Large Aspect Ratio Limit Research Articles

Mode Interactions in Large Aspect Ratio Convection

Mode interaction between odd and even modes in two-dimensional Boussinesq convection in a box is revisited. It is noted that in the large aspect ratio limit the structure of the amplitude equations depends on the boundary conditions applied at the sidewalls, however distant.With no-slip sidewall boundary conditions the equations approach those for an unbounded layer with periodic boundary conditions; this is not the case for free-slip boundary conditions. Thus only in the former case can the large aspect ratio system be considered a small perturbation of the unbounded system. The reasons for the different large aspect ratio limits are traced to the presence of “hidden” symmetries in the stress-free case. Homotopic continuation is used to extend these results to other types of boundary conditions.

Pattern formation in weakly damped parametric surface waves

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary–gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.

Oscillatory doubly diffusive convection in a finite container.

Oscillatory doubly diffusive convection in a large aspect ratio Hele-Shaw cell is considered. The partial differential equations are reduced via center-unstable manifold reduction to the normal form equations describing the interaction of even and odd parity standing waves near onset. These equations take the form of the equations for a Hopf bifurcation with approximate ${D}_{4}$ symmetry, verifying the conclusions of the preceding paper [A.S. Landsberg and E. Knobloch, Phys. Rev. E 53, 3579 (1996)]. In particular, the amplitude equations differ in the limit of large aspect ratios from the usual Ginzburg-Landau description in having additional nonlinear terms with $O(1)$ coefficients. Numerical simulations of the amplitude equations for experimental parameter values are presented and compared with the results of recent experiments by Predtechensky et al. [Phys. Rev. Lett. 72, 218 (1994); Phys. Fluids 6, 3923 (1994)].

A Singular Perturbation Analysis of Antipolarization Dialysis at High Aspect Ratio

Antipolarization dialysis (APD) has previously been proposed (Nitsche, L.C.Q. Appl. Math. 1994, 52, 83-102) as a concept for fractionation that combines countercurrent dialysis with longitudinal polarization of macrosolutes at the downstream end of each flow channel. The paper applies singular perturbation theory to the practically important (but numerically problematical) limit of large aspect ratio, subject only to very general assumptions about the flow field. At leading order, APD is shown to yield the same basic functional form of the membrane selectivity curve as does conventional countercurrent dialysis - but without the requirement of additional clean solvent (dialysate) for selective removal of macrosolutes. Changing the second species-selective parameter, Peclet number, shifts the curve horizontally without changing its shape. This means that the macrosolute-macrosolute selectivity of one membrane could be optimized for different regimes of conductivity simply by adjusting the flow rate. Asymptotic concentration profiles and/or selectivity curves compare favorably with previous numerical results for (1) a simplified plug-flow model (Ibid.) and (2) a more realistic flow field associated with a practical plate-spacer design (Nitsche, L.C.; Zhuge, S. Chem. Eng. Sci. 1995, 50, 2731-2746)

Excitation of Alfvén cyclotron instability by charged fusion products in tokamaks

The spectrum of ion cyclotron emission (ICE) observed in tokamak experiments shows narrow peaks at multiples of the edge cyclotron frequency of background ions. A possible mechanism of ICE based on the fast Alfvén Cyclotron Instability (ACI) resonantly excited by high energy charged products (α-particles or protons) is presented here. Two-dimensional eigenmode analysis of ACI mode structure and eigenfrequency are obtained in the large tokamak aspect ratio limit. The ACI is excited via wave-particle resonances in phase space by tapping the fast ion velocity space free energy. The instability growth rates are computed perturbatively from the perturbed fast particle distribution function, which is obtained by integrating the high frequency gyrokinetic equation along the particle orbit. Numerical examples of ACI growth rates are presented for Tokamak Fusion Test Reactor (TFTR) [World Survey of Activities in Controlled Fusion Research [Nuclear Fusion special supplement 1991] (International Atomic Energy Agency, Vienna, 1991)] supershot plasmas. The fast ion distribution function is assumed to be singular in pitch angle near the plasma edge. The results are employed to understand the ICE in Deuterium-Deuterium (DD) and Deuterium-Tritium (DT) tokamak experiments.

