Direct Interaction Approximation Research Articles

Turbulence and Rossby Wave Dynamics with Realizable Eddy Damped Markovian Anisotropic Closure

The theoretical basis for the Eddy Damped Markovian Anisotropic Closure (EDMAC) is formulated for two-dimensional anisotropic turbulence interacting with Rossby waves in the presence of advection by a large-scale mean flow. The EDMAC is as computationally efficient as the Eddy Damped Quasi Normal Markovian (EDQNM) closure, but, in contrast, is realizable in the presence of transient waves. The EDMAC is arrived at through systematic simplification of a generalization of the non-Markovian Direct Interaction Approximation (DIA) closure that has its origin in renormalized perturbation theory. Markovian Anisotropic Closures (MACs) are obtained from the DIA by using three variants of the Fluctuation Dissipation Theorem (FDT) with the information in the time history integrals instead carried by Markovian differential equations for two relaxation functions. One of the MACs is simplified to the EDMAC with analytical relaxation functions and high numerical efficiency, like the EDQNM. Sufficient conditions for the EDMAC to be realizable in the presence of Rossby waves are determined. Examples of the numerical integration of the EDMAC compared with the EDQNM are presented for two-dimensional isotropic and anisotropic turbulence, at moderate Reynolds numbers, possibly interacting with Rossby waves and large-scale mean flow. The generalization of the EDMAC for the statistical dynamics of other physical systems to higher dimension and higher order nonlinearity is considered.

Random coupling model of turbulence as a classical Sachdev-Ye-Kitaev model.

We point out that a classical analog of the Sachdev-Ye-Kitaev (SYK) model, a solvable model of quantum many-body chaos, was studied long ago in the turbulence literature. Motivated by the Navier-Stokes equationin the turbulent regime and the nonlinear Schrödinger equationdescribing plasma turbulence, in which there is mixing between many different modes, the random coupling model has a Gaussian-random coupling between any four of a large number N of modes. The model was solved in the 1960s, before the introduction of large-N path-integral techniques, using a method referred to as the direct interaction approximation. We use the path integral to derive the effective action for the model. The large-N saddle gives an integral equationfor the two-point function, which is very similar to the corresponding equationin the SYK model. The connection between the SYK model and the random coupling model may, on the one hand, provide new physical contexts in which to realize the SYK model and, on the other hand, suggest new models of turbulence and techniques for studying them.

Cross-helicity effect onα-type dynamo in non-equilibrium turbulence

Turbulence is typically not in equilibrium, i.e. mean quantities such as the mean energy and helicity are typically time-dependent. The effect of non-stationarity on the turbulent hydromagnetic dynamo process is studied here with the use of the two-scale direct-interaction approximation, which allows one to explicitly relate the mean turbulent Reynolds and Maxwell stresses and the mean electromotive force to the spectral characteristics of turbulence, such as the mean energy, as well as kinetic and cross-helicity. It is demonstrated that the non-equilibrium effects can enhance the dynamo process when the magnetohydrodynamic turbulence is both helical and cross-helical. This effect is based on the turbulent infinitesimal-impulse cross-response functions, which do not affect turbulent flows in equilibrium. The evolution and sources of the cross-helicity in magnetohydrodynamic turbulence are also discussed.

Jackson R. Herring and the Statistical Closure Problem of Turbulence: A Review of Renormalized Perturbation Theories

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, in terms of its elegance and its pedagogical value. We then concentrate on the assessment of these theories and take the historical route of Kraichnan’s diagnosis of the failure of DIA, followed by Edwards’s analysis of the failure of his self-consistent theory, when compared to the Kolmogorov spectrum. As all three theories are closely related, these analyses also shed light on Herring’s theory. The second-generation theories that grew out of this assessment are then discussed. First, there were the Lagrangian theories, initially stemming from the work of Kraichnan and Herring, and later the purely Eulerian local energy-transfer (LET) theory. The latter is significant because its development exposes the underlying problems with the pioneering theories in terms of the basic physics of the inertial energy transfer. In particular, later work allows us to assign a unified explanation of the incompatibility of all three pioneering theories with the Kolmororov spectrum, in that they are all Markovian approximations (in wavenumber) to the non-Markovian phenomenon of fluid turbulence. In the interests of completeness, we briefly review the formalisms of Wyld and Martin, Siggia, and Rose. More recent developments are also discussed, in order to bring the subject up to the present day.

