## Publications

163 results found

Gibbon JD, Vincenzi D, 2022, How to extract a spectrum from hydrodynamic equations, *Journal of Nonlinear Science*, Vol: 32, Pages: 1-25, ISSN: 0938-8974

The practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum E(k)∼k−q and a cutoff wavenumber kc. For example, the values q=5/3 and ℓkc∼Re3/4 are intimately associated with Kolmogorov’s 1941 theory. To extract such spectral information from the Navier–Stokes equations, Doering and Gibbon (Phys. D 165, 163–175, 2020) introduced the idea of forming a set of dynamic wavenumbers κn(t) from ratios of norms of solutions. The time averages of the κn(t) can be interpreted as the 2nth moments of the energy spectrum. They found that 1<q⩽8/3, thereby confirming the earlier work of Sulem and Frisch (J. Fluid Mech. 72, 417–423, 1975) who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless q⩽8/3. Since the κn(t) are based on Navier–Stokes weak solutions, this approach connects empirical predictions of the energy spectrum with the mathematical analysis of the Navier–Stokes equations. This method is developed to show how it can be applied to many hydrodynamic models such as the two dimensional Navier–Stokes equations (in both the direct- and inverse-cascade regimes), the forced Burgers equation and shell models.

Gibbon JD, Dubrulle B, 2022, A correspondence between the multifractal model of turbulence and the Navier-Stokes equations, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 380, ISSN: 1364-503X

The multifractal model of turbulence (MFM) and the three-dimensional Navier–Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum C(h), which appears naturally in the Large Deviation formulation of the MFM, and on h the standard scaling parameter. These bounds respectively take the form: (i) C(h)≥1−3h, which is consistent with Kolmogorov’s four-fifths law ; and (ii) h≥−23. The latter is significant as it prevents solutions from approaching the Navier–Stokes singular set of Caffarelli, Kohn and Nirenberg.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

Gibbon JD, 2021, Variable density model for the Rayleigh-Taylor instability and its transformation to the diffusive, inhomogeneous, incompressible Navier-Stokes equations, *Physical Review Fluids*, Vol: 6, Pages: 1-7, ISSN: 2469-990X

It is shown how the variable density model that governs the Rayleigh-Taylor instability for the miscible mixing of two incompressible fluids can be transformed into a diffusive version of the inhomogeneous, incompressible Navier-Stokes equations forced by gradients of the composition density ρ of the mixing layer. This demonstrates how buoyancy-driven flows drive and enhance Navier-Stokes turbulence. The role of the potential vorticity q=ω⋅∇ρ is also discussed.

Vincenzi D, Gibbon JD, 2021, How close are shell models to the 3D Navier-Stokes equations?, *Nonlinearity*, Vol: 34, Pages: 5821-5843, ISSN: 0951-7715

Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier–Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier–Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.

Gibbon J, 2020, Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence, *Europhysics Letters: a letters journal exploring the frontiers of physics*, Vol: 131, Pages: 64001-p1-64001-p5, ISSN: 0295-5075

Visual manifestations of intermittency in computations of three dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or ‘thin’ filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension D= 1,2,3 and, in the light of the occurrence of thin sets, to discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

Gibbon JD, 2019, Weak and strong solutions of the 3D Navier–Stokes equations and their relation to a chessboard of convergent inverse length scales, *Journal of Nonlinear Science*, Vol: 29, Pages: 215-218, ISSN: 0938-8974

Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the 3D Navier–Stokes equations, on a periodic domain =[0,L]3 an infinite ‘chessboard’ of estimates for these inverse length scales is displayed in terms of labels (n,m) corresponding to n derivatives of the velocity field in L2m(). The (1,1) position corresponds to the inverse Kolmogorov length Re3/4. These estimates ultimately converge to a finite limit, Re3, as n,m→∞, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by (n,m). In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by (n,m), the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi–Serrin conditions for n≥1.

