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Buoyancy driven turbulence with free-slip top and bottom plates in a confined box is studied via direct numerical simulations (DNS). The Rayleigh number is fixed to Ra=108 and the Prandtl number is Pr=10. The length/height aspect ratio Γy is fixed to 2, while the depth/height aspect ratio Γx varies from 0.125 to 0.3. With the variation of Γx, the hysteresis-like behavior of the heat and momentum transfer is found due to the emergence of two stable states of flow organization. For small Γx, the sheared flow occurs with disordered plumes occupying the entire bulk region, while it is replaced by convection rolls with increasing Γx. The striking feature is that the heat and momentum transport is greatly suppressed in the shear flow (SF) state, compared with that in the convection roll (CR) state. The turbulent kinetic energy budget is performed to delineate the relative contributions of physical processes that govern energy transport. It is found that the kinetic energy in the SF state is mainly produced by the buoyancy force, while it is dominated by the shear production near the plates in the CR state. During energy transport, it seems that the buoyancy force plays a more important role in the SF state than in the CR state, particularly in the bulk. Furthermore, through the analysis of dynamic mode decomposition of temperature fields, we illustrate that the flow in the SF state is dominated by the cloud-like spatially coherent structures with low frequencies, while the low-frequency roll structures play a leading role in the CR state. The flow state transition from SF to CR corresponds to the process that the cloud-like coherent structures form roll structures.

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We report the results of the direct numerical simulations of two-dimensional Rayleigh-Bénard convection(RBC) in order to study the influence of the periodic(PD) and confined(CF) samples on the heat transport Nu. The numerical study is conducted with the Rayleigh number(Ra) varied in the range 106≤Ra≤109 at a fixed Prandtl number Pr=4.3 and aspect ratio Γ=2 with the no-slip(NS) and free-slip(FS) plates. There exists a zonal flow for Ra≥3×106 with the free-slip plates in the periodic sample. In all the other cases, the flow is the closed large-scale circulation(closed LSC). The striking features are that the heat transport Nu is influenced and the temperature profiles do not be influenced when the flow pattern is zonal flow.

Thermally driven vertical convection (VC)-the flow in a box heated on one side and cooled on the other side, is investigated using direct numerical simulations with Rayleigh numbers over the wide range of 10 7 ≤ Ra ≤ 10 14 and a fixed Prandtl number Pr = 10 in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient S on Ra shows three different regimes: in regime I (Ra 5 × 10 10), S is almost independent of Ra; in the newly identified regime II (5 × 10 10 Ra 10 13), S first increases with increasing Ra (regime II a), reaches its maximum and then decreases again (regime II b); and in regime III (Ra 10 13), S again becomes only weakly dependent on Ra, being slightly smaller than in regime I. The transition from regime I to regime II is related to the onset of unsteady flows arising from the ejection of plumes from the sidewall boundary layers. The maximum of S occurs when these plumes are ejected over approximately half of the area (downstream) of the sidewalls. The onset of regime III is characterized by the appearance of layered structures near the top and bottom horizontal walls. The flow in regime III is characterized by a well-mixed bulk region owing to continuous ejection of plumes over large fractions of the sidewalls, and, as a result of the efficient mixing, the mean temperature gradient in the centre S is smaller than that of regime I. In the three different regimes, significantly different †

Turbulent convection processes in nature are often found to be organized in a hierarchy of plume structures and flow patterns. The gradual aggregation of convection cells or granules to a supergranule which eventually fills the whole horizontal layer is reported and analyzed in spectral element direct numerical simulations of three-dimensional turbulent Rayleigh-Bénard convection at an aspect ratio of 60. The formation proceeds over a time span of more than 104 convective time units for the largest accessible Rayleigh number and occurs only when the turbulence is driven by a constant heat flux which is imposed at the bottom and top planes enclosing the convection layer. The resulting gradual inverse cascade process is observed for both temperature variance and turbulent kinetic energy. An additional analysis of the leading Lyapunov vector field for the full turbulent flow trajectory in its high-dimensional phase space demonstrates that turbulent flow modes at a certain scale continue to give rise locally to modes with a longer wavelength in the turbulent case. As a consequence, successively larger convection patterns grow until the horizontal extension of the layer is reached. This instability mechanism, which is known to exist near the onset of constant heat flux-driven convection, is shown here to persist into the fully developed turbulent flow regime, thus connecting weakly nonlinear pattern formation with the one in fully developed turbulence. We discuss possible implications of our study for observed, but not yet consistently numerically reproducible, solar supergranulation which could lead to improved simulation models of surface convection in the Sun.