Path integral description and direct interaction approximation for elastic plate turbulence

In this work, we apply the Martin–Siggia–Rose path integral formalism to the equations of a thin elastic plate. Using a diagrammatic technique, we obtain the direct interaction approximation (DIA) equations to describe the evolutions of the correlation function and the response function of the fields. Consistent with previous results, we show that DIA equations for elastic plates can be derived from a non-markovian stochastic process and that in the weakly nonlinear limit, the DIA equations lead to the kinetic equation of wave turbulence theory. We expect that this approach will allow a better understanding of the statistical properties of wave turbulence and that DIA equations can open new avenues for understanding the breakdown of weakly nonlinear turbulence for elastic plates.

On the Bimodality of the Wind-Wave Spectrum: Mean Square Slopes and Azimuthal Overlap Integral

Abstract We present an investigation of the azimuthal bimodality of the wind-wave spectrum for waves shorter than the dominant scale comparing numerical model solutions of developing waves from idealized experiments using WAVEWATCH III (WW3). The wave solutions were forced with the “exact” Webb–Resio–Tracy (WRT) nonlinear energy fluxes and the direct interaction approximation (DIA) with three different combinations of wind input and breaking dissipation parameterizations. The WRT gives larger azimuthal bimodal amplitudes compared to the DIA regardless of wind input/dissipation. The widely used wind input/dissipation parameterizations (i.e., ST4 and ST6) generally give narrow directional distributions with relatively small bimodal amplitudes and lobe separations compared to field measurements. These biases are significantly improved by the breaking dissipation of Romero (R2019). Moreover, the ratio of the resolved cross- to downwind mean square slope is significantly lower for ST4 and ST6 compared to R2019. The overlap integral relevant for the prediction of microseisms is several orders of magnitude smaller for ST4 and ST6 compared to R2019, which nearly agrees with a semiempirical model. Significance Statement Spectral gravity wave models generally cannot accurately predict the directional distribution which impacts their ability to predict the resolved down- and crosswind mean square slopes and the generation of microseisms. Our analysis shows that a directionally narrow spectral energy dissipation, accounting for long-wave–short-wave modulation, can significantly improve the directional distribution of the wind-wave spectrum by coupling to the nonlinear energy fluxes due to wave–wave interactions, which has important implications for improved predictions of the mean square slopes and the generation of microseisms.

Statistical Dynamics of Mean Flows Interacting with Rossby Waves, Turbulence, and Topography

Abridged statistical dynamical closures, for the interaction of two-dimensional inhomogeneous turbulent flows with topography and Rossby waves on a beta–plane, are formulated from the Quasi-diagonal Direct Interaction Approximation (QDIA) theory, at various levels of simplification. An abridged QDIA is obtained by replacing the mean field trajectory, from initial-time to current-time, in the time history integrals of the non-Markovian closure by the current-time mean field. Three variants of Markovian Inhomogeneous Closures (MICs) are formulated from the abridged QDIA by using the current-time, prior-time, and correlation fluctuation dissipation theorems. The abridged MICs have auxiliary prognostic equations for relaxation functions that approximate the information in the time history integrals of the QDIA. The abridged MICs are more efficient than the QDIA for long integrations with just two relaxation functions required. The efficacy of the closures is studied in 10-day simulations with an easterly large-scale flow impinging on a conical mountain to generate rapidly growing Rossby waves in a turbulent environment. The abridged closures closely agree with the statistics of large ensembles of direct numerical simulations for the mean and transients. An Eddy Damped Markovian Inhomogeneous Closure (EDMIC), with analytical relaxation functions, which generalizes the Eddy Dampened Quasi Normal Markovian (EDQNM) to inhomogeneous flows, is formulated and shown to be realizable under the same circumstances as the homogeneous EDQNM.

Edward E. O'Brien contributions to reactive-flow turbulence

Professor Edward Ephraim O'Brien (“Ted”) has made lasting contributions to the theory and modeling of scalar mixing and reaction in turbulent flows. With a doctoral dissertation at The Johns Hopkins University in 1960, entitled “On the Statistical Behavior of a Dilute Reactant in Isotropic Turbulence,” supervised by the legend Stanley Corrsin, and in the company of notable pioneer of turbulence, John Leask Lumley, Ted's academic training propelled him through a prolific career. In the opening article of this Special Issue, we provide a review of some of Ted's contributions. First, a summary is presented of his work on the examination of the failure of the cumulant discard approximation for the scalar mixing. This is followed by a highlight of his impacts on other spectral theories of turbulence including Kraichnan's direct interaction approximation. His contributions to more modern theoretical/computational description of reactive turbulence are discussed next, including the transported probability density function (pdf) formulation, scalar-gradient pdf transport equation, scalar interfaces, and the filtered density function. Finally, some of his research on Direct Numerical Simulation of compressible turbulence is reviewed.

