plasma only in space). For this reason in the years several space missions have been launched aiming at collecting data about fundamental processes in plasmas. Among them we cite MMS and CLUSTER missions. It is worth to notice that the study of space plasmas became more and more important for direct application to a recent developed branch of physics called
space weather. Indeed, because of a plasma instability called magnetic reconnection, fast solar
wind particles, ejected from the Sun, enter into the Earth’s magnetosphere and cause beautiful phenomena such as auroras or influence man-made systems. Space-based telecommunications, broadcasting, navigation, power distribution, spacecraft performances and lifetimes, aviation are all examples of sectors that are potentially affected by space weather.
Our work focuses on the interaction of the solar wind with the Earth’s magnetosphere. The Solar Wind (SW) is the fully ionized magnetized plasma originated in Sun, which streams through the Solar System, while the magnetosphere is a structure which surrunds our planet and protects it from the hot charged Sun particles. The first models which tried to describe the SW-magnetosphere system were closed ones where no solar wind plasma entry into the mag-netosphere was admitted (Chapman and Ferraro, 1930). In these models the magmag-netosphere was seen as an impenetrable barrier: since the particles must follow the magnetic field lines (through important plasma theorems) and there was no possibility to change magnetic connec-tions among lines of force (reconnection was unknown), magnetospheric plasma and solar wind one could not mix. However later (around the 1970s) data provided by satellites disproved this closed model showing evident injections of hot particles into the magnetosphere. Luckily in the meantime other models was proposed in order to explain this interaction and they admitted the possibility to interconnect, thanks to the magnetic reconnection instability, solar wind magnetic field lines to magnetospheric ones, so to provide the mixing between different type of plasmas.
The first to propose this open model was Dungey in the early 1960s. From that moment a grow-ing interest for this research field has brought generations of physicists to dedicate themselves to this problem, developing more and more sophisticated models. In particular, concering the periods in which the magnetic field associated to the solar wind is directed northward near the Earth (which is the case of our thesis work), the role of a classical hydrodynamic instability, the Kelvin-Helmholtz one, has been deeply investigated in order to understand its possible role in enhancing the process of magnetic reconnection. Kelvin-Helmholtz instability is of hydrody-namic nature and it develops in the presence of a velocity variation (shear) across the interface between two types of fluids. It leads to the formation of fully rolled-up vortices. In nature it’s very common: for example it can develop in a river after an obstacle or in the lower atmosphere when two laminar sections of wind stream at different velocity (in this case it can be made visible by clouds) or at the interface between wind and water in the sea. A general way to modelize plasmas consists in treating them as a fluid (under appropriate conditions). Thus we expect the developing of KH instability at the flanks of the Earth’s magnetosphere because of the shear ve-locity between the solar wind plasma and the magnetospheric one at rest. In last years satellites have provided clear evidence of Kelvin-Helmholtz vortices at the boundary of the magneto-sphere. Thus researchers have started to improve numerical codes able to simulate the complex dynamics of the system, encouraged by the possibility to compare their results with satellite data. The numerical tool is essential for this problem because of its strong non-linearity which makes difficult to develop fully analytical models. The modelling of the whole magnetosphere requires an enormous computational effort because of the number of scales involved for the full system (separated by several orders of magnitude) and of the different physical regimes at play connected one to each other. Thus, computational resources did not allow (and do not allow still today) to simulate the whole magnetosphere from the smallest to the biggest scales. More-over, the main mechanism allowing for the mixing at the magnetosphere boundary is magnetic reconnection, a multi-scale multi-physics process acting at small kinetic scales but impacting on the large fluid scales. Thus when researchers want to improve a numerical simulation they have to choose whether to neglect smaller scales or to reduce the portion of magnetosphere to simulate, risking in both cases to miss some important physical effects. Waiting for more pow-erful machines the only way is to make a choice and to proceed by gradual steps and so they did. The course of research which has led to our thesis work chose to neglect small-scale kinetic effects employing a fluid tractation. We retrace the main steps. First the researchers treated separately the problem of magnetic reconnection and that of Kelvin-Helmholtz instability, then they began to produce simplified 2D simulations able to analyze the interaction between the two phenomena. In the last decade they deeply analyze the 2D case adding more and more realistic details, such as magnetic shears or density inhomogeneities, finding one first classification for the process of Kelvin-Helmholtz mediated magnetic reconnection (Type I and II Vortex Induced Reconnection) and deeply investigate the effects of vortex pairing and secondary instabilities in reducing or increasing the efficiency of the process of mixing. After these 2D studies the re-searchers began to improve more realistic 3D simulations from which they observed a new kind of Kelvin-Helmholtz mediated magnetic reconnection namely double mid-latitude reconnection because it develops at higher latitudes with respect to the KH vortices region. This model was the first able to take into account the high latitude stabilization of the Kelvin-Helmholtz instabil-ity due to the increase of the magnetic tension. This stabilization causes a differential advection of magnetic field lines with latitude. Thus, so far from the equatorial plane the lines, strongly distorted, undergo reconnection. The first simulations of this kind were simplified. Indeed they did not take into account the presence of a magnetic shear (and of a density inhomogeneity) in the equalibrium configuration. And here we come to the point of our thesis work. Encouraged by satellite observations of remote reconnection, i.e. far away from the equatorial plane, we have improved the 3D model by adding a pre-esisting magnetic shear and inhomogeneity of density and pressure. Doing this we have choosen the physical parameters such to reproduce at best the
large scale configuration recorderd by satellites. The present thesis work consists in the analysis of the results of this 3D simulation aimed at reproducing the dynamics at the magnetosphere flanks. We give in the following an outline of this work.
In chapter 2 we open introducing the typical parameters important in plasma physics and giving an overview on the fluid models adopted to reproduce the large-scale plasma behaviour: two fluid and MHD one. The two-fluid model is the one used in our work. We discuss also about the approximations allowing one to use one model rather than another, and in particular the conditions under which we can neglect the kinetic-small scale effects.
In chapter 3 we deal with the process responsible for the injection of solar wind particles into the magnetosphere, magnetic reconnection instability, the only one able to change magnetic connections among magnetic field lines. We start by listing the theorems which would prohibit the change of the magnetic topology and by underlining under what conditions their hypothesis break. Two main type of magnetic reconnection exist: resistive reconnection and inertial recon-nection, depending on the term which breaks the ideal Ohm’s law. We present both, focusing in particular on the former because in our simulation we use a small but finite resistivity (∼ 10−3)
in order to allow magnetic reconnection to occur.
In chapter 4 we introduce the classical theory of the magnetized Kelvin-Helmholtz instabil-ity in the limit of a pure discontinuinstabil-ity separating the two fluids. Then, we discuss about the limitations of this theory, underlining in particular the effects of the finite thickness of the shear layer, of compressibility and of density inhomogeneity.
In chapter 5 we give an overview on the system we want to simulate, which is the space surrounding the Earth, in particular, the interaction between the solar wind and the magne-tosphere at the magnetospheric flanks. We retrace the main results obtained in last decades about the mechanisms which lead the enhancing of magnetic reconnection via Kelvin-Helmholtz instability. We consider both the 2D and the 3D case. We summarize the characteristics of Type I and II Vortex Induced Reconnection and of double-mid latitude reconnection.