Gibbon JD, Rao P, Caulfield CCP, 2019, The Rayleigh-Taylor instability in buoyancy-driven variable density turbulence, Partial Differential Equations in Fluid Mechanics, Pages: 50-67, ISBN: 9781108460965

This paper reviews and summarizes two recent pieces of work on the Rayleigh-Taylor instability. The first concerns the 3D Cahn-Hilliard-Navier-Stokes (CHNS) equations and the BKM-type theorem proved by Gibbon, Pal, Gupta, & Pandit (2016). The second and more substantial topic concerns the variable density model, which is a buoyancy-driven turbulent flow considered by Cook & Dimotakis (2001) and Livescu & Ristorcelli (2007, 2008). In this model Ρ*(x, t) is the composition density of a mixture of two incompressible miscible fluids with fluid densities Ρ*2 > Ρ*1 and Ρ*0 is a reference normalisation density. Following the work of a previous paper (Rao, Caulfield, & Gibbon, 2017), which used the variable θ = lnΡ*/Ρ*0, data from the publicly available Johns Hopkins Turbulence Database suggests that the L2-spatial average of the density gradient θ can reach extremely large values at intermediate times, even in flows with low Atwood number At = (Ρ*2 - Ρ*1)/(Ρ*2 + Ρ*1) = 0.05. This implies that very strong mixing of the density field at small scales can potentially arise in buoyancy-driven turbulence thus raising the possibility that the density gradient θ might blow up in a finite time.

Gibbon JD, Gupta A, Pal N,
et al., 2018, The role of BKM-type theorems in 3D Euler, Navier-Stokes andCahn-Hilliard-Navier-Stokes analysis, *Physica D: Nonlinear Phenomena*, Vol: 376-377, Pages: 60-68, ISSN: 0167-2789

The Beale–Kato–Majda theorem contains a single criterion that controls the behaviour of solutions of the 3D incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the 3D incompressible Euler and Navier–Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the 3D Cahn–Hilliard–Navier–Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energydissipation rate that, remarkably, reproduces the Re³⁄⁴ upper bound on the inverse Kolmogorov length normally associated with the Navier–Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a 128³ grid.

Plan ELCM, Gupta A, Vincenzil D,
et al., 2017, Lyapunov dimension of elastic turbulence, *Journal of Fluid Mechanics*, Vol: 822, ISSN: 0022-1120

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

Gibbon JD, Holm DD, 2017, Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations, *Nonlinearity*, Vol: 30, Pages: R1-R24, ISSN: 0951-7715

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members ${{ \Omega }_{m}}(t)$ ($1\leqslant m<\infty $ ) are made up from the respective sum of the L 2m -norms of vorticity and the density gradient. Each ${{ \Omega }_{m}}(t)$ has a lower bound in terms of the inverse Rossby number, Ro −1, that turns out to be crucial to the argument. For convenience, the ${{ \Omega }_{m}}$ are also scaled into a new set of variables D m (t). By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the D m (t) in terms of Ro −1 and the Reynolds number Re. These upper bounds vary across bands in the $\left\{{{D}_{1}},\,{{D}_{m}}\right\}$ phase plane. The boundaries of these bands depend subtly upon Ro −1, Re, and the inverse Froude number Fr −1. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${{ \Omega }_{1}}$ deviates from Re 3/4 as a function of $R{{o}^{-1}},\,Re$ and Fr −1.

Gibbon JD, Pal N, Gupta A,
et al., 2017, Regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations (vol 94, 603103, 2016), *PHYSICAL REVIEW E*, Vol: 95, ISSN: 2470-0045

Gibbon JD, Pal N, Gupta A,
et al., 2016, Regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations, *Physical Review E*, Vol: 94, ISSN: 2470-0045

We consider the three-dimensional (3D) Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter ϕ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the 3D incompressible Euler equations [J. T. Beale, T. Kato, and A. J. Majda, Commun. Math. Phys. 94, 61 (1984)]. By taking an L∞ norm of the energy of the full binary system, designated as E∞, we have shown that ∫t0E∞(τ)dτ governs the regularity of solutions of the full 3D system. Our direct numerical simulations (DNSs) of the 3D CHNS equations for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with 1283 to 5123 collocation points and over the duration of our DNSs confirm that E∞ remains bounded as far as our computations allow.