Towards a universal theory for natural patterns

Our goal is to find a macroscopic description of patterns that both unifies and simplifies classes of externally stressed, dissipative, pattern forming systems, such as convecting fluids, liquid crystals, wideband lasers, that are seemingly unrelated at the microscopic level. We construct an order parameter equation which provides a controlled approximation of the original microscopic field in the limit of large aspect ratios. It is built from, and is a regularization of, the Cross-Newell phase diffusion equation obtained by averaging over the local periodicity of the pattern. Unlike the latter, it is valid for all wavenumbers and can correctly capture the nucleation, shape and nontrivial properties of the far fields of disclinations, dislocations and grain boundaries. It reduces to the Cross-Newell equation away from pattern singularities and to the Newell-Whitehead-Segel equation near onset. As a consequence, it correctly determines all the long wave instability boundaries (zig-zag, Eckhaus-skew-varicose) of the Busse balloon. Far from onset, the order parameter is a real variable but its equation involves a functional corresponding to its local amplitude. The local amplitude and phase, required for the order parameter equation and the reconstruction of the approximation to the original field respectively, are extracted from the order parameter field by wavelet analysis. Numerical comparisons between solutions of the original equation and the regularized equation are carried out. We also explore a new class of singular and weak solutions of the Cross-Newell equation which take account of the energetics of defects as well as their topologies. These solutions correspond to convex and concave disclinations and their composites, including saddles, vortices, targets, dislocations and two new objects, handles and bridges. Finally, we show that phase grain boundaries, lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation.

Thermally driven tall cavity flows in porous media: the convective régime

Thermally driven flows in a two-dimensional rectangular cavity filled with a fluid-saturated porous medium are considered when the applied temperature difference is perpendicular to the gravity vector. The flow depends on two non-dimensional parameters, the Darcy-Rayleigh number, A , and the cavity aspect ratio, h (height / width). Steady motion is generated by maintaining the vertical sidewalls at different constant temperatures and the horizontal boundaries are assumed to be thermally insulating. The present study is concerned with the limit of large aspect ratio, h → ∞. and Darcy-Rayleigh numbers A of order h , such that convection is important throughout the cavity. The main properties of the flow and heat transfer are obtained by considering a vertical boundary-layer approximation to the motion in the main body of the cavity. This leads, in particular, to the prediction of a position of minimum heat transfer across the cavity, of interest in the thermal insulation of buildings and other areas of technology.

Numerical study of thermal convection in tall laterally heated cavities

Two-dimensional convective flows in tall cavities with adiabatic or conducting horizontal boundaries and driven by differential heating of the two vertical sidewalls, are studied numerically over a range of Rayleigh numbers and Prandtl numbers. By using a Dufort-Frankel-Multigrid method, numerical results for the relevant end-zone problems in the limit of large aspect ratio are obtained for a range of Rayleigh numbers from A = 500 to 70000 and for different Prandtl numbers σ. The flow patterns show that the parallel flow in the core region is destroyed at A c ≈ 7880σ , in good agreement with stability analysis. Nusselt number predictions are compared with those of previous numerical and experimental studies of the overall cavity flow.

Thermally driven tall cavity flows in porous media

Thermally driven flows in a two-dimensional rectangular cavity filled with a fluid-saturated porous medium are considered when the applied temperature difference is perpendicular to the gravity vector. The flow depends on two non-dimensional parameters, the Darcy–Rayleigh number A and the cavity aspect ratio h (height/length). Steady motion is generated by maintaining the vertical sidewalls at different constant temperatures and the present study is concerned with the limit of large aspect ratio, h –> ∞, where nonlinear convective flow occurs near each end of the cavity. This flow is analysed by asymptotic methods for small and large values of A and is computed numerically for other values of A . Predictions of the heat transfer across the cavity are obtained and are compared with the results of full numerical simulations.

Some accurate solutions of the lifting surface integral equation

AbstractThis note describes a simple numerical method for solution of the lifting surface integral equation of aerodynamics, and provides benchmark computations of up to 7 figure accuracy for flat rectangular wings of arbitrary aspect ratio. The nature of the large aspect ratio limit is also investigated numerically and asymptotically. This enables determination of the limiting behaviour near the wing tips, which is compared to the predictions of lifting line theory. Generalisations to non-rectangular wings are discussed.

