Flow Damping Research Articles

Propondo medidas compensatórias de drenagem como ferramenta de governança participativa

This paper addresses the problem of flooding in the Tiburtino Stream Watershed, focusing on a desirable adaptation towards urban resilience oriented towards environmentally fragile areas of the basin. It is assumed that the construction of small parallel reservoirs located upstream of the critical points of event occurrences would considerably mitigate their severity and intensity, thus indicating a feasible possibility of adaptation that must be taken into consideration by the competent authorities when planning. of future urban drainage actions in the region. The method used was the convolution of the SCS Unitary Hydrogram from upstream to downstream of the watershed, and Lapa Market was defined as the measurement point – a critical location in terms of occurrences – for a quantitative verification of the damping of the peak flow in two scenarios: a single reservoir, of larger proportion; distributed smaller reservoirs. The results validated the initial hypothesis, demonstrating that small reservoirs located below the road system are as efficient or more efficient than a single reservoir of more robust proportions with a high potential for environmental impact.

Transient dynamic stress behavior analysis of the axial flow pump as turbine at part loads

To explore the structural dynamic characteristics of the impeller and shaft system of the axial flow pump as turbine, the variation law of impeller transient equivalent stress, deformation, and the impeller and shaft system mode are analyzed by the transient fluid-structure interaction method. Findings indicate that the equivalent stress is centralized at the blade root around the hub and tends to decrease as it moves towards the shroud. The maximum equivalent stress at 1.0Qbep and 1.1Qbep is 1.26 times and 2.0 times higher than that at 0.8Qbep. Blade deformation is significantly influenced by flow rate. The maximum deformation of the blade at 1.0Qbep and 1.1Qbep is 1.6 times and 3.6 times higher than that at 0.8Qbep. The impeller and shaft system with prestress in dry mode has an average natural frequency of the first 10 orders that is 16.7 Hz higher than without prestress. However, with the fluid adding mass and flow damping, the average natural frequency of the first 10 orders of the impeller and shaft system in wet mode is reduced by 278.08 Hz compared with dry mode. In fact, the impeller and shaft system of the axial flow pump as turbine is not prone to vibration.

Turbulent flow characteristics over rough permeable and impermeable gravel-bed stream- an experimental study

Abstract The study aims to presents flow heterogeneity over rough permeable and impermeable gravel-bed stream. The rough permeable stream is prepared by laying multiple layers of gravel, whereas its impermeable counterpart is presented by a resin-casted gravel-bed. In general, a common approach can be found in literature to mimic an impermeable bed by laying single layer of gravels, therefore some results are compared with single layer gravel-bed stream keeping the hydraulic conditions same. An acoustic Doppler velocimeter was used for flow measurements whereas; double averaging (DA) methodology was adopted for data analysis. The larger flow penetration depth and intense flow mixing in permeable gravel-bed infer sufficient impetus for organized flow turbulence and damping of DA Reynolds shear stresses whereas; the wall-blocking in resin-casted gravel-bed prevents fluid infiltration which leads to follow the linear stress profile away from the crest level. The damping of DA Reynolds shear stress (RSS) is compensated by enhanced DA form-induced shear stress (FISS). The results are further analysed under the light of the energy budget to characterize the mass-momentum exchange as it penetrates the subsurface layers. The energy budget indicates negative pressure energy diffusion rates corroborating gain in turbulence production in the permeable gravel-bed stream.

Safety factor influence on the edge E × B velocity establishment in tokamak plasmas

This study is motivated by experiments on Tore Supra and WEST tokamaks where a deepening of the E × B velocity—governed by the radial electric field Er —near the edge is observed when the safety factor decreases in L-mode plasmas. Flux-driven global simulations of ion temperature gradient turbulence recover qualitatively the trend observed in the experiments, i.e. the E × B velocity increases when decreasing the safety factor. From these simulations, multiple clues point out the role of turbulence in the establishment of the radial electric field even though the turbulent intensity increases with the safety factor. The proposed mechanism to elucidate this phenomenon, backed up by a reduced model, is that the damping of the poloidal flow, governed by the neoclassical friction, increases more strongly with the safety factor than the turbulent drive for Er , due to the (r,θ) component of the Reynolds stress.