Rayleigh-Bénard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers Ra from 10 7 to 10 9 and Prandtl numbers Pr from 1 to 100. The effect of the width-to-height aspect ratio Γ is investigated for 1 Γ 128. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360-385) for Γ = 2, is only stable when Γ is smaller than a critical value, which depends on Ra and Pr. The regime in which only zonal flow can exist is called the first regime in this study. With increasing Γ , we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger Γ , in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on Ra and Pr. For instance, for Ra = 10 8 and Pr = 10, the aspect ratio Γ r of an individual, statistically stable convection roll can vary in a large range between 16/11 and 64. A convection roll with a large aspect ratio of Γ r = 64, or more generally already with Γ r 10, can be seen as 'localized' zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix Ra = 10 7 and Pr = 0.71, and compare the flow for Γ = 8 and Γ = 16. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number Nu and the Reynolds number Re increasingly behave like in the 2-D case. We then demonstrate that with increasing † Email address for correspondence: d.lohse@utwente.nl https://doi.org/10.1017/jfm.2020.793 Downloaded from https://www.cambridge.org/core. IP address: 130.89.108.130, on 28 Oct 2020 at 13:42:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. 905 A21-2 Q. Wang and others aspect ratio Γ , zonal flow, which was observed for small Γ = 2π by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for Γ = 16. For such large Γ , only convection roll states are statistically stable. In-between, here for medium aspect ratio Γ = 8, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.

Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different
transport properties, even for the very same values of the control parameters. What state the system takes
depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio (Γ) twodimensional
turbulent Rayleigh-B´enard flow with no-slip plates and horizontally periodic boundary
conditions as model system. We determine the number n of convection rolls, their mean aspect ratios
Γr ¼ Γ=n, and the corresponding transport properties of the flow (i.e., the Nusselt number Nu), as function
of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent β in Nu ∼ Raβ
is found to depend on the realized state and thus Γr, with a larger value for the smaller Γr. By making use of
a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping
determine the Γr window for the realizable turbulent states. The theoretical results are in excellent
agreement with our numerical finding 2=3 ≤ Γr ≤ 4=3, where the lower threshold is approached for the
larger Ra. Finally, we show that the theoretical approach to frame Γr also works for free-slip boundary
conditions.

We perform a numerical study of the heat transfer and flow structure of Rayleigh-Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity φ in the flow properties during the transition process from the traditional RB convection with φ = 1 (so no obstacles included) to Darcy-type porous-media convection with φ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number Ra from 10 5 to 10 10 and varying porosities φ, at a fixed Prandtl number Pr = 4.3, and we restrict ourselves to the two-dimensional case. For fixed Ra, the Nusselt number Nu is found to vary non-monotonically as a function of φ; namely, with decreasing φ, it first increases, before it decreases for φ approaching 0. The non-monotonic behaviour of Nu(φ) originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed φ, depending on Ra, two different heat transfer regimes are identified, with different effective power-law behaviours of Nu versus Ra, namely a steep one for low Ra when viscosity dominates, and the standard classical one for large Ra. The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

For rapidly rotating turbulent Rayleigh--B\'enard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like $Ra^{1/4}Ek^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.

In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface roughness, the scaling exponent $\unicode[STIX]{x1D6FD}$ in the relationship between the Nusselt number $Nu$ and the Rayleigh number $Ra$ , $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ can be ${\approx}1/2$ locally, provided that $Ra$ is large enough to ensure that the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is comparable to the roughness height. However, at even larger $Ra$ , $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ becomes thin enough to follow the irregular surface and $\unicode[STIX]{x1D6FD}$ saturates back to the value for smooth walls (Zhu et al. , Phys. Rev. Lett. , vol. 119, 2017, 154501). In this paper, we prevent this saturation by employing multiscale roughness. We perform direct numerical simulations of two-dimensional RB convection using an immersed boundary method to capture the rough plates. We find that, for rough boundaries that contain three distinct length scales, a scaling exponent of $\unicode[STIX]{x1D6FD}=0.49\pm 0.02$ can be sustained for at least three decades of $Ra$ . The physical reason is that the threshold $Ra$ at which the scaling exponent $\unicode[STIX]{x1D6FD}$ saturates back to the smooth wall value is pushed to larger $Ra$ , when the smaller roughness elements fully protrude through the thermal boundary layer. The multiscale roughness employed here may better resemble the irregular surfaces that are encountered in geophysical flows and in some industrial applications.

The coexistence of multiple turbulent states was reported in several recent studies in different flows. We present in this work that multiple turbulent states also exist for thermal convection in two-dimensional tilted cells with large aspect ratios (\Gamma= width/height) through direct numerical simulations for the Rayleigh number Ra = 10^7 and the Prandtl number Pr = 0.71. The considered \Gamma ranges from 1 to 16. The tilt angle β varies from 0◦ to 180◦. Multiple states are identified for small β with \Gamma>= 2, where the different flow states are reflected in different numbers of convection rolls. The corresponding Nu is generally higher for the flow state with more convection rolls. Moreover, flow mode
transitions between different roll states are observed for large \Gamma>= 8 when β is larger than a critical value. The effect of cell tilting on Nu and Re is also investigated. It is found that for \Gamma<= 4, Nu first increases with increasing β and then declines after reaching its local maximum. However, Nu generally decreases monotonically with increasing β for \Gamma = 8, 12, and 16. This indicates that the idea to enhance heat transfer by tilting the cell can be realized only for relatively small \Gamma for the present system. It is also found that the previous finding that Re decreases monotonically with increasing β for large β with \Gamma = 1 does not
hold for large-\Gamma cases.