Force-free magnetic field in the axisymmetric torus of arbitrary aspect ratio

The equation ∇×B=αB is solved analytically in a toroidal coordinates system for the axisymmetric torus of arbitrary aspect ratio without the approximation of large aspect ratios.The solution is derived from the Helmholtz equation and is expressed in the series that contains the associated Legendre functions. A set of fundamental solutions is presented, each of which approaches asymptotically the solution for cylindrical geometry, whose azimuthal dependence is cos lψ, in the limit of large aspect ratios. The force-free magnetic field is given in the form of the infinite series of these solutions. By using the set of fundamental solutions, the contour of the magnetic surface, eigenvalues, and F-Θ curves of the circular cross section are calculated.

Lagrangian dynamics of a charged particle in a tokamak magnetic field

The Euler–Lagrange equations for a charged particle in a tokamak magnetic field, in the limit of large aspect ratio, are obtained in terms of toroidal coordinates (r,θ,φ,ṙ,θ̇,φ̇) from the exact Lagrangian, which is expressed in terms of the toroidal and poloidal magnetic fluxes. It differs fundamentally from the standard guiding-center approach, where the local magnetic-field direction defines a spatial axis, and where there is no direct link between the velocity coordinates and the time derivatives of the spatial coordinates, so that application to the particular magnetic geometry of the tokamak is often difficult. In contrast, the tokamak magnetic field is specified at the outset in the present approach. For example, cyclotron motion in the combined poloidal and toroidal fields appears as a libration in real space, superimposed on motion composed of linear nonoscillatory and averaged quadratic oscillatory terms. The primary application that is considered concerns the plasma response to high-frequency waves, using both cold plasma and kinetic treatments. A kinetic expression for cyclotron resonance is obtained that agrees with the gyrokinetic result [see, for example, C. N. Lashmore-Davies and R. O. Dendy, Phys. Fluids B 1, 1565 (1989) and references therein], which differs from the standard expression.

The generation of spatio-temporal chaos in large aspect ratio hydrodynamics

By studying typical hydrodynamic systems in the large aspect ratio limit, the authors show that spatio-temporal chaos is a natural consequence of the availability of secondary hydrodynamic instabilities. The study of such systems beyond the onset of secondary instabilities shows that, within perturbation theory, there exist stationary, spatially biperiodic solutions. Going beyond perturbation theory, it is found that a KAM mechanism is responsible for creating stationary, spatially chaotic solutions. Second, these solutions are shown to have stable and unstable manifolds in function space. They therefore conjecture that, typically, the time evolution is attracted to one of these spatially chaotic states and is then repelled along an unstable direction. In this second stage, topological defects are created if the system is sufficiently wide and couples to two-dimensional modes. Thus, they identify spatio-temporal chaos with a mechanism of generating a random array of defects in an already spatially disordered system.

Eigenmodes for electromagnetic waves propagating in a toroidal cavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Toroidal internal kink stability in tokamaks with ultra flat q profiles

Linear stability properties of the ideal internal kink mode in toroidal geometry are calculated for equilibria in which the q(r) profile is very flat and q ≈ 1 in the core region of the plasma. Marginal stability criteria and growth rates are calculated analytically in the large aspect ratio limit and compared with numerical results from the FAR code. The theory is developed for m = n = 1 modes and for higher m (= n) modes. The temperature perturbation due to adiabatic expansion in the linear phase of the 1/1 mode is calculated and compared with experimental data, showing that the predictions from ideal MHD theory cannot account for the observed temperature increase on JET. Comparison with collisionless theory suggests that the observed temperature increase can be accounted for by compression of the trapped particles.

Kinetic equilibrium properties of relativistic non-neutral electron flow in a cylindrical diode with applied magnetic field.

