Hasegawa-Wakatani Equations Research Articles

Adding a Vorticity Source to the Hasegawa-Wakatani Equations

Instabilities in fusion reactor plasma are an area of ongoing research as they represent the primary hurdle in producing an ongoing fusion reaction for fusion energy production. To study one possible source of these instabilities, computational simulations to solve the modified Hasegawa-Wakatani Equations and the evolution of edge plasmas through time are required. This work shows that the addition of a source of vortices leads to edge instabilities using the TOKAM2D turbulence code. This is an important step into developing our understanding of how instabilities form in plasmas and maintaining a fusion reaction.

Lagrangian Chaotic Mixing in a Nonlinear Model of Resistive Drift-wave Turbulence in Tokamak Plasmas

In fusion plasma, numerical simulations are commonly employed to investigate the confinement properties of plasma in the bulk region of tokamaks. The modified Hasegawa-Wakatani (MHW) equations are used to model the behavior of the plasma, which enables us to understand the radial transport in two-dimensional numerical simulations of electrostatic resistive drift-wave turbulence. By utilizing the MHW equations, we have gained insights into the low-to-high confinement (L-H) transitions that occur spontaneously in the plasma when it moves from a low confinement stage, characterized by turbulent flow, to a turbulence-suppressed regime known as zonal flow. To investigate these transitions, we vary the value of a control parameter \(\alpha\), which is related to adiabaticity, in numerical simulations, and observe the transition between the two regimes. This simplified model of L-H transitions can provide valuable information for tokamaks. To identify the Lagrangian coherent structures (LCS) and to better characterize the chaotic mixing during the L-H transition, we computed the finite-time Lyapunov exponent (FTLE) of the calculated velocity field derived from the electrostatic potential. We further compared the statistics of the chaotic mixing of the two regimes. The results of our study offer insights into the turbulent transport processes in magnetic confinement fusion plasmas.

Elucidating plasma dynamics in Hasegawa–Wakatani turbulence by information geometry

The impact of adiabatic electrons on drift-wave turbulence, modeled by the Hasegawa–Wakatani equations, is studied using information length. Information length is a novel theoretical method for measuring distances between statistical states represented by different probability distribution functions (PDFs) along the path of a system and represents the total number of statistically different states that a system evolves through in time. Specifically, the time-dependent PDFs of turbulent fluctuations for a given adiabatic index A are computed. The changes in fluctuation statistics are then quantified in time by using information length. The numerical results provide time traces exhibiting intermittent plasma dynamics, and such behavior is identified by a rapid change in the information length. The effects of A are discussed.

Global-in-time existence results for the two-dimensional Hasegawa–Wakatani equations

In order to describe the resistive drift wave turbulence appearing in nuclear fusion plasma, the Hasegawa–Wakatani (HW) equations were proposed in 1983. We consider the two-dimensional HW equations, which have numerous structures (that is, they explain the branching phenomenon in turbulent and zonal flow in a two-dimensional plasma) and the generalized HW equations that include temperature fluctuation. We prove the global-in-time existence of a unique strong solution to both the HW equations and the generalized HW equations in a two-dimensional domain with double periodic boundary conditions.

An almost-periodic solution of Hasegawa–Wakatani equations with vanishing resistivity

We consider the zero-resistivity limit for Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov almost-periodic in the axial direction. First, we prove the existence of a solution to Hasegawa–Wakatani equations with zero resistivity; second, we obtain uniform a priori estimates with respect to resistivity. Such estimates can be obtained in the same way as for our previous results; therefore, the most important contribution of this paper is the proof of the existence of a local-in-time solution to Hasegawa–Wakatani equations with zero resistivity. We apply the theory of Bohr–Fourier series of Stepanov almost-periodic functions to such a proof.

Almost-periodic solutions to an initial boundary value problem for model equations of resistive drift wave turbulence

In this paper we are concerned with the drift wave turbulence in a strong magnetic field. The existence and the uniqueness of a strong Stepanovalmost- periodic solution to the initial boundary value problems are established both for the model equations of the resistive drift wave turbulence and for the three-dimensional Hasegawa–Wakatani equations when the initial data are Stepanov-almost-periodic in the magnetic field direction.

