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DOE OSTI2021

Machine learning assisted modeling of mixing timescale for LES/PDF of high-Karlovitz turbulent premixed combustion

Accurate modeling of mixing in the transported probability density function (PDF) method remains a great challenge, especially for turbulent premixed combustion under extreme conditions such as high Karlovitz number Ka. Recently, a power-law based mixing timescale model was developed for the large-eddy simulations (LES)/PDF modeling of high-Ka number turbulent premixed flames. It is found in this work that the power-law mixing timescale model is highly sensitive to the model parameters. It is thus critically needed to develop accurate calibration of these model parameters. The empirical specification of the model parameters developed in Zhang et. al. is found to be inadequate for accurate modeling of the mixing timescale. Here, machine learning is introduced as an attractive alternative in this work for the specification of the model parameters. A high-Ka number DNS jet flame is used as the training and validation of the machine learning models. The choices of the input parameters are discussed and compared for the machine learning models. The effect of differential molecular diffusion on mixing is examined by including the effect of the Lewis number in the training of the machine learning models. The performance of different machine learning algorithms is compared for the specification of the mixing model parameters. Overall, excellent performance of the machine learning models is observed for assisting the mixing modeling. The feasibility, interpretability, applicability, generality, and portability of using machine learning are discussed in general to provide a perspective on applying data-driven machine learning for turbulent combustion modeling studies.

42 ENGINEERING↗

DOE OSTI2024

Numerical Mixing Suppresses Submesoscale Baroclinic Instabilities Over Sloping Bathymetry

Abstract The impacts of spurious numerical salinity mixing on the larger‐scale flow and tracer fields are characterized using idealized simulations. The idealized model is motivated by realistic simulations of the Texas‐Louisiana shelf and features oscillatory near‐inertial wind forcing. can exceed the physical mixing from the turbulence closure in frontal zones and within the mixed layer. This suggests that simulated mixing processes in frontal zones are driven largely by . Near‐inertial alongshore wind stress amplitude is varied to identify a base case that maximizes the ratio of to in simulations with no prescribed horizontal mixing. We then test the sensitivity of the base case with three tracer advection schemes (MPDATA, U3HC4, and HSIMT) and conduct ensemble runs with perturbed bathymetry. Instability growth is evaluated using the volume‐integrated eddy kinetic energy and available potential energy . While all schemes have similar total mixing, the HSIMT simulations have over double the volume‐integrated and 20% less relative to other schemes, which suppresses the release of and reduces the by roughly 25%. This results in reduced isohaline variability and steeper isopycnals, evidence that enhanced suppresses instability growth. Differences in and between the MPDATA and U3HC4 simulations are marginal. However, the U3HC4 simulations have 25% more . Experiments with variable horizontal viscosity and diffusivity coefficients show that small amounts of prescribed horizontal mixing improve the representation of the ocean state for all advection schemes by reducing the and increasing the .

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Transient mixing behavior of a supercritical fluid injected into supercritical and subcritical environments

Transient, near-field mixing of a supercritical fluid injected into supercritical and subcritical environments is investigated experimentally. Fuel (propane) from a 100 μm diameter Bosch diesel injector is injected into a high-pressure test rig with a steady flow of heated, low-velocity nitrogen to control the ambient pressure and temperature. The fuel supply pressure is approximately 190 bar, allowing short injections lasting 3–4 ms to be performed in quick succession to acquire statistically significant datasets to represent the unsteady fuel–air mixing. Two test cases with the same ambient density and injector inlet pressure and temperature were chosen in an attempt to isolate effects of mixture phase behavior. Microscopic rainbow Schlieren deflectometry (RSD) is used for visualization and quantitative data acquisition. In the case of the supercritical ambient, the injector inlet and exit and ambient conditions are all supercritical. The RSD analysis reveals a mixing process similar to the single phase fluid mixing, though condensation is also observed locally, and can be explained by the adiabatic mixing theory presented in the paper. For the subcritical ambient case, the injector inlet is supercritical, but the injector exit and ambient conditions are both subcritical. The results show a rapid expansion in the near-field promoting mixing between propane and nitrogen. At downstream locations, significant condensation is also observed as predicted by the adiabatic mixing theory. As a result, the supercritical ambient case exhibited longer and narrower jet penetration compared to the subcritical ambient case.

