Fluid Drift Research Articles

Tokamak edge turbulence: background theory and computation

The basic scales of motion and computational requirements for low frequency fluid drift turbulence are summarized in tutorial fashion, with emphasis on the tokamak edge region. Parameters are given by experimental observations, but the computations are otherwise done from first principles. Edge turbulence is fundamentally electromagnetic and nonlinear, not treatable by standard linear or secondary instability analysis. Energetic character is determined by diagnosis of the terms in the energy theorem within the fully developed saturated phase. The spectra of the fluctuations and transport always extend to below the ion gyroradius scale. Direct coupling of pressure fluctuations and E-cross-B eddies through the parallel current is always active. Edge turbulence derives its character from steep gradients, with a parallel/perp scale ratio larger than 100, rather than from collisional effects. Collisionality is neither absent nor strongly dominant for electrons, but very weak for ions. Fluctuations in the axisymmetric component, including the Pfirsch–Schlüter currents, are dynamically integrated into the turbulence. Time scales are one to two orders of magnitude shorter than the ion collision time, hence significant delays occur in the response of heat fluxes and viscosity to temperature gradients and flows. Hence the need for a trans-collisional gyrofluid model to treat cases with comparable ion and electron temperature. Two orders of magnitude in spatial scales and three in time scales are typically involved.

The operational phase-space of the edge plasma and its sensitivity to magnetic topology in Alcator C-Mod

The range of ‘edge plasma states’ (i.e., the combination of densities, temperatures and gradients in the scrape-off layer near the separatrix) accessible to a tokamak are highly restricted. Experiments in Alcator C-Mod suggest that this restriction is a consequence of electromagnetic fluid drift turbulence (EMFDT), which sets the cross-field transport levels; plasma states are constrained to lie in a narrow region of EMFDT ‘phase-space’ defined by poloidal beta gradient (αMHD) and normalized plasma collisionality. Recent experiments have investigated this behavior over an increased parameter range in lower- and upper-null (LSN/USN) discharges. L-mode edge pressure gradients are found to exhibit clear sensitivity to magnetic topology, favoring higher αMHD for LSN compared to USN. These variations may be caused, in part, by the different edge plasma flow patterns in LSN versus USN. Such flows arise from a ballooning-like transport drive and can produce a topology-dependent modulation in the toroidal plasma rotation.

Advances in measurement and modeling of the high-confinement-mode pedestal on the Alcator C-Mod tokamak

Edge transport barrier (ETB) studies on the Alcator C-Mod tokamak [Phys. Plasmas 1, 1511 (1994)] investigate pedestal scalings and the radial transport of plasma and neutrals. Pedestal profiles show trends with plasma operational parameters such as total current IP. A ballooning-like IP2 dependence is seen in the pressure gradient, despite calculated stability to ideal ballooning modes. A similar scaling is seen in the near scrape-off layer for both low-confinement (L-mode) and H-mode discharges, possibly due to electromagnetic fluid drift turbulence setting transport near the separatrix. Neutral density diagnosis allows an examination of D0 fueling in H-modes, yielding profiles of effective particle diffusivity in the ETB, which vary as IP is changed. Edge neutral transport is studied using a one-dimensional kinetic treatment. In both experiment and modeling, the C-Mod density pedestal exhibits a weakly increasing pedestal density and a nearly invariant density pedestal width as the D0 source rate increases. Identical modeling performed on pedestal profiles typical of DIII-D [Nucl. Fusion 42, 614 (2002)] reveal differences in pedestal scalings qualitatively similar to experimental results.

Note on the induced Lagrangian drift and added-mass of a vortex

Darwin (1953) introduced a simple heuristic that relates the Lagrangian fluid drift induced by a solid body propagating in irrotational flow to its virtual- or added-mass. The force required to accelerate the solid body must also overcome this added-mass. An extension of Darwin's (1953) method to the case of vortices propagating in a real fluid is described here. Experiments are conducted to demonstrate the existence of an added-mass effect during uni-directional vortex motion, which is analogous to the effect of solid bodies in potential flow. The definition of the vortex added-mass coefficient is modified from the solid body case to account for entrainment of ambient fluid by the vortex. This modified coefficient for propagating vortices is shown to be equal in magnitude to the classical coefficient for a solid body of equivalent boundary geometry. An implication of these results is that the vortex added-mass concept can be used as a surrogate for the velocity potential, in order to facilitate calculations of the pressure contribution to forces required to set fluid into unsteady vortical motion. Application of these results to unsteady wake analyses and fluid–structure interactions such as vortex-induced vibrations is suggested.