Wave-number spectrum of dissipative drift waves and a transition scale.

We study the steady state spectrum of the Hasegawa-Wakatani (HW) equations that describe drift wave turbulence. Beyond a critical scale k_{c}, which appears as a balance between the nonlinear time and the parallel conduction time, the adiabatic electron response breaks down nonlinearly and an internal energy density spectrum of the form F(k_{⊥})∝k_{⊥}^{-3}, associated with the background gradient, is established. More generally a dual power law spectrum, approximately of the form F(k_{⊥})∝k_{⊥}^{-3}(k_{c}^{-2}+k_{⊥}^{-2}) is obtained, which captures this transition. Using dimensional analysis, an expression of the form k_{c}∝C/κ is derived for the transition scale, where C and κ are normalized parameters of the HW equations signifying the electron adiabaticity and the density gradient, respectively. The results are numerically confirmed using a shell model developed and used for the Hasegawa-Wakatani system.

On the almost-periodic solution of Hasegawa–Wakatani equations

In order to describe the resistive drift wave turbulence appearing in nuclear fusion plasma, the Hasegawa–Wakatani equations were proposed in 1983. In this paper, we consider the zero-resistivity limit for the Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov-almost-periodic in the axial direction. We prove two results: one is the existence and uniqueness of a strong global in time Stepanov-almost-periodic solution to the initial boundary value problem for the Hasegawa–Mima-like equation; another is the convergence of the solution of the Hasegawa–Wakatani equations to that of the Hasegawa–Mima-like equation established at the first stage as the resistivity tends to zero. In order to obtain a priori estimates of the Stepanov-almost-periodic solutions to our problems, we have to overcome some difficulties. In the proof, we prepare some lemmas for the Stepanov-almost-periodic functions and then obtain a priori estimates.

Almost-periodic solution of linearized Hasegawa–Wakatani equations with vanishing resistivity

In this paper we consider the zero-resistivity limit for linearized Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov-almost-periodic to the axial direction. We prove two results: one is the existence and uniqueness of a strong Stepanov-almost-periodic solution to the initial boundary value problem for linearized Hasegawa–Wakatani equations with zero resistivity; another is the convergence of the solution of linearized Hasegawa–Wakatani equations established in [24] to the solution of the problem studied at the first stage as the resistivity tends to zero. In the proof we obtain two useful lemmas for Stepanov-almost-periodic functions.

Nonadiabatic electron response in the Hasegawa-Wakatani equations

Tokamak edge turbulence is strongly influenced by parallel electron physics, which relaxes density and potential fluctuations towards electron adiabatic response. Beginning with the paradigmatic Hasegawa-Wakatani equations (HWEs) for resistive tokamak edge turbulence, a unique decomposition of the electric potential (φ) into adiabatic (a) and nonadiabatic (b) portions is derived, based on the requirement that a neither drive nor respond to the parallel current j∥. The form of the decomposition clarifies that, at perpendicular scales large relative to the sound radius, the electron adiabatic response controls the nonzonal φ, not the fluctuating density n. Simple energy balance arguments allow one to rigorously bound the ratio of rms nonzonal nonadiabatic fluctuations (b̃) relative to adiabatic ones (ã). The role of the vorticity nonlinearity in transferring energy between adiabatic and nonadiabatic fluctuations aids intuitive understanding of self-sustained turbulence in the HWEs. When the normalized parallel resistivity is weak, b̃ becomes effectively slaved, allowing the reduction to an approximate one-field model that remains valid for strong turbulence. In addition to guiding physical intuition, the one-field reduction should greatly ease further analytical manipulations. Direct numerical simulation of the 2D HWEs confirms the convergence of the asymptotic formula for b̃.