We report an experimental observation of a flow topology transition via global bifurcation in a turbulent Rayleigh-Bénard convection. This transition corresponds to a spontaneous symmetry breaking with the flow becomes more turbulent. Simultaneous measurements of the large-scale flow (LSF) structure and the heat transport show that the LSF bifurcates from a high heat transport efficiency quadrupole state to a less symmetric dipole state with a lower heat transport efficiency. In the transition zone, the system switches spontaneously and stochastically between the two long-lived metastable states.

We report the observation of superstructures, i.e., very large-scale and long living coherent structures in highly turbulent Rayleigh-Bénard convection up to Rayleigh Ra=109. We perform direct numerical simulations in horizontally periodic domains with aspect ratios up to Γ=128. In the considered Ra number regime the thermal superstructures have a horizontal extend of six to seven times the height of the domain and their size is independent of Ra. Many laboratory experiments and numerical simulations have focused on small aspect ratio cells in order to achieve the highest possible Ra. However, here we show that for very high Ra integral quantities such as the Nusselt number and volume averaged Reynolds number only converge to the large aspect ratio limit around Γ≈4, while horizontally averaged statistics such as standard deviation and kurtosis converge around Γ≈8, the integral scale converges around Γ≈32, and the peak position of the temperature variance and turbulent kinetic energy spectra only converge around Γ≈64.

Turbulence is ubiquitous in nature and engineering applications. Although Kolmogorov’s ( C. R. Acad. Sci. URSS , vol. 30, 1941 a , pp. 301–305; Dokl. Akad. Nauk URSS , vol. 30, 1941 b , pp. 538–540) theory suggested a unique turbulent state for high Reynolds numbers, multiple states were reported for several flow problems, such as Rayleigh–Bénard convection and Taylor–Couette flows. In this paper, we report that multiple states also exist for turbulent plane Couette flow with spanwise rotation through direct numerical simulations at rotation number $Ro=0.2$ and Reynolds number $Re_{w}=1300$ based on the angular velocity in the spanwise direction and half of the wall velocity difference. With two different initial flow fields, our results show that the flow statistics, including the mean streamwise velocity and Reynolds stresses, show different profiles. These different flow statistics are closely related to the flow structures in the domain, where one state corresponds to two pairs of roll cells, and the other shows three pairs. The present result enriches the studies on multiple states in turbulence.

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number $Pr=6.4$ , corresponding to water, in a cylinder with a diameter-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=2$ ; second, a fluid with $Pr=0.8$ , corresponding to $\text{SF}_{6}$ or air, confined in a slender cylinder with $\unicode[STIX]{x1D6E4}=0.5$ ; and third, the main focus of this paper, a fluid with $Pr=0.025$ , corresponding to a liquid metal, in a cylinder with $\unicode[STIX]{x1D6E4}=1.87$ . The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate $Pr$ a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number $Ra$ , the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For $Pr=0.8$ , DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low- $Pr$ fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher $Ra$ , the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate $Pr$ , the quasi-steady Taylor vortices, is entirely absent in low- $Pr$ flows.

The AFiD code, an open source solver for the incompressible Navier-Stokes equations ({\color{blue}\burl{http://www.afid.eu}}), has been ported to GPU clusters to tackle large-scale wall-bounded turbulent flow simulations. The GPU porting has been carried out in CUDA Fortran with the extensive use of kernel loop directives (CUF kernels) in order to have a source code as close as possible to the original CPU version; just a few routines have been manually rewritten. A new transpose scheme, which is not limited to the GPU version only and can be generally applied to any CFD code that uses pencil distributed parallelization, has been devised to improve the scaling of the Poisson solver, the main bottleneck of incompressible solvers. The GPU version can reduce the wall clock time by an order of magnitude compared to the CPU version for large meshes. Due to the increased performance and efficient use of memory, the GPU version of AFiD can perform simulations in parameter ranges that are unprecedented in thermally-driven wall-bounded turbulence. To verify the accuracy of the code, turbulent Rayleigh-B\'enard convection and plane Couette flow are simulated and the results are in good agreement with the experimental and computational data that published in previous literatures.

The strong east–west jet flows on the gas giants, Jupiter and Saturn, have persisted for hundreds of years. Yet, experimental studies cannot reach the planetary regime and similarly strong and quasi-steady jets have been reproduced in numerical models only under simplifying assumptions and limitations. Two models have been proposed: a shallow model where jets are confined to the weather layer and a deep model where the jets extend into the planetary molecular envelope. Here we show that turbulent laboratory flows naturally generate multiple, alternating jets in a rapidly rotating cylindrical container. The observed properties of gas giants’ jets are only now reproduced in a laboratory experiment emulating the deep model. Our findings demonstrate that long-lived jets can persist at high latitudes even under conditions including viscous dissipation and friction and bear relevance to the shallow versus deep models debate in the context of the ongoing Juno mission.

