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DOE OSTI2022

Comment on Comment on “Anomalous structural recovery in the near glass transition range in a polymer glass: Data revisited in light of temperature variability in vacuum oven‐based experiments”*

Abstract Cangialosi, Alegría, and Colmenero have made a comment on a paper of ours [Polym. Eng. Sci. 2022:1–13], in which we discussed the concern that the enthalpy recovery data reported by Cangialosi and co‐workers [Phys. Rev. Lett. 2013;111(9):095701] for polystyrene aged up to 15 K below glass transition temperature was anomalous and contradicted existing experimental results from the literature over a similar range of aging conditions. Their response shifts the focus away from the raised questions about their experimental results and attempts to invalidate the data that we cited in support of our argument. Here we respond to the comment and add additional analysis that suggests the structural recovery response of glassy materials exhibits smooth behavior over the full range of measurements, up to 7 or 8 logarithmic decades. We do this by referring to the work on the intrinsic isotherm down‐jump and memory responses of poly(vinyl acetate) between 40°C and 15°C over six logarithmic decades by Kovacs [Fortsch. Hochpolym. Fo. 1963;3(1/2):394–508], the small‐strain tensile creep of poly(vinyl chloride) quenched from 90°C to 40°C (approximately 40°C below T g ) over seven logarithmic decades from Struik [Polym Eng. Sci., 1977;17:165–173], and the volume recovery behavior for aging times up to 3 months in a temperature range between 95°C and −50°C by Greiner and Schwarzl [Rheol. Acta. 1984;23(4):378–395]. We also add discussion that an isothermal aging procedure using a vacuum oven is highly vulnerable to temperature errors due to the problem of good temperature control when the heat transfer mechanism is primarily radiative.

Jin, Shuang↗

DOE OSTI2025

Spinning black hole scattering at $ \mathcal{O} $(G 3 S 2 )(G 3 S 2 ): Casimir terms, radial action and hidden symmetry

We resolve subtleties in calculating the post-Minksowskian dynamics of binary systems, as a spin expansion, from massive scattering amplitudes of fixed finite spin. In particular, the apparently ambiguous spin Casimir terms can be fully determined from the gradient of the spin-diagonal part of the amplitudes with respect to S 2 = −s(s+1)ħ 2 , using an interpolation between massive amplitudes with different spin representations. From two-loop amplitudes of spin-0 and spin-1 particles minimally coupled to gravity, we extract the spin Casimir terms in the conservative scattering angle between a spinless and a spinning black hole at $ \mathcal{O} $(G 3 S 2 ), finding agreement with known results in the literature. This completes an earlier study [Phys. Rev. Lett. 130 (2023), 021601] that calculated the non-Casimir terms from amplitudes. We also illustrate our methods using a model of spinning bodies in electrodynamics, finding agreement between scattering amplitude predictions and classical predictions in a root-Kerr electromagnetic background up to $ \mathcal{O} $(α 3 S 2 ). For both gravity and electrodynamics, the finite part of the amplitude coincides with the two-body radial action in the aligned spin limit, generalizing the amplitude-action relation beyond the spinless case. Surprisingly, the two-loop amplitude displays a hidden spin-shift symmetry in the probe limit, which was previously observed at one loop. We conjecture that the symmetry holds to all orders in the coupling constant and is a consequence of integrability of Kerr orbits in the probe limit at the first few orders in spin.

Classical Theories of Gravity↗

DOE OSTI2022

Type II Seesaw leptogenesis

The Type II Seesaw Mechanism provides a minimal framework to explain the neutrino masses involving the introduction of a single triplet Higgs to the Standard Model. However, this simple extension was believed to be unable to successfully explain the observed baryon asymmetry of the universe through Leptogenesis. In our previous work (Phys. Rev. Lett. 128 (2022) 141801), we demonstrated that the triplet Higgs of the Type II Seesaw Mechanism alone can simultaneously generate the observed baryon asymmetry of the universe and the neutrino masses while playing a role in setting up Inflation. This is achievable with a triplet Higgs mass as low as 1 TeV, and predicts that the neutral component obtains a small vacuum expectation value $v_Δ$ < 10 keV. We find that our model has very rich phenomenology and can be tested by various terrestrial experiments as well as by astronomical observations. Particularly, we show that the successful parameter region may be probed at a future 100 TeV collider, upcoming lepton flavor violation experiments such as Mu3e, and neutrinoless double beta decay experiments. Additionally, the tensor-to-scalar ratio from the inflationary scenario will be probed by the LiteBIRD telescope, and observable isocurvature perturbations may be produced for some parameter choices. In this article, we present all the technical details of our calculations and further discussion of its phenomenological implications.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Continuous and discrete modeling of HIV-1 decline on therapy