Zonal flow generation and its feedback on turbulence production in drift wave turbulence

Plasma turbulence described by the Hasegawa-Wakatani equations is simulated numerically for different models and values of the adiabaticity parameter C. It is found that for low values of C turbulence remains isotropic, zonal flows are not generated and there is no suppression of the meridional drift waves and particle transport. For high values of C, turbulence evolves towards highly anisotropic states with a dominant contribution of the zonal sector to the kinetic energy. This anisotropic flow leads to a decrease of turbulence production in the meridional sector and limits the particle transport across the mean isopycnal surfaces. This behavior allows to consider the Hasegawa-Wakatani equations a minimal PDE model, which contains the drift-wave/zonal-flow feedback loop mechanism.

Reduced-order model based feedback control of the modified Hasegawa-Wakatani model

In this work, the development of model-based feedback control that stabilizes an unstable equilibrium is obtained for the Modified Hasegawa-Wakatani (MHW) equations, a classic model in plasma turbulence. First, a balanced truncation (a model reduction technique that has proven successful in flow control design problems) is applied to obtain a low dimensional model of the linearized MHW equation. Then, a model-based feedback controller is designed for the reduced order model using linear quadratic regulators. Finally, a linear quadratic Gaussian controller which is more resistant to disturbances is deduced. The controller is applied on the non-reduced, nonlinear MHW equations to stabilize the equilibrium and suppress the transition to drift-wave induced turbulence.

On the Hasegawa–Wakatani equations with vanishing resistivity

In this paper, we are concerned with the drift wave turbulence in a strong magnetic field. We prove the existence and uniqueness of a strong global in time solution to the initial boundary value problem for the model equations of drift wave turbulence similar to Hasegawa–Mima equation. Then we prove that the solution of Hasegawa–Wakatani equations established in [5] converges to that of Hasegawa–Mima like equation established at the first stage as the resistivity tends to zero on some time interval.

Initial Boundary Value Problem for Model Equations of Resistive Drift Wave Turbulence

In this paper we consider the drift wave turbulence in a strong magnetic field. Here we first establish the existence and uniqueness of a strong global solution to the initial boundary value problem for the model equations of the resistive drift wave turbulence. Second, by the standard method of passing to the limit, we establish the existence and uniqueness of a strong solution to the initial boundary value problem for three-dimensional Hasegawa–Wakatani equations.

The effects of nonuniform magnetic field strength on density flux and test particle transport in drift wave turbulence

The extended Hasegawa–Wakatani equations generate fully nonlinear self-consistent solutions for coupled density n and vorticity ∇2ϕ, where ϕ is electrostatic potential, in a plasma with background density inhomogeneity κ=−∂ ln n0/∂x and magnetic field strength inhomogeneity C=−∂ ln B/∂x. Finite C introduces interchange effects and ∇B drifts into the framework of drift turbulence through compressibility of the E×B and diamagnetic drifts. This paper addresses the direct computation of the radial E×B density flux Γn=−n∂ϕ/∂y, tracer particle transport, the statistical properties of the turbulent fluctuations that drive Γn and tracer motion, and analytical underpinnings. Systematic trends emerge in the dependence on C of the skewness of the distribution of pointwise Γn and in the relative phase of density-velocity and density-potential pairings. It is shown how these effects, together with conservation of potential vorticity Π=∇2ϕ−n+(κ−C)x, account for much of the transport phenomenology. Simple analytical arguments yield a Fickian relation Γn=(κ−C)Dx between the radial density flux Γn and the radial tracer diffusivity Dx, which is shown to explain key trends in the simulations.

Bifurcation in electrostatic resistive drift wave turbulence

The Hasegawa-Wakatani equations, coupling plasma density, and electrostatic potential through an approximation to the physics of parallel electron motions, are a simple model that describes resistive drift wave turbulence. Numerical analyses of bifurcation phenomena in the model are presented, that provide new insights into the interactions between turbulence and zonal flows in the tokamak plasma edge region. The simulation results show a regime where, after an initial transient, drift wave turbulence is suppressed through zonal flow generation. As a parameter controlling the strength of the turbulence is tuned, this zonal-flow-dominated state is rapidly destroyed and a turbulence-dominated state re-emerges. The transition is explained in terms of the Kelvin-Helmholtz stability of zonal flows. This is the first observation of an upshift of turbulence onset in the resistive drift wave system, which is analogous to the well-known Dimits shift in turbulence driven by ion temperature gradients.

