INSTABILITY OF THE IONOSPHERIC PLASMA: MODELING AND ANALYSIS · INSTABILITY OF THE IONOSPHERIC...

INSTABILITY OF THE IONOSPHERIC PLASMA: MODELING ANDANALYSIS ∗

CHRISTOPHE BESSE† , JEAN CLAUDEL‡ , PIERRE DEGOND§ , FABRICE DELUZET¶,

GERARD GALLICE‖, AND CHRISTIAN TESSIERAS∗∗

Abstract. This paper is concerned with the theory and modeling of plasma instabilities in theionosphere. We first consider the so-called ’striation model’ which consists of balance equations forthe density and momenta of the plasma species, coupled with an elliptic equation for the potential.The linearized instability of this model is analyzed in the framework of Fourier theory, both forsmooth and discontinuous steady-states. Then, we show that the dissipation mechanisms at work inthe more refined ’dynamo model’ allow to stabilize high wave-number perturbations. We also analyzeturbulence as a possible source of additional dissipation (in a similar way as in fluid mechanics). Tothis aim, we use the statistical approach to turbulence and derive a so-called ’turbulent striationmodel’, of which we analyze the stability properties. Numerical experiments are used to support ourinvestigations.

Key words. Euler-Maxwell system, dynamo model, striation model, ionospheric plasma, stria-tions, turbulence, statistical approach, linearized stability analysis

AMS subject classifications. 82D10, 76N99, 76X05, 76W06, 78M35

1. Introduction. This paper is concerned with the modeling and analysis ofplasma instabilities in the ionosphere, at altitudes ranging between a few hundredand a thousand kilometers (F region). The plasma may be created, either by thenatural ionization of the atmosphere, or by possible artificial causes (such as e.g.thermonuclear explosions [31], [43], [19]). The ionospheric plasma is strongly struc-tured by the earth magnetic field. Indeed, the mobility of the ionized species (i.e.their velocity in response to an external electric field) is strongly anisotropic: whilefield-aligned mobilities (i.e. mobilities in the direction of the magnetic field) are large,transverse mobilities (also called Pedersen mobilities) are quite small. Additionally,a component of the plasma velocity orthogonal to both the electric and magneticfields appears as a result of the Hall effect. This component is the major actor in theso-called E × B drift instability which we are going to discuss in the present paper.

At lower altitudes, the density of the neutral atmosphere is large and the plasmais dragged by the motion of the neutral molecules (or neutral wind). As a result, anet electrical current flows across the magnetic field lines and generates an inducedelectric field. This is the so-called ionospheric dynamo effect [1]. The reader can referto [37], [21], [9] and [2] for reviews about ionospheric physics. In the presence of agradient in the plasma density, the neutral wind can trigger the E×B drift instability,which bears strong analogies with the Rayleigh-Taylor instability in fluid mechanics[10]. This instability produces strong inhomogeneities (the ionospheric striations)

∗Supports by the ’Centre d’Etudes Scientifiques d’Aquitaine’ of the ’Commisariat a l’EnergieAtomique’ and by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282 are gratefully acknowledged.

†Mathematiques pour l’Industrie et la Physique, CNRS UMR 5640, Universite Paul Sabatier, 118Route de Narbonne, 31062 Toulouse Cedex 4, France ([email protected]).

‡Commissariat a l’Energie Atomique, Centre d’Etudes Scientifiques et Techniques d’Aquitaine,BP2, 33114 Le Barp, France ([email protected]).

§Mathematiques pour l’Industrie et la Physique ([email protected]).¶Mathematiques pour l’Industrie et la Physique ([email protected]).‖Commissariat a l’Energie Atomique ([email protected]).

∗∗Commissariat a l’Energie Atomique ([email protected]).

1

2 C. BESSE, J. CLAUDEL, P. DEGOND, F. DELUZET, G. GALLICE AND C. TESSIERAS

which soon propagate over hundreds of kilometers along the magnetic field lines. Thegeneration of plasma irregularities is reviewed in [13], [14], [36].

Our goal is to discuss some aspects of the mathematical and numerical modeling ofthis instability. Striations as well as related instability phenomena of the ionosphericplasma have been the subject of a wide literature (see e.g. discussions of the ’Spread F’in [44], [26], [34], of the equatorial electrojet in [8], [41], [38], [39] and of Barium releasesexperiments in [12]). The well-accepted mathematical model for these phenomena isthe ’dynamo’ model [44], [12] which consists of mass and momentum balance equationsfor the plasma species. A simpler model, the ’striation’ model, is obtained when thefield-aligned mobilities are supposed infinite. The derivation of these models and theirinterrelations are reviewed in [3], and will be briefly recalled in section 2.

The E×B drift instability is well-described in the framework of the striation model(see section 3). A linear stability analysis indeed shows that exponential densityprofiles are unstable (see the review in [14] and section 3.2). Exponential densityprofiles are the only non-constant smooth stationary states which allow analyticalcomputations (via Fourier analysis). However, they are quite unrealistic and a bettertheory should consider discontinuous density profiles. We consider this problem insection 3.3 and show that the striation model is also unstable in this case for certainconfigurations of the neutral wind. In a companion work [6], we show that smoothdensity profiles which are linearly unstable are nonlinearly unstable. However, theproof of [6] does not extend to discontinuous solutions. Similarly, we do not know,even for smooth density profiles, if linear stability implies nonlinear stability.

In practice, the instability saturates and cascades towards smaller scales by non-linearity [13], [39], [41], [25], [33], until it is ultimately damped by dissipation. In thestriation model however, all dissipation mechanisms have been removed. In section4, we reintroduce dissipation effects by considering the dynamo model, where bothfinite temperature and finite conductivity effects are retained. A linearized stabilityanalysis shows that high wave-number perturbations are damped. However, in prac-tice, the magnitude of the dissipation is too small and we must consider other sourcesof dissipation.

In this paper, we investigate the possible influence of fluid turbulence. In fluidmechanics, it is a well-known fact that turbulence may enhance dissipation ([32] andreferences therein). The statistical approach to fluid turbulence considers averagesof the Navier-Stokes equations over various approximate realizations of the same so-lution. The chains of resulting statistical equations are closed by various types ofphenomenological assumptions which are still mathematically unjustified except invery simple cases, such as that considered in [27]. The obtained models (such asthe K-ǫ model) involve additional terms compared with the standard Navier-Stokesequations which describe the enhancement of diffusion by turbulence.

In section 5, we develop a similar statistical framework to model turbulence withinthe striation model (see also [28] for an application to MHD theory). We first de-rive an averaged striation model, for which we propose a closure Ansatz inspired by[27]. This leads to a diffusive version of the striation model, the ’turbulent striationmodel’. To find the value of the turbulent diffusion constant, a stability analysis of themodel is performed. It allows us to relate the threshold wave-number for instability(i.e. the typical size of the finest persisting structures in the plasma, which can beexperimentally observed) with the value of this constant.

