The Magnetopause Back in 1930 Chapman and Ferraro foresaw that a planetary magnetic field could provide an effective obstacle to the solar-wind plasma.

The Magnetopause

• Back in 1930 Chapman and Ferraro foresaw that a planetary magnetic field could provide an effective obstacle to the solar-wind plasma.

• The solar-wind dynamic pressure presses on the outer reaches of the magnetic field confining it to a magnetospheric cavity that has a long tail consisting of two antiparallel bundles of magnetic flux that stretch in the antisolar direction.

• The pressure of the magnetic field and plasma it contains establishes an equilibrium with the solar wind.

• The solar wind is usually highly supersonic before it reaches the planets. The wind velocity exceeds the velocity of any pressure wave that could act to divert the flow around the obstacle and a shock forms.

Lecture 8

Magnetopause Magnetosheath

Bow shockFore Shock

A Digression on the Dipole Magnetic Field

• To a first approximation the magnetic field of the Earth can be expressed a that of the dipole. The dipole moment of the Earth is tilted ~110 to the rotation axis with a present day value of 8X1015Tm3 or 30.4x10-6TRE

3where RE=6371 km (one Earth radius).

• In a coordinate system fixed to this dipole moment

where is the magnetic colatitude, and M is the dipole magnetic moment.

21

)cos31(

sin

cos2

23

3

3

MrB

MrB

MrBr

The Dipole Magnetic Field

• Alternately in cartesian coordinates

• The magnetic field line for a dipole. Magnetic field lines are everywhere tangent to the magnetic field vector.

• Integrating where r0 is the distance to equatorial crossing of the field line. It is most common to use the magnetic latitude instead of the colatitude

where L is measured in RE.

522

5

5

)3(

3

3

rMrzB

ryzMB

rxzMB

zz

zy

zx

B

dr

B

dr

r

0d

20 sinrr

2cosLr

Properties of the Earth’s Magnetic Field

• The dipole moment of the Earth presently is ~8X1015T m3

(3 X10-5TRE3).

• The dipole moment is tilted ~110 with respect to the rotation axis.

• The dipole moment is decreasing. – It was 9.5X1015T m3 in 1550 and had decreased to 7.84X1015T m3 in 1990.

– The tilt also is changing. It was 30 in 1550, rose to 11.50 in 1850 and has subsequently decreased to 10.80 in 1990.

• In addition to the tilt angle the rotation axis of the Earth is inclined by 23.50 with respect to the ecliptic pole.

– Thus the Earth’s dipole axis can be inclined by ~350 to the ecliptic pole. – The angle between the direction of the dipole and the solar wind varies

between 560 and 900.

The MagnetosphereThe Magnetopause

• In the simples t approximation themagnetopause can be considered to be theboundary between a vacuum magnetic fieldand a plasma.

• Charged particles in the solar wind approachthe Earth’s magnetic field B which is pointedupward in the equatorial plane

• The Lorentz force q(V x B) on the particlesdeflects protons to the right (left handgyration), and electrons to the left (right hand)

• The opposite motion of the charges produces asheet current from left to right (dawn to dusk)

• Magnetic perturbations from this currentreduce the Earth’s field Sunward of the currentand increase the field Earthward

• Above the pole the field points in the oppositedirection so the current does as well. This is thereturn current

Magnetopause

Dusk

Chapman-FerraroCurrent

ReturnCurrent

North

SolarWind

F=q(VxB)

V

B

T h e M a g n e t o s p h e r e A P a r t i c l e V i e w o f t h e M a g n e t o p a u s e

• W h e n a n e l e c t r o n o r i o n p e n e t r a t e s t h e b o u n d a r y t h e y s e n s e a f o r c e . A f t e r h a l f a n o r b i t t h e y e x i t t h e b o u n d a r y .

