1

ESS200 CThe Magnetosphere

Lectures 10, 11, 12C.T. Russell

2

The Dipole Magnetic Field• For many purposes, the Earth’s magnetic field can be approximated by

a dipole. In a spherical coordinate system with the polar axis along the magnetic dipole axis, the magnetic field can be expressed as:

Where θ is the angle from the pole, r is the distance from the center of the dipole, and there is no azimuthal component of the magnetic field.

• In a cartesian system with the magnetic moment along Z, the field is:

522

5

5

)3(

3

3

−

−

−

−=

−

=

rMrzB

ryzMB

rxzMB

zz

zy

zx

2/123

3

3

)cos31(

sin

cos2

θ

θ

θ

θ

+=

−=

=

−

−

−

MrB

MrB

MrBr

3

L-Value and Invariant Latitude• The distance of the magnetic field line

from the center of the Earth at its most distant point (the magnetic equator) is called the L value.

• The latitude at which the field line touches the Earth is called the invariant latitude.

• In a planetary magnetosphere, particles with small gyroradii are guided by the magnetic field and move back and forth along a field line while drifting around the Earth. Hence, they maintain a constant L-value.

• Planetary magnetic fields are more complicated than a dipole, but the more complex fields (quadrupole, octupole, etc.) fall off more rapidly with distance so eventually the field will be mostly dipolar in a vacuum.

4

Mirror Dipole Magnetosphere

Image Dipole Magnetosphere Dipole Field in a Vacuum with Superconducting Shell

• If you bring a flat superconducting sheet close to a magnetic dynamo, it mirrors the magnetic field to produce a very simple magnetosphere with two neutral points and a doubled magnetic field strength at the nose.

• If you wrap a superconducting sheet around the magnetosphere like the solar wind envelops the Earth, you change the subsolar field to 2.4 times the dipole strength and preserve the topology.

5

Pressure Exerted by the Solar Wind on the Magnetosphere

npnuu ˆ)ˆ( +⋅ρ• The momentum flux and thermal pressure in the solar wind confine the size of the

magnetosphere.

• The magnetosheath causes streamlines to diverge so there is a pressure drop across the magnetosheath. This depends on Mach number and the polytropic index.

• The magnetic field pushes back with the magnetic pressure. The field falls off in strength as r-3 and the magnetic pressure as r-6

• The field is enhanced by a factor, a, at the nose where a depends on the geometry or the curvature of the boundary.

• The pressure balance is:

where K is determined from Bernoulli’s law, Bo is the field at the equator on the surface of the planet, μo is the magnetic permeability of free space and Lmp is the distance to the magnetopause in planetary radii.

• For the Earth,

where nsw is the solar wind proton number density in cm-3 and the usw is the solar wind bulk speed in kms-1.

160

20 )2()(2 −

∞∞ = mpLaBuK μρ

167.02 )(4.107 −= swswmp unL

6

Pressure Balance of a Tangential Discontinuity

• At a magnetopause in the absence of reconnection, the discontinuity is a tangential discontinuity.

• The magnetic pressure on one side equals the plasma pressure on the other if this boundary separates a pure magnetic field region from a pure plasma region.

• The forces that balance are a gradient in magnetic field pressure pushing against a gradient in plasma thermal pressure.

• The gradient in plasma thermal pressure produces a current that flows across the magnetic field. This is the J x B force.

7

The Shape of the Magnetic Cavity

ψψρ 222 sincos ∞∞∞ + PuK

Empirical magnetosphere of Tsyganenko (1989) with realistic boundary shape and implicit plasma

content

• The pressure of the solar wind is applied to the magnetosphere along the normal to its surface, the magnetopause.

• The direction of the magnetopause normal varies with position and the pressure applied drops as one moves away from the subsolarpoint.

• There is always some pressure applied even when the boundary is aligned with the asymptotic flow. A good approximation to the pressure is:

Where Ψ is the angle of themagnetopause normal to the solar windflow, P∞ is the thermal pressure at ∞ andK accounts for stream tubeexpansion.

8

Extreme Solar Wind Conditions• The magnetopause size is also

affected by the north-south component of the interplanetary magnetic field.

• When the IMF is southward, magnetospheric magnetic flux is carried from the dayside to the nightside allowing the magnetopause to move inward on the dayside.

• This contour plot shows how the magnetopause distance at noon changes as both the dynamic pressure and IMF Bz change.

• The distance 6.6 RE is important because this is the location of synchronous orbit where many communication and monitoring spacecraft reside.