In chapter 6 we introduce the code used to simulate the magnetosphere flank, discussing the simulation box features, the plasma model implemented, the normalization used and the equilibrium from which the simulation starts. Then we present the results of the linear analysis of Kelvin-Helmholtz instability obtained with a Fourier transform procedure. Thanks to this analysis we get the most unstable modes of KHI, their growth rates in the linear stage and the angle with respect to the equatorial plane for the maximal growth rate. We will find that the Kelvin-Helmholtz waves propagate with wavefront which tilts with respect to the equatorial plane and we will compare this experimental angle with the analytical prediction given by the model of chapter 4. Then we analyze the vortices developing during the non-linear phase and we discuss on how the plane where the vortices are most developed drifts northward. We talk about the developing of secondary instabilities. Finally we concentrate on magnetic reconnec-tion events, discussing the methods thanks to which we individuated them, their evolureconnec-tion in time and the latitudinal distribution of the reconnection sites.
Fluid Equations and MHD
We begin our work with an overview on the non-relativistic fluid models used to reproduce the large-scale behaviour of plasmas, systems of many charged particles where collective electro-magnetic long-range interactions are predominant rather than binary ones ("collisions"). We will start by discussing the regimes where fluid models can be adopted, then we will expose the two-fluid model and the MHD one (one-fluid model). Finally we will show which kind of normal modes exist in a MHD context.
2.1
Validity regimes of non-relativistic fluid equations
We define L as the scale length of the typical fluctuations and τ as their characteristic dynamic time. The first hyphothesis is the non-relativistic one: all the characteristic velocities of the system are small as compared to c, which is the speed of light in the vacuum:
L/τ c (2.1)
An important hyphothesis that must be satisfied in order to adopt a fluid model concerns the possibility of defining a fluid element. The latter must be smaller than all the characteristic scale lengths of the phenomena at play
lf e L (2.2)
(lf e is the scale length of the fluid element) and must diffuse in a characteristic time slower
than that of the evolution of the system. We can extrapolate the parameters of the diffusion coefficient (D) by considering a continuity equation for the number density of a fluid element:
∂tn+ ∇ · ~Γ = 0, where ~Γ = n < ~v >= −D∂xnis the flux of particles per unit area per unit time.
Thanks to a dimensional analysis we get D ∼ l2
mf p/tc, where lmf p is the mean free path and tc
the time between two collisions. Because lmf p = vth/νcoll (with vth the thermal velocity of the
particles and νcoll the collision frequency), D ∼ vth2 /νcoll. Thus we find the fluid element is well
defined if
ω νcoll
1 ←→ lmf p
L 1 (2.3)
The latter hypothesis for the use of fluid equations tell us that we can use a fluid model only in a "strongly" collisional plasma. Thus it could be surprising to find that these equations can reproduce the plasma dynamics also in a quasi-collisionless medium (such as the Earth’s magnetosphere and magnetosheath) that should be described with the adoption of a kinetic approach. However, in presence of a background electromagnetic field and looking at motions at scale larger than the characteristic kinetic ones, the plasma exhibits a single fluid behaviour because the ions and the electrons drift at nearly the same velocity ( ~E ∧ ~B-drift), which doesn’t
depend on particle charge or mass. A two-fluid behaviour emerges at intermediate scales, in between the one-fluid (MHD) ones and the small kinetic scales. In this context the models we are
going to analyze are defined “fluid” because of their similarity with the classical hydrodynamics equations, but they are actually obtained as the velocity moments of the Vlasov equation. Indeed, as the velocity moments of the distribution function give the mascroscopically observable quantities of the plasma, such as density, averaged velocity, pressure, heat flux. . . , the moments of Vlasov equation give fluid equations. Now we introduce some quantities in order to complete this overview about the validity regimes of fluid equations:
• the cyclotron frequency: Ωc= qB/mc
• the Larmor radius: ρL= vth/Ωc
The physical meaning of these quantities is due to the fact that in a uniform magnetic field, in absence of collisions, a charged particle moves, in the plane perpendicular to ~B, on a circular
orbit of radius ρL at frequency Ωc. The 3D orbit is helical, being the k velocity unaffected. In
presence of both a magnetic and an eletric field the orbit slightly changes and the particle, while rotating, gradually drifts in the direction ⊥ to both ~B and ~E at a velocity independent of either
its mass or charge. The fluid models describe the dynamics at scales larger than the Larmor radius:
lf e ρL (2.4)
L ρL (2.5)
2.2
Two-fluid model
Let’s recover the fluid description starting from the kinetic one. The statistical description of a system of N particles is given in terms of a distribution function F (~x1, . . . ~xN, ~v1. . . ~vN, t) which
obeys the Liouville equation1
∂F ∂t + X i ∂F ∂~xi · ~vi+ ∂F ∂~vi · ~aTi = 0 (2.6) where ~aT
i is the total acceleration of particle i due to external and interparticles forces. From
Equation (2.6) it is possible to obtain the equation for the one-particle distribution function f(1)
including a collissional term (see for example [1] for the well-known procedure):
∂fα(1) ∂t + ~v1· ∂fα(1) ∂~x1 + qα mα ~ E+~v1∧ ~B c ·∂f (1) α ∂~v1 = ∂fα(1) ∂t c (2.7) where α indicates the species of the particles. Eq. (2.7) becomes the Vlasov equation if we neglect the collision term on the right-hand side which takes into account the contribute of binary interactions. We can do it when the so-called plasma parameter g = 1/nλ3
D 1,
where λD = KBTe/4πnee2 is the Debye lenght and represents the spatial scale at which an
excess of charge is shielded, KB = 8.61x10−5eV/K is the Boltzmann costant, Te is the electron
temperature, ne is the electron number density, e is the electron charge. Thus, nλ3D represents
the number of plasma particles in a Debye sphere, while g may be interpreted as a measure of the degree to which collective effects dominate over single-particle behaviour (see [1]). Eq. (2.7) describes a "mean field" dynamics as the collective contribution of all particles is retained in the force term qα < ~E+ ~v1∧ ~B/c > /mα where < ~E > and < ~B >are the sum of the external and
internal average fields and satisfy the average Maxwell equations: • ∇ · ~E= 4π < ρα>
• ∇∧ < ~B >= 1c∂< ~∂tE> +4πc < ~J >
From the velocity moments of the one-particle distribution function we find the macroscopically observable quantities for particles of species α:
• the particles density: nα=
R fαd~v
• the mass and charge density, respectively ρmα = nαmα and ρcα= nαqα
• the average velocity: ~uα =R~vfαd~v/R fαd~v
• the current density: ~Jα = nαqα~uα
• the pressure tensor: Pα,ij = mαR(~v − ~uα)i(~v − ~uα)jfαd~v. This can be divided into two
parts: Pα,ij = Pαδij + Πα,ij where Πα,ij = mαR((~v − ~uα)i(~v − ~uα)j− |~v − ~uα|2δij/3)fαd~v
and Pα= mαR |~v − ~uα|2fαd~v is the scalar pressure.
