## 2. Nonlinear equations for self-generated magnetic fieldsWe 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 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 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 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 |