6. Renormalized Perturbation Theory

6. Renormalized Perturbation Theory 

6.1. Perturbation expansion of the Navier-Stokes equation 

Kraichnan (1959): direct-interaction approximation 

Wyld (1961): partial summation for a simplified scalar model 

Lee (1965): partial summation for Navier-Stokes and MHD 

6.1.1. The zero-order isotropic propagators 

What is a plasma ? 

Kinetic description 

Fluid description 

Navier-Stokes turbulence 

Closure Theories 

Renormalized . . . 

λ: bookkeeping parameter, D αβ : projector on divergence free part 

zero-order Green tensor: 

Renormalization Group 

Passive Scalar 

Dynamic Functional 

Magnetic Reconnection 

Mean Field . . . 

with 

Magneto-Rotational . . . 

Interstellar Medium (ISM) 

so that 

Turbulence in Fusion 

Literature

zero-order velocity: 




Fourier transform in time: 




=⇒ Navier-Stokes 





zero-order propagator is given by 





Literature

6.1.2. The primitive perturbation expansion 

Stirring force 


after Fourier-trafo 




zero-order velocity field after in Fourier-space 



which is the zero-order term in expansion 




correlation Q αβ (k; ω, ω ′ ) 






Literature

homogeneous isotropic turbulence 

and 

Perturbation expansion for Q 












Take Fourier transformed NS-equation, invert linear part and substitute 

forcing by u (0) 




Literature

Now substitute perturbation expansion for u 







equating coefficients: 






u (2) can be expressed by u (0) 




Literature

stirring Gaussian =⇒ u (0) Gaussian 

=⇒ 〈 u (0) u (0) u (0) u (0)〉 can be factored as in Quasi-Normal approximation 

second-order correlation tensor: 















Literature

6.1.3. Graphical representation of the perturbation series 

all orders can be expressed by zero-order terms, but divergent series 

three main constituents: u (0) , G 0 and M 






zero-order: 




first-order: wavenumber conservation 







Literature

second-order: two M factors: 








third-order: three M factors: 








Literature

graphical expansion for correlation tensor 

zero-order: 



second-order: 









this is middle second-order term: 





Literature




this is the last second-order term: 









The third is a mirrow image of this one. 




Literature

fourth-order showing four of the 29 fourth-order diagrams: 








Now resummation (renormalisation): new diagram elements 






Write correlation tensor as: 



Literature

6.1.4. Class A diagram: the renormalized propagator 

Wyld (1961): 

Class A diagrams are those diagrams which can be split into 

two pieces by cutting a single Q 0 line. 

zero-order: Q 0 can be expressed in terms of two zero-order propagators 

acting on the spectrum of the stirring forces w(k; ω, ω ′ ) 

This looks graphically like 







Now second-order: 







Let’s summarize: at zero order, we have w with a G 0 on each side. At 

second order, w has a G 0 on one side and a diagram which connects like a 

G 0 on the other. This holds for all orders. Thus we have a generalization 



Literature

of 

which reads 

where G(k, ω) is the renormalized propagator. 

Graphically, this corresponds to 















Literature

6.1.5. Class B diagrams: renormalized perturbation series 

Class B diagrams can’t be split into two by cutting a single Q 0 line. 

In the class A diagrams, certain diagram parts were propagator like, that is, 

they connected like G 0 : renormalize G 0 by adding up all diagrams which 

connect like G 0 . 

Renormalize vertex: add up all diagrams which connect like a vertex 

Example: consider fourth-order diagram 









The part 





connects like a point vertex =⇒ renormalized vertex 



Literature

eplace vertex by renormalized vertex: 






Therefore the key to the class B diagrams is as follows: 

1. Find those diagrams which cannot be reduced to a lower order by 

replacing diagram parts. 

2. Call these the irreducible diagrams. 

3. Replace all elements in the irreducible diagrams by their renormalized 

forms. 

4. Write down all these modified diagrams in order, thus generating a 

renormalized perturbation expansion. 










Literature

Result for Q(k; ω, ω ′ ) 








This is an integral equation for Q(k; ω, ω) 



Combine vertex and propagator expansions: 

Integral equation for the renormalized vertex 






Literature

Integral equation for the renormalized propagator G(k, ω) 







Pecularity of this diagram: 

unrenormalized propagator emerging from the left !!! 

Reason for this: symbolic form of Navier-Stokes 

L 0 u(k) = λM(k)u(j)u(k − j) , L 0 = ∂ t + νk 2 

and renormalize r.h.s., then invert L 0 which results in G 0 









Literature

6.1.6. Second-order closures 

What have we done: 

We replaced a wildly divergent series with one of unknown properties ! 

We have hope that it might be assymptotic, but we simple don’t know ! 

Well known examples recovered from this Wyld (1961) formulation: 

Example 1: 

correlation tensor: truncate at second order (in number vertices) 

vertex: truncate at first order (unrenormalized vertex) 

propagator: truncate at zero order (unrenormalized propagator) 

This is Chandrasekhar’s theory (1955) which is the two-time analog of 

quasi-normality. 















Literature

Example 2: 

correlation tensor: 

vertex: 

propagator: 

truncate at second order 

truncate at first order 

truncate at second order 

This is the pioneering direct-interaction approximation (DIA) by Kraichnan 

(1959): second-order closure with line and with no vertex renormalization. 















Literature

6. Renormalized Perturbation Theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?