Solution: We are given the original dispersion equation:
F(w) = (me/mi)/ ( w / w e)2 + 1/ [(w / w e ) 2 - (k Vo/ w e)2 ]
Let: x = w/ w e And: y = k Vo/ w e
Then:
1= F(x, y) = (me/mi)/ x 2+ 1/ (x2 - y2)
Now focus on Fm for F(x,y) whereby Fm > 1 and there will be two real roots (See e.g. Fm defined in the graph below and xm in relation to it)
Now, take:
¶ x F(x,y) = -2(me/mi)/ x3 - 2/ (x - y) 3
Or: ¶ x F(xm, y) = 0
And we note: ¶ 2 x F(xm, y) = 6 (me/mi)/ x4 - 6/ (x - y) 4 is always +ve
Whence: Fm ~ (me/mi) 1/3 / y 2+ 1/ y2
So Fm > 1 requires:
y < [ (me/mi) 1/3 + 1 ] 1/2
y < [( 0.08) 1/3 + 1 ] 1/2
i.e. y < 1.03
So in the long wavelength limit, the two real roots can be deduced to approach
x = y + 1 and x = y - 1 Or ( in terms of w ) :
w =
k Vo + w e
k Vo - w e
2) Explain how the plasma bifurcation above occurs mathematically
The plasma bifurcation occurs because of a counter streaming plasma flow (i.e. for ions and electrons) in velocity space, which leads to a split symmetrical solution as well as a symmetrical one - but in a different directions relative to the coordinate axes. "Instability" then occurs when the (dispersion) equation has complex roots, i.e. .when the local minimum ( Fm ). is greater than 1,
1= F(x, y) = (me/mi)/ x 2+ 1/ (x2 - y2)
Now focus on Fm for F(x,y) whereby Fm > 1 and there will be two real roots (See e.g. Fm defined in the graph below and xm in relation to it)
Now, take:
¶ x F(x,y) = -2(me/mi)/ x3 - 2/ (x - y) 3
Or: ¶ x F(xm, y) = 0
And we note: ¶ 2 x F(xm, y) = 6 (me/mi)/ x4 - 6/ (x - y) 4 is always +ve
Whence: Fm ~ (me/mi) 1/3 / y 2+ 1/ y2
So Fm > 1 requires:
y < [ (me/mi) 1/3 + 1 ] 1/2
Or:
y < [( 0.08) 1/3 + 1 ] 1/2
i.e. y < 1.03
x = y + 1 and x = y - 1 Or ( in terms of w ) :
w =
k Vo + w e
2) Explain how the plasma bifurcation above occurs mathematically
The plasma bifurcation occurs because of a counter streaming plasma flow (i.e. for ions and electrons) in velocity space, which leads to a split symmetrical solution as well as a symmetrical one - but in a different directions relative to the coordinate axes. "Instability" then occurs when the (dispersion) equation has complex roots, i.e. .when the local minimum ( Fm ). is greater than 1,
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