1ESS200 CThe MagnetosphereLectures 10, 11, 12C.T. Russell2The 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:52255)3(33−−−−=−=rMrzBryzMBrxzMBzzzyzx2/12333)cos31(sincos2θθθθ+=−==−−−MrBMrBMrBr3L-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.4Mirror Dipole MagnetosphereImage 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.5Pressure Exerted by the Solar Wind on the Magnetospherenpnuuˆ)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, Bois the field at the equator on the surface of the planet, μois the magnetic permeability of free space and Lmpis the distance to the magnetopause in planetary radii.• For the Earth, where nswis the solar wind proton number density in cm-3and the uswis the solar wind bulk speed in kms-1.16020)2()(2−∞∞=mpLaBuKμρ167.02)(4.107−=swswmpunL6Pressure 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.7The Shape of the Magnetic Cavityψψρ222sincos∞∞∞+ PuKEmpirical 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.8Extreme 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 Bzchange. • 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)9Effect 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)10Subsonic 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.11Gasdynamic 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