From the zero and first order velocity moments of the Vlasov equation (see [1]) we obtain the continuity equation and the momentum equation for the density nαand the average velocity uα
of particles of species α:
• mass continuity equation:
∂tnα+ ∇ · (nαu~α) = 0 (2.8) • momentum equation: ∂(nα~uα) ∂t + ∇ · (nα~uα~uα) = −∇ ↔ Pα+ qαnα mα ~ E+qαnα mαc ~ uα∧ ~B (2.9)
where the electrostatic force can be neglected in the non-relativistic approximation. These equations do not constitute a closed set because they require the velocity moment of second order (P↔α). If we include the pressure equation, obtained as the second order moment
of the Vlasov equation, the new set will not constitute a closed one because it would require the moment of third order and so on. A complete description requires all velocity moments because any finite set of equations does not completely specify the plasma behaviour. Thus, it is necessary to truncate or "close" such infinite chain of equations at some order, in general by an assumption about the pressure. Before specifying the closures usually adopted it is worth to remember that there are two general approaches to a macroscopic description of a plasma: the ions and the electrons can be treat as separate but interacting fluids otherwise the plasma is considered as a single fluid with an averaged density, velocity and current. In the two-fluid model, equations (2.8) and (2.9) are coupled with Maxwell’s equations:
• the Gauss law:
∇ · ~E = 4πX
α
qαnα (2.10)
• the Faraday law:
∇ ∧ ~E = −1 c
∂ ~B
∂t (2.11)
• the Gauss law for magnetism:
∇ · ~B = 0 (2.12)
• the Ampere equation:
∇ ∧ ~B = 4π
c J~ (2.13)
where the current is defined as ~J = P
αqαnα~uα and the displacement current ∂tE~ is
The simpler choice for the closure is to assume a scalar pressure which obeys to a polytropic relation of the form:
Pα/nγα = costante (2.14)
where γ is a number called polytropic factor: γ = 0 expresses an isobaric transformation, γ = 1 an isothermal one, γ = 5/3 an adiabatic one.
2.3
Two fluid model in a one-fluid framework and Magnetohydrohynamics
We define ω2
pα= 4πqα2nα/mα, which is the plasma frequency, i.e. the typical electrostatic
oscil-lation frequency of a given species in response to a small charge separation. In the limit where
L ρL,e, L ρL,i, lf e λD, λD lmf p, i.e. long wavelength limit, and ω Ωc,i,Ωc,e,
ω ωp,e, ωp,i, i.e. low frequency limit, we can semplify the two-fluid model and rewrite its
equations in a one-fluid framework, derived adding and subtracting appropriately equations (2.8) and (2.9). Thanks to the assumption lf e λD, it is possible to use the quasineutrality
ap-prossimation (ne= ni). The macroscopical variables for this model are obtained by opportunely
combining the density and velocity of the species (note that in our case the ions are protons): • the mass density: ρm= neme+ nimi
• the charge density: ρc= e(ni− ne) = 0
• the center-of-mass velocity: ~u = neme~ue+nimi~ui
nemi+nime =
me~ue+mi~ui
mi+me ;
me
mi 1
• the current density: ~J = en(ui− ue)
• the thermal pressure: P = Pe+ Pi
The one-fluid equations are obtained by adding and subtracting opportunely the equations that describe the dynamics of electrons and protons:
• the mass continuity equation:
∂ρ
∂t + ∇ · (ρ~u) = 0 (2.15)
• the momentum equation2
ρ ∂~u ∂t + (~u · ∇)~u = −∇P +1 cJ ∧ ~~ B (2.16) • Faraday’s law: ∂ ~B ∂t = −c∇ ∧ ~E (2.17)
• Generalized Ohm’s law3
~ E+1 c~u ∧ ~B = me ne2 d ~J dt + 1 necJ ∧ ~~ B − 1 ne∇ · Pe (2.18) 2
It is worth to notice that actually there would be small terms ∝ d2e, but they are negligible when u ∼ ui
(me/mi 1). The complete equation will be discussed in section 3.3.2, Eq. (3.94).
3
By subtracting Eq. (2.9) for protons and for electrons we obtain:
∂t(n(~ui− ~ue)) + ∇(n~ui~ui− n~ue~ue) = −∇Pi/mi+ ∇Pe/me+ ne(1/mi+ 1/me) ~E + ne(~ui/mi+ ~ue/me) ∧ ~B
Remembering that ~uiand ~uecan be written as functions of ~J and ~u (~ui= ~u + (me/mi) ~J /ne and ~ue= ~u − ~J /ne),
we can manipulate the previous expression in order to obtain Eq. (2.18), neglecting all terms ∝ d4
e or higher
but keeping the ones ∝ d2e, indeed, as we will show in chapter 3, they can be as important as the others if the
• Ampere’s law:
~ J = c
4π∇ ∧ ~B (2.19)
Usually in the Ohm’s law is added the artificial term η ~J, called resistivity term, with the aim of
taking into account all the possible contributions due to binary interactions (collisions). Anyway, we will discuss about the meaning of each term of generalized Ohm’s law in chapter 3. Here we just recall that the first term on the right-hand side of Eq. (2.18), the electron inertia, becomes important when L ∼ deand the second, the Hall term, when L ∼ di, where dα = c/ωpαis called
inertial lenght for the species α. Moreover we recall the so-called ideal Ohm’s law: ~
E+1
c~u ∧ ~B = 0 (2.20)
When the latter holds the evolution of magnetic field lines is linked to the fluid motion, as we will show in chapter 3. This means that the fluid velocity advects in the same way plasma fluid elements and magnetic field lines. Finally, in this model the closure is in general achieved by assuming an equation of state as in the two-fluid one. An equation for the pressure term is obtained by using P n−γ = costante ↔ d
t(P n−γ) = 0 which means that the quantity P n−γ is
perfectly advected by the fluid motion, thus we get:
∂
∂t+ ~u · ∇
P = −γP ∇ · ~u (2.21)
2.3.1 Magnetohydrodynamics (MHD)
If we add the condition L di dewe are in the so-called magnetohydrodynamics regime. The
ideal MHD equations are (2.15)-(2.21) but in the Ohm’s law the right-hand side is neglected (so that Eq. (2.20) is valid). This fact will be justified later, in chapter 3. In this regime the evolution equation for the magnetic field is obtained by combining Eq. (2.20) and (2.17), and can be written in this form:
∂ ~B
∂t = ∇ ∧ (~u ∧ ~B) = −(~u · ∇) ~B+ ( ~B · ∇)~u − ~B(∇ · ~u) (2.22)
In order to underline the role of the magnetic field we can manipulate Eq. (2.16), indeed, using vector identities: ~J ∧ ~B/c= (∇ ∧ ~B) ∧ ~B/4π = −∇( ~B2/2)/4π + ( ~B · ∇) ~B/4π. Thus, Eq. (2.16)
becomes: ρ ∂~u ∂t + (~u · ∇)~u = −∇P −4π1 ∇ B~2 2 +( ~B · ∇4π) ~B (2.23) The second term on the right-hand side is the magnetic pressure, while the last term on the same side is called magnetic tension.
