Role of the Pressure Tensor on the Evolution of the Non-Linear Kelvin-Helmoltz instability

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Universit`

a di Pisa

DIPARTIMENTO DI FISICA Corso di Laurea Magistrale in Fisica

Role of the Pressure Tensor on the

Evolution of the Non-Linear

Kelvin-Helmoltz Instability

Examinee:

Alessandro Moirano

Supervisor:

Prof. Francesco Califano

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Introduction

Space matter is typically found in a state called plasma, a collection of charged particles, globally neutral and dominated by electromagnetic forces where col-lisions are very weak. With the advent of space exploration it was discovered that actually the entire space of the Solar System is filled by charged particles at high temperature and rarefied, mainly composed by protons and electrons. This pushed plasma research to sketch a map of the plasma environment surrounding the Sun and planets and to study the physical processes involved in the dynam-ics. This effort led to the overall mapping of the terrestrial magnetosphere and to the identification of a time-dependant flux of charged particles streaming away from the Sun. In 1958 Eugene Parker [1] theorised that in the solar corona, where temperature is of the order of 106K and matter is almost completely ionized, particles are ejected with velocity higher than the ”escape” velocity. Because a plasma has the tendency of being tied to an external magnetic field1, this stream, known as solar wind, drags and warps the stellar magnetic field, explaining the temporal and spatial variability observed in the Interplanetary Magnetic Field (IMF).

The solar wind is free to expand in the Solar System except where it impacts a planetary magnetosphere. Here the magnetic field acts as a shield because of the frozen-in law and thus forbids plasma mixing between the magnetosphere and the solar wind. Around the Earth, the magnetic field is warped by the ram pressure of the solar wind, being compressed on the dayside and stretched and stirred from the Sun on the nightside, but no mixing can occurr in principle.

The above model of magnetosphere shows to be inconsistent with observa-tions showing the injection of solar wind plasma into the Earth’s magnetosphere even at the flanks near the equatorial plane. To solve this issue, the frozen-in law must be relaxed, at least locally, to allow diffusion of plasma through the magnetic field (or slipping of magnetic lines through the plasma, as can be alternatively viewed). If the plasma has some degree of diffusion, no matter how small, and the magnetic field has an inversion point2_{, it’s possible that the}

1

This is a consequence of the frozen-in law, which state that two volumes of plasma on the same magnetic line at a certain time t0remain linked one with the other by that line for every

t > t0. For the conditions of validity of this law, see section 2.5

2_{Actually, the inversion point is a necessary condition in a 2D model. In a 3D model, it is}

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4 CHAPTER 1. INTRODUCTION magnetic lines re-connect in a different topology. This phenomenon is known as magnetic reconnection and may be the ultimate mechanism to allow plasma exchanges between the solar wind and the magnetosphere. Indeed, this is what happens when the IMF points southward [2] on the sunward side where the magnetosphere reconnects with the IMF and plasma can flow along field lines. Nevertheless, injection of solar plasma is also detected by spacecraft during northward IMF configuration, when dayside reconnection is suppressed. One possible explanation is given by the development of a Kelvin-Helmoltz vortex chain, a fluid-like ideal instability that develops in the presence of a velocity shear. Indeed, the shocked solar wind flowing in the magnetosheath around the Earth’s magnetosphere creates a velocity shear with respect to the stand-ing magnetospheric plasma. As pointed out in many textbooks (e.g: [3]), this configuration is unstable and eventually forms a trail of vortices after the insta-bility saturates and the non-linear regime starts. It’s worth noticing that the Kelvin-Helmoltz instability is inhibited at high latitudes because the magnetic field is more and more parallel to the flow and so it acts as a stabilizing force.

The Kelvin-Helmoltz instability alone is not sufficient to explain plasma mixing, because the corresponding vortices are on a MHD scale lenght and thus evolves in accordance with the frozen-in law. Indeed, vortices with twisted elongated structures of different plasma are formed, but there is no mixing, so that each plasma components remains linked to its original plasma. Neverthe-less, the Kelvin-Helmoltz vortices in the non-linear phase may set up conditions favourable to the development of secondary instabilities on the shoulder of such vortex configuration, which can be considered as a new equilibrium state. For example, the rolling-up of the vortices stretches and enrolls the field lines creat-ing inversion layers potentially unstable to reconnection. Further, if the density of the two plasma is different, as it is at the magnetospheric flanks, the cen-tripetal force of a vortex can push heavier layers on lighter ones, giving raise to secondary Rayleigh-Taylor instabilities. Finally, velocity shears are formed be-tween the alternating vortex arms and secondary Kelvin-Helmoltz can develop. Secondary instabilities like the ones discussed above can disrupt the well-ordered vortex chain, leading to the injection of blobs of plasma or to the formation of a mixing layer of thickness comparable to the vortex width (about 104km just after the dusk and dawn meridian, as recorded in [4]).

In order to fully understand the dynamics of the boundary layer between magnetosphere and magnetosheath, the interplay between primary and sec-ondary instabilities must be investigated. This dictates observational and the-oretical constraints because of the wide range of scale lenghts involved; for example, it’s impossibile to simulate or observe a whole magnetosphere (∼ 10 − 100REarth) retaining electron gyration effects (which became relevant on

scales of less than a meter). This last requires high resolution in order to inves-tigate the small scale development of the secondary insabilities and the feedback of small scales on the large scale system.

required that singular surfaces where ~k · ~B = 0 must be present, where ~B is the magnetic field and ~k is the wavevecctor of the perturbation.

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5 In this Thesis we analyze the role of the pressure tensor on the evolution of the dynamics of the system and investigate the development of the primary as well as the secondary instabilities. In the literature a scalar pressure following a polytropic law is in general assumed, but this choice appears to be inconsistent when a magnetic field is embedded into the plasma. Indeed, particles are forced to gyrate around the magnetic field, while they can move freely along field lines, so the response of the system to compression and rarefaction in the direction along and transverse to the magnetic field are different. As pointed out by Chew, Goldberger and Low (CGL,[5]), pressure in a plasma must be regarded as a non-isotropic tensor which account for the different response of the plasma in the direction parallel and perpendicular to the magnetic field, instead as a simple scalar field.

Taking into account the full pressure tensor, which is obtained from the kinetic theory, is quite complicated, thus we begin with a simplified model. Indeed, following the CGL theory, the pressure tensor has the form Πij =

p⊥(δij − bibj) + pkbibj3 and two scalar equations for p⊥ and pk are obtained

to determine the evolution of Πij.

The CGL model account for a different behaviour of the plasma. For exam-ple, the anisotropy |pk−p⊥| can grow locally, even if it is initially negligible, and

this difference may lead to the development of instabilities such as the firehose and the mirror instability. These instabilities can act as secondary instabilities like the previous one discussed before and interfere with other processes like reconnection and vortex merging, potentially leading to turbulent mixing layer development. Thus, it is worthwhile investigating how the system evolves, when allowing for a separate parallel and perpendicular pressure and what quantities are involved where a strong anisotropy develops.

However, the CGL theory has its own limits. It takes into account the different response of the plasma with respect to the direction of the magnetic field, but neglects the effect of a finite radius of gyration for the particles (also known as Larmor radius rL) around a field line. In other words, the CGL

model assumes rL = 0. In order to obtain a more realistic model which takes

into account the effect of a finite Larmor radius (FLR), the pressure tensor must be expanded in series of a small parameter given by the ratio of the Larmor radius to the characteristic lenght of the fluctuations. We will also see that such an expansion can be seen as a correction to the CGL theory which is valid when and where the Larmor radius rL is small compared to the characteristic lenght

Lf luidof the fluctuations.

The Thesis is divided in the following chapters: in Chapter 2 we give a general theoretical background for plasmas within a fluid modelling. Thus we won’t consider single ions or electrons interacting via electromagnetic forces (this approach is known as kinetic theory), but we make use of small volumes containing many charged particles, similarly to the classical approach of the fluidodynamics (and for this reason this approach is known as

magnetohydrody-3

δij= 1 if i = j and zero otherwise, bi= Bi/B, Bibeing the i -component of the magnetic

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6 CHAPTER 1. INTRODUCTION namic theory). We require that each volume is homogeneous, thus no gradient are present across each volume, and it is overall neutral. As we will show, the two theories are closely related because of the moments of the distribution func-tion which statistically describe an ensamble of charged particels. Furthermore, we will show that a set of equation can be obtained for each quantity of every species (electrons and ions), but they can be manipulated in order to get the so-called single fluid model. Furthermore, we present in detail different form of the pressure tensor and explain the physics they intend to represent. In Chapter 3 we perform the stability analysis of different sheared velocity configuration. In particular we investigate the stability of the a single, double and triple shear layer across which the velocity change considerably. We begin with discontinu-ous interfaces separating layer of incompressible plasma with uniform velocity; then we analyze the role of compressibility and of a shear layer with finite thick-ness. In Chapter 4 we present four simulations of the magnetosphere-solar wind boundary, which is an unstable equilibium. The simulations are different in the assumed pressure tensor form, but they have the same initial condition in order to point out the role of the model adopted. We focus on the pressure, velocity and magnetic field evolution, in particular on small scale structures which are developed during the non-linear phase of the primary Kelvin-Helmoltz instabil-ity. In Chapter 5 a brief summary and conclusion are given.

