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Open AccessArticle

Turbulence with Magnetic Helicity That Is Absent on Average

by

Axel Brandenburg

and

Gustav Larsson

Atmosphere 2023, 14(6), 932; https://doi.org/10.3390/atmos14060932 - 26 May 2023

Viewed by 307

Abstract

Magnetic helicity plays a tremendously important role when it is different from zero on average. Most notably, it leads to the phenomenon of an inverse cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as well as some less common examples of magnetic evolution [...] Read more.

Magnetic helicity plays a tremendously important role when it is different from zero on average. Most notably, it leads to the phenomenon of an inverse cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as well as some less common examples of magnetic evolution under the Hall effect and ambipolar diffusion, as well as cases in which the magnetic field evolution is constrained by the presence of an asymmetry in the number density of chiral fermions, whose spin is systematically either aligned or anti-aligned with its momentum. In all those cases, there is a new conserved quantity: the Hosking integral. We present quantitative scaling results for the magnetic integral scale as well as the magnetic energy density and its spectrum. We also compare with cases were a magnetic version of the Saffman integral is initially finite. Rotation in MHD turbulence tends to suppress nonlinearity and thereby also inverse cascading. Finally, the role of the Hosking and magnetic Saffman integrals in shell models of turbulence is examined. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Small-Scale Anisotropy in Stably Stratified Turbulence; Inferences Based on Katabatic Flows

by

Eliezer Kit

and

Harindra J. S. Fernando

Atmosphere 2023, 14(6), 918; https://doi.org/10.3390/atmos14060918 - 24 May 2023

Viewed by 288

Abstract

The focus of the current study is on the anisotropy of stably stratified turbulence that is not only limited to large scales and an inertial subrange but also penetrates to small-scale turbulence in the viscous/dissipation subrange on the order of the Kolmogorov scale. [...] Read more.

The focus of the current study is on the anisotropy of stably stratified turbulence that is not only limited to large scales and an inertial subrange but also penetrates to small-scale turbulence in the viscous/dissipation subrange on the order of the Kolmogorov scale. The anisotropy of buoyancy forces is well-known, including ensuing effects such as horizontal layering and pancakes structures. Laboratory experiments in the nineties by Van Atta and his students showed that the anisotropy penetrates to very small scales, but their experiments were performed only at a relatively low Re_λ (i.e., at Taylor Reynolds numbers) and, therefore, did not provide convincing evidence of anisotropy penetration into viscous sublayers. Nocturnal katabatic flows having configurations of stratified parallel shear flows and developing on mountain slopes provide high Reynolds number data for testing the notion of anisotropy at viscous scales, but obtaining appropriate time series of the data representing stratified shear flows devoid of unwarranted atmospheric factors is a challenge. This study employed the “in situ” calibration of multiple hot-film-sensors collocated with a sonic anemometer that enabled obtaining a 90 min continuous time series of a “clean” katabatic flow. A detailed analysis of the structure functions was conducted in the inertial and viscous subranges at an Re_λ around 1250. The results of DNS simulations by Kimura and Herring were employed for the interpretation of data. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Review of Jackson Herring’s Early Work on Thermal Convection

by

Robert M. Kerr

Atmosphere 2023, 14(6), 907; https://doi.org/10.3390/atmos14060907 - 23 May 2023

Viewed by 333

Abstract

Jack Herring had three mid-1960s numerical papers on Rayleigh-Bénard thermal convection that might seem primitive by today’s standards, but already encapsulated many of the questions that are still being asked. All of them use severely truncated versions of the incompressible Navier–Stokes–Boussinesq equations with [...] Read more.

Jack Herring had three mid-1960s numerical papers on Rayleigh-Bénard thermal convection that might seem primitive by today’s standards, but already encapsulated many of the questions that are still being asked. All of them use severely truncated versions of the incompressible Navier–Stokes–Boussinesq equations with only one, or just a few, horizontal Fourier modes. In the first two papers, 1963 and 1964, the presented results used only one Fourier mode

α

and three variables. The single mode’s variables are its vertical velocity profile

w_{α} (z, t)

, its temperature profile

θ_{α} (z, t)

and the horizontally uniform vertical profile of the background temperature

ψ (z, t)

. All of the second- and third-order terms are ignored except the convective heat flux

\bar{w θ}

. The objective was to find asymptotic steady-state solutions. Each paper found evidence for the one-third Nusselt versus Rayleigh scaling of

