Turbulence Spreading Research Articles

Taming turbulent fronts by bending pipes

The flow of fluid through a pipe has been instrumental in illuminating the subcritical route to turbulence typical of many wall-bounded shear flows. Especially important in this process are the turbulent–laminar fronts that separate the turbulent and laminar flow. Four years ago Michael Graham (Nature, vol. 526, 2015, p. 508) wrote a commentary entitled ‘Turbulence spreads like wildfire’, which is a picturesque but also accurate characterisation of the way turbulence spreads through laminar flow in a straight pipe. In this spirit, the recent article by Rinaldi et al. (J. Fluid Mech., vol. 866, 2019, pp. 487–502) shows that turbulent wildfires are substantially tamed in bent pipes. These authors find that even at modest pipe curvature, the characteristic strong turbulent–laminar fronts of straight pipe flow vanish. As a result, the propagation of turbulent structures is modified and there are hints that the route to turbulence is fundamentally altered.

Localized modulation of turbulence by m/n = 1/1 magnetic islands in the HL-2A tokamak

Modulation of turbulent electron temperature fluctuations () and density fluctuations () by an m/n = 1/1 tearing mode island was observed in the core plasma region of the HL-2A tokamak. High spatiotemporal resolution two-dimensional images of show the first evidence that the turbulence modulation occurs only when the island width exceeds a certain threshold value ( cm) and the modulation is localized merely in the inner area of the island due to significant alteration of local profiles and turbulence drives. Evidence also reveals that for large islands turbulence spreading takes place across the island region. The results are generally consistent with theories and simulations.

Uncovering what triggers avalanches in plasma

A new, unified model of turbulence spreading and avalanching is the first to propose a mechanism for plasma avalanches.

Subcritical turbulence spreading and avalanche birth

In confined plasmas, a localized fluctuation in a marginal or weakly damped region will propagate and generate an avalanche if it exceeds a threshold. In this letter, a new model for turbulence spreading based on subcritical instability in the turbulence intensity is introduced. We derive a quantitative threshold for spreading from a seed in a stable region, based on a competition between diffusion and nonlinear growth of the turbulence intensity. The model resolves issues with the established Fisher equation model for turbulence spreading, which is supercritical and cannot support the stationary coexistence of multiple turbulence levels. Implications for turbulence spreading are discussed, including the dynamics of ballistic penetration of turbulence into the stable zone. Tests of the theory are suggested.

Effects of RMP-induced changes of radial electric fields on microturbulence in DIII-D pedestal top

Gyrokinetic simulations of DIII-D tokamak with axisymmetric equilibrium show that the reduction in the radial electric field shear at the top of the pedestal during edge localized mode (ELM) suppression with the n = 2 resonant magnetic perturbations (RMPs) leads to enhanced drift-wave turbulence and extended turbulence spreading to the top of the pedestal relative to ELMing plasmas with similar RMP and pedestal parameters. The simulated turbulent transport at the top of the pedestal in ELM suppressed conditions is consistent with experimental observations of enhanced turbulence at the top of the pedestal during ELM suppression by the RMPs. These results imply that enhanced drift-wave turbulence due to reduced shear at the pedestal top can contribute to the additional transport required to prevent the pedestal growing to a width that is unstable to ELMs.

Compressible mixing layer in shock-induced separation

Unsteadiness in separated shock–boundary layer interactions have been previously analysed in order to propose a scenario of entrainment–discharge as the origin of unsteadiness. It was assumed that the fluid in the separated zone is entrained by the free shear layer formed at its edge, and that this layer follows the properties of the canonical mixing layer. This last point is addressed by reanalysing the velocity measurements in an oblique shock reflection at a nominal Mach number of 2.3 and for two cases of flow deviation ($8^{\circ }$ and $9.5^{\circ }$). The rate of spatial growth of this layer is evaluated from the spatial growth of the turbulent stress profiles. Moreover, the entrainment velocity at the edge of the layer is determined from the mean velocity profiles. It is shown that the values of turbulent shear stress, spreading rate and entrainment velocity are consistent, and that they follow the classical laws for turbulent transport in compressible shear layers. Moreover, the measurements suggest that the vertical normal stress is sensitive to compressibility, so that the anisotropy of turbulence is affected by high Mach numbers. Dimensional considerations proposed by Brown & Roshko (J. Fluid Mech., vol. 64, 1974, 775–781) are reformulated to explain this observed trend.