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Chapter 2

Theorical background

2.1 Fluid equations for collisionless magnetized

plas-mas

A plasma is an ensemble of many weakly collisional charged particles, typically ions and electrons, interacting among them via electromagnetic forces. This feature represents the main difference between a neutral fluid and a plasma: while in the former particles interact mainly via binary collisions, which is a short-range coupling, in the latter every particle experiences the influence of the whole system thanks to the long-range electromagnetic force. This difference leads to important different behaviour. For example, in a neutral fluid the collisional frequency increases as the temperature increases, while in a plasma the opposite happens. This can be understood by noticing that the temperature represents the mean velocity of the particles of the system: increasing that parameter means that a neutral particle can hit more particles in a given amount of time. On the other hand, a charged particle which approaches a target experiences a higher deflection if it spends more time close to the target, i.e: if it’s slow1. Keeping in mind for a plasma such long-range interaction, one can ask how such an ensemble evolves. In principle, knowing the properties of all species, the external field and the interparticle forces it is possible to solve the equation of motion for each particle with the suitable initial condition. This is a system of 3N equations ,where N is the number of particles, almost unsolvable in practice.

For this reason, a statistical approach is usually choosen by defining a dis-tribution function in a (6N+1)D phase space (3N for space, 3N for velocity and one for time), which gives the probability of finding N particles at locations (~x1, . . . , ~xN) with velocity (~v1, . . . , ~vN) at time t. Without taking a specific

pro-file, the Liouville theorem states that the distribution function is constant along

1_{Note that here we’re referring as collision to two quite different phenomena. In the case of}

a neutral fluid every contact between two particle is regarded as a collision, while in a plasma particles never touch one the others: in that case we define a collision as an encounter that produce an appreciable deflection.

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8 CHAPTER 2. THEORICAL BACKGROUND a path in phase space:

DFN Dt = ∂FN ∂t + X i vi ∂FN ∂xi +X i ai ∂FN ∂vi = 0. (2.1)

where FN = FN(~xα1, ~xα2, . . . , ~vα1, ~vα2, . . . , t), α being the species considered.

Up to here there is no great advantage because we still have a (6N+1)D description of the system, which is hardly manageable. But in a plasma the thermal energy of the particles is much higher than the Coulomb potential be-tween particles, so the position and velocity of each particle is almost unaffected by binary interaction with the other particles. This allows one to average the distribution function over the phase space of k < N particles, so that it’s pos-sible to obtain a simplified description of the system without committing a big error. In other words, the reduced distribution function (d~Γ = d~xd~v)

fα(~xα1, ~vα1, . . . ~xαN −k, ~vαN −k, , t) = V

Z

FNd~ΓαN −k+1, d~ΓαN −k+2, . . . , d~ΓαN

(2.2) give the statistical representiation of the subsystem composed by N-k particles in a (3(N-k)+1 )D phase space. If we put k = N − 1, the one particle distribu-tion is obtained, which is the first real simplificadistribu-tion towards a mathematically tractable equation, even though we have given up a lot of information about the whole system.

The Liouville equation can be then integrated over all the phase space of the entire system except for one particle, obtaining

∂fα ∂t + ~v · ∂fα ∂~x + ~aT · ∂fα ∂~v = 0 (2.3)

In Eq.2.3 ~aT is the sum of external and internal forces (i.e: electric and

mag-netic field generated by the system itself). However, it’s possible to distinguish two contributions: one from few near particles inside the Debye sphere (i.e: collisions, in a sense as in footnote 1) and a second one from the remainder of the system (i.e: the mean internal field). The latter can be incorporated in the external field, so Eq.2.3 can be written as

∂fα ∂t + ~v · ∂fα ∂~x + qα mα D ~_{E +}~v × ~B c E ·∂fα ∂~v = ∂f_α ∂t collision (2.4)

where the third term now represents the acceleration due to external plus av-erage self-generated field. The term on the rhs represents the interaction of the particle with its neighbours - the collisions - so it involves the two-particle distribution function. This introduce another common issue in plasma physics: the closure problem. Indeed, if one tries to evaluate this term integrating again the Liouville equation over all but two particles phase space, it is left with an equation involving the two-, but also the three-particle distribution. There are different way to ”solve” the closure problem, depending on the physics one wants to investigate. One of the most employed is to completely neglect collisions, so

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2.1. FLUID EQUATIONS FOR COLLISIONLESS MAGNETIZED PLASMAS9 that the rhs of Eq.2.4 is zero. This assumption is actually quite realistic: if νc and ωp are the collision and plasma frequency respectively 2, then it can be

shown that νc ωp =Wij kBT 3₂ 1 (2.5)

where Wij is the Coulomb interaction between particles, which is dominated

by the thermal energy in a plasma kBT . In this case Eq.2.4 is known as the

Vlasov equation, the foundamental equation for the study of the dynamics of collisionless plasmas.

Despite the simplification made so far, one still has a 6D+1 description of the system, which continue to be difficult to handle. It is possible to further reduce the dimension of the phase space by averaging the Vlasov equation and the one-particle distribution function over the velocity space: in this way one is left with a 3D+1 description of the system. As will become clear soon, this is a fluid description of the plasma: after having taken moments of Eq.2.2 and of Eq.2.4, one is left with the typical hydrodynamic quantity - such as numerical density and pressure - and the governing fluid equation - like the continuity and the momentum equation.

Indeed, it’s easily seen that integrating the one-particle distribution function over the phase space of the particle gives the total number of particles, while integrating it over the velocity space gives the number of particles at a given space location (in other words, the numerical density):

nα(~x, t) =

Z ¯

nαfα(~x, ~v, t)d~v, (2.6)

where ¯nα is the mean plasma density. Eq.2.6 multiplied by the suitable

ele-mental quantity leads to the mass or charge density of each species. Taking the average of the velocity multiplied by the mean density leads to the particle flux:

~

Φα(~x, t) =

Z ¯

nα~vfα(~x, ~v, t)d~v = nα(~x, t)~Vα(~x, t) (2.7)

from which the current can by obtained by multiplying by the elemental charge. The higher order moments such as the pressure tensor and the heat flux are given by Pα= mα¯nα Z (~v − ~Vα)(~v − ~Vα)fα(~x, ~v, t)d~v, (2.8) Qα = 1 2mαn¯α Z ~v(~v~v)fα(~x, ~v, t)d~v, (2.9)

which represent the second and third moments of the distribution function. To obtain the set of equations for the macroscopic quantities, moments of the Vlasov equation has to be taken, giving the following set of equations:

∂nα

∂t + ∇ · (nαV~α) = 0 (2.10)

2

The plasma frequency is defined by ωα,p =p(4πnαe2/mα), where nα and mα are the

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10 CHAPTER 2. THEORICAL BACKGROUND nαmα ∂ ~Vα ∂t + nαmαV~α· ∇~Vα= nαqα D ~_{E +} V~α× ~B c E + ∇ · Pα (2.11) dPα dt + Pα(∇ · ~Vα) + ∇ · Qα+ h Pα· ∇ ~Vα i S= qα mαc h Pα× ~B i S (2.12)

where [ ]S stands for the symmetrization with respect to the free indices3.

Eq.2.10 - 2.12 are known as fluid equations. However, this system is not closed. To determine the density evolution, the momentum is needed, which in turn needs the pressure to be specified. The pressure tensor requires the knowledge of the heat flux, which again depends on higher order quantities.

