ESS 154/200C Lecture 7 Physics of Plasmas

ESS 154/200CLecture 7Physics of Plasmas

ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial PhysicsM/W/F 10:00 – 11:15 AM Geology 4677 Instructors:C.T. Russell (Tel. x-53188; Office: Slichter 6869)R.J. Strangeway (Tel. x-66247; Office: Slichter 6869) • DateDayTopicInstructorDue • 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: UnmagnetizedPlanets YM • 1/29 F CollisionlessShocks 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 • 2/8M The Inner Magnetosphere II CTR PS5 • 2/10W Planetary Magnetospheres CTR • 2/12F The Auroral Ionosphere RJS • 2/17W 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

The Plasma State • A plasma is an electrically neutral ionized gas. • The Sun is a plasma • The space between the Sun and the Earth is filled with a plasma. • The Earth is surrounded by a plasma. • A stroke of lightning forms a plasma • Over 99% of the Universe is a plasma. • Although neutral a plasma is composed of charged particles – electric and magnetic forces are critical for understanding plasmas.

Maxwell’s equations • Gauss’s Law • is the electric field • is the charge density • is the electric permittivity of free space (8.85 x 10-12 Farad/m) • Absence of magnetic monopoles • is the magnetic field • Faraday’s Law • Ampere’s Law • is the current density, • is the permeability of free space, H/m • c is the speed of light,

The Motion of Charged Particles • Equation of motion • stands for non-electromagnetic forces (e.g., gravity) – often ignored • Last term is change in momentum from collisions – important in the ionosphere • SI Units • mass (m) - kg • length (l) - m • time (t) - s • electric field (E) - V/m • magnetic field (B) - T • velocity (v) - m/s

B acts to change the motion of a charged particle only in directions perpendicular to the motion. • Set electric field equal to zero, define perpendicular direction with respect to magnetic field: • Taking time derivative of this equation: • This is the equation of motion for a simple harmonic oscillator: where is the particle gyro-frequency, , with units of radians/sec.

Solution is circular motion dependent on initial conditions.Letting the magnetic field define the z-axis of a cartesian coordinate system then from the x-component of the momentum equation: with the upper sign corresponding to a positively charged particle. • Assuming , then . • On integration we find where x0 are y0 constants of integration. • Equations of circular motion with angular frequency  (cyclotron frequency or gyro frequency). Upper signs are for positive charge, lower signs are for negative charge. • If q is positive particle gyrates in left-handed sense • If q is negative particle gyrates in a right-handed sense

Radius of circle - cyclotron radius or Larmor radius (rL) or gyro radius (rg). • The gyro radius is a function of energy. • Energy of charged particles is usually given in electron volts (eV) • Energy that a particle with the charge of an electron gets in falling through a potential drop of 1 Volt – 1 eV = 1.6x10-19 Joules (J). • Energies in space plasmas go from electron Volts to kiloelectron Volts (1 keV = 103 eV) to millions of electron Volts (1 meV = 106 eV) • Cosmic ray energies go to gigaelectron Volts ( 1 GeV = 109 eV). • The circular motion does no work on a particle Only the electric field can energize particles Particle energy remains constant in absence of E

Next Step – Include an Electric Field • For parallel motion: • This implies parallel electric fields are localized or of short duration, otherwise particles would be accelerated to relativistic energies. In general assume , and hence is constant. • has a major effect on motion. • As a particle gyrates it moves along the electric field and gains energy • Later in the gyration it losses energy. • This causes different parts of the orbit to have different radii - the orbit doesn’t close on itself • Results in a net “ExB” drift • No charge dependence, therefore no currents

Take time derivative of momentum equation: • Assume , where is a time-varying term and is constant. Then: which is the same equation for a uniform magnetic field, resulting in gyration, and • This is the ExB drift, which is independent of mass and charge. All particles drift with this velocity.

Ions and electrons gyrate in the opposite sense, but ExB drift in the same direction, with the same velocity Trajectory depends on gyrational speed versus ExB speed

Note that VExB is: • Independent of particle charge, mass, energy • VExB is frame dependent, as an observer movingwith same velocity will observe no drift. • The electric field has to be zero in that moving frame. • Consistent with transformation of electric field in moving frame: E’=(E+VxB). Ignoring relativistic effects, electric field in the frame moving with V= VExB is E’=0. • Mnemonics: • E is 1mV/m for 1000km/s in a 1nT field • V[1000km/s] = E[mV/m] / B [nT] • Thermal velocities (kT=1/2 m vth2): • 1keV proton = 440 km/s • 1eV electron = 600 km/s, or a 2.5 keV electron has a velocity of c/10 • Gyration: • Proton gyro-period: 66 s * (1nT/B) • Electron: 28 Hz * (B/1nT) • Gyroradii: • 1keV proton in 1nT field: 4600 km*(m/mp)1/2*(W/keV)*(1nT/B) • 1eV electron in 1nT field: 3.4 km *(W/eV)*(1nT/B)

Any force capable of accelerating and decelerating charged particles can cause an average (over gyromotion) drift: • This is functionally the same as the momentum equation with E replaced by F/q. • Example, gravity: • If the force is charge independent the drift motion will depend on the sign of the charge and can form perpendicular currents. • Forces resembling the above gravitational force can be generated by centrifugal acceleration of orbits moving along curved fields. This is the origin of the term “gravitational” instabilities which develop due to the drift of ions in a curved magnetic field (not really gravity). • In general 1st order drifts develop when the 0th order gyration motion occurs in a spatially or temporally varying external field. To evaluate 1st order drifts we have to integrate over 0th order motion, assuming small perturbations relative to , g. This leads to the concept of guiding center motion.