• T h e e l e c t r o n s a n d i o n s m o v e i n o p p o s i t e d i r e c t i o n s a n d c r e a t e a c u r r e n t . T h e i o n s m o v e f a r t h e r a n d c a r r y m o s t o f t h e c u r r e n t . T h e n u m b e r o f p r o t o n s p e r u n i t l e n g t h i n t h e z - d i r e c t i o n t h a t e n t e r t h e b o u n d a r y a n d c r o s s y = y 0 p e r u n i t o f t i m e i s 2 r L p n u . ( P r o t o n s i n a b a n d 2 r L p i n y c r o s s t h e s u r f a c e a t y = y 0 . ) S i n c e e a c h p r o t o n c a r r i e s a c h a r g e e t h e c u r r e n t p e r u n i t l e n g t h i n t h e z - d i r e c t i o n c r o s s i n g y = y 0 i s

w h e r e

• A p p l y i n g A m p e r e ’ s l a w a n d n o t i n g

Bu

222 u

B

nmnuerI

z

pLp

)()( zpLp eBumr

jdxI

IB z 022

0

2

2 swswpz uunm

B

A Fluid Picture of the Magnetopause

• The location of the boundary can be calculated by requiring the pressure on the two sides of the boundary to be equal. The pressure in the magnetosphere which is mostly magnetic must match the pressure of the magnetosheath which is both magnetic and thermal.

• The magnetosheath pressure is determined by the solar wind momentum flux or dynamic pressure.

• The current on the boundary must provide a force sufficient to change the solar wind momentum (divert the flow).

• The change in momentum flux into the boundary is

(we are assuming perfect reflection at the boundary)

2swswu

Bj

2sw swu

22

0sw sw

Bu I B

Current Continuity

e 0

0 0 0

From

= /

We have

0

In long time scales (for MHD) is very small.

0

Current has no source nor sink. Current lines are continuous.

e

e

t

t

t

E

EB J

J

J

The MagnetosphereCurrents on the Magnetopause

• Near the pole there is a singular point inthe field where |B| = 0. This is calledthe neutral point.

• The Chapman-Ferraro (C-F) currentcirculates in a sheet around the neutralpoint

• This current is symmetric about theequator with a correspondingcirculation around the southern neutralpoint

• The C-F current completely shields theEarth’s field from the solar windconfining it to a cavity called themagnetosphere

Field Line

MagnetopauseNorth

NeutralPoint

Dusk

SolarWind

Chapman-FerraroCurrent

The MagnetosphereThe Location of the Magnetopause

• The standoff distance to the subsolarmagnetopause is determined by a balancebetween the solar wind dynamic pressure andthe magnetic field inside the boundary

• The collisions of particles with the boundarymay not be completely elastic hence a factor kis introduced

• The magnetic field inside the boundary is thetotal field from dipole and boundary current.For an infinite planar sheet current the fieldwould be exactly doubled. Inside a sphericalboundary the multiplication factor is 3. Thefactor f must lie in this range .

• Equate and substitute for the dipole strengthvariation with distance

• Solve for the dimensionless standoff distanceLs.

0

2

2

2D

B

dyn

Bdyn

fBp

kmnup

pp

Where k is the elasticity of particle collisionsand f is the factor by which the magnetosphericmagnetic field is enhanced by the boundarycurrent. Rs is the subsolar standoff distance.

3

0

0

222

2

s

ED

D

RRBB

Bfkmnv

6

1

20

20

2

2

mnu

B

k

f

R

RL

E

ss

The MagnetosphereThe Shape of the Magnetopause

• Half of the noon-midnight meridianplane is shown above the axis and halfof the equatorial plane is shown below

• Dashed lines show different solutionswhile the solid line shows the finalshape obtained by iteration

• The equatorial section is quite simplewith no indentations. The subsolarpoint is at ~10 Re for the most probablesolar wind conditions

• The equatorial boundary crosses theterminators at 15 Re

• The meridian boundary is indented atthe neutral points where the Earth’smagnetic field is too weak to stand offthe solar wind

The MagnetosphereThe Effect of the Magnetopause Currents

• Close to the Earth the dipole fielddominates and there is little distortion

• Further away there is a significantchange in the shape of the field lineswith all field lines passing through theequator closer to the Earth than dipolefield lines from the same latitude.

• All dipole field lines that originallypassed through the equator more than10 Re sunward of the Earth are bentback and close on the night side

• The neutral point separates the twotypes of field lines

The MagnetosphereThe Shape of the Nightside Magnetosphere

• At every point along the magnetopause the component of dynamic pressure normal to the boundary must be balanced by the pressure of the tangential magnetic field interior to the boundary

• Far downstream the solar wind velocity becomes parallel to the magnetopause and the normal component of dynamic pressure becomes zero

• This would lead to a cylindrical tail• But both the thermal and magnetic pressure of

the solar wind exert a transverse pressure that eventually becomes important

• At the distance where the dipole field pressure equals the sum of the solar wind thermal and magnetic pressure the magnetosphere should close giving it a tear-drop shape

•The solution in the previous view graph treated the normal stresses correctly but did not include tangential stresses.

