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We study a conceivable two-fluid system with a layer of one fluid between a solid support and a semi-infinite layer of the other fluid in which the molecular van der Waals (vdW) forces act like gravity in well-studied cases with similar spatial arrangements. The solid support consists of two layers so that vdW forces are, similar to the gravity systems (GS), finite at the solid–fluid interface. In this Part 1, we focus on the Young–Laplace (YL) equilibrium drops of the “light” liquid with a zero angle between the solid–fluid and fluid–fluid interfaces and compare them with the YL drops in the analogous GS. We point out that there is essentially just a single GS, in contrast to a continuum of vdW systems corresponding to different values of a dimensionless Hamaker constant (DHC). We find that when the latter is above a certain threshold value, the YL drop volume cannot exceed a certain terminal value, just as it is known to be the case for GS. In contrast, in vdW systems with the smaller DHC, the YL drops may have arbitrarily large volumes, which is a major difference from GS. We introduce a certain “non-classical particle” kinetic energy and a corresponding generalized phase plane. These “energy” considerations predict interfaces with vertical tangents, which are in an excellent agreement with equilibrium drop interfaces found by numerical continuation methods. For small DHC, numerical simulations on a neutral-wave-long interval confirm the lubrication-approximation evolutions from initially flat films toward the—perpetually approached—equilibrium YL drops of the same volume. We point out that the evolution for larger DHC may be approached by the boundary integral methods suitable for the Stokes flows, which we will further investigate in the future Part 2 of this work.

The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.

A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances in the two layers. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected the velocity amplitudes are the linear combinations of certain hyperbolic functions, and a quadratic dispersion equation for the increment, that is the complex growth rate, is obtained, where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number $Ma$ that measures the effects of surfactant and the Bond number $Bo$ that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. Since the dispersion equation is quadratic in the growth rate, there are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that, to the contrary, vanishes when $Ma\downarrow 0$. Regimes have been uncovered with crossings of the two dispersion curves, their reconnections at the point of crossing and separations as $Bo$ changes. Due to the availability of the explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics.

A horizontal channel flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant monolayer present at the interface, is investigated. The base Couette flow is driven by the uniform horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The (inertialess) instability is a combined result of the surfactant action characterized by the Marangoni number $Ma$ and the gravitational effect corresponding to the Bond number $Bo$ that ranges from $-\infty$ to $\infty$. The other parameters are the top-to-bottom thickness ratio $n$, which is restricted to $n\geqslant 1$ by a reference frame choice, the top-to-bottom viscosity ratio $m$ and the base shear rate $s$. The linear stability is determined by an eigenvalue problem for the normal modes, where the complex eigenvalues (determining growth rates and phase velocities) and eigenfunctions (the amplitudes of disturbances of the interface, surfactant, velocities and pressures) are found analytically by using the smallness of the wavenumber. For each wavenumber, there are two active normal modes, called the surfactant and the robust modes. The robust mode is unstable when $Bo/Ma$ falls below a certain value dependent on $m$ and $n$. The surfactant branch has instability for $m, and any $Bo$, although the range of unstable wavenumbers decreases as the stabilizing effect of gravity represented by $Bo$ increases. Thus, for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow. The correlations of vorticity-thickness phase differences with instability, present when gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. This is in marked contrast to the dynamical role of vorticity in the mechanism of the well-known Yih instability due to effects of inertia, and is contrary to some earlier literature. Unlike the semi-infinite case that we previously studied, a small-amplitude saturation of the surfactant instability is possible in the absence of gravity. For certain $(m,n)$-ranges, the interface deflection is governed by a decoupled Kuramoto–Sivashinsky equation, which provides a source term for a linear convection–diffusion equation governing the surfactant concentration. When the diffusion term is negligible, this surfactant equation has an analytic solution which is consistent with the full numerics. Just like the interface, the surfactant wave is chaotic, but the ratio of the two waves turns out to be constant.

We describe a new version of GBS, a 3D global, flux-driven plasma turbulence code to simulate the turbulent dynamics in the tokamak scrape-off layer (SOL), superseding the code presented by Ricci et al. (2012) [14]. The present work is driven by the objective of studying SOL turbulent dynamics in medium size tokamaks and beyond with a high-fidelity physics model. We emphasize an intertwining framework of improved physics models and the computational improvements that allow them. The model extensions include neutral atom physics, finite ion temperature, the addition of a closed field line region, and a non-Boussinesq treatment of the polarization drift. GBS has been completely refactored with the introduction of a 3-D Cartesian communicator and a scalable parallel multigrid solver. We report dramatically enhanced parallel scalability, with the possibility of treating electromagnetic fluctuations very efficiently. The method of manufactured solutions as a verification process has been carried out for this new code version, demonstrating the correct implementation of the physical model. (C) 2016 Published by Elsevier Inc.

