Slow evolution of magnetic potential fields in barotropic

Transcription

1 Slow evolution of magnetic potential fields in barotropic ideal MHD flows Dieter Nickeler Astronomical Institute, Ondřejov in collaboration with Marian Karlický

2 Overview Motivation: Magnetic structures in pre-flare stages

3 Overview Motivation: Magnetic structures in pre-flare stages Assumptions and basic (evolution) MHD equations: Is it possible to find barotropic flows which enable a sequence of magnetic potential fields?

4 Overview Motivation: Magnetic structures in pre-flare stages Assumptions and basic (evolution) MHD equations: Is it possible to find barotropic flows which enable a sequence of magnetic potential fields? Possible solution methods and preliminary results: Characteristics and restriction to general solution of ideal Ohm s law

5 Overview Motivation: Magnetic structures in pre-flare stages Assumptions and basic (evolution) MHD equations: Is it possible to find barotropic flows which enable a sequence of magnetic potential fields? Possible solution methods and preliminary results: Characteristics and restriction to general solution of ideal Ohm s law Problems and Outlook

6 What we want: Aims: (i) We would like to find expressions for restrictions for the (general) solution of the ideal Ohm s law using characteristic method (problem of correct boundary and initial conditions)

7 What we want: Aims: (i) We would like to find expressions for restrictions for the (general) solution of the ideal Ohm s law using characteristic method (problem of correct boundary and initial conditions) (ii) We want to find stable/asymptotical time independent and/or unstable/collapsing solutions which could serve as models for pre-flare stages

8 What we want: Aims: (i) We would like to find expressions for restrictions for the (general) solution of the ideal Ohm s law using characteristic method (problem of correct boundary and initial conditions) (ii) We want to find stable/asymptotical time independent and/or unstable/collapsing solutions which could serve as models for pre-flare stages (iii) First attempt to to tackle this problem analytically and dynamically and not only magnetically or kinematically

9 Interesting example: Example of magnetic structures (potential fields) in the solar corona: Plasma β is small: p Pa, B 10 2 T, thus β = p/(b 2 /2µ 0 ) thus pressure gradient is zero, therefore Euler equation can be neglected, but unfortunately if we are in the vicinity of magnetic neutral points...??? plasma β pressure gradient cannot be neglected

10 Assumptions and basic MHD equations Assumptions new: The nonlinear part of the Euler equation should be neglected but not the partial derivative, implying v v S, v 0 and / t 0 assuming a barotropic law p = p(ρ) Solving the problem in pure 2D, / z = 0, but all variables are in principle time dependent ( parametrically time dependent)

11 Assumptions and basic MHD equations Basic MHD equations I ρ t + ρ v = 0, ρ v = t p, A t + v A = 0, A = 0, where B = (A e z ) = A e z = φ m p = p(ρ) = v = 0 = v = ϕ

12 Assumptions and basic MHD equations Basic MHD equations and assumptions Neglecting the nonlinear term in the mass continuity equation: ϕ t 2 ϕ ϕ t ( ϕ ) ϕ, Thus it is at least no contradiction to the neglection of the ( v ) v-term in the Euler equation, as v ( ) t v v ϕ t ( ϕ ) ϕ.

13 Assumptions and basic MHD equations Basic MHD equations and assumptions But is it then really justified to reduce the mass continuity equation to: ρ t + ρ v = ρ t + ρ ϕ = 0? We assume that it is justified with respect to the hydrodynamical equations.

14 Assumptions and basic MHD equations Basic MHD equations II ρ t + ρ ϕ = 0, ϕ t = f ; f (ρ) := ϕ A = A t, A = 0, Z p (ρ) ρ dρ

15 Assumptions and basic MHD equations Basic MHD equations III ρ t + ρ ϕ = 0, f (ρ) = ϕ t ρ = 1 f ϕ A = A t, A = 0, ( ϕ t ) (.ϕ ) g

16 Assumptions and basic MHD equations Basic MHD equations IV g (. ϕ).. ϕ +g(. ϕ) ϕ = 0, ρ = g(. ϕ), ϕ A = A t, A = 0.

17 Example: The standard form of a potential field in the vicinity of a magnetic neutral point With A(x,y,t) = A 0 (t)xy we find the general solution of ideal Ohm s law by applying the characteristic method with ξ = x 2 y 2. A x ϕ x + A y ϕ y = A t A 0 y ϕ x + A 0x ϕ y = Ȧ 0 xy ϕ = f (ξ,t) G 2 x2, G :=. A0 A 0

