# Presentation: 246264

## A stepped pressure profile model for internal transport barriers

- M. J. Hole1, S. R. Hudson2, R. L. Dewar1, M. McGann1 and R. Miills1

- [1] Research School of Physical Sciences and Engineering, Australian  National University, ACT 0200, Australia
- [2] Princeton Plasma Physics Laboratory, New Jersey 08543, U.S.A.

*Supported by Australian Research Council Grant DP0452728*

- _Acknowledgements :_ Useful discussion / input from Brian Taylor (FRS), Robert MacKay (FRS), Chris Gimblett, Allan Boozer

## Contents

- Motivation
-   - MHD equilibria in 3D
-   - Proposed solution for 3D MHD equilibria: stepped Beltrami  equilibria
-   - Model applications to internal transport barriers
-   - Project aims
- Variational principle for a stepped pressure profile model
-   - Equilibria
-   - Stability
- Generalised Taylor cylindrical plasmas
- 	- Equivalence to tearing mode treatment
- 	- Ideal/Taylor cylindrical plasmas
- 4. 	A model for internal transport barriers
- 5. 	Summary

## 1. Toroidal plasma equilibrium in 3D

- Good model for toroidal fusion plasma steady state is force balance for total pressure *p* combined with Ampère’s law relating magnetic field **B** and current density **J**:

- and Tokamaks, (*also* 3D due to coil ripple or instabilities):

- EG Stellarators—intrinsically 3D, i.e. no continuous symmetry:

-  Magnetic fields are in general fully 3D

## 3D Equilibrium Problems

**Problems:**
- A. In 3D plasmas, magnetic islands form on rational flux surfaces, destroying flux surface
- Field is chaotic within magnetic islands, and ergodically fills island volume
- Fortunately... not all flux surfaces are destroyed
- B. 3D MHD equilibria have *current singularities* if *p*  0

## A. 3D MHD field is, in general, chaotic

-  Field lines can be described as a 11⁄2 DOF Hamiltonian

*H = * ** ** *, t = **** *
*p = **,   q = ***

-  If axisymmetric, Hamiltonian is *autonomous*
-   (ie. * * ** **    **)

- If non axis-symmetric , Hamiltononian is *non-autonomous*   ** **  and the field is in general *non-integrable*.
- Magnetic islands form at rational **(), in which field is stochastic or chaotic, ergodically filling the island volume.
- B*p*=0  confinement lost in these islands.

- = irrational:  **B **ergodically passes through all points in magnetic surface.
** = rational (m/n) :  **B** lines close on each other.

- Eg. **B***** ***Poincaré sections in H-1, courtesy S. Kumar

## Some sufficiently irrational magnetic flux surfaces survive 3D perturbation

- Perturbs an integrable Hamiltonian p within a torus (flux surface) by a periodic functional perturbation p1:

**KAM theory :** if flux surface are sufficiently far from resonance (*q *sufficiently irrational), **some **flux surfaces** survive for ******** < ********c**

-  Kolmogorov Arnold Moser (KAM) Theory (c. 1962)

- Maximising number of good flux surfaces is the topic of advanced stellarator design S. R. Hudson *et al* Phys. Rev. Lett. 89, 275003, 2002

## Standard map captures breakdown of surfaces

- Properties of 3D field captured by standard map:

- For N’th toroidal orbit:

- 3D perturbation parameter

- k=0

- 

- s

- 

- 

- s

- s

- k=0.9

- k=1.1

*Increasing destruction of flux surfaces*

-  For k>0.98 no surfaces (extending over all ) exist

## B. Current singularities exist in 3D equilibria

-  General Case :

- 

- For rational , (**B**. ) in 3D is a singular operator   *J**||*  
- To remove singular currents we require . **J****** =0 *p*=0

- With solution

- &  constant on field lines

- To see singularity of (**B**. ) operator, Fourier expand (**B**. )  in magnetic coordinates (, ***m**, ***)

- Radial coordinate
- (const  surface)

- Toroidal angle**
- (** = const locus on constant  torus)

- Poloidal angle**
- (** = const locus on constant  torus)

-  = *average* *toroidal* flux but field lines do *not* necessarily lie within this torus.

