Numerical study of rotating turbulence with external forcing

The modification of decaying homogeneous turbulence due to its interaction with dispersed small solid particles (d/η<1), at a volumetric loading ratio φv≤5×10−4, is studied using direct numerical simulation. The results show that the particles increase the fluid turbulence energy at high wave numbers. This increase of energy is accompanied by an increase of the viscous dissipation rate, and, hence, an increase in the rate of energy transfer T(k) from the large-scale motion. Thus, depending on the conditions at particle injection, the fluid turbulence kinetic energy may increase initially. But, in the absence of external sources (shear or buoyancy), the turbulence energy eventually decays faster than in the particle-free turbulence. In gravitational environment, particles transfer their momentum to the small-scale motion but in an anisotropic manner. The pressure-strain correlation acts to remove this anisotropy by transferring energy from the direction of gravity to the other two directions, but at the same wave number, i.e., to the small-scale motion in directions normal to gravity. This input of energy in the two directions with lowest energy content causes a reverse cascade. This reverse cascade tends to build up the energy level at lower wave numbers, thus reducing the decay rate of energy as compared to that of either the particle-free turbulence or the zero-gravity particle-laden flow.

We analyze spectra from five sets of hourly vertical profiles of upper‐ocean horizontal velocity collected with the Cartesian Diver velocity recorder. Each set consists of 11 to 27 profiles over depth ranges of 25–220 m to 25–340 m. The vertical wave number (m) power spectra of velocity and shear have no reproducible power law form in the band [0.02–0.1] cpm. Shear spectra tend to rise then fall with increasing wave number, with the transition wave number mc varying within the above band. For shear spectra, steepness at m near 0.1 cpm ranges from m−1/2 to m−2. The data sets fall into two categories. For one group, inverse Richardson functions Ri−1(mc) have values near 2. Despite variability of velocity variance, these sets have Ri−1(0.1 cpm) ≈ 3, suggesting that shear is near saturation. Low variability of Ri−1(0.l), despite high kinetic energy variability for this group, suggests that energy at high wave number (and thus shear) is suppressed. This suggests that instability mechanisms attenuate spectra at high wave number, with attenuation becoming more broadband and influencing progressively lower vertical wave numbers (in addition to consistently influenced high wave numbers) at higher energy. For the other group, Ri−1(0.1) ≈ 1, shear spectra are less steep, and shear is concentrated at high wave number, suggesting less attenuation at the small‐scale end of our measurement band. For both groups, predicted wave numbers of spectral steepening, estimated from the data using two techniques, are related approximately inversely with horizontal kinetic energy in the band [0.005–0.01] cpm, quantifying the shear spectral roll‐off.

We investigate properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence, by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M = 1/3 and M = 1, and with moderate Reynolds numbers, R_lambda ~ 100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Karman-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector k and the spatial separation length l as k l ~ 3^1/2. For a given time the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach whereas the ST terms become dominant on small scales; this is owing to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems there exists an inertial range for the kinetic energy cascade ...

This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By "large scale" it is meant that the spectral support of the flows is confined to the wave-number region k<kd , where kd is relatively small compared with the diffusion wave number kkappa. Such flows mediate couplings between neighboring wave numbers within kd of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number k>kd is bounded from above by UkdkTheta(k,t) , where U denotes the maximum fluid velocity and Theta(k,t) is the spectrum of the scalar variance, defined as its average over the shell (k-kd,k+kd) . For a given flux, say vartheta>0 , across k>kd , this bound requires Theta(k,t)> or =(varthetaUkd)k(-1) . This is consistent with recent numerical studies and with Batchelor's theory that predicts a k(-1) spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with (kd,kkappa) . Thus, Batchelor's formula for the variance spectrum is recovered by the present method in the form of a critical lower bound. The present result applies to a broad range of large-scale advection problems in space dimensions > or =2 , including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, Theta(k,t) and vartheta are the kinetic energy spectrum and flux, respectively.