02 PETROLEUM↗

DOE OSTI2025

Mixed-valence halide perovskites

Mixed-valence compounds—which feature an element in at least two different oxidation states—can display emergent optical and transport phenomena stemming from electron transfer between the different valences (intervalence charge-transfer; IVCT). As halide perovskites show promise as active materials in numerous optoelectronic devices, it is an opportune moment to incorporate and study the effects of mixed-valence in this versatile materials family, to access tunable electronic structures ranging from insulators to semiconductors to metals. Herein, we introduce the basic concepts of mixed-valence in molecules and discuss how these concepts may be extended to mixed-valence in extended solids. We then review the few studies of mixed valence in 3D and 2D halide perovskites and halide perovskites with mixed-valence impurities, ranging from studies in the early 1900s to the present day. In conclusion, through judicious choice of metal ion, its coordinating ligands and their geometry, and overall structural dimensionality, chemists can exert powerful synthetic control over electronic delocalization in mixed-valence perovskites, and we hope to see this intriguing materials class expand to encompass new compositions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Numerical study of multi-component flow and mixing in a scaled fission product venting system

Numerical simulation of isothermal multi-component gas flow was conducted to study fluid flow and mixing in a scaled low-pressure, low-temperature test facility for the fission product venting system (FPVS) of a gas cooled fast reactor (GFR). The geometry of FPVS test facility was an open loop including gradual expansion coupling and two 90°pipe elbows. First, the validation of large eddy simulation (LES) wall-adapting local eddy-viscosity (WALE) turbulent model was performed for transitional flow in a circular pipe with gradual expansion. The reactingFOAM solver in OpenFOAM v8 was employed. By introducing appropriate turbulence disturbance at the flow inlet, such as turbulence intensity, the numerical results showed good agreement in (1) the velocity profile downstream of the pipe expansion measured by particle image velocimetry (PIV) in this study, and (2) the reattachment length reported in the literature. Using this validated model, the numerical simulation of flow and component mixing was performed for the FPVS with an inlet flowrate corresponding to a Reynolds number of 2,400 to investigate the flow behavior and component mixing. Standard deviation of the mass fraction of component, referred to as absolute mixing index (AMI), was calculated to quantify the mixing of components, showing that the component mixing length determined by AMI is related to the reattachment length downstream of the expansion. Additionally, this study was able to identify the best location in the FPVS test facility to measure the velocity and concentration profies where multicomponent flow is well mixed.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

Models and Algorithms for Equilibrium Analysis of Mixed-Material Nucleic Acid Systems

Dynamic programming algorithms within the NUPACK software suite enable analysis of equilibrium base-pairing properties for complex and test tube ensembles containing arbitrary numbers of interacting nucleic acid strands. Currently, calculations are limited to single-material systems that are either all-RNA or all-DNA. Here, to enable analysis of mixed-material systems that are critical for modern applications in vitro, in situ, and in vivo, we develop physical models and dynamic programming algorithms that allow the material of the system to be specified at nucleotide resolution. Free energy parameter sets are constructed for both RNA/DNA and RNA/2'OMe-RNA mixed-material systems by combining available empirical mixed-material parameters with single-material parameter sets to enable treatment of the full complex and test tube ensembles. New dynamic programming recursions account for the material of each nucleotide throughout the recursive process. For a complex with N nucleotides, the mixed-material dynamic programming algorithms maintain the O(N 3 ) time complexity of the single-material algorithms, enabling efficient calculation of diverse physical quantities over complex and test tube ensembles (e.g., complex partition function, equilibrium complex concentrations, equilibrium base-pairing probabilities, minimum free energy secondary structure(s), and Boltzmann-sampled secondary structures) at a cost increase of roughly 2.0-3.5×. The results of existing single-material algorithms are exactly reproduced when applying the new mixed-material algorithms to single-material systems. Accuracy is significantly enhanced using mixed-material models and algorithms to predict RNA/DNA and RNA/2'OMe-RNA duplex melting temperatures from the experimental literature as well as RNA/DNA melt profiles from new experiments. In conclusion, mixed-material analyses can be performed online using the NUPACK web app (www.nupack.org) or locally using the NUPACK Python module.

