Modeling of striations instabilities and numerical simulations

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1 1 Modeling of striations instabilities and numerical simulations C. Besse MIP, Université Paul Sabatier, 118 route de Narbonne, Toulouse cedex, France

2 Joint work with 2 Model derivation, turbulence modeling and numerical simulations F. Deluzet, P. Degond (MIP, Toulouse) J. Claudel, G. Gallice and Ch. Tessieras CEA-CESTA, Le Barp, France Nonlinear instability P. Degond, R. Poncet (MIP, Toulouse) H.-J. Hwang (Duke University)

3 Outline 3 Physical motivations Derivation of a hierarchy of models Striation model in uniform and non uniform magnetic field Numerical simulations Instabilities of the Striation model

4 Physical context and motivations 4 Ionosphere long wave UHF VHF AM radio space wave ground wave sky wave FM radio, TV TV, phones short wave sky wave ground wave Ionosphere : 90 to 1500 km Reflects radio waves : useful for AM radio, long range communications... Many irregularities (aurora, solar eruptions, striations... ) : disturbances of high frequency communications earth-satellites. Focus on striations : long time (many hours) and big scale (100km) irregularities. Striations increase the attenuation and scintillation encoutered by satellite to ground systems

Physical context and motivations Northern Hemisphere Evolution of a natural or artificial plasma bubble at altitude around 700 km m k 00 1 5 km 15 wind Ionosphere Troposphere Sea level Cosmic Ray

5 Physical context and motivations Northern Hemisphere Evolution of a natural or artificial plasma bubble at altitude around 700 km m k km 15 wind Ionosphere Troposphere Sea level Cosmic Ray Time The plasma bubble is stretched ments Southern Hemisphere along the magnetic field lines. In a plane orthogonal, the plasma bubble is under the influence of the E B instability. The plasma is bent in this plane. Ionospheric Plasma Instabilities 6th MAFPD, Kyoto September 2004

6 Hierarchy of models : euler-maxwell model 6 The modeling is based on the Euler-Maxwell equations with simplifications Only two type of charged particles : O + and electrons Euler equations. No gravity, no energy equations (replace by p e,i = p e,i (n e,i )), no chemical reactions, low electron-ion collisions. Magnetic field B and electric field (dynamo effect) E Maxwell equations. Some parameters are small : m e /m i, u n /c... rescaling Scaling of the Euler-Maxwell equations six dimensionless parameters remain ε α κ τ η β Electron to ion mass ratio squared reciprocal of light speed number of e-n or i-n collisions per rotation period in B-field Mean-time between i-n collisions (dimensionless) Measure of the thermal energy Drift energy relative to magnetic energy measures the strength of the magnetic field perturbation

7 Hierarchy of models : scaled euler-maxwell system 7 t n e + (n e u e ) = 0, τε (Inertia ek ) = η xk p e κ 1 n e (E k + (u e B) k ) ν e n e (u ek u nk ), t n i + (n i u i ) = 0, τ (Inertia ik ) = η xk p i + κ 1 n i (E k + (u i B) k ) ν i (u ik u nk ). α t E B = βj, t B + E = 0, κα β E = ρ, B = 0, ρ = n i n e, κj = n i u i n e u e. Typical values of the parameters for density n i,e = m 3 ε = 10 4, τ = 10 1, η = 10 1, κ = 10 4 α = 10 12, β =

8 Hierarchy of models : scaled euler-maxwell system 8 Euler-Maxwell ε 0, α 0 MHD hierarchy Hall-MHD Dynamo hierarchy κ 0 τ 0 Finite conductivity-mhd Massless Hall-MHD τ 0 β 0 Massless MHD Dynamo β 0 κ 0 Striation MHD hierarchy : valid for large density perturbations Dynamo hierarchy : valid for standard situations

9 MHD hierarchy : massless mhd 9 Plasma is quasi-neutral, ν = ν e + ν i, u := u i = u e, n := n i = n e n t + (nu) = 0, η p(n) = j B νn(u u n ), E + u B = 0, B = βj, B t + E = 0, B = 0. E, u and j can be eliminated from the system gives rise to a system of two equations for n and B.

