John C. Bowman

The realizable Markovian closure. I. General theory, with application to three‐wave dynamics

A type of eddy‐damped quasinormal Markovian (EDQNM) closure is shown to be potentially nonrealizable in the presence of linear wave phenomena. This statistical closure results from the application of a fluctuation–dissipation (FD) ansatz to the direct‐interaction approximation (DIA); unlike in phenomenological formulations of the EDQNM, both the frequency and the damping rate are renormalized. A violation of realizability can have serious physical consequences, including the prediction of negative or even divergent energies. A new statistical approximation, the realizable Markovian closure (RMC), is proposed as a remedy. An underlying Langevin equation that makes no assumption of white‐noise statistics is exhibited. Even in the wave‐free case the RMC, which is based on a nonstationary version of the FD ansatz, provides a better representation of the true dynamics than does the EDQNM closure. The closure solutions are compared numerically against the exact ensemble dynamics of three interacting waves.

On the dual cascade in two-dimensional turbulence

We study the dual cascade scenario for two-dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range [kmin, kmax] (with kmin > 0) such that the respective energy and enstrophy injection rates ǫ and η satisfy k 2 ǫ ≤ η ≤ k 2ǫ. The classical Kraichnan–Leith–Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scaleselectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For two-dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the acrosschannel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy β > 5 and α + β ≥ 8, where −α (−β) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analyzed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.

Statistical theory of resistive drift-wave turbulence and transport

Resistive drift-wave turbulence in a slab geometry is studied by statistical closure methods and direct numerical simulations. The two-field Hasegawa–Wakatani (HW) fluid model, which evolves the electrostatic potential and plasma density self-consistently, is a paradigm for understanding the generic nonlinear behavior of multiple-field plasma turbulence. A gyrokinetic derivation of the HW model is sketched. The recently developed Realizable Markovian Closure (RMC) is applied to the HW model; spectral properties, nonlinear energy transfers, and turbulent transport calculations are discussed. The closure results are also compared to direct numerical simulation results; excellent agreement is found. The transport scaling with the adiabaticity parameter, which measures the strength of the parallel electron resistivity, is analytically derived and understood through weak- and strong-turbulence analyses. No evidence is found to support previous suggestions that coherent structures cause a large depression of sa...

On inertial range scaling laws

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorovs k −5/3 scaling is derived for the energy inertial range. A related modification is found to Kraichnans logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

Exactly conservative integrators

Traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants. In this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly. We illustrate the method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator-prey model, and the Kepler problem. The ideas are discussed in the context of symplectic (phase--space-conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.

The realizable Markovian closure and realizable test-field model. II. Application to anisotropic drift-wave dynamics

The test-field model is shown to be potentially nonrealizable in the presence of linear waves such as those frequently encountered in models of plasma and geophysical turbulence. A new statistical closure, the realizable test-field model (RTFM), is proposed as a remedy. Both the damping rate and frequency are renormalized to account for nonlinear damping and frequency shifts. Like the realizable Markovian closure (RMC), the RTFM is based on a modified fluctuation-dissipation ansatz. Numerical solutions of the RTFM, RMC, and direct-interaction approximation for the Hasegawa–Mima equation are presented; rough agreement with direct numerical solution is found. The number of retained Fourier modes is dramatically reduced with an anisotropic generalization of a recently developed wave-number partitioning scheme.

Resistive drift-wave plasma turbulence and the realizable Markovian closure

Abstract The realizable Markovian closure (RMC) developed by Bowman et al. [Phys. Fluids B 5 (1993) 3558] is employed to study the Hasegawa-Wakatani model, a paradigm for resistive drift-wave turbulence. Turbulent transport, spectral properties, and energy-transfer directions are discussed. For second-order statistics, the closure predictions are in excellent agreement with direct numerical simulations. A marked depression of the hydrodynamic flux from its quasilinear value is well predicted by the RMC and is explained without reference to “coherent structures”.

Efficient Dealiased Convolutions without Padding

Algorithms are developed for calculating dealiased linear convolution sums without the expense of conventional zero-padding or phase-shift techniques. For one-dimensional in-place convolutions, the memory requirements are identical with the zero-padding technique, with the important distinction that the additional work memory need not be contiguous with the input data. This decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher-dimensional in-place convolutions. The technique also allows one to dealias the higher-order convolutions that arise from Fourier transforming cubic and higher powers. Implicitly dealiased convolutions can be built on top of state-of-the-art fast Fourier transform libraries: vectorized multidimensional implementations for the complex and centered Hermitian (pseudospectral) cases have been implemented in the open-source software FFTW++.

Modeling Sediment Deposition Patterns Arising From Suddenly Released Fixed‐Volume Turbulent Suspensions

Models presented in several recent papers [1–3] dealing with particle transport by, and deposition from, bottom gravity currents produced by the sudden release of dilute, well-mixed fixed-volume suspensions have been relatively successful in duplicating the experimentally observed long-time, distal, areal density of the deposit on a rigid horizontal bottom. These models, however, fail in their ability to capture the experimentally observed proximal pattern of the areal density with its pronounced dip in the region initially occupied by the well-mixed suspension and its equally pronounced local maximum at roughly the one-third point of the total reach of the deposit. The central feature of the models employed in [1–3] is that the particles are always assumed to be vertically well-mixed by fluid turbulence and to settle out through the bottom viscous sublayer with the Stokes settling velocity for a fluid at rest with no re-entrainment of particles from the floor of the tank. Because this process is assumed from the outset in the models of [1–3], the numerical simulations for a fixed-volume release will not take into account the actual experimental conditions that prevail at the time of release of a well-mixed fixed-volume suspension. That is, owing to the vigorous stirring that produces the well-mixed suspension, the release volume will initially possess greater turbulent energy than does an unstirred release volume, which may only acquire turbulent energy as a result of its motion after release through various instability mechanisms. The eddy motion in the imposed fluid turbulence reduces the particle settling rates from the values that would be observed in an unstirred release volume possessing zero initial turbulent energy. We here develop a model for particle bearing gravity flows initiated by the sudden release of a fixed-volume suspension that takes into account the initial turbulent energy of mixing in the release volume by means of a modified settling velocity that, over a time scale characteristic of turbulent energy decay, approaches the full Stokes settling velocity. Thereafter, in the flow regime, we assume that the turbulence persists and, in accord with current understanding concerning the mechanics of dense underflows, that this turbulence is most intense in the wall region at the bottom of the flow and relatively coarse and on the verge of collapse (see [22]) at the top of the flow where the density contrast is compositionally maintained. We capture this behavior by specifying a “shape function” that is based upon experimental observations and provides for vertical structure in the volume fraction of particles present in the flow. The assumption of vertically well-mixed particle suspensions employed in [1–5] corresponds to a constant shape function equal to unity. Combining these two refinements concerning the settling velocity and vertical structure of the volume fraction of particles into the conservation law for particles and coupling this with the fluid equations for a two-layer system, we find that our results for areal density of deposits from sudden releases of fixed-volume suspensions are in excellent qualitative agreement with the experimentally determined areal densities of deposit as reported in [1, 3, 6]. In particular, our model does what none of the other models do in that it captures and explains the proximal depression in the areal density of deposit.

Advances in the analytical theory of plasma turbulence and transport: Realizable Markovian statistical closures

Explaining anomalous plasma transport in magnetic confinement devices requires a deeper understanding of the underlying turbulent processes than presently exists. In this work, Markovian closures are built by imposing the constraints of realizability, conservation of quadratic invariants, and covariance to arbitrary linear transformations. One such closure is solved numerically. The results compare favorably to the data available from numerical simulations.