The Concept of the Guiding Center • Separates the motion of a particle into motion perpendicular ( ) and parallel ( ) to the magnetic field. • To a good approximation the perpendicular motion can consist of a drift ( ) and the gyro-motion ( ) • Over long times the gyro-motion is averaged out and the particle motion can be described by the guiding center motion consisting of the parallel motion and drift. This is very useful for distances l such that and time scales  such that • Closely related concept – first adiabatic invariant (or magnetic moment), as this is related to particle gyration.

Time-varying magnetic fields – magnetic moment • Time-varying magnetic field results in induction electric field around gyro-orbit: • Positively charged particles move in the same direction as the azimuthal electric field, negatively charged particles in the opposite direction. So both increase in energy for increasing magnetic field: • Or, defining the perpendicular energy as ,

Magnetic moment – current loop • The current in the loop defined by one gyration of the particle =q/t, where t is the gyro-period, or, in vector notation: • Dipole moment of a current loop is given by IA, where A is the area of the current loop: • Negative sign indicates that particles in a plasma are diamagnetic; their magnetic moment opposes the applied field. • Magnetic moment conserved for slowly time-varying magnetic fields. Also the case for magnetic fields with large scale spatial variation.

Magnetic mirror • Consider cylindrical geometry, from : • Therefore: • Azimuthal component of momentum equation gives since • Hence • m is conserved for both temporal and spatial changes in B.

Magnetic mirror – force on a magnetic dipole • Same cylindrical geometry • Again • Axial component of momentum equation gives • Hence force on a magnetic dipole is: • Derived for axial field and gradient, but this is a general result.

Inhomogeneous magnetic fields cause grad-B drift. • If B changes over a gyro-orbit rg will change. • rg gets smaller when particle goes into stronger B,results in a drift perpendicular to grad-B. • Use general force drift to derive grad-B drift • From force on magnetic dipole: • From generalized drift, grad-B drift given by: • vG depends on charge, yields perpendicular currents (but see later, when we discuss MHD).

Curvature of the magnetic field also causes a drift • The curvature of the magnetic field line introduces a drift motion. • As particles move along the field they undergo centrifugal acceleration. • Centrifugal force (rc is radius of curvature – positive outwards): • In terms of the magnetic field: • Hence • Both gradient and curvature drift can also be derived explicitly using Taylor expansion of the particle momentum equation.

At Earth’s dipole are in the same direction and comparable: • ~ 1RE/min=100km/s for 100keV particle at 5RE, at 100nT • The drift around Earth is: 0.5hrs for the same particle at the same location • At Earth’s magnetotail current sheet are opposite each other: • Curvature drift dominates at center of the current sheet, gradient drift dominates further away from center of current sheet

The first adiabatic invariant – again • As a particle moves to a region of stronger (weaker) B it gains (loses) perpendicular energy. • Conservation of m results in betatron acceleration if the magnetic field changes in time (slowly). • Pitch angle is defined as • Particles with pitch angle less than will be lost from the mirror – loss cone (Bmax is the maximum field in the mirror).

The second adiabatic invariant – particle bounce motion • The integral of the parallel momentum over one complete bounce between mirrors is constant (as long as B does not change much in a bounce). Since Then • Bm is the magnetic field at the mirror point • Note the integral depends on the field line, not the particle • If the length of the field line decreases, will increase • Fermi acceleration

The total particle drift in static E and B fields is: • From conservation of m • For equatorial particle and static electric field • Particle conserves total (potential plus kinetic) energy • Rewriting centrifugal force in terms of a potential : • Bounce-averaged motion for particles with finite J • Particle’s equatorial trace conserves total energy

As particles bounce they also drift because of gradient and curvature drift motion and in general gain/lose kinetic energy in the presence of electric fields. • If the field is a dipole and no electric field is present, then their trajectories will take them around the planet and close on themselves. • The third adiabatic invariant • As long as the magnetic field does not change much in the time required to drift around a planet the magnetic flux inside the orbit must be constant. • Note it is the total flux that is conserved including the flux within the planet.

Drift paths for equatorially mirroring (J=0) particles, or • Drift paths for bounce-averaged particles, equatorial tracesin a realistic magnetosphere. Note, corotation electric field ignored

Limitations on the invariants • is constant when there is little change in the field’s strength over a cyclotron path. • All invariants require that the magnetic field not change much in the time required for one cycle of motion where is the cycle period.

ESS 154/200C Lecture 7 Physics of Plasmas