Theoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions

Transcription

1 Theoretical Foundation of 3D Alfvén Resonances: Time Dependent Solutions Tom Elsden 1 Andrew Wright 1 1 Dept Maths & Stats, University of St Andrews DAMTP Seminar - 8th May 2017

2 Outline Introduction Coordinates Model Results Introduction to Alfvén waves and resonances. Motivation for study - applications in solar/earth plasma physics. Numerical considerations. Main 3D results.

3 Magnetohydrodynamics (MHD) MHD describes a coupling of the electromagnetic and hydrodynamic equations.

4 Magnetohydrodynamics (MHD) MHD describes a coupling of the electromagnetic and hydrodynamic equations. Key assumptions: 1 Typical plasma velocities much less than speed of light. 2 Generally large length and time scales (low frequencies) compared to particle gyroradii and gyrofrequencies.

5 Magnetohydrodynamics (MHD) MHD describes a coupling of the electromagnetic and hydrodynamic equations. Key assumptions: 1 Typical plasma velocities much less than speed of light. 2 Generally large length and time scales (low frequencies) compared to particle gyroradii and gyrofrequencies. Describes plasma behaviour in terms of large scale fields, without the complicated kinetic description involving particle distribution functions.

6 Equations of MHD B t = (u B) + η 2 B, Induction (1) ρ u + ρ (u ) u t = p + j B + ρg, Motion (2) ρ + (ρu) t = 0, Mass continuity (3) B = 0, Solenoidal constraint (4) p + u p t = γp u, Energy (5) p = ρrt, Gas law (6) j = 1 µ 0 B, Ampère s law (7)

7 MHD Waves Consider small perturbations about an equilibrium can linearise equations.

8 MHD Waves Consider small perturbations about an equilibrium can linearise equations. Simplified case: uniform medium, B = B 0 ẑ. Leads to the magnetoacoustic dispersion relation (e.g. Roberts, 1985): ( ω 2 kz 2 VA 2 ) ( ω 4 ω 2 ( cs 2 + VA 2 ) k 2 + k 2 kz 2 cs 2 VA 2 ) = 0.

9 MHD Waves Consider small perturbations about an equilibrium can linearise equations. Simplified case: uniform medium, B = B 0 ẑ. Leads to the magnetoacoustic dispersion relation (e.g. Roberts, 1985): ( ω 2 kz 2 VA 2 ) ( ω 4 ω 2 ( cs 2 + VA 2 ) k 2 + k 2 kz 2 cs 2 VA 2 ) = 0. 3 solutions here for 3 wave modes: fast, slow and Alfvén.

10 MHD Waves Consider a magnetically dominated plasma (low β, cold), dispersion relations are: ω f = kv A, Fast mode ω A = k z V A. Alfvén mode

11 MHD Waves Consider a magnetically dominated plasma (low β, cold), dispersion relations are: ω f = kv A, Fast mode ω A = k z V A. Alfvén mode Fast: associated with magnetic pressure, propagates in all directions, causes density variations (compressible).

12 MHD Waves Consider a magnetically dominated plasma (low β, cold), dispersion relations are: ω f = kv A, Fast mode ω A = k z V A. Alfvén mode Fast: associated with magnetic pressure, propagates in all directions, causes density variations (compressible). Alfvén: associated with magnetic tension, motions transverse to magnetic field, no density variations (incompressible), energy along field.

13 Role in Earth s Magnetosphere Hartinger et al Ultra-low frequency (ULF - 1 mhz to 1 Hz) waves observed on the ground for 150 years. Couple different magnetospheric regions. Important in radiation belt dynamics, auroral processes.

14 Role in a Solar Context Waves believed to be an important heating mechanism in the solar corona. Hundreds of studies of coronal loop oscillations, both observational and numerical.

15 Research Developments over 50 years - 1D In a uniform medium, MHD wave modes are decoupled. However, non-uniformity can couple the modes. Consider a 1D density variation, say ρ = ρ(x): d 2 b z dx 2 ω2 dv 2 A /dx ( ) db z ω 2 ω 2 /VA 2 k2 z dx + VA 2 ky 2 kz 2 b z = 0. (Southwood, 1974) Singular point where ω 2 /V 2 A k2 z = 0 - Alfvén wave resonance. Turning point where ω2 k 2 VA 2 y kz 2 = 0.

16 Alfvén waves - 1D resonances 1D resonance simulations (Mann et al., 1995). Energy accumulation at resonance site x = Phase mixing to smaller scales over time.

17 Relation to Magnetosphere Figure: Mann et al., 2008

18 2D Alfvén Resonances - Cartesian Cartesian geometry, 2D variation in the density (Alfvén speed V A (x, y)), B = B 0 ẑ. Natural Alfvén frequency ω A (x, y). Figure: Energy density showing resonance path [Russell and Wright, 2010] Resonant field lines where ω A (x, y) = ω d. Alfvén frequency is independent of polarisation.

19 2D Alfvén Resonances - Curvilinear Curvilinear field-aligned coordinates (α, β, γ). Resonance has toroidal polarisation, such that ω Aβ = ω d. Poloidal and toroidal frequencies are not the same! ( ) 1 U β + 1 ( ln γ h γ γ h γ γ ( hβ h α )) Uβ γ + ω2 A β VA 2 h γ U β = 0. e.g., Singer et al, 1981

20 Purpose of Study To develop a formalism for resonances in 3D, when the Alfvén frequency varies in 3D, and to study their development and dominant characteristics.