Magnetopause position (Shue, et al., 1998)

9

Effect of Magnetic Flux Transport on the Tail

• This two-dimensional model of Atkinson and Unti (1968) illustrates the effect of magnetic flux transport on the dayside magnetosphere and the magnetotail.

• Five tail states are indicated with different amounts of magnetic flux carried from the dayside to the tail (1 –most; 5 – least)

• Increasing tail flux causes its boundary to flare more so the solar wind presses harder on the tail. This is observed by spacecraft in the tail as an increase in the magnetic pressure in the tail lobes.

• At the same time, the current sheet in the tail moves inward. This is observed by spacecraft in the night magnetosphere as the magnetic field changing from dipolar to tail-like.

Two-dimensional magnetosphere with arbitrary tailmagnetic flux content (Atkinson and Unti, 1968)

10

Subsonic Versus Supersonic Interaction

• If a flowing magnetized plasma encounters an obstacle to that flow such as a magnetosphere or ionosphere, it is deflected around the obstacle by a standing wave.

• In a gas-dynamic flow, a pressure gradient forms that slows and deflects the flow. This is possible because the thermal speed (temperature) of the particles is large enough that the sound speed is greater than the flow speed. The Mach number is less than 1.

• When the particles are cold, the flow speed exceeds the sound speed, a shock forms heating and slowing the flow so that a pressure gradient can develop sufficient to deflect the flow.

11

Gasdynamic Simulations of the Solar Wind Interaction

• Gasdynamics has only one wave mode, the compressionalwave to slow and deflect the incoming flow.

• Since there is no magnetic force, the obstacle cannot be a magnetosphere.

• This diagram shows streamlines around a cylindrically symmetric magnetospherically shaped obstacle for a Mach number of 8, and a polytropic index of 5/3.

12

Gasdynamic Interaction Continued

• Density contours show that density jumps close to a factor of 4 across the shock. It increases behind the shock in the subsolar region and decreases elsewhere.

• In gasdynamics, the velocity and temperature ratios are proportional. The velocity increases along the flank and the temperature drops.

13

Gasdynamics Concluded

• The mass flux is equal to the product of the density ratio and the velocity ratios and parallels the streamlines. The bow shock positions itself so that all the shocked mass can flow past the obstacle. The mass flux is highest near the shock than near the obstacle.

• The magnetic field cannot exert a force in gasdynamics, but it can be approximated as threads being carried by the flow. Here we show the magnetic field lines behind the shock for two upstream orientations.

14

Magnetohydrodynamics• When there is a magnetic force as there is in a magnetized

plasma, the interaction becomes much more complicated but we can solve for more quantities such as the size and shape of the magnetosphere.

• These equations can provide a solution that allows the flow to go by the obstacle, the magnetic field to be convected around and over the obstacle, and the magnetic field to stretch. This is done with three standing waves: fast compressional, Alfven or shear, and slow compressional modes.

)()(

)()(/

)()(/

)(//)()(/

)()(/

2

AmpereBBJ

FaradayBButB

pressureupputp

momentumBJpuutu

continuityut

d−×∇=

∇+××∇=∂∂

⋅∇−∇⋅−=∂∂

×+∇−∇⋅−=∂∂

⋅−∇=∂∂

η

γ

ρρ

ρρ

15

MHD Forces Affecting the Flow• Here we show the thermal

pressure gradient force and the magnetic force separately on a background contour plot of density.

• At the shock, the pressure gradient and magnetic forces (arrows) both act to slow the flow.

• Well away from the shock, the pressure gradient forces turnaround as the plasma density and pressure is lower, close to the boundary.

• The magnetic forces still push away from the magnetosphere.

• The net result is a total force that slows and turns the flow parallel to the magnetopause.

16

Hybrid Simulations: Importance of Scale Size

• MHD simulations ignore the underlying motions of the charged particles. The plasma is treated as a fluid. Hybrid simulations treat the ion motion but not the electrons.

• If the solar wind (top) encounters a magnetic obstacle much smaller than the ion inertial length, only a whistler mode wake is formed.

• If the obstacle is larger, it causes a pile-up in density ahead of the obstacle and decreases the density in the wake.

• For obstacles larger than the ion inertial length, a compressional wave forms upstream that heats, compresses, and diverts the flow. A plasma sheet forms downstream

• For obstacles the size of Mercury, you obtain a shock and a magnetosphere and tail similar to those of Earth.

17

Hybrid Simulations: Quasi-Parallel Shock

• When the interplanetary field is near the Parker spiral direction, it creates a quasi-parallel shock on one side and a quasi-perpendicular shock on the other.