Section 6 is devoted to 2-dimentional numerical simulations. Their goal is toprovide a numerical and quantitative evidence of the features predicted in section 5,

TURBULENCE MODELING OF THE IONOSPHERIC STRIATION MODEL 3

namely that the turbulent striation model produces persisting structures whose typicalsizes are related to the magnitude of the diffusion. Three-dimensional simulations ofthe striation model are shown in [4]. A review collecting material from [3], [4], [6] aswell as from the present paper is presented in [5].

Turbulence modeling in ionospheric plasmas has been widely investigated in theliterature. Most of the approaches rely on nonlinear Fourier analysis [39], [22], [23]and bear similarities with the spectral approach to turbulence in fluid mechanics [29](see also [15] for applications of these ideas to other plasma physics contexts). Inusing the statistical approach, we have chosen a slightly different route.

2. The ’dynamo’ and ’striation’ models of the ionospheric plasma. Weconsider two different species of particles: electrons and one ion species. They areassumed so dilute that they have no influence on the dynamics of the neutrals, thevelocity of which un(x, t) (also called the neutral wind) is supposed known. In [3],a hierarchy of models for the ionospheric plasma has been derived. Of particularinterest in the present study are the ’dynamo’ and ’striation’ models. The dynamomodel is written as follows:

∂tn + ∇ · (nui) = 0 , (2.1)

−∇φ + ue,i × B = κqe,i [νe,i(ue,i − un) + η∇ log n] . (2.2)

∇ · j = 0 , κj = n(ui − ue) , (2.3)

where we denote by n(x, t) the density of the plasma, ue(x, t), ui(x, t), the electronand ion velocities, j(x, t) the plasma current, φ(x, t), the electric potential, B(x), theearth magnetic field and νe(x), νi(x), the electron-neutral and ion-neutral collisionfrequencies. These quantities depend on the 3-dimensional position coordinate x andon the time t ≥ 0. The parameters η and κ are dimensionless and defined below. Eq.(2.2) actually consists of two equations, one for the electrons (with the index ’e’ choseneverywhere) and one for the ions (with the index ’i’). We let qi = 1, qe = −1. Wesuppose that the geomagnetic field B(x) is unperturbed by the presence of the plasmaand is known. Similarly, the collision frequencies νe(x), νi(x), which primarily dependon the neutral density, are supposed known. The plasma is supposed quasi-neutral,i.e. the electron and ion densities coincide with n. Despite the quasi-neutrality, theelectron and ion velocities can be different, giving rise to a non-zero plasma currentj. We have supposed that the electron and ion gases are isothermal with the sameuniform temperature, which seems a valid physical hypothesis for the earth ionosphere[7]. The ionization-recombination terms which should appear at the r.h.s. of (2.1)have been omitted as well. Typical ionization-recombination times are of the orderof several hours, which is about the typical growth time of the instability. Therefore,these terms would only make a small correction to the analysis below and have beenomitted for the sake of simplicity.

System (2.1)-(2.3) is written in dimensionless units. The scaling units and theirtypical values in the situations of interest are summarized in table 2.1 below. Theparameters η and κ are given by:

η =kB T

miu2

1

νi t, κ =

meνe

eB=

miνi

eB,

(where kB is the Boltzmann constant) and respectively measure the ratio of the ther-mal energy to the ion drift energy and the typical number of electron-neutral or


Quantity Scaling unit ValueTime t 103 sLength x 105 mVelocity u = x/t 102 ms−1

Density n 1012 m−3

Temperature T 103 KMagnetic field B 10−5 TElectric potential φ = uBx 102 Ve-n collision frequency νe 102 s−1

i-n collision frequency νi = me

mi

νe 10−2 s−1

Table 2.1

Scaling units

ion-neutral collisions per rotation period in the geomagnetic field. These two parame-ters have typical values (according to table 2.1): η ∼ 101, κ ∼ 10−4. Since κ is small,it is meaningful to investigate the limit of the dynamo model when κ → 0. This leadsto the so-called striation model that we give in more detail below.

Before doing so, we rewrite the dynamo model in a more appropriate form. In alocal reference frame in which the last basis vector is aligned with the magnetic field,the ion and electron mobility matrices Me and Mi are given by:

Me =

µPe −µH

e 0µH

e µPe 0

0 0 µ‖e

, Mi =

µPi µH

i 0−µH

i µPi 0

0 0 µ‖i

,

where the electron and ion Pedersen, Hall and field-aligned mobilities are respectivelydefined by

µPe,i =

κνe,i

(κνe,i)2 + |B|2 , µHe,i =

|B|(κνe,i)2 + |B|2 , µ

‖e,i =

1

κνe,i.

In the situation κ → 0, the electron or ion field-aligned mobilities tend to infinity.

Thanks to the mobility matrices, equations (2.2) and (2.3) may be rewritten as

ue,i = Me,i (−qe,i∇φ + κ(νe,iun − η∇ log n)) , (2.4)

−∇ · (n(Mi + Me)∇φ)) =

= −κ∇ · (n[Mi(νiun − η∇ log n) − Me(νeun − η∇ log n)]) . (2.5)

It is clear that the conductivity matrix n(Mi + Me) is positive definite (provided thatνi or νe is positive and finite). Therefore, (2.5) is a three-dimensional elliptic equationfor φ.

Now, we assume that the magnetic field is constant and uniform (see Fig. 2.1).An extension to the non-uniform B case is given in [3] and [4]. Let us denote by(x1, x2, x3) the orthonormal coordinate basis, with x3 aligned with B. We can choosethe scaling units such that |B| = 1, so that B = x3. We denote by x = (x1, x2)the position vector in the 2-dimensional plane orthogonal to B and ∇ = (∂x1

, ∂x2)

the 2-dimensional gradient. For any 3-dimensional vector a = (a1, a2, a3), we definea = (a1, a2) its projection onto this plane.


When κ → 0, the dynamo model reduces to the the so-called striation model [3]:

∂tn + ∇ · (nu) = 0 , (2.6)

u = −∇φ × B + ((un − ην−1∇ log n) · B)B , (2.7)

∇ · ((−σ(x)∇φ + (Un − 2η∇N) × B) = 0 , (2.8)

with φ = φ(x), σ(x) =∫

nνdx3, Un =∫

nνundx3, N =∫

ndx3, ν = νi + νe and ui =ue = u. The striation model couples a 3-dimensional convection-diffusion equation(2.6), (2.7) for the density n with a 2-dimensional elliptic equation (2.8) for the electricpotential φ. The coefficients of the elliptic equation (2.8) involve integrals of n overx3 i.e. along the magnetic field lines. The infinite conductivity of the plasma alongthe magnetic field lines constrains the electric potential to be constant along theselines, i.e. to depend only on the 2-dimensional coordinate x.

eart

h

xy

x

yz

B

B

B

B

B

n

Fig. 2.1. Geometry of the earth environment and reduction to a cartesian geometry

If we additionally suppose that un is orthogonal to B and that all data andunknowns are independent of x3, the striation model reduces to the following mono-layer striation model:

∂tn + ∇ · (nu) = 0 , u = −∇φ × B , (2.9)

∇ · (nh) = 0 , h = ν(

−∇φ + (un − 2ην−1∇ log n) × B)

, (2.10)

where now, all variables and vectors are 2-dimensional (except B = x3) and theunderlying of 2-dimensional vectors has been omitted. The quantity h represents theelectron-ion relative velocity. We remark that ∇ · u = 0. Therefore, we can writerelation (2.9)

∂tn + (u · ∇)n = 0 . (2.11)

As we will next see, the pressure term ∇ log n does not change the linearized stabilityproperties of the striation model. When η = 0, (2.10) becomes:

∇ · (nh) = 0 , h = ν (−∇φ + un × B) . (2.12)

In the next section, we analyze the linearized stability of this model.

