Active Control of Drift Wave Turbulence Christian Brand PhD Thesis

Here I am writing down notes and important concepts that I have learned reading Christian Brandt’s PhD thesis on active drift wave turbulence control.

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The paper investigates control of drift waves and drift wave turbulence experimentally in the linear magnetized helicon experiment VINETA.

Legend

$\vec{B}$ : Magnetic Field $\nabla n$ : Density gradient

Notes

Introduction

• Drift waves are characterized by currents parallel to the ambient magnetic field, that are tightly coupled to a coherent mode structure rotating in the perpendicular plane.
• If the imposed mode number as well as the rotation direction match those of the drift waves, classical synchronization effects like, e.g., frequency locking, frequency pulling, and Arnold tongues are observed.
• Turbulence itself decays into an array of smaller turbulent spots with it’s spectral energy that is approximately relative to Kolmogorov’s power law $E(k)\propto k^{5/3}$.
• In general, turbulence can be simply influenced by changing global parameters. One parameter from hydrodynamics is the Reynolds Number
• Chaotic and turbulent dynamics are very sensitive to tiny perturbations. (The butterfly effect) In chaos control unstable periodic orbits are stabilized by small feedback corrections for the control parameter pushing the system to an attractor.
• Vortex structures and turbulent cascades, as known from hydrodynamics, are present in plasmas as well.
• The isotropy of particle motion in a plasma is often broken by an ambient magnetic field, which results in inhomogeneous 3D turbulence and makes predictions based on homogenous and “controlled” chaos inaccurate.
• Drift waves belong to the most prominent plasma edge instabilities, and are assumed to cause much of the anomalous transport. Including nonlinear interaction schemes such as Alfvén driftwaves, drift Alfvén turbulence, perpendicular transport and zonal flows.
• Røssby waves (concept of fluid dynamics strongly related to driftwaves in plasma) and driftwaves can be described by the Hasegawa-Mima Equation. It can reproduce zonal flows

Drift waves

Drift Wave Instability

• In the course of this thesis, drift waves will be exclusively examined in a collision dominated plasma driven by a density gradient in cylindrical geometry, the so-called resistive drift waves.
• A plasma is considered in a homogenous magnetic field with a density gradient perpendicular to it. This orthogonality is interpreted as: moving along the field lines, the density remains constant; moving perpendicular to them, it changes.
• The equilibrium density gradient is perturbed by $\tilde{n}/n=exp(i(k\_\perp y+wt))$, ia sinusoidal density oscillation.
• Due to the much higher mobility of electrons compared to ions, they diffuse out of density maxima into minima along the magnetic field lines and give rise to parallel currents while remaining ions create a positive space charge in regions of positive density perturbation. The emerging magnetic field $\vec{E}$ is thus perpendicular to the magnetic field and cause local ExB drifts.

Turbulent Transport

• If the $\nabla n\perp \vec{B}$ induced parallel current (electron motion) wouldn’t be disturbed, drift waves would have practically no effect of the plasma stability. The electron non-adiabaticity (which causes the resistive effects of drift waves) is generated by plasma resistivity (primary source) , magnetic induction, particle-wave interactions, and electron-inertia.
• The electrostatic part of the transport is usually dominant compared to the magnetic field fluctuations $\tilde{B}$
• The radial net transport depends on the phase shift between density and potential perturbation. The net transport of coherent drift waves is relatively low with a cross-phase close to zero. Drift wave turbulence exhibits larger cross-phases and thus the net transport is considerably higher.

Creation Process of Drift Waves

• The non-adiabacity of electrons causes a delay of the potential perturbation relative to the propagation of the density perturbation. The cross-phase between density and potential gets larger than zero and ExB Drift are phase shifted by less than $\pi /2$ (the equilibrium in which drift waves won’t affect the plasma stability) relative to the density perturbation. On this account the Drifts amplify density perturbation and the propagating wave gets unstable.

Drift Wave Models

• Hasegawa-Mima describes the dynamics of nonlinear driftwaves parallel to the ambient magnetic field: their propagation but not their growth rate

• Hasegawa-Wakatani ehibits a dispersion relation containing a growth rate by including resistivity of the plasma

• Variierende lokale Feldstärke