The Quantitative Formulation Of Nonlinear Alfven Waves: Part I Using MHD Eqns.

 In this post I show a quantitative formulation of nonlinear Alfven waves.    We begin with the standard equations of magnetohydrodynamics (MHD):

1)    ¶r/ ¶t  +  Ñ ·(r v) =  0

      2) r ¶v / ¶t  +  r(v·Ñ )v =   J X B - Ñ p

      3)  E  +  v x B   =  0

      4) P/ r^γ  = const.

 (Where the exponent for density in (4) is the ratio of specific heats)Also include two of the Maxwell equations:

5) Ñ x E  =  - ¶B / ¶ t 

 6) Ñ x B = m ₀  J  

From (3) and (5) We get:   Ñ x (v x B)  =  ¶B / ¶ t 

Or:  Ñ X  (Ñ X B) =   (B ·Ñ  ) v  -   (v·Ñ)  B   - B (Ñ ·v ) 

Assuming incompressibility: (Ñ·v )  =  0

Then:   ¶r/ ¶t  +  v·rÑ    +  r Ñ v =  0

(N.B.  For an incompressible flow dr / dt = 0  and r is constant along a stream line.)

Introduce a fluctuation such that: B  =  B ₀  +  b

Where:  b >  B ₀ 

Step II.  

Assume  B ₀    = const., also assume: r   = r ₀  =  const.  (uniform plasma)

Then:  (B ₀ ·Ñ ) v   - ¶ b / ¶t   =  (v·Ñ)b -  (b ·Ñ  ) v 

 Now sub substitute equation (6) into (1):

r ¶v / ¶t  +  r(v·Ñ )v =   1/m ₀   ( Ñ x B) x B -  Ñp

B x (Ñ x B)  =   ½ Ñ B ²   -   (B ·Ñ  ) B 

=   ½ Ñ (2 B ₀ ·b) + ½ Ñ b ²   - (B ₀ · Ñ ) b – (b · Ñ ) b

Combine last 3 terms:

 ½ Ñ b ²   - (B ₀ · Ñ ) b – (b · Ñ ) b  =

 Ñ ( B ₀ ·b) - (B ₀ ·Ñ ) b + b x (Ñ  x  b)

Use the momentum equation:

r ¶v / ¶t  +  r (v·Ñ )v + 1/m ₀  Ñ (B ₀ ·b) =

1/m ₀  (B ₀ · Ñ ) b  - 1/m ₀  b  x (Ñ  x  b)  -  Ñp

 Now apply the identity:

B x (Ñ x B)  =   ½ Ñ B ²   -   (B ·Ñ  ) B 

 

à    1/m ₀ r  (B ₀ · Ñ ) b  - ¶v / ¶t  =

 1/m ₀ r Ñ (B ₀ ·b) + 1/m ₀ r  x (Ñ  x  b) 

+   1/ r   Ñp   +   Ñ v ² /2  -   v  x (Ñ ·v)

Whence we arrive at two nonlinear equations:

(A)           ( B ₀ · Ñ ) v -    ¶b / ¶t  =  (v·Ñ )b  -  (b·Ñ )v 

(B)           1/m ₀ r  (B ₀ · Ñ ) b  - ¶v / ¶t  = 1/m ₀ r  b x (Ñ  x  b)  -  v x (Ñ  x  v)  + 1/ r   Ñ [(p +  1/m ₀  ( B ₀   ·b)]  + Ñ v ² /2 

Step 1b towards a solution requires noting the relation of v to b in the preceding equations. 

Then:  v  =  +   b/ Ö mr ₀

This leads to Step (2), substituting v into the nonlinear equations (A) and (B).

1/ r ₀   Ñ [(p +  1/m ₀  ( B ₀   ·b)]  + Ñ ( b ² /2 mr ₀  )  =

1/ r ₀   Ñ [(p +  1/m ₀  ( B ₀  b)  + b ² /2 m ₀  

=   1/ r ₀   Ñ [(p +  1/ 2m ₀  ( B ₀  + b) ²  -  B ₀  ² /2 m ₀  ]

 

Note that a key part of the solution is the pressure balance condition:

P +  B ² /2 m ₀    = const.

Since Pr ^{- g}  = const.,  then r is constant then P is constant.

à   B ² =  const.,   b ²  = const. 

 This means that nonlinear Alfven waves must be circularly polarized. We have for the phase velocity:

 v _f     =  +  B ₀ / Ö m ₀ r ₀

i.e.   For  v  =  +  b/ Ö m ₀ r ₀

Then if  B ₀ , b  are parallel the directions of propagation must be anti-parallel.

Suggested Problem:

Show that:   v _f     =  +  B ₀ / Ö m ₀ r ₀

Hint: Check solutions for dz/dt.

Consider: b =  b ₀  f (z  -   v _f   t)

And:  v _f      >   0

 Let:  x =    (z  -   v _f   t)

0 =   dz/dt   - v _f          And:    v _f       = dz/dt  

You will also need the following partials:

¶b / ¶z ,   ¶b / ¶t, ¶r / ¶z  and   ¶b / ¶x

See Also:

• Revisiting Linear Alfven Waves