Dust distribution study at the blast furnace top based on k–S –up model

The dust distribution law acting at the top of a blast furnace (BF) is of great significance for understanding gas flow distribution and mitigating the negative influence of dust particles on the accuracy and service life of detection equipment. The harsh environment inside a BF makes it difficult to describe the dust distribution. This paper addresses this problem by proposing a dust distribution k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model based on interphase (gaspowder) coupling. The proposed model is coupled with a k-Sε model (which describes gas flow movement) and a model up (which depicts dust movement). First, the kinetic energy equation and turbulent dissipation rate equation in the k-Sε model are established based on the modeling theory and single-Green-function two-scale direct interaction approximation (SGF-TSDIA) theory. Second, a dust particle movement u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model is built based up on a force analysis of the dust and Newton's laws of motion. Finally, a coupling factor that describes the interphase interaction is proposed, and the k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model, with clear physical meaning, rigorous mathematical logic, and adequate generality, is developed. Simulation results and on-site verification show that the k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model not only has high precision, but also reveals the aggregate distribution features of the dust, which are helpful in optimizing the installation position of the detection equipment and improving its accuracy and service life.

Temperature dependence of the interphase interaction energy of core/shell nanoparticles

Within the framework of the Ising model, in the direct exchange interaction approximation by the “average spin” method, the constant of the interphase exchange interaction A(T), which determines the exchange bias field, is estimated. It was shown that A(T) ~ e(−T/T0), where the characteristic temperature T0 is determined by the constant of the exchange interaction between the atoms of the core and the shell.

Direct interaction approximation for non-Markovianized stochastic models in the turbulence problem.

The purpose of this paper is to consider the application of the direct interaction approximation (DIA) [R. H. Kraichnan, J. Math. Phys. 2, 124 (1961); Phys. Fluids 8, 575 (1965)] to the non-Markovianized stochastic models in the turbulence problem. For illustration, a linear damped stochastic oscillator described by a stochastic equation is considered. This process is shown to lead to a functional equation, and construction of solutions of this equation is addressed within the framework of a continued fraction representation. The relation of the DIA solution to the perturbative solution is discussed in the Laplace transform framework, and qualitative differences between the time evolutions of the two solutions are pointed out. The perturbative solution is shown to exhibit an oscillatory behavior indefinitely, in contradiction to the fact that the linear stochastic oscillator in question is damped. The DIA solution, on the contrary, is shown to exhibit a decaying behavior in accord with the latter property. The DIA procedure is applied to the problem of wave propagation in a random medium, which is described by a stochastic differential equation, with the characteristics of the medium represented by stochastic coefficients. The results are compared with those given by the perturbative procedure.

Analysis of destruction term in transport equation for turbulent energy dissipation rate

The K–\(\varepsilon \) model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier–Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number \(\textit{Re}\) has suggested that the two dominant \(O(\textit{Re}^{1/2})\) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected \(O(\textit{Re}^{-1/2})\) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the \(-5/3\) law was responsible for the \(O(\textit{Re}^{-1/2})\) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation.

Mass and internal-energy transports in strongly compressible magnetohydrodynamic turbulence

Turbulent mass and internal-energy transports in strongly compressible magnetohydrodynamic (MHD) turbulence are investigated in the framework of the multiple-scale direct-interaction approximation, an analytical closure scheme for inhomogeneous turbulence at very high Reynolds numbers. Utilising the analytical representations for the turbulent mass and internal-energy fluxes and their transport coefficients, which are expressed in terms of the correlation and response functions, turbulence models for these fluxes are proposed. In addition to the usual gradient-diffusion transports, cross-diffusion transports mediated by the density variance and the transports along the mean magnetic field mediated by the compressional or dilatational turbulent cross-helicity (velocity–magnetic-field correlation coupled with compressive motions) are shown to arise. These compressibility effects are of fundamental importance since they provide deviations from the usual gradient-diffusion transports. Analogies of the dilatational cross-helicity effects to the magnetoacoustic waves are also argued.

Markovian inhomogeneous closures for Rossby waves and turbulence over topography

Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized $\unicode[STIX]{x1D6FD}$-plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the correlation FDT. Markov equations for the triad relaxation functions are derived that carry similar information to the time-history integrals of the non-Markovian QDIA closure but become relatively more efficient for long integrations. Far from equilibrium processes are studied, where the impact of a westerly mean flow on a conical mountain generates large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Excellent agreement between the evolved mean streamfunction and mean and transient kinetic energy spectra are found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. In all cases mean Rossby wavetrain pattern correlations between the closures and the DNS ensemble are greater than 0.9998.

