1. ESS 154/200C Lecture 9 Solar Wind Interactions: Magnetized Planets

2. Date Day Topic Instructor Due
ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
Date Day Topic Instructor Due
1/4 M A Brief History of Solar Terrestrial Physics CTR
1/6 W Upper Atmosphere / Ionosphere CTR
1/8 F The Sun: Core to Chromosphere CTR
1/11 M The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares CTR PS1
1/13 W The Solar Wind and Heliosphere, Part 1 CTR
1/15 F The Solar Wind and Heliosphere, Part 2 CTR
1/20 W Physics of Plasmas RJS PS2
1/22 F MHD including Waves RJS
1/25 M Solar Wind Interactions: Magnetized Planets YM PS3
1/27 W Solar Wind Interactions: Unmagnetized Planets YM
1/29 F Collisionless Shocks CTR
2/1 M Mid-Term PS4
2/3 W Solar Wind Magnetosphere Coupling I CTR
2/5 F Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I CTR
2/8 M The Inner Magnetosphere II CTR PS5
2/10 W Planetary Magnetospheres CTR
2/12 F The Auroral Ionosphere RJS
2/17 W Waves in Plasmas 1 RJS PS6
2/19 F Waves in Plasmas 2 RJS
2/26 F Review CTR/RJS PS7
2/29 M Final

3. Dipole Magnetic Field
Gauss showed that the magnetic field of the Earth could be described as the gradient of a scalar potential in the region in which no currents were flowing
The solution consists of a series of multipole moments: dipole, quadrupole, octupole, etc.
There are 3 dipole moments with axes along the 3 coordinate axes x, y, z; five quadrupole, 7 octupole, etc.
Earth’s dipole is tilted about 10.2° (or more correctly, 169.8°) to the rotation axis with a moment of 7.8 x 1015 Tm3 or 30.2μ T · RE3
In spherical coordinates

4. Dipole Field Lines
In a coordinate system with Z along the dipole axis of symmetry, the angle up from the equator, λ, is the latitude, and the angle, θ, down from the polar axis is the co-latitude. The field line crosses the equator at a distance, L, from the center of the planet. The latitude at which this field line reaches the planet is called the invariant latitude, . The invariant latitude

5. Dipoles in Cartesian Coordinates
We can express the field of a dipole with its magnetic axis along Z as
Or we can generalize it for an arbitrarily oriented dipole as
Given a series of observations, Bi, at locations ri (xi, yi, zi), we can solve for the dipole moment using standard matrix-inversion techniques.

6. Generalized Potential Field Models
For a generalized field of arbitrary configuration in a current-free region, we can express the scalar potential of the field in terms of associated Legendre polynomials
Here, a is the planetary radius, θ the co-latitude, is east longitude, are the associated Legendre functions with Schmidt normalization.
For the Earth, a consensus list of and are regularly published as the IGRF.
The dipole moment is
The tilt of the dipole is

7. A Hierarchy of Dipoles
A dipole is two charges separated by a small distance.
A magnetic dipole can be thought of as two magnetic charges separated by a distance, but such charges do not exist. We must create it from a ring of electric current.
We can make quadrupoles from dipoles in 5 different ways by stacking them near each other with a separation between them.
We can make octupoles in 7 different ways by stacking quadrupoles with separations.
This slides shows the resultant configuration of dipoles.

8. Simple Magnetospheres: Image Dipole Model
The simplest magnetosphere can be made by putting a flat superconducting plate in front of the dipole. Two cusps are formed on the ‘front’ and the field becomes stretched in the tail region. This model shows the effect of the solar wind dynamic pressure squeezing the magnetosphere.

9. Simple Magnetospheres: Models of Tsyganenko
If the superconductor instead wraps around the magnetosphere in the form of a paraboloid of revolution then the field looks more like the observations but the tail is not realistic. This model contains no plasma.
If one makes a model of the magnetosphere based on observations, then the magnetosphere looks more realistic. The differences are caused by the presence of warm plasma in the real magnetosphere.
Tsyganenko’s vacuum magnetosphere
Tsyganenko’s empirical magnetosphere

10. Size of the Magnetosphere
The pressure of the solar wind on the magnetopause is exerted principally by the dynamic pressure or momentum flux ρu2 but at the magnetopause itself the flow is along the boundary, tangential to it, and is conveyed totally by thermal and magnetic pressure.
The pressure applied by the solar wind is slightly less than the pressure of the gas approaching from infinity because the flow is diverted around the magnetosphere.
The factor expressing the drop in pressure depends on the ratio of specific heats, γ, and the Mach number and is for γ = 5/3 and M = .
The field just inside the magnetopause at the nose is shape-dependent. For the planar magnetopause, the field is doubled; for a spherical magnetopause, the factor is 3. For a realistic magnetopause the factor is 2.44.
The distance to the magnetopause from the center of the Earth can then be expressed as
where nsw is in proton masses per cm3 and usw is the bulk speed in km·sec-1

11. Shape of the Magnetopause
Tangential stress along the surface of the magnetopause can also alter its shape. In particular, when the IMF is southward, the magnetic flux is eroded by reconnection and transported to the magnetotail.
The top diagram shows contours of constant magnetopause location at the subsolar point as a function of IMF BZ and dynamic pressure. The northward IMF results in a much more distant magnetopause.
The bottom diagram shows how this transport of flux by reconnection adds to the tail while removing flux from the dayside allowing it to move inward.
As the magnetic flux is moved to the tail, the inner edge of the tail current system moves inward and the cross section of the tail increases. As the shape changes, the pressure of the solar wind on the tail changes.
The magnetic pressure in the tail at its asymptotic width balances the thermal pressure in the solar wind.