Measurement and control of turbulence spreading in the scrape-off layer of TJ-II stellarator

In this paper, we report measurements of turbulence drive and spreading in the edge region and scrape-off layer (SOL) of the TJ-II stellarator for plasmas heated by ECRH and NBI. In addition, we show how these and various other relevant quantities can be modulated actively via edge biasing, i.e. by establishing an edge flow shear layer. When the edge shear reaches values close to the inverse of the turbulence correlation time, turbulence spreading and edge-SOL coupling are reduced significantly. The resulting control of SOL spreading via the edge shear is important in view of the issue of understanding and controlling the SOL width in fusion-relevant devices.

Childhood of turbulent spots in a shear flow

We numerically investigate temporal aspects of turbulent spot spreading in plane Couette flow for transitional Reynolds numbers, focusing on streamwise spreading. From the topology of turbulent spots and associated large-scale flows, we propose a decomposition of the streamwise growth rate.

Mesoscopic Transport Events and the Breakdown of Fick’s Law for Turbulent Fluxes

This paper presents a pedagogical review of the physics of mesoscopic transport events and their role in the breakdown of Fick’s Law for turbulent transport in magnetically confined plasma. It is now clear that the conventional picture of localized turbulence and quasi-linear calculation of fluxes fails to address and account for the phenomenology of tokamak transport. One key issue is the observed departure from the expected gyro-Bohm transport scaling. The causes of this breakdown of Fickian thinking include turbulent avalanching and pulse propagation (turbulence spreading). Both are mesoscopic transport events, and both tend to de-localize the flux–gradient relation. Turbulence spreading is the process of self-scattering and expansion of a slug or other local exciton of turbulence. Spreading is described by theoretically-motivated, phenomenological reaction–diffusion models for the turbulence activity (intensity) field, much in the spirit of Ginzburg–Landau theory. Such models imply that spreading will occur by propagation of intensity fronts. After discussing the basic theory, this paper presents several critical tests of turbulence spreading models using gyrokinetic simulation. Applications include rho-star scaling, penetration of transport barriers and core-edge coupling. Relevant experiment–theory comparisons are addressed, as well. Avalanching refers to a process whereby correlated topplings of nearby localized cells overturn sequentially and drive a burst of transport. Avalanching is a process intrinsic to systems that support a broad range of scales l between a cell size Δ and system size L, i.e. Δ < l < L. Avalanching is also a natural way to produce transport events on scales that exceed the cell size or correlation length. Therefore, the PDF (probability distribution function) of avalanches as a function of l is a crucial quantity, necessary for predicting confinement in a system like ITER, with a very large-scale separation between L and Δ. Avalanching emerged from the theory of selforganized criticality but is a more general phenomenon. The paper traces the intellectual prehistory of avalanching through the advent of self-organized criticality. Special focus is devoted to reduced continuum models of avalanching. The physics of avalanching in confined plasma is discussed in detail, via several multi-faceted comparisons to flux-driven fluid and gyrokinetic simulations. The dominance of bursty, large transport events in the flux is identified. Evidence for avalanching in basic and confinement experiments is summarized. The paper concludes with sections on selected special topics, a discussion of the relation between turbulence spreading and avalanching, and a list of possible future directions. Throughout the paper, an effort is made to set fusion theory and phenomenology in the context of ideas discussed in the broader scientific community.

Hysteresis Relation between Turbulence and Temperature Modulation during the Heat Pulse Propagation into a Magnetic Island in DIII-D.

The hysteresis relation between turbulence and temperature modulation during the heat pulse propagation into a magnetic island is studied for the first time in toroidal plasmas. Lissajous curves of the density fluctuation (n[over ˜]/n) and the electron temperature (T_{e}) modulation show that the (n[over ˜]/n) propagation is faster than the heat pulse propagation near the O point of the magnetic island. This faster n[over ˜]/n propagation is experimental evidence of the turbulence spreading from the X point to the O point of the magnetic island.

Impact of bootstrap current and Landau-fluid closure on ELM crashes and transport

Results presented here are from 6-field Landau-Fluid simulations using shifted circular cross-section tokamak equilibria on BOUT++ framework. Linear benchmark results imply that the collisional and collisionless Landau resonance closures make a little difference on linear growth rate spectra which are quite close to the results with the flux limited Spitzer-Härm parallel flux. Both linear and nonlinear simulations show that the plasma current profile plays dual roles on the peeling-ballooning modes that it can drive the low-n peeling modes and stabilize the high-n ballooning modes. For fixed total pressure and current, as the pedestal current decreases due to the bootstrap current which becomes smaller when the density (collisionality) increases, the operational point is shifted downwards vertically in the Jped – α diagram, resulting in threshold changes of different modes. The bootstrap current can slightly increase radial turbulence spreading range and enhance the energy and particle transports by increasing the perturbed amplitude and broadening cross-phase frequency distribution.

