# Presentation: 614801

## The effect of cantori on transport in chaotic magnetic fieldsStuart Hudson

*Cantori are the invariant sets remaining after destruction of the KAM surfaces and create partial barriers to transport in chaotic fields. **This talk shall ..**          1) give an explicit construction of cantori for magnetic fields,**          2) give some review of cantori properties,**          3) show that cantori affect advective-diffusive transport,       and    4) suggest “ghost coordinates” for chaotic fields . . . *

## Ideally the magnetic field is integrable, but non-axisymmetry destroys integrability.

- For 2-D (axisymmetric) systems, field lines lie on nested toroidal flux surfaces
- rational surfaces : a continuous family of periodic orbits
- irrational surfaces : closure of single irrational curve
- Stellarators are intrinsically 3-D (non-axisymmetric), and therefore chaotic . .
- though they are designed to be as close to integrable as possible
- . .  and some flux surfaces are destroyed.
- the O and X periodic orbits form robust invariant periodic skeleton for chaos
- the irrational _*curves*_ survive, though not necessarily the irrational _*surfaces*_
- KAM : sufficiently irrational surfaces survive sufficiently small perturbation
*[Kolmogorov 1954, Arnol’d 1963, Moser 1962]*
- Greene : existence of KAM surfaces determined by stability of nearby periodic orbits *[Greene, 1979]*
- Cantorus : remnant invariant irrational set; KAM surface with gaps
*[Percival 1979; Aubry 1983; Mather 1982; MacKay . . .]*
*cantori play the dominant role in restricting transport in irregular regions*

## Cantori restrict transport in chaotic regions

*KAM torus*
*(complete barrier)*

*cantor+torus=cantorus*
* **(partial barrier)*

* **cantori are leaky, but can severely restrict transport (depending on degree of destruction)**                              *

*near critical cantori*

*super critical cantori*
*(black dots)*

*near critical cantori*

*near critical cantori*

*near critical cantori*

*10**5** iterations*

*10**5** iterations*

*connected chaos  *
*(**no KAM)*
*cantori restrict transport*

*e.g. Poincare Plot*

## Cantori are identified by their (irrational) transform, approximated by periodic orbits. 

- Cantori are identified by their (irrational) transform, approximated by periodic orbits.

*Farey Tree*
*(spanned by 1⁄2, **1⁄3)** *

* **tails of 1,1,. = noble irrationals = alternating path;*
* **noble irrational KAM surfaces most robust;*
* **noble irrational cantori most important barriers;*

## Outline of Talk

- Construction of cantori for magnetic field (Hamiltonian) flow
- cantori approximated by (high-order) unstable periodic orbits
- need to construct high-order periodic orbits, in chaotic regions, for continuous time magnetic field line flow (rather than discrete time mapping)
- Lagrangian variational methods are robust, efficient, and field-line-flux across cantori easily quantified
- Investigation of effect of cantori on *diffusive* Hamiltonian flow
- the advection-diffusion equation, with a chaotic flow, a discrete-time model, is solved numerically
- graphical evidence is given that cantori (and the unstable manifold) have an important impact on the steady state distribution
- Describe ghost coordinates for chaotic fields
- ghost curves are curves that connect the stable & unstable periodic orbits
- these possibly may provide a convenient framework for understanding chaos . .

## Magnetic field lines are determined from Lagrangian variational principles

*sensitive to exponential increase of integration error*

*robust, efficient;*
*need to specify trial curve*

## The simplest representation of a trial curve is piecewise-linear

*Action integral integrated piecewise analytically*

*Hessian cyclic tridiagonal;*
*easily inverted O(N);*
*initial guess by tracking;*
*2**nd** order convergence;*

*if ****0** constrained, *
*Hessian is tridiagonal*

## Unstable periodic orbits are action minimizing curves;stable periodic orbits are action minimax curves

*unstable X curve;*
*action minimizing*

*surface*

*stable O curve;*
*action minimax*

*flux independent of surface*

- Unstable periodic orbits are action *minimizing* curves;stable periodic orbits are action *minimax* curves

## Analysis of chaotic magnetic field; construction of critical function, flux Farey tree

*Critical Function*
*(application of Greene’s residue criterion)*

*Flux Farey Tree*
*(for case k=2.1**×10**-3**)*

*transform*

*transform*

## The Lagrangian variational approach provides an efficient, robust method of constructing high-order periodic orbits ≈ cantori

*periodic orbits, with periodicity ~ 10**5**, are located, even in strongly chaotic regions, for continuous time flows*

## Higher periodic orbits approximate the cantori;the gap structure becomes clear.

*k < k**c **, KAM surface exists*
*high-order periodic orbits tend to fill curve*

