Negative Eddy Diffusivity Research Articles

Countergradient turbulent transport in a plume with a crossflow

AbstractDirect numerical simulation of a turbulent forced buoyant plume in a crossflow is performed at a source Reynolds number $${\text{ Re }}_0=1000$$ Re 0 = 1000 , Richardson number $$\mathrm{{Ri}}_0=1$$ Ri 0 = 1 , Prandtl number $$\mathrm{{Pr}}=1$$ Pr = 1 and source-to-crossflow velocity ratio $$R_0=1$$ R 0 = 1 . The instantaneous and temporally averaged flow fields are assessed in detail, providing an overview of the flow dynamics. The velocity, temperature and pressure fields are used together with enstrophy fields to describe qualitatively the evolution of the plume as it is swept downstream by the crossflow, and the mechanisms involved in its evolution are outlined. The plume trajectory is determined quantitatively in a number of ways, and it is shown that the central streamline and the centre of buoyancy of the plume differ significantly—as with jets in crossflow, the central streamline is seen to follow the top of the plume, whereas the centre of buoyancy, by definition, describes the plume as a whole. We then investigate the turbulence properties inside the plume; in particular the eddy viscosity and diffusivity are presented, which are significant parameters in turbulence modelling. Assessment of turbulence production demonstrates the presence of regions where turbulence kinetic energy is redistributed to the kinetic energy of the mean flow, implying a negative eddy viscosity within certain regions of the domain. Similarly, the observation that the buoyancy flux and buoyancy gradient are anti-parallel in specific regions of the flow implies a negative eddy diffusivity in said regions, which must be realised in models of such flows in order to capture the countergradient transport of thermal properties. A characteristic eddy viscosity and diffusivity are presented, and shown to be approximately constant in the fully developed regime, resulting in a constant characteristic turbulent Prandtl number, in turn signifying self-similarity.

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. III. Asymptotics of branching

The previous paper of this series presented the results of a numerical investigation of the dependence of the dominant growth rates of Bloch eigenmodes on the diffusivity parameters (the molecular viscosity ν and molecular magnetic diffusivity η) in three linear stability problems: the kinematic dynamo problem, and the hydrodynamic and MHD stability problems for steady spaceperiodic flows and MHD states. The dominant eigenmodes (i.e., the stability modes, whose growth rates are maximum over the wave vector q of the planar wave involved in the Bloch modes) comprise branches. In some branches, the dominant growth rates are attained for constant half-integer q. In all the three stability problems for parity-invariant steady states, offshoot branches, stemming from the branches of this type, were found, in which the dominant growth rates are attained for q depending on ν and/or η. We consider now such a branching of the dominant magnetic modes in the kinematic dynamo problem, where an offshoot stems from a branch of neutral eigenmodes for q = 0, and construct power series expansions for the offshoots and the associated eigenvalues of the magnetic induction operator near the point of bifurcation. We show that the branching occurs for the molecular magnetic diffusivities, for which the two eigenvalues of the eddy diffusivity operator become imaginary, and magnetic field generation by the mechanism of the negative eddy diffusivity ceases. The details of branching in the other linear stability problems under consideration are distinct.

Implications of Reynolds Averaging for Reactive Tracers in Turbulent Flows

RANS simulations have been broadly used to investigate turbulence in the oceans and atmosphere. Within these environments there are a multitude of tracers undergoing reactions (e.g., phytoplankton growth, chemical reactions). The distribution of these reactive tracers is strongly influenced by turbulent mixing. With a 50 member ensemble of two-dimensional Rayleigh–Taylor-induced turbulent mixing, we show that the dynamics of a reactive tracer growing according to Fisher’s equation are poorly captured by the ensemble mean. A fluctuation-dependent sink introduced by Reynolds averaging Fisher’s equation transfers tracer concentration from the mean to the fluctuations. We compare the dynamics of the reactive tracer with those of a passive tracer. The reaction increases the reactive tracer’s concentration thereby increasing Fickian diffusion and allowing the reactive tracer to diffuse into turbulent structures that the passive tracer cannot reach. A positive feedback between turbulent mixing and fluctuation growth is identified. We show that eddy viscosity and diffusivity parameterizations fail to capture the bulk trends of the system and identify a need for negative eddy diffusivities. One must, therefore, be cautious when interpreting RANS results for reactive tracers.

