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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 shock Fore Shock Homework: 6.5, 6.10, 6.11*, 8.1, 8.3, 8.7, 8.2* 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 30.4x10-6TRE3where RE=6371 km (one Earth radius). • In a coordinate system fixed to this dipole moment Br 2Mr 3 cos B Mr 3 sin B Mr (1 3 cos ) 3 2 1 2 where is the magnetic colatitude, and M is the dipole magnetic moment. The Dipole Magnetic Field • Alternately in cartesian coordinates Bx 3xzM z r 5 By 3 yzM z r 5 Bz (3z 2 r 2 ) M z r 5 • The magnetic field line for a dipole. Magnetic field lines are everywhere tangent to the magnetic field vector. dr d d 0 r Br B • Integrating r r0 sin 2 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 r L cos 2 where L is measured in RE. 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 Magnetosphere The Magnetopause • • • • • • In the simples t 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 B which 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) Solar Magnetic perturbations from this current Wind 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. This is the return current Return Current North Magnetopause B Dusk V F=q(VxB) Chapman-Ferraro Current The Magnetosphere A Particle View of the Magnetopause • When an electron or ion penetrates the u B 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=y0 per 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=y0 is 2nm p 2 I 2rLp nue u rLp (um p ) ( eBz ) Bz where • Applying Ampere’s law and noting I jdx Bz 0 I Bz2 2 nm p u 2 sw u sw 2 0 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. 2 swu sw • The current on the boundary must provide a j B force sufficient to change the solar wind momentum (divert the flow). • The change in momentum flux into the boundary is 2 swu sw2 (we are assuming perfect reflection at the boundary) 2 2 u I B B 2 sw sw 0 Current Continuity From E = e / 0 B 0 J 0 0 E t We have e J 0 t In long time scales (for MHD) e is very small. t J 0 Current has no source nor sink. Current lines are continuous. The Magnetosphere Currents on the Magnetopause • • • • Near the pole there is a singular point in the field where |B| = 0. This is called the neutral point. The Chapman-Ferraro (C-F) current circulates in a sheet around the neutral point This current is symmetric about the equator with a corresponding circulation around the southern neutral point The C-F current completely shields the Earth’s field from the solar wind confining it to a cavity called the magnetosphere North Magnetopause Field Line Neutral Point Solar Wind Dusk Chapman-Ferraro Current The Magnetosphere The Location of the Magnetopause • • • • • The standoff distance to the subsolar magnetopause is determined by a balance between the solar wind dynamic pressure and the magnetic field inside the boundary The collisions of particles with the boundary may not be completely elastic hence a factor k is introduced The magnetic field inside the boundary is the total field from dipole and boundary current. For an infinite planar sheet current the field would be exactly doubled. Inside a spherical boundary the multiplication factor is 3. The factor f must lie in this range . Equate and substitute for the dipole strength variation with distance Solve for the dimensionless standoff distance Ls. pdyn pB pdyn kmnu2 pB f 2 BD2 kmnv 2 0 2 2 ( fBD ) BD B0 æç RE ö÷ è Rs ø 3 2 0 Where k is the elasticity of particle collisions and f is the factor by which the magnetospheric magnetic field is enhanced by the boundary current. Rs is the subsolar standoff distance. æR Ls çç s è RE öù ö éæ f 2 öæ B02 ÷ ÷÷ êçç ÷÷çç 2 ÷ú ø ëè k øè 2 0 mnu øû 1 6 The Magnetosphere The Shape of the Magnetopause • • • • • Half of the noon-midnight meridian plane is shown above the axis and half of the equatorial plane is shown below Dashed lines show different solutions while the solid line shows the final shape obtained by iteration The equatorial section is quite simple with no indentations. The subsolar point is at ~10 R e for the most probable solar wind conditions The equatorial boundary crosses the terminators at 15 R e The meridian boundary is indented at the neutral points where the Earth’s magnetic field is too weak to stand off the solar wind The Magnetosphere The Effect of the Magnetopause Currents • • • • Close to the Earth the dipole field dom inates and there is little distortion Further away there is a