Eq.2.10 - 2.12 must be coupled with the Maxwell equations that determine the evolution of the electromagnetic fields. The Ampere law and the Faraday law read ∇ × ~B =4π c J +~ 1 c ∂ ~E ∂t (2.13) ∇ × ~E = −1 c ∂ ~B ∂t (2.14)

For low frequency dynamics, Eq.2.13 is simplified by neglecting the displacement current. This is allowed if the typical velocity of the system is much less than the speed of light. Indeed, if L and τ are the characteristic lenght and time over which the system evolves4, the magnetic and electric field can be estimated from Eq.2.14 as E/B ∼ L/τ c ∼ U/c. Comparing the order of magnitude of the displacement current and the ∇ × ~B term in the Ampere law, togheter with the previous estimate, leads to the conclusion that ∇ × ~B/∂tE ∼ EU /Bc ∼ U~ 2/c2,

which is very small in the non relativistic limit, so that one can neglect the displacement current.

2.2 The closure problem

The set of equation 2.10 - 2.12 is not solvable, because there are more uknown function than equations. One may be tempted to evaluate higher order moments of the Vlasov equation in order to find an equation able to close the system. But the system of fluid equations cannot be closed in this way, as can be foreseen by looking at the equations themselves: the i -order moment of the Vlasov equation describes the evolution of the i -order moment of the distribution function, but it requires the knowledge (i + 1)-order moment of the distribution function. For example, the first order moment of the Vlasov equation - which is the momentum

3 e.g: h Pα· ∇ ~Vα i S,ij= Pik ∂uj ∂xk + Pjk ∂ui ∂xk and h Pα× ~B i S,ij = iklPjkBl+ jklPikBl 4

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2.2. THE CLOSURE PROBLEM 11 equation - require the evolution for the pressure - which is the second order moment.

In order to close the system, an ”external” relation which relates different moments of the distribution function must be choosen, i.e: an equation of state. There is no a priori prescription about the choice of such an equation, but there are some reasonable choice, depending on the physics one wants to investigate5: • Cold plasma approximation. This choice is appropriate if the phase speed of a perturbation is higher than the thermal velocity. In this case each particle can be considered as motionless, so the evaluation of the pressure tensor from Eq.2.8 gives

Pα= 0

This approximation can also be useful when the fluid pressure of the plasma is small compared to the magnetic pressure.

• Isothermal closure. This choice is suitable for a dynamics governed by perturbations moving with a phase velocity lower than the thermal speed. Indeed, if the typical time of a process is shorter than the time scale of heat transfert, then the system can be considered in thermal equilibrium. Choosing an isothermal closure allows to relate pressure and density with a proportional law, i.e:

Pα = nαTα (2.15)

• Adiabatic closure. In this case the perturbation acts on a time scale shorter than the heat trasfert, so the fluid element is not able to exchange energy with its surroundings. In this case a power law between pressure and density is appropriate:

Pαn−γα = const (2.16)

where γ = 5₃ is the adiabatic index.

For every species in the system one closure equation must be choosen, depending on the regime one is interested in. For example, if we consider an ensamble of protons and electrons with different thermal velocity and a perturbation with phase speed that satisfies vth,i vph vth,e, the protons follow an adiabatic

closure, while the electrons follow an isothermal one.

Sometimes the incompressibility conditions (∇ · ~uα = 0) is used instead of

an equation of state.

5

Actually there is another possibility, which is going back to Eq.2.4 and using it to determine the evolution of the one-particle distribution function: this approach is known in general as kinetic theory. It’s worth noticing that there is a similar issue with the closure: indeed, Eq.2.4 requires the knowledge of the two-particle distribution function, which in turn requires the three-particle distribution and so on, so one is again left with a closure problem. But contrary of the fluid theory there is no need for an equation of state. Generally to close the chain of equations in kinetic theory, the neglect of the rhs is assumed, i.e: there are no collisions in the plasma.

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12 CHAPTER 2. THEORICAL BACKGROUND

2.3 The pressure tensor

Despite the common approach of the literature which assume a scalar pressure, in general it is a full tensor quantity, as it is clear if we use the index notation:

Pijα = mαn¯α

Z

(vi− Viα)(vj− Vjα)fα(~x, ~v, t)d~v, (2.17)

If the distribution function is isotropic, then the off-the-diagonal terms vanish and Piiα= 1₃nαmαv2α for i = x, y, z, so that the pressure tensor is proportional

to the identity matrix and thus it can be viewed as a scalar. But this is true if collisons are frequent enough to randomize the velocity of the particles or, in other words, if the collisional time is lower than the time over which dynamical processes generate anisotropy.

In many plasma contexts - both astrophysical and in laboratory - collisions are very infrequent, so any initial or growing anisotropy cannot be smoothed by collisions. Then, the choice of an isotropic pressure could be misleading, if not completely unacceptable.

Further, in many interesting cases an external magnetic field is present, for example, to confine the plasma in a tokamak or as an intrinsic property of the system, such as the solar wind. Charged particles tend to gyrate around magnetic field lines, while they are free to move along the lines: this suggest that an isotropic distribution function is not appropriate, as pointed out by Chew, Goldberger and Low ([5], CGL hereafter), and one of the form

fα(~x, vk, v⊥, t) (2.18)

should be choosen (where vk and v⊥are the velocity parallel and perpendicular

to the magnetic field). Substituting this into the definition of the pressure tensor (Eq.2.8) leads to the form

Pijα= pα⊥(δij − bibj) + pαkbibj (2.19)

where bi is the i -component of the versor of the magnetic field, δij is the

Kro-neker delta and p⊥and pk are two different scalar. It’s immediately evident that

such a tensor is not similar to the identity matrix, although it can be written in a diagonal form in a frame of reference with one axis along the magnetic field.

With this description, Eq.2.12 for the pressure tensor evolution gives two equation for p⊥ and pk:

∂pα⊥ ∂t + ∇ · (pα⊥~uα) = −pα⊥(∇ · ~uα) + pα⊥ˆb · [(ˆb · ∇)~uα] − − ∇ · (q_α⊥ˆb) − qα⊥(∇ · ˆb) (2.20) ∂pαk ∂t + ∇ · (pαk~uα) = −2pαk ˆ b · [(ˆb · ∇)~uα] − − ∇ · (q_αkˆb) + 2qα⊥(∇ · ˆb) (2.21)

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2.4. TOWARDS A MORE ACCURATE MODEL: THE FLR EXPANSION13 Here, q_αk and qα⊥ are the heat fluxes of parallel and perpendicular thermal

energy along the magnetic field lines. As one should expect, these terms leave the set of equation open, so a closure equation for the heat fluxes is required. One of the possible assumption is that there are no heat fluxes, which is the case for a system in which there are no variations along the field lines6. This is a strong assumption, because it means that the phase velocity of a perturbation which propagates along the magnetic field line must be grater than both the ion and electron thermal speed (_kω

k vα,thk).

With this closure, Eq.2.20 and 2.21 can be cast in the following conservative form: d dt p_α⊥ nB = 0 (2.22) d dt p_αkB2 n3 = 0 (2.23)

It’s easy to show that these two equations corrispond to two polytropic laws. Indeed, making use of the Faraday as well as the continuity equation, they can be re-written as ndp⊥ dt − 2p⊥ dn dt = np⊥bibj∂jui (2.24) ndpk dt − pk dn dt = −2npkbibj∂jui (2.25) where the rhs can be neglected if the magnetic field is mainly along the direction of invariance of the system (i.e: the direction along which the derivatives are small). If this is true, Eq.2.20 and 2.21 can be written in a polyropic conservative form:

p⊥n−γ⊥= const⊥ (2.26)

pkn−γk = constk (2.27)

with γ⊥ = 2 and γk = 1. These can be explained noting that along a magnetic

field line there are no heat fluxes, so an isothermal closure with γ = 1 is appro-priate, while in the perpendicular direction kinetic theory and the equipartition theorem state that the polytropic index is γ = 1 + d+2_d , where d is the degree of freedom of the particle: in this case d = 2 lead to γ = 2 7_.

2.4 Towards a more accurate model: the FLR

ex-pansion

In the previous section we have seen that a more realistic model is given by the CGL theory, which expects the plasma compressibility to vary with respect to the direction of the magnetic field. Actually, it can be shown that Eq.2.12 can be expanded in power of a small parameter α and that the CGL theory is the

result from the zeroth-order term. We anticipate that the small parameter we

6_{This is why the CGL model is also known as double-adiabatic theory} 7

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14 CHAPTER 2. THEORICAL BACKGROUND will find is the ratio of the Larmor radius8_r

Lto the characteristic lenght L, thus

the zero-order CGL term represents the fluid modeling of the gyration motion of the particles in the limit rL= 0, while the first-order term will represent the

correction for rL << L to the CGL term (leading to the so-called CGL model

with FLR corrections).

