Kinetic equilibrium for an asymmetric tangential layer

1. Kinetic equilibrium for an asymmetric tangential layer Gérard BELMONT and Nicolas Aunai LPP EcolePolytechnique, Palaiseau, France Belmont_Alpbach2011

2. Why one needs a kinetic model At least two good reasons : 1. Theoretical Much interest in thin layer stability (atkineticscales). Anystudy of the stability of an equilibriumdemands first to know thisequilibrium 2. Practical Initializing a PIC or hybrid simulation withsuch a layer alsodemands to know itsequilibrium to avoidspuriousevolutionfrom the verybeginning Belmont_Alpbach2011

3. Tangentialcurrentlayers Bn= 0 Equilibriumwhich possible profiles ? (B, n, u, p, …) In fluidtheories(e.g.resistive MHD), onlyconstraint on the pressure profile: p+B2/2 = cst. In kinetictheory, Vlasovmuch more constraining f(v) profiles to bechosen for verifyingit No generaltheory for building a kineticequilibrium solution (Harris = simple but particular) Models of a tangential layer(magnetopause-like) B2 n B1 Belmont_Alpbach2011

4. B reversal  tearinginstability In MHD, easy to buildequilibriummodels analytical and numericalstudies of the growth rate as a function of width, beta, etc., In kinetics, similarstudies for a Harris sheet, but almostnothingelse startingfrom a trueequilibrium, in particular for magnetopause-likelayers How does the instabilitydepends on the profiles? !All stability analyses questionable if not startingfrom a truekineticequilibrium. Theory of currentsheetstability B2 n B1 Belmont_Alpbach2011

5. Whennothingbetter Initializingwith local Maxwellians (from a fluidequilibrium) Not a kineticequilibrium wavesemitted by the layer, from the first gyroperiod Initialization of a kinetic simulation How do thesewavesinterferwith the triggering of an instability? Belmont_Alpbach2011

6. Harris = symmetric solution withvacuum on bothsides Harris equilibriumis not adaptedfor magnetopause-likelayers Harris n duz Magnetopause j n y duz=cst currentlocalized if and only if the densityislocalized Currentlocalized becauseduzislocalized The densityisneitherzeronorsymmetric on bothsides duz j Belmont_Alpbach2011 y

7. Modified Harris sheets Possible to add a constant particle population everywhere to avoidhaving vacuum on bothsides,  anotherequilibrium, but… for a given y f n no v y • stillsymmetric • with a possibly non realistic distribution function (stable?) Belmont_Alpbach2011

8. Use the invariants of the particle motion In 1-D: E=mvx2/2+mvy2/2+mvy2/2 and pz = mvz+qA (and px=mvx) Takef(vx, vy, vz) = g(E, pz)  grantst(f)=0  : Vlasovequilibriumwhatever the functiong Fill full (E, pz) spacewith an "arbitrary" analyticalfunctiong (univaluate) with few free parameters. Onlygeneralconstraint in the choice of g(E, pz)  : it must tend to a givenMaxwelliango(E) when pz tends toinfinity. Do it for electrons and protons separately (generallywith the sameanalyticalfunctiong)  full kineticequilibrium. Usualmethod for building a kineticequilibrium(Harris, Channel, Mottez, etc.) • Example of Harris equilibrium : • g = e-E/T e-bpz • for bothspecies Belmont_Alpbach2011

9. 1. Profiles of B, p, n,etc… cannotbeprescribed a priori Input = functiong(E, pz)  : which one?? Profiles = output (unveiledat the end of the calculation) Contrary to what one needs: givencurrentsheet model function 2. All profiles are alwayssymmetric Drawbacks of the usualmethod(whynothingelsethan Harris isneverused in practice,neither for theorynor for simulation) Belmont_Alpbach2011

10. Ateachgiveny, all the distribution functionf(vx, vy, vz) corresponds to the interior of a parabola centered in qAzin the (E, pz) spacesince E> mvz2/2 = (pz-qAz)2/2. Explainswhy: f can change with y (displacement of the parabola due to the change in A) if g univaluatefunction of (E,p)  f(v)=symmetric  go to multivaluateg for building asymmetricequilibria Change from (vx, vy, vz) to (E, pz)  accessibility qAz Belmont_Alpbach2011

11. Method: Fixthe B profile; choose an analyticalform for g of ions withenoughfree parameters; choosethe parametersdifferent on eachside for granting : _theright pressure profile (derivingfrom the j profile and an hypothesis on je) _theright asymptoticMaxwellians _thesame value in the central parabola (withcontinuity) Building an asymmetricequilibrium for ions • Shared central population • One different distribution on each side ! • Maxwellian boundaries LEFT • Local symmetry inevitable if continuity RIGHT Belmont_Alpbach2011

12. Analyticalformchosen This form has the advantage to allow semi-analyticalworksince the moments such as P(A) have the sameanalyticaldependence The coefficients gpsi are the parameters to bedetermined. Theirnumberis free Some are determined by an easylinear system from the twonecessary conditions : • Total pressure profile imposed • Equality of the distribution functions of bothsides and of their first derivatives in the center of the layer • The others are determined by a minimizationprocess to get a solution as close as possible to anyprescribed one (extra-criterion) • An infinity of solutions canbefoundso. Belmont_Alpbach2011

13. Infinity of solutions, but withsomecommoncharacteristics: Density profiles obtained Oscillatingdensity "gyrotropy" around the normal y (perp to B): Pxx=Pzz Belmont_Alpbach2011

14. Initial distribution= local Maxwellians Initial density profile=odd Evolution  waves, but: appearance of a local maximum atB=0 and oscillations around Pxx=Pzz  Similarproperties as the model Evolution of the profiles after a non equilibriumMaxwellianinitialization Belmont_Alpbach2011

15. Kinetic solution: density and total pressure profiles = stationary The calculated solution as initial condition of an hybrid code Same initial profiles for the macroscopic moments, but Maxwellian distributions: profiles = evolving Belmont_Alpbach2011

16. New method to buildkinetic solutions for the magnetopause-liketangentialcurrentlayers. Current profile fixed. Density profile = result (many solutions) Density profile = in agreement withnaturalevolution. Theorydemandsmultivaluatefunctions for g(E,p) to getasymmetricprofiles Conclusions Belmont_Alpbach2011