An In Depth Treatment Of The Landau Contour - And The Role Of the Inverse Laplace Transform. (Part 2)

Recall at the end of Part 1, I noted Landau's contribution:

What Landau did was to extend e (k,w) to the entire w- plane. In velocity space the pole can be anywhere in the upper right quadrant (since w _i >  0). To make e (k,w) an entire function one needs to follow an analytic continuation process, i.e. from the upper half plane to the lower half plane.  

Thus we need:

i) I =   ò ^¥_-¥  ^du f_o /(u  -  w/ k)                              (w >0)

ii) I =   ò ^¥_-¥  ^du f_o/(u   -  w/ k)  + 2pi (Res)      (w _i > 0)

These will feature later on when we determine the Cauchy principal value P for the quantity:   e(k, wr) 

We look next at the dielectric function to see if any damping is associated with normal modes. We find:

e  =1 - w_e ²/ k^{2   ò  du} f_o  / ( u   -  w/ k)    

Note the dielectric function is equivalent to the Poisson equation: 

ik E  =  4pr

Now consider:

E (w ) =  4pe/ k² e (k,w)   ò  dv f₁  (k, v, t= 0)/  v _x-  (w/ k)

analytically continued to the entire complex   w -plane.  Then the L-contour will be deformed as shown in the graphic below:

No half plane attempt is made to close the contour, i.e. with a semi-circle in the lower  w-plane since it isn't clear that E (w) falls fast enough at large negative  w_i  for this to be done. The alternative method is to treat each part of the contour separately. In this case four types of contributions emerge but we focus on one in this post.

 That is, the poles of  E (w) such as the one at A.  Assume the pole comes from a zero of E (w)  instead of the factor on the right of the equation. We can denote the frequency at A by w_A  - assuming it originates from the zero at E (w).  Then the pole contributes a term to E(t) with time dependence, exp (-iw_A t).

These types of contributions are called normal modes and since their frequencies are given by  E (w)= 0 they correspond to waves found by solving the dispersion equation for electrostatic waves in unmagnetized plasma, i.e.:

1 + w_e²/ k^{2   ò  du  d} u g(u)/  u   -  w/ k = 0

The dominant normal mode contribution comes from the mode with the largest imaginary part of the frequency, e.g. A. if positive the normal mode is unstable and grows with time. If negative, the mode is damped and decays with time. Typically, one looks at the dielectric function to see if any damping is associated with the normal modes, i.e.

 e(k, w)   = 1 -  w_e²/ k^{2   ò  du  d} u g(u)/  u   - ( w/ k) 

We rewrite the dispersion relation (which we will also use at the end) as:     e (k, w) =   er   +  i er    = 0

Or:   e (k, w) =  0

We have:  w_i =  e i (k, wr)/ {¶ e r (k, w )/¶ w] w= wr   

When w_i = 0, the Landau contour appears as shown below:

  It is instructive to perform the integrations from just above the pole to just below to show the explicit form for the Cauchy principal value, P.

From just above the pole:

I ⁺ =  ò ^¥_-¥  ^du f_o /  (u  -  w/ k -  id) 

From just below:

I ^- =  ò ^¥_-¥  ^du f_o /  (u   -  w/ k -  id) +  2pi (Res)

Take the average:   

½ [ I ⁺ +  I ^- ]  Þ 

lim d _{® 0}   [ò ^¥_-¥  ^du f_o / (u  -  w/ k -  id) + ò ^¥_-¥  ^du f_o /  (u  -  w/ k + id)] +pi (Res)

Where the (purple) bracketed quantity is P the Cauchy Principal Value.   Use of the Cauchy principal value allows us to obtain:

e(k, wr)    = 

1 -  w_e²/ k^2    P ò ^¥_-¥ ^{du  (du} u g(u))/  (u   -  w/ k) 

-  pi  w_e²/ k^2    du u g(u)

 The 2nd term on the right (shown directly above) comes from integrating around the 'mini' semicircle for the previous Landau contour.  This yields one half of 2 pi  times the residue.

In normal mode (steady state) behavior, we only need Landau's prescription - so there's no need to go back to the Laplace transforms- which actually contain more information than needed.

From the previous equation:

 er(k, wr) = 0

and:

e(k, w)   = 1 -  w_e²/ k²   ò  ^du  d u g(u)/  u   - ( w/ k) 

We can determine the real part of the frequency  wr  from:

e(k, wr)    = 

1 -  w_e²/ k^2    P ò ^¥_-¥ ^{du  (du} u g(u))/  u   - ( w/ k)  

-  pi w_e²/ k^2    du u g(u)|u= /k 

 For the purposes of this integration we take: 

^du u g(u) = 0  at   u   =  w_r/ k

Then integrate by parts to get:

e(k, wr)    =   1 -  w_e²/w_r² - {3k²   v_0e² w_e² /w_r⁴ }  =  0

Where:  v_0e  =  v_thermal   - which for the Langmuir wave dispersion relation:

w²   = w_e² + 3k²  v_oe² 

yields:  w_r²   = w_e² + 3k²  v_oe² 

with the assumption:

 k²   v_oe²     <<  w_r²

Taking a Taylor expansion about  w_r yields:  

e (k, wr) + (w  -  wr )¶ er/ ¶ w] wr + i e i + i (w  -  wr ) ¶ e i/ ¶ w

Solving the non-zero part:

Þ

i er (k, wr)  +  i wi ¶wi  ¶ w] wr + i e  = 0

er= 0     Þ   wr  =  wi   =  -  ei  /¶ er/ ¶ w] wr   

=    pi  w_e²/ k^2   (df_o^/du) 1/(¶ er/ ¶ w) wr   

w_r²  = w_e² + 3k²  v_oe²

Which returns us to the Langmuir dispersion relation obtained earlier via integration by parts. 

Note that:  

(¶er/ ¶ w) ] wr 

 »

  2 w_e²/w_r3  

 »  

2 /w_e

Then by:    w_i =  e i (k, wr)/ {¶ e r (k, w )/¶ w] w= wr  

And: e(k, wr)   = 1 -  w_e²/ k^{2   P} ò ^¥_-¥ ^{du  (du} u g(u))/ (u  -  w/ k) 

-  pi  w_e²/ k^2    du u g(u)]u= w/k

Þ

w_i   = p w_e³/ 2k^2   (du u g(u)]u= w/k

And the total normal mode frequency is given by:

 w   = w_e (1 + 3/2  k²   l_e²) + i p w_e³/ 2k^2  (du u g(u)]u= w/k

Assume now that the phase velocity is large compared to the thermal velocity, i.e.      vf ² >>  v_oe²

Then:  er  = 1 -  w_e²/w²

Þ     (¶er/ ¶ w)   =  2 w_e²/w_r3    +  6k²   v_0e²/w_r³

  k²   v_0e²/w_r³     =  k²   l_D²/w_e

 w_i   =  -  ei  /¶ er/ ¶ w] wr   =  p w_e²/ k^2   (df_o^/du) (1/ 2 w_e²/w_r3 )

=  {p/2}  w_e³/k^2   df_o/du]u= v f

Note from the preceding that the damping rate depends significantly on the slope of the distribution function, i.e.

If then the phase velocity ( vf) or f_o (u) shows negative slope you get damping.  Hence, if there are more slower than faster plasma particles one gets a net energy transfer from wave -> particle.

In Part 3 I look at the transient (energy) effects on normal modes for the three-way deformed Landau contour and also obtaining the plasma dispersion function.