## Singularity of (B.) operator 

-  Fourier expand (**B**. ) in magnetic coordinates (, ***m**, ***)

**() = rotational transform (=1/*q*)
-        = poloidal / toroidal transit of field

-  If **() *=n/m*, then (**B**. )=0, and so

- We require *p*=0 for rational **(),
*J**||* singularity removed

- Eg. ITER field lines

- NB: For a tokamak, only 0m is nonzero, & problem vanishes

## Recap: 3D Equilibrium Problems

**Problems:**
- A. In 3D plasmas, magnetic islands form on rational flux surfaces, destroying flux surface
- Field is chaotic within magnetic islands, and ergodically fills island volume
- Fortunately... not all flux surfaces are destroyed
- B. 3D MHD equilibria have *current singularities* if *p*  0

## Proposed solution:  Stepped-pressure Beltrami equilibria

*Pros*
- Beltrami eqn. is a linear elliptic PDE, solvable by variety of methods even if **B** has chaotic regions
- Has already been partially investigated mathematically [e.g. Bruno & Laurence, Comm. Pure Appl. Math. **XLIX**, 717 (1996)]
*Cons* (?)
- Pressure profile not differentiable (**but** may approximate a smooth profile arbitrarily closely, limited only by existence of invariant tori)

- To ensure a mathematically well-defined *J*, we set *p* = 0 over finite regions  B = **B, *** *= const (*Beltrami field*)* *separated by assumed* invariant tori*.

- Different *** *in each region

## Force balance on invariant tori

- Pressure discontinuous: [[*p*]]  0 (where [[]] is jump across an invariant torus), but *total* pressure, magnetic plus kinetic, is continuous: [[*p*+*B*2/2]] = 0
-  -function *p*
-  sheet current **J**
-  discontinuity in **B **(both magnitude & direction)
-  winding number ** not necessarily same on either side of invariant torus (*not* standard KAM problem)

- If constructed as a variational problem, may have other applications...

- NB: Beltrami or force free field initially inspired by Astrophysical research, Chandrasekhar, Woltjer et al.

## Internal Transport Barriers

- Plasmas with radially localised regions of improved confinement with steep pressure gradients1. Typically,
- non-monotonic q profile
- q rationality plays a role (eg. appearance of q=2 can promote ITB formation in JET)
- Most theoretical models rely on suppression of micro-instability induced transport by sheared **E****B** flows.. *however *do not offer an energetic reason for ITB formation

- 1*J. W. Connor et al, Nuc. Fus. 44, R1-R49, 2004.*

## ETB

- Also in MAST ...M. J. Hole *et al*., PPCF, 2005

- ITB

- #7085 at 290ms

## Variational Model for ITB’s

- _Idea:_ perhaps a variational formulation (min. energy states) of a relaxed plasma-vacuum system may offer insight into *why* the ITB forms

- Trial pressure profile, field in each region Beltrami

## Builds on recent variational models of  ETB’s

- A relaxed plasma-vacuum model has been recently applied to Edge Transport Barriers to describe the ELM cycle:
- Cylindrical geometry assumed
- Toroidal peeling modes initiate Taylor relaxation
- Taylor relaxation flattens torodial current profile, further destabilising peeling mode, *but...*
- Stabilising edge skin current also formed by relaxation
- Balance between destabilising and stabilising terms determines width of the ELM.

- 1 C. G. Gimblett et al PRL, 035006, 96, 2006

## Project Aims  

- design a convergent algorithm for constructing 3D equilibria,
- solve a 50-year old fundamental mathematical problem
- quantify relationship between magnitude of departure from axisymmetry and existence of 3D equilibria
- provide a better computational tool for rapid design and analysis

- (2)  explore relationship between ideal MHD stability of multiple interface model and internal transport barrier formation

- NB: Group is investigating different methods to construct 3D equilibria:
- Variational Approach, method of lines/shooting method1 , Hamilton–Jacobi equation for surface magnetic potentials*, *Hamiltonian trial function. This talk will focus on an **energy variational approach**.

- 1 S.R. Hudson, M.J. Hole and R.L. Dewar, Phys. Plasmas **14**, 052505 (2007)

##  In 1974, Taylor argued that turbulent plasmas with small resistivity, and viscosity relax to a Beltrami field 

-  i.e. solutions to W=0 of functional :

- Internal energy:

- Taylor solved for minimum *U* subject to fixed *H*

- Total Helicity :

*I*

*P*

*V*

- Zero pressure gradient regions are **force-free** magnetic fields:

**2. A Stepped Pressure Profile Model**

## Generalised Taylor Relaxation 1/2

- Require constraints to be a **subset** of the **ideal-MHD** constraints, so generated states are ideal equilibria:

- Ideal MHD: infinity of constraints

- Relaxed MHD: *finite* no. of constraints

- Spaces of allowed variations:

**Idea: ** Extremize total energy

- Kruskal–Kulsrud equilibria — *include* Taylor states

- Generalized Taylor equilibria

- cf. A. Bhattacharjee and R.L. Dewar, Phys. Fluids **25**, 887 (1982)* Energy principle with global invariants*

**Generalised Taylor Relaxation 1/2**

## Generalised Taylor Relaxation 2/2

- helicity functional:

- mass functional:

- New system comprises:
-  *N* plasma regions *P**i* in relaxed states.
-  Regions separated by ideal MHD barrier *I**i*.
-  Enclosed by a vacuum *V*,
-  Encased in a perfectly conducting wall *W*

*W*

**Generalised Taylor Relaxation 2/2**

- toroidal and poloidal fluxes:  ***i* and ***i*

- Assume each invariant tori  *I**i* act as ideal MHD barriers to relaxation, so that Taylor constraints are localized to subregions.