We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio Γ , 1/128 Γ 1, and Rayleigh number Ra, 3 × 10 4 Ra 1 × 10 11 , at a fixed Prandtl number of Pr = 4.38 by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing Γ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number Nu, exhibits the scaling Nu − 1 ∼ Ra 0.61 over three decades of Ra at Γ = 1/128, which contrasts sharply with the classical scaling Nu − 1 ∼ Ra 0.31 found at Γ = 1. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, Ra * = (29.37/Γ) 3.23 , such that the scaling transition in Nu and Re can be clearly demonstrated when plotted against Ra/Ra * .

Coherent structures are ubiquitous in turbulent flows and play a key role in transport. The most important coherent structures in thermal turbulence are plumes. Despite being the primary heat carriers, the potential of manipulating thermal plumes to transport more heat has been overlooked so far. Unlike some other forms of energy transport, such as electromagnetic or sound waves, heat flow in fluids is generally difficult to manipulate, as it is associated with the random motion of molecules and atoms. Here we report how a simple geometrical confinement can lead to the condensation of elementary plumes. The result is the formation of highly coherent system-sized plumes and the emergence of a new regime of convective thermal turbulence characterized by universal temperature profiles and significantly enhanced heat transfer. It is also found that the universality of the temperature profiles and heat transport originate from the geometrical properties of the coherent structures, i.e., the thermal plumes. Therefore, in contrast to the classical regime, boundary layers in this plume-controlled regime are being controlled, rather than controlling.

We simulate three-dimensional, horizontally periodic Rayleigh-B\'enard
convection between free-slip horizontal plates, rotating about a horizontal
axis. When both the temperature difference between the plates and the rotation
rate are sufficiently large, a strong horizontal wind is generated that is
perpendicular to both the rotation vector and the gravity vector. The wind is
turbulent, large-scale, and vertically sheared. Horizontal anisotropy,
engendered here by rotation, appears necessary for such wind generation. Most
of the kinetic energy of the flow resides in the wind, and the vertical
turbulent heat flux is much lower on average than when there is no wind.

We present a numerical scheme geared for high performance computation of
wall-bounded turbulent flows. The number of all-to-all communications is
decreased to only six instances by using a two-dimensional (pencil) domain
decomposition and utilizing the favourable scaling of the CFL time-step
constraint as compared to the diffusive time-step constraint. As the CFL
condition is more restrictive at high driving, implicit time integration of the
viscous terms in the wall-parallel directions is no longer required. This
avoids the communication of non-local information to a process for the
computation of implicit derivatives in these directions. We explain in detail
the numerical scheme used for the integration of the equations, and the
underlying parallelization. The code is shown to have very good strong and weak
scaling to at least 64K cores.

We report on direct numerical simulations of two-dimensional, horizontally
periodic Rayleigh-B\'enard convection, focusing on its ability to drive
large-scale horizontal flow that is vertically sheared. For the Prandtl numbers
($Pr$) between 1 and 10 simulated here, this large-scale shear can be induced
by raising the Rayleigh number ($Ra$) sufficiently, and we explore the
resulting convection for $Ra$ up to $10^{10}$. When present in our simulations,
the sheared mean flow accounts for a large fraction of the total kinetic
energy, and this fraction tends towards unity as $Ra\to\infty$. The shear helps
disperse convective structures, and it reduces vertical heat flux; in parameter
regimes where one state with large-scale shear and one without are both stable,
the Nusselt number of the state with shear is smaller and grows more slowly
with $Ra$. When the large-scale shear is present with $Pr\lesssim2$, the
convection undergoes strong global oscillations on long timescales, and heat
transport occurs in bursts. Nusselt numbers, time-averaged over these bursts,
vary non-monotonically with $Ra$ for $Pr=1$. When the shear is present with
$Pr\gtrsim3$, the flow does not burst, and convective heat transport is
sustained at all times. Nusselt numbers then grow roughly as powers of $Ra$,
but the growth rates are slower than any previously reported for
Rayleigh-B\'enard convection without large-scale shear. We find the Nusselt
numbers grow proportionally to $Ra^{0.077}$ when $Pr=3$ and to $Ra^{0.19}$ when
$Pr=10$. Analogies with tokamak plasmas are described.

The ubiquity of turbulent flows in nature and technology makes it of utmost importance to fundamentally understand turbulence. Kolmogorov's 1941 paradigm suggests that for strongly turbulent flows with many degrees of freedom and large fluctuations, there would only be one turbulent state as the large fluctuations would explore the entire higher dimensional phase space. Here we report the first conclusive evidence of multiple turbulent states for large Reynolds number, Re(10(6)) (Taylor number Ta(10(12))) Taylor-Couette flow in the regime of ultimate turbulence, by probing the phase space spanned by the rotation rates of the inner and outer cylinder. The manifestation of multiple turbulent states is exemplified by providing combined global torque- and local-velocity measurements. This result verifies the notion that bifurcations can occur in high-dimensional flows (that is, very large Re) and questions Kolmogorov's paradigm.