We introduce the anti-symmetric representation of the extended magnetohydrodynamic (MHD) equations. In this representation, the use of the anti-symmetric flux operator ( ∇ · v + v · ∇ ) results in conservation theorems with discrete analogs. Inherently robust numerical applications are achieved with little effort, and conservation to machine precision is possible with simple numerical schemes. Starting from the two-fluid equations, we construct a single-fluid MHD model based on generalized center-of-mass variables for the mass ( ρ), momentum ( ρ v), and pressure ( p). This model is shown to possess identical conservation properties to the two-fluid system, with the only restriction being the use of a single temperature. Common approximations to the Braginskii heat fluxes and to the gyroviscous stress tensor are cast into our representation for convenience. The discrete conservation properties are verified using the classic Orszag–Tang vortex problem. In addition to the favorable mass, momentum, and energy conservation properties, the time reversibility of the simulations is demonstrated.

This paper presents two discretisation schemes for the parallel gradient operator used in scrape-off layer plasma turbulence simulations. First, a simple model describing the propagation of electrostatic shear-Alfven waves, and retaining the key elements of the parallel dynamics, is used to test the accuracy of the different schemes against analytical predictions. The most promising scheme is then tested in simulations of limited scrape-off layer turbulence with the flux-driven 3D fluid code GBS (Ricci et al., 2012): the new approach is successfully benchmarked against the original parallel gradient discretisation implemented in GBS. Finally, GBS simulations using a radially varying safety profile, which were inapplicable with the original scheme are carried out for the first time: the well-known stabilisation of resistive ballooning modes at negative magnetic shear is recovered. The main conclusion of this paper is that a simple approach to the parallel gradient, namely centred finite differences in the poloidal and toroidal direction, is able to simulate scrape-off layer turbulence provided that a higher resolution and higher convergence order are used.

We investigate the question of how plasma currents circulate and close in the scrape-off-layer (SOL) of convection-limited tokamak plasmas. A simplified two-fluid model describes how currents must evacuate charge at the sheaths due to cross-field currents that are not divergence-free. These include turbulence-driven polarization currents and poloidally asymmetric equilibrium diamagnetic currents. The theory provides an estimate for the radial profile of the floating potential, which reveals a dipolar structure like the one observed experimentally. Simulations with a fluid turbulence code provide evidence for the predicted behaviour of currents and floating potential.

A numerical and an analytical study of the triggering of resistive MHD modes in tokamak plasmas with low magnetic shear core is presented. Flat q profiles give rise to fast growing pressure driven MHD modes, such as infernal modes. It has been shown that infernal modes drive fast growing islands on neighbouring rational surfaces. Numerical simulations of such instabilities in a MAST-like configuration are performed with the initial value stability code XTOR-2F in the resistive frame. The evolution of magnetic islands are computed from XTOR-2F simulations and an analytical model is developed based on Rutherford’s theory in combination with a model of resistive infernal modes. The parameter Δ ′ is extended from the linear phase to the non-linear phase. Additionally, the destabilising contribution due to a helically perturbed bootstrap current is considered. Comparing the numerical XTOR-2F simulations to the model, we find that coupling has a strong destabilising effect on (neoclassical) tearing modes and is able to seed 2/1 magnetic islands in situations when the standard NTM theory predicts stability.

The properties of plasma turbulence in a poloidally limited scrape-off layer (SOL) are addressed, with focus on ISTTOK, a large aspect ratio tokamak with a circular cross section. Theoretical investigations based on the drift-reduced Braginskii equations are carried out through linear calculations and non-linear simulations, in two-and three-dimensional geometries. The linear instabilities driving turbulence and the mechanisms that set the amplitude of turbulence as well as the SOL width are identified. A clear asymmetry is shown to exist between the low-field and the high-field sides of the machine. While the comparison between experimental measurements and simulation results shows good agreement in the far SOL, large intermittent events in the near SOL, detected in the experiments, are not captured by the simulations., United States. Department of Energy (Grant DE-FG02-91ER54109)

We derive a set of fluid moment equations with inherent consistency and numerical stability, conceived by exploiting the anti-symmetric nature of the plasma flow operator (∇·v+v·∇). The obtained equations can be interpreted as an alternative to the traditional Eulerian and Lagrangian representations—one in which plasma flows generate infinitesimal rotations of generalized fluid moments n, nv, and p. The continuous model has a discrete analog with exact mass, momentum, and energy conservation, which is achieved by construction through vanishing integrals of the anti-symmetric flow terms. Positivity preservation is obtained through the use of the generalized moment quantities. The conservation properties of the approach are illustrated in simulations of seeded blob propagation, where we verify numerical conservation to machine accuracy.

The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.

In this work, we theoretically explore the characteristics of electroosmostic flow (EOF) in a microcavity with nonuniform surface charges. It is well known that a uniformly charged EOF does not give rise to flow separation because of its irrotational nature, as opposed to the classical problem of viscous flow past a cavity. However, if the cavity walls bear nonuniform surface charges, then the similitude between electric and flow fields breaks down, leading to the generation of vorticity in the cavity. Because this vorticity must necessarily diffuse into the exterior region that possesses a zero vorticity set by a uniform EOF, a new flow structure emerges. Assuming Stokes flow, we employ a boundary element method to explore how a nonuniform charge distribution along the cavity surface affects the flow structure. The results show that the stream can be susceptible to flow separation and exhibits a variety of flow structures, depending on the distributions of zeta potentials and the aspect ratio of the cavity. The interactions between patterned EOF vortices and Moffatt eddies are further demonstrated for deep cavities. This work not only has implications for electrokinetic flow induced by surface imperfections but also provides optimal strategies for achieving effective mixing in microgrooves.

