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Magnetohydrodynamics

Conservative form of MHD equations

Covection and diffusion

Frozen-in field lines

Magnetohydrostaticequilibrium

Magnetic field-aligned currents

Alfvn waves

Quasi-neutral hybrid approach

Basic MHD

(or energy equation)

(isotropic pressure)

( )

Relevant Maxwellsequations; displacementcurrent neglected

Note that in collisionless plasmas and may need to be

introduced to the ideal MHDs Ohms law before

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Some hydrodynamicsConservation of mass density:

Navier-Stokes equation(s):

Energy equation:

we can write these in the conservation formIn case

is the momentum density

is a tensor with components

thermal energy density

kinetic energy density

primitive variables

conserved variables

viscosity, if

these are calledEuler equation(s)polytropic index

MHD: add Ampres force

Now the energy density contains also the magnetic energy density

In MHD we neglect the displacement current, thus

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In case of the diffusion becomes very slowand the evolution of B is completely determinedby the plasma flow (field is frozen-in to the plasma)

convection equation

The measure of the relative strengths of convectionand diffusion is the magnetic Reynolds numberRm

Let the characteristic spatial and temporal scales be

& and the diffusion time

In fully ionized plasmasRm is often very large. E.g. in the solar windat 1 AU it is 1016 1017. This means that during the 150 million kmtravel from the Sun the field diffuses about 1 km! Very ideal MHD:

The order of magnitude estimates for the terms of the induction equation are

and the magnetic Reynolds number is given by

This is analogous to the Reynolds number in hydrodynamics

viscosity

Example: Diffusion of 1-dimensional current sheet

in the frame co-moving with the plasma (V = 0)

Initially:

has the solution

Total magnetic flux remains constant = 0,but the magnetic energy density

decreases with time

Now

Ohmic heating, or Joule heating

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Photosphere partiallyionized, collisions with neutrals cause resisitivity:

Photospheric granules

Observations of the evolution of magnetic structuressuggest a factor of 200 larger diffusivity, i.e., smallerRm

Explanation: Turbulence contributes to the diffusivity:

Example: Conductivity and diffusivity in the Sun

This is an empirical estimate,

nobody knows how to calculate it!

Corona above 2000 km the atmosphere becomes fully ionized

Spitzers formula for the effective electron collision time

numerical estimate for the conductivity

Estimating the diffusivity becomes

For coronal temperature

Home exercise: EstimateRm in the solar wind close to the Earth

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Frozen-in field linesA concept introduced by Hannes Alfvn but laterdenounced by himself, because in the Maxwelliansense the field lines do not have physical identity.

It is a very useful tool, when applied carefully.

Assume ideal MHD and consider two plasma elementsjoined at time tby a magnetic field line and separated by

In time dtthe elements move distances udt and

In Introductionto plasma physics it was shown that the plasma elementsare on a common field line also at time t + dt

In ideal MHD two plasma elements that are on a common field line remain ona common field line. In this sense it is safe to consider moving field lines.

Another way to express the freezing is to showthat the magnetic flux through a closed loopdefined by plasma elements is constant

B

CS

S

C dl

dl x VDt

Closed contour Cdefined by plasma elements at time t

The same (perhaps deformed) contour C at time t + t

Arc element dl moves in time tthe distance Vtandsweeps out an area dl x Vt

Consider a closed volume defined by surfaces S, S andthe surface swept by dl x Vtwhen dl is integratedalong the closed contour C.

As B = 0, the magnetic flux through the closed surfacemust vanish at time t + t, i.e,

Calculate d/dtwhen the contour Cmoves with the fluid:

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The integrand vanishes, if

Thus in ideal MHD the flux through a closed contour defined byplasma elements is constant, i.e., plasma and the field move together

The critical assumption was the ideal MHD Ohms law. This requires that theExB drift is faster than magnetic drifts (i.e., large scale convection dominates).As the magnetic drifts lead to the separation of electron and ion motions (J),the first correction to Ohms law in collisionless plasma is the Hall term

so-calledHall MHD

It is a straightforward exercise to show that in Hall MHD the magnetic field

is frozen-in to the electron motion and

When two ideal MHD plasmas with different magnetic field orientations flowagainst each other, magnetic reconnection can take place. Magneticreconnection can break the frozen-in condition in an explosive way andlead to rapid particle acceleration

Example of reconnection:

a solar flare

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Magnetohydrostatic equilibriumAssume scalar pressure and

i.e., B and J are vector fields onconstant pressure surfaces

Write the magnetic force as:

magnetic

pressure

magnetic

stressand torsion

Magnetohydrostatic equilibrium

after elimination of the current

Assume scalar pressure andnegligible (BB)

Plasma beta:

Diamagneticcurrent: this is the way how B reacts on the presenceof plasma in order to reach magnetohydrostatic equilibrium.