10 MHD hierarchy : massless mhd 10 Massless MHD : equivalent formulation n t + (nu) = 0, B + (ub Bu) = 0, t with the constraint B = 0 and u given by 1 β (BB) + ( ηp(n) + 1 β ) B 2 2 = νn(u u n ).

11 Dynamo hierarchy : dynamo model 11 Plasma is quasi-neutral n := n i = n e B verifies B = 0 and B = 0 B = B earth B/ t = 0 n t + (nu i) = 0, [ E + u i B = κ ν i (u i u n ) + η ] n p i(n). [ E + u e B = κ ν e (u e u n ) + η ] n p e(n). j = 0, E = 0, κj = n(u i u e ).

12 Multi-Layer Striation model : uniform B field : 12 MHD or Dynamo model always depend on κ or β : are always 3D : limits κ 0 or β 0 Uniform B : B = B x 3 x = (x 1, x 2 ), = ( x1, x2 ), A = (A 1, A 2, A 3 ), A = (A 1, A 2 ). n, u depend on the 3-D coord x n = n(x, t), u = u(x, t) E is orthogonal to B and derives from a potential V : E = (E, 0), E = V V depends on the 2-D coord x V = V (x, t)

13 Multi-Layer Striation model : uniform B field : 13 Suppose p = 0 (for simplicity) n t + (nu) = 0 u = E B ( ) B B + u B 2 n B B ; E = V (x) J = 0, J = 1 B ( σ(x) V + U 2 n B), σ(x) = nν dx 3, U n = nνu n dx 3. σ/ B 2 : field-integrated Pedersen conductivity 3D transport, 2D elliptic

14 Uniform B field : proof 14 Equivalent formulation of the Dynamo model t n + (nu e ) = 0, t n + (nu i ) = 0, u e = M e ( E + κ(ν e u n η p e n )), u i = M i (E + κ(ν i u n η p i n )), E = V, j = 0, κj = n(u i u e ), M e = µ P e,i = µ P e µ H e 0 µ P i µ H i 0 µ H e µ P e 0, M i = µ H i µ P i µ e 0 0 µ i κν e,i (κν e,i ) 2 + B 2, µh e,i = µ e,i = 1 κν e,i. B (κν e,i ) 2 + B 2, M e (resp. M i ) : electron (resp. ion) mobility matrix, We also have j B = n(ν i u i + ν e u e ) nu n (ν i + ν e ).

15 Uniform B field : proof 15 The limit κ 0 : κj = n(u i u e ) u i = u e lim µ P i = lim µ P e = 0, lim µ H i = lim µ H e = 1 κ 0 κ 0 κ 0 κ 0 B, u 1 = x1 V/ B, u 2 = x2 V/ B. third component of the relation leads to kν i u i3 = B x3 V + κν i u n3 x3 V = 0 and u 3 = u n,3. ν = ν i + ν e j B = nν(u u n ) gives the components of j orthogonal to B j = 1 B nν x 1 V + u n2 B 2 x2 V u n1 B

16 Uniform B field : proof 16 j = 0 j 1 x 1 + j 2 x 2 + j 3 x 3 = 0. j 1 and j 2 known a first order differential equation for j 3 integration on the bounded interval [x 3,min, x 3,max ] x 3,min and x 3,max taken inside the neutral atmosphere j 3 vanishes

17 Monolayer striation model in uniform B : 17 Suppose all quantities only depend only on x Suppose u n3 = 0 2D model n t + ( (nu) ) = 0, n V = B 2 ( n u ) n B B 2. with u = E B B 2, E = V u = 0.