Tangential Stresses on the Boundary

• Tangential stresses (drag) transfers momentum to the magnetospheric plasma and causes it to flow tailward. The stress can be transferred by diffusion of particles from the magnetosheath, by wave process on the boundary, by the finite gyroradius of the magnetosheath particles and by reconnection. Reconnection is thought to have the greatest effect.

• Assume that one tail lobe is a semicircle, then the magnetic flux in that tail lobe is where RT is the lobe radius, and BT is the magnetic field strength.

• The asymptotic radius of the tail is given by where psw included both the thermal and magnetic pressure of the solar wind.

TT

T BR

2

2

41)(2 022

swTT pR

The Tail (Magnetopause) Current

• The stretched field configuration of the magnetotail is naturally generated by a current system.

– The relationship between the current and the magnetic field is given by Ampere’s law

where C bounds surface with area A

where I is the total sheet current density (current per unit length in the tail)

– For a 20nT field I=30 mA/m or 2X105A/RE

AdjsdBc

0

IBT 02

The MagnetosphereObserving the Magnetopause

• Boundary normal coordinates arefrequently used to study the magnetopause

• The boundary normal coordinates have onecomponent normal to the boundary ( )and two tangential ( nothward andazimuthal).

• The dayside m agnetopause can beapproximated as a tangential discontinuitywhen IMF Bz >0. In this case there will beno field normal to the boundary on eitherside and the normalized cross product ofthe two fields defines the normal.

• When IMF Bz < 0 the boundary is arotational discontinuity with a small normalcomponent.

• In this case minimum variance analysisdefines the directions of maximum,intermediate, and m inimum variance withthe m inimum variance determining thenormal.

n̂L̂ M̂

The Magnetosphere Observing the Magnetopause

• Data from two spacecraft show two crossings of the boundary.

• Initially both spacecraft are inside the magnetosphere (strong field).

• The boundary moves inward and crosses first the ISEE-1 spacecraft (thick line) and later the ISEE-1 spacecraft (thin line).

• Some time later the boundary reverses and moves outward appearing first at ISEE-2 and later at ISEE-1.

• Assume a planar boundary moving with constant velocity along the average normal during each crossing.

• The spacecraft separation along the average normal divided by the time delay gives the boundary velocity.

• The time profile scaled by the velocity gives the spatial profile of the boundary

• The thickness of the magnetopause varies from 200 to 1800 km with a most probably thickness of 700 km.

Structure of Magnetopause(theory)

Structure of the MagnetopauseNorthward IMF Southward IMF

Magnetopause Crossings

Magnetopause Shape Model

Bow shock and magnetosheath divert the solar wind flow around the magnetosphere: computer simulation

Formation of Sonic Shock

• A shock is a discontinuity separating two different regimes in a continuous media.– Shocks form when velocities exceed the signal speed in the medium.– A shock front separates the Mach cone of a supersonic jet from the undisturbed air.

• Characteristics of a shock :– The disturbance propagates faster than the signal speed. In gas the signal speed is the speed

of sound, in space plasmas the signal speeds are the MHD wave speeds.– At the shock front the properties of the medium change abruptly. In a hydrodynamic shock,

the pressure and density increase while in an MHD shock the plasma density and magnetic field strength increase.

– Behind a shock front a transition back to the undisturbed medium must occur. Behind a gas-dynamic shock, density and pressure decrease, behind an MHD shock the plasma density and magnetic field strength decrease. If the decrease is fast a reverse shock occurs.

• A shock can be thought of as a non-linear wave propagating faster than the signal speed.

– Information can be transferred by a propagating disturbance.– Shocks can be from a blast wave - waves generated in the corona.– Shocks can be driven by an object moving faster than the speed of sound.

• The Shock’s Rest Frame– In a frame moving with the shock the

gas with the larger speed is on the left and gas with a smaller speed is on the right.

– At the shock front irreversible processes lead the the compression of the gas and a change in speed.

– The low-entropy upstream side has high velocity.

– The high-entropy downstream side has smaller velocity.