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a β plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system.
Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability.
Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

We report an experimental study of structures and dynamics of the large-scale mean flow in Rayleigh–Bénard convection cells with aspect ratio (Γ)1, 1/2, and 1/3. It is found that both a single circulating roll flow structure and one with two vertically stacked counter-rotating rolls exist in the three aspect ratio cells. The average percentages of time that the large-scale mean flow spends in the single-roll mode (SRM) and the double-roll mode (DRM) are 87.1% and 0.8% for Γ = 1, 69.5% and 7.9% for Γ = 1/2, and 26.7% and 34.1% for Γ = 1/3. Several routes of transitions among the different flow modes are identified. In addition, different structures for the DRM are found and their relative weights are determined. We also show direct evidence that the SRM is more efficient for heat transfer than the DRM. Although the difference is very small, it shows how changes in internal flow state can manifest in the global transport properties of the system. It is also found that the time interval between successive flow mode transitions has an exponential distribution, suggesting a Poisson process for the underlying dynamics. The duration of the flow mode transition is found to be log-normally distributed.

We report an experimental and numerical study of the effect of spatial confinement in turbulent thermal convection. It is found that when the width of the convection cell is narrowed, the heat-transfer efficiency increases significantly despite the fact that the overall flow is slowed down by the increased drag force from the sidewalls. Detailed experimental and numerical studies show that this enhancement is brought about by the changes in the dynamics and morphology of the thermal plumes in the boundary layers and in the large-scale flow structures in the bulk. It is found that the confined geometry produces more coherent and energetic hot and cold plume clusters that go up and down in random locations, resulting in more uniform and thinner thermal boundary layers. The study demonstrates how changes in turbulent bulk flow can influence the boundary layer dynamics and shows that the prevalent mode of heat transfer existing in larger aspect ratio convection cells, in which hot and cold thermal plumes are carried by the large-scale circulation along opposite sides of the sidewall, is not the most efficient way for heat transport.

The extraction of dynamically relevant structures from time-resolved flow data has commonly be restricted to numerically generated flow fields. Equivalent structures, however, could not be obtained from experimental measurements, since the commonly used mathematical techniques required the explicit or implicit availability of an underlying model equation. A numerical scheme based on a Krylov subspace method for the extraction of dynamic modes directly from flow fields --- without the need to resort to a model equation --- will be introduced. This technique can be applied equally to numerically generated or experimental data and thus provides a means to decompose time-resolved measurements into dynamically dominant structures. The treatment of subdomains, spatially evolving flows, PIV data and simple flow visualizations will be demonstrated; a connection to the proper orthogonal decomposition (POD) technique, which is a byproduct of the dynamic mode decomposition, will be pointed out.

The progress in our understanding of several aspects of turbulent
Rayleigh-Benard convection is reviewed. The focus is on the question of how the
Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the
Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic
boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the
dynamics of the large-scale convection-roll are addressed as well. The review
ends with a list of challenges for future research on the turbulent
Rayleigh-Benard system.

A finite-difference scheme for the direct simulation of the incompressible time-dependent three-dimensional Navier-Stokes equations in cylindrical coordinates is presented. The equations in primitive variables (vr,vθ,vzandp) are solved by a fractional-step method together with an approximate-factorization technique. Cylindrical coordinates are singular at the axis; the introduction of the radial fluxqr=r·vron a staggered grid simplifies the treatment of the region atr= 0. The method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory, experiments, and other numerical results. The formation of a tripolar vortex, where the highest vorticity is atr= 0, is also considered. Finally to emphasize the accurate treatment near the axis, the motion of a Lamb dipole crossing the origin is simulated.

The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Bénard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

Non-Oberbeck–Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh–Bénard turbulence are numerically analysed. The working fluid is water. We focus on the temperature profiles, the centre temperature, the Nusselt number and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 108) for fixed temperature difference Δ between top and bottom plates and as functions of Δ (‘non-Oberbeck–Boussinesqness’) for fixed Ra with Δ up to 60K. All results are consistent with the available experimental NOB data for the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB (the label OB meaning that the Oberbeck–Boussinesq conditions are valid). For the temperature profiles we find – due to plume emission from the boundary layers – increasing deviations from the extended Prandtl–Blasius boundary layer theory presented in Ahlers et al. (J. Fluid Mech., vol. 569, 2006, p. 409) with increasing Ra, while the centre temperature itself is surprisingly well predicted by that theory. For given non-Oberbeck–Boussinesqness Δ, both the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB only weakly depend on Ra in the Ra range considered here.Beyond Ra ≈ 106 the flow consists of a large diagonal centre convection roll and two smaller rolls in the upper and lower corners, respectively (‘corner flows’). Also in the NOB case the centre convection roll is still characterized by only one velocity scale. In contrast, the top and bottom corner flows are then of different strengths, the bottom one being a factor 1.3 faster (for Δ = 40K) than the top one, due to the lower viscosity in the hotter bottom boundary layer. Under NOB conditions the enhanced lower corner flow as well as the enhanced centre roll lead to an enhancement of the volume averaged energy based Reynolds number of about 4% to 5% for Δ = 60K. Moreover, we find , with β the thermal expansion coefficient and Tm the arithmetic mean temperature between top and bottom plate temperatures. This corresponds to the ratio of the free fall velocities at the respective temperatures. By artificially switching off the temperature dependence of β in the numerics, the NOB modifications of ReE is less than 1% even at Δ = 60K, revealing the temperature dependence of the thermal expansion coefficient as the main origin of the NOB effects on the global Reynolds number in water.