18 Non uniform B field : 18 A uniform B field corresponds to cut the earth and make an artificial rectification of a magnetic field tube Extension to a non uniform B differential geometry

19 Non uniform B field : 19 In spherical coordinates, axisymmetric B = (B r (r, ϕ), 0, B ϕ (r, ϕ)) Since B = B = 0, we have (rb ϕ ) r (B r) ϕ = 0 and (r2 sin ϕb r ) r + (r sin ϕb ϕ ) ϕ = 0. β(r, ϕ), γ(r, ϕ), s.t. r β = r sin ϕb ϕ, ϕ β = r 2 sin ϕb r, r γ = B r, ϕ γ = rb ϕ. Transform to (α, β, γ) coordinate system α = θ Orthogonal curvilinear local coordinate system associated to B y z B B ϕ ϕ r B r earth x L earth x r θ y z FIG. 1: The spherical coordinates associated to the earth surrounded by a magnetic field line. (α, β, γ) are known as Euler potentials

Non uniform B field : 20 4 β=constant γ=constant 3 2 ts 1 α β γ 0 1 2 3 4 0 1 2 3 4 5 6 7 8 9 10 FIG.

20 Non uniform B field : 20 4 β=constant γ=constant 3 2 ts 1 α β γ FIG. 2: Generalized coordinates associated to the magnetic field. FIG. 3: Local coordinates associated to a magnetic field tube.

21 In the (α, β, γ) coordinate system, we have Non uniform B field : 21 ds 2 = 1 B 2 (r2 sin 2 ϕ B 2 dα 2 + dβ2 r 2 sin 2 ϕ + dγ2 ), ( ) 1 f f =, r sin ϕ B f r sin ϕ α β, B f, γ ( ( ) A = B 2 A α + ( ) r sin ϕ A α r sin ϕ B 2 β + ( )) 1 β B γ B A γ ( ( ) fγ r sin ϕ ( )) f β β B γ r sin ϕ B ( A = 1 r sin ϕ B γ (r sin ϕf α) ( )) fγ ( ( α B ) 1 f β ( )) fα B α r sin ϕ B β B,

22 Non uniform B field : 22 γ corresponds to x 3 B-field lines = {(α, β) = Constant } n, u depend on the 3-D coord (α, β, γ) n = n(α, β, γ, t), u = u(α, β, γ, t) E is orthogonal to B and derives from a potential V : E γ = 0, E = V V is constant along magnetic field lines : V = V (α, β, t)

23 Non uniform B field : striation model 23 n t + (nu) = 0, u α = r sin ϕ V β, u β = 1 r sin ϕ B V α, u γ = u nγ, J α α + J β β = 0, J α = A α V α U nβ, J β = A β V β + U nα, A α = U nα = γmax γ min γmax γ min nν dγ r 2 sin 2 ϕ B 4, A β = nνu nα r sin ϕdγ B 2, U nβ = γmax γ min γmax nν r2 sin 2 ϕdγ B 2, γ min nνu nβ dγ r sin ϕ B 3.

24 About the models : 24 Massless MHD : valid for large density perturbations Striation model : Valid for standard situations Striation model : widely used by physicists e.g. Ronchi, Similon, Farley (1989) Zalesak, Ossakov, Chaturvedi (1982) Extension to non uniform B-fields possible

25 Numerical simulations : euler potentials 25 Computations of the Euler potentials Earth dipole field B = µ 0 M (2 cos(ϕ)ˆr + sin(ϕ) ˆϕ), 4π r3 with M = A m 2, µ 0 the vacuum permeability β = µ 0M sin 2 ϕ, γ = µ 0M cos ϕ 4π( r ) 4π r 2 2 ( B 2 µ0 M 4γ 2 = 4π r + β ) 2 r 5.

26 Numerical simulations : scheme 26 Let c = ( α, β ) t and assume u nγ = 0. The multi-layer striation model can be rewritten as : n t + v c n = ns n, c (A c V ) = c J n, with V v = β V α A = A α 0, J n = 0 A β, S n = B 2 β ( 1 U nβ U nα B 2. ) V α,

27 Numerical simulations : scheme 27 n t + v c n = ns n, c (A c V ) = c J n. linear part : a classical convection velocities computed with the electric potential V S n : source term including the magnetic field curvature effects If ν > 0, elliptic equation for the electric potential V with a source term.