The interplay between Reynolds stress and zonal flows: direct numerical simulation as a bridge between theory and experiment

We describe the results of a measurement campaign on the CASTOR tokamak where the drive of flows and zonal flows by Reynolds stress was investigated by means of a dual probe head system allowing us to measure the properties of the electrostatic turbulence and the rotation velocities at the same location and at the same moment. We compare these experimental results with a turbulence model linked to a one dimensional fluid model describing the electrostatic turbulence and its influence on the background flow. The turbulence is simulated locally on the basis of the Hasegawa–Wakatani equations, completed with magnetic inhomogeneity terms. In the fluid model the toroidal geometry is correctly taken into account, while various sources and sinks like viscosity, interaction with neutrals, Reynolds stress and electric current induced by biasing are included. The good agreement of the predicted flow with the measured one demonstrates that in a pure cylindrical geometry the modelled strength of Reynolds stress acceleration of flow is overestimated.

Turbulent flux and the diffusion of passive tracers in electrostatic turbulence

The connection between the diffusion of passive tracer particles and the anomalous turbulent flux in electrostatic drift-wave turbulence is investigated by direct numerical solutions of the 2D Hasegawa–Wakatani equations. The probability density functions for the point-wise and flux surface averaged turbulent particle flux are measured and compare well to a folded Gaussian, respectively a log-normal distribution. By following a large number of passive tracer particles we evaluate the diffusion coefficient based on the particle dispersion. It is found that the particle diffusion coefficient is in good agreement with the one derived from the turbulent E×B-flux by using Fick’s law. Employing the Lagrangian conservation of the “Potential Vorticity” in the Hasegawa–Wakatani equations, the analytical support for this result is obtained. The transport estimated by passive tracer dispersion and turbulent plasma flux are found to coincide.

Particle diffusion in anisotropic turbulence

We have investigated the diffusion of test particles in anisotropic two-dimensional turbulence. To be specific we have considered electrostatic drift-wave turbulence in magnetized plasma, described by the so-called Hasegawa–Wakatani equations, which produce a two-dimensional turbulent flow-field with a self-consistent instability drive. However, our results are also relevant for particle dispersion in beta-plane turbulence, due to the similarity between drift waves in plasmas and Rossby waves on the beta-plane. In general, we observe that the particle dispersion is strongly anisotropic, with the diffusion in the zonal direction much larger than the diffusion in the direction parallel to the background density gradient––the radial direction (meridional direction for the beta-plane analogy). Trapping of particles in and subsequent displacement with vortical structures leads to an intermediate anomalous diffusion with superdiffusive behaviour in the zonal direction and subdiffusive behaviour in the radial direction. However, the time asymptotic behaviour always approaches a normal diffusive process. By including inertial effects in the particle velocity we observe a tendency for increased superdiffusive behaviour.

Self-consistent modeling of turbulence and transport

We describe an efficient procedure for coupling a turbulent system to a transport equation which evolves the equilibrium fields that drive and are driven by the turbulence. As an example, we apply the procedure to the coupling of turbulence simulations of the two-dimensional Hasegawa–Wakatani equations to a one-dimensional transport equation for the density n. Our coupling scheme uses implicit temporal discretization of the transport equation, rendering it stable for arbitrarily large time steps. This allows the computation of turbulence with self-consistent steady-state equilibrium profiles in a single time step of the transport equations and with a total computational time comparable to that required for the turbulence code alone to reach a statistical steady state with fixed equilibrium profiles. Results are presented for both local and non-local turbulence simulations. In the former, which requires running a separate turbulence simulation for each transport grid cell, the transport flux Γ( x) depends on only local values of n( x) and n ′( x); for this case, Γ is expressed using Fick’s law, Γ=− Dn ′, with D>0 prescribed by the turbulence. In the non-local simulations, Γ( x) depends on the form of n over the entire domain. For such simulations we present two methods for representing Γ that allow for anomalous flux transport, i.e., regions where the flux flows up the local gradient of n.