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# Time evolution plots of radial particle flux from oHW (black), MHW (green), MHW w/o ZF (green), and MHW w/o ZD (red) for a ¼ 2.

Source publication

The generations of zonal flow (ZF) and density (ZD) and their feedback on the resistive drift wave turbulent transport are investigated within the modified Hasegawa-Wakatani model. With proper normalization, the system is only controlled by an effective adiabatic parameter, α, where the ZF dominates the collisional drift wave (DW) turbulence in the...

## Contexts in source publication

**Context 1**

... all the cases, we observe that the fluctuations first grow linearly due to the resistive DW instability, where the modifications will not affect the growth of the DW in the linear regime (e.g., see Fig. 1 for a ¼ 2). Note that, in the late linear stage, the simulations of oHW and MHW are dominated by the mode ðk x ; k y Þ ¼ ð0:2; 1Þ, while those of MHW w/o ZF/ZD are governed by ðk x ; k y Þ ¼ ð0:2; 0:6Þ. The growth rates from the simulations are, respectively, 0.025 and 0.057, which agree with the analytic result of resistive DW in the ...

**Context 2**

... (as an example, the growth rate of ZF can be $0:1 in our normalized unit 22,23 for a ¼ 2, i.e., c zf $ 0:1jx ci ). The ZF and/or ZD will, in turn, affect the DW, and the system will be saturated eventually when the particle flux C n approximately balances the dissipation, D a ¼ a À1 Ð ½^ að~ n À ~ /Þ 2 dx, due to the parallel resistivity. From Fig. 1 we see that, the saturated particles fluxes are quite different, where the time-averaged C n at the saturated state from t ¼ 400 to t ¼ 1000 are 0.36 (oHW), 7:4 Â 10 À3 (MHW), 2:2 Â 10 À2 (MHW w/o ZD), and 0.4 (MHW w/o ZF), respectively. This is consistent with the standard recognition that the ZF can largely suppress the anomalous ...

**Context 3**

... when the amplitudes of DWs become the order of unity as a result of growth from the linear state, the secondary instabilities will lead to the generation of the ZF and/or ZD, which will, in turn, suppress the DW turbulence and thus reduce the radial particle flux. We find that the ZF itself can significantly reduce the particle flux as shown in Fig. 1, whereas the ZD can reduce the transport only through its synergy with ...

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## Citations

... Hasegawa-Mima model does not [5]), ii) finite frequency (so that resonant interactions are possible [6]), and iii) a proper treatment of zonal flows [7]. The model is well known to generate high levels of large scale zonal flows, especially for C 1 [8][9][10]. It has been studied in detail for many problems in fusion plasmas including dissipative drift waves in tokamak edge [11,12], subcritical turbulence [13], trapped ion modes [14], intermittency [15,16], closures [17][18][19], feedback control [20], information geometry [21] and machine learning [22]. ...

... and Ω ± k is given in (10). Note that these coefficients are complex, and have different phases in general. ...

Hasegawa-Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However when an additional triad, which is a reflection of the original one with respect to the $y$ axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a 'collective mean field', individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation is discussed for the same parameters and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.

... For example, if n e and ϕ are connected via Ohm's law, one is led to a set of two-field equations known as the modified Hasegawa-Wakatani equation. [50][51][52][53][54] This model can be applied to resistive DWs at the tokamak edge. Similar two-field models have also been proposed for core plasmas, including ones that describe the ion-temperature-gradient (ITG) mode, 9,43,55,56 trapped-electron modes, 57 etc. ...

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... For example, if n e and ϕ are connected via Ohm's law, one is led to a set of two-field equations known as the modified Hasegawa-Wakatani equation. [45][46][47][48][49] This model can be applied to resistive DWs at the tokamak edge. Similar two-field models have also been proposed for core plasmas, including ones that describe the ion-temperature-gradient (ITG) mode, 8,38,50,51 trapped-electron modes, 52 etc. ...

Basic physics of drift-wave turbulence and zonal flows has long been studied within the framework of wave-kinetic theory. Recently, this framework has been re-examined from first principles, which has led to more accurate yet still tractable "improved" wave-kinetic equations. In particular, these equations reveal an important effect of the zonal-flow "curvature" (the second radial derivative of the flow velocity) on dynamics and stability of drift waves and zonal flows. We overview these recent findings and present a consolidated high-level picture of (mostly quasilinear) zonal-flow physics within reduced models of drift-wave turbulence.

... As we can see, for smallerŝ and thus larger ZF, the fluctuations and the particle flux are more localized near the extrema of V zf (e.g., see [20,21]), where the flow shearing is negligible. We note that the characteristic wavelength of trapped RDW near the maximum of ZF is much larger than that near the minimum of ZF [22] and thus the kinetic energy E f k (x) is localized near the minimum of ZF as shown in Fig. 4. Such localization of turbulence will reduce the particle flux [22] since the large |V zf | on both sides of the localized positions will set a transport barrier. ...

... We note that the characteristic wavelength of trapped RDW near the maximum of ZF is much larger than that near the minimum of ZF [22] and thus the kinetic energy E f k (x) is localized near the minimum of ZF as shown in Fig. 4. Such localization of turbulence will reduce the particle flux [22] since the large |V zf | on both sides of the localized positions will set a transport barrier. In fact, even forŝ>10 −2 , no obvious localization is found as the peak of ZF is below the order of unity. ...

The impact of neutrals on the anomalous edge plasma transport is examined in detail within the resistive drift wave (RDW) turbulence and zonal flow (ZF) system. It is shown that the neutral impact on the RDW turbulence itself is weak, but it can largely damp the zonal flow and thus lead to an enhancement of the anomalous transport. Such an impact is stronger for the system with a larger adiabatic parameter, where the ZF is more dominant in the absence of neutrals. It is shown that the enhancement of the anomalous transport is related to the detrapping of fluctuations from the vicinity of the extrema of ZF when the neutrals weaken the ZF effect.

The Dimits shift, an upshift in the onset of turbulence from the linear instability threshold, caused by self-generated zonal flows, can greatly enhance the performance of magnetic confinement plasma devices. Except in simple cases, using fluid approximations and model magnetic geometries, this phenomenon has proved difficult to understand and quantitatively predict. To bridge the large gap in complexity between simple models and realistic treatment in toroidal magnetic geometries (e.g. tokamaks or stellarators), the present work uses fully gyrokinetic simulations in a Z-pinch geometry to investigate the Dimits shift through the lens of tertiary instability analysis, which describes the emergence of drift waves from a zonally dominated state. Several features of the tertiary instability, previously observed in fluid models, are confirmed to remain. Most significantly, an efficient reduced-mode tertiary model, which previously proved successful in predicting the Dimits shift in a gyrofluid limit (Hallenbert & Plunk, J. Plasma Phys. , vol. 87, issue 05, 2021, 905870508), is found to be accurate here, with only slight modifications to account for kinetic effects.

The Hasegawa–Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices, is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking, where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However, when an additional triad, which is a reflection of the original one with respect to the y axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered, including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a “collective mean field,” individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation are discussed for the same parameters, and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.