Observation of reduced-turbulence regime with tungsten injection in HL-2A edge plasmas

A reduced-turbulence regime has been observed in HL-2A NBI-heated deuterium plasmas. The transition to this regime is achieved by injecting a certain amount of tungsten into the plasma based on the laser blow-off technique. It has been found that the amplitude of geodesic acoustic mode (GAM) zonal flow and turbulent vortex size together with eddy tilting angle are all significantly increased in the edge region after tungsten injection. However, the frequency of GAM zonal flow remains nearly unchanged. Measurement shows the nonlinear coupling degree of turbulence dramatically increases while the collisional damping of GAM zonal flow drops slightly. We conclude that the increased nonlinear coupling is the main cause of the excitation of GAM zonal flow, which consequently results in the reduction in turbulent transport as observed in this experiment. These results indicate that tungsten ions play an active role in turbulence-GAM dynamics through a symmetry-breaking mechanism, which could help us to better understand the inherent physical mechanisms governing turbulent transport in the presence of high-Z impurity ions in fusion plasmas.

Low-cost UAV monitoring: insights into seasonal volumetric changes of an oyster reef in the German Wadden Sea

This study aims to quantify the dimensions of an oyster reef over two years via low-cost unoccupied aerial vehicle (UAV) monitoring and to examine the seasonal volumetric changes. No current study investigated via UAV monitoring the seasonal changes of the reef-building Pacific oyster (Magallana gigas) in the German Wadden Sea, considering the uncertainty of measurements and processing. Previous studies have concentrated on classifying and mapping smaller oyster reefs using terrestrial laser scanning (TLS) or hyperspectral remote sensing data recorded by UAVs or satellites. This study employed a consumer-grade UAV with a low spectral resolution to semi-annually record the reef dimensions for generating digital elevation models (DEM) and orthomosaics via structure from motion (SfM), enabling identifying oysters. The machine learning algorithm Random Forest (RF) proved to be an accurate classifier to identify oysters in low-spectral UAV data. Based on the classified data, the reef was spatially analysed, and digital elevation models of difference (DoDs) were used to estimate the volumetric changes. The introduction of propagation errors supported determining the uncertainty of the vertical and volumetric changes with a confidence level of 68% and 95%, highlighting the significant change detection. The results indicate a volume increase of 22 m³ and a loss of 2 m³ in the study period, considering a confidence level of 95%. In particular, the reef lost an area between September 2020 and March 2021, when the reef was exposed to air for more than ten hours. The reef top elevation increased from -15.5 ± 3.6 cm NHN in March 2020 to -14.8 ± 3.9 cm NHN in March 2022, but the study could not determine a consistent annual growth rate. As long as the environmental and hydrodynamic conditions are given, the reef is expected to continue growing on higher elevations of tidal flats, only limited by air exposure. The growth rates suggest a further reef expansion, resulting in an increased roughness surface area that contributes to flow damping and altering sedimentation processes. Further studies are proposed to investigate the volumetric changes and limiting stressors, providing robust evidence regarding the influence of air exposure on reef loss.

Study of Gap Flow Effects on Sound Transmission Through a Micro-Perforated Sandwich Cylindrical Shell Excited by a Plane Wave

This paper presents a theoretical model to study sound transmission through a micro-perforated double-walled sandwich cylindrical shell with emphasis on the potential of internal gap flow to improve the sound insulation performance. This model adopts Donnell’s thin shell theory to govern the shell motions and a simplified model based on Biot’s theory to describe sound propagation in the porous lining. The mean acoustic particle velocity model with the grazing flow effect is applied to define the coupling condition between the fluid medium and the perforated shell. Transmission loss (TL) through the configuration with one gap flow is numerically calculated using the mode superposition method when subjected to an oblique plane wave in the presence of an external mean flow. Results reveal that the gap flow in the opposite direction to the external flow would elevate the mid-frequency TL. Gap depth has a complicated effect on TL since it influences the aerodynamic damping of the gap flow and sound dissipation in the porous layer. The average TL in 20–20[Formula: see text]000[Formula: see text]Hz is then optimized with a genetic algorithm, and an increase of 17.82 dB is finally achieved. The possibility of tuning the structural sound insulation performance by the gap flow has been verified.