Observation of multi-channel non-local transport in J-TEXT plasmas

In cold pulse experiments in J-TEXT, not only are rapid electron temperature increases in the core observed, but also steep rises in the inner density are found. Moreover, some evidence of acceleration of the core toroidal rotation is also observed during the non-local transport process of electron temperature. These new findings of cold pulse experiments in J-TEXT suggest that turbulence spreading is a possible mechanism for the non-local transport dynamics.

Thermal Transients in District Heating Systems

Heat fluxes in a district heating pipeline systems need to be controlled on the scale from minutes to an hour to adjust to evolving demand. There are two principal ways to control the heat flux - keep temperature fixed but adjust velocity of the carrier (typically water) or keep the velocity flow steady but then adjust temperature at the heat producing source (heat plant). We study the latter scenario, commonly used for operations in Russia and Nordic countries, and analyze dynamics of the heat front as it propagates through the system. Steady velocity flows in the district heating pipelines are typically turbulent and incompressible. Changes in the heat, on either consumption or production sides, lead to slow transients which last from tens of minutes to hours. We classify relevant physical phenomena in a single pipe, e.g. turbulent spread of the turbulent front. We then explain how to describe dynamics of temperature and heat flux evolution over a network efficiently and illustrate the network solution on a simple example involving one producer and one consumer of heat connected by “hot” and “cold” pipes. We conclude the manuscript motivating future research directions.

Multi-shell transport model for L-H transition

A coupled model of transport, turbulence, and mesoscale flows is proposed, including turbulence spreading. The model consists of transport equations for plasma density and pressure coupled to a shell model of drift wave turbulence, which incorporates coupling to mesoscale flows via disparate scale interactions. The model can describe the turbulent cascade and its dynamical interplay with zonal and mean shear flows as well as the profile evolution (including the profiles of turbulence intensity itself) due to these self-consistent turbulent fluxes. This simple system of equations is shown to capture the low to high confinement (L-H) transition. It is also observed that as the heating is increased, the system goes through an intermediate phase that displays oscillations between zonal flows and turbulence. The transition towards the H mode, which is characterized by the presence of a strong mean shear flow at the edge, is triggered by the mesoscale dynamics due to the action of zonal flows, with turbulence spreading playing an important role in the H to L back transition.

Enhancement and suppression of turbulence by energetic-particle-driven geodesic acoustic modes

We propose a novel mechanism of enhancement of turbulence by energetic-particle-driven geodesic acoustic modes (EGAMs). The dynamics of drift-wave-type turbulence in the phase space is investigated by wave-kinetic equation. Spatially inhomogeneous turbulence in the presence of a transport barrier is considered. We discovered that trapping of turbulence clumps by the EGAMs is the key parameter that determines either suppress or enhance turbulence. In regions where turbulence is unstable, EGAM suppresses the turbulence. In contrast, in the stable region, EGAM traps clumps of turbulence and carries them across the transport barrier, so that the turbulence can be enhanced. The turbulence trapped by EGAMs can propagate independent of the gradients of density and temperature, which leads to non-Fickian transport. Hence, there appear a new global characteristic velocity, the phase velocity of GAMs, for turbulence dynamics, in addition to the local group velocity and that of the turbulence spreading. With these effect, EGAMs can deteriorate transport barriers and affect turbulence substantially. This manuscript provides a basis to consider whether a coherent wave breaks or strengthen transport barriers.

Bistable dynamics of turbulence spreading in a corrugated temperature profile

We present a new model of turbulence spreading in magnetically confined plasma. A basic question in turbulence spreading is how to sustain finite amplitude fluctuations in a stable subcritical region, where linear dissipation of the turbulence is strong? The answer to this question relies on a consistent treatment of mesoscale temperature profile corrugation and microscale turbulence. We argue that inhomogeneous mixing of the turbulence corrugates the mean temperature profile and that the temperature corrugation then induces subcritical bifurcation of the turbulence. Thus, the system will transition from a metastable “laminar” state to an absolutely stable, excited state. Incorporating spatial coupling of the locally excited turbulent regions, a front forms. This front connects the excited and laminar states and penetrates the linear stable region efficiently. We argue that such bistable turbulence spreading can explain observations of hysteresis in the intensity of L-mode core turbulence.