We start with the full pressure tensor equation9 (retaining for completeness the heat flux terms):

∂Pαij ∂t + ∂ ∂xk (Pαijuαk) + (Pαik ∂uαj ∂xk + Pαjk ∂uαi ∂xk ) +∂Qαijk ∂xk = (2.28) = eα mαc (ilmPαlj+ jlmPαli)Bm

where Pαij is the ij -component of the pressure tensor of the α-species, Qαijk is

the ijk -component of the heat flux tensor, eαand mαare the charge and

elemen-tal mass, Bm the m-component of the magnetic field and uk the k -component

of the velocity.

The previous equation can be recast in adimensional form with the charac-teristic units for lenght, time, speed, density and magnetic field L, τ, vth, n0 and

B0: xk= L· ∼ xk , t = τ · ∼ t , uk= vth· ∼ uk , u = n0· ∼ n , Bm = B0· ∼ Bm ,

(in the remainder of the section we drop the tilde but we continue to refer to adimensional quantities).

With these definitions, the pressure tensor equation is (ilmPαlj + jlmPαli)Bm= = σαα h L vαthτ ∂Pαij ∂t + ∂ ∂xk (Pαijuαk) + (Pαik ∂uαj ∂xk + Pαjk ∂uαi ∂xk ) + = ∂Qαijk ∂xk i (2.29) where α = rL L 1

is expected if the Larmor radius is much smaller than the characteristic fluid lenght, and σα = sign(eα) takes into account the change of sign due to the

charge of the species.

The pressure and heat flux tensors can be expanded in power of the small parameter (dropping the species index):

Pij = ∞ X r=0 rP_ij(r) , Qijk = ∞ X r=0 rQ(r)_ijk (2.30) 8

The Larmor radius is defined as the radius whose circumference is travelled by a charged particle with velocity vth, thus rL= vth/Ωc, where Ωc is the cyclotron frequency.

9

The full calculus for the components of the gyroviscous terms can be found in [6]. Here we give only the results and the important steps for the power expansion of the pressure equation.

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2.4. TOWARDS A MORE ACCURATE MODEL: THE FLR EXPANSION15 Substituting Eq.2.30 into Eq.2.29 and gathering terms of the same order (i.e: term of the same degree of r) lead to a recursive formula of the form:

LB[P_ij(n)] = Ru[P_ij(n−1)] + D[Q(n−1)_ijk ] (2.31) where LB[P_ij(n)] = (ilmP_αlj(n)+ jlmP_αli(n))Bm Ru[Pij(n−1)] = σα h L vαthτ ∂P_αij(n−1) ∂t + ∂ ∂xk(P (n−1) αij uαk) + + (P_αik(n−1)∂uαj ∂xk + P (n−1) αjk ∂uαi ∂xk) i D[Q(n−1)_ijk ] = σα ∂Q(n−1)_αijk ∂xk (2.32)

The zeroth-order equation for Eq.2.31 is

(ilmP_lj(0)+ jlmP_li(0))Bm = 0 (2.33)

because Ru[P_ij(−1)] = 0 and D[Q! (−1)_ijk ] = 0. It’s easy to see that this equation!

leads to the following components for the zeroth-order pressure tensor: P_ij(0) = 0 i 6= j

Pxx(0) = Pyy(0)= P⊥

Pzz(0) = Pk

where P⊥ and Pk are two independent scalar quantities. Thus, the pressure

tensor can be written as

P_ij(0) = P⊥(δij− bibj) + Pkbibj

which is the usual form of the CGL gyrotropic10 _{pressure tensor.}

In order to obtain the FLR corrections, the n = 1 term of Eq.2.31 must be evaluated, which read:

LB[Pij(1)] = Ru[Pij(0)] + D[Q (0)

ijk] (2.34)

thus the convective derivative of the zeroth-order pressure tensor must be eval-uated and is given by

dP_ij(0) dt = ηij dP⊥ dt + bibj dPk dt + Pk− P⊥ B2 × ×hBj dB_i dt + Bi dB_j dt −2BiBj B dB dt i (2.35) 10

i.e: it is isotropic in the plane perpendicular to the magnetic field, but anisotropic with respect to the direction parallel and perpendicular to the magnetic field.

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16 CHAPTER 2. THEORICAL BACKGROUND where ηij = δij−bibj is the projector on the plane perpendicular to the magnetic

field. The term in square brackets takes into account the time variation of direction of the magnetic field: in order to evaluate this term, the curl of the Ohm’s law in the zero mass-ratio limit is used neglecting the resistivity (see Eq.2.43 with the substitution ν = 0), togheter with the Maxwell’s equations (neglecting the displacement current) and the vector identity ∇ × ( ~U × ~B) = ( ~B · ∇) ~U + ~U (∇ · ~B) − ( ~U · ∇) ~B − ~B(∇ · ~U ). Thus, the magnetic field variation is given by (in adimensional units):

dBi dt = Bk ∂ui ∂xk − Bi ∂uk ∂xk + vth,e vth,α rL Lijk ∂ ∂xj h 1 n ∂ ∂xl 1 βe0P (mag) lk + P (0) e,lk i ≈ ≈ Bk_∂x∂u_ki − Bi∂u_∂xk_k (2.36)

where βe0 = 8πmen0vth2/B₀2. The approximation carried between the first and

second line of Eq.2.36 is justified by the multiplication of the last term on the rhs by rL/L = , thus it can be neglected becasue we are interested only in the

time variation of P_ij(0), which is of order 0. Thus, noticing that 1 B dB dt = Bl B2 dBl dt

the term between square brackets in Eq.2.35 can be substituted by Eq.2.36, leading to the determination of the components of the gyroviscous tensor11 Gαij = αPαij listed below (the species index is re-introduced):

Gαxx = −Gαyy = −αPα⊥σ₂α ∂uαy ∂x + ∂uαx ∂y Gαxy = Gαyx= αPα⊥σ₂α ∂uαx ∂x − ∂uαy ∂y Gαxz = Gαzx = ασα h (Pα⊥− Pαk) ∂uαy ∂z − Pα⊥ ∂uαz ∂y i − _ασα∂q_∂yα⊥ Gαyz = Gαzy = −ασα h (Pα⊥− Pαk)∂u∂zαx − Pα⊥ ∂uαz ∂x i + ασα∂q_∂xα⊥

The Gαzz component of the gyroviscous tensor is not specified by the FLR

corrections, becasue we have carried on an approximation in the plane perpep-ndicular to the magnetic field which doesn’t hold for the parallel direction, thus it is reasonable that the zz -component cannot be expanded in power of α and

it is not determined by Eq.2.34. We then assume that all the information about the parallel pressure Pk resides in the zeroth-order pressure tensor, thus

P_αzz(0) = Pαk , Pαzz(n) = 0 ∀n > 0

11

For sake of simplicity we make the simplification ~B = Bzeˆz in the following calculus.

Nevertheless, in Chapter 4 we will relax this assumption in the adiabatic limit in order to point out the role of the magnetic field on the pressure evolution.

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2.5. FROM A TWO TO A SINGLE FLUID FORMALISM 17 The components of the gyroviscous tensor can be recast in dimensional form, which reads Gαxx = −Gαyy = −σ_2ΩαP_αcα⊥ ∂uαy ∂x + ∂uαx ∂y Gαxy = Gαyx= σ_2ΩαP_αcα⊥ ∂uαx ∂x − ∂uαy ∂y Gαxz = Gαzx= _Ωσ_αcα h (Pα⊥− Pαk) ∂uαy ∂z − Pα⊥ ∂uαz ∂y i − σα Ωαc ∂qα⊥ ∂y Gαyz = Gαzy = −_Ωσ_αcα h (Pα⊥− Pαk)∂u∂zαx − Pα⊥ ∂uαz ∂x i + σα Ωαc ∂qα⊥ ∂x

In order to obtain an expression for both Pkand P⊥ the conditions Gαzz = 0

and the symmetric property of the pressure tensor are employed. Indeed, we have Gαxx = −Gαyy, thus

Gαxx+ Gαyy+ Gαzz = 0

Adding the condition Gαzz = 0, we are left with the following equations dPk dt = −Pk∇ · ~u − 2Pk∂zuz− (∂zq⊥− q⊥ B∂zB) dP⊥ dt = −P⊥∇ · ~u − P⊥(∂xux+ ∂yuy) − (∂zq⊥+ 2q⊥−qk B ∂zB)

In our simulation the adiabaic limit is taken (q⊥ = qk = 0), thus the previous

equations are simplified and the last term in parentheses is neglected.