• The quasi-parallel shock is highly fluctuating.

• The parallel shock is even more fluctuating.

18

Hybrid Simulations: Upstream Waves

• Ions can move back upstream from the bow shock in the quasi-parallel region of the bow shock.

• The counter-streaming of the solar wind and the back streaming ions provides free energy that generates waves.

• The waves scatter the particles and thermalize them.

• Thus the foreshock preprocesses the plasma, altering it upstream of the main shock.

19

The Magnetic Cavity• The magnetosphere deflects the

solar wind leading to a large magnetic cavity with a long extended tail.

• The magnetosphere contains much solar wind plasma; it is stirred by the solar wind; and it is energized by the solar wind.

• Current research addresses these issues. Missions currently active include Polar, Wind, and Geotail, as well as the 4-spacecraft cluster mission. Future missions include Magnetosphere Multiscale.

• This diagram shows some of the couplings of the system: solar wind to magnetosphere and magnetosphere to ionosphere.

20

Momentum Transfer• There are several possible ways

for the solar wind to transfer momentum to the magnetosphere, i.e. to stir the plasma and cause it to circulate:

– Diffusion of flowing particles across the boundary

– Waves causing boundary motion– Tubes of magnetic field entering

the magnetosphere• None of these purely particle or

wave particle interactions fully explain what we see in the magnetosphere.

• The momentum transfer appears to be controlled by the interplanetary magnetic field.

21

Momentum Transfer by Reconnection

• In 1961 and 1963, J.W. Dungeyproposed a reconnecting model of the magnetosphere where the magnetic field lines of the solar wind became connected to the Earth’s magnetic field, whether the field was northward or southward. This connection led to a circulation of the plasma.

• The circulation went from the dayside to the nightside over the poles when the interplanetary field was southward and returned in the equatorial plane.

• The circulation for northward IMF goes around the magnetosphere in the equatorial plane and then returns sunward over the polar caps.

22

IMF Control of Geomagnetic Activity

• There is much indirect evidence in favor of the Dungey models for the role of the IMF in applying tangential stress to the magnetosphere.

• Arnoldy examined the integrated southward magnetic flux in the solar wind and cross correlated it with AE and got a predictive measure of geomagnetic activity.

• Murayama used the AL index normalized by the square of the solar wind velocity and found that the magnetosphere behaved as a half-wave rectifier. Energy entered only when the IMF was southward.

23

Energy, Transfer, and Storage• Given that reconnection occurs that

links the solar and terrestrial fields, the energy transfer from the solar wind to the tail occurs quite naturally.

• On the dayside, the field lines straighten and accelerate the solar wind plasma.

• On the nightside, the field lines in the tail stretch and the solar wind plasma slows down. Hence, energy is removed from the flow and stored in the magnetic field in the tail.

• The rate of energization can be calculated from the Poynting vector integrated over the surface of the tail.

• The magnetic energy, flux and field strength of the tail increase.

24

Reconnection at the Magnetopause 1

• Rapid reconnection of oppositely directed magnetic fields of necessity occurs at neutral points.

• Flow enters from the left and right sides and exits top and bottom.

• In this process, magnetic energy stored in the magnetic field is converted to bulk flow and heat.

• The field and flow may be different on the two sides of the boundary

• If there is a guide field (into or out of the page), the fields are no longer exactly anti-parallel.

25

Reconnection at the Magnetopause 2

• Half-wave rectification implies a very strong dependence of the rate of reconnection on the relative direction of the two magnetic fields.

• In a resistive plasma, reconnection can occur over a range of relative orientations.

• In a collisionless plasma, reconnection can take place at a variety of angles once initiated, but its start is delayed unless the fields are anti-parallel (no guide field).

• The location of anti-parallel directions depend very much on the IMF orientation. This can explain the half-wave rectification.

Luhmann et al, (19**)

26

Physics of Reconnection in a Collisionless Plasma

• Reconnection in a collisionlessplasma must be a fully kinetic process involving the motions of both electrons and ions.

• The magnetic field line concept is valid if electrons do not get scattered as they move parallel to the magnetic field.

• Problem is to determine what causes electrons to become confused and how to make the region of confusion sufficiently large that it has a high rate.

• Kinetic model of Daughton et al, produces a large electric field that makes the electron pressure agyrotropic over a region much larger than an electron gyro radius or electron inertial length.

Daughton et al, (2007)

27

Magnetospheric Potential Drop• A thin slab of solar wind

plasma merges with the magnetospheric magnetic field at the magnetopause.

