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Astron. Astrophys. 364, 785-792 (2000)

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2. Nonlinear equations for self-generated magnetic fields

We start from Vlasov's equation

[EQUATION]

[EQUATION]

where [FORMULA] is the particle distribution function

[EQUATION]

where [FORMULA] is the particle density and [FORMULA] is the electromagnetic force. One can divide [FORMULA] and [FORMULA] into two parts:

[EQUATION]

where [FORMULA] is unperturbed and [FORMULA] is perturbed. As [FORMULA] and [FORMULA] are closely coupled through Maxwell's equation and the current density equation

[EQUATION]

and the field [FORMULA] is assumed weak, so that the energy density excited is much smaller than the thermal one of the plasma, i.e.,

[EQUATION]

we may expand [FORMULA] in powers of the perturbed field [FORMULA]

[EQUATION]

where the index a indicates that [FORMULA] is proportional to the a-th power of [FORMULA]. Put [FORMULA] and expand [FORMULA] in a Fourier series

[EQUATION]

Taking Eqs. (5),(7) and the Maxwell equations into consideration, we can find the following field equation for the transverse mode from Eq. (1)

[EQUATION]

and the nonlinear currents (to the second and third order)

[EQUATION]

[EQUATION]

where [FORMULA] is the transverse dielectric constant

[EQUATION]

where [FORMULA] is the particle energy, [FORMULA] the polarization vector for the transverse mode;

[EQUATION]

[EQUATION]

the term [FORMULA] in the denominators of integrated functions arises from the Landau rule. For the low-frequency fields [FORMULA] and high frequency field [FORMULA], we obtain the following equations from Eq. (9):

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA]and [FORMULA]are matrix elements of interaction between high frequency fields and low frequency fields. The upper indicets "+" and "-" denote the positive and negative frequency parts for the high-frequency perturbation, and [FORMULA] is different from [FORMULA] by a phase factor. Substituting Eq. (15) into Eq. (16) and retaining only the dominant contributions of electrons, we derive

[EQUATION]

[EQUATION]

where [FORMULA] is the perturbed density caused by the low-frequency fields

[EQUATION]

and [FORMULA] is the envelope of the high-frequency field

[EQUATION]

If the characteristic scale L of the low-frequency field is much larger than Debye length [FORMULA], in coordinate representation, we can get the low-frequency field [FORMULA] from Eq. (15)

[EQUATION]

Through the substitutions

[EQUATION]

[EQUATION]

[EQUATION]

we can now write Eqs. (17),(18) and (21) in the following form (Li & Ma 1993)

[EQUATION]

[EQUATION]

[EQUATION]

We restrict ourselves to the static limit for the low-frequency magnetic field and so Eq. (25) is reduced to

[EQUATION]

Substituting Eq. (28) into Eq. (26) yields

[EQUATION]

Eq. (27) is reduced to

[EQUATION]

Therefore it may be seen clearly that a self-generated magnetic field with very low-frequency is not an oscillatory field but a quickly collapsing field, which is completely determined by the closed Eqs. (29) and (30).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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