2.5 From a two to a single fluid formalism

The quantities 2.6 - 2.8 can be used to cast Eq.2.10 - 2.11 in a single fluid theory: instead of taking the set of equations for every species, it’s possibile to define center-of-mass quantities which lead to a new set of equations12. Explicitly, we are doing the following substitution:

n = mini+ mene mi+ me (2.37) ρc= e(ni− ne) ~ U = mini~ui+ mene~ue mini+ mene ~ J = e(ni~ui− ne~ue)

Indeed, note that ~U represents the velocity of the center of mass of the plasma, while ~J is the current from the relative motion of the two species. In the same way, n is the mean density, while ρc is the charged density, which depends on

the different concentration of ions and electrons.

12_{This is conceptually the same as solving a system of two particles using the center-of-mass}

equation along with the relative motion equation instead of the equation of motion of the two particles.

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18 CHAPTER 2. THEORICAL BACKGROUND With this foreword, it’s now possible to cast the one-fluid set of equations. Multiplying Eq.2.10 for mα or qα and adding up one obtain the total mass and

charge continuity, respectively: ∂n

∂t + ∇ · (n ~U ) = 0 (2.38) ∂ρc

∂t + ∇ · ~J = 0 (2.39)

Note that with the quasi-neutrality assumption (i.e: ρc ≈ 0), the second

equa-tion implies that the current is a solenoidal field. The same can be done with Eq.2.11: summing the equation for the two species, one obtains:

∂ ∂t(n ~U ) + ∇ · h A1n ~U ~U + A2 e ( ~U ~J + ~J ~U ) + A3 e2_nJ ~~J i = = − 1 M∇ · P + ρc eE +~ ~ J × ~B M c (2.40)

which can actually be greatly simplified (note that P = Pi+ Pe). Indeed, if one

assumes quasi-neautrality (i.e: ni ≈ ne ≈ n), the contribution of the electric

field vanishes (ρc≈ 0), while A1 = 1, A2 = 0 and A3 = mime/M2 13. Thus we

obtain the simplified form of the single-fluid equation of motion ∂ ∂t(n ~U ) + ∇ · (n ~U ~U ) = − 1 mi ∇ · P +J × ~~ B mic (2.41) One last equation can be obtained by subtracting the momentum equation for the two species, leading to the fundamental generalized Ohm’s law, which has the most general form

C1E + C~ 2 ~ U × ~B c + 4π ω2 pe ∂ ~J ∂t = C3 ~ J × ~B enc − 1 en∇ · Pe− me mi Pi − − 4π ω2 pe ∇ ·hC4en ~U ~U + C5( ~U ~J + ~J ~U ) − C6 enJ ~~J i (2.42) where, except for C2 and C3, the coefficients Ci assume different values in the

zero mass-ratio limit or in the quasi-neutrality assumption. For the explicit form of the coefficient Ai and Ci see [6].

As can be easily seen, Eq.2.40 and 2.42 are quite cumbersome, so zero mass-ratio and quasineutrality are usually both assumed, leading to the known full magnetohydrodynamics (Full-MHD). In this case the Ohm’s law reads

~ E + U × ~~ B c = η ~J + ~ J × ~B enc − 1 en∇ · Pe (2.43) where η ~J ≈ ∂tJ .~ 13

One can also assume that the ratio between the electron and ion masses is zero: in this case A1= 1 and A2= 0, while A3= 0.

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2.5. FROM A TWO TO A SINGLE FLUID FORMALISM 19 The relevance of every term in Eq.2.42 depends on various parameter of the system (for example, the typical scale of the gradients or the competition between fluid and magnetic pressure). Hereafter, quasi-neutrality is assumed, so that C4 = 0 and the terms in the generalized Ohm’s law have the following

ordering14: |4π ω2 pe ∂ ~J ∂t| |U × ~~_cB| ∼ d 2 e L2 |J × ~~_necB| |U × ~~ _cB| ∼ diωcic LωpiU = divA LU |∇Pe ne | |U × ~~ B c | ∼ vth,eλD,eωpe U Lωce (2.44) |_ω12 pe∇ · ( ~U ~J + ~J ~U )| |U × ~~ B c | ∼ d 2 e L |_ω12 pe∇ · ( ~J ~J )| |J × ~~ B c | ∼ d 2 e L

where λD,e is the electron Debye lenght, dα is the electron or ion skin depth

and vA=p(B2/4πnmi) is the Alfv´en speed. From the last two estimates, it’s

worth pointing out that the ∇ · ( ~J ~J ) term is smaller than the Hall term ~J × ~B, which in turn is smaller than the ~U × ~B term, provided that U ∼ vA.

It’s possible to show that the electron inertial terms ∂tJ and ∇ · ( ~~ U ~J + ~J ~U )

are of the same order of magnitude and both are smaller than the Hall term, provided that perturbations with wavelenght longer than the Larmor radius are considered: |_ω12 pe∇ · ( ~U ~J + ~J ~U )| |J × ~~ B nec | ∼ ω ωce

Further, the relative importance of the pressure term with respect to the Hall term is given by the ratio of the fluid to the magnetic pressure, i.e:

|∇·Pe ne | |J × ~~_necB| ∼ pe B2 8π ∼ β (2.45)

With this hierarchy it’s possible to make Ohm’s law more manageable. The most simple form is achieved in low-β plasma in the low frequency, large-scale regime. In this case all the terms in the rhs of Eq.2.42 can be neglected and the ideal Ohm’s law is found:

~

E + U × ~~ B

c = 0 (2.46)

14

From Eq.2.13 and 2.14 we get that that the ~E and ~U × ~B/c have the same order of magnitude.

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20 CHAPTER 2. THEORICAL BACKGROUND Eq.2.46 is one of the hypoteses which leads to the frozen-in law : without going into details, putting Eq.2.46 into the Farady equation leads to:

∂ ~B

∂t = ∇ × ( ~U × ~B) (2.47) which states that the magnetic field is advected by the plasma without diffu-sion. Historically, violation of the frozen-in law was achieved by assuming that collisions are relevant, so Eq.2.46 is replaced by

~

E + U × ~~ B

c = η ~J (2.48)

where η is the resistivity. In this case the Faraday equation has the form ∂ ~B

∂t = ∇ × ( ~U × ~B) + ηc2

4π∇

2_B_~ _(2.49)

where ∇2 is the Laplace operator. The new term in Eq.2.49 lead to diffusion of magnetic field across plasma, no matter how small is η (but it must be non-zero). Indeed, if one takes ~U = 0, it’s left with an equation formally equal to the heat diffusion equation. This way, it’s possibile to change the topology of the magnetic field through reconnection, provided that there is a singular surface, i.e: a surface where ~k · ~B = 0 (~k is the wavevector of the perturbation).

Because one has to take the curl of Ohm’s law to substitute into Faraday equation, the inclusion of curl-free terms - such as the pressure gradient or the Hall term - doesn’t allow for reconnection to occur. Nevertheless, the inclusion of these terms affects the evolution of the system, so favorable condition for such instability may be established more or less easily.

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Chapter 3

Stability of sheared flows

3.1 Stability analysis

To calculate the stability of a given equilibrium, one can make use of the normal mode analysis. It consists in linearizing a closed set of equation, assuming a zero order equilibrium field and a first order ”small” perturbations:

A → A0+ δA , |δA| |A0| (3.1)

Then one must neglect all terms of the order of δA2in the equations. We assume that the perturbations are in the form of normal modes. In the presence of an inhomogeneous direction, here the x - direction, the amplitudes varies in that direction, so it can be written as

A(x, y, z, t) = ˆA(x)eikyy+ikzz−iωt _(3.2)

This allows one to make the following changes into the equations: ∂t→ −iω , ∇ → (

d

dx, ikyy, ikzz) (3.3) and the set of partial differential equations is reduced to a set of ordinary dif-ferential equations, the time, y and z derivatives being replaced by algebraic multiplication by −iω, iky and ikz. By giving explicitly the form of the

equilib-rium field it is possible in general to solve the system with the proper boundary conditions and to obtain the temporal and spatial evolution of the perturba-tions. Then it is possible to infer the stability of the equilibrium by looking at the values of ω and of the wavevector ~k: if they are real, then the perturba-tion leads to oscillaperturba-tions. On the contrary, if they are complex, their imaginary part leads to damped and growing perturbations depending on the sign of the exponent.