N u

∼

R a^{1 / 3}

, originally derived from Malkus’ maximum flux principle. The 1963 paper uses free-slip upper and lower boundaries, with magnitudes of

N u

that are a factor of three larger than the experiments. In the 1964 paper, by introducing no-slip/rigid boundary conditions, the magnitude of

N u

dropped to within 20% of the experimental values. Both

N u (R a)

relations are in good agreement with circa-1990 direct numerical simulations (DNS). This dependence upon the boundary condition at the walls suggests that to obtain physically realistic scaling, no-slip boundary conditions are necessary. The third paper is discussed only in terms of what it might have been aiming to accomplish and its relation to the earlier free-slip results. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Potential Vorticity Generation in Breaking Gravity Waves

by

Michael L. Waite

and

Nicholas Richardson

Atmosphere 2023, 14(5), 881; https://doi.org/10.3390/atmos14050881 - 18 May 2023

Viewed by 437

Abstract

Potential vorticity (PV) is an important quantity in stratified flows because it is conserved following the flow in the absence of forcing and viscous and diffusive effects. However, as shown by previous work for unstratified turbulence, viscosity and diffusion, when present, are not [...] Read more.

Potential vorticity (PV) is an important quantity in stratified flows because it is conserved following the flow in the absence of forcing and viscous and diffusive effects. However, as shown by previous work for unstratified turbulence, viscosity and diffusion, when present, are not purely dissipative and can create potential vorticity even when none is present initially. In this work, we use direct numerical simulations to investigate the viscous and diffusive generation of potential vorticity and potential enstrophy (integrated square PV) in stratified turbulence. Simulations are initialized with a two-dimensional standing internal gravity wave, which has no potential vorticity apart from some low-level random noise; as a result, all potential vorticity and enstrophy comes from viscous and diffusive effects. Significant potential enstrophy is found when the standing wave breaks, and the maximum potential enstrophy increases with increasing Reynolds number. The mechanism for the initial PV generation is spanwise diffusion of buoyancy perturbations, which grow as the standing wave three-dimensionalizes, into the direction of spanwise vorticity. The viscous and diffusive terms responsible are small-scale and are sensitive to under-resolution, so high resolution is required to obtain robust results. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Fog Intermittency and Critical Behavior

by

,

,

,

,

and

Harindra J. S. Fernando

Atmosphere 2023, 14(5), 875; https://doi.org/10.3390/atmos14050875 - 17 May 2023

Viewed by 413

Abstract

The intermittency of fog occurrence (the switching between fog and no-fog) is a key stochastic feature that plays a role in its duration and the amount of moisture available. Here, fog intermittency is studied by using the visibility time series collected during the [...] Read more.

The intermittency of fog occurrence (the switching between fog and no-fog) is a key stochastic feature that plays a role in its duration and the amount of moisture available. Here, fog intermittency is studied by using the visibility time series collected during the month of July 2022 on Sable Island, Canada. In addition to the visibility, time series of air relative humidity and turbulent kinetic energy, putative variables akin to the formation and breakup conditions of fog, respectively, are also analyzed in the same framework to establish links between fog intermittency and the underlying atmospheric variables. Intermittency in the time series is quantified with their binary telegraph approximations to isolate clustering behavior from amplitude variations. It is shown that relative humidity and turbulent kinetic energy bound many stochastic features of visibility, including its spectral exponent, clustering exponent, and the growth of its block entropy slope. Although not diagnostic, the visibility time series displays features consistent with Pomeau–Manneville Type-III intermittency in its quiescent phase duration PDF scaling (

- 3 / 2

), power spectrum scaling (

- 1 / 2

), and signal amplitude PDF scaling (

- 2

). The binary fog time series exhibits properties of self-organized criticality in the relation between its power spectrum scaling and quiescent phase duration distribution. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Turbulent Convection at Very High Rayleigh Numbers and the Weakly Nonlinear Theory

by

Katepalli R. Sreenivasan

and

Joseph J. Niemela

Atmosphere 2023, 14(5), 826; https://doi.org/10.3390/atmos14050826 - 04 May 2023

Viewed by 526

Abstract

To provide insights into the challenging problem of turbulent convection, Jack Herring used a greatly truncated version of the complete Boussinesq equations containing only one horizontal wavenumber. In light of later observations of a robust large-scale circulation sweeping through convecting enclosures at high [...] Read more.