Mathematical models have shed light on the dynamics of HIV- 1 infection in vivo. In this paper, we generalize continuous mathematical models of drug therapy for HIV-1 by Perelson et al. (Science 271:1582–1586, 1996) and Perelson and Nelson (SIAM Rev 41:3–44, 1999) on time scales, i.e., a nonempty closed subset of real numbers in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. Here, one of our main goals is to compare discrete mathematical models with the continuous model in Perelson et al. (1996) where HIV infected patients were given protease inhibitors and sampled frequently thereafter. For the comparison, we use experimental data collected in Perelson et al. (1996) and estimate the parameters such as the virion clearance rate and the rate of loss of infected cells by fitting the total concentration of plasma virus to this data set. Our results show that discrete systems describe the best fit. In the previous models of this study, the efficacy of protease inhibitor is assumed to be perfect. Motivated by Perelson and Nelson (1999), we end the paper with a mathematical model of imperfect protease inhibitor and reverse transcriptase (RT) inhibitor combination therapy of HIV-1 infection on time scales with its stability analysis.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2022

Devices for Thermal Conductivity Measurements of Electroplated Bi for X-ray TES Absorbers

Electroplated bismuth (Bi) is commonly used in transition-edge sensors (TESs) for X-rays because of its high stopping power and low heat capacity (Collan in Phys Rev B 1:2888, 1970, Yan in Appl Phys Lett 111:192602, 2017). Electroplated Bi is usually grown on top of another metal that acts as seed layer, typically gold (Au), making it challenging to extrapolate its thermoelectric properties. In this work, we present four-wire resistance measurement structures that allow us to measure resistance as a function of temperature of electroplated Bi independently of Au. The results show that the thermal conductivity of the Bi at 3 K is high enough to ensure the correct thermalization of X-ray photons when used as an absorber for TESs.

Bismuth↗

DOE OSTI2023

On the question of quark confinement in the Abelian U(1) QED gauge interaction

If we approximate light quarks as massless and apply the Schwinger confinement mechanism to light quarks, we will reach the conclusion that a light quark q and its antiquark ${\bar q}$ will be confined as a $q\bar q$ boson in the Abelian U(1) QED gauge interaction in (1+1)D, as in an open string. From the work of Coleman, Jackiw, and Susskind, we can infer further that the Schwinger confinement mechanism persists even for massive quarks in (1+ 1)D. Could such a QED-confined $q\bar q$ one-dimensional open string in (1 + 1) D be the idealization of a flux tube in the physical world in (3+l)D, similar to the case of QCD-confined $q\bar q$ open string? If so, the QED-confined $q\bar q$ bosons may show up as neutral QED mesons in the mass region of many tens of MeV [Phys. Rev. C 81, 064903 (2010) & J. High Energy Phys. 2020(8), 165 (2020)]. Is it ever possible that a quark and an antiquark be produced and interact in QED alone to form a confined QED meson? Is there any experimental evidence for the existence of a QED meson (or QED mesons)? The observations of the anomalous soft photons, the XI7 particle, and the E38 particle suggest that they may bear the experimental evidence for the existence of such QED mesons. Further confirmation and investigations on the XI7 and E38 particles will shed definitive light on the question of quark confinement in QED in (3+1)D. Finally, implications of quark confinement in the QED interaction are discussed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Local transport dynamics of cold pulses in tokamak plasmas

Abstract For over two decades, our fundamental understanding of energy transport dynamics in the core of tokamak plasmas had been challenged by the striking observation of temperature perturbation reversals following the injection of cold pulses at the plasma edge. These phenomena were first discovered by Gentle et al. (Phys. Rev. Lett. 74(18):3620–3623, 1995) in 1995 and had long been suggested to be evidence of nonlocal transport effects. In recent years, a new explanation to these phenomena has emerged, fully consistent with the theory of turbulent transport in magnetized plasmas and in remarkable agreement with experiment. This article reviews the experimental observation of temperature reversals in tokamak plasmas and presents the explanation based on local transport physics.

