Search

Russell James Donnelly (b. 1930) was an exceptionally creative physicist with many other interests: art, music, history, and scientific societies and their scholarly journals. He reinvigorated the maturing field of low temperature physics by linking it strongly to fluid turbulence through bold and optimistic leadership at the intersection of the two fields. Immediately after achieving his Ph.D. at Yale University with C.T. Lane and L. Onsager, Russ joined the University of Chicago in 1956, where he became a professor at the first possible opportunity. After some ten years at University of Chicago, where he worked for a time with S. Chandrasekhar, he moved to the University of Oregon and led a vigorous life until his death in 2015. Russ was an excellent organizer of scientific meetings and an enthusiastic expositor of his science. He had a profound sense of service to the community, both civic and scientific, and showed exceptional scientific openness and generosity to his colleagues. He was greatly admired by his community.

Universal properties of turbulence have been associated traditionally with very high Reynolds numbers, but recent work has shown that the onset of the power-laws in derivative statistics occurs at modest microscale Reynolds numbers of the order of 10, with the corresponding exponents being consistent with those for the inertial range structure functions at very high Reynolds numbers. In this paper we use well-resolved direct numerical simulations of homogeneous and isotropic turbulence to establish this result for a range of initial conditions with different forcing mechanisms. We also show that the moments of transverse velocity gradients possess larger scaling exponents than those of the longitudinal moments, confirming past results that the former are more intermittent than the latter.

Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers as in stellar convection and in Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh-B\'{e}nard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $\Gamma =L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$, and Rayleigh numbers $10^5 \le Ra \le 10^7$, obtained by massively parallel computations on grids of up to $5.36\times 10^{11}$ points. The low end of this $Pr$-range cannot be accessed in controlled laboratory measurements. We report the essential properties of the flow and their trends with Rayleigh and Prandtl numbers, in particular the global transport of momentum and heat -- the latter decomposed into convective and diffusive contributions -- across the convection layer, mean vertical profiles of the temperature and temperature fluctuations, and the kinetic energy and thermal dissipation rates. We also explore the degree to which the turbulence in the bulk of the convection layer resembles classical homogeneous and isotropic turbulence in terms of spectra, increment moments, and dissipative anomaly, and find close similarities. Finally, we show that a characteristic scale on the order of the mesoscale seems to saturate to a wavelength of $\lambda\gtrsim 3H$ for $Pr\lesssim 0.005$. We briefly discuss possible implications of these results for the development of subgrid scale parameterization of turbulent convection., Comment: 27 pages, 14 figures, 4 tables

To understand turbulent convection at very high Rayleigh numbers typical of natural phenomena, computational studies in slender cells are an option if the needed resources have to be optimized within available limits. However, the accompanying horizontal confinement affects some properties of the flow. Here, we explore the characteristics of turbulent fluctuations in the velocity and temperature fields in a cylindrical convection cell of aspect ratio 0.1 by varying the Prandtl number $Pr$ between 0.1 and 200 at a fixed Rayleigh number $Ra = 3 \times 10^{10}$, and find that the fluctuations weaken with increasing $Pr$, quantitatively as in aspect ratio 25. The probability density function (PDF) of temperature fluctuations in the bulk region of the slender cell remains mostly Gaussian, but increasing departures occur as $Pr$ increases beyond unity. We assess the intermittency of the velocity field by computing the PDFs of velocity derivatives and of the kinetic energy dissipation rate, and find increasing intermittency as $Pr$ decreases. In the bulk region of convection, a common result applicable to the slender cell, large aspect ratio cells, as well as in 2D convection, is that the turbulent Prandtl number decreases as $Pr^{-1/3}$., 13 pages, 16 figures

Scalar dissipation rate plays an important role in combustion, especially for fast chemistry in non-premixed flames. Even for finite-rate chemistry, it is a parameter of major importance in models ...

The physical complexity and the large number of degrees of freedom that can be resolved today by direct numerical simulations of turbulent flows, and by the most sophisticated experimental techniqu...