2′OMe-RNA↗

DOE OSTI2026

Geologic stress modulates fluid mixing at fracture intersections

Fracture intersections are critical links that enable flow and transport in subsurface fracture networks, and their behavior strongly influences fluid mixing in a network. Although all subsurface fractures are subjected to geological stress, we lack a fundamental understanding of how fracture intersection geometry evolves under stress and how these changes influence fluid mixing. Here, we combine 3D printing, 3D X-ray tomographic imaging, and 3D pore-scale numerical simulations to reveal stress-induced changes in intersection geometry and their impact on mixing. Mixing is found to be strongly affected by partial closure of an intersection under stress. As an intersection closes, the void area for fluid flow and diffusion decreases leading to substantial deviations between conventional mixing models and full pore-scale modeling. To address this, we propose a modified mixing model that accounts for intersection deformation, which is essential for accurate modeling of solute transport and mixing through fracture networks.

15 GEOTHERMAL ENERGY↗

DOE OSTI2020

Gibbs mixing of partially distinguishable photons with a polarising beamsplitter membrane

For a thought experiment concerning the mixing of two classical gases, Gibbs concluded that the work that can be extracted from mixing is determined by whether or not the gases can be distinguished by a semi-permeable membrane; that is, the mixing work is a discontinuous function of how similar the gases are. Here we describe an optomechanical setup that generalises Gibbs’ thought experiment to partially distinguishable quantum gases. Specifically, we model the interaction between a polarisation dependent beamsplitter, that plays the role of a semi-permeable membrane, and two photon gases of non-orthogonal polarisation. We find that the work arising from the mixing of the gases is related to the potential energy associated with the displacement of the microscopic membrane, and we derive a general quantum mixing work expression, valid for any two photon gases with the same number distribution. The quantum mixing work is found to change continuously with the distinguishability of the two polarised gases. In addition, fluctuations of the work on the microscopic membrane become important, which we calculate for Fock and thermal states of the photon gases. Our findings generalise Gibbs’ mixing to the quantum regime and open the door for new quantum thermodynamic (thought) experiments with quantum gases with non-orthogonal polarisations and microscopic pistons that can distinguish orthogonal polarisations.

distinguishability↗

DOE OSTI2024

Assessment of the literature about Be-W mixed material layer formation in the fusion reactor environment

Abstract All plasma facing surfaces in a fusion reactor, whether initially pure or an alloy, will rapidly evolve into a mixed material due to plasma-induced erosion, migration and redeposition. Beryllium (Be) erosion from the main chamber, and its transport and deposition on to a tungsten (W) divertor results in the growth of mixed Be-W layers, which can evolve to form beryllides. These Be-W mixed materials exhibit generally less desirable properties than pure tungsten or pure beryllium, such as lower melting points. In order to better understand the parameter space for growth of these alloys, this paper reviews the literature on Be-W mixed material formation experiments—in magnetically confined fusion reactors, in linear plasma test stands, and during thin-film deposition—and on computational modeling of Be-W interactions, as well as briefly assesses the Be-W growth kinetics. We conclude that the following kinetic steps drive the material mixing: adsorption of the implanted/deposited ion on the metal surface; diffusion of the implanted/deposited ion from surface into the bulk, which is accelerated by defects; and loss of deposited material through erosion. Adsorption dominates (or prevents) material mixing in thin-film deposition experiments, whereas diffusion drives material mixing in plasma exposures due to the energetic ion implantation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Disruption of Brachypodium lichenase alters metabolism of mixed‐linkage glucan and starch

Mixed-linkage glucan, which is widely distributed in grasses, is a polysaccharide highly abundant in cell walls of grass endosperm and young vegetative tissues. Lichenases are enzymes that hydrolyze mixed-linkage glucan first identified in mixed-linkage glucan-rich lichens. In this study, we identify a gene encoding a lichenase we name Brachypodium distachyon LICHENASE 1 (BdLCH1), which is highly expressed in the endosperm of germinating seeds and coleoptiles and at lower amounts in mature shoots. RNA in situ hybridization showed that BdLCH1 is primarily expressed in chlorenchyma cells of mature leaves and internodes. Disruption of BdLCH1 resulted in an eight-fold increase in mixed-linkage glucan content in senesced leaves. Consistent with the in situ hybridization data, immunolocalization results showed that mixed-linkage glucan was not removed in chlorenchyma cells of lch1 mutants as it was in wild type and implicate the BdLCH1 enzyme in removing mixed-linkage glucan in chlorenchyma cells in mature vegetative tissues. We also show that mixed-linkage glucan accumulation in lch1 mutants was resistant to dark-induced degradation, and 8-week-old lch1 plants showed a faster rate of starch breakdown than wild type in darkness. Here our results suggest a role for BdLCH1 in modifying the cell wall to support highly metabolically active cells.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Improving the representation of isopycnal mixing in E3SM (Final Report)