Rodriguez-Fernandez, P. (ORCID:0000000273611131)↗

DOE OSTI2022

Simulation of multi-shell fullerenes using Machine-Learning Gaussian Approximation Potential

Multi-shell fullerenes ”buckyonions ” were simulated, starting from initially random configurations, using a density-functional-theory (DFT)-trained machine-learning carbon potential within the Gaussian Approximation Potential (GAP) Framework [Volker L. Deringer and Gábor Csányi, Phys. Rev. B 95, 094203 (2017)]. Fullerenes formed from seven different system sizes, ranging from 60 ~ 3774 atoms, were considered. The buckyonions are formed by clustering and layering starting from the outermost shell and proceeding inward. Inter-shell cohesion is partly due to interaction between delocalized π electrons protruding into the gallery. The energies of the models were validated ex post facto using density functional codes, VASP and SIESTA , revealing an energy difference within the range of 0.02 - 0.08 eV/atom after conjugate gradient energy convergence of the models was achieved with both methods.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2020

Tracer particle transport dynamics in the diffusive sandpile cellular automaton

The confinement properties of the diffusive running sandpile are characterized by tracking the motion of a population of marked grains of sand. It is found that, as the relative strength of the avalanching to the diffusive transport channel is varied, a point is reached at which the particle global confinement time and the probability density functions of the jump-sizes and waiting-times of the tracked grains experience a sudden change, thus revealing a dynamical transition, that is consistent with previous studies (Newman DE et al., Phys Rev Lett 2002;88(20):204304). Across this transition, the sandpile moves from a regime characterized by self-similarity and memory, where avalanches of all possible sizes dominate transport across the system, to another regime where transport is taken over by near system-size, quasi-periodic avalanches. Values for the fractional transport exponents that quantify effective transport across the sandpile prior to the transition are also obtained.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Influence of gradation on shear behavior and pore water response of sand

Here, this paper examines the influence of particle size distribution (PSD) on the shearing behavior of sand using axisymmetric triaxial compression (ATC) testing in conjunction with 3D imaging using synchrotron micro-computed tomography (SMT). Angular sand was sieved to obtain a narrow gradation (uniform) and a wide gradation (non-uniform) with uniformity coefficients (C u ) of 1.2 and 3.0, respectively. Experiments were conducted under drained conditions and a constant effective cell pressure (σ$^{'}_3$) of 50 kPa at low and high back pressures (BP). Results of paired ATC-SMT were analyzed in terms of global constitutive behavior, strain localization, and localized flow characteristics. A new particle tracking algorithm is developed to expose strain localization. Relative particle translation gradient (RPTG) results revealed that gradation affects how early the shear band develops and a tendency for narrow gradation to exhibit varying degrees of diffused shear band at high axial strain (ε 1 ) even though a clear single shear band was observed at the peak strength. Geometric tortuosity (τ) is used as a quantitative parameter to assess flow changes and the efficiency of the flow path. Representative elementary volumes (REVs) inside and outside the shear band were analyzed and the results showed a variation in flow path characteristics where wide gradation specimens had preferential flow paths.

Computed tomography↗

DOE OSTI2023

Pore-scale simulation of multiphase flow and reactive transport processes involved in geologic carbon sequestration