Electromagnetic fluid drift turbulence in static ergodic magnetic fields

Numerical simulations of three-dimensional nonlinear electromagnetic fluid drift turbulence in a tokamak plasma with externally applied stochastic magnetic-field perturbations are presented. The contributions to the radial particle transport due to nonlinearities arising from E×B advection and magnetic flutter are investigated for perturbation fields of varying strengths in the cases of low and high collisionalities. The perturbation strength is varied to study the physics for Chirikov parameters above 1. In all the cases considered a significant increase of E×B transport is found. A static contribution in the density and velocity perturbations contributes significantly to the total radial E×B transport. For low collisionality, the external perturbation leads to enhanced density and velocity fluctuations over a broad range in the toroidal wave-number spectrum, resulting in an enhanced turbulent flux. For high collisionality, the density fluctuations stay roughly the same and the velocity fluctuations are increased in an intermediate range of the toroidal wave number spectrum, separated from the maximum of the density fluctuations, thus leaving the turbulent flux almost unchanged.

Evidence for electromagnetic fluid drift turbulence controlling the edge plasma state in the Alcator C-Mod tokamak

Plasma profiles across the separatrix and scrape-off layer (SOL) in Alcator C-Mod are examined for a range of plasma densities, currents and magnetic fields in Ohmic L-mode discharges and for a subset of conditions in Ohmic H-mode discharges. In all plasmas, electron pressure gradient scale lengths () exhibit a minimum value just outside the separatrix (i.e. in the near SOL), forming the base of a weak (strong) pedestal in L-mode (H-mode) plasmas. Over a wide range of conditions in Ohmic L-mode discharges, at this location are found to track with a monotonic function of electron collision frequency, when this quantity is normalized according to the framework of electromagnetic fluid drift turbulence (EMFDT) theory. Moreover, at fixed values of normalized collisionality parameter (characterized as the ‘diamagnetic parameter’, αd), electron pressure gradients in the near SOL increase with plasma current squared, holding the MHD ballooning parameter, αMHD, unchanged. Thus, the state of the near SOL is restricted to a narrow region within this two-parameter phase-space. An implication is that cross-field heat and particle transport are strong functions of these parameters. Indeed, as αd is decreased below ∼0.3, cross-field heat convection increases sharply and competes with parallel heat conduction along open field lines, making high plasma density regions of αMHD–αd space energetically inaccessible. These observations are consistent with the idea that the operational space of the edge plasma, including boundaries associated with the tokamak density limit, is controlled by EMFDT.

Nonlocal nonlinear electrostatic gyrofluid equations

Building on Lagrangian field theory methods of fluid dynamics, we construct a set of equations for an electrostatic gyrofluid model which can treat arbitrarily nonlinear situations. Noether’s theorem is used to find the exact energy theorem satisfied by the equations. The exchange of energy between the E×B fluid drift and thermal/kinetic parts of the dynamics is recovered rigorously. Diamagnetic cancellations are inserted manually.

Two-dimensional fast reconnection in a fluid drift model

A simple scenario for two-dimensional reconnection is reexamined in the context of a fluid drift model which treats the pressure and parallel ion velocity independently. The rapid nonlinear establishment of an arbitrarily thin current sheet is unaffected for expected values of the plasma β of a few percent. Eventually the sound wave dynamics should enter to alter the results, but remains insignificant for reasonable β values (gas/magnetic pressure up to 0.1).

On Lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows

As intermediate quantities available from various existing numerical computations, the fluid Lagrangian time scales are of primary importance in the development of probability density function models for turbulent flows. Similarly, the time scales of the fluid seen by discrete particles in two-phase flows are essential for the development of dispersion models based on stochastic differential equations. Such time scales are obviously depending not only on the particle properties but also on the fluid Lagrangian and Eulerian time scales. A model is proposed here to estimate the directional dependence of the fluid Lagrangian time scales and drift coefficients in one-directional equilibrium turbulent shear flows, based on a local homogeneity assumption in the frame of the generalized Langevin model. Through comparison with available direct and large eddy simulation predictions in channel flows and in a homogeneous shear flow, the model is shown to lead to significant improvements in the streamwise and spanwise directions, where the existing empirical laws for the Lagrangian time scales are far from being satisfactory. We examine the way this model can be used to build a suitable stochastic process for the fluid seen by inertial particles in such basic turbulent shear flows.