• The potential drop across this slab of plasma in steady state appears across the polar cap and across the return flow in the equatorial plane.

• Thus -VswBsLsw = VpcBpcLpc

• And -VswBsLsw = VmBeqLm

• If uniform Em = -VswBsLsw/Lm

28

Formation of the Plasmasphere• The cold ionospheric plasma

can move along field lines and fill them to a saturation density of about 10,000 cm-3.

• The circulation of magnetospheric plasma stirred by reconnection will carry this plasma to the magnetopause if it is on open drift paths.

• A sharp density boundary can form between the open and closed drift paths.

• The high-density region is called the plasmasphere. Its boundary is called the plasmapause.

29

Plasma Regions of the Magnetosphere

• From inside moving outward first is the cold (<1 ev) plasmasphere with density up to 104 cm-3.

• Trapped radiation belt penetrates the plasmasphereand extends outside it.

• The plasma sheet sits in the distant equatorial region and the center of the tail.

• The tail lobe has little plasma.• The boundary layer of the

magnetosphere and tail can contain mantle or boundary layer plasma that may be a mixture of magnetosheath and magnetospheric plasma.

30

Radiation Belt

• The flux of low energy particles is quite variable at all distances.• The very energetic particle fluxes in the inner magnetosphere are

rather stable and change only under unusual circumstances.• At high latitudes or L-values, the fluxes of very energetic particles

are quite variable.

31

Gyroradii of Radiation Belt Particles• The gyroradius of a charged

particle is

where V is the perpendicular velocity, A is the mass in amu, E is the perpendicular energy and B is the magnetic field strength.

• When the particle gyroradii are small compared to the curvature of the field lines, the particles remain trapped.

)100()1

()46( 21

21

BnT

keVEAkmor

qBmv ⊥⊥

⊥

⊥

32

Time Scales of Radiation Belt Motions

211

322

ˆ2BqR

BrWqB

BBWqB

BFB

BEVc

cextD

×+

∇×+

×+

×= ⊥

ABnTs

qBmT

gg )100)(66.0(22

==Ω

=ππ

21

21

)1()10

.)(min5(2WkeVA

Rl

vl

E

bbB ≈

><≈τ

)1)(100

()5

)(56(~22 22

WkeV

nTB

Rrh

WqBr

Vr

EgcD ≈=

πτ

Gyro Period

Bounce Period

Drift Perpendicular to B

Drift Period

33

Time Scales Continued• Three motions can be studied separately

through the use of adiabatic invariants:pdq is conserved under slow changes of the system where q is the coordinate and p is the momentum

• The first adiabatic invariant is

• The energy of a particle is conserved

constant

• Particle must mirror when particle reaches a critical value of

• Second adiabatic

is conserved on periods long compared to the bounce period.

)(2 2

Bmv

qmTvmdtvp gxxx

⊥=><=∫π

Bmv2

2⊥=μ

=+=+= ⊥ BmvVVmW μ222

21)(

21

μWBm =

∫∫ −==2

1

)(22m

m

dssBWmdspJ μ

∫

34

Dessler-Parker-Sckopke Relation• The magnetic field at the center of the Earth caused by the gradient drift of an ion in

the equatorial plane is

• The particle also has a dipole moment that produces a northward field at the center of the Earth

• Since ∆B does not depend on L, we may write

• The total energy in the dipole field above the surface of the Earth is

• So we may write

• We measure the Dst index on the surface of a conducting Earth. Taking account of the exclusion of the field from the interior we obtain

zoE

ograddrift e

BRWB ˆ

43

3μ

π−=Δ

zoE

omagmom e

BRWB ˆ

41

3.μ

π=Δ

zoE

totalototal e

BRWB ˆ

21

3

μπ

−=Δ

oEo

mag BRW 3

34μπ

=

zmag

total

o

total eWW

BB ˆ

32

−=Δ

)100/(108.2 15 nTBJoulesWtotal Δ××=

35

Magnetic Storms• The Dessler-Parker-Sckopke relation allows

us to monitor the energy contained in particles in the magnetosphere.

• This energy varies continually; most severely in events known as magnetic storms.

• There are three stages to a storm:– Sudden impulse, where the

magnetosphere is compressed but little energy flows in

– Main phase, where the energizationtakes place

– Recovery phase, where the energy decay exponentially from the magnetosphere

• These properties reflect the character of ICMEs

– Sudden impulse marks the arrival of the shock in front of the ICME

– Main phase marks the period of strong southward interplanetary magnetic field (IMF)

– Recovery phase marks the period when the IMF is northward or the ICME has passed

• ICMEs may have leading or trailing southward field.