28 Numerical simulations : scheme 28 Discretization of the transport equation : Strang splitting method in time and directional splitting method in space On a half time step t n + v c n = 0, On a time step t n = ns n, On a half time step t n + v c n = 0, to ensure the positivity of the density n k+1 = n ( k 1 + t Sn) k, if S k n > 0, n k+1 = n [ k 1 t Sn] k 1, otherwise. linear transport equation : a TVD upwind flux limiter scheme (Ultrabee) Computation of the potential : big linear system to solve

29 Numerical simulations : data 29 computational domain : magnetic field tube immersed in the earth ionosphere the tube is divided into 61 layers central layer located on the equator spreads over 1000 km in the α direction lower and higher altitudes : 700 km and 1900 km each plan is meshed with a Cartesian grid : nodes The neutral wind blows eastward in the α direction with a constant velocity of 45 m s 1.

30 Numerical simulations : data 30 ts Ionosphere upper layer Ionosphere lower layer Magnetic field tube Computational domain FIG. 4: Configuration of a test case. FIG. 5: Mesh of a field tube.

Numerical simulations : data 31 eplacements x 10 11 4.5 PSfrag replacements 4 3.5 3 2.5 2 1.5 1 0.5 0 Layer 57 Layer 39 Layer 25 Layer 1 Layers 2, 4,... 58. FIG.

31 Numerical simulations : data 31 eplacements x PSfrag replacements Layer 57 Layer 39 Layer 25 Layer 1 Layers 2, 4, FIG. 6: Slices of a 61 layers discrete field tube. Height (from the earth ground in km) Day plasma density Night plasma density x Plasma density (m 3 ) FIG. 7: Ionospheric ambient plasma.

32 Numerical simulations : data 32 FIG. 8: Initial datum

33 Instabilities of the striation model : 33 Stability analysis of the striation model : phenomenological point of view We consider the monolayer striation model in a uniform B field, B = 1 n t + (nu) = 0 u = V, V = ( x2 V, x1 V ) (n V ) = (nu n ) v = u n = neutral wind (given) after π/2 rotation (n V ) = (nu n ) (nh) = 0 with h = V u n n, x 2 < 0, discontinuous density n(x) = n > n, x 2 > 0, Stable state : V = 0, u n = (0, u n2 ), h = (u n2, 0) x 2 is defined as the direction of the gradient of n.

34 Instabilities of the striation model : 34 Recall that u = V and h = V u n. x 2 u n u n n > x 2 u n u n n > u E u E E u E E u x 1 n < Since u = E B B 2 u E, striations = E B instability. u x 1 n <

35 Instabilities of the striation model : 35

36 Some mathematical results : 36 Local existence in H k, k > 3 Th. Let n 0 H k (R 2 ), n 0 κ > 0. T, a solution (n, u) in C([0, T ], H k ) Lip([0, T ], H k 1 ) Tools : Banach fixed point theorem Energy estimates Linear stability analysis Assumption : u n = (U(x), 0) U of constant sign, smooth, bounded Stationary solution ( n(x), ū = 0) n smooth, bounded, n κ > 0 Instability hypothesis : x 0 such that (U x n) x0 > 0

37 Linearized system Some mathematical results : 37 n = n + η, u = ψ t η + x n y ψ = 0 u = ψ ( n ψ) = U y η Plane wave solutions η(x, y) = η(x) exp(iky) exp(λ k t) ψ(x, y) = ψ(x) exp(iky) exp(λ k t) Growth rate of linear system The growth rate λ k is bounded by a constant Λ. Th. Λ controls the growth rate of solutions of the linearized system : η(t) H s + ψ(t) H s C η(0) H s (1 + t s )e Λt

38 Some mathematical results : 38 Nonlinear instability k 3, ( n(x), 0) steady state. v = (U(x), 0) s.t. x 0, (U x n) x0 > 0 ε 0, δ > 0 small, (n δ, V δ ) solution, T δ > 0 s.t. n δ (0) n H k δ and n δ (T δ ) n L 2,L + V δ (T δ ) L 2,L ε 0 Proof : Follows Grenier, Guo-Strauss, Hwang-Guo,...

Asymptotic-Preserving Schemes for anisotropic diffusion problems related to magnetized plasma simulation

Asymptotic-Preserving Schemes for anisotropic diffusion problems related to magnetized plasma simulation Fabrice Deluzet 1 joint work with C. Besse 2, S. Brull 3, A. Crestetto 1, P. Degond 1, A. Lozinski