arXiv subjects

Eleftherios Gkioulekas

Publications and source records attributed to Eleftherios Gkioulekas.

14 recordsLinked to original sources

arXiv2020

Solving parametric radical equations with depth 2 rigorously using the restriction set method

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to eliminate all radicals appearing in the equation. We state and prove theorems for all three equation forms with depth 2 that give the solution set of all real-valued solutions. The theorems are shown via the restriction set method that uses inequality restrictions to decide whether to accept or reject candidate solutions. We distinguish between formal solutions that satisfy the original equation in a formal sense, where we allow some radicals to evaluate to imaginary numbers during verification, and strong solutions, where all radicals evaluate to real numbers during verification. Our theorems explicitly identify the set of all formal solutions and the set of all strong solutions for each equation form. The theory underlying radical equations with depth 2 is richer and more interesting than the theory governing radical equations with depth 1, and some aspects of the theory are not intuitively obvious. It is illustrated with examples of parametric radical equations.

math.HO↗

arXiv2020

The role of the asymmetric Ekman dissipation term on the energetics of the two-layer quasi-geostrophic model

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We present theoretical results that show that in either the standard or the extrapolated configurations, the Ekman term dissipates energy at large scales, but does not dissipate potential enstrophy. It also creates an approximately symmetric stable distribution of potential enstrophy between the two layers. The behavior of the Ekman term changes fundamentally at small scales. Under the standard formulation, the Ekman term will unconditionally dissipate energy and also dissipate, under very minor conditions, potential enstrophy at small scales. However, under the extrapolated formulation, there exist small "negative regions", which are defined over a two-dimensional phase space, capturing the distribution of energy per wavenumber between baroclinic energy and barotropic energy, and the distribution of potential enstrophy per wavenumber between the upper layer and the lower layer, where the Ekman term may inject energy or potential enstrophy.

physics.ao-ph↗

arXiv2016

Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence

Using the fusion rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous non-perturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We shall call this generalization of non-perturbative locality to multiple applications of the nonlinear interactions operator "multilocality". The resulting cross-terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to non-perturbative locality of generalized structure functions on all orders, and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.

nlin.CD↗

arXiv2013

Energy and potential enstrophy flux constraints in quasi-geostrophic models

We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant $k^{-3}$ scaling is overtaken by a subdominant $k^{-5/3}$ contribution beyond a transition wavenumber $k_t$ situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber $k_t$ implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier-Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier-Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. etc.

nlin.CD↗

arXiv2013

Energy and potential enstrophy flux constraints in the two-layer quasi-geostrophic model

We investigate an inequality constraining the energy and potential enstrophy flux in the two-layer quasi-geostrophic model. This flux inequality is unconditionally satisfied for the case of two-dimensional Navier-Stokes turbulence. However, it is not obvious that it remains valid under the multi-layer quasi-geostrophic model. The physical significance of this inequality is that it decides whether any given model can reproduce the Nastrom-Gage spectrum of the atmosphere, at least in terms of the total energy spectrum. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized multi-layer model. We then specialize these results for the case of the two-layer quasi-geostrophic model under dissipation configurations in which the dissipation terms for each layer are dependent only on the streamfunction or potential vorticity of that layer. We derive sufficient conditions for satisfying the flux inequality and discuss the possibility of violating it under different conditions.

nlin.CD↗

arXiv2013

Effect of differential diffusion and extrapolated Ekman dissipation on flux constraints on the two-layer quasi-geostrophic model

We continue our investigation of an inequality constraining the energy and potential enstrophy flux spectra in the two-layer quasi-geostrophic model. Its physical significance is that it can diagnose whether any given model that allows coexisting downscale cascades of energy and potential enstrophy can reproduce the Nastrom-Gage spectrum, in terms of the total energy spectrum. This inequality holds unconditionally for two-dimensional turbulence, however it is far from obvious that it generalizes to multi-layer quasi-geostrophic models. In previous work we considered the case of a two-layer quasi-geostrophic model in which the dissipation terms for each layer are dependent only on the streamfunction field of the corresponding layer. We now generalize this configuration as follows: First, following a 1980 paper by Salmon, we use an extrapolated Ekman term at the bottom layer which uses the streamfunction field of both layers to approximate the streamfunction field at the surface boundary layer. Second, for reasons explained in detail in the paper itself, we use small-scale dissipation terms with different hyperviscosity coefficients. Sufficient conditions for satisfying the flux inequality are given under this more general dissipation configuration, and we discuss the potential role of extrapolated Ekman dissipation and differential small-scale dissipation in violating the flux inequality.