2.3.2 MHD normal modes
We want now to look at the normal modes that characterize a MHD plasma4. In order to do
that we linearize the MHD equations (2.15), (2.21), (2.22) and (2.23) by writing each quantity as a sum of two terms:
• ρ(~x, t) = ρ0(~x) + ρ1(~x, t)
• ~u(~x, t) = ~u0(~x) + ~u1(~x, t)
• P (~x, t) = P0(~x) + P1(~x, t)
• ~B(~x, t) = ~B0(~x) + ~B1(~x, t)
where the subscript 0 denote the equilibrium quantities and the subscript 1 small perturbations, |ρ1|/ρ0 ∼ || ~u1||/|| ~u0|| ∼ |P1|/P0 ∼ || ~B1||/|| ~B0|| ∼ 1. The equilibrium quantities fulfill the
MHD equations with ∂t = 0 (steady state condition). At the first order the equations (2.15),
(2.21), (2.22) and (2.23) become: ∂ρ1 ∂t + ~u1· ∇ρ0+ ρ0∇ · ~u1+ ~u0· ∇ρ1+ ρ1∇ · ~u0= 0 (2.24) ρ0 ∂~u 1
∂t + (~u1· ∇)~u0+ (~u0· ∇)~u1 +ρ1(~u0· ∇)~u0 = = −∇P1−4π1 ∇ B~ 0· ~B1+ ~B1· ~B0 2 +( ~B0· ∇) ~B1 4π + ( ~B1· ∇) ~B0 4π (2.25) ∂ ~B1
∂t = −(~u1· ∇) ~B0−(~u0· ∇) ~B1+ ( ~B0· ∇)~u1+ ( ~B1· ∇)~u0− ~B0(∇ · ~u1) − ~B1(∇ · ~u0) (2.26) ∂P1
∂t + ~u0· ∇P1+ ~u1· ∇P0+ γP1∇ · ~u0+ γP0∇ · ~u1 = 0 (2.27)
From the "linearized" equations (2.24)-(2.27) we can find the waves of ideal MHD. We assume the unpertubed plasma as static and homogenous, ~u0 = 0, and ρ0, P0 costant values. We also
take an unidirectional uniform magnetic field ~B = B0ˆz. We make use of the dispacement vector
~
ξ, so that ~ξ(~x, t) =Rt
0~u1(~x, t0)dt0. We perform a Fourier normal mode analysis in space and time
assuming ~ξ(~x, t) =P
~k,ω ˜ξ(~k, ω)e−i(~k·~x−ωt) where the wavevector is given by ~k = k⊥ˆy + kkˆz. We
get: ω2− k2 kvA2 0 0 0 ω2− k2 ⊥c2s− k2v2A −k⊥kkc2s 0 −k⊥kkc2s ω2− kk2c2s · ˜ξx ˜ξy ˜ξz = 0 (2.28) where v2
A= B02/4πρ is the Alfvén velocity and c2s = γP0/ρ0is the sound velocity. From (2.28) we
see the decoupling between the mode referred to ˜ξx and those related to ˜ξy and ˜ξz. The former
gives the dispersion relation of the shear Alfvén wave, which is transverse, incompressible and propagates at the Alfvén speed,
ω2− k2kvA2 = 0 (2.29)
instead the latter leads to
ω2 = 1
2k2(c2s+ vA2)[1 ± (1 − δ)1/2] (2.30)
where δ = 4k2
kc2svA2/k2(c2s + v2A)2. The two corresponding modes are called the magnetosonic
waves, fast and slow, for the plus and minus signs respectively, which are compressible. For
perpendicular propagation (~k = kˆy) the slow wave disappears and we obtain
ω2= k2(c2s+ vA2) (2.31)
which propagates in direction perpendicular to ~B0 at magnetosonic speed (vf2 = c2s+ v2A). For
parallel propagation (k⊥ = 0) the magnetosonic waves are divided into compressional Alfvén
2.3.3 Mach numbers
Finally we define the Mach numbers: • sonic Mach number: Ms = ∆u/cs
• alfvénic Mach number: MA= ∆u/vA
• magnetosonic Mach number: MF = ∆u/
q
v2A+ c2
s
where ∆u is the typical velocity variation due to equilibrium configuration or perturbations. The magnetosonic Mach number is primarily used to determine the approximation with which a flow, in a plasma, can be treated as an incompressible flow (for which ∇·~u = 0). The lowest MF
is, the less compression we have. Instead in a gas the degree of compressibility is determined by the sonic Mach number (Ms): in the same way, the lowest Ms is, the less compression we
have. Finally the alfvenic Mach number gives the importance of the inertia with respect to the magnetic tension.
Magnetic Reconnection
3.1
Introduction
Magnetic reconnection is a non-ideal instability with a great impact on fusion plasmas and astrophysical plasmas1. In particular it is the only process able to reorganize the large-scale
magnetic topology by a rearrangement of the connections of the magnetic field. Changing the global topology allows the system to reach lower energy states that would otherwise be forbidden in an ideal MHD evolution, where topology is preserved. Doing so, reconnection converts large amounts of magnetic energy into kinetic energy, thermal energy and particle acceleration. In order to break the ideal MHD constraints small-scale effects are required allowing for a local decoupling between the plasma motion and the magnetic field. These small-scale effects, such as resistivity or electron inertia, are important only locally where the plasma current is large, i.e. in the so-called current sheets. An important characteristic of magnetic reconnection is that it acts on time scale longer than that of the ideal MHD dynamics because the system must generate sufficiently small scales before this process can act. In Figure 3.1 we show a cartoon of a simple magnetic reconnection process. Vertical yellow arrows represent the plasma motion. The "in-flows" advect magnetic field lines toward the central current sheet. There non-ideal effects are at play and violate the ideal Ohm’s law, so that the "breaking" and rearrangement of magnetic field lines occur. It is worth to notice that the plasma behaves as an ideal one everywhere except around a very thin boundary layer where the ideal assumption breaks down and magnetic fields can diffuse across the plasma. In the next sub-sections we will discuss the
1For example, in the solar corona it is responsible for a rapid release of energy in the context of a pheonomenon
called “solar flare”, and for the outflow of solar wind in the outer space. In the Earth’s magnetosphere it is the main cause for the mixing of solar wind and magnetospheric plasma during southward periods, as we will discuss in chapter 5.
Figure 3.1: process of magnetic reconnection. 17
meaning of ideal MHD conservation laws and the conditions for their validity as well as the necessary conditions for magnetic reconnection to develop.
3.1.1 Conservation theorems and magnetic reconnection
In ideal MHD plasma motion and magnetic field lines are subjected to constraints because of the validity of the following theorems:
• Alfven Theorem: in ideal MHD the magnetic flux through any surface S bounded by a closed contour C moving with the fluid is constant2:
d
dtΦ = 0 (3.3)
Sometimes it is also called “frozen-in law”.
• Connection theorem: in ideal MHD if two plasma elements are initially connected at ~x1
and ~x2 by a magnetic line, then for every following time there will be a magnetic line
connecting them3. Formally, if, for t = 0, d~l∧ ~B = 0, then
d dt d~l ∧ ~B = 0, ∀t and d~l ∧ ~B = 0, ∀t (3.7)
The main consequences of these theorems are: 1. global topology is conserved;
2. lower energy states with different topology are forbidden;
3. regions of plasma initially not connected will remain unconnected forever.
Fundamental hypothesis for the validity of the previous thorems is that the ideal Ohm’s law is valid:
~ E+1
c~u ∧ ~B = 0 (3.8)
2
Proof: using the Leibnitz’s theorem: d dt Z S(t) ~ B(~r, t) · d ~S = Z S(t) ∂ ∂tB · d ~~ S + I C(t) ~ B · ~u ∧ d~l (3.1)
where S(t) is the surface, C(t) is the contour and d~l is the line element. The first term of the right-hand side represents the change of flux due to the time rate of change of ~B. The second term, instead, represents the change
in the surface area due to the movement of the contour C(t). Using ~B · ~u ∧ d~l = −~u ∧ ~B · d~l and the Stokes’
theorem, we obtain: d dt Z S(t) ~ B(~r, t) · d ~S = Z S(t) ∂ ∂tB − ∇ ∧ (~~ u ∧ ~B) ·d~S = 0 (3.2)
Indeed in ideal MHD: ∂t∂B − ∇ ∧ (~~ u ∧ ~B) = 0, because the ideal Ohm’s law is valid. QED.
