# Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

## Abstract

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. As a result, applying these results to a given rotational-transform profile in cylindrical geometry, we find magnetic field and current density profiles compatible with the fractal pressure.

- Authors:

- Princeton Univ., Princeton, NJ (United States)
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1395556

- Alternate Identifier(s):
- OSTI ID: 1395590

- Grant/Contract Number:
- AC02-09CH11466

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 9; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

### Citation Formats

```
Kraus, B. F., and Hudson, S. R. Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles. United States: N. p., 2017.
Web. doi:10.1063/1.4986493.
```

```
Kraus, B. F., & Hudson, S. R. Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles. United States. doi:10.1063/1.4986493.
```

```
Kraus, B. F., and Hudson, S. R. Fri .
"Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles". United States. doi:10.1063/1.4986493. https://www.osti.gov/servlets/purl/1395556.
```

```
@article{osti_1395556,
```

title = {Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles},

author = {Kraus, B. F. and Hudson, S. R.},

abstractNote = {In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. As a result, applying these results to a given rotational-transform profile in cylindrical geometry, we find magnetic field and current density profiles compatible with the fractal pressure.},

doi = {10.1063/1.4986493},

journal = {Physics of Plasmas},

number = 9,

volume = 24,

place = {United States},

year = {2017},

month = {9}

}

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Works referenced in this record:

##
Fractal structures in the chaotic motion of charged particles in a magnetized plasma under the influence of drift waves

journal, March 2017

- Mathias, A. C.; Viana, R. L.; Kroetz, T.
- Physica A: Statistical Mechanics and its Applications, Vol. 469

##
A universal instability of many-dimensional oscillator systems

journal, May 1979

- Chirikov, Boris V.
- Physics Reports, Vol. 52, Issue 5

##
Fat-fractal scaling exponent of area-preserving maps

journal, February 1987

- Hanson, James D.
- Physical Review A, Vol. 35, Issue 3

##
Self-Affine Fractals and Fractal Dimension

journal, October 1985

- Mandelbrot, Benoit B.
- Physica Scripta, Vol. 32, Issue 4

##
Physics of magnetically confined plasmas

journal, January 2005

- Boozer, Allen H.
- Reviews of Modern Physics, Vol. 76, Issue 4

##
Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields

journal, July 2017

- Hudson, S. R.; Kraus, B. F.
- Journal of Plasma Physics, Vol. 83, Issue 4

##
Fractal dimensionality for different transport modes in the turbulent boundary plasma of TEXTOR

journal, March 1993

- Budaev, V.; Fuchs, G.; Ivanov, R.
- Plasma Physics and Controlled Fusion, Vol. 35, Issue 3

##
Proof of a Theorem of a. n. Kolmogorov on the Invariance of Quasi-Periodic Motions Under Small Perturbations of the Hamiltonian

journal, October 1963

- Arnol'd, V. I.
- Russian Mathematical Surveys, Vol. 18, Issue 5

##
Contributions of plasma physics to chaos and nonlinear dynamics

journal, September 2016

- Escande, D. F.
- Plasma Physics and Controlled Fusion, Vol. 58, Issue 11

##
Toroidal Containment of a Plasma

journal, January 1967

- Grad, Harold
- Physics of Fluids, Vol. 10, Issue 1

##
Fractal structures in nonlinear dynamics

journal, March 2009

- Aguirre, Jacobo; Viana, Ricardo L.; Sanjuán, Miguel A. F.
- Reviews of Modern Physics, Vol. 81, Issue 1

##
A method for determining a stochastic transition

journal, June 1979

- Greene, John M.
- Journal of Mathematical Physics, Vol. 20, Issue 6

##
Ideal magnetohydrodynamic equilibrium in a non-symmetric topological torus

journal, February 2014

- Weitzner, Harold
- Physics of Plasmas, Vol. 21, Issue 2

##
On Euler's totient function

journal, January 1932

- Lehmer, D. H.
- Bulletin of the American Mathematical Society, Vol. 38, Issue 10

##
Fat Fractals on the Energy Surface

journal, August 1985

- Umberger, David K.; Farmer, J. Doyne
- Physical Review Letters, Vol. 55, Issue 7

##
Smoothing and Differentiation of Data by Simplified Least Squares Procedures.

journal, July 1964

- Savitzky, Abraham.; Golay, M. J. E.
- Analytical Chemistry, Vol. 36, Issue 8

##
Symplectic maps, variational principles, and transport

journal, July 1992

- Meiss, J. D.
- Reviews of Modern Physics, Vol. 64, Issue 3