The normal mode analysis provides additional informations besides stability. Indeed, it is possible to find out a dispersion relation of the form ω(~k) which allows to evaluate how the perturbation evolves. The dispersion relation gives the phase velocity as well as the group velocity of the waves, which are defined

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22 CHAPTER 3. STABILITY OF SHEARED FLOWS as ω k = vph , ∂ω ∂~k = vgr

The drawback of this method is that it is valid only if the amplitude of the perturbation is ”small”, i.e: when it is possible to neglect second and higher order terms. If the initial perturbation doesn’t excite unstable modes, the os-cillations have a bounded amplitude which satisfies Eq.3.1 at any time and position. On the contrary, if an unstable mode is excited (i.e: there is an insta-bility), the amplitude becomes larger and larger, until the linear approximation breaks down. This can be physically undestood by noticing that the energy trasported by a wave is proportional to the squared amplitude: if it continues to grow, then the carried energy diverges. So, there must be some mechanism which saturates the growth of the amplitude of the instability.

This allows us to distinguish two main phases of the KelvHelmoltz in-stability: the linear phase, when the perturbation grows exponentially, and the non-linear phase, which requires the solution of the full system of equations. Simulations are run in order to investigate the dynamics during the non linear evolution of the instability, when moreover secondary instabilities may develop, thus giving rise to a much more complex evolution.

As discussed in the introduction, the solar wind impacts the Earth’s magne-tosphere on the dayside. Fluctuations then propagate along the magnemagne-tosphere- magnetosphere-magnetosheath boundary. The Kelvin-Helmoltz instability is triggered and the oscillations grow moving towards the magnetotail until they become a ”chain” of vortices which continue to propagate away from the Sun. Simulating such an evolution is very difficult because of the number of scales involved, from the sys-tem size to the small scale fluctuation generated by the occurence of secondary instabilities.

For these reasons, the simulations we will present in section 4 investigate a relatively small portion of the boundary layer as it evolves under the perturba-tion. In other words, the simulation box represents what is seen by an observer moving along the boundary layer with the phase velocity of the growing modes of the perturbing oscillations.

3.2 Kelvin-Helmoltz instability

The Kelvin-Helmoltz instability may raise whenever there are two fluids in rela-tive motion one with respect to the other. It’s worth to note that this instability is of fluid nature and develops in plasmas as well as in neutral fluids. No viscos-ity is needed to make a sheared flow unstable, so the Kelvin-Helmoltz instabilviscos-ity can be included among the ideal instabilities.

Following the analysis made by Chandrasekhar [3], here we show that in the presence of a sheared flow the system of fluid equations leads to a dispersion relation from which one can determine the growth rate of the instability. We follow the standard linearization procedure around an equilibrium configura-tion, where all quantities are written as A = A0 + δA, where δA is a small

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3.2. KELVIN-HELMOLTZ INSTABILITY 23 perturbation (δA A). In the system all terms of order (δA)2 _{or higher will}

be neglected. Furthermore, we consider a system with periodicity along two di-rection (say, y and z ) and the disomogenuous didi-rection along x. This allows us to Fourier-transform the equations, so that y-, z - and t -derivatives are replaced by algebraic multiplication iky, ikz, −iω and we are left with a system with only

the x -derivative of the perturbation.

In the following, we use the subscript 0 and 1 for the equilibrium and per-turbed quantities respectively. The equilibrium configuration has a velocity field ~U0 along the y-direction which varies along x, a density profile ρ0 which

also varies only on x and a magnetic field ~B0 with no x -component and which

ensures pressure equilibrium. Further, we add a constant external force field ~

g directed along x in order to account for the role of an external acceleration on the development of the instability. The system of linearized equations then reads ∂tρ1+ ∇ · (ρ0~u1+ ρ1~u0) = 0 (3.4) ρ0[∂t~u1+ (~u1· ∇) ~U0+ ( ~U0· ∇)~u1] + (3.5) + ∇ h P1+ ~b1· ~B0 4π i − 1 4π[(~b1· ∇) ~B0+ ( ~B0· ∇)~b1] + ~gρ1 = 0 ∂t~b1+ ∇ × [ ~B0× ~u1+ ~b1× ~U0] = 0 (3.6) ∇ · ~u1 = 0 (3.7)

These equations are the continuity, the momentum, the Faraday equation plus the incompressibility condition respectively. The incompressibility condition is used for the sake of mathematical simplicity, but it also corresponds to the most unstable modes in general.

We take perturbations of the form

A1(x, y, z, t) = A(x)exp(ikyy + ikzz − iωt) (3.8)

allowing to substitute ∂t → −iω and ∇ → (∂x, iky, ikz). The linearized system

Eq.3.4 - 3.7 then becomes

i(kyU0y− ω)ρ1+ u1x∂xρ0 = 0 (3.9) iρ(kyU0y− ω)u1x+ ∂x h P1+ ~ B0· ~b1 4π i − ib1x (~k · ~B0) 4π − gρ1= 0 (3.10) iρ(kyU0y− ω)˜u1y+ iky h P1+ ~ B0· ~b1 4π i − 1 4π[ib1y(~k · ~B0) + b1x∂xB0y] = 0 (3.11) iρ(kyU0y− ω)u1z+ ikz h P1+ ~ B0· ~b1 4π i − 1 4π[ib1z(~k · ~B0) + b1x∂xB0z] = 0 (3.12) i(kyU0y− ω)b1x− i[~k · ~B0]u1x= 0 (3.13) i(kyU0y− ω)b1y− i[~k · ~B0]u1y+ u1x∂xB0y+ b1x∂xU0y= 0 (3.14)

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24 CHAPTER 3. STABILITY OF SHEARED FLOWS i(kyU0y− ω)b1z− i[~k · ~B0]u1z+ u1x∂xB0z = 0 (3.15)

∂xu1x+ ikyu1y+ ikzu1z = 0 (3.16)

In deriving the previous set of equation we used the identity ∇ × (U × B) = (B · ∇)U − (U · ∇)B (thanks to the solenoidal property of U and B) and defined

˜

u1y = u1y− i

u1x∂xU0y

kyU0y− ω

(3.17) in the y-component of the momentum equation Eq.3.11. The aim is to write a differential equation for the x -component of the perturbed velocity. The Faraday equation can be used to obtain b1x, ∂xB0y and ∂xB0z, so that the momentum

equation can be re-casted as ihρ(kyU0y− ω) − (~k · ~B0)2 4π(kyU0y− ω) i u1x+ ∂x h P1+ ~ B0· ~b1 4π i (3.18) ihρ(kyU0y− ω) − (~k · ~B0)2 4π(kyU0y− ω) i ˜ u1y+ iky h P1+ ~ B0· ~b1 4π i (3.19) ihρ(kyU0y− ω) − (~k · ~B0)2 4π(kyU0y− ω) i u1z+ ikz h P1+ ~ B0· ~b1 4π i (3.20) Eq.3.19 and Eq.3.20 can be multiplied by kz and ky respectively and summed

to get rid of the fluid plus magnetic pressure. The result is h ρ(kyU0y− ω) − (~k · ~B0)2 4π(kyU0y− ω) i [kzu˜1y− kyu1z] = 0 (3.21)

which can be satisfied if one of the two factors is set equal to zero. Because the first may be zero only for particular choice of the equilibrium field, one has to solve the second one; using the incompressibility condition Eq.3.7 we get:

kzu˜iy− kyu1z = 0 ⇒ ikzu1y− ikyu1z+ kz

u1x∂xU0y

kyU0y− ω

= 0 (3.22) This equation and the incompressibility condition can be multiplied by ky and

±k_z and summed to obtain

u1y = i k2 h ky∂xu1x+ k2_z kyU0y− ω u1x∂xU0y i (3.23) u1z = i k2 h kz∂xu1x+ kzky kyU0y− ω u1x∂xU0y i (3.24) which allows us to write the perturbed velociy field ~u1if we know its x -component.

Instead, multipling the y and the z component of the momentum equation by ky and kz and summing one obtains

[K − B]∂x h u_1x kyU0y− ω i − ik2h_P 1+ ~ B0· ~b1 4π i = 0 (3.25)

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3.2. KELVIN-HELMOLTZ INSTABILITY 25 which can be substituted in the x -component of the momentum equation to obtain ∂x n [K − B]∂x h u_1x kyU0y− ω io − k2[K − B + g∂xρ0] h u_1x kyU0y− ω i = 0 (3.26) where K = ρ0(kyU0y−ω)2 and B = (~k· ~B0)2/4π depends only on the equilibrium

fields.

It’s worth noticing that the quantity u1x/(kyU0y− ω) must be a continuous

function across any surface of discontinuity. Indeed, if we require that the fluid displacement ~ξ1 must be continuous at a surface of discontinuity, we get

u1x= dξ1x dt = ∂tξ1x+ U0y∂yξ1x= i(kyU0y− ω)ξ1x → → ξ1x= −i u1x kyU0y − ω (3.27) thus we require that the unknown function between square brackets in Eq.3.26 must be continuous.

