Simulations of MHD waves in Earth s magnetospheric waveguide

Transcription

1 Simulations of MHD waves in Earth s magnetospheric waveguide Tom Elsden A.N. Wright University of St Andrews 26th May 2015

2 Overview Introduction to Ultra Low Frequency (ULF) waves in Earth s magnetosphere - why are we interested in them? Examples of satellite observations to motivate the theory. Treating ULF waves with an MHD waveguide model. Simulation results and conclusions.

3 Introduction to ULF waves ULF waves are a type of MHD wave that propagate in the magnetosphere. They manifest on the ground as small oscillations (a few nt) in Earth s magnetic field. Specified frequency range of 1 mhz - 1 Hz or periods of 1s s. Important for: Transport of energy throughout the magnetosphere. Coupling together of different regions. Contribute to the magnetospheric current system.

4 Generation of ULF waves Variety of processes: Changes in solar wind dynamic pressure, buffeting the magnetopause. Kelvin-Helmholtz instability in the magnetospheric flanks. Upstreaming ions at the bow shock. We focus here on a specific kind of ULF wave - a waveguide or cavity mode. This wave form can be thought of as one of the natural modes of oscillation of the waveguide.

5 Observational Motivation - Cluster b z b y b x E z E y E x S z S y S x Clausen et al., [2008] ˆx - radial, ŷ - azimuthal, ẑ - field aligned. Cluster location: magnetic lat 12, near plasmapause, downtail of source region. Frequency 17.2mHz. Strong b y, b z and E x. Purely tailward S y.

6 Observational Motivation - THEMIS Frequency 6.5mHz. Hartinger et al., [2012] Dominant b z and E y. Radially inward S x.

7 Model & Theory We model the magnetosphere using a waveguide based on the hydromagnetic box implemented by Kivelson and Southwood [1986]. Wright and Rickard, [1995] Uniform background magnetic field B = Bẑ. ˆx radially outwards, ŷ azimuthal coordinate. Let ρ = ρ(x) V A = V A (x).

8 Model & Theory - MHD Equations Ideal low beta plasma (low pressure). (Cold plasma equations) B t = (u B), ρ u t + ρ (u ) u = 1 µ j B p + ρg, ρ + (ρu) t = 0.

9 Model & Theory - System of PDEs For numerical modeling we express the system as 5 first order PDEs: b x t b y t b z t u x t u y t = k z u x, = k z u y, ( ux = x + u ) y, y = 1 ( k z b x b ) z, ρ x = 1 ( k z b y b ) z, ρ y which we solve using a Leapfrog-Trapezoidal finite difference scheme. [Zalesak, 1979]

10 Model & Theory - Boundary Condition for Driven Boundary Implement a new boundary condition on the driven magnetopause boundary - different to previous works. Drive with b z perturbation to mimick pressure driving, the main source of ULF waves [Takahashi and Ukhorskiy, 2008]. Can prove this gives a node of b z (antinode u x ) at driven boundary. Yields a quarter wavelength fundamental radial mode. Can reduce fundamental eigenfrequencies without resorting to unphysical higher plasma densities [Mann et al., 1999].

11 Reminder - Cluster Observation b z b y b x E z E y E x S z S y S x Clausen et al., [2008] ˆx - radial, ŷ - azimuthal, ẑ - field aligned. Cluster location: magnetic lat 12, near plasmapause, downtail of source region. Frequency 17.2mHz. Strong b y, b z and E x. Purely tailward S y.

12 Results - Cluster Simulation b z b y b x u y E x u x E y S z S y S x

13 Results - What have we shown? Evident similarities between our model results and the data. Signal mostly dependent on satellite location. Have an interpretation of the signals as a fast waveguide mode, rather than a field line resonance (FLR) as hypothesized in the original paper. Indications: Strong b z throughout event. b y correlates with b z, rather than being persistent post driving. Satellite position relative to driven region gives S y stand out feature.

14 Reminder - THEMIS observation Frequency 6.5mHz. Hartinger et al., [2012] Dominant b z and E y. Radially inward S x.

15 Results - components - THEMIS Simulation b z b y b x u y E x u x E y S z S y S x

16 Results - Phase Shifts - THEMIS Simulation Difference between driving and post driving phase shifts. Can be used in observations to infer the end of the driving phase. Correlated with the shape of S x, i.e. ratio of in to out signal. Can infer incident and reflection coefficients given phase shift and shape of S x.

17 Results - What have we shown? Just as with Cluster observation - a good match to the observed data. Description matches as a fast waveguide mode interpretation. Can infer source location relative to the spacecraft from the Poynting vector components i.e. THEMIS spacecraft must be within azimuthal extent of the driven region. Length of driving phase - we know when we stop driving.

18 Results - Satellite Position Position A models the position of THD. Locations B-D display the change in S x signature from inward to outward. S y further downtail is always tailward. C B D

19 Conclusions Designed a model for ULF waves in Earth s outer magnetosphere. Developed a new boundary condition to effectively drive with pressure. Have modeled two different observations from Cluster and THEMIS satellites. Our simple simulation could match to the main features of the observations. Could infer information about the driving phase, source location and wave mode. Hence can use as a test of observational hypotheses.

20 Finite Difference Method Can express equations in the form where U t = F u x (k z b x b z, x) /ρ u y U = b x b y, F = (k z b y b z, y) /ρ k z u x k z u y b z (u x, x + u y, y)

21 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.