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Solar Wind Heating as a Non-Markovian Process: L₫vy Flight, Fractional Calculus, and

Ú

-functions

Robert Sheldon 1 , Mark Adrian 2 , Shen-Wu Chang 3 , Michael Collier 4

UAH/MSFC, NRC/MSFC, CSPAAR/MSFC, GSFC

May 29, 2001

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The Solar Coronal Issue

• (I apologize for repeating the obvious, please bear with me) • Q: Corona is 2MK, photosphere only 5.6kK, so how does heat flow against the gradient?

• A: Non-equilibrium heat transport • a) coherence (waves) • b) topology (reconnection) • c) velocity filtration + non-Maxwellians • Each solution has its pros/cons, I want to present some mathematical results that may support (c)

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Problems, Opportunities

• 1) Coherent phenomena (waves), need to randomize to heat. Dissipation at 1-2 Rs turns out to be a problem.

[Parker 92]

• 2) Topology tangles, like a comb through long hair, collect in clumps or boundaries where they would heat unevenly. In contrast, all observational evidence shows even heating. Nor is there direct observation of nanoflares

[Zirker+Cleveland 92].

• 3) Velocity filtration is not itself a heating mechanism, it requires a non-Maxwellian distribution as well. Thus it postpones the problem to one of non-equilibrium thermodynamics in the highly collisional photosphere.

[Scudder 92]

• 4) Heat conduction too low to smooth out hot spots

[Marsden 96]

• 5) Non-Kolmogorov spectrum, non-turbulent heating.

[Gomez93]

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Characteristics needed

• Robust (nearly independent of magnetic polarity, geometry, location on sun, etc.) • Fine-grained (no evidence of clumps) • Non-turbulent (wrong spectrum) • Non-equilibrium statistical mechanics (it still transports heat the wrong way.) • What we need is a mechanism so fine grained that it escapes observation, yet so macroscopic that it does not rely on fickle micro/meso-scale physics.

Velocity Filtration

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• A Maxwellian maximizes the entropy for a fixed energy, so switching velocity space via collisions, or removing part of the distribution has no effect on the temperature. • But as Scudder shows graphically, the power-law tail of a kappa function has a higher temperature than the core, so removing the core leaves hi-T, unlike a Maxwellian which is self-similar.

Pin a tail on the distribution?

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• What sort of stochastic processes generate tails?

• For the sake of transparency, we will discuss spatial distributions that depart from Gaussian, and later apply these distributions to velocity space Maxwellians.

• If we consider a random stochastic process, such as a drunk staggering around a lamppost, we can plot the resulting spatial distribution of a collection of drunks.

• For some simple restrictions, the Central Limit Theorem predicts the distribution will converge to a Gaussian.

• Even worse, the distribution that maximizes entropy while conserving energy is a Maxwellian= Gaussian in v-space.

• How can we get a tail without violating the physics?

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Central Limit Theorem

• Paul Levy [1927] generalized the C.L.T.

• Variance: s 2 = < x 2 > - < x > 2 = 2Dt •

Diffusion: D = (

< x 2 > - < x > 2 ) / 2T •

Probability Distribution Function,

P

: =

•

n P(x)

•

We just need a different PDF to get a fat tail.

•

P(x)~x

-m •

if

m < 3, then < x 2 > = â and s 2 ~

t

1

well, we lost the 2

nd moment, but we have a tail.

•

What does this do to the physics? What happened to the entropy (or is the energy)?

PDF and spatial diffusion

P(x)

m=2 .2

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m=3 .8

x

• A slight change in the PDF can change diffusion radically.

• Levy-flights • Self-similar

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Self-similarity

• Paul L₫vy generalized the Central Limit Theorem, by looking for distributions that were "stable", having 3 exclusive properties: • Invariant under addition: X

1

+X

2

+...+X

n

= c

n

X + d

n

• Domain of attraction: c

n

= n

1/ Ñ ; 0< Ñ< 2

• Canonical characteristic function:

H-

or Fox-fcns.

