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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
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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 .
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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Observational Motivation - THEMIS
Hartinger et al., [2012]
Frequency 6.5mHz.
Dominant bz and Ey .
Radially inward Sx .
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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) .
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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]
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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].
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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 .
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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Results - Cluster Simulation
bz
by
bx
uy∼−Ex
ux∼Ey
Sz
Sy
Sx
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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Reminder - THEMIS observation
Hartinger et al., [2012]
Frequency 6.5mHz.
Dominant bz and Ey .
Radially inward Sx .
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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Results - components - THEMIS Simulation
bz
by
bx
uy∼−Ex
ux∼Ey
Sz
Sy
Sx
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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 .
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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.
Tom Elsden, A.N. Wright MHD Simulations in magnetospheric waveguides
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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
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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.
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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)
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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 MHD Simulations in magnetospheric waveguides