Now we take a particular configuration in which two fluids of uniform equi-librium density ρ1 and ρ2 slip one over the other across a piecewise uniform

magnetic field (say, B1 and B2) with a velocity shear ∆U = |U2 − U1| (where

both U1 and U2 are constant). In this configuration our equilibrium field can

be written as

A0 =

1

2[(A2− A1)σ + (A1+ A2)] (3.28) where A1 and A2 are the value of the equilibrium field on the sides of the

discontinuity and σ is the sign function defined as

σ(x − x0) =    −1 x < x₀ 0 x = x0 1 x > x0

With this equilibrium profile Eq.3.26 can be integrated around the disconti-nuity. Before going further into a detailed calculus, it’s worth noticing that the continuity of the fluid displacement doesn’t imply the continuity of its derivative (as will become clear once we get u1x), thus some care should be taken when

integrating Eq.3.26.

First, any function f (x) with a finite jump corresponding to a discontinuity at a point x0 and analytical everywhere except in x0 can be written as a sum

of a function g(x) continuous at x = x0 plus a sign function. Thus

f (x) = g(x) −1

2∆f [σ(x − x0) + 1] (3.29) where g(x) is a function continuous in x0 defined as1

g(x) =    f−(x) x < x0 lim_x→x− 0 f −_(x) _{x = x} 0 f+(x) + ∆f x > x0 (3.30)

1_{The superscripts - and + refer to the left and right part of f(x) with respect to x = x} 0.

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26 CHAPTER 3. STABILITY OF SHEARED FLOWS where ∆f is the jump between the left and right limit for x → x0. So, our

concern is solving the following integral: lim →0 Z x0+ x0− f0(x)dx = lim →0 Z x0+ x0− g0(x)dx − 1 2∆f Z x0+ x0− 2δ(x − x0)dx (3.31) where the prime denotes the derivative with respect to x, while the last term on the rhs takes into account the discontinuity of the sign function at x = x02.

Using the property of the δ-functionRx0+

x0− δ(x − x0)dx = 1 for every we obtain

the jump across the discontinuity. The first integrand g0(x) on the rhs may be in general a discontinuous function so one has to check under which condition is integrable. If we suppose that g0(x) has a finite jump, we can exploit the same method used above to write it as a continuous function plus a sigma-function:

g0(x) = h(x) −1 2∆(g

0_{)[σ(x − x}

0) + 1] (3.32)

where h(x) is again a function continuous across x = x0 and ∆(g0) is the jump

of the derivative of g(x) across the discontinuity in x0.

This ensures that the first term on the rhs of Eq.3.31 is bounded in [−, ], thus this integral vanish in the limit → 0 and we are left with

lim →0 Z x0+ x0− f0(x)dx = −∆f Z x0+ x0− δ(x − x0)dx = −∆f

Applying this argument to Eq.3.26 with the substitution f (x) → [K − B]∂x

u_1x kyU0y− ω

, x0 → 0

we are allowed to carry out the integration of the first term of Eq.3.26 and write lim →0 Z − ∂x n [K −B]∂x h u_1x kyU0y− ω io dx = ∆ n [K −B]∂x h u_1x kyU0y− ω io dx (3.33) The second term of Eq.3.26 can also be integrated and gives

lim →0 Z − k2[K − B]h u1x kyU0y− ω i dx = 0 (3.34)

because both K and B are bounded, while u1x/(kyU0 − ω) is continuous, so

their product is a bounded function and the integral vanish in the limit → 0. The last term to evaluate is the gravitational one which leads to

lim →0 Z − g∂xρ0 h u_1x kyU0y− ω i dx = g∆ρ lim →0 Z − δ(x)h u1x kyU0y− ω i dx = (3.35) = g∆ρ h u_1x kyU0y− ω i 0

2_{In other words, we use the relation}

dσ(x − x0)

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3.2. KELVIN-HELMOLTZ INSTABILITY 27 Putting togheter all the integrals one obtains the jump condition at discon-tinuity which must be satisfied by u1x:

∆nhρ0(kyU0y− ω)2− (~k · ~B0)2 4π i ∂x u_1x kyU0y− ω − gρ0 u_1x kyU0y− ω 0 o = 0 (3.36) Once the jump condition is found, one has to solve Eq.3.26 for the two plasmas: doing so, two solution are obtained and they must be matched with the jump condition Eq.3.36.

In the two media all the derivatves of the equilibrium field vanish, so Eq.3.26 become [∂_x2− k2] u_1x kyU0y− ω = 0 (3.37)

and the solution is given by u1x=

A1(kyU1− ω)exp(kx) x < 0

A2(kyU2− ω)exp(−kx) x > 0

Imposing the continuity of the fluid displacement we see that A1 = A2, while

substituting into the jump equation Eq.3.36 we are left with a relation which depends only on k, ω and the equilibrium fields. The dispersion relation after some manipulation reads:

ω = −ky(α1U1+α2U2)± s (~k · ~B1)2+ (~k · ~B2)2 4π(ρ1+ ρ2) − k2 yα1α2(U1− U2)2− g(α1− α2)k (3.38) where αi = ρi/(ρ1+ ρ2).

Such a dispersion relation gives us important information about the devel-opment of the instability. First, provided that ρ1 = ρ2 and U1 = −U2, for any

real wavenumber with ky 6= 0 the real part ωrof the frequency is zero. Secondly,

ω may exhibit a non-zero imaginary part ωi if the discriminant under square

root is negative, i.e. if

(~k · ~B1)2+ (~k · ~B2)2

4π(ρ1+ ρ2)

< k_y2α1α2(U1− U2)2+ g(α1− α2)k (3.39)

In Eq.3.38 the magnetic term is always positive and thus acts as a stabilizing term. This can be understood by considering that the Kelvin-Helmoltz insta-bility tries to warp the interface between the two plasmas, but the frozen-in law states that the magnetic field lines must be equally bent. This gives rise to a restoring magnetic tension which couteracts the instability, delaying or suppressing it completely.

On the contrary, the second term under square root is always negative, thus leading to destabilization. Therefore if the magnetic field has no component along the flow and no external acceleration g is assumed, a sheared configuration is always unstable disregard how small the velocity difference is (in the limit of a discontinuity separating the two fluids).

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28 CHAPTER 3. STABILITY OF SHEARED FLOWS The last term acts as a stabilizing or a destabilizing contribution depending on the sign of α1−α2with respect to the external acceleration g: if a plasma with

density ρ1is pushed into another plasma with density ρ2 < ρ1, then α1−α2 > 0,

thus instability is enhanced. In other words if we consider two superposed plasmas at rest one with respect to the other, we obtain ωr= 0 again. On the

other hand, if a denser plasma is on top of a lighter one, we have ωi 6= 0, thus

the instability can develop: this is the well known Rayleigh-Taylor instability.

3.2.1 Effect of compressibility and of a finite width of the shear layer

In the previous section we have assumed the incompressibility condition, which corresponds to the closure P = const, thus the kinetic energy of the initial shear cannot be converted into internal energy of the plasma and it’s all available for the development of instabilities. In other words, we are looking for the modes that do not compress the plasma. Furthermore, the initial shear is treated analytically as a tangential discontinuity instead as a finite-width layer. This leads to a dispersion relation which gives a growth rate that increases linearly with the wavevector, which is unphysical for wavelenght smaller than the shear characteristic lenght. In our simulations the plasma is compressible and the shear has a finite width, thus in this section the impact of relaxing these two assumptions on the dispersion relation is analyzed.

In order to understand the effects of compressibility, let’s assume a polytropic law

P n−γ = const

with a pressure fluctuation of the form P0+ δP and look for the corresponding

density variation. Making a first order expansion, we get δn n0 = (1 + δP P0 )1γ ≈ 1 γ δP P0

Thus, the smaller the polytropic index, the greater the compression δn associ-ated with a fluctuation δP . As a consequence, the kinetic energy of the bulk motion of the fluid is converted into internal energy and it is not available for driving the instability.

The effect of a finite shear layer impacts first the dependance of the growth rates on the wavevector: perturbations with small waveleght as compared to the shear layer thickness Ls the do not ”see” the shear and thus are just damped

by the inhomogeneity. On the other hand, when long wavelenght perturbations are considered, the shear layer can be viewed as a discontiuous surface and we make use of Eq.3.38 to determine the growth rate of the instability. Thus, when we take into account a finite shear layer we have that in the limit k → 0 the growth rate vanishes (in agreement with Eq.3.38) and the same occurs for kLs 1. As consequence, the imaginary part of the dispersion relation for

the Kelvin-Helmoltz instability must exhibit a maximum which corresponds the most unstable wavelegnt.