Rao P, Caulfield CP, Gibbon JD, 2016, Nonlinear effects in buoyancy-driven variable-density turbulence, *Journal of Fluid Mechanics*, Vol: 810, Pages: 362-377, ISSN: 0022-1120

We consider the time dependence of a hierarchy of scaled L2m-norms Dm,ω and Dm,θof the vorticity ω = ∇ × u and the density gradient ∇θ, where θ = log(ρ∗/ρ∗0), ina buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. FluidMech., vol. 591, 2007, pp. 43–71). Here, ρ∗(x, t) is the composition density of amixture of two incompressible miscible fluids with fluid densities ρ∗2 > ρ∗1, and ρ∗0is a reference normalization density. Using data from the publicly available JohnsHopkins turbulence database, we present evidence that the L2-spatial average of thedensity gradient ∇θ can reach extremely large values at intermediate times, evenin flows with low Atwood number At = (ρ∗2 − ρ∗1)/(ρ∗2 + ρ∗1) = 0.05, implying thatvery strong mixing of the density field at small scales can arise in buoyancy-driventurbulence. This large growth raises the possibility that the density gradient ∇θ mightblow up in a finite time.

Arnaudon A, Gibbon J, 2016, Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates, *Physical Review A*, ISSN: 1050-2947

We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB) which acts as a simplified model for the dynamics of strongly correlated Bose-Einstein condensates. A proof of their integrability is found by the derivation of a Lax pair which is valid for both the hyperbolic and standard cases of the reduced Maxwell-Bloch equations. The origin of the latter lies in quantum optics. We derive explicit solutions of the HRMB equations that correspond to kinks propagating on the Bose-Einstein condensate (BEC). These solutions are different from Gross-Pitaevskii solitons because the nonlinearity of the HRMB equations arises from the interaction of the BEC and excited atoms.

Gibbon JD, Gupta A, Krstulovic G,
et al., 2016, Depletion of nonlinearity in magnetohydrodynamic turbulence: insights from analysis and simulations, *Physical Review E*, Vol: 93, ISSN: 1539-3755

Gibbon JD, 2015, High-low frequency slaving and regularity issues in the 3D Navier-Stokesequations, *IMA Journal of Applied Mathematics*, Vol: 81, Pages: 308-320, ISSN: 0272-4960

The old idea that an infinite dimensional dynamical system may have its high modes or frequenciesslaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A setof dimensionless frequencies {Ω˜ m(t)} are used which are based on L2m-norms of the vorticity. To avoidusing derivatives a closure is assumed that suggests that the Ω˜ m (m > 1) are slaved to Ω˜1 (the globalenstrophy) in the form Ω˜ m = Ω˜1Fm(Ω˜1). This is shaped by the constraint of two Holder inequalities ¨and a time average from which emerges a form for Fm which has been observed in previous numericalNavier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation isparametrized by a set of functions 1 6 λm(τ) 6 4, where curves of constant λm form the boundariesbetween tongue-shaped regions. In regions where 2.5 6 λm 6 4 and 1 6 λm 6 2 the Navier-Stokesequations are shown to be regular : numerical simulations appear to lie in the latter region. Only in thecentral region 2 < λm < 2.5 has no proof of regularity been found.