3. Stability analysis of the striation model.


3.1. Introduction and phenomenology. The striation model exhibits an in-stability, the gradient-drift or E × B drift instability [13], [14]. In a recent work [6],local-in-time existence and uniqueness of solutions for this model have been provenand, following the methodology of [18], [20], [11], [24] smooth stationary density pro-files which are linearly unstable have been shown to be non-linearly unstable. How-ever, it is still open whether the converse is true. In [6], a variational formulation forthe instability growth rate is given. In the present work, we are aiming at a morequantitative result for certain specific classes of stationary profiles.

We restrict ourselves to two particular kinds of steady-state profiles. The firstones are smooth with an exponentially increasing density in one direction; they havealready been investigated [13], [14] and we shall only summarize the results. Thesecond ones are discontinuous density profiles; their analysis is, to the best of ourknowledge, new. In passing, we shall have to show that it is meaningful to considerdiscontinuous solutions of the striation model.

We first give a phenomenological view of the instability of the striation model.We consider a steady state consisting of a discontinuous density n(x) = n for x2 < 0and n = n > n for x2 > 0, with ∇φ = 0 and un = (0, U). We slightly perturb theinterface which is now represented by the graph of the function x2 = ε sin(ξx1) whereε represents the magnitude of the perturbation (ε ≪ 1) and ξ is its spatial frequency.

The term un ×B in (2.12) creates a charge modulation along the interface whichis alternately positive and negative. A non-zero electric field −∇φ parallel to theinterface with a similar sign modulation is generated according to (2.12). Then, by(2.9) a non-zero component of the velocity u in the direction normal to the interface iscreated with again an alternating sign. According to the sign of un, this component ofthe velocity tends to either damp the modulation of the interface or increase it. Theformer case is a stable one while the latter is an unstable one. The precise geometricconfiguration is depicted on Fig. 3.1 and 3.2.

un

x1

x2

u u

u

E E

E

un × B

n

n

Fig. 3.1. Stable configuration

un

x1

x2

u

u

u

E E

E

un × Bn

n

Fig. 3.2. Unstable configuration

This behavior can be recovered by the linear stability analysis. We first turn tothe analysis of the exponential density profile. In the remainder of this section, weassume that ν is a uniform constant and by a convenient choice of the scaling units,we let ν = 1.

3.2. Linear stability analysis : the exponential density profile . Let usdenote by (n0, u0, h0, φ0) the unperturbed state, i.e. a time-independent solutionof the striation system (2.9), (2.12). We consider an exponential density profile in


the x2-direction, i.e. n0 = N exp(x2/λ), where λ > 0 is the gradient length, while(u0, h0,∇φ0) are uniform constants. We suppose that the neutral wind is uniform aswell and has components un = (V, U). In this configuration, a necessary and sufficientcondition for (n0, u0, h0, φ0) to form a steady-state is that u0 = (V, 0), h0 = (νU, 0),−∇φ0 = (0, V ). In this analysis, we suppose that η = 0, unless otherwise specified.

Then, we introduce the perturbation n = n0(1 + εn1 + O(ε2)), (u, h, φ) = (u0, h0,φ0)+ε(u1, h1, φ1)+O(ε2), with ε ≪ 1 in the striation model and neglect the terms oforder higher than ε. The neutral wind being a datum remains unperturbed. We notethat ∇n0 = (0, n0/λ). An easy computation gives the linearized system governing thefirst order perturbation:

∂tn1 + λ−1∂x1

φ1 + V ∂x1n1 = 0 , (3.1)

−λ−1∂x2φ1 − ∆φ1 + U∂x1

n1 = 0 , (3.2)

Remark 3.1. If the pressure gradient terms are retained in the model (i.e. ifη 6= 0), the steady-state is only modified through the expression of h0 which should betaken as h0 = (U − 2ηλ−1, 0). However, the first order perturbation equations are thesame as (3.1), (3.2). The details are left to the reader.

We develop the solution of (3.1), (3.2) into plane waves, i.e. (dropping the su-perscripts ’1’ for clarity) (n, φ) = (n, φλ|U |) exp

(

iλ−1 (ξ1x1 + ξ2x2 − ωt|U |))

whereξ = (ξ1, ξ2) is the (normalized) wave-vector of the perturbation and ω its frequency.Introducing this Ansatz into (3.1), (3.2), we get:

−ωn + ξ1φ = 0 ; iσξ1n + (ξ21 + ξ2

2 − iξ2)φ = 0 , (3.3)

with σ = sign(U) ∈ −1, 1. This system has a non trivial solution iff its determinantis non vanishing. This condition yields the dispersion relation

ω =−iσξ2

1

(ξ21 + ξ2

2)2 + ξ22

(ξ21 + ξ2

2 + iξ2) , (3.4)

We now recall the following standardDefinition 3.2. The perturbation is stable if n and φ stay bounded for all times

t ≥ 0 and unstable in the converse situation. Therefore, a perturbation is stable iffℑm(ω) ≤ 0 and unstable iff ℑm(ω) > 0. A stationary state is called stable if all itsperturbations are stable for all wave vectors ξ. It is unstable as soon as there exists awave vector ξ giving rise to an unstable perturbation.

Thanks to (3.4), we have sign(ℑm(ω)) = −σ. It follows:Proposition 3.3. The steady-state configuration with an exponential density

profile is stable if and only if U ≥ 0, i.e. if the x2-component of the neutral windpoints in the same direction as the density gradient. Furthermore, in the case U < 0,all wave vectors ξ 6= 0 are unstable and for ξ2 = 0 the growth rate is independent ofξ1.

As seen above, exponential density profiles allow explicit computations. However,they are fairly unrealistic, as the density tends to infinity on one side and degeneratesto zero (and the elliptic problem (2.12) as well) on the other side. In order to study amore realistic situation, we extend our analysis to the case of a discontinuous densityprofiles in the next section.