## 1st variation “relaxed” equilibria

- Energy Functional *W*:

- Setting *W*=0 yields:

**n** = unit normal to interfaces *I*, wall *W*

- Poloidal flux ***pol*, toroidal flux ***t* constant during relaxation:

## 2nd variation stable equilibria

- + expressions for perturbed fluxes, pol , t in each region.

- . Yields

-  Minimize 2*W, *wrt fixed constraint. Two possible choices are

- NB :
**b** = **B**
- n = ******·****B**
****** **= interface displacement vector

- Find solutions of

- with

## 3.Generalized Taylor Relaxed Cylindrical Plasmas

- ...

- R

- rw

- rN=1

- r1

- rN-1

- Each region Pi has Lagrange multiplier i, pressure pi
- At interface *I*, safety factor on inner and outer sides is i, o

- I1

- IN-1

- IN

## B solutions produce eigenvalue problem

- Cylindrical solutions are Bessel functions:

- with unknowns: coeffs. ki, di; interface radii ri; vacuum field.
- 4N+1 unknowns can be cast as constraint set:

- This is an eigenvalue1 problem for i.

- with

- 1 S. R. Hudson, M. J. Hole, R. L. Dewar Phys. Of Plas., 14, 052205, 2007

- or

## Eigenvalues for Beltrami multiplier

- Eg. Constrain 1=2, r1=0.5, r2=1  solutions of 2 are multi-valued in 2

- Fundamental: continuous  in  *r**1**<r< r**2*

- 1st harmonic: (r) one pole in *r**1**<r< r**2*

- 2nd harmonic: (r) two poles in *r**1**<r< r**2*

- Eigenvalues versus 2

- Rotational transform radial profile

## Equilibria with positive shear exist

- Eg. 5 layer equilibrium solution

- Contours of poloidal flux p

*q* profile smooth in plasma regions,
- core must have some reverse shear
- Not optimized to model tokamak-like equilibria

- M. J. Hole, S. R. Hudson and R. L. Dewar, J. Plasma Phys., 72, 1167, 2006

## Approximating continuous pressure gradients 

- Eg. 5 layer equilibrium solution

- Contours of poloidal flux p

* **p* is piecewise smooth.

## Spectral Analysis  eigenvalue problem

- Fourier decompose perturbed field **b** and interfaces i

-  In *P**i*,* V*,  ODE’s solved: eg in *P**i*

- 2(N + 1) unknown constants *c**i1*, *c**,i2*

- - BC’s at wall and core eliminate 2 unknowns

- - Apply 1st interface condition 2N times (inside + outside)

- 2nd interface condition reduces to form

-  choice of N2 has shifted Alfvén continuum to 
-  # solutions for lambda = *N*

## Cylindrical Example: Stability 

-  For N interfaces reduces to eigenvalue equation

-  For *N=1, *reproduces calculations of Kaiser and Uecker, Q. Jl of Appl. Math. 57(1), 2004

- m=1

- m=2

- Marginal stability boundaries (=0) for m=1 and m=2

- 

- = jump in pitch across interface

- 

- 

- M. J. Hole, S. R. Hudson, R. L. Dewar, Nuc. Fusion, **47**, 2007

- Stable region interior to locii

- Stable region exterior to locii

## Benchmarking multi-interface stability

**Benchmarking multi-interface stability**

- Consider two similar plasma regions, with 1= 2

- Dispersion curves match for *N=1* and *N=2* in limit of vanishing interface separation ***r = r**2** – r**1*

*r**1*

*r**2*

*r**w*

- Unstable configuration studied

- 2-02

- 02=0

- 

- 

- m=1

## N=2: Stability differs if q is C0, even if r0

***N=2: *****Stability differs if *****q***** is C****0****, even if *****r****0*

**Aim:** Investigate stability of plasma with *q* continuous across second plasma region (adjust ***2** to *match* q *at r1, r2)
**Motivation:** Eliminate singular current on interface (infinite current density) by removing jump in *q*
**Result:** For ***r ****0,* *N=2* unstable, while *N=1* stable. Physically, r 0 requires 2    new current sheet in *P**2*.