3
Proof : using d
dt(d~l) = (d~l·∇)~u (see Figure 3.2, d~l+d(d~l) = d~l+(~u+d~u)dt−~udt → d(d~l) = d~udt → d dt(d~l) = d~u and d~u = (d~l · ∇)~u) and d dtB =~ ∂ ∂tB + (~~ u · ∇) ~B we obtain: d dt d~l ∧ ~B = d dt(d~l) ∧ ~B + d~l ∧ d dt( ~B) = (d~l · ∇)~u ∧ ~B + d~l ∧ (∇ ∧ (~u ∧ ~B)) + d~l ∧ (~u · ∇) ~B (3.4) Using: ∇ ∧ (~u ∧ ~B) = − ~B(∇ · ~u) + ( ~B · ∇)~u − (~u · ∇) ~B (3.5) Eq. (3.4) becomes: d dt d~l ∧ ~B = (d~l · ∇)~u ∧ ~B + d~l ∧ ( ~B · ∇)~u − d~l ∧ ~B(∇ · ~u) (3.6) ~
B k d~l, thus the first term in the right-hand side of Eq. (3.6) is equal to |d~l|| ~B|[(ˆb · ∇)~u ∧ ˆb], the second term is
Figure 3.2: evolution in time of a line connecting two plasma elements (d~l).
However we know that exists an extended form (see [1],[2]) of the Ohm’s law (generalized Ohm’s
law), given by:
~ E+1 c~u ∧ ~B = η ~J+ me ne2 d ~J dt + 1 necJ ∧ ~~ B − 1 ne∇ · ↔ Pe (3.9)
Not all terms that violate the ideal Ohm’s law, given by the extended form, break magnetic connections. Let’s see the effect of each of them. The first term of the right-hand side of Eq. (3.9), which is resistivity term, breaks magnetic connections and the phenomenon is called
resistive reconnection. Indeed by taking only the resistivity term in the right-hand side of the
Ohm’s law and using the electric field of Eq. (3.9) the Faraday law reads:
∂ ~B
∂t = ∇ ∧ (~u ∧ ~B) + c2η
4π∇2B~ (3.10)
The first term on the right-hand side of the latter equation describes the convection of the magnetic field by the plasma flow, while the second one describes the resistive diffusion of the field through the plasma. In general, or better almost always, the first term is much bigger than the second, so magnetic flux is “frozen” into the plasma, and the connections between field lines cannot be changed; viceversa in the regions where the first term more or less vanishes there is little coupling between the field and the plasma flow, and the connections of the magnetic field lines are allowed to change.
The second term of Eq. (3.9) is the electron inertia and breaks magnetic connections at the electron scale. The third term (Hall term) decouples electrons and ions dynamics (indeed:
~
u ∧ ~B/c − ~J ∧ ~B/nec= ~ue∧ ~B/c) but keeps valid the connection theorem but only for electrons.
The fourth term does not break magnetic connections when the electron pressure is assumed as scalar (P↔e = pe
↔
I) and a polytropic equation of state of the form pe = pe(n) is used. We
can decide which of the previous terms to keep by considering the physics of the system we are studying and the scales at which we have to analyze phenomena. In order to understand what terms are to keep in Ohm’s equation we can consider the following order of magnitude extimations: • |η ~J | |1c~u ∧ ~B| ∼ ηcB 4πL c√4πρ B2 = ωR ωA = S −1 = η (3.11) where ωA= 1/τA= uA/L= p B2/4πρ/L with u AAlfven velocity, ωR= 1/τR= c2η/4πL2
and S is the Lundquist number (a dimensionless ratio which compares the timescale of an Alfvén wave crossing to the timescale of resistive diffusion); at scales L where ωR/ωA∼1
the resistive term becomes important. From (3.11) we see also that the two timescales associated with the resistive reconnection process are the Alfvén time (which is the time of the dynamics evolution) and the resistive time4 (which is the one on which we have
4
It is worth to notice that the ratio S = (τR/τA) is very large in most plasmas of interest because the resistivity
the equilibrium diffusion). The resistive reconnection processes acts on a hybrid timescale that is much shorter than τR but much longer that τA.
• | 1 necJ ∧ ~~ B| |1c~u ∧ ~B| ∼ cB2 4πLnec c√4πmin B2 = c ωpi 1 L = di L (3.12) where di = c/ωpi = c/ p 4πne2/m
i is the ion inertial length. We see that, starting on
length scales of order of the ion skin depth, the Hall term can not be neglected, so there is the decoupling between ion and electron dynamics.
• |ω4π2 pe d ~J dt| |1 c~u ∧ ~B| ∼ 4πcB ω2 pe4πLτA c uAB = c2 ω2 pe 1 LτAuA = d e L 2 (3.13) where de = c/ωpe = c/ p 4πne2/m
e is the electron inertial length; thus on length scales
of order of the electron skin depth we have to take into account the electron inertia term (inertial reconnection). • |∇ ↔ Pe ne | |1c~u ∧ ~B| ∼ nT Lne c uB = mu2 2Le c uB = mcu 2LeB = u 2ΩiL = ρL 2L ∼ di L p β (3.14) where ρL = di √
β and β = 8πP/B2 is the ratio between kinetic pressure and magnetic
pressure. For low-β plasmas the pressure term is negligible.
3.1.2 Conditions for magnetic reconnection
Magnetic reconnection can take place only if the resonant condition
~k · ~B0= 0 (3.15)
is satisfield somewhere in the spatial domain, where ~k is the instability wavevector and ~B0 the
equilibrium magnetic field5. Ideal Ohm’s law requires the electric field to be balanced by the
magnetic Lorentz force acting on the fluid plasma. This is not possible at locations where the resonance condition is satisfied: in those locations some of the terms of the right-hand side of Ohm’s equation become important and cause the breaking of the connection theorems (and so magnetic reconnection can occur)6. In Figure 3.3 it is shown a simple example of configuration
that can lead to the develpment of the magnetic reconnection processes. The lines of force in this case are directed in opposite directions separated by the dotted “neutral line” where the magnetic field becomes zero. As we have discussed, there are two main types of magnetic reconnection: the resistive and the inertial one. In the next sections we will deal mainly with the resistive reconnection because in our case of interest, i.e. 3D simulation of the interaction between the solar wind and the magnetosphere in the Earth’s flank magnetopause (see chapters 5 and 6), the non ideal contribute in the Ohm’s law is given by the resistive term. It is worth to notice that the analytical model we will handle in this chapter is the simplest one, because it uses a slab 2D geometry. The case which we analyze in our simulation will be more complex for many reasons, first of all because the 3D geometry. We don’t have an analytical model able to reproduce it. Despite its simplifications this basic slab resistive theory is useful in order to understand the main characteristics of the magnetic reconnection process. For the sake of completeness we will discuss also the main characteristic of the inertial reconnection. Moreover, at the end of the section 3.3 for the intertial reconnection, we will underline the importance and explain the role of the Hall term.