A model fractal forcing for direct numerical simulations (DNS) of stationary, homogeneous and isotropic turbulence is proposed. The power spectrum of this forcing is a power-law function of wavenumber with a positive exponent that is an increasing function of $D_{f}$, the fractal dimension of the fractal stirrer. The following results are obtained for DNS turbulence subjected to fully self-similar fractal forcing of fractal dimension $D_f$. (i) The Taylor and Kolmogorov microscales are proportional to each other and to the smallest length scale of the fractal forcing. (ii) The integral length scale is much smaller than the size $L_b$ of the DNS box and a decreasing function of the extent of the fractal forcing range because fractal forcing generates very irregular velocity fields. (iii) In qualitative (but not quantitative) agreement with renormalization group (RG) theories of turbulence, higher values of $D_f$ lead to increased energy at the highest wavenumbers. (iv) The energy and energy input rate spectra and the inter-scale energy transfer $T(k)$ all scale with the turbulence r.m.s. velocity $u'$ and the Taylor microscale $\lambda$. (v) More than 80% of the total dissipation occurs in the fractal forcing range of scales which extends from $L_b$ to about one to two times the Taylor microscale. In that range, $T(k)$ is negligible. (vi) $\lambda \,{\sim}\, \nu/u'$ (where $\nu$ is the kinematic viscosity), the kinetic energy dissipation rate per unit mass $\epsilon \,{\sim}\, u'^{3}/\lambda \,{\sim}\,u'^{4}/\nu$ and the velocity derivative skewness $S$ is independent of Reynolds number in the limit where the Reynolds number and the fractal forcing range are increased together. (vii) The intermediate eigenvalue of the strain rate tensor is on average positive, and the negligible values of $T(k)$ in the fractal forcing range are accompanied by lower values of $S$ and a significant reduction in local compression by comparison to turbulence forced only at the large scales. (viii) The geometrical alignments between vorticity, strain rate tensor eigenvectors and vortex stretching vector are qualitatively as in turbulence forced only at the large scales but significantly weakened. (ix) The p.d.f.s of velocity increments are approximately, though not exactly, Gaussian at all scales between the Kolmogorov and integral length scales. A few preliminary DNS results are also given for the case of turbulence generated by a fractal forcing that is discretely, as opposed to fully, self-similar.

The spectrum of two-dimensional incompressible motion is studied with particular attention paid to non-linear interactions. Conditions for the absence of direct interactions between high and low wave numbers are developed. The non-linear tendency of an initial spectrum is computed to illustrate these points. The importance of the angular distribution of the spectrum is illustrated in a spectral stability problem for an isotropic stream function spectrum. Finally, a special inertial equilibrium spectrum is derived exhibiting the properties of an energy barrier in wave number space. DOI: 10.1111/j.2153-3490.1962.tb00126.x

Large-scale atmospheric turbulence is examined through the application of a wave-number frequency Fourier analysis of the velocity distributions on latitude circles. The primary advantage of the wave-number frequency energy spectra is that it permits analysis of the transient eddies in terms of their length scale and in terms of the speed and direction of their motion. Thus, the relative importance of retrogressing waves may be considered in the analysis. The kinetic energy spectra of the meridional component of velocity is found to contain a distinct band of energy for each wave number which shifts to higher negative frequencies as wave number increases from 4 through 10, at 40° N during winter season 1964. This indicates that the motion of these waves is predominantly from west to east. During the summer season, waves moving in the opposite direction carry a relatively large portion of the transient eddy kinetic energy. The terms of the component form of the kinetic energy equations in wave-number frequency space are evaluated with horizontal nondivergent velocities. The form of the equations used involves nonlinear interaction terms, which represent an inertial transfer process; ageostrophic terms, which represent the work done by the pressure and Coriolis forces; and the eddy frictional terms. Of the nonlinear interaction terms, the one which provides the largest positive or negative contribution to the spectral energy is the inertial term involving the longitudinal derivative. The largest interaction combination within each of these terms involves the spectral components of the kinetic energy production terms. This condition causes a positive contribution to the energy of waves moving from west to east and a negative contribution to waves moving from east to west. The contributions to the spectral energy through the interaction terms involving latitudinal derivatives are smaller with less consistency in sign. The interaction combinations representing the kinetic energy production in these terms have the same magnitude as several other interaction combinations. The contributions from the ageostrophic terms serve to balance the contributions from the interaction terms. The magnitude of these balancing contributions is from 2 to 5 times the magnitude of the spectral energy. DOI: 10.1111/j.2153-3490.1969.tb00484.x

The hydrodynamics of marginal seas exhibit internal variability unprovoked by external forcing. To date, the role of tides in reducing this “noise” has not been evaluated. We investigated the effect of tides on internal variability in the Bohai and Yellow Sea. To do so, we conducted three ensembles of numerical experiments using the three-dimensional Finite-Volume Coastal Ocean Model (FVCOM) with tidal forcing, with “half-tidal” forcing, and without tidal forcing, while everything else was unchanged, and determined the intensity of the signal-to-noise ratio (hereinafter referred to as the S/N ratio), with the “signal” represented by the variance of the coherent variations of the different simulations subject to the same atmospheric variability and the noise represented by the intra-ensemble variance. The S/N ratio was determined for depth-averaged velocities, surface temperature, and surface salinity. The first result was that in all three ensembles, noise emerged but with different intensities. In the ensemble with tidal forcing, unprovoked variability emerged primarily at smaller scales. When the tides were weakened or turned off, the S/N ratios were reduced, more so in the Yellow Sea than in the Bohai. The increase in the S/N ratio was largest for large scales and for depth-averaged velocity. The reduction in tidal forcing resulted in an approximately 30% increase in S/N ratios in the Bohai at large scales. Thus, the absence of tidal forcing favored the emergence of increased unprovoked variability at large and medium scales but not at small scales. A hypothesis for explaining this scale-selective effect of tides, based on the stochastic climate model, is suggested.