3.3. Linear stability analysis : discontinuous density profiles. We con-sider a density profile which is piecewise constant and discontinuous across a parame-trized curve C(t) given by the equation x2 = f(x1, t), where f ∈ C1(R× [0, +∞[) i.e.


(see Fig. 3.3):

n(x, t) =

n for x2 < f(x1, t) ,n for x2 > f(x1, t) .

(3.5)

n

n

x1

x2

x2 = f(x1, t)

Fig. 3.3. Discontinuity curve of n

First, we must give a meaning to discontinuous solutions of this kind. To thisaim, we use the notion of weak solution of (2.11).

Definition 3.4. let u ∈ C1(R2 × [0,∞[). A function n ∈ L∞loc

(R2 × [0,∞)) is aweak solution of (2.11) with initial data n0 if and only if n verifies

∫

R2×[0,∞)

n

(

∂ϕ

∂t+ ∇ · (uϕ)

)

dx dt +

∫

R2

n0 ϕ(x, 0) dx = 0 , (3.6)

for all functions ϕ(x, y, t) ∈ C1c (R2 × [0,∞)), where C1

c defines the space of functionsof class C1 with compact support.

The solution of (3.6) can be obtained through the method of characteristics. Inparticular, it satisfies the maximum principle. Therefore, if there exist two constantsn∗, n∗ such that 0 < n∗ < n0(x) < n∗, this inequality is satisfied at all times:0 < n∗ < n(x, t) < n∗.

This notion has to be extended to the case of discontinuous velocities. Supposethat u = (u1, u2) is taken in the space L1

loc([0,∞), Hdiv), with Hdiv(R2) = u ∈

L2(R2), s.t. ∇ ·u ∈ L2(R2). Then, ∇ · (uϕ) ∈ L1loc(R

2 × [0,∞)) for all test functionsϕ and the expression (3.6) still has a meaning. Now, in the striation model, u is agiven by u = −∇φ × B where φ is a solution of (2.12). To solve (2.12), we use thefollowing (classical) proposition:

Proposition 3.5. Let un ∈ L2(R2) and n be such that there exist two constantsn∗, n∗ with 0 < n∗ < n(x, t) < n∗. Then, eq. (2.12), which can be written

∇ · (n∇φ) = ∇ · (nun × B) ,

has a solution in the space L1loc

([0,∞), H), with H = φ ∈ D′(R2), ∇φ ∈ L2(R2),unique up to an additive constant.

Since u satisfies ∇ · u = 0, this proposition guarantees that u belongs to L1loc

([0,∞), Hdiv(R2)). For such velocities, this allows us to define n as a weak solution

of (2.11) in the sense of (3.6). Therefore, it is meaningful to look for solutions withdiscontinuous densities. Of course, we have not shown the actual existence of suchsolutions, which will be the subject of future work. Now, we recall the followingclassical trace property [16]:

Lemma 3.6. Let C be a regular orientable curve of R2. Then, the mapping

γN : v → (v · N)|C (with N the unit normal vector to C) defined on D(R2) can be


extended by continuity to a linear and continuous mapping, still denoted by γN , fromHdiv(R

2) into H−1/2(C).We are now ready to determine the conditions that f must fulfill for n to be a

weak solution. We haveProposition 3.7. Let u belong to L1

loc([0,∞), Hdiv(R

2)). A function n definedby (3.5) is a weak solution to (2.11) if and only if f is a smooth solution to theequation

∂tf = (u · N)(

1 + (∂x1f)2

)1/2, (x1, t) ∈ R × [0,∞) , (3.7)

where N is the unit normal vector to the curve of discontinuity C(t) (pointing towardsx2 > 0) and (u · N) is the trace along C(t) as defined by Lemma 3.6. We can write

(u · N)(

1 + (∂x1f)2

)1/2= [u2 − u1∂x1

f ] |C , where the index C indicates that thisquantity is the common limit of the bracket as x2 → f(x1, t) from above and below.

Proof. We insert the expression of n in (3.6). We get

0 = n

∫

x2<f, t≥0

(∂tϕ + ∇ · (uϕ)) dx dt + n

∫

x2>f, t≥0

(∂tϕ + ∇ · (uϕ)) dx dt .

Since ϕ is compactly supported, we have∫

R2×[0,∞)(∂tϕ + ∇ · (uϕ)) dx dt = 0. We

regard ϕ as compactly supported in R2 × (0,∞) since the treatment of the initial

condition at t = 0 is standard. We deduce that

0 = (n − n)

∫

x2<f, t≥0

(∂tϕ + ∇ · (uϕ)) dx dt . (3.8)

In order to apply the Green formula, we use Lemma 3.6. We define the surface Σ =(x, t) ∈ R

2 × [0,∞), x ∈ C(t) and the open sets O(t) = x ∈ R2, x2 < f(x1, t)

and Ω = (x, t) ∈ R2 × [0,∞), x ∈ O(t). Let N = (N1, N2, Nt) be the outgoing unit

normal to Ω at (x, t) of Σ and Nx = (N1, N2). Thanks to Lemma 3.6, we can applythe Green formula and get:

∫

Ω

(∂tϕ + ∇ · (uϕ)) dx dt =

∫

Σ

ϕ(

Nt + u · Nx

)

dΣ(x, t) ,

where the integrals on Σ should be understood as the duality L∞([0,∞), H1/2(C(t)))against L1([0,∞), H−1/2(C(t))). Now, we have NdΣ(x, t) = (−∂x1

f, 1, −∂tf) dx1 dtwhich implies NxdΣ(x, t) = (1 + (∂x1

f)2)1/2N dx1 dt. Assuming that n 6= n, (3.8)gives

∫

R×[0,∞)

ϕ(

(u · N)(

1 + (∂x1f)2

)1/2 − ∂tf)

dx1 dt = 0 , (3.9)

Since (3.9) has to be verified for all test functions ϕ, we deduce (3.7).Then, the striation model (2.9) (2.12) for weak solutions can be written:

∂tf = [∂x2φ∂x1

f + ∂x1φ] |C(t) , (3.10)

−∇ · ((nχf + n(1 − χf ))(∇φ − un × B)) = 0 , (3.11)

with χf = 1 if x2 < f(x1, t) and χf = 0 otherwise.We now turn to the stability analysis of the striation model with discontinuous

initial density. A steady state of this model is given by f0 = 0, ∇φ0 = (0,−V ),


un = (V, U). We define the small perturbations of order ε as f = εf1, φ = φ0 + εφ1,with ε ≪ 1. We introduce this Ansatz in (3.10), (3.11) and keep only the terms oforder ε. We get

(

∂tf1 + V ∂x1

f1 − ∂x1φ1

)

|(x1,0,t) = 0 , (3.12)

−∇ · (n0∇φ1) = −U limε→0

ε−1∂x1(nχ(εf1) + n(1 − χ(εf1))) . (3.13)

where n0 = nχ0 +n(1−χ0) is the unperturbed density profile. A simple computationleads to ∂x1

χ(εf) = ε(∂x1f)δx2=0 +O(ε2), where the distribution g(x1)δx2=0 is defined

through the relation 〈g(x1)δx2=0, ϕ〉 =∫

Rϕ(x1, 0)g(x1) dx1 with ϕ(x) ∈ C∞

c . Then,(3.13) reads

−∇ · (n0∇φ1) = U(n − n)∂x1f1 δx2=0 . (3.14)

Like in the exponential density profile case, we develop the solution as a plane wavein the x1 direction: (f1, φ1) = (f , φ(x2)) exp i(ξx1 − ωt), where f and φ(x2) must bedetermined.