A numerical study of pressure driven magnetohydrodynamic (MHD) instabilities in a low-shear tight aspect ratio configuration is presented. When the magnetic shear is sufficiently small over an extended region in the core, enhanced instability occurs due to the coupling to poloidal sidebands, which itself occurs due to toroidicity. Numerical simulations have been performed with the initial value code XTOR-2F both in the ideal and resistive MHD frame. Two-fluid effects (plasma diamagnetic flows) have been retained as well. The predictions of the XTOR-2F code on the amplitude of the growth rate, and on the rotation frequency of the modes, have been compared with analytic linear theory of infernal modes. Qualitative agreement has been found between numerical and analytical results, in spite of the tight aspect ratio configuration. The intermediate scaling gamma similar to S-3/8, predicted by the linear theory (Brunetti et al 2014 Plasma Phys. Control. Fusion 56 075025), is recovered by the numerical results. A study of the nonlinear evolution of the magnetic island of the tearing sideband has been performed and the results from the simulations are compared with Rutherford's theory.

© 2015 IOP Publishing Ltd. We carry out a quantitative comparison between gas puff imaging (GPI) turbulence measurements in Alcator C Mod inner wall limited discharges (Zweben et al 2009 Phys. Plasmas 18 082505) and 3D flux driven drift reduced Braginskii turbulence simulations of scrape off layer dynamics. The comparison is carried out for a series of inner wall limited discharges where the magnetic field and the density are varied. The comparison between GPI data and non linear simulations yields overall good agreement for several observables such as the Da emission levels and intermittency the radial and poloidal correlation lengths and propagation velocities and the power and frequency spectral density.

We investigate the unsteady motion of a long bubble advancing under either prescribed pressure p(b) or prescribed volume flux q(b) into a fluid-filled flexible-walled channel at zero Reynolds number, an idealized model for the reopening of a liquid-lined lung airway. The channel walls are held under longitudinal tension and are supported by external springs; the bubble moves with speed U. Provided p(b) exceeds a critical pressure p(crit) the system exhibits two types of steady motion. At low speeds, the bubble acts like a piston, slowly pushing a column of fluid ahead of itself, and U decreases with increasing p(b). At high speeds, the bubble rapidly peels the channel walls apart and U increases with increasing p(b.) Using two independent time-dependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed p(b) > p(crit), peeling motion is stable and the steady pushing solution is unstable; for p(b) > p(crit) the system ultimately exhibits unsteady pushing behaviour for which U continually diminishes with time. When q(b) is prescribed, peeling motion (with large q(b)) is again stable, but pushing motion (with small q(b)) loses stability at long times to a novel instability leading to spontaneous relaxation oscillations of increasing amplitude and period, for which the bubble switches abruptly between slow unsteady pushing and rapid quasi-steady peeling. This stick-slip motion is characterized using a third-order lumped-parameter model which in turn is reduced to a nonlinear map. Implications for the inflation of occluded lung airways are discussed.

Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette-Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number M, the viscosity ratio m, the interfacial velocity shear s, and the thickness ratio n (≥ 1). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If M > 0, s ¬= 0, m 1, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between

We consider two problems in which a semi-infinite bubble moves into a uniformly convergent two-dimensional channel under creeping-flow conditions. In the first (steady) problem, the bubble is stationary with respect to the channel walls, which move away from the channel vertex parallel to themselves, each entraining a liquid film of uniform thickness; the appropriate flux is provided by a line source at the channel vertex. In the second (unsteady) problem, the channel walls are fixed and the bubble is drawn at a constant speed towards the channel vertex by a line sink. In this case the bubble interface assumes a self-similar shape, and wedge-shaped films are deposited on the channel walls. The boundary-element method, supported by asymptotic approximations, is used to compute flows over a range of bubble speeds (measured by a capillary number Ca) and wedge angles α. In the steady problem, the deposited film thickness increases monotonically with α at low Ca, but diminishes with increasing α at sufficiently high Ca. In the unsteady problem, the film thickness was found always to increase with both α and Ca. In both cases, the dimensionless pressure drop across the bubble tip can be nonmonotonic in α. Implications of these results in modeling coating and peeling flows are discussed.

The steady motion at zero Reynolds number of a semi-infinite bubble through a fluid-filled, flexible-walled channel, a model for the reopening of a collapsed lung airway, is described using asymptotic and numerical methods. The channel walls are membranes that are supported by external springs and are held under large longitudinal tension. An asymptotic analysis is presented under the assumption that membrane slopes are uniformly small. Near the bubble tip, the flow is equivalent to that of a semi-infinite bubble in a weakly tapered channel. Key features of this two-dimensional flow are matched to long-wavelength approximations describing the remainder of the solution domain. The analysis is valid for a wide range of bubble speeds, and it takes a particularly simple form when the bubble peels apart the channel walls as it advances. Predictions of bubble pressure as a function of bubble speed are validated by comparison with existing computations, new boundary-element simulations describing bubble motion in a channel with one rigid and one deformable wall, and experiments.