Collective dynamic properties of different kind of binary liquid mixtures have been investigated by molecular dynamics simulation. The study includes both the longitudinal and the transverse current spectra in simple liquid alloys, 1:1 molten salts and liquid binary mixtures of neutral particles with an ionic-like structure. These systems were chosen as representative of binary liquids with different static structures in order to analyse the effects of structural ordering on the mechanisms of dynamic collective properties. The effect of the mass asymmetry between the two species in the mixture has been also discussed from the results for two different mass ratios for each kind of structure. Two length scales have been considered. On the one hand, the hydrodynamic scale (low wave numbers), where the modes for the partial currents of the two species are characterised by very close frequencies. On the other hand, the molecular scale (higher wave numbers), where the characteristic frequencies for the two species show noticeable differences. Vibrational concentration current modes (optic modes) have been found in neutral mixtures though their influence is rather weak, being the collective dynamic properties of this kind of systems dominated by the mass current modes (acoustic modes). On the contrary, in mixtures of charged particles such as molten salts the contribution of the concentration (charge) currents to the collective dynamics is important and optic modes can be characterised by a well-defined frequency for a wide range of wave numbers. It has been observed that heavy particles have a more relevant role on the mass current correlations whereas light particles play a dominant role on the concentration current correlations. The overall results for the three kinds of liquid mixtures analysed in this paper show that both the longitudinal and transverse current spectra are little dependent on the static structure of the system whereas marked differences are revealed when the particles in the system are either neutral or carry an electric charge.

In this paper, we present an overview of concepts and data concerning inverse cascades of excitation towards scales larger than the forcing scale in a variety of contexts, from two-dimensional fluids and wave turbulence to geophysical flows in the presence of rotation and stratification. We briefly discuss the role of anisotropy in the occurrence and properties of such cascades. We then show that the cascade of some invariant, for example, the total energy, may be transferred through nonlinear interactions to both the small scales and the large scales, with in each case a constant flux. This is in contrast to the classical picture, and we illustrate such a dual cascade in the context of atmospheric and oceanic observations, direct numerical simulations, and modeling. We also show that this dual cascade of total energy can in fact be decomposed in some cases into separate cascades of the kinetic and potential energies, provided the Froude and Rossby numbers are small enough. In all cases, the potential energy flux remains small, of the order of 10% or less relative to the kinetic energy flux. Finally, we demonstrate that, in the small-scale inertial range, approximate equipartition between potential and kinetic modes is obtained, leading to an energy ratio close to one, with strong departure at large scales due to the dominant kinetic energy inverse cascade and piling-up at the lowest spatial frequency and at small scales due to unbalanced dissipation processes, even though the Prandtl number is equal to one.

The clustering of galaxies in ongoing and upcoming galaxy surveys contains a wealth of cosmological information, but extracting this information is a nontrivial task since galaxies and their host haloes are stochastic tracers of the nonlinear matter density field. This stochasticity is usually modeled as the Poisson shot noise, which is constant as a function of wave number with amplitude given by $1/\overline{n}$, where $\overline{n}$ is the number density of galaxies. Here we use dark matter haloes in $N$-body simulations to show evidence for deviations from this simple behavior and develop models that explain the behavior of the stochasticity on large scales. First, haloes are extended, nonoverlapping objects, i.e., their correlation function needs to go to $\ensuremath{-}1$ on small scales. This leads to a negative correction to the stochasticity relative to the Poisson value at low wave number $k$, decreasing to zero for wave numbers large compared to the inverse exclusion scale. Second, haloes show a nonlinear enhancement of clustering outside the exclusion scale, leading to a positive stochasticity correction. Both of these effects go to zero for high $k$, making the stochasticity scale dependent even for $k&lt;0.1h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. We show that the corrections in the low-$k$ regime are the same in Eulerian and Lagrangian space, but that the transition scale is pushed to smaller scales for haloes observed at present time (Eulerian space), relative to the initial conditions (Lagrangian space). These corrections vary with halo mass, and we present approximate scalings with halo mass and redshift. We also discuss simple applications of these effects to galaxy samples with nonvanishing satellite fraction, where the stochasticity can again deviate strongly from the fiducial Poisson expectation. Overall, these effects affect the clustering of galaxies at a level of a few percent even on very large scales and need to be modeled properly if we want to extract high precision cosmological information from the upcoming galaxy redshift surveys.

