tom elsden a.n. wright - university of st andrewstelsden/talks/elsden_buks... · 2015-06-01 · tom...

Simulations of MHD waves in Earth’s magnetosphericwaveguide

Tom Elsden A.N. Wright

University of St Andrews

26th May 2015

Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides

Overview

Introduction to Ultra Low Frequency (ULF) waves in Earth’smagnetosphere - 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.


Introduction to ULF waves

ULF waves are a type of MHD wave that propagate in themagnetosphere.

They manifest on the ground as small oscillations (a few nT) inEarth’s magnetic field.

Specified frequency range of 1 mHz - 1 Hz or periods of 1s - 1000s.

Important for:

Transport of energy throughout the magnetosphere.Coupling together of different regions.Contribute to the magnetospheric current system.


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 ofoscillation of the waveguide.


Observational Motivation - Cluster

bz

by

bx

Ez

Ey

Ex

Sz

Sy

Sx

Clausen et al., [2008]

x - radial, y -azimuthal, z -field aligned.

Cluster location:magnetic lat∼ 12◦, nearplasmapause,downtail of sourceregion.

Frequency17.2mHz.

Strong by , bz andEx .

Purely tailwardSy .


Observational Motivation - THEMIS

Hartinger et al., [2012]

Frequency 6.5mHz.

Dominant bz and Ey .

Radially inward Sx .


Model & Theory

We model the magnetosphere using a waveguide based on thehydromagnetic box implemented by Kivelson and Southwood [1986].

Wright and Rickard, [1995]

Uniform background magnetic field B = B z.

x radially outwards, y azimuthal coordinate.

Let ρ = ρ(x) ⇒ VA = VA(x) .


Model & Theory - MHD Equations

Ideal low beta plasma (low pressure). (Cold plasma equations)

∂B

∂t= ∇× (u× B) ,

ρ∂u

∂t+ ρ (u ·∇) u =

1

µj× B−��*∇p +��*ρg,

∂ρ

∂t+∇ · (ρu) = 0.


Model & Theory - System of PDEs

For numerical modeling we express the system as 5 first order PDEs:

∂bx

∂t= −kzux ,

∂by

∂t= −kzuy ,

∂bz

∂t= −

(∂ux

∂x+∂uy

∂y

),

∂ux

∂t=

1

ρ

(kzbx −

∂bz

∂x

),

∂uy

∂t=

1

ρ

(kzby −

∂bz

∂y

),

which we solve using a Leapfrog-Trapezoidal finite difference scheme.[Zalesak, 1979]


Model & Theory - Boundary Condition for DrivenBoundary

Implement a new boundary condition on the driven magnetopauseboundary - different to previous works.

Drive with bz perturbation to mimick pressure driving, the mainsource of ULF waves [Takahashi and Ukhorskiy, 2008].

Can prove this gives a node of bz (antinode ux) at driven boundary.

Yields a quarter wavelength fundamental radial mode.

Can reduce fundamental eigenfrequencies without resorting tounphysical higher plasma densities [Mann et al., 1999].


Reminder - Cluster Observation

bz

by

bx

Ez

Ey

Ex

Sz

Sy

Sx

Clausen et al., [2008]

x - radial, y -azimuthal, z -field aligned.

Cluster location:magnetic lat∼ 12◦, nearplasmapause,downtail of sourceregion.

Frequency17.2mHz.

Strong by , bz andEx .

Purely tailwardSy .


Results - Cluster Simulation

bz

by

bx

uy∼−Ex

ux∼Ey

Sz

Sy

Sx


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 theoriginal paper.

Indications:

Strong bz throughout event.

by correlates with bz , rather than being persistent post driving.

Satellite position relative to driven region gives Sy stand out feature.


Reminder - THEMIS observation

Hartinger et al., [2012]

Frequency 6.5mHz.

Dominant bz and Ey .

Radially inward Sx .


Results - components - THEMIS Simulation

bz

by

bx

uy∼−Ex

ux∼Ey

Sz

Sy

Sx


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 Sx , i.e. ratio of in to out signal.

Can infer incident and reflection coefficients given phase shift andshape of Sx .


Results - What have we shown?

Just as with Cluster observation - a good match to the observeddata.

Description matches as a fast waveguide mode interpretation.

Can infer source location relative to the spacecraft from thePoynting vector components i.e. THEMIS spacecraft must be withinazimuthal extent of the driven region.

Length of driving phase - we know when we stop driving.


Results - Satellite Position

Position A models the positionof THD.

Locations B-D display thechange in Sx signature frominward to outward.

Sy further downtail is alwaystailward.

B

C

D


Conclusions

Designed a model for ULF waves in Earth’s outer magnetosphere.

Developed a new boundary condition to effectively drive withpressure.

Have modeled two different observations from Cluster and THEMISsatellites.

Our simple simulation could match to the main features of theobservations.

Could infer information about the driving phase, source location andwave mode.

Hence can use as a test of observational hypotheses.


Finite Difference Method

Can express equations in the form

∂U

∂t= F

where

U =

ux

uy

bx

by

bz

, F =

(kzbx − bz , x) /ρ(kzby − bz , y) /ρ

−kzux

−kzuy

− (ux , x + uy , y)


Finite Difference Method

Assuming we know U at times t and t −∆t, then the scheme is

U† = Ut−∆t + 2∆tFt

F∗ =1

2

(Ft + F†

),

Ut+∆t = Ut + ∆tF∗.

Use centered finite differences to calculate the spatial derivatives.

Scheme is second order accurate in time and space.


tom elsden a.n. wright - university of st andrewstelsden/talks/elsden_buks... · 2015-06-01 · tom...

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