5In section 3.2.3 we will give a physical interpretation which clarifies the meaning of this statement and the
role of the resonant condition.
Figure 3.3: An example of configuration in which magnetic reconnection can develop. Figure from [7].
3.2
Resistive reconnection
3.2.1 Linearized reconnection equations
In resistive reconnection the term of Ohm’s law that violate the connection theorems is the resistive one (η ~J). Before discussing the equations for this kind of reconnection we want to
dis-tinguish between a spontaneous and a forced process. "Spontaneous" means that the instability develops from an equilibrium configuration which evolves in a characteristic time much greater that the typical reconnection rate. Thus, it resembles a stationary equilibrium on the time scales of the reconnection process. We could instead consider a formulation where the instability is "forced" by the large scale plasma motion, that means by an ideal configuration which is not an equilibrium and which develops at a rate faster than the reconnection one (for exemple, the configuration for Kelvin-Helmholtz instability, which leads to the formation of vortices). In the simplest case of forced reconnection the large scale plasma motion pushes the magnetic field lines as in Figure 3.1. The current increases because of the advection of magnetic field lines caused by the motion, moreover the so-called current sheet becomes more and more thin. When the current sheet becomes so thin that the term cη∇2B/~ 4π is comparable with the other terms
of Eq. (3.10), reconnection acts on a typical time rate comparable with that imposed by the large scale motion. If resistivity is too small the inertial term of Ohm’s law can become impor-tant. Thus, in the spontaneous problem the reconnection rate is determined by the spontaneous development of small non-ideal scales (in between τA and τR), instead in the forced problem
the time scales are determined essentially by the large scale dynamic time evolution. Thus, for example, the reconnection rate for a process forced by the Kelvin-Helmholtz instability is that of the development of the KHI itself. From a mathematical point of view, at least in the resistive case, the two problems, the spontaneous and the forced one, are analogous7.
Let’s present the resistive MHD equations (which are equations (2.15)-(2.19) with only the resistive term in the generalized Ohm’s law):
• the mass continuity equation:
∂ρ
∂t + ∇ · (ρ~u) = 0 (3.16)
• the momentum equation:
ρ ∂~u ∂t + (~u · ∇)~u = −∇P +1 c ~ J ∧ ~B (3.17) 7
Indeed the difference between the two process is analogous to the difference between an initial value problem and a boundary value one.
Figure 3.4: Initial configuration. Bsh is represented by the green arrows while the blue one
represents the background component (guide field) B0 along z.
• Faraday’s law:
∂ ~B
∂t = −c∇ ∧ ~E (3.18)
• Generalized Ohm’s law:
~ E+1 c~u ∧ ~B = η ~J (3.19) • Ampere’s law: ~ J = c 4π∇ ∧ ~B (3.20)
where the displacement current has been neglected in the non-relativistic limit.
We consider an equilibrium configuration where the density is uniform, the equilibrium velocity is zero and where there is a large magnetic field component B0 along the z direction and a
smaller “sheared field” Bsh(x) in the plane (x,y) where the dynamics develops. In Figure 3.4 we
show a cartoon of the equilibrium configuration. Bsh is along y, depends on x only and changes
sign across x = 0. Moreover, we take an incompressibile closure:
∇ · ~u= 0 (3.21)
The latter assumption is justified if B0 Bsh since ∆u ∼ vA,sh and vf ∼ vA,0, thus MF =
∆u/vf ∼ vA,sh/vA,0 1. It is worth to notice that the use of the incompressibility closure
selects the Alfven waves as the unique normal mode plasma response. We can introduce the flux function ψ(x, y, t) as a function that is constant along magnetic field lines ( ~B · ∇ψ= 0), so
that the magnetic field can be written as a function of ψ:
~
B = B0eˆz+ ∇ψ(x, y, t) ∧ ˆez (3.22)
Writing ψ(x, y, t) = ψ0(x) + ψ1(x, y, t), where ψ0(x) is the equilibrium flux, while ψ1(x, y, t) is
the small perturbation, we get:
~ B = ∂yψ −∂xψ B0 = 0 −∂xψ0 B0 + ∂yψ1 −∂xψ1 0 (3.23)
The same procedure can be adopted for the perturbed velocity by defining the stream function
φ(x, y, t) = φ1(x, y, t) (only perturbed because the equilibrium velocity is zero):
~ u= ∇φ ∧ ˆez = ∂yφ −∂xφ 0 = ∂yφ1 −∂xφ1 0 (3.24)
We manipulate the momentum equation and Faraday’s law by using (3.22) and (3.24). Replacing the generalized Ohm’s law in Faraday’s equation and using vector identities8 we obtain:
∂ ~B ∂t = −c∇∧ ~E = ∇∧(~u∧ ~B)−ηc∇∧ ~J = ∇∧(~u∧ ~B)+ ηc2 4π∇2B~ = ( ~B · ∇)~u−(~u·∇) ~B+ ηc2 4π∇2B~ (3.25) Using equations (3.23) e (3.24) in Eq. (3.25) we obtain for the x-component:
∂(∂yψ) ∂t = ∂yψ ∂ ∂x − ∂xψ ∂ ∂y ∂yφ − ∂yφ ∂ ∂x − ∂xφ ∂ ∂y ∂yψ+ ηc2 4π ∂2 ∂x2 + ∂2 ∂y2 ∂yψ (3.26)
We want to linearize Eq. (3.26). It is not convenient to perform a Fourier transform of pertur-bation along all directions and time, because ψ0 depends on x only. Instead we consider small
perturbations of the form ψ1(x, y, t) = ˜ψ1(x)eikyy−iωt and φ1(x, y, t) = ˜φ1(x)eikyy−iωt (handling
each frequency and wavevector separately). Moreover let’s consider the following adimensional quantities:
• t → ¯t= t τA,
• ω → ¯ω = ω
ωA = ωτA,
• ∂xφ → ∂x¯φ = ∂cxAφ (idem for ∂yφ) with c2A= B
2 4πρ = L2 τ2 A, • x → ¯x = x
L (idem for y),
• ∂xψ → ∂xψ¯= ∂Bxψ (idem for ∂yψ),
• k → ¯k = kL, • τR= 4πL
2
c2η , η = S−1 = ττAR 1.