Chemical and colloidal transport processes in partially saturated porous and fractured media are dependent on liquid flow along a solid surface, which may be chaotic even for small Reynolds numbers. This paper presents the derivation of a one-dimensional evolution equation describing the slow motion (small Reynolds numbers, R<

Tokamaks with weak to moderate reversed central shear in which the minimum inverse rotational transform (safety factor) qmin is in the neighbourhood of unity can trigger bifurcated magnetohydrodynamic equilibrium states, one of which is similar to a saturated ideal internal kink mode. Peaked prescribed pressure profiles reproduce the ‘snake’ structures observed in many tokamaks which has led to a novel explanation of the snake as a bifurcated equilibrium state. Snake equilibrium structures are computed in simulations of the tokamak `a configuration variable (TCV), DIII-D and mega amp spherical torus (MAST) tokamaks. The internal helical deformations only weakly modulate the plasma–vacuum interface which is more sensitive to ripple and resonant magnetic perturbations. On the other hand, the external perturbations do not alter the helical core deformation in a significant manner. The confinement of fast particles in MAST simulations deteriorate with the amplitude of the helical core distortion. These three-dimensional bifurcated solutions constitute a paradigm shift that motivates the applications of tools developed for stellarator research in tokamak physics investigations.

The narrow power decay-length ($\lambda_q$), recently found in the scrape-off layer (SOL) of inner-wall limited (IWL) discharges in tokamaks, is studied using 3D, flux-driven, global two-fluid turbulence simulations. The formation of the steep plasma profiles measured is found to arise due to radially sheared $\vec{E}\times\vec{B}$ poloidal flows. A complex interaction between sheared flows and outflowing plasma currents regulates the turbulent saturation, determining the transport levels. We quantify the effects of sheared flows, obtaining theoretical estimates in agreement with our non-linear simulations. Analytical calculations suggest that the IWL $\lambda_q$ is roughly equal to the turbulent correlation length., Comment: 5 pages, 5 figures

The International Tokamak Physics Activity Topical Group on scrape-off layer and divertor physics has amassed a database comprising hundreds of reciprocating Langmuir probe measurements of the main scrape-off layer heat-flux width lambda(q) in inner-wall limited discharges. We have carried out an analysis, based on turbulent transport theory, of the variation of lambda(q) with respect to the dimensionless plasma parameters. Restricting our analysis to circular plasmas, we find that a model based on non-linearly saturated turbulence can well reproduce the lambda(q) values found in the database.

Particle transport and deposition associated with flow over a wedge is investigated as a model for particle transport and flow at the carina of an airway bifurcation during inspiration. Using matched asymptotics, a uniformly valid solution is obtained to represent the high Reynolds number flow over a wedge that considers the viscous boundary layer near the wedge and the outer inviscid region and is then used to solve the particle transport equations. Sometimes particle impaction on the wedge is prevented due to the boundary layer. We call this boundary layer shielding (BLS). This effect can be broken down into different types: rejection, trapping and deflection that are described by what happens to the particle’s initial negative velocity normal to the wall either changing sign, reaching zero, or remaining negative in the boundary layer region. The deposition efficiency depends on the critical Stokes number but exhibits a weak dependence on Reynolds number. Deposition efficiency for Sc in the range 0 < Sc < 0.4 yields the following relationship De ≈ (1.867 Sc1.78− 0.016) sin(βπ/2) at large Reynolds numbers, where βπ is the wedge angle. For a specific deposition efficiency, Sc decreases as βπ increases. The distribution of impacted particles was also computed and revealed that particles primarily impact within one airway diameter of the carina, consistent with computational fluid dynamics approaches. This work provides a new insight that the BLS inherent to the wedge component of the structure is the dominant reason for the particle distribution. This finding is important in linking aerosol deposition to the location of airway disease as well as target sites for therapeutic deposition.

The long-term dynamics of the internal kink mode are studied using the non-linear, two-fluid magnetohydrodynamics (MHD) code XTOR-2F (Lutjens et al 2010 J. Comput. Phys. 229 8130). The dynamics of the internal kink are studied in resistive MHD plus transport, and, in addition, also including some two-fluid effects. The simulation results yield a pattern of kink cycles or stationary states with m/n = 1/1 helicity. The threshold for sustained (non-decaying) kink cycles, which characterize sawtooth oscillations, is shown to depend on the plasma pressure, the current and energy diffusion times, and the ion and electron diamagnetic drifts. In particular, the diamagnetic flow stabilization extends the cyclic kink regime into a parameter space that is adequate to carry out studies of sawtooth oscillations in ohmic tokamak discharges.