Significance The heat transport law in turbulent convection remains central to current research in the field. Our present knowledge of the heat transport law for Ra>1012 is inconclusive, where the Rayleigh number Ra is a measure of the strength of convection. Massively parallel simulations of the three-dimensional convection have progressed to Ra=1015 in slender cells. We resolve velocity gradients inside thin boundary layers and show that the turbulent heat transport continues to follow the classical 1/3 scaling law with no transition to the so-called “ultimate” state that is variously argued to have 1/2 scaling. Our work suggests that the boundary layers remain marginally stable and continue to act as the bottleneck for global heat transport., The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh number Ra—the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells for Ra≳1012 have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh–Bénard convection flows in a slender cylindrical cell of aspect ratio 1/10, that the Nusselt number—the dimensionless measure of heat transport—follows the classical power law of Nu=(0.0525±0.006)×Ra0.331±0.002 up to Ra=1015. Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at all Ra considered here, increasing with increasing Ra, and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

No abstract available.

Auroras are emissions in a planetary atmosphere caused by its interactions with the surrounding plasma environment. They have been observed in most planets and some moons of the solar system. Since their first discovery in 2005, Mars auroras have been studied extensively and is now a rapidly growing area of research. Since Mars lacks an intrinsic global magnetic field, its crustal field is distributed throughout the planet and its interactions with the surrounding plasma environment lead to a number of complex processes resulting in several types of auroras uncommon on Earth. Martian auroras have been classified as diffuse, discrete and proton aurora. With new capability of synoptic observations made possible with the Hope probe, two new types of auroras have been observed. One of them, which occurs on a much larger spatial scale, covering much of the disk, is known as discrete sinuous aurora. The other subcategory is one of proton auroras observed in patches. Further study of these phenomena will provide insights into the interactions between the atmosphere, magnetosphere and the surrounding plasma environment of Mars. We provide a brief review of the work done on the subject in the past 17 years since their discovery, and report new developments based on observations with Hope probe., Comment: 14 pages, 7 figures

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

Turbulent flows contain occasional, well-separated sharp features, as illustrated in this picture by Bonn et al. (1993), reproduced with permission, surrounded mostly by low levels of activity. Both interacting structures contribute to the complexity of turbulence. A full theory requires the understanding of both, which is what our paper attempts to do. We predict quantitatively the multi-scaling exponents for these singular-like features of different strengths, while also showing that the background flow is Gaussian. The large spatial separation between the sharp structures enables one to treat them as a ``weakly interacting gas''.

An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called Area Rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the Area Rule, for both planar and non-planar loops, using high-resolution data from Direct Numerical Simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is further from unity, and when the moment-order increases. The differences do not exceed about $5\%$ for any condition examined here. The aspect-ratio dependence observed for the second-order moment are indistinguishable from results for the Gaussian Random Field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller $(< 2\%)$ but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence., 7 pages, 6 figures

Aim: To report the surgical technique and early outcomes of robot-assisted ureterocalicostomy (RAUC) with near-infrared fluorescence (NIRF) in the management of secondary pelvi-ureteric junction obstruction (PUJO) in adults. Materials and Methods: The data of 6 patients who underwent RAUC using Da Vinci Xi system with NIRF technology between November 2017 and April 2019 were retrospectively reviewed. Indication was secondary PUJO due to previous failed pyeloplasty. The outcome variables evaluated include operative time (total and console), length of hospital stay, blood loss, and complications. Success was defined as the absence of pain and radionuclide scan demonstrating nonobstructive drainage and improvement/stabilization of split function of the renal moiety. Results: The mean (range) patient age was 33.7 (18-41) years. Total mean (range) operating time was 178 (140-240) minutes and mean console time was 135 minutes. Estimated mean blood loss was 115 (50-200) mL and average hospital stay was 6.1 (5-8) days. There were no conversions to open or laparoscopic surgery. Clavien-Dindo complications occurred in 2 patients. (Grade I-1 and Grade II-1). After a median (range) follow-up of 15 (6-22) months, all 6 patients had successful clinical and radiological outcomes. Conclusion: RAUC is a safe and feasible technique with acceptable outcomes in the setting of secondary PUJO. The utility of NIRF in determining vascularity of the ureter seems to be promising in this setting. Multi-institutional prospective studies with large patient cohort are necessary to validate the role of robotic platform in ureterocalicostomy in the reoperative settings.

Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl numbers, $Pr$, as low as $10^{-6}$, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, to values as low as $10^{-4}$, of the turbulent Prandtl number, $Pr_t$, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying $Pr$ from $O(10)$ to $10^{-4}$ at fixed Grashof number $Gr \equiv Ra/Pr = 10^9$. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean $Pr$ varies in the bulk from 0.07 to $5 \times 10^{-4}$. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of $10^5$, and $0.005 \leq Pr \leq 7$. The principal finding is that $Pr_t$ increases with decreasing $Pr$ in both OB and NOB convection: $Pr_t \sim Pr^{-0.3}$ for OB convection and $Pr_t \sim Pr^{-1}$ for the NOB case. The $Pr_t$-dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence., 17 pages, 11 figures, 3 tables

A simulation tool which utilizes parallel processing is developed to describe molecular kinetics in 2D, single- and multi-component atmospheres on Callisto. This expands on our previous study on the role of collisions in 1D atmospheres on Callisto composed of radiolytic products (Carberry Mogan et al., 2020) by implementing a temperature gradient from noon to midnight across Callisto’s surface and introducing sublimated water vapor. We compare single-species, ballistic and collisional O 2 , H 2 and H 2 O atmospheres, as well as an O 2 +H 2 O atmosphere to 3-species atmospheres which contain H 2 in varying amounts. Because the H 2 O vapor pressure is extremely sensitive to the surface temperatures, the density drops several orders of magnitude with increasing distance from the subsolar point, and the flow transitions from collisional to ballistic accordingly. In an O 2 +H 2 O atmosphere the local temperatures are determined by H 2 O near the subsolar point and transition with increasing distance from the subsolar point to being determined by O 2 . When radiolytically produced H 2 is not negligible in O 2 +H 2 O+H 2 atmospheres, this much lighter molecule, with a scale height roughly an order of magnitude larger than that for the heavier species, can cool the local temperatures via collisions. In addition, if the H 2 component is dense enough, particles originating on the day-side and precipitating into the night-side atmosphere deposit energy via collisions, which in turn heats the local atmosphere relative to the surface temperature. Moreover, the difference between H 2 atmospheric escape rates in single-species and multi-species atmospheres is small: the H 2 only has to diffuse through a few hundred km of the heavier gases before it is the lone species in the atmosphere out to the Hill sphere. Finally, we discuss the potential implications of this study on the presence of H 2 in Callisto’s atmosphere and how the simulated densities correlate with expected detection thresholds at flyby altitudes of the proposed JUpiter ICy moons Explorer (JUICE) spacecraft.

The parameter space for the very uncertain composition of sublimated H2O and its photochemical products H and H2 in Callisto's atmosphere is examined using the Direct Simulaton Monte Carlo (DSMC) method. We focus on two significantly different versions of H2O production in which: (1) the ice and dark, non-ice/ice-poor material are intimately mixed and H2O sublimates at Callisto's warm day-side temperatures (e.g., as in most atmospheric modeling efforts at Callisto to date [1-4]); and (2) the ice and dark, non-ice/ice-poor material are segregated (e.g., consistent with interpretations of images of Callisto's surface taken by Voyager [5, 6] and Galileo [7]) and H2O sublimates at "ice" temperatures [8]. Our 2D molecular kinetic models track the motion H2O, whose sublimation yield varies several orders of magnitude depending on the description of Callisto's surface, its photochemical products H and H2, and a relatively dense O2 component. Whereas H is assumed to react in the regolith on return to the surface, H2 is assumed to thermalize and re-enter the atmosphere. We compare the simulated LOS column densities of H to the detected H corona at Callisto [9], which was suggested to be produced primarily by photodissociation of sublimated H2O. Our goal is to use the corona observations to help constrain the source rate for H2O from Callisto’s complex surface. References [1] Liang et al., 2005. Atmosphere of Callisto. Journal of Geophysical Research: Planets. [2] Vorburger et al., 2015. Monte-Carlo simulation of Callisto’s exosphere. Icarus. [3] Hartkorn et al., 2017. Structure and density of Callisto’s atmosphere from a fluid-kinetic model of its ionosphere: Comparison with Hubble Space Telescope and Galileo observations. Icarus. [4] Carberry Mogan et al., 2021 (under review). A tenuous, collisional atmosphere on Callisto. Icarus. [5] Spencer and Maloney, 1984. Mobility of water ice on Callisto: Evidence and implications. Geophysical Research Letters. [6] Spencer, 1987. Thermal segregation of water ice on the Galilean satellites. Icarus. [7] Moore et al., 1999. Mass movement and landform degradation on the icy Galilean satellites: Results of the Galileo nominal mission. Icarus. [8] Grundy et al., 1999. Near-infrared spectra of icy outer solar system surfaces: Remote determination of H2O ice temperatures. Icarus. [9] Roth et al., 2017. Detection of a hydrogen corona at Callisto. Journal of Geophysical Research: Planets.