This macroscopic current is a sum of dri ft currents and of the magnetizationcurrent due to inhomogeneous distribution of elementary magnetic moments(particles on Larmor motion)

dense

plasma

total

current

tenuous

plasma

The current perpendicular to B:

In magnetized plasmas pressure and temperature can be anisotropic

Define parallel and perpendicular pressures as and

Using these we can derive themacroscopic currents fromthe single particle drift motions

where

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The magnetization is

Summing up JG, JCand JMwe get the total current

Magnetohydrostatic equilibrium in anisotropic plasma is described by

In time dependent problems the polarization current must be added

In case of the pressure gradientis negligible and the equilibrium is

Now is constant along B

If is constant in all directions, the equationbecomes linear, the Helmholtz equation

Force-free fields a.k.a.

magnetic field-aligned currents

i.e., no Ampres force

The field and the currentin the same direction

Flux-rope:

is a non-linear equation

If and are two solutions, does not need to be another solution

Note that potential fields are trivially force-free

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Projection of field lines in thexy-plane are straigth linesparallel to each other

The solutionlooks like:

Arcs in thexz-plane

Case of so weak current that we can neglect it (= 0):Potential field (no distortion due to the current)

Solve the two-dimensional Laplace equation

Look for a separable solution

This is fulfilled by

Magnetic field lines are now arcs

without the shear in they-direction

Grad Shafranov equation

Oftena 3D structure isessentially 2D(at least locally)

or an ICME

e.g. the linear force-free arcade above

Consider translational symmetryB uniform inz direction

Assume balance btw. magneticand pressure forces

Thez component reduces to

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Now also

These are valid if

Total pressure

the Grad-Shafranov equation

Whilenon-linear, it is a scalar eq.

The GS equation is not force-free.

For a force-free configuration wecan set P = 0

Examples of using the Grad-Shafranovmethod in studies of thestructure of ICMEs, see, Isavnin et al., Solar Phys., 273, 205-219, 2011DOI: 10.1007/s11207-011-9845-z

Black arrows: Magnetic field measured by a spacecraft during the passage of an ICME

Contours: Reconstruction of the ICME assuming that GS equation is valid

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J E no energy dissipation!

q F v E v J E

EM energy is dissipated via currents J in a medium.

The rate of work is defined

Recall: Poyntings theorem

EJ > 0: magnetic energy converted into kinetic energy (e.g. reconnection)JE < 0: kinetic energy converted into magnetic energy (dynamo)

work performed

by the EM fieldenergy flux through

the surface of V

change of

energy in V

Poynting vector(W/m2) giving theflux of EM energy

How are magnetospheric currents produced?

Dayside reconnectionis a load ( J E > 0 ) anddoes not create current

Here ( J E < 0 ), i.e., the magnetopauseopened by reconnection acts as a generator(Poynting flux into the magnetosphere)

Solar wind pressuremaintainsJCFand the magnetopausecurrent

E

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Currents in the magnetosphere

Thus, if we know B everywhere, we can calculate the currents(in MHD J is a secondary quantity!).However, it is not possible to determine B everywhere,as variable plasma currents produce variable B

In large scale (MHD) magnetospheric currents are source-free

and obey Ampres law

Recall the electric current in (anisotropic) magnetohydrostatic equilbirum:

In addition, a time-dependent electric field drives polarization current:

These are the main macroscopic currents ( B) in the magnetosphere

A simple MHD generator

Now Ampres force J x B deceleratesthe plasma flow, i.e., the energy to thenewly created J (and thus B) in thecircuit is taken from the kinetic energy.

Kinetic energy EM energy: Dynamo!

Plasma flows across the magnetic field

F = q (u x B) edown, i+ up

The setting can be reversed by driving a currentthrough the plasma: J x B accelerates the plasma

plasma motor

Let the electrodes be coupled througha loadcurrent loop, where the currentJ through the plasma points upward

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Boundary layer generator

One scenario how an MHD generatormay drive field-aligned current fromthe LLBL:

Plasma penetrates to the LLBL throughthe dayside magnetopause (e.g. byreconnection.

The charge separation (electrodesof the previous example) is consistentwith E = V x B .

The FAC ion the inside part closes viaionospheric currents (the so-calledRegion 1 current system)

Note: This part of the figure

(dayside magnetopause)

is out of the plane

FAC from the magnetosphere

We start from the (MHD) current continuity equation

where

Sources of FACs:pressure gradients &

time-dependent

The divergence of the static total perpendicular current is

[ Note: ( M) = 0 ]

In the isotropic case this reduces to

The polarizationcurrent can havealso a divergence = b ( V)

is the vorticity

Integrating the continuity equation along a field line we get:

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FAC from the ionosphereWrite the ionospheric () current as a sum ofPedersen and Hall currents and integrate thecontinuity equation along a field line:

The sources and sinks of FACs in the ionosphereare divergences of the Pedersen current andgradients of the Hall conductivity

R1

R2R2

R1

R1: Region 1, poleward FAC

R2: Region 2, equatorward FAC

Approximate the RHS integral as

where P = hP and H= hare

height-integrated Pedersen and Hallconducitivities and h the thickness ofthe conductive ionosphere (~ the E-layer)