The motion of a rigid disk-shaped particle with rounded edges, which fits closely in the space between two parallel flat plates, and which is suspended in a viscous fluid subject to an imposed pressure gradient, is analysed. This problem is relevant to the squeezing of red blood cells through narrow slot-like channels which are found in certain tissues. Mammalian red cells, although highly flexible, conserve volume and surface area as they deform. Consequently, a red cell cannot pass intact through a channel which is narrower than some minimum width. In channels that are just wide enough to permit cell passage, the cell is deformed into its ‘critical’ shape: a disk with rounded edges. In this paper, the fluid mechanical aspects of such motions are considered, and the particle is assumed to be rigid with the critical shape. The channel cross-section is assumed to be rectangular. The flow of the suspending fluid is described using lubrication theory. Use of lubrication theory is justified by considering the motion of a circular cylinder between parallel plates. For disk-shaped particles, approximate solutions are obtained by applying lubrication equations throughout the flow domain. In the region beyond the particle, this is equivalent to assuming a Hele-Shaw flow. More accurate solutions are obtained by including effects of boundary layers around the particle and at the sides of the channel. Pressure distributions and particle velocities are computed as functions of geometrical parameters, and it is shown that the particle may move faster or slower than the mean velocity of the surrounding fluid, depending on the channel dimensions.

Ion temperature has been measured to be of the same order, or higher, than the electron temperature in the scrape-off layer (SOL) of tokamak machines, questioning its importance in determining the SOL turbulent dynamics. Here, we present a detailed analysis of finite ion temperature effects on the linear SOL instabilities, such as the resistive and inertial branches of drift waves and ballooning modes, and a discussion of the properties of the ion temperature gradient (ITG) instability in the SOL, identifying the gi 1⁄4 Ln=LTi threshold necessary to drive the mode unstable. The non-linear analysis of the SOL turbulent regimes by means of the gradient removal theory is performed, revealing that the ITG plays a negligible role in limited SOL discharges, since the ion temperature gradient is generally below the threshold for driving the mode unstable. It follows that the resistive ballooning mode is the prevailing turbulence regime for typical limited SOL parameters. The theoretical estimates are confirmed by non-linear flux-driven simulations of SOL plasma dynamics.

We study, for the case of the two-layer plane Couette flow, the effects of inertia on the recently found instability due to insoluble surfactants. The insoluble-surfactant instability takes place even when inertia is absent provided an interface or a free surface under a nonzero shear is laden with an insoluble surfactant. Considering a normal mode of the streamwise wave number alpha , the perturbation theory we construct is good for any alpha provided the Reynolds number is correspondingly small. Inertia is responsible for some notable effects, including the appearance of new regions of instability and stability. For long--and only for long--waves, the following growth-rate additivity property for the inertia and interfacial instabilities holds: the growth rate corresponding to some nonzero values of the Marangoni number M and the Reynolds number Re is the sum of two contributions, one corresponding to the same value of M but zero Re, and the other corresponding to the same (nonzero) Re but zero M. This violation of the additivity property is in contrast to the case of a surfactantless Couette flow where this property holds for all wave numbers. Thus these results provide a counterexample to a conjecture that this additivity property is a universal principle. Among other results, when the thinner layer is the less viscous one, there is a nonzero critical Marangoni number Mc for the onset of instability; this (long-wave) threshold Mc grows from zero with the Reynolds number. Also, varying the ratio of viscosities through certain characteristic values leads to changes in the topology of marginal-stability curves.

We develop a predictive theory applicable to the scrape off layer (SOL) of inner wall limited plasmas. Using the non linear flattening of the pressure profile as a saturation mechanism for resistive ballooning modes we are able to demonstrate and quantify the increase of the SOL width with plasma size connection length plasma ß and collisionality. Individual aspects of the theory such as saturation physics parallel dynamics and system size scaling are tested and verified using non linear 3D flux driven SOL turbulence simulations. Altogether very good agreement between theory and simulation is found. © 2014 IAEA Vienna.

The propagation of blobs structures of localized enhanced plasma pressure is studied in global three dimensional simulations of a simple magnetized torus. In particular we carry out single seeded blob simulations to explore the dependence of the blob velocity with respect to its size. It is found that the velocity scaling for two dimensional blobs is satisfied in the parameter space where polarization currents are the dominant damping mechanism. On the other hand three dimensional blobs propagate faster than their two dimensional counterparts in the parallel current damping regime. A detailed analysis of the charge and current balance reveals that in fact the difference in speed is due to an overestimation of the strength of the sheath current term in the two dimensional model compared to the self consistent three dimensional model. © 2014 Euratom.

The effect of the limiter position on the scrape-off layer (SOL) width, radial electric field and intrinsic flows is investigated via global, three-dimensional turbulence simulations in four different limiter configurations. The limiter position affects the SOL dynamics in a number of ways, for example by changing the effective connection length or by modifying the unstable modes present in the system. The simulations show that the SOL width is much smaller and less poloidally asymmetric when the plasma is limited on the low-field side than on the high-field side, which can be explained by a change in the turbulence regime between the two configurations. The radial electric field is determined by the combined effect of the sheath physics and the electron adiabaticity condition, and its poloidal structure depends on the limiter position, as it can be fairly well explained through an analytical model. Intrinsic parallel flows established in the SOL, typically leading to co-current toroidal rotation with a magnitude that strongly depends on the limiter position, can also be fairly well reproduced analytically for each limiter configuration.