- 2<0

- 2

- 2

*P**1*

*P**2*

*I**1*

*I**2*

## Resolving the edge pressure (R. Mills et al)

**Aim:** What is the effect of resolving edge pressure gradient?
**Path:** In *P**2* make plasma ideal with nonzero ramp in pressure
- need to solve explicitly for (r), as **b** = (******B**)
- produces Euler-Lagrange equation for  (r)
- resonances must be handled explicitly1

- 1Newcomb, Annals of Phys. 10, 232-267, 1960

-  Taylor-Taylor treatment allows resistive modes in *P**2* as *r*0.
-  Taylor-ideal, and Taylor do not.

| reson in P2 | N=2 |  | N=1 |
| --- | --- | --- | --- |
|  | Taylor-Taylor | Taylor-Ideal | Taylor |
|  | 2<0 | 2>0 | 
2>0 |
|  | 2>0 | 2>0 |  |

- Ideal: helicity per field line conserved
- Taylor: helicity in each volume conserved

**Result:**

## Modes of generalized Taylor relaxation

- What unstable modes allowed by these generalized Taylor states?
- 	- modes of ideal MHD, current driven except at interfaces (where pressure driven modes can exist)
- 	- tearing modes

- Recent work by Tassi *et al* investigate tearing mode stability of force-free equilibria:
**Motivation:  **MHD activity often observed in RFP’s with well defined value of m/n, equal to central resonance (quasi-singular helicity states). Taylor’s theory suggests entire plasma should relax via modes with multiple-helicity (m/n)

*Tassi , Hastie and Porcelli  Phys. Plas., 14, 092109, 2007*

## Tearing modes of force-free equilibria

*Tassi et al Phys. Plas., 14, 092109, 2007*

- We generalize model for multiple interfaces, inclusion of vacuum and to enable pressure jumps.

*Tassi et al*
- Assume two plasma regions, no vacuum, and no pressure
- Conclude presence of downward step in q (lamda) destablises innermost mode, and suggest possible cyclic mechanism for QSH relaxation

## Tearing mode vs. Taylor stability 

- generalize model for multiple interfaces, inclusion of vacuum and to enable pressure jumps.
- Does tearing mode and variational principle stability agree? *YES*

- q

- r

- 0

- 0.1

- 1

- 1.1

- -0.05

- ’→

- Variational eigenvalues

- tearing modes

- m=1

- m=1

- ’→

- external kink

- Internal kink

## A model for Internal Transport Barriers

- reverse shear profile cylindrically periodic plasma assumed
- Choose qmin, q0 and qedge chosen,
- Stability computed for these configurations
- Appeal to toroidal mode number discretisation to tune system to eliminate unstable modes.
- In limit that m→, n →. Is anything stable left?
- Scan over qmin, q0
- Investigate pressure profile dependence

- Equilibrium constraints:

## In progress: Exploration of ITB’s

- 2<q100<4 stable

- marginal stability

- internal mode

- external mode

- m=2

## Other approaches to constructing 3D equilibria

- Development of iterative algorithms to solve Beltrami fields in 3D geometry (S.Hudson *et al* )
- - noting ** is an eigenvalue, simultaneously solve for ** and B (method of lines / shooting method)
- - adjust interface to reduce force imbalance
- 	**Motivation**: Solve Beltrami fields in 3D configurations
- Surface current equilibria formulation. (M. McGann)
- 	Generalize treatment of Kaiser and Salat Phys. Plasmas 1(2), 1994, who
- parameterise an interface surface *S*.
- select ***B***=0 interior to *S,* and ***B*** =  exterior to *S,*
- substitute ***B*** into force balance  PDE for 
- search for S which allow an analytic solution for .
- 	**Motivation:** Provides method to construct toroidal flux surfaces, and establish robustness to pressure difference

## 4. Summary 1/2

- Explanation of problem:
- 	- flux surfaces wanted for good confinement
- 	- in general, they are rare in 3D geometry
- (2) Project aims :
- 	- design a convergent algorithm for constructing 3D equilibria
- 	- are ITB’s in a constrained minimum energy state?
- (3) Variational model developed – frustrated Taylor relaxation
- (4) Analytic solutions presented for a cylinder
- - Equilibria with positive shear exist
- - Stability: benchmarked for *N*=1, stable configurations for *N*>1 exist without jumps in *q* if appeal to discretisation
- - Need to be careful with resonances if trying to construct jumps in *q* from multi-interface with continuous *q*
- - stability of variational problem in multi-interfaces can be reduced to   stability of tearing + ideal modes

## 4. Summary 2/2

- (5) Exploration of ITB-like configurations underway. Seeks to scan over qmin, q0 and qedge, utilise toroidal/poloidal discretisation and determine whether any stable modes left in the limit of m→, n →
*(6) Various algorithms to solve in arbitrary 3D geometry based on stepped pressure profile now being developed. *