Experimental investigation of coherent structures in a low-energy electron beam

A sharp transition to a space-charge dominated regime is induced in a low-energy electron beam produced in a Malmberg–Penning trap by increasing the emission current of the source. The transition is characterized by the appearance of a region, around the axis of the beam, not accessible to beam electrons, and by the fast development of coherent structures in the remaining electron plasma, due to the sharp increase of local vorticity. The results are interpreted in the framework of a cold fluid drift–Poisson model, and using a three-dimensional particle-in-cell simulation code.

Computation of electromagnetic turbulence and anomalous transport mechanisms in tokamak plasmas

The basic scales of motion and computational requirements for low frequency fluid drift turbulence are summarized in tutorial fashion. Basic signatures for each of the competing eigenmode types are given, some of which are accessible experimentally. The emphasis is on edge turbulence since it is less intuitive. The difference in physical mechanism between linear instabilities and fully developed turbulence at the same parameters is shown. The difference in physical character from more familiar MHD models is also addressed. Experimentally, one needs to measure the fluctuating potential as well as the transported quantities in order to distinguish among physical mechanisms.

R‐Modes in the Ocean of a Magnetic Neutron Star

We study the dynamics of r-modes in the ocean of a magnetic neutron star. We model the star's ocean with a spherical rotating thin shell and assume that the magnetic field symmetry axis is not aligned to the shell's spin axis. In the magnetohydrodynamic approximation, we calculate the frequency of l = m r-modes in the shell of an incompressible fluid. Different r-modes with l and l ± 2 are coupled by the inclined magnetic field. Kinematical secular effects for the motion of a fluid element in the shell undergoing the l = m = 2 r-mode are studied. The magnetic-corrected drift velocity of a given fluid element undergoing the l = m r-mode oscillations is obtained. The magnetic field increases the magnitude of the fluid drift produced by the r-mode oscillations. The drift velocity is strongly modulated by the inclined magnetic field. We show that the magnetic field is distorted by the high-l magnetic r-modes more strongly than by the low-l modes. Furthermore, because of the shear produced by the r-mode drift velocity, the high-l modes in the ocean fluid will damp faster than the low-l ones.

Role of collisionless heat flux in magnetospheric convection

A fluid model of magnetospheric convection appropriate for the inner magnetosphere, including the effects of heat flux in collisionless plasma, is presented. The plasma is assumed to be isotropic, with the flow speed much less than the thermal speed, and parallel electric fields and loss cone effects are neglected; the effects of slow time variations of the magnetic field are included. The classical transport coefficients are considered and, except for the collisionless heat flux, shown to be negligible in plasma in the inner magnetosphere. Beginning with three‐dimensional two‐fluid equations, we derive two‐dimensional equations for transport of mass and energy mapped to the magnetospheric equator. The equation of mass transport, derived from the mass conservation equations, is equivalent to those obtained in previous studies [e.g., Peymirat and Fontaine, 1994]. The equation of energy transport contains the effects of collisionless heat conduction that represents the transport of energy in the rest frame of the species and has hitherto been neglected in magnetospheric fluid and MHD models. The energy transport equation is shown to be equivalent to that of Peymirat and Fontaine [1994] if the heat flux is neglected. The two equations are coupled first‐order partial differential equations; they can be uncoupled by taking linear combinations. The uncoupled equations show that the effect of the collisionless heat flux is to spread information across the fluid drift paths in a manner quite different from that of fluid flow neglecting heat flux.

Warm-ion drift Alfvén turbulence and the L-H transition

Computations of fluid drift turbulence treating ions and electrons on equal footing, including both temperatures, are conducted in a model toroidal geometry. The resulting `ion mixing mode' turbulence bears features of both electron drift-Alfvén and ion temperature gradient turbulence, and nonlinear sensitivity to the relative strengths of the density and temperature gradients provides a possible route to the bifurcation needed for the L-H transition.

The excitation of lower hybrid waves driven by the ∇B effect in the current sheet

We have investigated the excitation of electrostatic and electromagnetic lower hybrid waves in the magnetic field reversal region of the plasma sheet using a two‐fluid approach. We consider a two‐dimensional tail configuration that includes the Bx and Bz magnetic field components and examine the nonlocal properties of lower hybrid modes near the weak field region, where the field Bx(z) reverses sign. We have shown that an electrostatic lower hybrid mode is excited by the magnetic field gradient while electromagnetic modes are driven unstable by the combined effects of the magnetic field gradient and the electron fluid drift across the Bx field. We have shown that the electrostatic and electromagnetic lower hybrid modes grow in a few milliseconds and a few tens of milliseconds, respectively, the former for wavelengths shorter than the current sheet width (that is, for kyλ ≥ 1, where λ is the current sheet width) and the latter for long wavelengths (kyλ ≪ 1).