We report an experimental study on the onset of the large-scale coherent mean flow in Rayleigh–Bénard turbulent convection. Shadowgraph and particle image velocimetry techniques are used to visualize the motion of thermal plumes and measure the velocity of the plumes and of the ‘background’ flow field, as the fluid motion evolves from quiescent to steady state. The experiment reveals the dynamical origin of the initial horizontal motion required by the large-scale flow: the fluid entrainment caused by the plume's vertical motion generates vortices surrounding the plume itself. These vortices in turn generate the initial horizontal motion of the flow field. Two types of interactions have been identified: (i) direct plume–vortex interaction; and (ii) plume–plume interaction via vortices. These interactions and the interaction and merging of the vortices from neighbouring plumes lead to groupings and/or merging of plumes, which in turn generate vortices of even larger scale. As a result of these interactions, the convective flow evolves into a coherent rotatory motion consisting of mainly the plumes themselves and spanning the whole convection box. This study clearly demonstrates that it is the thermal plumes that initiate the horizontal large-scale flow across the top and bottom conducting plates.

A large-scale circulation velocity, often called the ‘wind’, has been observed in
turbulent convection in the Rayleigh–Bénard apparatus, which is a closed box with a
heated bottom wall. The wind survives even when the dynamical parameter, namely
the Rayleigh number, is very large. Over a wide range of time scales greater than
its characteristic turnover time, the wind velocity exhibits occasional and irregular
reversals without a change in magnitude. We study this feature experimentally in an
apparatus of aspect ratio unity, in which the highest attainable Rayleigh number is
about 1016. A possible physical explanation is attempted.

This paper reports measurements of Reynolds numbers Rep corresponding to the turnover time of thermal excitations ('plumes') and Re? corresponding to the twisting-oscillation period of the large-scale circulation (LSC) of turbulent Rayleigh?B?nard convection over the Rayleigh-number range and Prandtl-number range for cylindrical samples of aspect ratio ? = 1. For both periods, and hence both Reynolds numbers, were the same and scaled as Re~R?eff with . Here both the ?-?and R-dependences were quantitatively consistent with the Grossmann?Lohse (GL) prediction. For R>R* the results could be represented by Rep = 0.138???0.82R0.493 for the plume turnover time and Re? = 0.17???0.81R0.480 for the twisting oscillation, both of which differ significantly from the GL prediction as well as from each other. A relatively sharp transition at R* to the large-R regime and the separation of the two Reynolds numbers from each other suggest a qualitative and sudden change that renders the measured quantities inapplicable to the GL prediction.
Combining Rep and previously reported measurements of the Nusselt number yielded the kinetic energy-dissipation as a function of Rep. For these results were in excellent agreement with the corresponding GL prediction, and both approached closely to the (Re)?-dependence that is expected at large Re where the bulk contribution to u dominates. For R>R* the data were consistent with . This differs from the expected large-Re behavior and suggests that Rep no longer is the Reynolds number relevant to?u.

Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Bénard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Bénard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

The Sun is our nearest star; it is also the most important star that determines life on Earth. A large variety of phenomena observed on the Sun’s surface, with potential impact on Earth, is thought to arise from turbulent convection in Sun’s interior, this being the dominant mode of heat transport within the outer envelope at r≳0.715R⊙. However, convection in the Sun differs in most of its aspects from convection processes known on Earth, certainly those under controlled laboratory conditions, thus seriously challenging existing physical models of convective turbulence and boundary conditions in the Sun. Solar convection is a multiscale-multiphysics phenomenon including the transport of mass, momentum, and heat in the presence of rotation, dynamo action, radiation fluxes, and partial changes in chemical composition. Standard variables of state such as pressure, mass density, and temperature vary over several orders of magnitude within the convection region, thus introducing immense stratification. Although the Sun has been explored intensely, observational evidence on the structure and intensity of turbulent convection processes remains indirect and essentially limited to observations of the granular convection patterns at the surface and helioseismologic data that probe the propagation of sound waves in the interior. In this Colloquium characteristic scales and dimensionless parameters are discussed, particularly from the perspective of laboratory convection, a research field that has progressed significantly in the last few decades. The estimates and calculations of solar conditions given here are based mostly on the standard solar model S of Christensen-Dalsgaard et al., which is a mean field model of solar convection. Light is shed on existing results to gain a deeper understanding of dynamical aspects of solar convection.