18 Example: The isothermal approach Let Equation of change (state) p(ρ) = k B µ ρt 0 p (ρ) = k B µ T 0 f (ρ) = k ( ) BT 0 ρ µ ln ρ 0 [ g(. ϕ) = ρ 0 exp =. ϕ. µ ϕ 0. ϕ +. ϕ 0 k B T 0 ]

19 Example: The isothermal approach Resulting differential equation 1 v 2 S.. ϕ + ϕ = 0, which can be derived from [ g( ϕ).. ϕ +. ] ϕ 0 = ρ 0 exp µ k B T 0 µ g( ϕ). ϕ.. +g( ϕ) ϕ. = 0 k B T 0

20 Isothermal approach General solution of the differential equation Having in mind that G Ȧ 0 /A 0 = G 1 t + G 0 ( ) G1 A 0 = A 00 exp 2 t2 + G 0 t for a decaying magnetic field, i.e. G 1 < 0 and for large times (t ) ρ = f unction o f space and time exp ( t 2) magnetic field will decay, complete region around the null point will be evacuated

21 Polytropic approach With p = Kρ γ, γ = (C C p )/(C C V ) we get: Resulting differential equation.. ϕ +(γ 1). ϕ ϕ = 0. we insert the general solution ϕ = f (ξ,t) G 2 x2...

22 Preliminary and incomplete results. 1. G= 0 and f = 0, exponentially unstable (good for flares (?) ) if G 0 < 0 and γ < 1 or G 0 > 0 and γ > 1 delivers with respect to time unbounded solutions of the flow

23 Preliminary and incomplete results. 1. G= 0 and f = 0, exponentially unstable (good for flares (?) ) if G 0 < 0 and γ < 1 or G 0 > 0 and γ > 1 delivers with respect to time unbounded solutions of the flow. 2. G= 0, ḟ = 0 and f 0, ϕ = f (ξ) G 0 x 2 /2 (stationary flow)

24 Preliminary and incomplete results. 1. G= 0 and f = 0, exponentially unstable (good for flares (?) ) if G 0 < 0 and γ < 1 or G 0 > 0 and γ > 1 delivers with respect to time unbounded solutions of the flow. G= 0, ḟ = 0 and f 0, ϕ = f (ξ) G 0 x 2 /2 2. (stationary flow) 3.. G 0 and f = 0, (i) oscillating flux function, but diverging non-parallel parts of the flow (finite time singularity tan(t)) or (ii) finite time singularities of the parallel flow cosh(t), but bounded flux function tanh(t) (iii) flux function and non-parallel flow velocity obey finite time singularity 1/(t t 0 )

25 Conclusions: In the vicinity of magnetic null points there can exist several kinds of breakdown of slow or quasi-static approach: 1. Due to exponential instabilities

26 Conclusions: In the vicinity of magnetic null points there can exist several kinds of breakdown of slow or quasi-static approach: 1. Due to exponential instabilities 2. Due to finite time singularities

27 Problems and Outlook: Question: what determines the finite time intervall of the finite time singularities?

28 Problems and Outlook: Question: what determines the finite time intervall of the finite time singularities? How do the solutions look like if we allow the null point to move (at least linearly)?

29 Problems and Outlook: Question: what determines the finite time intervall of the finite time singularities? How do the solutions look like if we allow the null point to move (at least linearly)? How to connect the general solution with the characteristics for the velocity potential with the nonlinear mass continuity equation?

30 Problems and Outlook: Question: what determines the finite time intervall of the finite time singularities? How do the solutions look like if we allow the null point to move (at least linearly)? How to connect the general solution with the characteristics for the velocity potential with the nonlinear mass continuity equation? Finding solutions for the non-linear problem, extending the Syrovatskii solutions to systems with null points and non-vanishing pressure gradients

31 Problems and Outlook: Question: what determines the finite time intervall of the finite time singularities? How do the solutions look like if we allow the null point to move (at least linearly)? How to connect the general solution with the characteristics for the velocity potential with the nonlinear mass continuity equation? Finding solutions for the non-linear problem, extending the Syrovatskii solutions to systems with null points and non-vanishing pressure gradients More general barotropic law instead of a polytropic law