The Quantitative Formulation Of Nonlinear Alfven Waves: Part 2 a (Derived from two- fluid equations)

 We now look at the case of non-Linear Alfven waves derived from two- fluid equations:

 We use wave frame so that: ¶ / ¶t  =  0

 Also:

1)    Continuity equation: ¶ / ¶x (n_s, V_s ) = 0 

2)    Momentum eqn.  V_s ¶ / ¶x (V_s ) = q_s, /  m_s (E + v  x B)

V_s  ¶ V_s / ¶x  =  q_s, /_/m_s  [E + V x B]

 

3)    ¶ / ¶x  (B) = 4 p å q_s, V_s  n _s,

 

4)   ¶ / ¶x  ·  (B) = 0

 

(5) ¶ / ¶x  ·  (E) = 4 p å q_s,  n _s,

We consider first the solution in 1-dimension. Choosing the field to have a geometry such that:

B =  [ B ₀ ,  B _y  (x), B _z  (x)]

And:   E =  (E(x), 0 , 0)

V_s    =   [U_s (x),  V_s  (x) _,  W_s  (x)]

Assuming exact neutrality:   n _i (x) =  n _e (x)    =    N(x)

But:  ¶ E / ¶x  =  0    So no E

We have then:

(i)  ¶ (n_s,  U _s ) / ¶x  =  0     so:  n_s U _s   = const.   (Or:  Ñ · J  = 0 )

(ii) m _s  U _s  (d U _s  / dx)    = q_s,  (V_s  B _z -  W_s B _y )

(iii) m _s  U _s  (d V _s / dx)    = q_s,  (W_s  B ₀ -  U_s B _z )

(iv) m _s  U _s  (d W _s / dx)    = q_s,  (U_s  B _y -  V_s B ₀ )

 

Further:   U_i =  U _e

dB _y / dx  =   4 p å q_s, W_s  n _s

dB _z / dx  =   - 4 p å q_s, V_s  n _s

Now multiply:    å n_s      by Eqn. (ii)

Þ   å n_s  m_s U_s   (d U _s  / dx)    =    å n _s q_s, (V_s  B _z -  W_s B _y )

Þ  F  å m_s (d U _s  / dx)    =    å n _s q_s, V_s  B _z    - å n _s q_s,  W_s B _y

Then:   

F  m _z  (dU  / dx)    =  - 1/4 p  B _z (d B _z / dx) - 1/4 p  B _y (d B_y / dx)

 

Þ   d / dx  (F  U ) =  - 1/8 p   d/dx  (B _z  ²  +   B_y ² )

 

F  U  +   (B _z  ² +   B_y  ² ) /8 p     =    const.  = P

 

Where the first term on the left is the ram pressure and the 2^nd term is the magnetic pressure.

To be continued...