12. Formation of Bow Shocks
If the solar wind were subsonic then when the wind hit the magnetopause, a pressure gradient would build up that slowed and diverted the solar wind.
In a supersonic flow, a shock is generated that compresses, heats, and deflects the flow by making the post-shock flow subsonic.
We can produce a global numerical simulation of the 3D interaction with different degrees of sophistication of the model. In the following examples, we first ignore the magnetic field (gasdynamics), include the magnetic forces (magnetohydrodynamics) but ignore kinetic processes, and finally add kinetic process for the ions (hybrid simulations).

13. Standoff Distance of Shock
Shock fronts arise when an airplane flies through the air, as well as when the solar wind flow past the Earth. It is relative velocity that matters.
If the object is sharp-nosed the shock will begin close to the object.
For a blunt object the shock stands off some distance, to allow the shocked plasma to flow between the obstacle and the shock front. The radius of curvature of the obstacle is the quantitative controlling factor.
The needed distance depends on how much the gas or fluid can be compressed. The degrees of freedom of the gas and the Mach number control this compressibility.
Schlieren photograph of model plane in supersonic wind tunnel.

14. Gasdynamic Simulations 1
The convected magnetic field gasdynamic model solves a rotationally symmetric problem for a hard boundary and then traces field lines that are carried along with the flow but have no magnetic forces.
This diagram shows the streamlines of the flow past the magnetosphere.
The density jump across the shock is close to a factor of 4 for this model of the flow at a Mach number of 8 and a polytropic index of 5/3.
Downstream from the subsonic nose region, the flow accelerates so the density drops. Still at the shock jump there is a large increase in density.

15. Gasdynamic Simulations 2
Velocity is least in the subsolar region, increasing to approach the solar wind speed downstream.
The temperature is highest in the subsolar region and cools as it expands downstream.
As shown, the velocity jump and the temperature jump are related and can be drawn with the same contours.
The mass flux is greatest near the shock where the density is the highest.

16. Gasdynamic Simulations 3
The magnetic field does not exert any force in the gas dynamic simulation. The field is just carried along as massless threads with no tension.
When the IMF is perpendicular to the flow, the field lines are draped over the magnetopause.
When the field is oblique to the solar wind flow, there is asymmetry in the magnetosheath with weak fields on one side an strong fields on the other.
Field and plasma pile-up together at magnetopause in contrast to observations.

17. Magnetohydrodynamic Simulations 1
We can define average properties of a plasma, like:
Density: n (number); ρ (mass)
Temperature: T
Pressure: p
These are governed by conservation laws:
Continuity (mass conservation)
Momentum
Eq. of state (entropy conservation) pρ–γ= constant
The polytropic index, γ, is the ratio of the specific heat at constant pressure to the specific heat at constant volume
The value of γ is related to the degrees of freedom (f) of the gas: γ = (f+2)/f
In 3D, γ= 5/3; in 2D, γ = 2
Faraday’s Law
Ampere’s Law

18. Magnetohydrodynamic Simulations 2
Magnetic fields exert force perpendicular to the direction of the magnetic field. One component of force is proportional to the spatial gradient on the magnetic pressure and one is proportional to the curvature of the magnetic field line.
Magnetic field lines want to fill space uniformly and be straight.
Plasma pressure forces are aided by magnetic forces in a realistic fluid model. These magnetohydrodynamic (MHD) models are more time consuming to run and produce slightly different results especially along the magnetopause boundary.
Both field and plasma pressure forces decelerate the flow across the shock but near the boundary the magnetic forces push back on the flow while the plasma accelerates the flow along the streamlines. This decreases the density near the boundary, as observed.

19. Global Hybrid Simulations 1
Hybrid simulations follow the motion of individual ions and treat electrons as a massless fluid.
These simulations can be used for studying the global flow interaction when the size of the obstacle approaches or is less than the ion scale (ion inertial length or proton gyroradius). They will be more computationally intensive than the MHD simulations and have less spatial resolution.
These panels show Bx/B0 and N/N0 for obstacles of 4 different sizes.
The strength of the magnetic dipole moment has been altered to make the scale size of the interaction have different sizes.
(Top) Much smaller than a gyroradius/ion inertial length.
Here we get a whistler-mode wave produced.
(Second) Just below the ion scale.
Whistler and magnetosonic waves are produced.
(Third) Just above the ion scale.
Standing fast mode waves and slow waves are produced.
(Bottom) Much larger than the ion scale.
Bow shock, magnetosheath, plasma sheet and tail produced.

20. Global Hybrid Simulations 2
These hybrid simulations show the solar wind interaction for a scale size 64 times the ion scale with an IMF at 45° to the solar wind flow.
This is similar in scale to the magnetosphere of Mercury.
The shock are quasi-perpendicular (top) and quasi-parallel (bottom). The left panels show contour map for density and ion temperature. On the right, the top two panels show two crossings of the quasi-parallel shock with one closer to the nose and the other further in the flank. The bottom two panels show the different style of upstream waves seen ahead of the shock. 20 20

21. Summary
The solar-wind interaction with a magnetized obstacle is a complex process. The magnetosphere acts as a nearly impenetrable obstacle to the solar wind.
The solar wind is supersonic, and a standing shock must form in front of the obstacle.
The size of the magnetosphere mainly depends on the solar wind dynamic pressure and the strength of the dipole moment.
The problem can be examined using different numerical models: gasdynamic, MHD and hybrid. Each has its own advantages and disadvantages. 21 21