We introduce the plane-wave Ansatz in (3.12), (3.14), and we get

−iωf + iV ξf = iξφ(0) , (3.15)

−(

∂x2

(

n0∂x2φ)

− ξ2n0φ)

= U(n − n)iξfδx2=0 . (3.16)

If we solve (3.16) away from the point x2 = 0 and look for a bounded solution when|x2| → ∞, we find φ(x2) = φ(0)e−|ξ||x2|. Then, in the distributional sense on R, wehave:

−(

∂x2

(

n0∂x2φ)

− ξ2n0φ)

= −(n + n)|ξ|φ(0)δx2=0 . (3.17)

We introduce (3.17) in (3.16) and find

iξfδx2=0 −n + n

n − n

|ξ|U

φ(0)δx2=0 = 0 . (3.18)

Solving for f thanks to (3.15) and inserting it into (3.18) allows us to find the disper-sion relation:

ω = −V ξ − i Un − n

n + nξ (3.19)

We can easily deduce the stability result:Proposition 3.8. Let us assume that n > n. Steady-states defined by (3.5) with

f = 0 are stable if and only U ≥ 0. Furthermore, if U < 0, all wave-vectors ξ areunstable. The growth rate of the instability is given by

|ℑm(ω)| =n − n

n + nU |ξ| , (3.20)

This theorem makes the formal analysis of section 3.1 more quantitative. We notethat the growth rate increases linearly as a function of ξ, while it is was a constant inthe exponential density case (section 3.2). This feature prevents from extending thenon-linear instability theorem of [6] to the discontinuous case.


3.4. Conclusion of the stability analysis. The instability contributes to thedevelopment of smaller and smaller structures in the plasma. Quickly, the plasmabecomes chaotic (see section 6). In practice, the instability saturates after reachingsome level by the effects of physical dissipation mechanisms, which are not accountedfor so far in the model. We can think of three sources of physical dissipation: (i) finitetemperature effects; (ii) finite conductivity effects; (iii) turbulence effects. By remark3.1, we have seen that finite temperature effects alone do not change the results of thestability analysis. Therefore, we must simultaneously introduce finite temperatureand finite conductivity effects, i.e. we must go back to the full dynamo model (2.1)-(2.3). In section 4, we perform the stability analysis of the dynamo model and showthat large wave-vector perturbations are stable. However, the level of dissipation i.e.the threshold wave-number for stability is too large to match practical observations.Therefore, in section 5, we investigate the effects of fluid turbulence.

4. Stability analysis of the dynamo model (2.1)-(2.3). This analysis canbe found, in parts, in [13], [14]. We consider steady-states with exponential densityprofiles. Due to diffusion, discontinuous density profiles are not steady-states of thedynamo model any longer and there is no point in trying to analyze their stability. Tosimplify the analysis and to make it as close as that of the striation model in section3.2, we still consider a uniform magnetic field pointing in the x3-direction and wesuppose that un is orthogonal to B. All unknowns are independent of x3 and vectorsare contained in the plane (x1, x2). We assume that νe and νi are constants and suchthat ν = νe + νi = 1.

The steady state is given by

n0 = N exp (x2/λ) , un = (V, U) , h0 =(

U − 2ηλ−1)

x1 ,

u0i =

κνe(U − 2ηλ−1) + V

x1 , u0e =

−κνi(U − 2ηλ−1) + V

x1 ,

∇φ0 = (−κ2νiνe(U − 2ηλ−1),−κ(νe − νi)(U − ηλ−1) − V ) .

We proceed to the linear stability analysis as in section 3.2. We introduce theperturbation n = n0(1 + εn1 + O(ε2)), ui,e = u0

i,e + εu1i,e + O(ε2) and φ = φ0 + εφ1 +

O(ε2) in the dynamo model (2.1)-(2.3). We only keep order ε terms and develop thesolution as a plane wave according to the same Ansatz as in section 3.2. Let us define:

µH− = µH

i − µHe , µH

+ = µHi + µH

e , µP− = µP

i − µPe , µP

+ = µPi + µP

e ,

X = ξ1µHi − ξ2µ

Pi , Y = ξ1µ

H− − ξ2µ

P+ , Z = ξ1µ

H+ − ξ2µ

P− ,

AX = µPi |ξ|2 + iX , AY = µP

+|ξ|2 + iY , AZ = µP−|ξ|2 + iZ .

Then, we get the dispersion relation

ω =A∗

Y

|U ||AY |2(

ξ1u0ixAY − i

κη

λAXAY − ξ1h

0xAX + i

κη

λAXAZ

)

, (4.1)

where h = ui − ue and the star denotes the complex conjugate. The expression ofℑm(ω) may be simplified as follows:

ℑm(ω) =2κη

λ|U ||AY |2 µPi µP

e µP+|ξ|2P (ξ) , P (ξ) = −|ξ|4 +(a+1)ξ2

1 +2cξ1ξ2− ξ22 , (4.2)

with a = −Uλ ( 2κηνeνiµP+ )−1, c = µH

− (µP+)−1.


If κ → 0, the dynamo model reduces to the striation model with non-zero tem-perature, which has the same dispersion relation (3.4) as the striation model withzero-temperature (see remark 3.1). We can verify this property on the dispersionrelation (4.2). Indeed, as κ → 0, we have a ∼ −Uλ ( 2ηνeνi )−1κ−2, c ∼ κ(νe − νi).Therefore, P (ξ) ∼ −Uλ ( 2ηνeνi )−1ξ2

1κ−2 and we have P (ξ) < 0 if U > 0 (resp.P (ξ) > 0 if U < 0). Thus, we recover the results of section 3.2 for the striation model.

On the other hand, if we let the temperature go to zero (i.e. η → 0) in (4.2)while keeping κ finite, we find a = O(η−1) while c = O(1). Therefore, P (ξ) ∼−Uλ(2κνiνeµ

+P η)−1 and again the stability conditions for this model are the same as

those of the striation model, i.e. the model is unstable for all wave-vectors if U < 0and stable otherwise. Therefore, for the model to exhibit a stable range of wave-vectors, we need at the same time a finite conductivity and a finite temperature. Wenow show that this is indeed the case.