New procedure for analyzing processes in a plate precipitator

A new procedure to analyze particle motion in turbulent flows is presented. The existing Euler method does not accurately account particle charging, while the computational cost of the Lagrange method is high. A new approach to this problem is based on the distribution of the particle concentration in the turbulent jet spreading from a point source. This procedure makes an allowance for the dispersion of particle charges and has a low computational cost.

Edge turbulence and divertor heat flux width simulations of Alcator C-Mod discharges using an electromagnetic two-fluid model

The BOUT++ code has been exploited in order to improve the understanding of the role of turbulent modes in controlling edge transport and resulting scaling of the scrape-off layer (SOL) heat flux width. For the C-Mod enhanced (EDA) H-mode discharges, BOUT++ six-field two-fluid nonlinear simulations show a reasonable agreement of upstream turbulence and divertor target heat flux behavior: (a) the simulated quasi-coherent modes show consistent characteristics of the frequency versus poloidal wave number spectra of the electromagnetic fluctuations when compared with experimental measurements: frequencies are around 60–120 kHz (experiment: about 70–110 kHz), are around 2.0 which is similar to the phase contrast imaging data; (b) linear spectrum analysis is consistent with the nonlinear phase relationship calculation which indicates the dominance of resistive-ballooning modes and drift-Alfven wave instabilities; (c) the SOL heat flux width versus current Ip scaling is reproduced by turbulent transport: the simulations yield similar to experimental measurements within a factor of 2. However the magnitudes of divertor heat fluxes can be varied, depending on the physics models, sources and sinks, sheath boundary conditions, or flux limiting coefficient; (d) Simple estimate by the ‘2-point model’ for is consistent with simulation. Moreover, blobby turbulent spreading is confirmed for these relatively high Bp shots.

How vegetation in flows modifies the turbulent mixing and spreading of jets

While studies on vegetated channel flows have been developed in many research centers, studies on jets interacting with vegetation are still rare. This study presents and analyzes turbulent jets issued into an obstructed cross-flow, with emergent vegetation simulated with a regular array of cylinders. The paper presents estimates of the turbulence diffusion coefficients and the main turbulence variables of jets issued into a vegetated channel flow. The experimental results are compared with jets issued into unobstructed cross-flow. In the presence of the cylinder array, the turbulence length-scales in the streamwise and transverse directions were reduced, relative to the unobstructed crossflow. This contributed to a reduction in streamwise turbulent diffusion, relative to the unobstructed conditions. In contrast, the transverse turbulent diffusion was enhanced, despite the reduction in length-scale, due to enhanced turbulent intensity and the transverse deflection of flow around individual cylinders. Importantly, in the obstructed condition, the streamwise and transverse turbulent diffusion coefficients are of the same order of magnitude.

Confinement of the solar tachocline by a cyclic dynamo magnetic field

The surprising thinness of the solar tachocline is still not understood with certainty today. Among the numerous possible scenarios suggested to explain its radial confinement, one hypothesis is based on Maxwell stresses that are exerted by the cyclic dynamo magnetic field of the Sun penetrating over a skin depth below the turbulent convection zone. Our goal is to assess under which conditions (turbulence level in the tachocline, strength of the dynamo-generated field, spreading mechanism) this scenario can be realized in the solar tachocline. We develop a simplified 1D model of the upper tachocline under the influence of an oscillating magnetic field imposed from above. The turbulent transport is parametrized with enhanced turbulent diffusion (or anti-diffusion) coefficients. Two main processes that thicken the tachocline are considered; either turbulent viscous spreading or radiative spreading. An extensive parameter study is carried out to establish the physical parameter regimes under which magnetic confinement of the tachocline that is due to a surface dynamo field can be realized. We have explored a large range of magnetic field amplitudes, viscosities, ohmic diffusivities and thermal diffusivities. We find that, for large but still realistic magnetic field strengths, the differential rotation can be suppressed in the upper radiative zone (and hence the tachocline confined) if weak turbulence is present (with an enhanced ohmic diffusivity of $\eta > 10^{7-8} \, cm^2/s$), even in the presence of radiative spreading. Our results show that a dynamo magnetic field can, in the presence of weak turbulence, prevent the inward burrowing of a tachocline subject to viscous diffusion or radiative spreading.