Negative magnetic eddy diffusivity due to oscillogenic α-effect

We study large-scale kinematic dynamo action of steady mirror-antisymmetric flows of incompressible fluid, that involve small spatial scales only, by asymptotic methods of the multiscale stability theory. It turns out that, due to the magnetic $\alpha$-effect in such flows, the large-scale mean field experiences harmonic oscillations in time on the scale O($\varepsilon t$) without growth or decay. Here $\varepsilon$ is the spatial scale ratio and $t$ is the fast time of the order of the flow turnover time. The interaction of the accompanying fluctuating magnetic field with the flow gives rise to an anisotropic magnetic eddy diffusivity, whose dependence on the direction of the large-scale wave vector generically exhibits a singular behaviour, and thus to negative eddy diffusivity for whichever molecular magnetic diffusivity. Consequently, such flows always act as kinematic dynamos on the time scale O($\varepsilon^2t$); for the directions at which eddy diffusivity is infinite, the large-scale mean-field growth rate is finite on the scale O($\varepsilon^{3/2}t$). We investigate numerically this dynamo mechanism for two sample flows.

Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation.

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic α-effect or negative eddy diffusivity mechanisms, large-scale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of an incompressible electrically conducting fluid.

We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic α-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field for sufficiently small magnetic molecular diffusivity. These findings indicate that kinetic helicity and helicity spectrum are not the quantities controlling the dynamo properties of a flow regardless of whether scale separation is present or not.

Effects of submesoscale turbulence on ocean tracers

AbstractOcean tracers such as carbon dioxide, nutrients, plankton, and oil advect, diffuse, and react primarily in the oceanic mixed layer where air‐sea gas exchange occurs and light is plentiful for photosynthesis. There can be substantial heterogeneity in the spatial distributions of these tracers due to turbulent stirring, particularly in the submesoscale range where partly geostrophic fronts and eddies and small‐scale three‐dimensional turbulence are simultaneously active. In this study, a large eddy simulation spanning horizontal scales from 20 km down to 5 m is used to examine the effects of multiscale turbulent mixing on nonreactive passive ocean tracers from interior and sea‐surface sources. The simulation includes the effects of both wave‐driven Langmuir turbulence and submesoscale eddies, and tracers with different initial and boundary conditions are examined in order to understand the respective impacts of small‐scale and submesoscale motions on tracer transport. Tracer properties are characterized using spatial fields and statistics, multiscale fluxes, and spectra, and the results detail how tracer mixing depends on air‐sea tracer flux rate, tracer release depth, and flow regime. Although vertical fluxes of buoyancy by submesoscale eddies compete with mixing by Langmuir turbulence, vertical fluxes of tracers are often dominated by Langmuir turbulence, particularly for tracers that are released near the mixed‐layer base or that dissolve rapidly through the surface, even in regions with pronounced submesoscale activity. Early in the evolution of some tracers, negative eddy diffusivities occur co‐located with regions of negative potential vorticity, suggesting that symmetric instabilities or other submesoscale phenomenon may act to oppose turbulent mixing.

A mean field dynamo from negative eddy diffusivity

Using direct numerical simulations, we verify that Roberts-IV flow exhibits dynamo action dominated by horizontally averaged large-scale magnetic field. With the test-field method, we compute the turbulent magnetic diffusivity and find that it is negative and overcomes the molecular diffusivity, thus explaining quantitatively the large-scale dynamo for magnetic Reynolds numbers above ≈8. As expected for a dynamo of this type, but contrary to α-effect dynamos, the two horizontal field components grow independently of each other and have arbitrary amplitude ratios and phase differences. Small length-scales of the mean magnetic field are shown to be stabilized by the turbulent magnetic diffusivity becoming positive at larger wavenumbers. Oscillatory decaying or growing solutions have also been found in certain wavenumber intervals and sufficiently large values of the magnetic Reynolds number. For magnetic Reynolds numbers below ≈0.5, the turbulent magnetic diffusivity is confirmed to be positive, as expected for all incompressible flows. Earlier claims of a dynamo driven by a modified Taylor–Green flow through negative eddy diffusivity could not be confirmed.

Eddy diffusivity in convective hydromagnetic systems

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.

Effect of Molecular Diffusivities on Countergradient Scalar Transfer in a Strongly Stably Stratified Flow (Study on the Linear and Nonlinear Processes Using RDT)

The linear Rapid Distortion Theory (RDT) is applied to stable thermally stratified air (Pr=0.7), thermally stratified water (Pr=6) and salt-stratified liquid (Sc-600) flows without mean shear. Effects of diffusivity and viscosity are included in the analysis and turbulence quanitities such as turbulent scalar fluxes and their cospectra are obtained. The results are compared with previous laboratory measurements and direct numerical simulations (DNS). The results show that counter-gradient scalar transfer (CGST), which transports scalar counter to the mean gradient (i.e. negative eddy diffusivity), can be predicted by linear RDT as shown in the previous studies. However, the persistent downgradient scalar transfer (P-DGST) at small scales in air flows and the persistent CGST (P-CGST) at small scales in water flows can not be predicted by RDT. In a linear process, CGST occurs first at small scales and it spreads to all scales regardless ofPr orSc when the initial length scale of turbulence is sufficiently large. The results suggest that the turbulent scalar transfer at small scales in a strongly stably stratified flow is dominated by nonlinear processes and only the large-scale wave-like motions are controlled by linear processes.