We finally get from Eq. (3.26) (simplifying and rewriting bar-terms without bar for sake of clarity): − iω ˜ψ1+ iky∂xψ0˜φ1= +η ∂2 ∂x2 − k 2 y ˜ ψ1 (3.27)
We can do the same steps with momentum equation (3.17) in which we use vector identity
~ A ∧(∇ ∧ ~B) = (∇ ~B) · ~A −( ~A · ∇) ~B: ∂~u ∂t − ~u ∧(∇ ∧ ~u) = −∇ P ρ − 1 2u2 +1 cJ ∧~ ~ B ρ (3.28)
We take the z component of the curl of Eq. (3.28):
∂ ∂t∇ ∧ ~u ·eˆz−eˆz∇ ∧[~u ∧ (∇ ∧ ~u)] = + 1 ceˆz∇ ∧ ~ J ∧ ~ B ρ (3.29) Using again (3.23) and (3.24) in Eq. (3.29):
− ∂
∂t∇
2φ −(∂
yφ)∂x∇2φ+ (∂xφ)∂y∇2φ= − 1
4πρ((∂yψ)∂x∇2ψ −(∂xψ)∂y∇2ψ) (3.30)
Doing the same steps as before (considering small perturbations of the form ∼ ˜φ1(x)eikyy−iωt
and linearizing the equation) we obtain: − iω ∂2 ∂x2 − k 2 y ˜φ1 = −ikyψ00 ∂2 ∂x2 − k 2 y ˜ ψ1+ ikyψ˜1ψ0000 (3.31) 8 ∇ ∧ (~u ∧ ~B) = ~u(∇ · ~B) − ~B(∇ · ~u) + ( ~B · ∇)~u − (~u · ∇) ~B = +( ~B · ∇)~u − (~u · ∇) ~B
So, the two equations we are interested in are: − iω ˜ψ1+ ikyψ00˜φ1 = +η ∂2 ∂x2 − k 2 y ˜ ψ1 (3.32) − iω ∂2 ∂x2 − k 2 y ˜φ1 = −ikyψ00 ∂2 ∂x2 − k 2 y ˜ ψ1+ ikyψ˜1ψ0000 (3.33) where ψ000
0 is the equilibrium current density derivative (thus ψ0000 6= 0 means there is a current
density inhomogeneity), indeed:
~ J ∼ ∇ ∧ ~B = 0 0 −∂2 xψ − ∂y2ψ at the equilibrium −−−−−−−−−−−→= 0 0 −∂2 xψ0 (3.34)
The term on the right-hand side of Eq. (3.32) represents plasma resistivity, whereas the term on the left-hand side of Eq. (3.33) represents plasma inertia.
3.2.2 Boundary layer approach
The non ideal terms of equations (3.32) e (3.33) become important only in a thin region around
x= 0 in which the resonant condition holds (~k · ~B ∼0), being more and more negligible moving
far away from x = 0 where the system can be described by ideal magnetohydrodynamics. In other words, as stated in the introduction, the plasma is ideal everywhere except in a very thin internal layer where the ideal assumption breaks down and the magnetic field can diffuse across the plasma. Moreover we note that the non-ideal resistive term of Eq. (3.33), characterized by a very small coefficient, is associated to the term with the bigger order derivative. Thus we can use an internal (boundary) layer approach since η∇2ψ˜1∼ ηψ˜1/L2 is negligible everywhere except
in the internal layer where η∇2ψ˜1 ∼ ηψ˜1/δ2 ηψ˜1/L2, where δ is the width of the inner
region. We divide the plane (x,y) in two regions that overlap in a zone defined as “matching layer”: an inner region in which we have to use non-ideal MHD equations (and take into account the resistive term) and an external one in which ideal MHD holds. In Figure 3.5 we see the two regions just defined. We will find two solution for the different regions which will be matched in the overlapping zone (after imposing the boundary conditions at x → ±∞).
External region: outer solution
In the external region ideal conditions hold. First we neglect the resistive term η∇2ψ˜1, then we
consider ω ∼ 0. The two hypothesis are quite different: by neglecting η∇2ψ˜1 we are assuming
that the ideal conditions hold, instead by taking ω → 0 we are considering perturbations that propagate or grow at a frequency much lower than the dynamical frequency ωA(this assumption
about frequency is due to the fact that we expect magnetic reconnection to develop at time scales in between τAand τR, so that for the frequency we expect: ωR ω ωA, thus our normalized
frequency ω/ωA→09). We will justify these approximations in the next, a posteriori, thanks
to an ordering of the various terms in power of the width δ of the non ideal region10. The
interesting equation in this case is Eq. (3.33), which reads: − ikyψ00 ∂2 ∂x2 − k 2 y ˜ ψ1,out+ ikyψ˜1,outψ0000 = 0 (3.35)
For x → ±∞ we assume a constant magnetic field and neglect the current density inhomogeneity: − ikyψ00 ∂2 ∂x2 − k 2 y ˜ ψ1,out= 0 (3.36) 9
Rembember that we have written bar-terms (normalized) without bar for sake of clarity.
Figure 3.5: in this picture we see the two regions in the boundary layer approach: the ideal MHD region in yellow and the non-ideal one in dark orange, they overlap in a region called “matching layer”.
From Eq. (3.36) we get the asymptotic solution ˜ψ1,out= Ae−ky|x| for |x| 1, showing that the
mode is localized close to x = 0. If we consider also the closer regions in which we cannot neglect the current density inhomogeneity, we can solve directly Eq. (3.36) assuming ψ0
0 = tanh(x), so ψ0000= −2 tanh(x) 1 cosh2(x). ∂2 ∂2 x − ky2−ψ 000 0 ψ00 ˜ ψ1,out = 0 (3.37) i.e.: ˜ ψ1,out00 − ky2ψ˜1,out− ψ0000 ψ00 ψ˜1,out= 0 → ˜ψ 00 1,out− ky2ψ˜1,out+ 2 1 cosh2(x)ψ˜1,out= 0 (3.38)
The solution of Eq. (3.38) is (see Ref. [3]): ˜ ψ1,out= e−ky|x| 1 +tanh(|x|) ky (3.39) which is even and continuous in x = 0. The derivative of Eq. (3.39) is discontinous in x = 0, indeed: d dx( ˜ψ + 1,out(0 +)) = −k y+ 1 ky (3.40) and d dx( ˜ψ − 1,out(0 −)) = +k y− 1 ky (3.41)
where the superscripts + and - indicate respectively the solution for x > 0 and x < 0. The unphysical discontinuity will be "cured" by the resistive term in the internal layer. We can define the so-called stability index, ∆0, as:
∆0 = lim |xout|→0 d dx( ˜ψ1,out(|xout|)) − d dx( ˜ψ1,out(−|xout|)) ˜ ψ1,out(|xout|) (3.42) which in our case (Eq. (3.41)) is equal to ∆0 = 2[(1/k
y) − ky]. In the next sections we will find
that the instability growth rate depends on ∆0, which depends on the external solution only.
Thus the stability of the system is determined by ∆0, in particular the system is unstable only
if ∆0 >0. It is worth to notice that if ˜ψ
1,out were odd (rather than even) ∆0 would be equal to
0. In other words reconnection is driven by the current density inhomegeneity that provides the magnetic energy from the outer region.
Inner region: internal solution.
In the inner region ˜ψ1 varies more quickly in x than in y (because the lenght scale in x, δ, is
smaller than the typical scale lenght in y), so that ∂2ψ˜
1/∂x2 k2yψ˜1. Thus in equations (3.32)
and (3.33) we neglect the terms associated with k2
y. Moreover we neglect the current density
inhomogeneity term (ψ000
0 ). The reason for the latter choice will be discussed in the next section
where we order each term in power of δ. Equations (3.32) and (3.33) reduce to: − iω ˜ψ1,in+ ikyψ00˜φ1 = +η ∂2 ∂x2 ˜ ψ1,in (3.43) − iω ∂2 ∂x2 ˜φ1 = −ikyψ00 ∂2 ∂x2 ˜ ψ1,in (3.44)
Before ordering each term in power of δ, we introduce the following notes: • ω → +iγ where γ is the growth rate of the instability;
• ψ0
0 ∼ x, in the inner region defined as |x| 1 (for example, ψ00 = tanh(x) ∼ x for |x| 1);
• ˜φ1(x) = −ω ˜ξx(x)/ky where ˜ξx(x) is the Fourier transform of the x component of the
displacement vector (~ξ(x, y, t) = Rt 0~udt
0 and ~ξ
x(x, y, t) = ˜ξx(x)eikyy−iωtˆx), indeed: ξx =
Rt
0uxdt = R0t∂yφ1dt → ˜ξ(x)eikyy−iωt = R0tiky˜φ1(x)eikyy−iωtdt = −iky˜φ1/iω (the relation
between φ and ~ux is provided by Eq. (3.24)).