When fluid stratification is induced by the vertical gradients of two scalars with different diffusivities, double-diffusive convection (DDC) may occur and play a crucial role in mixing. Such a process exists in many natural and engineering environments. Especially in the ocean, DDC is omnipresent since the seawater density is affected by temperature and salinity. The most intriguing phenomenon caused by DDC is the thermohaline staircase, i.e., a stack of alternating well-mixed convection layers and sharp interfaces with very large gradients in both temperature and salinity. Here we investigate DDC and thermohaline staircases in the salt finger regime, which happens when warm saltier water lies above cold fresher water and is commonly observed in the (sub)tropic regions. By conducting direct numerical simulations over a large range of parameters, we reveal that multiple equilibrium states exist in fingering DDC and staircases even for the same control parameters. Different states can be established from different initial scalar distributions or different evolution histories of the flow parameters. Hysteresis appears during the transition from a staircase to a single salt finger interface. For the same local density ratio, salt finger interfaces in the single-layer state generate very different fluxes compared to those within staircases. However, the salinity flux for all salt finger interfaces follows the same dependence on the salinity Rayleigh number of the layer and can be described by an effective power law scaling. Our findings have direct applications to oceanic thermohaline staircases.

On non-Oberbeck–Boussinesq effects in Rayleigh–Bénard convection of air for large temperature differences - Volume 889 - Zhen-Hua Wan, Qi Wang, Ben Wang, Shu-Ning Xia, Quan Zhou, De-Jun Sun

In the past, dual states were reported in turbulent plane Couette flow with spanwise rotation at Rew=1300 and Ro=0.2 based on direct numerical simulations at a computational box 10πh×2h×4πh, where the flow would evolve to a state with three or two pairs of roll cells if the simulation started with the initial flow field ui3p or ui2p. In the present work, the influence of an initial flow field on the dual states is investigated. Nine new simulations with an initial flow field constructed by ui=αui2p+(1−α)ui3p (α=0.1,0.2,...,0.9) are carried out, and it turns out that the flow will evolve to a state with three pairs of roll cells if α≤0.6, while the flow will have only two pairs of roll cells when α≥0.7, which documents the robustness of two states. The turbulent statistics is also revisited, and it is found that the turbulent kinetic energy and related dissipation are larger in the state with more roll cells. The terms in the transport equations of the Reynolds stresses are generally of the same shape at two different states, but the state with more roll cells has a larger value. The energy transfer between the secondary and the residual fields shows that the secondary flows are more energetic at the state with more roll cells while the residual field is more energetic in the other state. Furthermore, a local inverse energy cascade is observed in the near wall region at the latter state with fewer roll cells where the energy is transferred from the residual field to the secondary flow field. Our results support the conjecture that the large-scale secondary flows play a very important role in the dual states of spanwise rotating plane Couette flows.

The hysteresis behavior of spanwise rotating plane Couette flow is investigated by direct numerical simulations, where the simulations are carried out at a computational box 8πh×2h×4πh (here h is the half-channel height) along two opposite directions in rotation number (Ro=2Ωh/Uw, with Ω the constant angular velocity in the spanwise direction and Uw half of the wall velocity difference) space. When rotation speed increases, structures with two pairs of roll cells appear in 0.02≤Ro≤0.3. However, structures with three pairs of roll cells are observed for 0.03≤Ro≤0.3 when Ro decreases from 0.5. Turbulent statistics on the two branches, such as the friction Reynolds number, turbulent kinetic energy, and kinetic energy from the secondary flows, exhibit similar hysteresis behaviors during 0.03≤Ro≤0.3. Intuitive discussions are also made to explain the hysteresis behavior.

The bifurcations of penetrative Rayleigh-Bénard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4°C, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θm. The relationship between the normalized critical Rayleigh number (Rac(θm)/Rac(0)) and θm is formulated, which is in good agreement with the stability results within a large range of θm. The aspect ratio has a minor effect on Rac(θm)/Rac(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

Rough surfaces have been widely used as an efficient way to enhance the heat-transfer efficiency in turbulent thermal convection. In this paper, however, we show that roughness does not always mean a heat-transfer enhancement, but in some cases it can also reduce the overall heat transport through the system. To reveal this, we carry out numerical investigations of turbulent Rayleigh–Bénard convection over rough conducting plates. Our study includes two-dimensional (2D) simulations over the Rayleigh number range $10^{7}\leqslant Ra\leqslant 10^{11}$ and three-dimensional (3D) simulations at $Ra=10^{8}$ . The Prandtl number is fixed to $Pr=0.7$ for both the 2D and the 3D cases. At a fixed Rayleigh number $Ra$ , reduction of the Nusselt number $Nu$ is observed for small roughness height $h$ , whereas heat-transport enhancement occurs for large $h$ . The crossover between the two regimes yields a critical roughness height $h_{c}$ , which is found to decrease with increasing $Ra$ as $h_{c}\sim Ra^{-0.6}$ . Through dimensional analysis, we provide a physical explanation for this dependence. The physical reason for the $Nu$ reduction is that the hot/cold fluid is trapped and accumulated inside the cavity regions between the rough elements, leading to a much thicker thermal boundary layer and thus impeding the overall heat flux through the system.