Internal wave band eddy fluxes above a continental slope

Three weeks of velocity and temperature measurements from the bottom 45 m above the continental slope in the Bay of Biscay are used to evaluate the role of the internal wave band in boundary mixing near a sloping bottom. Utilizing acoustic Doppler current profilers and thermistor strings, internal wave band eddy fluxes of momentum and heat are estimated. The instrumentation is specifically designed to resolve internal wave band processes. Due to unresolved Doppler shifting, this wave band may include turbulence as well as internal waves. A very energetic and highly variable near-bottom environment is found. Periods of mixing and restratification alternate at the M 2 tidal frequency. Interpreting the observations in an Ekman sense, the three-week mean current is downwelling-favorable, which is consistent with existing boundary layer theories. However, a bi-directional flow associated with sloping boundary mixing is not found in the near-bottom layer, possibly due to observed strong stratification all the way to the bottom. We evaluated boundary layer dynamics and the effect of internal wave-band fluxes from two frequency ranges (σ ≥ 15 cpd and σ ≥ 1.9 cpd, including tides) on the three-week mean flow. The high-frequency range (σ ≥ 15 cpd) of the internal wave band supports significant momentum and buoyancy fluxes while the low-frequency range (σ ≥ 1.9 cpd) only supports significant momentum fluxes. Mean bottom-normal eddy diffusivities associated with anisotropic, nonlinear internal waves, are negative and O(-10 -2 m 2 s -1 ). Interpreting these negative eddy diffusivities as indication of a restratification process, high mixing efficiencies are expected throughout the mixing layer, which extends typically 20 m above the bottom. Mean eddy viscosities are positive in cross-slope direction and negative in alongslope direction, implying a strong anisotropy in the interaction between internal wave band eddies and the mean flow. Alongslope momentum is transferred from the internal tide to the mean flow. Buoyancy and pressure gradient forces, which we could not determine directly, may generate a buoyancy-driven secondary flow. The buoyancy equation is dominated by advection, possibly balanced by divergence of cross-slope and alongslope internal wave band fluxes.

Dynamo effect in parity-invariant flow with large and moderate separation of scales

It is shown that non-helical (more precisely, parity-invariant) flows capable of sustaining a large-scale dynamo by the negative magnetic eddy diffusivity effect are quite common. This conclusion is based on numerical examination of a large number of randomly selected flows. A few outliers with strongly negative eddy diffusivities are also found, and they are interpreted in terms of the closeness of the control parameter to a critical value for generation of a small-scale magnetic field. Furthermore, it is shown that, for parity-invariant flows, a moderate separation of scales between the basic flow and the magnetic field often significantly reduces the critical magnetic Reynolds number for the onset of dynamo action.

On modeling the turbulent transport of passive biological variables in aquatic ecosystems

Abstract Plankton models must incorporate time-dependent flow fields to properly resolve the advective and eddy transports which shape the distribution of many biological and chemical variables in the sea. Models which exclusively use coefficients of eddy diffusivity to parameterize mixing processes are unable to achieve the same detail as those models in which variable flow can lead to an implicit turbulent flux. Using a model of the Oregon upwelling ecosystem, it is possible to calculate the complex eddy diffusivity which would have to be included in the model in lieu of realistic advection, although the authors do not recommend such a substitute. A demonstration of how an apparent negative eddy diffusivity can arise in advective plankton models and act to create gradients in biological fields is also included.

Negative eddy diffusivities for asymmetric turbulent velocity profiles?

It appears from the experimental results obtained by quite a number of authors that for asymmetric turbulent velocity profiles the locations of maximum velocity and zero shear stress are not identical. This report deals with the importance of this effect for the calculation of momentum, heat, and mass transfer in non-circular channels as well as for the discussion on universal velocity profiles. The report is providing a survey of the works referring to this effect. These informations are supplemented by unpublished results derived from the critical interpretation of measurements carried out by several authors. An empirical relation is indicated which allows to estimate the magnitude of the effect studied.