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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