6. Renormalized Perturbation Theory

Transcription

1 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 The zero-order isotropic propagators λ: bookkeeping parameter, D αβ : projector on divergence free part zero-order Green tensor: with so that

2 zero-order velocity: Fourier transform in time: = Navier-Stokes zero-order propagator is given by

3 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 )

4 homogeneous isotropic turbulence and Perturbation expansion for Q Take Fourier transformed NS-equation, invert linear part and substitute forcing by u (0)

5 Now substitute perturbation expansion for u equating coefficients: u (2) can be expressed by u (0)

6 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:

7 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

8 second-order: two M factors: third-order: three M factors:

9 graphical expansion for correlation tensor zero-order: second-order: this is middle second-order term:

10 this is the last second-order term: The third is a mirrow image of this one.

11 fourth-order showing four of the 29 fourth-order diagrams: Now resummation (renormalisation): new diagram elements Write correlation tensor as:

12 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

13 of which reads where G(k, ω) is the renormalized propagator. Graphically, this corresponds to

14 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

15 replace 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.

16 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

17 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

18 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.

19 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.