4.2.5 Non-Uniform Electric Field

4.2.5 Non-Uniform Electric Field • Equation for the particle motion

Non-Uniform Electric Field (II) • Need to evaluate Ex(x) (Exat the particle position) • Use undisturbed orbit approximation: and in to obtain

Exercise 7 • Explain why the 1st order Taylor expansion for cos and sin requires krL<<1

Non-Uniform Electric Field (IV) • Use orbitaveraging: expecting a drift perpendicular to both E and B • Velocity along x averages to zero • Oscillating term of velocity along y averages to zero: • 1st order Taylor expansion for cos and sin for krL<<1 yields

Non-Uniform Electric Field (IV) • The orbitaveraging makes the sin term to vanish • The average of the cos term yields: • This expression was obtained as special case with E non-uniformity perpendicular to y and z • The general expression for the ExB modified by the inomogeneity is:

Exercise 8 • Why has a minus sign factored while the does not?

Non-Uniform Electric Field (V): Physics Understanding • The modification of the ExB due to the inomogeneity is decreasing the ExB drift itself for a cos(kx) field • If an ion spends more time in regions of weaker |E|, then its average drift will be less than the pure ExBamount computed at the guiding center • If the field has a linear dependence on x, that is depends on the first derivative dE/dx, itwill cause contributions of weaker and larger E to be averaged out and the drift correction (in this case depending on E and dE/dx) will be zero • Then drift correction must have a dependence on the second derivative for this reduced drift to take place

Non-Uniform Electric Field (VI): Physics Understanding • The 2nd derivative of a cos(kx) field is always negative w.r.t. the field itself, as required in • An arbitrary field variation (instead of cos shaped) can be always expressed as a harmonic (Fourier) series of cos and sin functions (or exp(ikx) functions) • For such a series or in a vector form

Non-Uniform Electric Field (VII): Physics Understanding • Finally, the expression can be then rewritten for an arbitrary field variation as where the finite Larmor radius effect is put in evidence • This drift correction is much larger for ions (in general) • It is more relevant at large k, that is at smaller length scales

4.2.6 Time-Varying Electric Field • Equation for the particle motion

Time-Varying Electric Field (II) • Define an oscillating drift • The equation for vy has been previously found as • It can be verified that solutions of the form apply in the assumption of slow E variation:

Time-Varying Electric Field (III) • The polarization drift is different for ion and electrons: in general • It causes a plasma polarization current: • The polarization effect is similar to what happens in a solid dielectric: in a plasma, however, quasineutrality prevents any polarization to occur for a fixed E

4.2.7 Time-Varying Magnetic Field • A time-varying magnetic field generates an electric field according to Faraday’s law: • To study the motion perpendicular to the magnetic field: or, considering a vector l along the perpendicular trajectory,

Time-Varying Magnetic Field (II) • By integrating over one gyration period the increment in perpendicular kinetic energy is: • Approximation: slow-varying magnetic field • For slow-varying B the time integral can be approximated by an integral over the unperturbed orbit • Apply Stoke’s theorem

Time-Varying Magnetic Field (III) • The surface S is the area of a Larmor orbit • Because the plasma diamagnetismB·dS<0 for ions and vice-versa for electrons. Then: • Define the change of B during the period of one orbit as: • Recalling the definition of the magnetic moment m:

Time-Varying Magnetic Field (IV) • The slowly varying magnetic field implies the invariance of the magnetic moment • Slowly-varying B cause the Larmor radius to expand or contract loss or gain of perpendicular particle kinetic energy • The magnetic flux through a Larmor orbit is is then constant when the magnetic moment m is constant

Time-Varying Magnetic Field (V): Adiabatic Compression • The adiabatic compression is a plasma heating mechanism based on the invariance of m • If a plasma is confined in a mirror field by increasing B through a coil pulse the plasma perpendicular energy is raised (=heating)

4.3 Particle Motion Summary • Charge in a uniform electric field: • Charge in an uniform magnetic field: yields the Larmor orbit solution where

Particle Motion Summary (II) • Charge in Uniform Electric and Magnetic Fields produces the ExB drift of the guiding center • Charge Uniform Force Field and Magnetic Field produces the (1/q)FxB drift of the guiding center

Particle Motion Summary (III) • Charge in Motion in a Gravitational Field produces a drift of the guiding center (normally negligible)

Particle Motion Summary (IV) • Charge Motion in Non Uniform Magnetic Field: Grad-B Perpendicular to the Magnetic Field the orbit-averaged solution gives a grad B drift of the guiding center

Particle Motion Summary (V) • Charge Motion in Non Uniform Magnetic Field: Curvature Drift due to Curved Magnetic Field • The particles in a curved magnetic field will be then always subjected to a gradB drift • An additional drift is due to the centrifugal force

Particle Motion Summary (VI) • Charge Motion in Non Uniform Magnetic Field: Grad-B Parallel to the Magnetic Field:in a mirror geometry, defining the magnetic moment the orbit-averaged solution of provides a force directed against the gradB

Particle Motion Summary (VII) • Charge Motion in Non-Uniform Electric Field the orbit-averaged solution produces

Particle Motion Summary (VIII) • Charge Motion in a Time-Varying Electric Field: the solution in the assumption of slow E variation yields a polarization drift that is different for ions and electrons

Particle Motion Summary (IX) • Charge Motion in a Time-Varying Magnetic Field: solution of in the perpendicular (w.r.t. B) plane and under the assumption of slow B variation shows a motion constrained by the invariance of the magnetic moment

4.2.5 Non-Uniform Electric Field