Electromotive force in strongly compressible magnetohydrodynamic turbulence

Fully compressible magnetohydrodynamic (MHD) turbulence is investigated in the framework of the multiple-scale direct-interaction approximation. With the aid of the propagators (correlation and Green’s functions), fluctuating fields are solved, and turbulent correlations are estimated in highly compressible turbulence. We focus on the expression of the turbulent electromotive force (EMF). Obliqueness between the mean magnetic field and the mean-density gradient, the mean internal density gradient and the non-equilibrium mean velocity contributes to the EMF in the presence of the density variance, which is ubiquitous in turbulence in strongly variable density flows such as the shock-front region. This density-variance effect is expected to locally enhance the turbulence intensity across the shock front, leading to a fast reconnection.

Quasi-diagonal inhomogeneous closure for classical and quantum statistical dynamics

The Quasi-diagonal Direct Interaction Approximation (QDIA) closure equations are formulated for inhomogeneous classical and quantum fields interacting through dynamical equations with quadratic nonlinearity and with first or second-order time derivatives. Associated more complex inhomogeneous DIA and Self-energy closure equations are expounded as part of the derivation. The QDIA employs a bare vertex approximation and is only a few times more computationally intensive than the homogeneous DIA. Examples of applications to turbulent classical geophysical and Navier Stokes fluids, including non-Gaussian noise, to classical and quantum Klein-Gordon equations with gϕ3 Lagrangian interaction, and to coupled field-auxiliary field equations associated with λϕ4 Lagrangian interaction are presented.

An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept for Geophysical Turbulence

There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models.

How turbulence fronts induce plasma spin-up.

A calculation which describes the spin-up of toroidal plasmas by the radial propagation of turbulence fronts with broken parallel symmetry is presented. The associated flux of parallel momentum is calculated by using a two-scale direct-interaction approximation in the weak turbulence limit. We show that fluctuation momentum spreads faster than mean flow momentum. Specifically, the turbulent flux of wave momentum is stronger than the momentum pinch. The scattering of fluctuation momentum can induce edge-core coupling of toroidal flows, as observed in experiments.

A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’sPlasma Turbulenceand the resonance-broadening theory of Dupree and Weinstock

In honour of the 50th anniversary of the influential review/monograph on plasma turbulence by B. B. Kadomtsev as well as the seminal works of T. H. Dupree and J. Weinstock on resonance-broadening theory, an introductory tutorial is given about some highlights of the statistical–dynamical description of turbulent plasmas and fluids, including the ideas of nonlinear incoherent noise, coherent damping, and self-consistent dielectric response. The statistical closure problem is introduced. Incoherent noise and coherent damping are illustrated with a solvable model of passive advection. Self-consistency introduces turbulent polarization effects that are described by the dielectric function${\mathcal{D}}$. Dupree’s method of using${\mathcal{D}}$to estimate the saturation level of turbulence is described; then it is explained why a more complete theory that includes nonlinear noise is required. The general theory is best formulated in terms of Dyson equations for the covariance$C$and an infinitesimal response function$R$, which subsumes${\mathcal{D}}$. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov’s theorem to develop an$\boldsymbol{x}$-space approach to the DIA that is complementary to the original$\boldsymbol{k}$-space approach of Kraichnan. A dielectric function is defined for arbitrary quadratically nonlinear systems, including the Navier–Stokes equation, and an algorithm for determining the form of${\mathcal{D}}$in the DIA is sketched. The independent insights of Kadomtsev and Kraichnan about the problem of the DIA with random Galilean invariance are described. The mixing-length formula for drift-wave saturation is discussed in the context of closures that include nonlinear noise (shielded by${\mathcal{D}}$). The role of$R$in the calculation of the symmetry-breaking (zonostrophic) instability of homogeneous turbulence to the generation of inhomogeneous mean flows is addressed. The second-order cumulant expansion and the stochastic structural stability theory are also discussed in that context. Various historical research threads are mentioned and representative entry points to the literature are given. Some outstanding conceptual issues are enumerated.

Modal time correlation functions for homogeneous isotropic turbulence in a projection operator method

Applying a projection operator method to homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications of the flow field, we obtain a closure equation, which is called the similarity approximation (SA) equation, for the time correlation function. We compare its solutions with the time correlation function obtained from direct numerical simulation in the Eulerian specification and with that obtained from direct interaction approximation in the Lagrangian specification. Results of the comparison show that the SA equation can express time correlation for homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications.