An Introduction to Waves in Earth's Magnetospheretelsden/talks/Solar_Seminar_3.pdf · An Introduction to Waves in Earth’s Magnetosphere Tom Elsden A.N. Wright University of St Andrews

An Introduction to Waves in Earth’s Magnetosphere

Tom Elsden A.N. Wright

University of St Andrews

18th March 2015

Tom Elsden, A.N. Wright An Introduction to Waves in Earth’s Magnetosphere

Overview

Introduction to the Magnetosphere.

Ultra Low Frequency (ULF) Waves.

What we do with these waves.


What I’d like you to know by the end...

Have an appreciation for the overall structure of the magnetosphere.

ULF Waves - theory, generation, observation.

Modeling - how to model ULF waves in the magnetosphere.


Magnetosphere


Magnetosphere - Structure


So now you know...

Bow Shock

Magnetosheath

Magnetopause

Van Allen Belts

Plasmasphere/pause

Magnetotail


Magnetic Waves/Pulsations

Low frequency oscillations in Earth’s magnetic field.

First discovered in the mid 1800s using a microscope and a verylong compass needle.

Typical magnetic oscillations of around a few nT .

ULF waves cover frequencies from 1 mHz to 1 Hz .

Importance: can drive currents in the ionosphere.

Understanding the dynamics of the magnetosphere.


Where do they come from?

Solar wind- originate at sun, or through variations in the solar winddensity, causing dynamic pressure fluctuations.

Kelvin-Helmholtz Instability- Heightened solar wind flow speeds leadto increased magnetosheath flow and the magnetopause can becomeKH unstable on the flanks.

Other types e.g. internally generated waves.


Cavity/Waveguide Modes - Discrete Frequencies

The outer magnetosphere can be thought of as a natural waveguide.

The magnetopause provides an outer boundary, with theplasmapause as a possible internal boundary.

Samson et al. [1992] found that similar discrete frequencies keptoccurring in the outer magnetosphere.

This developed the idea of the cavity selecting the frequencies basedon it’s size and shape, just like an instrument.

Wright and Rickard [1995] showed discrete frequencies excited by abroadband frequency driver.


Wave Coupling and Resonance

In a cold uniform plasma, Alfven and fast waves don’t couple.

In the magnetosphere, the nonuniformity couples these modes.

Where the Alfven frequencyequals the fast modefrequency, can excite a fieldline resonance (FLR).

This has detectableobservational features.


Theory - Hydromagnetic Box Model

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

Wright & Rickard 1995

Uniform background magnetic field B = B z, picture straighteningdipole fieldlines.

x radially outwards, y azimuthal coordinate (around earth).

Let ρ = ρ(x).


Theory - MHD Equations

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

⇒ Energy equation is zero.

∂B

∂t= ∇× (u× B) ,

ρ∂u

∂t+ ρ (u ·∇) u =

1

µj× B−��*∇p +��*ρg,

∂ρ

∂t+∇ · (ρu) = 0.


Theory - Governing Equations Simplified

Linearise and assume perturbations as bz(x)e i(kyy+kzz−ωt)

d2bz

dx2−ω2dV−2

A /dx

ω2/V 2A − k2

z

· dbz

dx+

(ω2

V 2A

− k2y − k2

z

)bz = 0

Second order ordinary differential equation for bz(x), the zcomponent of the magnetic field.

Have assumed a density variation only in x .

Singularity when ω2 = V 2Ak2

z , at a specific location in x .

Represents location where energy from longitudinal waves isconverted into transverse waves, like waves along a string.


Theory - System of PDEs

We’re interested in numerical modeling so express as

∂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]


Waveguide & Boundary Conditions

Waveguide of length 1 in x , and 10 in y . 1 unit = 10Re .

Perfectly reflecting boundary at x = 0 (plasmapause).

Driven boundary at x = 1 (magnetopause).

Symmetry condition at y = 0 ⇒ uy = 0.

Open ended in y → wave never reaches end of box in y .


Driving with bz .

Drive with bz perturbation, rather than displacement as previously,to mimic pressure driving.

Changes the boundary condition at the magnetopause: closer to anode of bz than ux .

Can provide a more realistic range of waveguide eigenfrequencies(natural guide frequencies) [Mann et al., 1999].

Figure shows radialwaveguide harmonicsfor a uniform densitymedium.


Applying the Model - Aims

Take an observation from a paper.

Attempt to model the equilibrium from the given parameters.

Match satellite positions.

Ask ourselves: Can we match to their results?

Can we question their conclusions?

What new information/explanation do we have?


Applying the Model - The Observation

[Hartinger et al., 2012]

Observation fromTHEMIS

Frequency 6.5mHzglobal mode.

Dominant bz and Ey

perturbations.

Radially inwards Sx .

THD location:magnetic latitude ∼ 3◦,near plasmapause,radially aligned withsource region.


Applying the Model - THEMIS Results

bz

by

bx

uy∼−Ex

ux∼Ey

Sz

Sy

Sx

[Elsden and Wright, 2015]Tom Elsden, A.N. Wright An Introduction to Waves in Earth’s Magnetosphere

Applying the Model: Phase Shifts

Difference between driving and post driving phase shifts.

Can be used in observations to infer the end of the driving phase.



Wanted to analyse the phase shift together with changes in Sx .

Sx = uxbz .

We’re a theoretical group, so time for some theory!

Consider an inward propagating wave + smaller amplitude reflectedwave.

ux = cos(ωt + kxx − kyy) cos(kzz)

+ R cos(ωt − kxx − kyy) cos(kzz).




Sx = uxbz .








Sx = uxbz .








Sx = uxbz .








Sx = uxbz .







Let ky = 0, as an approximation for a global mode.

Close to magnetic equator at z = 0.

Substitute form for ux into equations.

ux = cos(ωt + kxx) + R cos(ωt − kxx),

bz = −A′ cos(ωt + kxx) + A′R cos(ωt − kxx),

for A′ = kx/ω.



Let ky = 0, as an approximation for a global mode.

Close to magnetic equator at z = 0.

Substitute form for ux into equations.

ux = cos(ωt + kxx) + R cos(ωt − kxx),

bz = −A′ cos(ωt + kxx) + A′R cos(ωt − kxx),

for A′ = kx/ω.



Try to express components in terms of a single sinusoid.

ux = G cos(ωt + ψ),

bz = G ′ cos(ωt + ψ′),

G ,G ′, ψ, ψ′ all functions of kxx and R.

Then we can define the phase shift as

φ = π + tan−1

(2R

1− R2sin(2kxx)

).



Try to express components in terms of a single sinusoid.

ux = G cos(ωt + ψ),

bz = G ′ cos(ωt + ψ′),

G ,G ′, ψ, ψ′ all functions of kxx and R.

Then we can define the phase shift as

φ = π + tan−1

(2R

1− R2sin(2kxx)

).



Now to compare this to the Poynting vector.

Sx = uxbz ,

= −A′ cos2(ωt + kxx) + R2A′ cos2(ωt − kxx).

Again express as a single sinusoid.

Sx = γ + C sin(2ωt + δ),

for γ, C and δ all functions of kxx and R.



Now to compare this to the Poynting vector.

Sx = uxbz ,

= −A′ cos2(ωt + kxx) + R2A′ cos2(ωt − kxx).

Again express as a single sinusoid.

Sx = γ + C sin(2ωt + δ),

for γ, C and δ all functions of kxx and R.



Consider the ratio of positive to negative Poynting vector signal,defining the ’shape’.

∆s =

∣∣∣∣∣R2 − 1 +√

R4 + 1− 2R2 cos(4kxx)

R2 − 1−√

R4 + 1− 2R2 cos(4kxx)

∣∣∣∣∣ .