Proposition 4.1. Suppose that η > 0 and κ > 0. Then,(i) the dynamo model (2.1)-(2.3), linearized about the above-defined steady-states, isstable iff Uλ > 2ηκ2νeνi.(ii) If Uλ < 2ηκ2νeνi, there exists R0(η, κ) > 0, such that if ξ is an unstable wave-vector, then |ξ| < R0. Furthermore, R0 = O((

√ηκ)−1) as

√ηκ → 0.

We note that the stability criterion for the dynamo model is more restrictive thanthat of the striation model. The quantity Uλ needs to be, not only positive, but alsolarge enough. But, in the unstable case, when Uλ is not large enough, the instabilityregion is a bounded domain in wave-vector space (by contrast with the case of thestriation model, in which the instability domain is unbounded). The instability regiongrows as η or κ decrease to 0 and ultimately fills the entire wave-vector space in thelimit.

Proof. We introduce polar coordinates ξ1 = r cos θ and ξ2 = r sin θ. We can writeP = −r2Q with Q = r2 − F (θ) with

F (θ) = (a + 1) cos2 θ + 2c cos θ sin θ − sin2 θ = δ cos(2θ − α) + (a/2) ,

and δ = (((a/2)+1)2+c2)1/2, α = tan−1(a/(a+2)). Therefore, Q = 0 iff cos(2θ−α) =δ−1(r2 − (a/2)). For this equation to have roots, we need that −1 ≤ δ−1(r2 −(a/2)) ≤ 1. Therefore, if −a/(2δ) > 1, this equation cannot have any root, forany value of r > 0. Conversely, if −a/(2δ) ≤ 1, this equation has roots as long as(a/2) − δ ≤ r2 ≤ (a/2) + δ. Therefore, the case −(a/(2δ)) > 1 characterizes thestable cases. This condition is equivalent to a + 1 + c2 < 0, or, after some easycomputations, to Uλ > 2ηκ2νeνi. In the unstable case, the instability domain i.e.the set of wave-vectors ξ leading to unstable modes is contained in the ball centeredat 0 and of radius R0 with R2

0 = (a/2) + δ. As κ or η tend to 0, we notice thatR2

0 ∼ |Uλ|(2νeνi)−1(ηκ2)−1, which ends the proof of proposition 4.1.

The fact that the model is stable apart from a bounded region of wave-vectorscan be seen as a favorable feature. Indeed, in such a case, small wave-vector (i.e. longwave-length) perturbations first grow exponentially due to the instability, but alsoundergo a mode cascade towards higher wave-numbers due to nonlinearity. Once thewave-vectors are large enough to reach the stability region, they are damped by thedissipation. We therefore expect that only structures of typical size R−1

0 will remainfor long times.

However, the values of the physical parameters in the dynamo model are toosmall to ensure a viable stabilization process. Indeed, we see that R0 = O((κ2η)−1/2)when κ or η → 0. This is too large compared with the observations (see e.g. [13],


[14]). Therefore, another dissipation mechanism must be present. In this paper,we postulate that the turbulence of the plasma induced by the instability modifiesthe dissipation constants in a way similar to what happens in fluid mechanics (seee.g. [32] and references therein). To make this assumption more quantitative, in thenext section, we develop a statistical approach to turbulence adapted to the striationmodel.

5. A ’turbulent’ striation model.

5.1. Derivation of the ’turbulent’ striation model. To produce this newmodel, we follow the statistical approach to turbulence [32] (see also [28] for an ap-plication to MHD). We suppose that the unknowns (n, u, φ, h) in the striation model(2.9), (2.12) are random variables representing the possible realizations of the flow.Any of these quantities a can be decomposed according to a = a + a′ where a is itsmean value and a′ is a random fluctuation about this average. Since the random-ness concerns the realization of the flow, the mean value operator commutes with thespace and time derivatives. Therefore, we have: (a) = a, a′ = 0, ∇a = ∇a + ∇a′,∂ta = ∂ta + ∂ta

′. If b is a non-fluctuating quantity, we have ba = ba and for two ran-dom quantities a and b, ab 6= ab unless they are statistically independent. However,

we note that ab = ab.We assume that un and ν are non-fluctuating quantities. Under this assumption,

by averaging the striation model (2.9)-(2.12), we obtain:

∂tn + ∇ · (nu) = 0 , u = −∇φ , (5.1)

∇ · (nh) = 0 , h = −ν(u − un) × B . (5.2)

We can write nu = nu + n′u′ with n′u′ 6= 0, since n′ and u′ are in general notindependent random variables. In a same way, we have nh = −ν(nu+n′u′−nun)×B 6=nh.

To close the model, we need a prescription for the correlation n′u′ as a function ofthe mean quantities. As in fluid turbulence (see e.g. [32]), we model this correlationby means of a diffusion term acting on the density, i.e. n′u′ = −D∇n, where D > 0is a diffusion coefficient. The use of this Ansatz can be formally justified by invokingKesten-Papanicolaou’s theorem [27] (see also [32] for a review and [35] for a relatedresult). For simplicity, we assume that D is a constant. Under this assumption, andnoting that ∇ · (n′u′ × B) = −D∇ · (∇n × B) = 0, system (5.1)-(5.2) reduces to thefollowing (turbulent striation) model (dropping the bars):

∂tn + ∇ · (nu) −∇ · (D∇n) = 0 , u = −∇φ × B , (5.3)

∇ · (nh) = 0 , h = ν (−∇φ + un × B) . (5.4)

The difficulty is now to find the correct value for the diffusion coefficient D. Forthis purpose, we again proceed to a stability analysis, in a similar fashion as whatwas done in sections 3.2 and 4.

5.2. Stability analysis of the turbulent striation model. We again choose asteady state characterized by an exponential density profile and uniform neutral windun = (V, U) and electric field. The unperturbed state is defined by n0 = Nex2/λ,u0 = (V, 1/λ) = (−∂x2

φ0, ∂x1φ0). We introduce D such that D = |U |λD. We proceed

as in sections 3.2 and 4 and we get the following imaginary part of the dispersionrelation (with σ = sign(U)):

ℑm(ω) =ND , N = −D|ξ|2(|ξ|4 + ξ2

2) − (σ − D)ξ21 |ξ|2 , D = |ξ|4 + ξ2

2 , (5.5)


Since D ≥ 0, we just have to discuss the sign of N . We introduce the polar coordinatesξ1 = r cos θ and ξ2 = r sin θ. Then, N = −Dr4(r2 + sin2 θ − (1 − σD−1) cos2 θ). Thedomain I defined in polar coordinates by r ≤ r(θ) := (1 − σD−1) cos2 θ − sin2 θ1/2

(for all θ such that the expression inside the square root is positive) characterizes the(bounded) instability domain. We can summarize the results in the following

Proposition 5.1. (i) If σ = 1 (stable case for the original striation model) andD < 1, the turbulent striation model (5.3), (5.4) linearized about the above definedstationary states is stable.(ii) If (σ = 1 and D > 1) or σ = −1, the turbulent striation model is unstable forwave-vectors lying in the instability domain I. I is bounded and contained in the ballcentered at the origin and of radius (1 − σD−1)1/2.

