2004/12/22 | 发电方程[Dynamo Equation]
类别(Ω〖物理〗) | 评论(0) | 阅读(35) | 发表于 12:41
Consider an electrically conducting fluid (i.e., a plasma). Ampère's law, ignoring the Maxwell displacement current term, states
(1)

where B is the magnetic field, is the permeability of free space, and J is the current density. Ohm's law, including the induction electric field, is
(2)

where is the electrical conductivity, E is the electric field, and v is the fluid velocity. Finally, Faraday's law states that
(3)

Taking the curl of (2) gives
(4)

Plugging in (1) and (3),
(5)

Now, if is constant, it can be pulled out, giving
(6)

Now define the magnetic diffusivity
(7)

and use the Maxwell equation gives
(8)

to obtain
(9)

Plugging (7) and (9) into (6) gives
(10)

(11)

This is the dynamo equation, also known as the hydromagnetic equation. From this equation, it can be shown that fluid motions cannot generate an exact dipole field or any other field with rotational symmetry, a result known as Cowling's theorem.
For incompressible plasmas (note that in reality, most plasmas are compressible), conservation of mass requires that the velocity be divergenceless, so
(12)

Using the vector identity
(13)

and plugging (12) and (13) into (11) then gives
(14)

so the resulting equation is
(15)

These terms physically correspond to
(16)

(17)

(18)

In order for the field to be nondecaying, the magnetic Reynolds number
(19)

must be larger than a critical value . The frozen flux approximation assumes that the magnetic field is not changing in space, so
(20)

and
(21)
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