nlin.CD↗

arXiv2011

The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

In the Nastrom-Gage spectrum of atmospheric turbulence we observe a $k^{-3}$ energy spectrum that transitions into a $k^{-5/3}$ spectrum, with increasing wavenumber $k$. The transition occurs near a transition wavenumber $k_t$, located near the Rossby deformation wavenumber $k_R$. The Tung-Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length-scale range. We show that, in a temperature forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber $k_t$ near $k_R$. We also show that if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behavior contrary to what occurs in two-dimensional turbulence. As a result, the ratio $\gn/\gee$ of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber $k_t$ further. Using a random Gaussian forcing model, we reach the same conclusion, under the modeling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung-Orlando theory can account for the approximate coincidence between $k_t$ and $k_R$. We also identify certain open questions that require further investigation via numerical simulations.

nlin.CD↗

arXiv2010

Dissipation scales and anomalous sinks in steady two-dimensional turbulence

In previous papers I have argued that the \emph{fusion rules hypothesis}, which was originally introduced by L'vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper we show that the fusion rules hypothesis, combined with \emph{non-perturbative locality}, itself a consequence of the fusion rules hypothesis, dictates the location of the boundary separating the inertial range from the dissipation range. In so doing, the hypothesis that there may be an anomalous enstrophy sink at small scales and an anomalous energy sink at large scales emerges as a consequence of the fusion rules hypothesis. More broadly, we illustrate the significance of viewing inertial ranges as multi-dimensional regions where the fully unfused generalized structure functions of the velocity field are self-similar, by considering, in this paper, the simplified projection of such regions in a two-dimensional space, involving a small scale $r$ and a large scale $R$, which we call, in this paper, the $(r, R)$-plane. We see, for example, that the logarithmic correction in the enstrophy cascade, under standard molecular dissipation, plays an essential role in inflating the inertial range in the $(r, R)$ plane to ensure the possibility of local interactions. We have also seen that increasingly higher orders of hyperdiffusion at large scales or hypodiffusion at small scales make the predicted sink anomalies more resilient to possible violations of the fusion rules hypothesis.

nlin.CD↗

arXiv2008

Locality and stability of the cascades of two-dimensional turbulence

We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. A careful discussion of the subtle notion of locality is given at the end of the paper.

nlin.CD↗

arXiv2007

Winterberg's conjectured breaking of the superluminal quantum correlations over large distances

We elaborate further on a hypothesis by Winterberg that turbulent fluctuations of the zero point field may lead to a breakdown of the superluminal quantum correlations over very large distances. A phenomenological model that was proposed by Winterberg to estimate the transition scale of the conjectured breakdown, does not lead to a distance that is large enough to be agreeable with recent experiments. We consider, but rule out, the possibility of a steeper slope in the energy spectrum of the turbulent fluctuations, due to compressibility, as a possible mechanism that may lead to an increased lower-bound for the transition scale. Instead, we argue that Winterberg overestimated the intensity of the ZPF turbulent fluctuations. We calculate a very generous corrected lower bound for the transition distance which is consistent with current experiments.

quant-ph↗

arXiv2006

On the elimination of the sweeping interactions from theories of hydrodynamic turbulence

In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity. It is shown that both the elimination of the sweeping interactions and the derivation of the 4/5-law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity. The quasi-Lagrangian transformation, on the other hand, requires an even stronger homogeneity assumption which is many-time rather than one-time but still weaker than many-time global homogeneity. We argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi-Lagrangian transformation.

nlin.CD↗

arXiv2006

A New Proof on Net Upscale Energy Cascade in 2D and QG Turbulence

A general proof that more energy flows upscale than downscale in two-dimensional (2D) turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in Surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases.

nlin.CD↗

arXiv2006

Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?

In systems governing two-dimensional turbulence, surface quasi-geostrophic turbulence, (more generally $α$-turbulence), two-layer quasi-geostrophic turbulence, etc., there often exist two conservative quadratic quantities, one ``energy''-like and one ``enstrophy''-like. In a finite inertial range there are in general two spectral fluxes, one associated with each conserved quantity. We derive here an inequality comparing the relative magnitudes of the ``energy'' and ``enstrophy'' fluxes for finite or infinitesimal dissipations, and for hyper or hypo viscosities. When this inequality is satisfied, as is the case of 2D turbulence,where the energy flux contribution to the energy spectrum is small, the subdominant part will be effectively hidden. In sQG turbulence, it is shown that the opposite is true: the downscale energy flux becomes the dominant contribution to the energy spectrum. A combination of these two behaviors appears to be the case in 2-layer QG turbulence, depending on the baroclinicity of the system.

nlin.CD↗

arXiv2006

Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum

Two-dimensional turbulence appears to be a more formidable problem than three-dimensional turbulence despite the numerical advantage of working with one less dimension. In the present paper we review recent numerical investigations of the phenomenology of two-dimensional turbulence as well as recent theoretical breakthroughs by various leading researchers. We also review efforts to reconcile the observed energy spectrum of the atmosphere (the spectrum) with the predictions of two-dimensional turbulence and quasi-geostrophic turbulence.

nlin.CD↗