We note this strange feature that adding too large a diffusion can destabilizethe striation model in the case where the unperturbed striation model is stable (caseσ = 1 and D > 1).

Thanks to this stability result, we can return to the problem of finding the valuefor D. Suppose that we know (from experimental observations for instance) that nostructures finer than a certain scale ℓ can persist. This means that all perturbationswith a wavelength less than ℓ are stable (i.e. are damped by dissipation), or equiva-lently, that all wave-vectors ξ larger than 1/ℓ lie in the stability domain. To ensurethis property, it is enough to have 1/ℓ > (1 + D−1)1/2 (we take σ = −1 because inpractice, there are always regions where the density gradient and the neutral windhave configurations which trigger the instability, see e.g. the numerical results in sec-tion 6). This condition translates into D ≥ ℓ2(1 − ℓ2)−1. In practice, it is legitimateto assume that ℓ ≪ 1 (because the typical size of the ultimate permanent structuresis far smaller than the typical size of the observation domain). Going back to theunscaled value of the diffusion constant D, we get

D≥≈ ℓ2λ|U | . (5.6)

In the next section, we present numerical simulations which display the relationbetween the typical size of the persisting structures triggered by the instability andthe value of this diffusion coefficient.

6. Numerical experiments. In this section, we present some numerical simu-lations of the striation model (2.9), (2.12) and of the turbulent striation model (5.3),(5.4). The elliptic equation (2.12) or (5.4) is discretized by a conservative finite differ-ence method. The plasma velocity is computed by means of finite differences appliedto the second equation of (2.9) or (5.3) on staggered grids. The transport equation(first equation of (2.9) or (5.3)) is discretized thanks to a classical TVD-scheme [17],[42], [30]. In order to deal with steep density gradients, the diffusion operator in (5.3)is implicitly discretized and we make use of a Strang splitting for the overall timediscretization of this equation. A preconditioned gradient method [40] is applied tosolve the linear systems resulting from the discretization of the elliptic equation (2.12)or (5.4) and from the implicit discretization of the diffusion equation (5.3).

Our first test problem is intended to mimic that of [44]. The initial density is arandom perturbation of a uniform density in the x1-direction with a Gaussian profilein the x2-direction (cf. Fig. 6.1). The neutral wind un is directed along the x2-axisand has a value of 45 ms−1. Different mesh sizes listed at Table 6.1 are considered.


−10 −5 0 5 100

5

10

15

20

2

4

6

8

10

x1 (km)

x2

(km

)

Fig. 6.1. Initial plasma density (m−3)

Nb. of cells ∆x, ∆y (m)Mesh 1 200 × 200 0.1 103

Mesh 2 400 × 400 0.05 103

Mesh 3 800 × 800 0.025 103

Table 6.1

Number of cells and mesh sizes

When the turbulent striation model is considered, the diffusion length ℓ (which setsthe value of the diffusion coefficient through (5.6)) is equal to 0.1 103 m (a scaleresolved by all mesh steps). In practice, its value should be prescribed by comparingwith experimental measurements (see for instance [8]). However, our purpose here istowards qualitative rather than quantitative results.

−10 −5 0 5 100

5

10

15

20

2

4

6

8

10

x1 (km)

x2

(km

)

(a) t = 196 s.

−10 −5 0 5 100

5

10

15

20

1

2

3

4

5

6

7

8

9

10

x1 (km)

x2

(km

)

(b) t = 392 s.

−10 −5 0 5 100

5

10

15

20

1

2

3

4

5

6

7

8

x1 (km)x

2(k

m)

(c) t = 588 s.

Fig. 6.2. Plasma density at various times given by the striation model discretized on mesh 1.

We first consider the original striation model (2.9), (2.12). Fig. 6.2 displays thetime evolution of the plasma density as a function of the 2-dimensional coordinatex. Periodic boundary conditions are used. We see that the upper side (with respectto the orientation of the figure) of the density gradient is unstable, while the lowerside is stable. The instability produces finger-like structures which rise in the positivex2-direction and eventually (by periodicity) appear as originating from the lowerboundary. On Fig. 6.3 we represent the plasma density computed on the different

−10 −5 0 5 100

5

10

15

20

1

2

3

4

5

6

x1 (km)

x2

(km

)

(a) Mesh 1 (200 × 200).

−10 −5 0 5 100

5

10

15

20

1

2

3

4

5

6

7

8

x1 (km)

x2

(km

)

(b) Mesh 2 (400 × 400).

−10 −5 0 5 100

5

10

15

20

1

2

3

4

5

6

7

x1 (km)

x2

(km

)

(c) Mesh 3 (800 × 800).

Fig. 6.3. Plasma density at t = 804 s given by the striation model discretized on the threedifferent meshes of Table 6.1.

meshes (see Table 6.1) at time t = 804 s. The mesh-size is divided by a factor 2from Fig. (a) to Fig. (b) and from Fig. (b) to Fig. (c). One can notice that the


number of persisting structures grows with the number of cells while their typicalsize decreases with the mesh-size. This remark can be made more quantitative onFig. 6.6 where the spectral density (i.e. the modulus of the Fourier transform) of theplasma density is displayed for the coarsest and finest meshes (dashed line curves).We can see that high frequency modes (corresponding to space scales ranging from2 to 5 times the value of ℓ) have a significantly larger contribution when the finestmesh is used. Correspondingly, on Fig. 6.3, we notice that the structures are all themore tinier that the mesh is finer.

This behavior can be related with the instability of the model. Indeed, numericaldiffusion is the only damping mechanism and the numerical diffusivity is proportionalto the mesh size [17], [42], [30]. According to the stability analysis in section 5.2, thediffusive striation model becomes stable for wave-vectors of the order of 1/

√D, which

is proportional to 1/√

∆x. Therefore, the size of the typical persisting structuresmust be divided by a factor

√2 each time the mesh-size is divided by 2. This is

roughly speaking what we observe on Fig. 6.5 where cuts of the density along linesx2 = Constant are plotted. The calculation carried out on the coarsest mesh (plainline of Fig. 6.5(a)) exhibits 5 to 6 main structures (area where the density variessignificantly) in the last quarter of the x1 range. In the same interval approximately14 main structures are counted for the density profile computed with the finest grid(plain line of Fig. 6.5(b)). Note that small patterns can exist in addition to thepersisting structures. Indeed there are respectively 7 and more than 20 local maximafor meshes 1 and 3 respectively. The same ratio is observed on Fig. 6.7 where thetime evolution of the number of local maxima is displayed. For long time evolution(greater than 600 sec.) the number of local maxima can be estimated as 25 for thecoarsest mesh and 85 for the finest one. These results confirm the mesh dependenceof the simulations.

−10 −5 0 5 100

5

10

15

20

2

3

4

5

6

x1 (km)

x2

(km

)

(a) Mesh 1 (200 × 200).