The equilibrium properties of a relativistic non-neutral electron layer confined in a magnetically insulated cylindrical diode are investigated within the framework of the steady-state (\ensuremath{\partial}/\ensuremath{\partial}t=0) Vlasov-Maxwell equations. The analysis is carried out for an infinitely long cylindrical electron layer with axis of symmetry parallel to an applied magnetic field ${B}_{0}$e\ensuremath{\rightarrow}${^}_{z}$, which provides radial confinement of the electrons. The theoretical analysis is specialized to the class of self-consistent Vlasov equilibria ${f}_{b}^{0}$(x\ensuremath{\rightarrow},p\ensuremath{\rightarrow}) in which all electrons have the same canonical angular momentum (${P}_{\ensuremath{\theta}}$=${P}_{0}$=const) and the same energy (H=${\mathrm{mc}}^{2}$), i.e., ${f}_{b}^{0}$=(n${^}_{b}$${R}_{c}$/2\ensuremath{\pi}m)\ensuremath{\delta}(H-${\mathrm{mc}}^{2}$ )\ensuremath{\delta}(${P}_{\ensuremath{\theta}}$-${P}_{0}$). One of the most important features of the analysis is that the closed analytic expressions for the self-consistent electrostatic potential ${\ensuremath{\varphi}}_{0}$(r) and the \ensuremath{\theta} component of vector potential ${A}_{0}$(r) are obtained. Moreover, all essential equilibrium quantities, such as electron density profile ${n}_{b}^{0}$(r), total magnetic field ${B}_{0z}$(r), perpendicular temperature profile ${T}_{\ensuremath{\perp}b}^{0}$(r), etc., can be calculated self-consistently from these potentials. As a special case, the equilibrium properties of a planar diode are investigated in the limit of large aspect ratio, further simplifying the functional form of the electrostatic and vector potentials. Detailed equilibrium properties are investigated numerically for a cylindrical diode over a broad range of system parameters, including diode voltage ${V}_{0}$, cathode electric field, electron density n${^}_{b}$ at the cathode, diode polarity, and applied magnetic field ${B}_{0}$.

A general method for calculating the self-induced magnetic force of an axisymmetric toroidal current

A method is presented for computing the class of axisymmetric current distributions flowing in a torus whose peripheral surface is a flux surface for the magnetic field produced by the current itself. The method allows the correct calculation of the ‘self-induced’ magnetic forces arising from the interaction between these currents and their own field. The general expression for the self-induced force is given and an approximate formula is presented in the large aspect-ratio limit.

Thermoconvective instabilities in a vertical cylinder of water with maximum density effects part 2. Theory: Onset of convection

Abstract The onset of convection in a vertical cylinder of water with regard to the density anomaly near 4°C is studied using linear stability analysis. Since the maximum density is located at the top of the cylinder, making the whole height of the cell gravitationally unstable, the Boussinesq equations are solved by the Galerkin method. Also for the special case of axisymmetrical flow pattern, direct numerical procedure and analytical method have been used. Special emphasis is given to the large aspect ratio limit (height of cylinder much larger than radius) for which the combined influence of the sidewalls and the non linear density gradient gives rise to a non symmetrical axial velocity profile. The maximum of the velocity amplitude is shifted towards the bottom of the cylinder as the aspect ratio increases and consequently the critical Rayleigh number is lower than when a linear density gradient is assumed.

The shear Alfvén continuous spectrum of axisymmetric toroidal equilibria in the large aspect ratio limit

Analytic forms for the MHD continuous spectra in axisymmetric tokamak equilibria are derived using an expansion scheme in powers of the small inverse aspect ratio. Resulting formulae for the shear Alfvén continuum have been evaluated for the shear Alfvén wave heating experiment on the Tokapole II device at the University of Wisconsin-Madison. Mode coupling, induced by toroidicity or non-circularity, is shown to occur, in first order, on and near rational surfaces for particular poloidal and toroidal mode numbers. The coupling is shown to produce gaps in the spectrum, analogous to the gaps which appear in the energy spectrum of a single electron in a periodic potential lattice.

Effects of toroidal coupling on the stability of tearing modes

The time evolution of tearing modes in toroidal geometry is studied in the low-β and large aspect ratio limit. An initial value three-dimensional computer code which numerically advances the reduced set of resistive magnetohydrodynamic equations is employed. Toroidicity has, in general, a destabilizing effect on tearing modes in this limit. A generalization of Δ′ formalism can be used to study the linear regime. The results obtained in this way are in very good agreement with the results from the initial value code. The nonlinear phase of the evolution is also followed numerically. In the case of strong interaction of different helicities, a larger region of stochastic magnetic field lines results than in the cylindrical geometry case.