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

2. Nonlinear equations for self-generated magnetic fields

We start from Vlasov's equation

where is the particle distribution function

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

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

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

we may expand in powers of the perturbed field

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

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

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

where is the transverse dielectric constant

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

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

where , and 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 is different from by a phase factor. Substituting Eq. (15) into Eq. (16) and retaining only the dominant contributions of electrons, we derive

where is the perturbed density caused by the low-frequency fields

and is the envelope of the high-frequency field

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

Through the substitutions

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

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

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

Eq. (27) is reduced to

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).

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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