We systematically study dissipative anomaly in compressible turbulence using a direct numerical simulations (DNS) database spanning a large parameter space, and show that the classical incompressible scaling does not hold for the total dissipation field. We assess the scaling for the solenoidal and dilatational parts separately. The solenoidal dissipation obeys the same scaling as incompressible turbulence when rescaled on solenoidal variables. We propose new scaling laws for total dissipation that predict the transition between regimes dominated by the solenoidal and dilatational components, and confirm them by the DNS data. An analysis of dilatational dissipation shows that dissipative anomaly may hold if properly scaled for certain regimes; on this empirical basis, we propose a new criterion for the energy cascade in the dilatational component.

Quantum turbulence—the stochastic motion of quantum fluids such as 4 He and 3 He-B, which display pure superfluidity at zero temperature and two-fluid behavior at finite but low temperatures—has been a subject of intense experimental, theoretical, and numerical studies over the last half a century. Yet, there does not exist a satisfactory phenomenological framework that captures the rich variety of experimental observations, physical properties, and characteristic features, at the same level of detail as incompressible turbulence in conventional viscous fluids. Here we present such a phenomenology that captures in simple terms many known features and regimes of quantum turbulence, in both the limit of zero temperature and the temperature range of two-fluid behavior.

Rossby waves are found at several levels in the Sun, most recently in its supergranule layer. We show that Rossby waves in the supergranule layer can be excited by an inverse cascade of kinetic energy from the nearly horizontal motions in supergranules. We illustrate how this excitation occurs using a hydrodynamic shallow-water model for a 3D thin rotating spherical shell. We find that initial kinetic energy at small spatial scales inverse cascades quickly to global scales, exciting Rossby waves whose phase velocities are similar to linear Rossby waves on the sphere originally derived by Haurwitz. Modest departures from the Haurwitz formula originate from nonlinear finite amplitude effects and/or the presence of differential rotation. Like supergranules, the initial small-scale motions in our model contain very little vorticity compared to their horizontal divergence, but the resulting Rossby waves are almost all vortical motions. Supergranule kinetic energy could have mainly gone into gravity waves, but we find that most energy inverse cascades to global Rossby waves. Since kinetic energy in supergranules is three or four orders of magnitude larger than that of the observed Rossby waves in the supergranule layer, there is plenty of energy available to drive the inverse-cascade mechanism. Tachocline Rossby waves have previously been shown to play crucial roles in causing seasons of space weather through their nonlinear interactions with global flows and magnetic fields. We briefly discuss how various Rossby waves in the tachocline, convection zone, supergranule layer, and corona can be reconciled in a unified framework.

Normal-mode coupling is a technique applied to probe the solar interior using surface observations of oscillations. The technique, which is straightforward to implement, makes more use of the seismic information in the wavefield than other comparable local imaging techniques and therefore has the potential to significantly improve current capabilities. Here, we examine supergranulation power spectra using mode-coupling analyses of intermediate-to-high-degree modes by invoking a Cartesian-geometric description of wave propagation under the assumption that the localized patches are much smaller in size than the solar radius. We extract the supergranular power spectrum and compare the results with prior helioseismic studies. Measurements of the dispersion relation and life times of supergranulation, obtained using near surface modes (f and p$_1$), are in accord with the literature. We show that the cross-coupling between the p$_2$ and p$_3$ acoustic modes, which are capable of probing greater depths, are also sensitive to supergranulation., 7 pages, 3 figures, Accepted into ApJ

Inspection of available data on the decay exponent for the kinetic energy of homogeneous and isotropic turbulence (HIT) shows that it varies by as much as 100\%. Measurements and simulations often show no correspondence with theoretical arguments, which are themselves varied. This situation is unsatisfactory given that HIT is a building block of turbulence theory and modeling. We take recourse to a large base of direct numerical simulations and study decaying HIT for a variety of initial conditions. We show that the Kolmogorov decay exponent and the Birkhoff-Saffman decay are both readily observed, albeit approximately, for long periods of time if the initial conditions are appropriately arranged. We also present, for both cases, other turbulent statistics such as the velocity derivative skewness, energy spectra and dissipation, and show that the decay and growth laws are approximately as expected theoretically, though the wavenumber spectrum near the origin begins to change relatively quickly, suggesting that the invariants do not strictly exist. We comment briefly on why the decay exponent has varied so widely in past experiments and simulations., Comment: 14 pages, 11 figures