The ultimate goal in seismic signal processing (pre- processing or main processing) is to enhance the signal to noise ratio (S-N-R). In this study, two high pass (low cut frequency) filters were designed in frequency - wave number (F-K) space and applied to onshore 3D seismic data acquired from a typical Niger Delta field to suppress dispersive ground roll in a bid to enhance S-N-R. The seismic shot records were initially displayed to show the seismic data ensembles in the conventional offset - time (X-T) domain. A frequency - wave number spectrum was then created using a VISTA software algorithm to analyze the spectral patterns of the signals and noise. Two high pass filters were then designed in the F-K space to mute the low frequency ground roll from the data. The filters were then applied to the entire seismic data ensemble for the filtering operation. The results obtained after the two filtering operations were compared to ascertain the optimal filter which was most effective for the suppression of the unwanted ground roll noise.

Aims Anthropogenic activities have drastically increased nutrient availability, resulting in declines in species richness in many plant communities. However, most previous studies have explored only species-loss patterns and mechanisms over small sampling areas, so their results might overestimate species loss at larger spatial scales. The aim of this research was to explore species diversity change patterns and speciesloss rates at multiple scales in alpine meadow communities following nutrient enrichment. Specifically, we asked two closely related questions: (i) do changes in species diversity and species-loss patterns differ among spatial scales? and (ii) how does community compositional dissimilarity and species turnover change among spatial scale? Methods This study was implemented in an alpine meadow community, which is regarded as one of the most sensitive and vulnerable terrestrial ecosystems to anthropogenic nutrient enrichment. We conducted a fertilization experiment that involved the addition of nitrogen (N), phosphorus (P) and a mixture of both to a series of quadrats ranging from 1 to 16 m~2 over 5 years to study the variations in the patterns of species diversity in response to nutrient additions at different spatial scales. Important Findings Our results showed that the changes in species diversity and species loss were dependent on the type of fertilization and the spatial scale. After N and NP fertilization, species diversity significantly decreased at the small scale but not at the large scale, and the rate of species loss decreased as the spatial scale increased. In contrast, the differences between the P addition and control communities were negligible at both the small and large spatial scales. N fertilization caused species to be lost from the small sampling scale, but because different species were lost from different samples, there was an increase in compositional dissimilarity at larger spatial scales, which reduced the total number of species lost when measured at larger scales. These findings highlight spatial scale in evaluating the biodiversity loss after fertilization and suggest that the compositional dissimilarity might play an important role in mediating species loss after fertilization. Our study significantly improved our understanding of changes in species diversity and species loss at different spatial scales under nutrient-enrichment scenarios.

The statistical dynamics of Rossby wave turbulence is examined by comparing direct numerical simulation of the vorticity form of the 2-D Navier--Stokes equation with a non-Markovian statistical closure theory for inhomogeneous flow over mean topography. The quasi-diagonal direct interaction approximation closure theory is formulated for the interaction of mean fields, Rossby waves and inhomogeneous turbulence over topography on a generalized ?-plane. The competing effects of nonlinear waves at the large scales and fully developed turbulence at the small scales is examined by comparing closure theory with ensemble averaged results from direct numerical simulation at resolution k=48 for circularly truncated wavenumber space. This work builds on the low resolution ?-plane studies of Frederiksen and O'Kane (2005) and extends the high resolution f-plane studies of O'Kane and Frederiksen (2004) to incorporate waves. We also examine the performance of a computationally efficient restart or cumulant update procedure at moderate Reynolds number in the presence of waves.

Insufficient numerical resolution of grid-based, direct numerical simulations (DNS) hampers the development of instability-driven turbulence at small (unresolved) scales. As an alternative to DNS, sub-grid models can potentially reproduce the effects of turbulence at small scales in terms of the resolved scales, and hence can capture physical effects with less computational resources. We present a new sub-grid model, the MHD-instability-induced-turbulence (MInIT) mean-field model. MInIT is a physically motivated model based on the evolution of the turbulent (Maxwell, Reynolds, and Faraday) stress tensors and their relation with the turbulent energy densities of the magnetorotational (MRI) and parasitic instabilities, modelled with two partial differential evolution equations with stiff source terms. Their solution allows obtaining the turbulent stress tensors through the constant coefficients that link them to the energy densities. The model is assessed using data from MRI in-box DNS and applying a filtering operation to compare the filtered data with that from the model. Using the L2-norm as the metric for the comparison, we find less than one order-of-magnitude difference between the two sets of data. No dependence on filter size or length scale of unresolved scales is found, as opposed to results using the gradient model (which we also use to contrast our model) in which the L2-norm of some of the stresses increases with filter size. We conclude that MInIT can help DNS by properly capturing small-scale turbulent stresses which has potential implications on the dynamics of highly magnetized rotating compact objects, such as those formed during binary neutron star mergers.