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. In this paper we present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through non-local spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta-plane and show that it is accurate even for a small number of large-scale modes. As this approximation is formally linear in the small zonal scales it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.

We investigate the existence of multiple turbulent states in highly turbulent
Taylor-Couette flow in the range of $\mathrm{Ta}=10^{11}$ to $9\cdot10^{12}$,
by measuring the global torques and the local velocities while probing the
phase space spanned by the rotation rates of the inner and outer cylinder. The
multiple states are found to be very robust and are expected to persist beyond
$\mathrm{Ta}=10^{13}$. The rotation ratio is the parameter that most strongly
controls the transitions between the flow states; the transitional values only
weakly depend on the Taylor number. However, complex paths in the phase space
are necessary to unlock the full region of multiple states. Lastly, by mapping
the flow structures for various rotation ratios in a Taylor-Couette setup with
an equal radius ratio but a larger aspect ratio than before, multiple states
were again observed. Here, they are characterized by even richer roll structure
phenomena, including, for the first time observed in highly turbulent TC flow,
an antisymmetrical roll state.

In this paper, the dynamic mode decomposition (DMD) is employed to study the database of a swirling jet simulated by large eddy simulation (LES), and the results are compared with its non-swirling counterpart in detail, in purpose of understanding swirl-induced changes of flow dynamics at transitional stage. The two jets have identical Mach number ( ) and Reynolds number ( ), and eigenmode forcing is applied at the inflow. For both jets, the neutrally/least stable dynamic modes are found illustrated by DMD-spectrum. The prevalent temporal frequencies from a global perspective are obtained by DMD, reasonably agree with that of LES. The dynamic modes capture the physical processes like vortex roll-up and merging accurately, including spatial structures and corresponding locations. It is found that the dominant mode is varied, more pertinent frequency contents are found, nonlinear growth rates are enhanced and nonlinear interactions are strengthened with the addition of swirl. Moreover, proper orthogonal decomposition (POD) presents some similar spatial structures to that of DMD, but differences between DMD and POD are also found in analyzing such transitional flows.

§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .

We report on a numerical study of the aspect-ratio dependency of Rayleigh-Bénard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers Ra for this study variate between 105 and 109, while the Prandtl number is Pr = 0.786. The main focus of the study concerns the dependency of the Nusselt number Nu and the Reynolds number Re on Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in Nu of up to 55% and in Re of up to 97% occur, which decrease for increasing Ra. In particular for small Γ sudden drops in the Ra-scaling of Nu and Re appear for Ra ≈ 106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields into two-dimensional modes. For Ra slightly above the onset of unsteady motion (i.e., Ra ≈ 106) for all considered Γ ⩽ 1/3 a four-roll structure is present, which corresponds to thermal plumes moving vertically through the domain's center. For Ra ≥ 107, also for small Γ, a single-roll structure is dominant, in agreement with two-dimensional simulations and experiments at larger Ra and Pr.

Stars can be fascinating settings in which to study intricate couplings among convection, rotation, magnetism, and shear, usually under distinctly nonlinear conditions that yield vigorous turbulence. The emerging flux and the rotation rates of stars can vary widely, yet there are common elements that must contribute to building and maintaining the vibrantly evolving magnetic activity they exhibit. Some of these elements, such as the rotational shear and meridional flows established by the coupling of convection with rotation, can now be studied in detail within our nearest star using helioseismology. Major three-dimensional numerical simulations help refine our intuitions about such interior dynamics, aided by rapid advances in supercomputing that are improving the fidelity of the modeling. These developments, combined with intense thrusts at new high resolution and continuous observations of solar magnetism and solar oscillations, herald a promising era for exploring such astrophysical fluid dynamics.

The role of stable shear flow in suppressing turbulence and turbulent transport in plasmas and neutral fluids is reviewed. Localized stable flow shear produces transport barriers whose extensive and highly successful utilization in fusion devices has made them the primary experimental technique for reducing and even eliminating the rapid turbulent losses of heat and particles that characterize fusion-grade plasmas. These transport barriers occur in different plasma regions with disparate physical properties and in a range of confining configurations, indicating a physical process of unusual universality. Flow shear suppresses turbulence by speeding up turbulent decorrelation. This is a robust feature of advection whenever the straining rate of stable mean flow shear exceeds the nonlinear decorrelation rate. Shear straining lowers correlation lengths in the direction of shear and reduces turbulent amplitudes. It also disrupts other processes that feed into or result from turbulence, including the linear instability of important collective modes, the transport-producing correlations between advecting fluid and advectants, and large-scale spatially connected avalanchelike transport events. In plasmas, regions of stable flow shear can be externally driven, but most frequently are created spontaneously in critical transitions between different plasma states. Shear suppression occurs in hydrodynamics and represents an extension of rapid-distortion theory to a long-time-scale nonlinear regime in two-dimensional stable shear flow. Examples from hydrodynamics include the emergence of coherent vortices in decaying two-dimensional Navier-Stokes turbulence and the reduction of turbulent transport in the stratosphere. (c) 2000 The American Physical Society.