To provide insights into the challenging problem of turbulent convection, Jack Herring used a greatly truncated version of the complete Boussinesq equations containing only one horizontal wavenumber. In light of later observations of a robust large-scale circulation sweeping through convecting enclosures at high Rayleigh numbers, it is perhaps not an implausible point of view from which to reexamine high-Rayleigh-number data. Here we compare past experimental data on convective heat transport at high Rayleigh numbers with predictions from Herring’s model and, in fact, find excellent agreement. The model has only one unknown parameter compared to the two free parameters present in the lowest-order least-squares power-law fit. We discuss why the underlying simplistic physical picture, meant to work at Rayleigh numbers slightly past the critical value of a few thousand, is consistent with the data when the single free parameter in it is revised, over some eleven decades of the Rayleigh number—stretching from about a million to about

10^{17}

. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Open AccessArticle

Two-Dimensional Flow on the Sphere

by

Rick Salmon

and

Nick Pizzo

Atmosphere 2023, 14(4), 747; https://doi.org/10.3390/atmos14040747 - 20 Apr 2023

Viewed by 627

Abstract

Equilibrium statistical mechanics predicts that inviscid, two-dimensional, incompressible flow on the sphere eventually reaches a state in which spherical harmonic modes of degrees

n = 1

and

n = 2

hold all the energy. By a separate theory, such flow is static in [...] Read more.

Equilibrium statistical mechanics predicts that inviscid, two-dimensional, incompressible flow on the sphere eventually reaches a state in which spherical harmonic modes of degrees

n = 1

and

n = 2

hold all the energy. By a separate theory, such flow is static in a reference frame rotating at angular speed

2 Ω / 3

with respect to the inertial frame. The vorticity field in the static frame is an accident of the initial conditions, but, once established, it lasts forever under the stated assumptions. We investigate the possibility of such behavior with a stereographic-coordinate model that conserves energy and enstrophy when the viscosity vanishes. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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Review

Jump to: Research

Open AccessReview

Jackson R. Herring and the Statistical Closure Problem of Turbulence: A Review of Renormalized Perturbation Theories

by

David McComb

Atmosphere 2023, 14(5), 827; https://doi.org/10.3390/atmos14050827 - 04 May 2023

Viewed by 649

Abstract

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, [...] Read more.

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, in terms of its elegance and its pedagogical value. We then concentrate on the assessment of these theories and take the historical route of Kraichnan’s diagnosis of the failure of DIA, followed by Edwards’s analysis of the failure of his self-consistent theory, when compared to the Kolmogorov spectrum. As all three theories are closely related, these analyses also shed light on Herring’s theory. The second-generation theories that grew out of this assessment are then discussed. First, there were the Lagrangian theories, initially stemming from the work of Kraichnan and Herring, and later the purely Eulerian local energy-transfer (LET) theory. The latter is significant because its development exposes the underlying problems with the pioneering theories in terms of the basic physics of the inertial energy transfer. In particular, later work allows us to assign a unified explanation of the incompatibility of all three pioneering theories with the Kolmororov spectrum, in that they are all Markovian approximations (in wavenumber) to the non-Markovian phenomenon of fluid turbulence. In the interests of completeness, we briefly review the formalisms of Wyld and Martin, Siggia, and Rose. More recent developments are also discussed, in order to bring the subject up to the present day. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

Open AccessReview

Beyond Scale-by-Scale Equilibrium

by

John C. Vassilicos

Atmosphere 2023, 14(4), 736; https://doi.org/10.3390/atmos14040736 - 19 Apr 2023

Viewed by 564

Abstract

Homogeneous turbulence and turbulence in scale-by-scale equilibrium, played a leading role in the turbulence research of the second half of the twentieth century, and Jack Herring was an important contributor to these developments. The research activity which has followed these developments over the [...] Read more.

Homogeneous turbulence and turbulence in scale-by-scale equilibrium, played a leading role in the turbulence research of the second half of the twentieth century, and Jack Herring was an important contributor to these developments. The research activity which has followed these developments over the past ten to fifteen years concerns turbulence, which is out of scale-by-scale equilibrium either because it is non-stationary or because it is non-homogeneous or both. This paper is a short review of recent progress in this relatively new direction of turbulence research. Full article

(This article belongs to the Special Issue Turbulence from Earth to Planets, Stars and Galaxies—Commemorative Issue Dedicated to the Memory of Jackson Rae Herring)

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