Multiphase flow and reactive transport are two essential physicochemical processes that govern the effectiveness of geological carbon sequestration (GCS). The interaction and feedback among different phases and components during intricate physicochemical processes hold great significance in understanding CO 2 sequestration. Pore-scale simulations can account for multiphase flow and reactive transport processes in porous media and obtain spatial distributions of parameters (density, velocity, concentration, etc.) in the pore space as well as their temporal evolutions. This proves especially valuable considering that experiments can be hindered by constraints in spatial and temporal resolution. The comprehensive insights garnered from pore-scale research can be leveraged for continuum modeling using the representative elementary volume (REV) concept. In this contribution, four sequential mechanisms of CO 2 -brine-rock interaction in three zones delineated by CO 2 saturation are elaborated to elucidate complicated physicochemical processes involved in GCS, which are followed by general descriptions of mathematical equations and pore-scale numerical methods. In addition, as interested and commonly encountered processes, leakage risks during GCS and CO 2 -enhanced oil recovery (CO 2 -EOR) processes are presented. The existing challenges and future directions are discussed for both the performance of the pore-scale models and the current gaps in the field of GCS. Importantly, we expect that this review will prove beneficial for researchers interested in pore-scale simulations, GCS, and related disciplines.

58 GEOSCIENCES↗

DOE OSTI2025

"Hidden" hydrothermal technical potential & technoeconomics: Revealing permeability & fluids with more data

Historical hydrothermal estimates have largely relied on temperature or heat flow estimates ignoring the need for natural flowing fluids. More accurate hydrothermal estimates require some indication of permeability and fluids that naturally exist in the subsurface. This paper describes a novel approach that includes proxies of permeability and fluids in hydrothermal estimates by leveraging the relatively data-rich Great Basin. Specifically, nameplate capacities (megawatts) of operating geothermal plants, negative (0 megawatt) locations and 48 geophysical and geologic features are used to used in eXtreme Gradient Boosting (XGBoost) regression to make hydrothermal capacity predictions. Additionally, this work inputs the XGBoost-based hydrothermal predictions into the Renewable Energy Potential (reV) model to quantify technical capacity, its uncertainty and techno-economics. Compared to historical hydrothermal estimates, these predictions adhere to the 37 operating geothermal plants and negative locations. We present a method for subsampling the negative sites to bring the labels into balance that uses the geologic domain knowledge to proportionally represent negatives. Overall, the distributions of the hydrothermal technical capacity and the site levelized cost of energy are respectively much tighter, lower and more accurate than the previous estimates for the Great Basin, as they include geological and geophysical surrogates for permeability and fluids. Percentile (50th and 90th, median and high estimate, respectively) models provide bookends for these metrics.

13 HYDRO ENERGY↗

DOE OSTI2023

Assessing pore network heterogeneity across multiple scales to inform CO2 injection models

Geologic heterogeneity is a key feature that must be considered when translations of scaled data are performed. This paper presents the assessment of geologic heterogeneity using a multiscale workflow that includes image analysis-based methods coupled with well log analysis to provide data in which fractals and machine learning methods estimate the carbon dioxide (CO 2 ) storage resource potential of a reservoir. The heterogeneity of rock properties of the complex Bell Creek reservoir in Montana, USA, was explored at the pore scale (~nm to mm), core scale (~mm to m), and well scale (~cm to m). The data used in this study included advanced image analysis of micro-CT (computed tomography) images (pore scale), thin sections (pore scale), plugs and core images (core scale) and well logs (well scale). The micro-CT images were segmented using a U-net segmentation approach into objects of pores and grains. Further, the segmented images were reconstructed into subvolumes of different sizes. Physical properties (porosity and permeability) and fractal dimensions were calculated for the various subvolumes, and Lorenz coefficient (Lc) values, a single parameter to describe the degree of heterogeneity within a pay zone section, were calculated from thin-section images and well logs. Porosity and fractal dimension values were used to estimate the 188-µm threshold of representative elementary volume (REV) in this study. Both the Lc and fractal dimension values were found to be negatively correlated. When these two parameters are combined, it is possible to discern differences in the complex porous networks of the samples analyzed in this study.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2023

Distance-dependent dielectric constant at the calcite/electrolyte interface: Implication for surface complexation modeling