The two roles of the SPH Kernel and dissipation on accretion disc modelling in semi-detached Low-Mass Close Binaries

This study addresses two aspects of the SPH technique. The first aspect regards the 3D (2D) numerical integration. The second aspect deals with the necessary damping for handling inviscid flow discontinuities. Accretion discs in Low-Mass Close Binaries (LMCBs) are a context pushing the limit of SPH codes since turbulence, shocks, and shear flows coexist. Thus, an LMCB with high mass-transfer from the inner Lagrangian point is considered. This choice is made for an in-depth understanding of the numerical and physical aspects related to pressure forces computed through SPH Kernel spatial gradients in high-speed collisional flows. In this regard, gas compressibility also plays a role. Firstly, we pay attention to the algebraic term 4πr2 coming from the cubic differential d3r and affecting any 3D SPH integration. This is made through a comparison of physically inviscid SPH structures referring to the same LMCB and mass-transfer conditions. Then, a reformulation of the non-viscous damping is also considered by adopting a Lagrangian-free physical formulation. Inviscid accretion discs should develop a steady toroidal structure because radial transport mechanisms should be excluded by missing any radial shear flow damping. Therefore, thinner discs, also including a steady toroidal ring, identify that SPH inviscid modelling free of any incorrect pressure gradient excess among thick disc structures. This task is simplified since initial conditions deliberately favour thick discs. The physically viscous hydrodynamics of the disc is also addressed by adopting a Prandtl-like turbulent kinematic viscosity coefficient in the Navier–Stokes equations, distinguishing the roles of the bulk and shear viscosities.

Viscosity impact on 3D non-linear MHD simulations of RFP fusion plasmas

Several studies pointed out the joint role of resistivity η and viscosity ν in determining the dynamics and the emergence of helical regimes of reversed-field pinch (RFP) plasmas. In this framework, the self-consistent time evolution of the η and ν coefficients still lacks of a fully satisfying modeling, being constrained by many approximations. In this work, the hypothesis of a flat viscosity profile is relaxed: A viscosity profile inspired by the Braginskii perpendicular viscosity is implemented in the code. This choice is motivated by the fact that the magnetohydrodynamics field instabilities relevant for the RFP configuration dynamics (resistive-kink/tearing modes) are active in the direction perpendicular to the magnetic field. Such a non-monotonous profile causes a localized damping of plasma flow in the regions, where the viscosity is stronger, close to the plasma edge. This results in the reduction of the flow shear, in turn allowing the enhancement of edge magnetic field modes amplitude. The impact on the magnetic topology and on connection length to the wall is also analyzed.

Toroidal modeling of plasma flow damping and density pump-out by RMP during ELM mitigation in HL-2A

Reduction of both the plasma density and toroidal flow speed, due to application of the predominantly n = 1 (n is the toroidal mode number) resonant magnetic perturbation (RMP) for controlling the edge localized mode in the HL-2A tokamak, is numerically investigated utilizing the quasi-linear initial-value code MARS-Q (Liu et al 2013 Phys. Plasmas 20 042503). Simulation results reveal that the neoclassical toroidal viscosity (NTV) due to three dimensional fields plays the key role in modifying the plasma momentum and particle transport in the HL-2A discharge. By comparing the modeling results with the measured density pump-out in the experiment, the electron NTV particle flux model, in combination with the free-boundary condition for the axisymmetric change of the density at the plasma edge, is found to yield the best agreement in terms of both the pump-out level and the overall time scale. Further sensitivity studies show that the simulated density pump-out level is reasonably robust against variations in the model assumptions, including the particle diffusion model and the non-ambipolar versus ambipolar NTV particle flux. The latter however affects the time scale for reaching the steady state solution. Finally, it is found that the plasma edge-peeling response, the NTV torque, as well as the plasma momentum and particle transport, all are sensitive to the toroidal phase difference between the upper and lower rows of the RMP coil currents in HL-2A, with the 30∘ coil phasing producing the minimal side effects on the plasma.