Gibbon JD, Donzis DA, Gupta A,
et al., 2014, Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations, *Nonlinearity*, Vol: 27, Pages: 2605-2625, ISSN: 0951-7715

The periodic 3D Navier–Stokes equations are analyzed in terms of dimensionless, scaled, L2m-norms of vorticity Dm (1 ≤ m < ∞). The first in this hierarchy, D1, is the global enstrophy. Three regimes naturally occur in the D1 − Dm plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime [1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

Gibbon JD, Titi ES, 2013, The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?, *JOURNAL OF NONLINEAR SCIENCE*, Vol: 23, Pages: 993-1000, ISSN: 0938-8974

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- Citations: 7

Gibbon JD, Holm DD, 2013, Stretching and folding processes in the 3D Euler and Navier-stokes equations, *Procedia IUTAM*, Vol: 9, Pages: 25-31, ISSN: 2210-9838

Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the contextof the vector B = ∇q×∇θ where, in atmospheric physics, θ is a temperature, q = ω ·∇θ is the potential vorticity, and ω = curluis the vorticity. These ideas are then discussed in the context of the full compressible Navier-Stokes equations where q is taken inthe form q = ω ·∇ f(ρ). In the two cases f = ρ and f = lnρ, q is shown to satisfy the quasi-conservative relation ∂tq+divJ = 0

Donzis DA, Gibbon JD, Gupta A,
et al., 2013, Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations, *JOURNAL OF FLUID MECHANICS*, Vol: 732, Pages: 316-331, ISSN: 0022-1120

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- Citations: 22

Gibbon JD, Holm DD, 2013, Enstrophy bounds and the range of space-time scales in the hydrostatic primitive equations, *Physical Review E*, Vol: 87, ISSN: 1539-3755

The hydrostatic primitive equations (HPEs) form the basis of most numerical weather, climate, and global ocean circulation models. Analytical (not statistical) methods are used to find a scaling proportional to (NuRaRe)1/4 for the range of horizontal spatial sizes in HPE solutions, which is much broader than is currently achievable computationally. The range of scales for the HPE is determined from an analytical bound on the time-averaged enstrophy of the horizontal circulation. This bound allows the formation of very small spatial scales, whose existence would excite unphysically large linear oscillation frequencies and gravity wave speeds.

Gibbon JD, 2013, Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations, IUTAM Symposium on Topological Fluid Dynamics II - Theory and Applications, Publisher: ELSEVIER SCIENCE BV, Pages: 39-48, ISSN: 2210-9838

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- Citations: 5

Gibbon JD, 2012, Conditional regularity of solutions of the three-dimensional Navier-Stokes equations and implications for intermittency, *JOURNAL OF MATHEMATICAL PHYSICS*, Vol: 53, ISSN: 0022-2488

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- Citations: 12

Gibbon JD, Holm DD, 2012, Quasiconservation laws for compressible three-dimensional Navier-Stokes flow, *Physical Review E*, Vol: 86, ISSN: 1539-3755

We formulate the quasi-Lagrangian fluid transport dynamics of mass density ρ and the projection q=ω⋅∇ρ of the vorticity ω onto the density gradient, as determined by the three-dimensional compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of q cannot cross a level set of ρ. That is, in this formulation, level sets of ρ (isopycnals) are impermeable to the transport of the projection q.

Gibbon JD, 2012, A HIERARCHY OF LENGTH SCALES FOR WEAK SOLUTIONS OF THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS, *COMMUNICATIONS IN MATHEMATICAL SCIENCES*, Vol: 10, Pages: 131-136, ISSN: 1539-6746

Bartuccelli MV, Gibbon JD, 2011, Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality, *JOURNAL OF MATHEMATICAL PHYSICS*, Vol: 52, ISSN: 0022-2488

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- Citations: 11

Gibbon JD, Holm DD, 2011, Extreme events in solutions of hydrostatic and non-hydrostatic climate models, *PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 369, Pages: 1156-1179, ISSN: 1364-503X

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- Citations: 5

Gibbon JD, Holm DD, 2011, The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles, *GEOPHYSICAL AND ASTROPHYSICAL FLUID DYNAMICS*, Vol: 105, Pages: 329-339, ISSN: 0309-1929

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- Citations: 1

Gibbon JD, 2010, Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 466, Pages: 2587-2604, ISSN: 1364-5021

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- Citations: 16

Gibbon JD, Holm DD, 2010, The dynamics of the gradient of potential vorticity, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 43, ISSN: 1751-8113

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- Citations: 13

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