Recall:

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Current closureThe closure of the current systems is a non-trivial problem.Lets make a simple exercise based on continuity equation

Let the system be isotropic. The magnetospheric current is

(the last term contains inertial currents)

we neglect this

Integrate this along a field line from one ionospheric end to the other

Assuming the magnetic field north-south symmetric, this reduces to

This must be the same as the FAC from the ionosphere and we get anequation for ionosphere-magnetosphere coupling

One more view on M-I coupling

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Magnetic helicityA is the vector potential

Gauge transformation:

Helicity is gauge-independent onlyif the field extends over all spaceand vanishes sufficiently rapidly

For finite magnetic field configurationshelicity is well defined if and only if

on the boundary

Helicity is a conserved quantity if the field is confined within a closed surface Son which

the field is in a perfectly conducting medium for which on S

To prove this, note that from

we get to within a gauge transformation

Calculate a little

and both B and V are ^ n on Sthus on S

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Example: Helicity of two flux tubes linked together

Calculate

For thin flux tubesis approximately normal to the cross section S

(note: here Sis not the boundary of V!)

Helicity of tube 1:

Thus

Similarly

If the tubes are woundNtimes around each other

Complex flux-tubes on the Sun

Helicity is a measure of structural complexity of the magnetic field

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Woltjers theoremFor a perfectly conducting plasma in a closed volume V0 the integal

is invariant and the minimum energy state is a linear force-free field

Proof: Invariance was already shown.To find the minimum energy state consider

Small perturbations:

on Sand added

transform to surface integral

that vanishes because

Thus if and only if

Magnetohydrodynamic waves

Dispersion equation for MHD waves

Alfvnwave modes

MHD is a fluid theory and there are similar wave modes as in ordinary fluidtheory (hydrodynamics). In hydrodynamicsthe restoring forces for perturbationsare the pressure gradient and gravity. Also in MHD the pressure force leads toacoustic fluctuations, whereas Ampres force (JxB) leads to an entirely newclassof wave modes, calledAlfvn (or MHD) waves.

As the displacement current is neglected in MHD, there are noelectromagnetic waves of classical electrodynamics. Of course EM waves canpropagate through MHD plasma (e.g. light, radio waves, etc.) and eveninteract with the plasma particles, but that is beyond the MHD approximation.

V1

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Nature, vol 150, 405-406 (1942)

Dispersion equation for ideal MHD waves

eliminateJ

eliminateP

eliminateE

Consider smallperturbations

and linearize

We are left with 7 scalar equations for 7 unknowns (m0, V, B)

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Find an equation for V1. Start by taking the time derivative of ()

()

()

()

Insert () and () and introduce theAlfvn velocity as a vector

Look for plane wave solutions

Using a few times we have

the dispersion equation for the waves in ideal MHD

Alfvn wave modes

Propagation perpendicular to the m agnetic field:

Now and the dispersion equation reduces to

clearly

And we have found the magnetosonic wave

This mode has many names in the literature:- Compressional Alfvn wave- Fast Alfvn wave- Fast MHD wave

Making a plane wave assumption also for B1 (very reasonable, why?)

(i.e., B1 || B0)

The wave electric field follows from the frozen-in condition

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As electrons and ions are separated, Ohms law is important.Recall the generalized Ohms law

0

(similar to Hall MHD)

Inclusion of pressure termrequires assumptions of

adiabatic/isothermal behaviorof electron fluid & Te

Note: The QN equations cannotbe casted to conservation form

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Quasi-neutralhybrid simulations of Titans interaction with the magnetosphereof Saturn during a flyby of the Cassini spacecraft 26 December 2005FromIlkka Sillanps PhD thesis, 2008.

Escapeof O+ and H+ from the atmosphere of Venus.Using the quasi-neutral hybrid simulationFromRiku Jrvinens PhD thesis, 2010.

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Beyond MHD: Kinetic Alfvn wavesAt short wavelengths kinetic effects start to modify the Alfvn waves.Easier than to immediately go to Vlasov theory, is to look for kineticcorrections to the MHD dispersion equation

For relatively large betae.g., in the solar wind, at magnetospheric boundaries, magnetotailplasma sheetthe mode is called oblique kinetic Alfvn wave and it has the phase velocity

For smaller beta ( ),e.g., above the auroral region and in the outer magnetosphere, the electronthermalspeed is smaller than the Alfvn speed electron inertial becomes important

inertial (kinetic) Alfvn wave orshear kinetic Alfvnwave

KineticAlfvn waves

carry field-aligned currents (important in M-I coupling)

are affected by Landau damping (althoughsmall as long as is small),

recall the Vlasov theory result

Full Vlasov treatment leads to the dispersion equation

modified Bessel function

of the first kind

For details, see Lysak and Lotko, J. Geophys. Res., 101, 5085-5094, 1996