An analytical study of free convective boundary layer flow in porous media: The effect of anisotropic diffusivity

The effect of an anisotropic thermal diffusivity tensor on the free convective boundary-layer flow in porous media is studied. Convection is induced by a generally inclined, uniformly heated surface embedded in a fluid-saturated medium. A third-order boundary-layer theory is presented in order to obtain accurate information on the effect of anisotropy on the rate of heat transfer into the porous medium. It is shown that the thickness of the resulting leading order boundary-layer flow depends on the precise nature of the anisotropy. On the other hand, the anisotropic diffusivity does not induce a fluid drift in the spanwise direction, a result which is different from that obtained in our earlier study of the effects of an anisotropic permeability. It is found that the second order temperature field does not contribute to the overall rate of heat transfer. Finally, we show that the third-order correction to the leading-order rate of heat transfer is given in terms of an explicit formula.

Displacement of inviscid fluid by a sphere moving away from a wall

We develop a theoretical analysis of the displacement of inviscid fluid particles and material surfaces caused by the unsteady flow around a solid body that is moving away from a wall. The body starts at position hs from the wall, and the material surface is initially parallel to the wall and at distance hL from it. A volume of fluid Df+ is displaced away from the wall and a volume Df- towards the wall. Df+ and Df- are found to be sensitive to the ratio hL/hs. The results of our specific calculations for a sphere can be extended in general to other shapes of bodies.When the sphere moves perpendicular to the wall the fluid displacement and drift volume Df+ are calculated numerically by computing the flow around the sphere. These numerical results are compared with analytical expressions calculated by approximating the flow around the sphere as a dipole moving away from the wall. The two methods agree well because displacement is an integrated effect of the fluid flow and the largest contribution to displacement is produced when the sphere is more than two radii away from the wall, i.e. when the dipole approximation adequately describes the flow. Analytic expressions for fluid displacement are used to calculate Df+ when the sphere moves at an acute angle α away from the wall.In general the presence of the wall reduces the volume displaced forward and this effect is still significant when the sphere starts 100 radii from the wall. A sphere travelling perpendicular to the wall, α = 0, displaces forward a volume Df+(0) = 4πa3hL/33/2hS when the marked surface starts downstream, or behind the sphere, and displaces a volume Df+(0) ∼ 2πa3/3 forward when it is marked upstream or in front of the body. A sphere travelling at an acute angle away from the wall displaces a volume Df+(α) ∼ Df+(0) cos α forward when the surface starts downstream of the sphere. When the marked surface is initially upstream of the sphere, there are two separate regions displaced forward and a simple cosine dependence on α is not found.These results can all be generalized to calculate material surfaces when the sphere moves at variable speed, displacements no longer being expressed in terms of time, but in relation to the distance travelled by the sphere.

The effect of anisotropic permeability on free convective boundary layer flow in porous media

The effect of an anisotropic permeability on thermal boundary layer flow in porous media is studied. The convective flow is induced by a vertical, uniformly heated surface embedded in a fluid-saturated medium. A leading-order boundary layer theory is presented. It is shown that the thickness of the resulting boundary layer flow is different from that obtained in an isotropic porous medium. In general, an anisotropic permeability induces a fluid drift in the spanwise direction, the strength of which depends on the precise nature of the anisotropy. Conditions are found which determine whether or not the boundary layer flow is three-dimensional.

Behavior of Drift Wood and Its Dam up Process

A numerical simulation model has been developed for computing the behavior of drift wood and its dam up process due to jamming between and/or in front of the houses when drift wood debouches into horizontally two-dimensional flow field. The calculation is accomplished by an interacting combination of Eulerian fluid and Lagragian drift wood equations, in which a turbulent diffusivity and rotational angular velocity of the drift wood are stochastically given. The positions, rotational angles in a group of drift wood, and time change of the number of dammed up drift wood in the experiments are well explained by the calculation.

Measurement of coherent drift-wave ion-fluid velocity field when ion dynamics are stochastic

In order to test the validity of the fluid drift approximation when ion dynamics are stochastic, the time-dependent ion-fluid velocity field in a coherent, drift-Alfven wave has been measured in Caltech’s Encore tokamak using plasma planar laser induced fluorescence. Many measured parameters of the wave are well described by the drift approximation above a threshold for stochastic ion dynamics, including the flow pattern in the single measured component of the ion-fluid velocity field. Surprisingly, the theory fails at a fundamental level by predicting flow speeds 10 times larger than those measured (‖umax‖≊10^5 cm/s).