*k > k**c **, KAM surface destroyed*
*high-order periodic show gaps*

## The flux across a cantorus is given as the limit of the flux across the convergents

*expressions from renormalization (MacKay)*

*expressions from MacKay for nobles in std. map.*

## Piecewise linear approximation gives 2nd order error

*The piecewise linear representation works well; *
*this is because the magnetic field lines are locally very smooth, being dominated by the strong toroidal field*

## Part I : Construction of cantori for flows : Summary

- Using the variational formulation, and piecewise-linear representation, cantori (high-order periodic orbits) are efficiently, and robustly, constructed for flows
- previously, cantori had only been calculated for maps (eg. standard map)
- periodic orbits, with periodicity ~ 104, calculated for chaotic fields (could probabaly be increased with some computational care)
- robustly here means immune to Lyapunov exponentiation of error
- The *‘flow’-*cantori share essential features of *‘discrete’-*cantori
- in particular, expressions for the flux across noble cantori are confirmed
- Cantori probably play an important role in chaotic transport
- cantori are clearly dominant restriction to Hamiltonian transport
- the remainder of this talk shall consider Hamiltonian+diffusive transport
- .

## Now, moving on to Part II of this presentation,do cantori play an important role in diffusive Hamiltonian systems ?

- Now, moving on to Part II of this presentation,do cantori play an important role in *diffusive* Hamiltonian systems ?

*L**D** – L**A** inverted using BiCGStab*

## The 2D advection-diffusion equation is solved in a chaotic layer surrounding an island chain

- The 2D advection-diffusion equation is solved in a chaotic layer surrounding an island chain

*lower boundary*
*T=1.0*

*upper boundary*
*T=0.0*

*stochastic sea*

*island flux surfaces*

*coordinate grid*

*flux surface*

*flux surface*

***Standard Map k=0.95***

## Steady state T contours are shown;the unstable manifold has an important effect

*D=10**-4*

*D=10**-5*

*D=10**-7*

*D=10**-6*

*unstable manifold*

* **the importance of the unstable manifold is well known in fluid mechanics*
*observed in DIIID heat footprint*
*as D decreases, invariant manifolds play an important role*

*T contours*

## The cantori have an important effect on advective-diffusive transport 

*cantori shown as black dots*

*T contours shown as lines*

*as D decreases,*
*correlation between cantori and T improves*

*D=10**-4*

*D=10**-5*

*D=10**-6*

*D=10**-7*

## Ghostcurves fill in the holes in the cantori,and align with Temperature contours  

- Ghostcurves fill in the holes in the cantori,and align with Temperature contours

*ghostcurves red lines*

* **the “holes” in cantori are  filled by ghostcurves,*
* **T contours, cantori and ghostcurves are related           **- depending on Diffusion, and criticality of cantorus*

|   [0,3,3,1,.] |   521/1707 |   11/36 |
| --- | --- | --- |
|   [0,3,2,1,.] |   377/1275 |   13/44 |
|   [0,3,1,1,.] |   233/843 |   21/76 |
|   [0,4,2,1,.] |   610/2673 |   13/57 |
|   [0,4,1,1,.] |   377/1741 |   13/60 |

*irrational : cantorus : ghost curve*

*cantori black dots*

*T blue lines*

*D=10**-6*

*D=10**-7*

## Ghostcurves do not intersect, and may beused to form a chaotic coordinate grid

- Ghostcurves do not intersect, and may beused to form a chaotic coordinate grid

* **different ghostcurves don’t intersect*
* **interpolation might be problematic                                                                                                    -- careful selection of (p,q) required*
* **can construct “chaotic-coordinates”*
* **coordinates cannot straighten chaos,                                                                                                  -- but coordinates that capture the                                                                                                                  invariant periodic sets come close *
* **a theory of diffusive-transport                                                                            across ghostcurves may                                                                                     complement numerical work*

## Further analysis is pending . . .

- Cantori, and other invariant sets such as the unstable manifold, clearly affect transport in chaotic, diffusive systems.
- Present work has given graphical evidence that shows cantori and *T* coincide, for small *D*
- Remains to quantify . . . .
- degree of correlation between cantori and *T*   (depends on degree of criticality, *D)*
- what is the diffusive flux across the ghostcurves ?   (depends on *D, *boundary conditions)
- can constructing a set of cantori/unstable-manifolds/ghostcurves (fast) provide
- a better initial guess for iterative solution of the steady-state Temperature ?
- optimal coordinates for analysis of chaotic flows ?
- Plan to extend to consider heat-diffusion in chaotic fields

## Poincare recurrence