The proper boundary conditions at the magnetic presheath entrance for plasma fluid turbulence models based on the drift approximation are derived, focusing on a weakly collisional plasma sheath with Ti≪Te and a magnetic field oblique to a totally absorbing wall. First, the location of the magnetic presheath entrance is rigorously derived. Then boundary conditions at the magnetic presheath entrance are analytically deduced for v||i, v||e, n, ϕ, Te, and for the vorticity ω = ∇⊥2ϕ. The effects of E × B and diamagnetic drifts on the boundary conditions are also investigated. Kinetic simulations are performed that confirm the analytical results. Finally, the new set of boundary conditions is implemented in a three-dimensional global fluid code for the simulation of plasma turbulence and, as an example, the results of a tokamak scrape-off layer simulation are discussed. The framework presented can be generalized to obtain boundary conditions at the magnetic presheath entrance in more complex scenarios.

The cycling dynamics of the internal kink mode, which drives sawtooth oscillations in tokamak plasmas, is studied using the three dimensional, non-linear magnetohydrodynamic (MHD) code XTOR-2F [H. Lütjens and J.-F. Luciani, J. Comput. Phys. 229, 8130 (2010)]. It is found that sawtooth cycling, which is characterized by quiescent ramps and fast crashes in the experiment, can be recovered in two-fluid MHD provided that a criterion of diamagnetic stabilization is fulfilled. The simulation results indicate that diamagnetic effects alone may be sufficient to drive sawteeth with complete magnetic reconnection in high temperature Ohmic plasmas.

The lungs consist of a network of bifurcating airways that are lined with a thin liquid film. This film is a bilayer consisting of a mucus layer on top of a periciliary fluid layer. Mucus is a non-Newtonian fluid possessing viscoelastic characteristics. Surface tension induces flows within the layer, which may cause the lung's airways to close due to liquid plug formation if the liquid film is sufficiently thick. The stability of the liquid layer is also influenced by the viscoelastic nature of the liquid, which is modeled using the Oldroyd-B constitutive equation or as a Jeffreys fluid. To examine the role of mucus alone, a single layer of a viscoelastic fluid is considered. A system of nonlinear evolution equations is derived using lubrication theory for the film thickness and the film flow rate. A uniform film is initially perturbed and a normal mode analysis is carried out that shows that the growth rate g for a viscoelastic layer is larger than for a Newtonian fluid with the same viscosity. Closure occurs if the minimum core radius, R(min)(t), reaches zero within one breath. Solutions of the nonlinear evolution equations reveal that R(min) normally decreases to zero faster with increasing relaxation time parameter, the Weissenberg number We. For small values of the dimensionless film thickness parameter epsilon, the closure time, t(c), increases slightly with We, while for moderate values of epsilon, ranging from 14% to 18% of the tube radius, t(c) decreases rapidly with We provided the solvent viscosity is sufficiently small. Viscoelasticity was found to have little effect for epsilon0.18, indicating the strong influence of surface tension. The film thickness parameter epsilon and the Weissenberg number We also have a significant effect on the maximum shear stress on tube wall, max(tau(w)), and thus, potentially, an impact on cell damage. Max(tau(w)) increases with epsilon for fixed We, and it decreases with increasing We for small We provided the solvent viscosity parameter is sufficiently small. For large epsilon approximately 0.2, there is no significant difference between the Newtonian flow case and the large We cases.

Electron thermal transport in DIII-D H-mode tokamak plasmas [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] is investigated by comparing predictive simulation results for the evolution of electron temperature profiles with experimental data. The comparison includes the entire profile from the magnetic axis to the bottom of the pedestal. In the simulations, carried out using the automated system for transport analysis (ASTRA) integrated modeling code, different combinations of electron thermal transport models are considered. The combinations include models for electron temperature gradient (ETG) anomalous transport and trapped electron mode (TEM) anomalous transport, as well as a model for paleoclassical transport [J. D. Callen, Nucl. Fusion 45, 1120 (2005)]. It is found that the electromagnetic limit of the Horton ETG model [W. Horton et al., Phys. Fluids 31, 2971 (1988)] provides an important contribution near the magnetic axis, which is a region where the ETG mode in the GLF23 model [R. E. Waltz et al., Phy...

Predictive simulations of ITER [R. Aymar et al., Plasma Phys. Control. Fusion 44, 519 (2002)], discharges are carried out for the 15MA high confinement mode (H-mode) scenario using PTRANSP, the predictive version of the TRANSP code. The thermal and toroidal momentum transport equations are evolved using turbulent and neoclassical transport models. A predictive model is used to compute the temperature and width of the H-mode pedestal. The ITER simulations are carried out for neutral beam injection (NBI) heated plasmas, for ion cyclotron resonant frequency (ICRF) heated plasmas, and for plasmas heated with a mix of NBI and ICRF. It is shown that neutral beam injection drives toroidal rotation that improves the confinement and fusion power production in ITER. The scaling of fusion power with respect to the input power and to the pedestal temperature is studied. It is observed that, in simulations carried out using the momentum transport diffusivity computed using the GLF23 model [R. Waltz et al., Phys. Plasm...