21 2D Dipole Coordinates - Field Aligned Let B = ψ = (A z ez) = ( A z ) z Solutions to 2D Laplace equation (st B = 0) in cylindrical coords yield ψ = B 0R 2 0 R A z = A 0R 0 R sin φ, cos φ.

22 2D Dipole Coordinates - Field Aligned (New!) α = R cos φ β = z γ = R g tan 1 ( Rg R sin φ ) h α = cos 2 φ h β = 1 ( ) R 2 h γ = + sin 2 φ R g

23 2D Dipole Coordinates - Why? Figure: (a) Classical dipole Figure: (b) New dipole Plotted are equally spaced contours of (a) ψ,a z and (b) α,γ.

24 Linearized low β MHD Equations U α t U β t B α t B β t B γ t ( 1 + Λ 2 ) 2 = VA 2 α 2 /Rg 2 + Λ 2 = VA 2 1 α 2 /Rg 2 + Λ 2 1 U α = α 2 /Rg 2 + Λ 2 γ, ( ) 1 + Λ 2 2 U β = α 2 /Rg 2 + Λ 2 γ, ( α 2 = R 2 g [ Bα γ B γ α [ Bβ γ B γ β ) [ + Λ 2 Uα α + U β β ] νu α, ] νu β, ]. U α = u α h β B, U β = u β h α B, B α = b α h α, B β = b β h β, B γ = b γ h γ, Λ = α R g tan γ R g.

25 2D Test Case Domain: α : , β : , γ : (φ = 0.2) Grid: (40, 40, 40), staggered. Essentially a weakly non-uniform cube like domain. Solve equations using the Leapfrog-Trapezoidal finite difference method (Zalesak, 1979).

26 Boundary Conditions Reflecting: α min, γ max Symmetry Condition at γ min. Reflecting at β min, β max. Driven at α max with perturbations of b γ (β, t).

27 Energy Conservation dw dt = ρνu 2 dv = L, W (t) + Ldt = c.

28 What happens in 3D? Extend domain: α : , β : , γ : (φ = 0.7) Grid: (160, 240, 60), staggered. Dissipative regions: β : , β : Alfvén speed dependent on β. Long wavelength driver, monochromatic (b γ ).

29 Introduction Coordinates Model Results 3D Alfve n Resonance Structures - Early Time (a) (b) (c) (d) Elsden and Wright, 2017 Tom Elsden 1 Andrew Wright 1 Theoretical Foundation of 3D Alfve n Resonances: Time Dependent Solutions

30 3D Alfvén Resonance Structures - Late Time (a) (b)

31 Why the chosen contour shape? β Resonance Map Resonant Zone Alfvén Wave Equation ( 1 γ + 1 h γ γ ) U β h γ γ ( ( )) hβ Uβ ln h α γ α + ω2 A VA 2 h γ U β = 0. 2D resonant regions favour particular contours. Can imagine stacking of resonance maps for various driving frequencies.

32 Resonance Time Signatures and FFTs u β u β (a) Time (c) FFT Power FFT Power (b) Cyclic Frequency (d) Time Cyclic Frequency

33 Introduction Coordinates Model Results Natural Alfve n Frequencies (a) (b) Field lines with natural frequencies close to the driven frequencies are initially excited before being damped by dissipation. Tom Elsden 1 Andrew Wright 1 Theoretical Foundation of 3D Alfve n Resonances: Time Dependent Solutions

34 Conclusions Introduction Coordinates Model Results Developed a theory for 3D Alfvén resonances. Resonances in 3D can form over a resonant zone: influenced by 2D resonant regions & boundary conditions. [Wright & Elsden, 2016]. Construct by solutions to the Alfvén wave equation. Time dependent case supported by normal mode (e iωt ) results. At early times, phase mixing leads to prominent ridges in energy density.

35 Thanks :) Introduction Coordinates Model Results

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37 Finite Difference Method Assuming we know U at times t and t t, then the scheme is U = U t t + 2 tf t F = 1 ( F t + F ), 2 U t+ t = U t + tf. Use centered finite differences to calculate the spatial derivatives. Scheme is second order accurate in time and space. May require another algorithm to get started if 2 timesteps are not known.

38 Analysis - Temporal Resonance Width X (τ 0 ) = ω(τ 0), Mann et al., [1995] ω A Confirmed that analysis from 1D carries over in 3D. Steady state width clearly reached.

39 Analysis - Spatial Resonance Width Dashed - simulation, solid - estimate.

40 Analysis - Resonance Amplitude Physically significant for local energy accumulation. Extend original 2D analysis which shows good agreement. b γ (α, β, γ) b γ0 (0, β, γ), ξ β (α, β, γ) β 0 α ξ β A (0, β, γ), Bh α ξ β (h A γ b γ0 )/ β β 0 = 1 2µ 0 ω d ρξβ 2 h A α h β h γ (0,β ) (0,β ) ( ) 1 dωa dα. (0,β )