Aortic stiffness and impairments in cerebrovascular function in cancer survivorship

An established consequence of anti-cancer treatment is increased aortic stiffness, which impairs the aortic damping of cardiac-generated pulsatile pressure and blood flow. These increased pulsations may result in vascular injury in the microcirculation of target end organs like the brain. The aim of the present study was to determine the ramifications of elevated aortic stiffness on cerebral macro- and microvascular pulsatility and autoregulatory function in cancer survivors. We hypothesized that cerebrovascular pulsatility would be enhanced, and cerebrovascular reactivity (CVR) to elevations in arterial CO 2 would be impaired, both in the large and small cerebral vessels in patients with a history of cancer and anti-cancer treatment versus healthy age- and sex-matched non-cancer controls. Aortic arch pulse wave velocity (aaPWV), a regional measure of stiffness in the proximal aorta, was calculated from echocardiographic views of the aortic valve and the descending aortic arch. Subjects then breathed room air for 60s followed by ~2 minutes of rebreathing with simultaneous collection of end-tidal pressure of CO 2 (P ET CO 2 ). Middle cerebral artery blood flow velocity (MCAv) was assessed using transcranial doppler. Oxygenated (oxy-[Hb]) and deoxygenated (deoxy-[Hb]) cerebral microvascular hemoglobin concentrations were monitored using two near-infrared spectroscopy probes placed over the prefrontal cortices. Pulsatility was assessed at baseline using Gosling’s Pulsatility Index (PI) for both the microvasculature and MCA. To account for changes in perfusion pressure, MAP was collected via finger plethysmography and cerebrovascular conductance indices were calculated for both oxy-[Hb] (CVCi oxy ) and MCAv (CVCi MCA ). CVR was calculated as the slope of CVCi oxy and CVCi MCA over P ET CO 2 during rebreathing. Six women were recruited (n = 3/group). Age was similar between the healthy non-cancer controls (HC) and patients with a history of cancer and anti-cancer treatment (CS) (31.0 ± 7 vs. 32.7 ± 13 years). Baseline ventilatory (P ET CO 2 ), central and cerebral hemodynamics were also similar (all p > 0.05). aaPWV was significantly elevated in CS compared to HC (5.3 ± 0.6 vs. 3.1 ± 1.1 m/s, p = 0.04). Cerebral microcirculation PI in CS was significantly elevated (0.04 ± 0.01 vs. 0.02 ± 0.01, p = 0.02), but MCA PI was not different (p > 0.05). CVR-CVCi oxy was impaired in CS (-3.6 ± 2.8 vs. 1.5 ± 0.5 nM·mmHg -1 /mmHg, p = 0.03), and CVR-CVCi MCA was similarly reduced (-0.05 ± 0.06 vs. 0.15 ± 0.01 mm/sec·mmHg -1 /mmHg, p < 0.01). The increase in MAP during hypercapnia was similar between groups. Consistent with our hypothesis, both CVR-CVCi oxy and CVR-CVCi MCA were attenuated in patients with a history of anti-cancer treatment. Additionally, the microvascular pulsatility was greater in CS. This suggests an impaired cerebrovascular function, possibly related to increased transmission of pulsatility into the brain, in the years following cancer diagnoses and treatment. This is the full abstract presented at the American Physiology Summit 2023 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

Effective Shear and Bulk Viscosities of the Quark–Gluon Plasma: QCD Versus Heavy-ion Data

In recent years, there has been a significant effort to extract the temperature-dependent shear ($\eta/s$) and bulk ($\zeta/s$) viscosity over entropy ratios of the quark-gluon plasma from a global comparison of heavy-ion data with results of hydrodynamic simulations. However, anisotropic flow, which is arguably the most sensitive probe of viscosity, is only sensitive to an {\it effective\/} viscosity over entropy ratio, which is obtained by taking a weighted average over the temperature, and summing the contributions of shear and bulk. We estimate this effective viscosity using existing first-principles calculations, which give $0.17<(\eta/s)_{\rm eff}<0.21$, and $(\zeta/s)_{\rm eff}<0.08$, implying that the damping of anisotropic flow at the LHC is mostly due to shear viscosity. The values extracted from global data analyses are compatible with these theory predictions.

Bidirectional Coupling Analysis of Support Pipeline for Shock Damping of Dilute Debris Flow

With the frequent occurrence of geological disasters and extreme weather, pipeline failure accidents and secondary disasters caused by debris flow continue to occur. Aiming at the impact of thin debris flow on support pipeline, a multi-physical coupling model of debris flow and support pipeline with vibration reduction is established based on the expansion flow model. The expansive fluid model is used to simulate the stress of the support pipeline under the action of thin debris flow. The results show that the impact effect of thin debris flow slurry on the support pipeline is concentrated at the inlet surface of the pipeline and the support, and gradually decreases to both ends. As the velocity of debris flow slurry increases, the stress acting on the support pipeline also increases. When the flow field tends to be stable, the load acting on the support pipeline maintains a stable value. The leading effect of debris flow slurry is the most serious damage to the support pipeline. The research results have a certain reference value for the impact of dilute debris flow in support pipeline.