•

In other words, For a given

Ñ

, there exists a self similar function, having a canonical form, that all random distributions will converge to.

•

L

₫ vy-stable probability distribution functions

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Stable Distributions

Lorentzian/Cauchy

Ñ

=1

Ñ

= 1.5

Gaussian/Normal

Ñ

= 2

L₫vy-stable distributions

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Linear Log -Self-similar, convergence to these stable distributions which are unimodal, and bell-shaped, but lack 2 nd moment -invariant under addition, domain of attraction, char. fn.

-Gaussian core, power-law tails (indistinguishable from Ú )

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Fractional Diffusion

• A completely separate mathematical technique has been found to describe L₫vy-stable distributions.

• Time-fractional Diffusion Equation 

d

 

da

  • d 2

Ý

f / d 2

Ý

t = D d 2 f / d 2 x • where D denotes positive constant with units of L 2 /T 2

Ý

•

Ý=

1 wave equation;

Ý=

1/2 diffusion equation (Gauss) • • • • Anomalous Diffusion

Ý<

1/2 = slow diffusion (Cauchy);

Ý>

1/2 = fast diffusion Solutions are Mittag-Leffler functions of order 2

Ý,

they are also

Levy-stable pdf.

We can identify

Ú -fcn with slow fractional diffusion

Fractional Derivatives (19

th

)

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note how neatly it interpolates between the lines

Linear=> Constant

note how the slope of the fractional deriv exceeds both.

It uses global info! (integro-differential)

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Physical Interpretation

• Collier [91] applied Levy-flight to velocity space diffusion and generated Cauchy tails as expected.

• Non-Markovian processes have "memory", or correlations in the time-domain. E.g., the lamppost is on a hill. That is, collisions have non-local information=> fractional calculus.

• Time-fractional & Space-fractional diffusion equations are equivalent (if there is a velocity somewhere).

• Non-local interactions, and/or non-Markovian interactions both produce fractional diffusion.

• Many physical systems exhibit super/sub diffusion with fat tails.

• Non-adiabatic systems need not conserve energy. E.g. if we maximize entropy holding log(E) constant => power law tails!

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W.A.G.

• 1) Non-local Coulomb collisions in the Earth's ionosphere are thought to put tails on upflowing ions. •

2) Equivalently, large changes in collision frequency are temporally correlated as plasma leaves Sun.

•

3) Fluctuations in energy are log-normal distributed, suggesting a pdf

already far from Gaussian.

•

4) Coulomb cross-section decreases with energy, so that E-field "runaway" modifies the power law of the P,

the step size between collisions (and energy gain).

•

5) "Sticky vortices" in Poincar₫ plots-Hamiltonian chaos.

Nomenclature

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Abstract

• Many space and laboratory plasmas are found to possess non-Maxwellian distribution functions. An empirical function promoted by Stan Olbert, which superposes a Maxwellian core with a power-law tail, has been found to emulate many of the plasma distributions discovered in space. These $\kappa$-functions, with their associatedpower-law tail induced anomalous heat flux, have been used by theorists$^1$\ as the origin of solar coronal heating of solar wind. However, the principle and prerequisite for the robust production of such a non equilibrium distribution has rarely been explained. We report on recent statistical work$^2$, which shows that the $\kappa$-function is one of a general class of solutions to a time-fractional diffusion equation, known as a L\'evy stable probability distribution. These solutions arise from time-variable probability distribution (or equivalently, a spatially variable probability in a flowing medium), which demonstrate that anomalously high flux, or equivalently, non equilibrium thermodynamics govern the outflowing solar wind plasma. We will characterize the parameters that control the degree of deviation from a Maxwellian and attempt to draw physical meaning from the mathematical formalism. $^1$Scudder, J. {\it Astrophys. J.}, 1992.\$^2$Mainardi, F. and R. Gorenflo, {\it J. Computational and Appl. Mathematics, Vol. 118}, No 1-2, 283 299 (2000).

Fractional Diffusion Examples

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•