Hypothesis: The electrical double layer formed at the mineral/electrolyte interface is often modeled using mean-field approaches based on a continuum description of the solvent whose dielectric constant is assumed to decrease monotonically with decreasing distance to the surface. In contrast, molecular simulations show that the solvent polarizability oscillates near the surface similar to the water density profile - as shown previously, for example, by Bonthuis et al. (D.J. Bonthuis, S. Gekle, R.R. Netz, Dielectric Profile of Interfacial Water and its Effect on Double-Layer Capacitance, Phys Rev Lett 107(16) (2011) 166102). Here, we showed that molecular and mesoscale pictures agree by spatially averaging the dielectric constant obtained from molecular dynamics simulations over the distances relevant to the mean-field representation. In addition, the values of capacitances used to describe the electrical double layer in Surface Complexation Models (SCMs) of the mineral/electrolyte interface can be estimated using molecularly informed spatially averaged dielectric constants and positions of hydration layers. Experiments: First, we used molecular dynamics simulations to model the calcite 101¯4/electrolyte interface. Next, by using atomistic trajectories, we calculated the distance-dependent static dielectric constant and water density in the direction normal to the. Finally, we applied spatial compartmentalization consistent with the model of parallel-plate capacitors connected in series to estimate SCM capacitances. Findings: Computationally expensive simulations are required to determine the dielectric constant profile of interfacial water near the mineral surface. On the other hand, water density profiles are readily assessable from much shorter simulation trajectories. Our simulations confirmed that dielectric and water density oscillations at the interface are correlated. Here, we parametrized linear regression models to estimate the dielectric constant directly from the local water density. This is a significant computational shortcut compared to slowly converging calculations relying on total dipole moment fluctuations. The amplitude of the interfacial dielectric constant oscillation can exceed the dielectric constant of the bulk water, suggesting an ice-like frozen state, but only if there are no electrolyte ions. The interfacial accumulation of electrolyte ions causes a decrease in the dielectric constant due to the reduction of water density and re-orientation of water dipoles in ion hydration shells. Finally, we show how to use the computed dielectric properties to estimate SCM's capacitances.

58 GEOSCIENCES↗

DOE OSTI2026

Analysis of attenuation data from the decommissioned ZIon unit 1 reactor pressure vessel beltline weld

In order to examine the attenuation of radiation damage through the thickness of an irradiated reactor pressure vessel (RPV), four segments were acquired from the Zion Unit 1 power plant RPV after the plant was decommissioned. The Zion Unit 1 RPV Beltline Weld Segment 1 was cut into seven blocks, consisting of five base metal and two beltline welds from the high fluence region of the segment. Through-wall test specimens were machined and tested. Specimens included those used for Charpy impact, Master Curve fracture toughness testing, and chemical analysis. The observed through-thickness ductile-to-brittle transition temperatures in the beltline weld deviated significantly from the expected behavior based on the attenuation of fast fluence as a function of depth into the RPV. Beginning at the inside surface, the 41-J Charpy transition temperature was either flat or slightly increasing until the ¾ -T location. The results of a simple, model-based analysis of the Zion beltline weld material that included the irradiation conditions and material chemistry were generally consistent with industry trend curves and the standard attenuation model, rather than the observed data. Although there was no archive material from the RPV available to permit measurement of the unirradiated properties, fracture toughness specimens fabricated from archive surveillance weld were used to obtain an estimate of the initial through-thickness values of the Charpy transition temperature. The Charpy shifts obtained using this approach were similarly in disagreement with the predictions of the US NRC Regulatory Guide 1.99, Rev. 2. However, testing of irradiated Charpy specimens taken from the RPV following post-irradiation annealing (10 hr. at 500 °C) provided a quite different estimate of the unirradiated properties which improved the agreement between the inferred through-thickness Charpy shifts and exponential attenuation model included in Regulatory Guide 1.99/2. In conclusion, the analysis of the Zion data and data obtained in previous post-mortem examinations of decommissioned RPVs indicates that more work is needed to understand the through-thickness properties of RPV materials in order to properly assess through-wall damage attenuation.

Charpy impact↗

DOE OSTI2026

Development and assessment of models for turbulent Rayleigh-Taylor mixing using the macroscopic forcing method

