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AN ANISOTROPIC ASPECT OF THE PLASMA TRANSPORT IN THE HASEGAWA-WAKATANI MODEL B. H . M in, C. Y. An a nd C.-B. K im Soongsil U nive rsit y, Se oul, Kore a 1 KSTAR Conference 2014 ABSTRACT Motivated by recent works on the advection excited in

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AN ANISOTROPIC ASPECT OF THE PLASMA TRANSPORT IN THE HASEGAWA-WAKATANI MODEL

  • B. H . M in, C. Y. An a nd C.-B. K im

Soongsil U nive rsit y, Se oul, Kore a

KSTAR Conference 2014

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ABSTRACT

Motivated by recent works on the advection excited in the noise- driven Hasegawa-Mima model, Hasegawa-Wakatani equations are studied as a self-consistent extension. Nonlinear energy fluxes due to the EΓ—B convective derivatives of both the vorticity and the plasma density are divided into two parts. One part, which is almost isotropic, is well known to engage in the transfer of the energies from the energy-producing scale where the phase mismatch between the density and the electric potential is large. The other part is found to be anisotropic and approximately advective in the Fourier space, with the directions of the advection of the kinetic and the particle energies being opposite. The advecting velocity in the Fourier space is derived after the comparison is made to the gradients of the

  • energies. Implications of the findings to the transport of the fusion

plasmas will be presented.

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CONTENTS

1. Hasegawa-Wakatani equations 2. Conservation laws

  • Global and local

3. Simulation results

  • Local energy balance
  • Density-potential coherence
  • Anisotropic part of nonlinear fluxes
  • adiabaticity = 1.0, 0.1, 0.01

4. Conclusions

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HASEGAWA-WAKATANI EQUATIONS

πœ– πœ–π‘’ πœ•

+ 𝑀 βƒ— βˆ™ 𝛼βŠ₯πœ• = 𝛽 𝜚 βˆ’ π‘œ + 𝐸𝜚

πœ– πœ–π‘’ π‘œ

+ 𝑀 βƒ— βˆ™ 𝛼βŠ₯π‘œ +

πœ–πœš

  • πœ–πœ– = 𝛽(𝜚

βˆ’ π‘œ ) + πΈπ‘œ

Adiabaticity parameter 𝛽 = βˆ’

π‘ˆ 𝑓2π‘œ0πœƒβˆ₯πœ•π‘‘π‘‘ πœ–2 πœ–π‘¨2

ExB velocity 𝑀

βƒ— = 𝑨̂ Γ— 𝛼βŠ₯𝜚

  • Vorticity πœ•

= 𝛼βŠ₯

2𝜚

  • Units: time π‘€π‘œ/𝑑𝑑, space πœπ‘‘, potential π‘ˆπœπ‘‘/π‘“π‘€π‘œ, density π‘œ0πœπ‘‘/π‘€π‘œ

𝐸𝜚 = πœ‰πœšπ›ΌβŠ₯

4πœ•

, πΈπ‘œ = πœ‰π‘œπ›ΌβŠ₯

4π‘œ

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EXB convecive term resistive coupling term comes from electron response

energy production term from π‘œ0 gradient

dissipative term

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CONSERVATION LAWS

Energy and enstrophy 𝐹 =

1 2 ∫ 𝑒2𝑦(π‘œ

2 + |𝛼βŠ₯𝜚 |2) 𝑉 =

1 2 ∫ 𝑒2𝑦(π‘œ

βˆ’ πœ• )2 Global conservation

πœ–πΉ πœ–π‘’ = ∫ 𝑒2𝑦 π‘œ

(βˆ’

πœ–πœš

  • πœ–πœ–) βˆ’ α∫ 𝑒2𝑦(π‘œ

βˆ’ 𝜚 )2 βˆ’ ∫ 𝑒2𝑦(π‘œ πΈπ‘œ βˆ’ 𝜚 𝐸𝜚)

πœ–π‘‰ πœ–π‘’ = ∫ 𝑒2𝑦 π‘œ

(βˆ’

πœ–πœš

  • πœ–πœ–) βˆ’ ∫ 𝑒2𝑦(π‘œ

βˆ’ πœ• )(πΈπ‘œ βˆ’ 𝐸𝜚)

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Local conservation law

Energy density

πœ– πœ–π‘’ 1 2 (|𝛼βŠ₯𝜚 |2 + π‘œ 2) βˆ’ 𝜚 𝑀 βƒ— βˆ™ 𝛼βŠ₯πœ• + π‘œ 𝑀 βƒ— βˆ™ 𝛼βŠ₯π‘œ

  • = π‘œ
  • βˆ’ πœ–πœš
  • πœ–πœ–

βˆ’ Ξ±(π‘œ βˆ’ 𝜚 )2 + (π‘œ πΈπ‘œ + 𝜚 𝐸𝜚)

Enstrophy density

πœ– πœ–π‘’ 1 2 (π‘œ

βˆ’ πœ• )2 + 𝑀 βƒ— βˆ™ 𝛼βŠ₯

1 2 (π‘œ

βˆ’ πœ• )2 = (π‘œ βˆ’ πœ• ) βˆ’

πœ–πœš

  • πœ–πœ– + (π‘œ

βˆ’ 𝜚 )(πΈπ‘œ βˆ’ 𝐸𝜚)

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Local energy balance

Adiabaticity parameter 𝜷 = 𝟐. 𝟏

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energy transfer energy production resistive loss dissipative loss

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Density-potential coherence

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Real Imaginary

  • 𝒍𝒛 βˆ‘

(𝐒𝐒𝐒𝐒 πͺ𝐒πͺπͺ)

π’π’š

  • πŸ“ βˆ‘

(𝐉𝐉𝐒𝐉. πͺ𝐒πͺπͺ)

π’π’š

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Anisotropic flux

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Sum over 𝒍𝒛 Adiabaticity parameter 𝜷 = 𝟐. 𝟏

  • sum over π’π’š
  • βˆ’πŸ. πŸπŸ’
  • βˆ’πŸ. πŸ‘πŸ‘

π’πŸ‘ 𝟐+π’πŸ‘

Fitting with advective flux

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Anisotropic flux

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Adiabaticity parameter 𝜷 = 0.01 Adiabaticity parameter 𝜷 = 0.1

  • sum over π’π’š
  • βˆ’πŸ. 𝟐
  • sum over π’π’š
  • βˆ’πŸ. πŸπŸ‘
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Anisotropic flux

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Sum over 𝒍𝒛 Adiabaticity parameter 𝜷 = 𝟐. 𝟏 Fitting with advective flux

  • sum over π’π’š
  • 0. πŸ‘
  • 0. πŸ‘
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Anisotropic flux

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Adiabaticity parameter 𝜷 = 0.01 Adiabaticity parameter 𝜷 = 0.1

  • sum over π’π’š
  • 0. πŸ–
  • 0. πŸ‘
  • sum over π’π’š
  • 0. πŸ‘πŸ‘
  • 0. πŸ“πŸ‘
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CONCLUSIONS

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Nonlinear energy fluxes excluding the parts that are responsible for the energy transfer are found to be: οƒΌ anisotropic; οƒΌ advective along the direction of π’˜π’†π’†; οƒΌ kinetic energy being advected; οƒΌ particle energy advection being determined by |𝜚 𝑙|2;  Work is still progressing to extend to larger values of Ξ±.