The magnetosphereThe Earth’s intrinsic magnetic fieldThe dipole magnetic field in Cartesian coordinatesProperties of the Earth’s Magnetic FieldA Fluid Picture of the MagnetopauseTangential Stresses on the BoundaryThe Tail CurrentThe Magnetotail - StructureReconnectionReconnectionThe Plasma MantleThe Plasma Sheet Boundary LayerThe Low Latitude Boundary LayerMagnetopause ReconnectionThe Radiation Belts and Ring CurrentThe Ring CurrentField Aligned CurrentsESS 200C Lectures 9, 10 and 11The Magnetosphere• The magnetosphere– 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.•The Earth’s intrinsic magnetic field– To a first approximation the magnetic field of the Earth can be expressed as that of the dipole. The dipole moment of the Earth is tilted ~110to the rotation axis with a present day value of 8X1015Tm3 or 30.4x10-6TRE3 where RE=6371 km (one Earth radius).– In a coordinate system fixed to this dipole momentwhere is the magnetic colatitude, and M is the dipole magnetic moment.21)cos31(0sincos22333θθθϕθ+====−−−MrBBMrBMrBrθ• The dipole magnetic field in Cartesian coordinates– Alternately in Cartesian coordinateswhere the z-axis is along the dipole magnetic moment.– This can easily be generalized to a dipole moment with an arbitrary orientation.52255)3(33−−−−===rMrzBryzMBrxzMBzzzyzx()()()rrrBxxr xy xzyx y r yzzx zy z rm()=−−−⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟33333 3333222222• Magnetic field lines and the L parameter– 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 colatitudewhere L is measured in RE.– A trapped particle conserving the adiabatic invariants will be confined to a surface specified by L.θθBdrBdrr=0=ϕdθ20sinrr =λλ2cosLr =• Generalized planetary magnetic fields– Although a dipole field is useful for many problems the actual internal fields of planets are more complex.– Gauss showed that the magnetic field of the Earth could be described by the gradient of a scalar potentialwhere Φiis the scalar potential due to sources within the Earth and Φe is due to external sources.– The scalar potentials can be expressed as a sum of associated Legendre polynomialswhere a is the planet’s radius, and θ and Φ are to colatitude and east longitude in planetographic coordinates. The Pnm(cos θ) are associated Legendre functions with Schmidt normalization.()eiB Φ+Φ−∇=Φ−∇=r()[]( ) () ()()ϕϕθϕθmhmgPararmnmnmnnnnmisincoscos,,110+=Φ∑∑∞=−−=()[]() () ()()ϕϕθϕθmHmGPararmnmnmnnnnmesincoscos,,10+=Φ∑∑∞==()()()()mnmnmmndPdNPmθθθθcoscoscos1cos22−=where Pn(cos θ) is the Legendre function, and Nn,m=1when m=0 and [2(n-m)!/(n+m)!]1/2otherwise.–The coefficients gnm, hnm, Gnm, and Hnmare chosen to minimize the difference between the model field and observations.–In this representation the dipole moment M=a3[(g10)2+(g11)2+(h11)2]1/2– The dipole tilt becomes α=cos-1(g10/M)• 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 ~110with respect to the rotation axis.• The dipole moment is decreasing.– It was 9.5X1015T m3in 1550 and had decreased to 7.84X1015T m3in 1990. – The tilt also is changing. It was 30in 1550, rose to 11.50in 1850 and has subsequently decreased to 10.80in 1990.• In addition to the tilt angle the rotation axis of the Earth is inclined by 23.50with respect to the ecliptic pole. – Thus the Earth’s dipole axis can be inclined by ~350to the ecliptic pole. – The angle between the direction of the dipole and the solar windvaries between 560and 900.• The Magnetopause– In the simplest approximation the magnetopause can be considered to be the boundary between a vacuum magnetic field and a plasma.– Charged particles in the solar wind approach the Earth’s magnetic field Bwhich is pointed upward in the equatorial plane.– The Lorentz force q(V x B) on the particles deflects protons to the right (left hand gyration), and electrons to the left (right hand).– The opposite motion of the charges produces a sheet current from left to right (dawn to dusk)– Magnetic perturbations from this current reduce the Earth’s field Sunward of the current and increase the field Earthward– Above the pole the field points in the opposite direction so the current does as well. Thisis the return currentMagnetopauseDuskChapman-FerraroCurrentReturnCurrentNorthSolarWindF=q(VxB)VB•A Particle View of the Magnetopause– When an electron or ion penetrates the boundary they sense a force. After half an orbit they exit the boundary. – The electrons and ions move in opposite directions and create a current. The ions move farther and carry most of the current. The number of protons per unit length in the z-direction that enter the boundary and cross y=y0per unit of time is 2rLpnu . (Protons in a band 2rLp in y cross the surface at y=y0.) Since each proton carries a charge e the current per unit length in the z-direction crossing y=y0is where – Applying Ampere’s law and notingBurr×222 uBnmnuerIzpLp==)()(zpLpeBumr =∫= jdxIIBz 0µ=22022swswpzuunmBρµ==• 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