Reynolds-Averaged Navier Stokes (RANS) simulations are a popular method for designing ICF experiments, and accurate mixing models are crucial for these simulations to give good predictions. To this end, the present work seeks to demonstrate the Macroscopic Forcing Method (MFM) as a tool for both improving existing RANS models as well as assessing RANS model forms. First, MFM analysis from Lavacot et al. (Phys. Rev. Fluids, 2025) is used to develop the k–L–F model, an extension of the k–L model of Dimonte and Tipton (Phys. Fluids, 2006) that incorporates nonlocality through addition of a turbulent species flux transport equation. MFM is then applied to the k–L–F model along with the k–L and BHR–4 models to assess their forms and compare the model-implied eddy diffusivity moments to those measured from high-fidelity simulations. Furthermore, the analysis reveals that models incorporating nonlocality (k–L–F and BHR–4) match the high-fidelity simulation data better than purely local models (k–L), both in terms of mean fields and eddy diffusivity moments. However, all of the considered RANS models struggle to match temporal moments at high Atwood numbers, highlighting the importance of temporal nonlocality in these regimes and the need for additional improvement even among models incorporating nonlocality.

general physics↗

DOE OSTI2025

Melting down a tetraquark: DD() interactions and T c c ( 3875 ) + in a hot environment

We discuss the modification of the properties of the tetraquark-like and its heavy quark spin partner, immersed in a hot bath of pions. We consider these exotic states as purely isoscalar DD* and D*D* S-wave bound states, respectively. Finite temperature effects are incorporated through the D and D* state-of-the-art thermal spectral functions calculated in [G. Montana et al., Phys. Rev. D, 102 (2020) 096020], using the imaginary-time formalism. We find important modifications of the DD* and D*D* scattering amplitudes already for MeV, and show that the hot-bath lineshapes of these tetraquark-like states strongly depend on their Weinberg molecular content. We find that the thermal and spectral functions change more rapidly with temperature for high molecular probabilities P0. For large values of P0, the widths significantly increase with temperature, leading to the melting of these exotic states for temperatures larger than 80 MeV. For small molecular components, the changes in the spectral functions of these states due to temperature become significantly less important. All these results show that any future experimental determination of the D(*)D* scattering amplitudes at finite temperature will provide valuable insights into the molecular content of the and exotics.

Montesinos, V. (ORCID:0000000261862777)↗

DOE OSTI2021

Boundary zonal flows in rapidly rotating turbulent thermal convection

Recently, in Zhang et al. ( Phys. Rev. Lett. , vol. 124, 2020, 084505), it was found that, in rapidly rotating turbulent Rayleigh–Bénard convection in slender cylindrical containers (with diameter-to-height aspect ratio $\varGamma =1/2$ ) filled with a small-Prandtl-number fluid ( ${Pr}\approx 0.8$ ), the large-scale circulation is suppressed and a boundary zonal flow (BZF) develops near the sidewall, characterized by a bimodal probability density function of the temperature, cyclonic fluid motion and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount ( ${>}60\,\%$ ) of the total heat transport for ${Pr} < 1$ , but decreases rather abruptly for larger ${Pr}$ to approximately $35\,\%$ . In this work, we show that the BZF is robust and appears in rapidly rotating turbulent Rayleigh–Bénard convection in containers of different $\varGamma$ and over a broad range of ${Pr}$ and ${Ra}$ . Furthermore, tirect numerical simulations for Prandtl number $0.1 \leq {\textit {Pr}} \leq 12.3$ , Rayleigh number $10^7 \leq {Ra} \leq 5\times 10^{9}$ , inverse Ekman number $10^{5} \leq 1/{\textit {Ek}} \leq 10^{7}$ and $\varGamma = 1/3$ , 1/2, 3/4, 1 and 2 show that the BZF width $\delta _0$ scales with the Rayleigh number ${Ra}$ and Ekman number ${\textit {Ek}}$ as $\delta _0/H \sim \varGamma ^{0} Pr^{\{-1/4, 0\}} {Ra}^{1/4} {\textit {Ek}}^{2/3}$ ( $\{{\textit {Pr}}<1, {\textit {Pr}}>1\}$ ) and with the drift frequency scales as $\omega /\varOmega \sim \varGamma ^{0} Pr^{-4/3} {Ra}\,{\textit {Ek}}^{5/3}$ , where $H$ is the cell height and $\varOmega$ the angular rotation rate. The mode number of the BZF is 1 for $\varGamma \lesssim 1$ and $2 \varGamma$ for $\varGamma = \{1,2\}$ independent of ${Ra}$ and ${Pr}$ . The BZF is quite reminiscent of wall mode states in rotating convection.

42 ENGINEERING↗