• Collisionless Shock Waves– In a gas-dynamic shock collisions

provide the required dissipation.– In space plasmas the shocks are

collision free.

• Microscopic Kinetic effects provide the dissipation.

• The magnetic field acts as a coupling device.

• MHD can be used to show how the bulk parameters change across the shock.

vu vd

Shock Front

Upstream(low entropy)

Downstream(high entropy)

• Shock Conservation Laws

– In both fluid dynamics and MHD conservation equations for mass, energy and

momentum have the form: where Q and are the density and flux of

the conserved quantity.

– If the shock is steady ( ) and one-dimensional or that

where u and d refer to upstream and downstream and is the unit normal

to the shock surface. We normally write this as a jump condition .

– Conservation of Mass or . If the shock slows the plasma then

the plasma density increases.

– Conservation of Momentum where the first term is the rate of

change of momentum and the second and third terms are the gradients of the gas and

magnetic pressure in the normal direction.

0

Ft

Q F

0 t 1

n

Fn

0ˆ)( nFF du

n̂

0][ nF

0)(

nvn 0][ nv

02 0

2

B

nn

p

n

vv n

n

02 0

22

Bpvn

– Conservation of momentum . The subscript t refers to

components that are transverse to the shock (i.e. parallel to the shock surface).

– Conservation of energy

The first two terms are the flux of kinetic energy (flow energy and internal

energy) while the last two terms come form the electromagnetic energy flux

– Gauss Law gives

– Faraday’s Law gives

00

t

ntn B

Bvv

01 00

22

21

nnn

BBv

Bv

pvv

0BE

0 B 0nB

tBE

0 tntn vBBv

• The jump conditions are a set of 6 equations. If we want to find the downstream quantities given the upstream quantities then there are 6 unknowns ( ,vn,,vt,,p,Bn,Bt).

• The solutions to these equations are not necessarily shocks. These conservations laws and a multitude of other discontinuities can also be described by these equations.

Types of Discontinuities in Ideal MHD

Contact Discontinuity , Density jumps arbitrary, all others continuous. No plasma flow. Both sides flow together at vt.

Tangential Discontinuity , Complete separation. Plasma pressure and field change arbitrarily, but pressure balance

Rotational Discontinuity , Large amplitude intermediate wave, field and flow change direction but not magnitude.

0nB

0nv

0nv

0nB

21

0nn Bv

0nv 0nB

Types of Shocks in Ideal MHD

Shock Waves Flow crosses surface of discontinuity accompanied by compression.

Parallel Shock

B unchanged by shock.

Perpendicular Shock

P and B increase at shock

Oblique Shocks

Fast Shock P, and B increase, B bends away from normal

Slow Shock P increases, B decreases, B bends toward normal.

Intermediate

Shock

B rotates 1800 in shock plane, density jump in anisotropic case

0nv

0tB

0nB

0,0 nt BB

•Configuration of magnetic field lines for fast and slow shocks. The lines are closer together for a fast shock, indicating that the field strength increases. [From Burgess, 1995].

Bow Shock and Magnetopause Crossings

Bow Shock Crossings with Location Front Orientation

Functions of Magnetosheath

Diverts the solar wind flow and bends the IMF around the magnetopause

Observations of Density Enhancements in the Sheath

Internal Structure of the Magnetosheath

Bow Shock

Magnetopause

Post-bow shock density

Slow Shock in the Magnetosheath

• Particles can be accelerated in the shock (ions to 100’s of keV and electrons to 10’s of keV).

• Some can leak out and if they have sufficiently high energies they can out run the shock. (This is a unique property of collisionless shocks.)

• At Earth the interplanetary magnetic field has an angle to the Sun-Earth line of about 450. The first field line to touch the shock is the tangent field line.

– At the tangent line the angle between the shock normal and the IMF is 900.

– Lines further downstream have

• Particles have parallel motion along the field line ( ) and cross field drift motion ( ).

– All particles have the same – The most energetic particles will move farther

from the shock before they drift the same distance as less energetic particles

• The first particles observed behind the tangent line are electrons with the highest energy electrons closest to the tangent line – electron foreshock.

• A similar region for ions is found farther downstream – ion foreshock.

Bn

090Bn

v

2/)( BBEvd

dv

Ion Foreshock

Upstream Waves

Summary of Foreshock:shock-field angle determines the features in the sheath and upstream