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3.2. KELVIN-HELMOLTZ INSTABILITY 29 Miura and Pritchett [7] investigated numerically a sheared configuration with a uniform magnetic field perpendicular to the flow within an MHD ap-proach, taking into account both the compressibility of the medium and the presence of a finite shear layer. In Fig.3.1 we show the growth rate resulting from their analysis as a function of the wavevector for various degree of com-pressibility. They found that including a finite shear width leads to a bell-shaped curve for the growth rate, thus a fastest growing mode is present. This curve must be considered as a correction to Eq.3.38 for kLs & 1. Further, different

degree of compressibility marked by different values of the fast magnetosonic number M_f2= U 2 0 c2 s+ v2A = M 2 AMs2 M2 s + MA2

where U0 is the typical fluid velocity, cs = γP/ρ is the sound speed and vA =

B2/4πρ is the Alfv´en speed, gives raise to a family of bell-shaped curves. For each curve, the maximum growth rate and the maximum unstable wavevector kmax decrease as Mf increases. Three observations deserve to be made. First,

there are no unstable modes for Mf > 2 because compression is very efficient

in suppressing the instability; second, the fastest growing mode always satisfies 0.5 . kLs . 1; third, disregarding of Mf, all the curves in Fig.3.1 converge

to the straight dashed line of the incompressible zero-thickness layer case as kLs→ 0, which thus can then be viewed as the limit for kLs 1.

The fast magnetosonic number can be related to the plasma compressibility in the case of thermal and magnetic pressure-dominated dynamics, respectively, by noticing that

∇ · ~u ≈ ~u·∇P_γP ≈ ~u·[mn~_γPu·∇~u] ≈ U 2L U cs 2 = _2LUM_s2 ∇ · ~u ≈ ~u·∇B_2B22 ≈ ~ u·[8πmn~u·∇~u] 2B2 ≈ _2LU U vA 2 = _2LUM_A2 (3.40)

The first of Eq.3.40 is obtained assuming a polytropic equation of state d(P nγ)/dt = 0 and making use of the equation of motion Eq.2.41 in the static case and ne-glecting the contribution of the magnetic field (i.e: we are considering a thermal pressure-dominated dynamics). The first approximation in the second row of Eq.3.40 is obtained by the scalar product of the Faraday equation Eq.2.47 in the static case with the magnetic field, while the second approximation is obtained by the equation of motion Eq.2.41 neglecting the thermal pressure. Thus, when both the magnetic and the thermal pressure are relevant (i.e: β ∼ 1), the proper Mach number is the fast magnetosonic one and the compressbility of the plasma increases as the Mach number is increased.

If the pressure and density are related by a polytropic law the fast magne-tosonic number can be written also as

M_f2 = M

2 A

1 +1₂γβ

which points out explicitly its dependance on the polytropic index: for a certain initial configuration (which sets the values of β = Pf luid/Pmag and MA), the

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30 CHAPTER 3. STABILITY OF SHEARED FLOWS

Figure 3.1: Normalized growth rates for the linear Kelvin-Helmoltz insta-bility for various value of M_f−1 as function of the normalized wavevector in a transversal magnetic field. The solid lines point out the growth rate for an initial configuration with a finite shear layer of thickness 2a with various degree of compressibility: corresponding to Mf = 0, the

upper-most curve represents the incompressible case, while the Mf = 2 curve

represents the highest compressibility which allows for the development of the instability. The dotted line is the growth rate for a discontinuity surface between two incompressible plasmas: for small wavevector, it can be considered as a good approximation in compressible plasmas. The picture is taken from [7].

closure relation determines the choice of γ, thus affecting the comprimibility of the plasma and the growth rates. In Chapter 4 three simulations with the same initial conditions will be analyzed. One of the analyzed simulations has a polytropic law for the pressure with γ = 5/3 ≈ 1, 667 (adiabatic closure), while the other two runs follow the CGL theory, thus the proper choice is γ⊥= 2. As

can be seen by looking at the curves of Fig.3.1, the different polytropic index among the simulations affects a little the fastest growing mode of the instability, thus we are confident that all the runs behave similarly during the linear phase.

3.3 Jet instability

In this section we aim at determining the stability of an incompressbile plasma streaming faster with respect to its surroundings, like a planar jet through a fluid at rest. In particular, we want to determine the stability of a single as well as a pair of opposite-streaming jets, as depicted in Fig.3.2. Indeed, as we will see in Chapter4, the evolution of the main Kelvin-Helmoltz instability in the non-linear phase may develop various profiles of velocity stratification on small scale which may be driven unstable, thus our aim is to investigate the stability

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3.3. JET INSTABILITY 31 of such configurations and eventually determine a dispersion relation.

(a) (b)

Figure 3.2: Sketch of a planar jet streaming with velocity U0 through a

plasma at rest. We assume that the magnetic field is perpendicular to the flow except between x = −a and x = +a, where a component parallel to the streaming is present. The density is assumed uniform. (a): single jet configuration. (b): double jet configuration.

We assume that a plasma with constant density streams along the y direc-tion, but only in a slab between x = −a and x = +a (which we will call ”internal region”), while it is static outiside (the ”external region”). Furthermore, we as-sume a magnetic field along the z -direction with an additional component along the flow in the internal region.

Thus, our magnetic and density field are given by ρ0(x) = ρ0 = const B0x= 0 B0y= B0 |x| < a 0 |x| > a B0z = B⊥ = const (3.41)

while the two models we want to investigate are represented by the single jet velocity profile, U0x= U0z = 0 U0y= U0 |x| < a 0 |x| > a (3.42)

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32 CHAPTER 3. STABILITY OF SHEARED FLOWS and by the double jets profile,

U0x= U0z = 0 U0y =    U0 −a < x < 0 −U0 0 < x < a 0 |x| > a (3.43)

For sake of simplicity, we assume that the perturbation propagates along the y-direction, thus kz= 0 and ky = k.

In order to determine the stability of a single jet we make use of the normal mode analysis for the Kelvin-Helmoltz instability, which have led to the jump condition Eq.3.36 and to the differential equation Eq.3.26. Because the equilib-rium fields Eq.3.41 and 3.42 are piecewise constant, Eq.3.26 can be simplified, thus our task is to find a solution to Eq.3.37 with the appropiate jump condition at x = −a, a.

The solution to Eq.3.37 with the equilibrium field Eq.3.42 have the form    A1ekx x < −a A2ekx+ B2e−kx −a < x < a A3e−kx x > a (3.44)

with the following matching condition given by the continuity of the fluid dis-placement at the surface of discontinuity:

A1e−ka= A2e−ka+ B2eka x = −a

A3e−ka= A2eka+ B2e−ka x = a

(3.45) Putting Eq.3.44 into the jump condition and making use of the matching con-ditions lead to the implicit dispersion relation

h ρ0(kU0−ω)2− (kB0y)2 4π +ρ0ω 2i2_e4ka₌h_ρ 0(kU0−ω)2− (kB0y)2 4π −ρ0ω 2i2 _(3.46)

This equation can be recasted in adimensional form multiplying by (2a/ρ0U0)2,

leading to

[(y − x)2− M y2_{+ x}2_]2_e2y _{= [(y − x)}2_{− M y}2_{− x}2_]2 _(3.47)

where x = 2aω/U0, y = 2ka and M = vA2/U02, vA= B0/

√

4πρ0being the Alfv´en

speed.

We remark that our analysis is in agreement with the literature on the stability of thin film of fluid streaming through an other fluid. For instance, we report the dispersion relation obtained by Squire [8] with a hydrodynamic approach for a thin slab of liquid: he took into account the presence of surface tension between the film and the surroundings, thus he found the following dispersion relation, recasted with our notation:

x = ypcoth(y/2)p1 − W · y · coth(y/2) + y

Role of the Pressure Tensor on the Evolution of the Non-Linear Kelvin-Helmoltz instability

Universit`

a di Pisa

Role of the Pressure Tensor on the

Evolution of the Non-Linear

Kelvin-Helmoltz Instability

Contents

Chapter 1

Introduction

Chapter 2

Theorical background

2.1

Fluid equations for collisionless magnetized

plas-mas

2.2

The closure problem

2.3

The pressure tensor

2.4

Towards a more accurate model: the FLR

ex-pansion

2.5

From a two to a single fluid formalism

Chapter 3

Stability of sheared flows

3.1

Stability analysis

3.2

Kelvin-Helmoltz instability

3.3

Jet instability