Eleftherios Gkioulekas

A new proof on net upscale energy cascade in two-dimensional and quasi-geostrophic 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.A general proof that more energy flows upscale than downscale in two-dimensional 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.

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.

On equivalent characterizations of convexity of functions

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition and the tangent line definition. Consequently, for differentiable functions, all three characterizations are logically equivalent.

Winterberg’s Conjectured Breaking of the Superluminal Quantum Correlations over Large Distances

Abstract 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.

Using restrictions to accept or reject solutions of radical equations

ABSTRACT The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equation. We propose alternative solution procedures that are rigorous and simple to execute where the extraneous solutions can be identified without verification against the original equation. In this article, we review previous literature, establish and illustrate rigorous solution procedures for radical equations of depth 1 (i.e. equations where all radicals can be eliminated in one step), and deal with an ambiguity concerning the definition of real-valued solutions to radical equations. An application to defining the inverse function, resulting in a parametric radical equation, is also explained.

On the Denesting of Nested Square Roots.

ABSTRACT We present the basic theory of denesting nested square roots, from an elementary point of view, suitable for lower level coursework. Necessary and sufficient conditions are given for direct denesting, where the nested expression is rewritten as a sum of square roots of rational numbers, and for indirect denesting, where the nested expression is rewritten as a sum of fourth-order roots of rational numbers. The theory is illustrated with several solved examples.

Energy and potential enstrophy flux constraints in quasi-geostrophic models

Abstract 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. We explore dissipation asymmetry further in the context of a two-layer quasi-geostrophic model and derive upper bounds on the asymmetry that will allow the flux inequality to continue to hold. Asymmetry is introduced both via an extrapolated Ekman term, based on a 1980 model by Salmon, and via differential small-scale dissipation. The results given are mathematically rigorous and require no phenomenological assumptions about the inertial range. Sufficient conditions for violating the flux inequality, on the other hand, require phenomenological hypotheses, and will be explored in future work.

Generalized local test for local extrema in single-variable functions

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test is local, in the sense that it is only necessary to evaluate all relevant derivatives at the critical point itself, and it is reasonably robust. We illustrate an application of the generalized test on a trigonometric function where the second derivative test fails to classify some of the critical points.

Zero-bounded limits as a special case of the squeeze theorem for evaluating single-variable and multivariable limits

Many limits, typically taught as examples of applying the ‘squeeze’ theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful for both single-variable limits and multidimensional limits. A comprehensive treatment of multidimensional limits and continuity is also outlined.