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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:

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

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