FEATURES OF DETERMINING THE CONVERSION OF ENERGY IN HYDRAULIC SYSTEMS

The analysis of the processes of transformation of kinetic energy in various types of devices providing both stabilizations of the flow and partial damping of energy is carried out. In the analysis, the features of the flow of a viscous fluid in diffusers, pulsation dampers, and shock absorbers are considered. In addition, it has been shown that for electrically conductive liquids, the transverse magnetic field method can be used. The materials of the analysis can be used to describe energy transport phenomena in liquids.

Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using discontinuous Galerkin method

A gyroaveraged nonlinear collision operator is formulated based on the Fokker-Planck operator in the Rosenbluth-MacDonald-Judd (RMJ) potential form and implemented for the gyrokinetic simulations with the discontinuous Galerkin scheme. The divergence structure of the original RMJ form is carefully preserved throughout the formulation to guarantee the density conservation while neglecting the finite Larmor radius effect. The B-spline finite element method is used to calculate the Rosenbluth potentials for the nonlinear collision operator. In addition to the nonlinear collision operator, linear and Dougherty collision models are also implemented to assess the benefits and drawbacks of each model. For the conservation of the parallel momentum and energy, we adopt a simple advection-diffusion model which numerically enforces the conservation of physical quantities. From bump-on-tail relaxation tests, the monotonically increasing entropy in time and conservation properties are demonstrated for the developed collision operator. Also, a few theoretical predictions for the neoclassical physics such as the neoclassical heat flux, poloidal flow and collisional damping of zonal flow are successfully reproduced by numerical simulations.

Numerical investigation of the fishbone instability effect on thermal pressure in EAST tokamak

The formation of the internal transport barrier (ITB) is observed after the emergence of fishbone instabilities on the Experimental Advanced Superconducting Tokamak (EAST). The kinetic–magnetohydrodynamic hybrid code M3D-K has been applied to investigate the fishbone instability effect on thermal pressure based on EAST Shot No. 71320. Without fluid nonlinearity, it is found that when the central gradient of the total pressure profile is above a threshold, the thermal pressure profile becomes more peaked due to the nonlinear evolution of the fishbone instability, which confirms that the fishbone instability can transport the thermal pressure radially inward and promote the ITB formation. When fluid nonlinearity is included, the poloidal zonal flow prevents the thermal pressure to become more peaked in the core region. As the neoclassical effect can cause the damping of the poloidal zonal flow and is neglected in our simulation, the actual promotion of ITB formation due to the fishbone instability is expected to be between that without fluid nonlinearity and with fluid nonlinearity.

Egg-speriments: Stretch, crack, and spin

Eggs are key ingredients in our kitchens because of their nutritional values and functional properties such as foaming, emulsifying, and gelling, offering a wide variety of culinary achievements. They also constitute ideal objects to illustrate a myriad of scientific concepts. In this article, we focus on several experiments (egg-speriments) that involve the singular properties of the liquids contained inside the eggshell, especially the egg white. We first characterize the rheology of an egg white in a rotational rheometer for constant and oscillatory shear stresses revealing its shear-thinning behavior and visco-elastic properties. Then, we measure the tendency of the fluid to generate very long filaments when stretched that we relate to the shear modulus of the material. Second, we explore the anisotropic crack pattern that forms on a thin film of egg white after it is spread on a surface and let dried. The anisotropy results from the long protein chains present in the egg white, which are straightened during film deposition. Finally, we consider the “spin test” that permits to distinguish between raw and hard-boiled eggs. To do so, we measure the residual rotation of a spinning raw egg after a short stop, which reflects the continuation of the internal flow. These observations are interpreted in terms of viscous damping of the internal flow consistently with the measurements deduced from rheology.

Electrostatic-Fluid-Structure 3D Numerical Simulation of a MEMS Electrostatic Comb Resonator.