The DOE-supported Energy Exascale Earth System Model (E3SM) is a major attempt by the Department of Energy to develop a new Earth System Model using unstructured grids, which allow for the model to concentrate resolution where it is most needed and avoid numerical artifacts experienced by previous Earth System Models (ESMs) using regular grids in which the lateral resolution becomes extremely fine as the resolution shrinks at the poles. This effort required rewriting many of the algorithms previously developed for regular grids. One algorithm that was problematic in the first version of E3SM was the representation of mixing due to turbulent ocean eddies with scales smaller than the model grid. These mesoscale eddies are the primary way by which tracers are stirred along density surfaces in the ocean interior. In particular, the process of isopycnal mixing exchanges fresher, more oxygenated waters from polar regions with saltier, nutrient-rich and oxygen poor waters from tropical regions. Previous work in our group has shown that this mechanism is vitally important for bringing oxygen into poorly ventilated tropical regions (Gnanadesikan et al., 2012, 2013) and plays a significant role in determining the uptake of anthropogenic carbon dioxide (Gnanadesikan et al., 2015). However, in E3SMv1 this process was turned off, as turning it on caused the model to become unstable. Additionally, the rate of isopycnal mixing is directly proportional to a mixing coefficient $A_{Redi}$ whose value varies from less than 400 m 2 /s to 2000 m 2 /s across contemporary climate models. The main thrusts of this proposal therefore were to: 1.) Improve the numerics of isopycnal mixing in E3SM; 2.) Identify ways in which climate models are sensitive to the isopycnal mixing coefficient; and 3.) Develop better representations of the isopycnal mixing coefficient.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Assessment of the literature about Be-W mixed material layer formation in the fusion reactor environment

All plasma facing surfaces in a fusion reactor, whether initially pure or an alloy, will rapidly evolve into a mixed material due to plasma-induced erosion, migration and redeposition. Beryllium (Be) erosion from the main chamber, and its transport and deposition on to a tungsten (W) divertor results in the growth of mixed Be-W layers, which can evolve to form beryllides. These Be-W mixed materials exhibit generally less desirable properties than pure tungsten or pure beryllium, such as lower melting points. In order to better understand the parameter space for growth of these alloys, this paper reviews the literature on Be-W mixed material formation experiments—in magnetically confined fusion reactors, in linear plasma test stands, and during thin-film deposition—and on computational modeling of Be-W interactions, as well as briefly assesses the Be-W growth kinetics. We conclude that the following kinetic steps drive the material mixing: adsorption of the implanted/deposited ion on the metal surface; diffusion of the implanted/deposited ion from surface into the bulk, which is accelerated by defects; and loss of deposited material through erosion. Adsorption dominates (or prevents) material mixing in thin-film deposition experiments, whereas diffusion drives material mixing in plasma exposures due to the energetic ion implantation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Aligning Standards Communities for Omics Biodiversity Data: Sustainable Darwin Core-MIxS Interoperability

The standardization of data, encompassing both primary and contextual information (metadata), plays a pivotal role in facilitating data (re-)use, integration, and knowledge generation. However, the biodiversity and omics communities, converging on omics biodiversity data, have historically developed and adopted their own distinct standards, hindering effective (meta)data integration and collaboration. In response to this challenge, the Task Group (TG) for Sustainable DwC-MIxS Interoperability was established. Convening experts from the Biodiversity Information Standards (TDWG) and the Genomic Standards Consortium (GSC) alongside external stakeholders, the TG aimed to promote sustainable interoperability between the Minimum Information about any (x) Sequence (MIxS) and Darwin Core (DwC) specifications. To achieve this goal, the TG utilized the Simple Standard for Sharing Ontology Mappings (SSSOM) to create a comprehensive mapping of DwC keys to MIxS keys. This mapping, combined with the development of the MIxS-DwC extension, enables the incorporation of MIxS core terms into DwC-compliant metadata records, facilitating seamless data exchange between MIxS and DwC user communities. Through the implementation of this translation layer, data produced in either MIxS- or DwC-compliant formats can now be efficiently brokered, breaking down silos and fostering closer collaboration between the biodiversity and omics communities. To ensure its sustainability and lasting impact, TDWG and GSC have both signed a Memorandum of Understanding (MoU) on creating a continuous model to synchronize their standards. These achievements mark a significant step forward in enhancing data sharing and utilization across domains, thereby unlocking new opportunities for scientific discovery and advancement.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS2005