Studies of strongly nonlinear dynamical systems such as turbulent flows call for superior computational prowess. With the advent of quantum computing, a plethora of quantum algorithms have demonstrated, both theoretically and experimentally, more powerful computational possibilities than their classical counterparts. Starting with a brief introduction to quantum computing, we will distill a few key tools and algorithms from the huge spectrum of methods available, and evaluate possible approaches of quantum computing in fluid dynamics., Comment: 20 pages, 17 figures, 2 Tables, To Appear in Pramana-Journal of Physics - Springer

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

In their comment on Iyer et al. (1), He et al. (2) appear to address two putative issues: 1) that Iyer et al. neglect to point out that the small aspect ratio of their simulation could have an impact on the outcome; 2) He et al. (2) use their own data on the Nusselt–Rayleigh scaling to attempt to quantify its dependence on the aspect ratio Γ = d / H , where d is the diameter and H is height of the convection cell. On issue 1, we point out that, in the Abstract, Introduction, and final summary, Iyer et al. (1) have stated that their aspect ratio of Γ … [↵][1]1To whom correspondence may be addressed. Email: katepalli.sreenivasan{at}nyu.edu. [1]: #xref-corresp-1-1

The Sun, Earth's star, is of fundamental interest for life on our planet and remains a source of many scientific mysteries. The motion in its interior is complex and involves diverse physical phenomena at many scales, from nuclear to astronomical. In this Colloquium the unusual flow of mass and energy inside the convective region of our yellow star is discussed.

As helioseismology matures and turns into a precision science, modeling finite-frequency, geometric and systematical effects is becoming increasingly important. Here we introduce a general formulation for treating perturbations of arbitrary tensor rank in spherical geometry using fundamental ideas of quantum mechanics and their extensions in geophysics. We include line-of-sight projections and center-to-limb differences in line-formation heights in our analysis. We demonstrate the technique by computing a travel-time sensitivity kernel for sound-speed perturbations. The analysis produces the spherical harmonic coefficients of the sensitivity kernels, which leads to better-posed and computationally efficient inverse problems.

Helioseismology reveals that turbulence in the Sun is suppressed on large scales and confined to equatorial regions., Convection in the Sun’s outer envelope generates turbulence and drives differential rotation, meridional circulation, and the global magnetic cycle. We develop a greater understanding of these processes by contrasting observations with simulations of global convection. These comparisons also enhance our comprehension of the physics of distant Sun-like stars. Here, we infer toroidal flow power as a function of wave number, frequency, and depth in the solar interior through helioseismic analyses of space-based observations. The inferred flows grow with spatial wave number and temporal frequency and are confined to low latitudes, supporting the argument that rotation induces systematic differences between the poles and equator. In contrast, the simulations used here show the opposite trends—power diminishing with increasing wave number and frequency while flow amplitudes become weakest at low latitudes. These differences highlight gaps in our understanding of solar convection and point to challenges ahead.

Context.Many moons have been detected around planets in our Solar System, but none has been detected unambiguously around any of the confirmed extrasolar planets.Aims.We test the feasibility of a supervised convolutional neural network to classify photometric transit light curves of planet-host stars and identify exomoon transits, while avoiding false positives caused by stellar variability or instrumental noise.Methods.Convolutional neural networks are known to have contributed to improving the accuracy of classification tasks. The network optimization is typically performed without studying the effect of noise on the training process. Here we design and optimize a 1D convolutional neural network to classify photometric transit light curves. We regularize the network by the total variation loss in order to remove unwanted variations in the data features.Results.Using numerical experiments, we demonstrate the benefits of our network, which produces results comparable to or better than the standard network solutions. Most importantly, our network clearly outperforms a classical method used in exoplanet science to identify moon-like signals. Thus the proposed network is a promising approach for analyzing real transit light curves in the future.

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution of up to $8192^3$ at high P\'eclet numbers to understand this anomaly as the scalar diffusivity, $D$, diminishes, or as the Schmidt number, $Sc = \nu/D$, increases; here $\nu$ is the kinematic viscosity of the fluid. The microscale Reynolds number varies from 140 to 650 and $Sc$ varies from 1 to 512. A simple model for the ramp-cliff structures is shown to characterize the scalar derivative statistics extremely well. It accurately captures how the small-scale isotropy is restored in the large-$Sc$ limit, and additionally suggests a slight correction to the Batchelor length scale as the relevant smallest scale in the scalar field., Comment: 5 pages, 8 figures