So we have expressions for the phase shift between ux and bz , andthe ’shape’ of the radial Poynting vector. How are they related?



Consider the ratio of positive to negative Poynting vector signal,defining the ’shape’.

∆s =

∣∣∣∣∣R2 − 1 +√

R4 + 1− 2R2 cos(4kxx)

R2 − 1−√

R4 + 1− 2R2 cos(4kxx)

∣∣∣∣∣ .So we have expressions for the phase shift between ux and bz , andthe ’shape’ of the radial Poynting vector. How are they related?



Well it turns out they have the same contours in (R, kxx) space.Awesome right!?

What does this mean? How can we show this? ∇φ×∇∆s = 0

This means that one can be expressed as a function of the other andhence shows how these two quantities are linked.

So can think of the phase difference determining the shape of thePoynting vector and vice versa.




What does this mean? How can we show this?

∇φ×∇∆s = 0






















Maple check


Maple check


Maple check


What was the point?

The driven and undriven phases can be distinguished either throughSx shape or the phase shift of ux and bz .

Know what phase shift / radial Poynting vector ratios are expectedfor global modes ⇒ useful for global mode identification in data.

Contour plots of the 2 quantities can constrain values of reflection Rand kxx . E.g. if phase shift = 120◦ then 0.6 < R < 1.


What was the point?





What was the point?





What have we learned from the model?

Confirmation of global mode signature.

Despite model simplifications still a very good match to the data.

Correlated phase shifts with Poynting vector signals.

Can infer satellite position in reference to the energy sourcelocationthrough the Poynting vector.

Displays the power of even very simplified models in revealinginformation not immediately apparent from the data.


Conclusions

So I said I’d like you to have learned something...

Magnetospheric structure

ULF waves: generation, theory, observations.

Wave modeling...

...and hopefully you had a good sleep for the last part!


Conclusions




Wave modeling...



Conclusions




Wave modeling...



Thanks!


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.


Bibliography I

T. Elsden and A. N. Wright. The use of the Poynting vector in interpreting ULFwaves in magnetospheric waveguides. Journal of Geophysical Research (SpacePhysics), 120:166–186, January 2015. doi: 10.1002/2014JA020748.

M. Hartinger, V. Angelopoulos, M. B. Moldwin, Y. Nishimura, D. L. Turner,K.-H. Glassmeier, M. G. Kivelson, J. Matzka, and C. Stolle. Observations ofa Pc5 global (cavity/waveguide) mode outside the plasmasphere by THEMIS.Journal of Geophysical Research (Space Physics), 117:A06202, June 2012.doi: 10.1029/2011JA017266.

I. R. Mann, A. N. Wright, K. J. Mills, and V. M. Nakariakov. Excitation ofmagnetospheric waveguide modes by magnetosheath flows. J. Geophys. Res,104:333–354, January 1999. doi: 10.1029/1998JA900026.

J. C. Samson, B. G. Harrold, J. M. Ruohoniemi, R. A. Greenwald, and A. D. M.Walker. Field line resonances associated with MHD waveguides in themagnetosphere. Geophys. Res. Lett, 19:441–444, March 1992. doi:10.1029/92GL00116.


Bibliography II

A. N. Wright and G. J. Rickard. A numerical study of resonant absorption in amagnetohydrodynamic cavity driven by a broadband spectrum. Ap. J., 444:458–470, May 1995. doi: 10.1086/175620.

S. T. Zalesak. Fully multidimensional flux-corrected transport algorithms forfluids. Journal of Computational Physics, 31:335–362, June 1979. doi:10.1016/0021-9991(79)90051-2.


Documents

An Introduction to Waves in Earth's Magnetospheretelsden/talks/Solar_Seminar_3.pdf · An Introduction to Waves in Earth’s Magnetosphere Tom Elsden A.N. Wright University of St Andrews