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.Comment: 25 pages, 24 figures, resubmitted to Physical Review

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.Comment: 28 pages, 3 figures, resubmitted to Physica

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.Comment: 29 pages; resubmitted to J. Fluid Mec

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.Comment: 23 pages; 0 figures; resubmitted to International Journal of Mathematical Education in Science and Technolog

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.Comment: v1: submitted to J. Fluid. Mech v2: revised and accepted by J. Fluid. Mech, 17 page

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 for- mal 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

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 $\alpha$-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.Comment: 23 pages; accepted at Discrete and Continuous Dynamical Systems B; Major revisio

On the denesting of nested square roots

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

Using restrictions to accept or reject solutions of radical equations

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