A comparison is made between two versions of the Weiland model for computing anomalous transport driven by drift modes such as the ion temperature gradient (ITG) and trapped electron mode (TEM) in tokamak plasmas. Both are quasilinear fluid models that include physical effects resulting from finite β, magnetic shear, electron-ion collisions, impurities, and fast ions. An outline of the derivation is presented for the newer Weiland19 model, which includes a more accurate description of the effects of finite β, low and negative magnetic shear, plasma elongation, varying correlation lengths, particle pinch, and momentum transport. It is shown that the two models produce nearly the same ion thermal diffusivity as a function of normalized temperature gradient in a circular plasma with moderate magnetic shear, low β, and moderately low density gradient. The models differ significantly at low magnetic shear and in elongated plasmas with high β. In addition, the two models differ significantly in the behavior of ...

The ISLAND module in the National Transport Code Collaboration (NTCC) Module Library computes the saturated widths of multiple magnetic islands due to neoclassical tearing modes by implementing a quasi-linear model that includes the effects of arbitrary aspect ratio. cross-sectional shape, and plasma beta. This paper presents a significantly improved version of the module available at the NTCC Library. The model was updated to include the distortion of islands caused by the radial variation of the perturbed magnetic field near the mode rational surface. The computer algorithm implemented by the module was redesigned to increase its speed, reliability and accuracy. The new module has been successfully interfaced with the BALDUR integrated modeling code, where it is used to compute the saturated widths of multiple islands in tokamak discharges.

A revised version of the ISLAND module [C. N. Nguyen et al., Phys. Plasmas 11, 3604 (2004)] is used in the BALDUR code [C. E. Singer et al., Comput. Phys. Commun. 49, 275 (1988)] to carry out integrated modeling simulations of DIII-D [J. Luxon, Nucl. Fusion 42, 614 (2002)], Joint European Torus (JET) [P. H. Rebut et al., Nucl. Fusion 25, 1011 (1985)], and ITER [R. Aymar et al., Plasma Phys. Control. Fusion 44, 519 (2002)] tokamak discharges in order to investigate the adverse effects of multiple saturated magnetic islands driven by neoclassical tearing modes (NTMs). Simulations are carried out with a predictive model for the temperature and density pedestal at the edge of the high confinement mode (H-mode) plasma and with core transport described using the Multi-Mode model. The ISLAND module, which is used to compute magnetic island widths, includes the effects of an arbitrary aspect ratio and plasma cross sectional shape, the effect of the neoclassical bootstrap current, and the effect of the distortion ...

Sound generated underwater by wind waves displays two sensitivities to wind speed depending on whether the waves are breaking. A historical review is presented of several experimental studies over the years, as well as new results obtained at a relatively quiet, deep equatorial Pacific site. The duration of the experiment, the multiple high frequency bands as well as the desirable acoustic attributes of the site are useful in establishing the wind sensitivity, and its variability. The results are compared to a recent theoretical model of sound generation by oscillating bubbles in breaking waves.

The drift-reduced Braginskii model describing turbulence in the tokamak scrape-off layer is written for a general magnetic configuration with a limiter. The equilibrium is then specified for a circular concentric magnetic geometry retaining aspect ratio effects. Simulations are then carried out with the help of the global, flux-driven fluid three-dimensional code GBS [Ricci et al., Plasma Phys. Controlled Fusion 54, 124047 (2012)]. Linearly, both simulations and simplified analytical models reveal a stabilization of ballooning modes. Nonlinearly, flux-driven nonlinear simulations give a pressure characteristic length whose trends are correctly captured by the gradient removal theory [Ricci and Rogers, Phys. Plasmas 20, 010702 (2013)], that assumes the profile flattening from the linear modes as the saturation mechanism. More specifically, the linear stabilization of ballooning modes is reflected by a 15% increase in the steady-state pressure gradient obtained from GBS nonlinear simulations when going from an infinite to a realistic aspect ratio. (C) 2014 AIP Publishing LLC.

Motivated by the wide range of physical parameters characterizing the scrape-off layer (SOL) of existing tokamaks, the regimes of low-frequency linear instabilities in the SOL are identified by numerical and analytical calculations based on the linear, drift-reduced Braginskii equations, with cold ions. The focus is put on ballooning modes and drift wave instabilities, i.e., their resistive, inertial, and ideal branches. A systematic study of each instability is performed, and the parameter space region where they dominate is identified. It is found that the drift waves dominate at high R/L-n, while the ballooning modes at low R/L-n; the relative influence of resistive and inertial effects is discussed. Electromagnetic effects suppress the drift waves and, when the threshold for ideal stability is overcome, the ideal ballooning mode develops. Our analysis is a first stage tool for the understanding of turbulence in the tokamak SOL, necessary to interpret the results of non-linear simulations. [http://dx.doi.org/10.1063/1.4758809]

The mechanism regulating the equilibrium electrostatic potential in the scrape-off layer (SOL) of magnetic confinement devices is elucidated. Based on a generalized Ohm's law and the boundary conditions at the magnetic presheath entrance, an analytical expression for the equilibrium electrostatic potential is derived. Results imply that the relative importance of the plasma dynamics at the sheath and far away from the wall in setting the value of the electrostatic potential depends on the density and temperature drops that are established between the two regions. Global, three-dimensional fluid simulations of tokamak SOL turbulence in a simple configuration are performed, confirming the validity of our predictions. The results presented here are general and can be applied to other open-field-line configurations, including linear devices and simple magnetized toroidal devices.