So equations (3.43) e (3.44) become:
γ( ˜ψ1,in+ x˜ξx) = ηψ˜1,in00 (3.45)
γ2˜ξx00= ky2x ˜ψ1,in00 (3.46)
Historically there are two different orderings (both in power of δ) used to obtain the reconnection rate in function of the parameter η. These are the resistive ordering and the costant-ψ one. The
difference between the two is the different ansatz about the general shape of the flux function. We discuss each of them.
Ordering in power of δ: resistive ordering.
In the so called resistive ordering the main assumption is that the flux function varies quickly in the inner region (this corresponds to ∆0 large: high jump of logarithmic derivative of ψ). In
this kind of ordering we take:
x ∼ δ ψ˜1,in∼ δ0 ψ˜1,in0 ∼ 1δ ψ˜
00
1,in∼ δ12 ˜ξx∼ 1δ ∂x∂ ∼ 1δ
that means the scale length is the width of the inner region, so that all x-derivatives depend on the factor 1/δ, included ˜ψ1,in0 . This choice essentially constitutes an ansatz about the flux
function: we are assuming ˜ψ1,in∼ ˜ψ1,in(x/δ). Moreover we take
δ ∼ αη γ ∼ βη
and we want to find the relation between α and β in order to get the growth rate in function of the term η. From Eq. (3.45), using the previous orderings, we have:
• ˜ψ1,in∼1 • x˜ξx∼ δ1δ ∼1 • η γψ˜ 00 1,in ∼ 1−β−2αη
Thus, in order to balance left-hand terms with the right-hand one, we obtain:
1 − β − 2α = 0 (3.47)
From Eq. (3.46), with similar steps: • γ2˜ξ00
x ∼ 2β−3αη
• k2
yx ˜ψ1,in00 ∼ −αη
2β − 2α = 0 (3.48)
So, combining equations (3.47) and (3.48) we obtain:
δ ∼ 1/3η ∼ S−1/3 (3.49)
γ ∼ 1/3η ∼ S−1/3 (3.50)
where S is the Lundquist number. The growth rate γ is intermediate between the frequency of diffusion processes (1/τR) and the Alfvén frequency (1/τA). There are some points worth to be
noticed: 1) with reference to the discussion about equations for the outer solution in section 3.2.2, we see that γ/η ∼
−2/3
η → ∞ for η → 0, so η is negligible with respect to γ, so that
the resistive term can be dropped; 2) with reference to the previous sub-section ("Inner region: internal solution") we have11ψ000
0 ∼ x, so that in Eq. (3.33) the ratio between the current density
inhomegeneity term and the first term of rhs is −ikyψ˜1,inψ0000 /ky2x ˜ψ001,in ∼ 2αη =
2/3
η → 0 when
η →0 , thus we can neglect the term ikyψ˜1,inψ0000 when we write equations for the inner region;
moreover ˜ψ001,in∼1/δ2 instead k2
yψ˜1,in∼ δ0, so the latter is negligible. In this way, thanks to the
ordering in power of δ we have justified a posteriori all the assumptions made in the previous paragraphs.
Ordering in power of δ: costant-ψ ordering.
The costant-ψ ordering, called also tearing mode ordering was proposed first by Furth et al [4] in 1963 aiming at fitting some simulation results in which the flux function turned out to be nearly constant in the inner region. This choice, ˜ψ1,in0 ∼ δ0, turns out to be reasonable when ∆0
is small (i.e. the flux function is nearly constant in the inner region, indeed, we remember, ∆0
represents the jump of the logarithmic derivative of ψ). Basic assumptions for this ordering are:
x ∼ δ ˜ξx∼ 1δ ∂x∂ ∼ 1δ
These are the same as in the previous case. On the other hand the ansatz for the flux function is different, for example, we can take ˜ψ1,in = ψc(1 + δf(x/δ)) = ψc+ ψcδf(x/δ) (where ψc =
ψ(0) = ψ0) which gives: ˜ψ1,in0 = ψ0 (indeed ∂xf ∼1/δ) and ˜ψ001,in= ψ0/δ(using again ∂x∼1/δ).
Moreover, as in the resistive ordering case:
δ ∼ αη γ ∼ βη
From Eq. (3.45) we have: • ˜ψ1,in∼1 • x˜ξx∼ δ1δ ∼1 • η γψ˜ 00 1,in ∼ 1−β−αη
11It is easy to see, for example, if ψ0
Thus, in order to balance left-hand terms with the right-hand one in Eq. (3.45), we obtain:
1 − β − α = 0 (3.51)
From Eq. (3.46), with similar steps: • γ2˜ξ00
x ∼ 2β−3αη
• k2
yx ˜ψ1,in00 ∼ δ1δ ∼1
2β − 3α = 0 (3.52)
So, combining equations (3.51) and (3.52) we get:
δ ∼ 2/5η ∼ S−2/5 (3.53)
γ ∼ 3/5η ∼ S−3/5 (3.54)
where S is the Lundquist number. Once again the growth rate is intermediate between the frequency of diffusion processes and the Alfvén frequency, but the exponent is different. A number of assumption previously made at the end of the section for resistive ordering are still valid in this case, in particular: 1) when we write equations for external region η is negligible
with respect to γ, indeed γ/η ∼
−2/5
η → ∞ for η → 0; 2) when we write equations for
inner region we can neglect the term ikyψ˜1,inψ0000 in Eq. (3.33), indeed ψ0000 ∼ x , so the ratio
−ikyψ˜1,inψ0000/ky2x ˜ψ001,in∼ ηα→0 when η →0; moreover ˜ψ1,in00 ∼1/δ instead kyψ˜1,in ∼ δ0, so the
latter is negligible in the passage from Eq. (3.33) to Eq. (3.44) for the inner region. Matching in the costant-ψ case: tearing mode.
Before matching the internal and external solution in the tearing mode limit and before deter-mining γ, we introduce some useful changes of variables. We substitute Eq. (3.46) into Eq. (3.45): ˜ ψ1,in+ x˜ξx= +ηγ1 x ˜ξ00 x k2 y (3.55) We make the following changes of variables:
x= (γη)1/4ζk−1/2 (3.56)
˜ξx = + ¯ψ(γη)−1/4χ(ζ)k1/2y (3.57)
with ¯ψ ∼ ˜ψ1,in. Thus Eq. (3.55) becomes:
χ00− χζ2= ζ2 (3.58)
which solution is given by ([3]):
χ= −ζ 2 Z 1 0 e−12ζ 2µ (1 − µ2)−1 4dµ (3.59)
In order to determine γ we can integrate Eq. (3.46) along the inner region, imposing as boundary conditions those determined by external solution (this means doing the match between external and internal regions: limx→±∞ψ˜1,in0 = limx→0ψ˜01,out):
P.V. Z +xp −xp γ2 k2 y 1 x ∂2˜ξx ∂x2 dx= Z +xp −xp ∂2ψ˜1,in ∂x2 dx (3.60)