The reliability and stability of MEMS electrostatic comb resonators have become bottlenecks in practical applications. However, there are few studies that comprehensively consider the nonlinear dynamic behavior characteristics of MEMS systems and devices in a coupled field so that the related simulation accuracy is low and cannot meet the needs of design applications. In this paper, to avoid the computational complexity and the uncertainty of the results of three-field direct coupling and take into the damping nonlinearity caused by coupled fields, a novel electrostatic-fluid-structure three-field indirect coupling method is proposed. Taking an actual microcomb resonant electric field sensor as an example, an electrostatic-fluid-structure multiphysics coupling 3D finite element simulation model is established. After considering the influence of nonlinear damping concerning the large displacement of the structure and the microscale effect, multifield coupling dynamics research is carried out using COMSOL software. The multiorder eigenmodes, resonant frequency, vibration amplitude, and the distribution of fluid load of the microresonator are calculated and analyzed. The simulated data of resonance frequency and displacement amplitude are compared with the measured data. The results show that the fluid load distribution of the microelectrostatic comb resonator along the thickness direction is high in the middle and low on both sides. The viscous damping of the sensor under atmospheric pressure is mainly composed of the incompressible flow damping of the comb teeth, which is an order of magnitude larger than those of other parts. Compared with the measured data, it can be concluded that the amplitude and resonance frequency of the microresonator considering the nonlinear damping force and residual thermal stress are close to the experimental values (amplitude error: 15.47%, resonance frequency error: 12.48%). This article provides a reference for studies on the dynamic characteristics of electrostatically driven MEMS devices.

Abstract 14119: Aortic Stiffness is Associated With Cognitive Decline and Cardiovascular Disease Manifestation in Cancer Survivors

Introduction: Cancer survivors experience a high prevalence of cognitive deficits and risk of CVD. Aortic stiffening, which increases following cancer diagnosis, inhibits pulsatile flow damping and potentiates microvascular damage within the cerebral circulation, which may contribute to associations observed between stiffness, cognitive decline, and mild cognitive impairment (MCI) in the general population. However, there is a paucity of evidence on the association between arterial stiffness and cognitive decline among cancer survivors. Methods: We evaluated dementia-free cancer survivors in the Framingham Original and Offspring Cohorts (n=277, 80±12.3 years old, 56.7% women, 9.3±8.8 years from first diagnosis) with carotid-femoral pulse wave velocity (cfPWV) measurements. Over a mean follow-up of 7.7±3.9 years, the Mini-Mental State Examination (MMSE; global cognitive function) and neuropsychological exams (NP; executive function, learning and memory) were completed. We noted cognitive decline and incident MCI per clinically relevant standards. We used linear and logistic regression models to determine the relationship between cfPWV and cognitive decline/MCI. Cox Regression related cfPWV and cognitive function to the risk of CVD. Multivariate models were adjusted for age, sex, depressive symptoms, and traditional CVD risk factors. Results: cfPWV was significantly associated with the rate of decline for global cognitive function (ΔMMSE). Higher cfPWV was significantly associated with clinically defined MCI, denoted by an impaired MMSE (OR(95%CI): 9.2(2.5-33.5)) and NP score (4.3 (1.4-13.0)) in univariate logistic models. We observed a 3.4 times higher risk of CVD in cancer survivors with high cfPWV (HR(95%CI): 3.45 (1.04-11.66)) in the final model. Changes in cognitive function were not associated with CVD. Conclusion: Our findings suggests an association between aortic stiffness, future cognitive decline, and an increased risk of CVD in a diverse cohort of cancer survivors. Our findings support the potential adverse consequences of a stiffening arterial vasculature following cancer diagnosis; specifically those related to cognitive function and long-term CVD outcomes.

Global gyrokinetic simulation with kinetic electron for collisionless damping of zonal flow in stellarators

Global gyrokinetic simulations with kinetic electrons for collisionless damping of zonal flows in LHD and W7-X stellarators show that the helical components of the equilibrium magnetic field responsible for helically trapped particles have significant impacts on zonal flow. Kinetic electrons reduce zonal flow residue and increase the frequency of low frequency oscillation (LFO). The LFO is induced by dominant helical harmonics of magnetic field strength. Furthermore, linear toroidal coupling of multiple toroidal n-harmonics barely affects the zonal flows, but can generate long wavelength toroidal harmonics with the same toroidal number as the helical magnetic field.