significant change in the s hape of the field lines with all field lines passing through the equator closer to the Earth than dipole field lines from the same latitude. All dipole field lines that originally passed through the equator more than 10 Re sunward of the Earth are bent back and close on the night side The neutral point separates the two types of field lines The Magnetosphere The 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 2semicircle, then the magnetic flux in that tail lobe is T RT BT where RT is the lobe 2 radius, and BT is the magnetic field strength. • The asymptotic radius of the tail is given by RT 2T2 ( 2 0 psw ) where psw included both the thermal and magnetic pressure of the solar wind. 1 4 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 B d s 0 j dA c where C bounds surface with area A 2 BT 0 I 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 The Magnetosphere Observing the Magnetopause • • • • • Boundary normal coordinates are frequently used to study the magnetopause The boundary normal coordinates have one component normal to the boundary ( n̂ ) and two tangential ( L̂ nothward and M̂ azimuthal). The dayside m agnetopause can be approximated as a tangential discontinuity when IMF Bz >0. In this case there will be no field normal to the boundary on either side and the normalized cross product of the two fields defines the normal. When IMF Bz < 0 the boundary is a rotational discontinuity with a small normal component. In this case minimum variance analysis defines the directions of maximum, intermediate, and m inimum variance with the m inimum variance determining the normal. 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 ISEE1. 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 18000 km with a most probably thickness of 700 km. Structure of Magnetopause (theory) Structure of the Magnetopause Northward 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 a 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 a 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. Shock Front Upstream (low entropy) vu Downstream (high entropy) vd • Shock Conservation Laws – In both fluid dynamics and MHD conservation equations for mass, energy and momentum have the form: Q F 0 where Q and F are the t density and flux of the conserved quantity. – If the shock is steady ( t 0 ) and one-dimensional Fn 1 or that n ( Fu Fd ) nˆ 0 where u and d refer to upstream and downstream and n̂ is the unit normal to the shock surface. We normally write this as a jump condition[ Fn ] 0. – Conservation of Mass ( vn ) 0 or [ vn ] 0. If the shock slows the n plasma then the plasma density increases. 2 æ ö v p B n – Conservation of Momentum vn ÷÷ 0 where the first term çç n n n è 2 0 ø 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. é 2 B2 ù ê vn p ú0 2 0 û ë é Bn ù v v B 0 . The subscript t refers – Conservation of momentum ê n t t ú 0 ë û to components that are transverse to the shock (i.e. parallel to the shock surface). é æ1 2 pö B 2 Bn ù ÷÷ vn – Conservation of energy ê vn çç 2 v v B ú 0 1 0 úû êë è ø 0 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 E B 0 – Gauss Law B 0 gives Bn 0 – Faraday’s Law E B t gives vn Bt Bnvt 0 • 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 vn 0 ,Bn 0 Density jumps arbitrary, all others continuous. No plasma flow. Both sides flow together at vt. Tangential Discontinuity vn 0 , Bn 0 Complete separation. Plasma pressure and field change arbitrarily, but pressure balance Rotational Discontinuity vn 0 , Bn 0 Large amplitude intermediate wave, field and flow change direction but not magnitude. vn Bn ( 0 )2 1 Types of Shocks in Ideal MHD Shock Waves vn 0 Flow crosses surface of discontinuity accompanied by compression. Parallel Shock Bt 0 B unchanged by shock. Perpendicular Shock Bn 0 P and B increase at shock Oblique Shocks Bt 0, Bn 0 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 •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 Magneto pause Postbow 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 Bn the angle between the shock normal and the IMF is 900. – Lines further downstream have Bn 90 0 • Particles have parallel motion along the field line ( v )and cross field drift motion ( v (E B) / B ). 2 d – All particles have the same vd – 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. Ion Foreshock Upstream Waves Summary of Foreshock: shock-field angle determines the features in the sheath and upstream