A theory based scaling for the characteristic length of a circular limited tokamak scrape off layer (SOL) is obtained by considering the balance between parallel losses and non linearly saturated resistive ballooning mode turbulence driving anomalous perpendicular transport. The SOL size increases with plasma size resistivity and safety factor q. The scaling is verified against flux driven non linear turbulence simulations which reveal good agreement within a wide range of dimensionless parameters including parameters closely matching the TCV tokamak. An initial comparison of the theory against experimental data from several tokamaks also yields good agreement. © 2013 IAEA Vienna.

We identify the turbulent regimes in the tokamak scrape-off layer as a function of plasma resistivity, electron to ion mass ratio, safety factor, and magnetic shear. We find that two main instabilities drive turbulence in the SOL: Drift Waves, with their resistive and inertial branches, and the Ballooning instabilities, with their resistive, ideal, and inertial branches. First, we identify how the linear growth rate, and the properties of these instabilities depend on the system physical parameters. Then, according to the gradient removal saturation mechanism, we use the fact that the transport is dominated by the mode with the highest growth rate divided by the poloidal wave number and the non-linear saturated pressure scale length is proportional to this ratio. This allows us to evaluate the non-linear saturated pressure gradient and the poloidal wavenumber of the dominating mode as a function of the resistivity, the mass ratio, the safety factor, and the magnetic shear . We can therefore define the non-linear instability phase space, locating the regions in which each instability is influencing transport the most. In order to validate our calculations, we run non-linear SOL simulations and we compare the plasma pressure scale length and the mode characteristics to the prediction provided by the phase space description. The non-linear simulations are performed using the GBS code, which solves the drift-reduced Braginskii equations evolving self-consistently equilibrium and fluctuations in three-dimensional geometry. The non-linear simulations are interpreted in light of our non-linear analysis and confirm its validity.

A drift-reduced Braginskii fluid model is used to carry out a linear and non-linear study of ideal ballooning modes in the tokamak scrape-off layer. First, it is shown that the scrape-off layer finite connection length and boundary conditions modify the ideal stability limit with respect to the closed flux-surface result. Then, in a two-fluid description, it is found that magnetic induction effects can destabilize long wavelength resistive ballooning modes below marginal ideal stability. Non-linear simulations confirm a gradual transition from small scale quasi-electrostatic interchange turbulence to longer wavelength modes as the plasma beta is increased. The transition to global ideal ballooning modes occurs, roughly, at the linearly obtained stability threshold. The transport levels and the pressure gradient as a function of plasma beta obtained in non-linear simulations can be predicted using the non-linear flattening of the pressure profile from the linear modes as a turbulent saturation mechanism.

The turbulence observed in the scrape-off-layer of a tokamak is often characterized by intermittent events of bursty nature, a feature which raises concerns about the prediction of heat loads on the physical boundaries of the device. It appears thus necessary to delve into the statistical properties of turbulent physical fields such as density, electrostatic potential and temperature, focusing on the mathematical expression of tails of the probability distribution functions. The method followed here is to generate statistical information from time-traces of the plasma density stemming from Braginskii-type fluid simulations, and check this against a first-principles theoretical model. The analysis of the numerical simulations indicates that the probability distribution function of the intermittent process contains strong exponential tails, as predicted by the analytical theory.

Based on the drift-reduced Braginskii equations, the Global Braginskii Solver, GBS, is able to model the scrape-off layer (SOL) plasma turbulence in terms of the interplay between the plasma outflow from the tokamak core, the turbulent transport, and the losses at the vessel. Model equations, the GBS numerical algorithm, and GBS simulation results are described. GBS has been first developed to model turbulence in basic plasma physics devices, such as linear and simple magnetized toroidal devices, which contain some of the main elements of SOL turbulence in a simplified setting. In this paper we summarize the findings obtained from the simulation carried out in these configurations and we report the first simulations of SOL turbulence. We also discuss the validation project that has been carried out together with the GBS development.

Bridging the gap between plasma physics and other scientific domains, in particular, the computational fluid dynamics community, a general, rigorous, and simple-to-apply methodology is presented for both the verification of the correct implementation of the model equations (code verification) and numerical error quantification (solution verification). The proposed code verification procedure consists in using the method of manufactured solutions and executing an order-of-accuracy test, assessing the rate of convergence of the numerical solution to the manufactured one. For the solution verification, the numerical error is quantified by applying the Richardson extrapolation, which provides an approximation of the analytical solution, and by using the grid convergence index to estimate the numerical uncertainty affecting the simulation results. The methodology is applied to verify the correct implementation of the drift-reduced Braginskii equations into the GBS code, and to estimate the numerical error affecting the GBS solutions. The GBS code is successfully verified, and an estimate of the numerical error affecting the simulation results is provided.