• Sometimes magnetic storms begin gradually.

36

Burton et al, 1975 Formula• The Dst index measures the deviation of the

worldwide surface field from the quiet day value.

• Two effects are important: compression of the magnetosphere by the solar wind, and injection of energy from the solar wind into magnetospheric particles.

• We know that the compressional ∆B is proportional to the square root of the dynamic pressure.

• We know empirically that the energy injected depends on the rate at which southward magnetic field is convected to the magnetopause, expressed here as the east-west component of the electric field.

• The decay time of the ring current is about 8 hours.

• This enables us to predict energy content of magnetosphere from the solar wind parameters

– We correct for solar wind dynamic pressure and quiet day ring current

– We calculate the injection and subtract the decay.

– Add change in field to previous (adjusted) Dst and then adjust value back to compressed state with quiet day ring current in order to compare with observed Dst index.

37

Structure of the Magnetotail

• The tail probably extends thousands of Earth radii in the anti-solar direction.

• The neutral point is beyond the moon at quiet times.• Plasma brought into the tail through reconnection can

sink into the tail.• Plasma sheet can extend from about 10RE to infinity.

38

Harris Current Sheet• This is the simplest self-consistent

analytical description of a one-dimensional current sheet

• The total pressure is constant

• The current is

• The pressure gradient is balanced by the j x b force

• Particles have gradient drift and serpentine drift in opposite directions.

)/(sec)(

ˆ)/tanh()(2 hzhpzp

xhzBzB

o

o

=

=

oooo BPpB μμ 2/2/ 22 ==+

)/(sec)/()()( 2 hzhhBzjB oyoy ==×∇ μ

Bjzhzhpdzdp

zhzhzhhBBj

o

oo

×==∇

=×

ˆ))/(sec(

ˆ)/tanh()/(sec)/(

2

22 μ

39

Current Sheet of Alfven• Alfven (1968) presented a self-

consistent model of a reconnecting current sheet that could sustain the current needed to maintain the magnetic field above and below the sheet.

• The tail boundaries are separated by a distance L with a potential drop Φ and an electric field (Φ/L)

• The particles drift to the center of the sheet with velocity E/Bproviding a current of 2 enLu

• Ampere’s law gives a current of 2B/μ so that u=B/(μoneL)

• The speed supplies the right amount of particles to provide the tail current to maintain the magnetic field.

y

40

Accelerating Particles in a Reconnecting Current Sheet

• If a current sheet is reconnecting, it has an electric field in the direction of the current.

• If there is a finite normal component of the magnetic field, the particles bouncing along the field line will drift in the direction of the electric field as they cross the current and gain energy.

• When the particle leaves the current sheet, it has added energy along the magnetic field.

41

Empirical Model of Substorms• An empirical model of the substorm was

developed by Russell and McPherron (1973) and McPherron et al (1973) based on the available ground and space data at that time.

• In response to a southward turning of the interplanetary magnetic field, reconnection begins on the front side of the magnetosphere, carries it to the tail expanding the distant tail and causing the cross section to flare.

• The tail field stretches, the current sheet thins and reconnection begins on closed field lines forming a magnetic island or plasmoid in the plasma sheet in what is known as the growth phase.

• Reconnection proceeds slowly to cut field lines until it reaches the open, low-density field lines in the tail lobes where reconnection becomes rapid.

• The rapid reconnection marks the onset of the expansion phase and the ejection of the plasmoid down the tail.

• After the expansion phase reduces the flux in the tail and the flaring of the tail, the recovery phase begins.

42

Magnetic Flux Inventory in Substorms

• The near-Earth neutral point model of substorms is simply a time-varying model of Dungey’s southward IMF magnetosphere.

• We can understand it by taking an inventory of the magnetic flux in each of three regions: the dayside magnetosphere, the plasma sheet and nightside magnetosphere, and the tail lobes as the rates of dayside reconnection, M; nighttime reconnection, R; and the convection to the dayside, C, vary.

43

Magnetic Flux Inventory with Triggering

• Substorms are often triggered by northward turnings of the IMF.

• To understand this behavior, we look at the role of the distant neutral point.

• Plasma is brought inward from the distant neutral point and closed magnetic field lines with the plasma thickening the region the near-Earth neutral point has to cut through to reach the tail lobe field lines.

• Thus the distant neutral point delays the onset of the expansion phase and the release of the plasmoid.

• A northward turning could shut off this plasma supply and allow rapid reconnection to begin.