−10 −5 0 5 100

5

10

15

20

2

3

4

5

6

7

x1 (km)

x2

(km

)

(b) Mesh 2 (400 × 400).

−10 −5 0 5 100

5

10

15

20

2

3

4

5

6

x1 (km)

x2

(km

)

(c) Mesh 3 (800 × 800).

Fig. 6.4. Plasma density at t = 804 s, given by the turbulent striation model discretized on thethree different meshes of Table 6.1.

We next consider the turbulent striation model (5.3), (5.4). Fig. 6.4 demonstratesthe stability brought by the diffusion : the number and size of the structures remainalmost the same when the mesh resolution increases. The dashed line curves of Fig. 6.5display cuts of the density on a line x2 = Constant for the coarsest and finest meshesrespectively. The small patterns which could be observed on the results computedwith the classical striation model (plain line curves) have disappeared. Moreover,the number of local maxima (6 for the coarsest mesh, 7 for the finest one) observedon Fig. 6.5 are now quite independent of the grid resolution. This invariance of thenumber of local maxima with respect to the grid resolution can be observed in thecourse of the time evolution (Fig. 6.7). Indeed, this number remains almost constant


10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

x1 (km)

Den

sity

(m−

3)

Striation ModelTurbulent S.M.

(a) Mesh 1 (second half of the x1 range).

10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

x1 (km)

Den

sity

(m−

3)

Striation ModelTurbulent S.M.

(b) Mesh 3 (second half of the x1 range).

Fig. 6.5. Plasma density profiles along the line x2 = 2 km at time t = 392 s.

in time, and equal to 9 when the mesh size varies. These results therefore show asignificant difference between the turbulent striation model and the original one. Thespectral densities computed with the turbulent striation model are displayed on Fig.6.6. The diffusion damps the high frequency modes out and the curves computed withthe two different mesh sizes are very similar whatever space scales are considered, bycontrast with the behavior of the original striation model. Note that the characteristicsize of the striations observed on Fig. 6.4 and 6.5 amounts to a few kilometers, whichsuits well with the experimental observations.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

−3

10−2

10−1

100

Mesh 1 no diffusion

Mesh 3 no diffusion

Mesh 1 with diffusion

Mesh 3 with diffusion

Magnitude

Dimensionless spatial frequency.

Fig. 6.6. Spectral densityof the plasma density n com-puted on the coarsest and thefinest meshes of table 6.1 (modemagnitude versus dimensionlessspatial frequency). The spatialfrequency scale is set to ℓ−1

where ℓ, given by (5.6) is equalto 0.1 103 m.

The second simulation is aimed at illustrating the results of the stability analysisdeveloped in the discontinuous density profile framework (see section 3.3). To thispurpose, we consider a similar set of simulation parameters as above, except for theinitial density and the neutral wind. The initial density consists of a plasma bubble(density equal to one) in a quasi-vacuum medium (very small density). This initialdata is perturbed by a random noise. The neutral wind is oriented along the x1 axis; its speed is set to 100 m.s−1. Simulations performed on mesh 2 (table 6.1) with theclassical striation model are displayed on Fig. 6.8(a), 6.8(b) and 6.8(c) respectivelyat time t = 0 , 281.4 and 562.8 seconds. The plasma bubble is set into motion bythe neutral wind and since periodic boundary conditions are used, the bubble seemsto go out of the domain through the right boundary of the frame and to re-enterthe computational box through the left boundary. The instability develops along theright edge of the bubble, the other edge being unaffected.


200 300 400 500 600 700 800 900 1000

20

22

24

26

28

30

32

34

10

11

12

13

14

15

Time (sec).

Turb

ule

nt

S.M

.Str

iation

Model

x2 = 18 kmx2 = 1 kmx2 = 6 km

(a) Mesh 1 (200 × 200).

200 300 400 500 600 700 800 900 1000

40

45

50

55

7

8

9

10

11

12

13

14

Time (sec).

Turb

ule

nt

S.M

.Str

iation

Model

x2 = 18 kmx2 = 1 kmx2 = 6 km

(b) Mesh 2 (400 × 400).

200 300 400 500 600 700 800 900 1000

70

80

90

100

8

9

10

11

12

13

Time (sec).

Turb

ule

nt

S.M

.Str

iation

Model

x2 = 18 kmx2 = 1 kmx2 = 6 km

(c) Mesh 3 (800 × 800).

Fig. 6.7. Number of local maxima alongthe lines x2 = 18, 1, 6 km as a function oftime (circles, squares and triangles respec-tively) for the turbulent striation model (topfigures) and the classical one (bottom figures)computed on the three meshes of Table 6.1.The number of local maxima is computed ashalf the number of sign changes in the densityderivative.

−10 −5 0 5 100

5

10

15

20

0.2

0.4

0.6

0.8

1

1.2

x1 (km)

x2

(km

)

(a) Initial Density.

−10 −5 0 5 100

5

10

15

20

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x1 (km)

x2

(km

)

(b) t = 281.4 s.

−10 −5 0 5 100

5

10

15

20

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x1 (km)

x2

(km

)

(c) t = 562.8 s.

Fig. 6.8. Plasma density at various times given by the classical striation model discretized onmesh 2 in the case of a discontinuous initial density.

The same simulation run on mesh 1 produces the result displayed on Fig. 6.9(a)and shows the sensitivity of the instability pattern with respect to the grid resolution.The last two pictures 6.9(b) and 6.9(c) show calculations performed with the turbu-lent striation model on mesh 2. The diffusion parameter used on Fig. 6.9(c) is fourtimes as big as the one considered for 6.9(b). When comparing Fig. 6.8(c) (with-out any diffusion) and Fig. 6.9(b) we get the same conclusion as before : diffusionbrings stability for small space scales, since the tiniest patterns have disappeared fromFig. 6.9(b). More diffusion can also brings stability for all space scales and prevents


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(a) Striation model on Mesh 1.

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(c) Turbulent SM with high dif-fusion.

Fig. 6.9. Plasma density at t = 562.8 s given by the classical striation model on mesh 1(left), the turbulent striation model with a moderate diffusion on mesh 2 (middle) and the turbulentstriation model with a large diffusion on mesh 2 (right) with the discontinuous initial density of Fig6.8 (left).

the growth of the instability, as demonstrated by the results of Fig. 6.9(c).

7. Conclusion. In this paper, we have been concerned with the modeling ofionospheric plasma instabilities. The first main point of this work was to remarkthat the ’striation model’ allows for discontinuous solutions and that discontinuoussteady-states may be unstable in a similar way as smooth ones. The second pointwas to propose that the turbulence induced by the instability may actually producediffusion, in a similar way as what occurs in fluid mechanics, and that this diffusionmay actually contribute to stabilize large wave-number perturbations. Following thestatistical approach to turbulence, we have derived and analyzed a ’turbulent striationmodel’. Numerical simulations have been produced in support to our analysis.

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