The Closure of the Ocean Mixed Layer Temperature Budget using Level-Coordinate Model Fields

Entrainment is an important element of the mixed layer mass, heat, and temperature budgets. Conventional procedures to estimate entrainment heat advection often do not permit the closure of heat and temperature budgets because of inaccuracies in its formulation. In this study a rigorous approach to evaluate the effect of entrainment using the output of a general circulation model (GCM) that does not have an explicit prognostic mixed layer model is described. The integral elements of the evaluation are 1) the rigorous estimates of the temperature difference between mixed layer water and entrained water at each horizontal grid point, 2) the formulation of the temperature difference such that the budget closes over a volume greater than one horizontal grid point, and 3) the apparent warming of the mixed layer during the mixed layer shoaling to account for the weak vertical temperature gradient within the mixed layer. This evaluation of entrainment heat advection is compared with the estimates by other commonly used ad hoc formulations by applying them in three regions: the north-central Pacific, the Kuroshio Extension, and the Nino-3 areas in the tropical Pacific. In all three areas the imbalance in the mixed layer temperature budget by the ad hoc estimates is significant, reaching a maximum of about 4 K yr(exp -1).

temperature gradient↗

DOE OSTI2021

A comparative study of machine learning models for predicting the state of reactive mixing

Mixing phenomena are important mechanisms controlling flow, species transport, and reaction processes in fluids and porous media. Accurate predictions of reactive mixing are critical for many Earth and environmental science problems such as contaminant fate and remediation, macroalgae growth, and plankton biomass evolution. Here, to investigate the evolution of mixing dynamics under different scenarios (e.g., anisotropy, fluctuating velocity fields), a finite-element-based numerical model was built to solve the fast, irreversible bimolecular reaction-diffusion equations to simulate a range of reactive-mixing scenarios. A total of 2,315 simulations were performed using different sets of model input parameters comprising various spatial scales of vortex structures in the velocity field, time-scales associated with velocity oscillations, the perturbation parameter for the vortex-based velocity, anisotropic dispersion contrast (i.e., ratio of longitudinal-to-transverse dispersion), and molecular diffusion. The outputs comprised concentration profiles of reactants and products. The inputs to and outputs from these simulations were concatenated into feature and label matrices, respectively, to train 20 different machine learning (ML) models intended to emulate system behavior. These 20 ML emulators, based on linear methods, Bayesian methods, ensemble learning methods, and multilayer perceptrons (MLPs), were trained to classify the state of mixing and predict three quantities of interest (QoIs) characterizing species production, decay (i.e., average concentration, square of average concentration), and degree of mixing (i.e., variances of species concentration). Unsurprisingly, linear classifiers and regressors failed to reproduce the QoIs; however, ensemble methods (classifiers and regressors) and the MLP model accurately classified the state of reactive mixing and the QoIs. Among ensemble methods, random forest and decision-tree-based AdaBoost faithfully predicted the QoIs. At run time, trained ML emulators produced results times faster than the finite-element simulations. Due to their low computational expense and high accuracy, ensemble and MLP models are excellent emulators for these numerical simulations and great utilities in uncertainty quantification exercises, which can require 1,000s of forward model runs.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A simple three-component mixing problem for the evaluation of a new reaction rate model

A simple computational mixing problem is presented which can be utilized to assess the behavior of Reynolds-averaged reaction rate models in a problem with temporally varying mixedness. In this problem, three mixing components are homogeneously distributed but initially separated in a triply periodic domain. Further, these components are initialized within a Taylor–Green-like velocity field, which creates a mixing history evolving from the so-called “no-mix limit” to a well-mixed state. Large-eddy simulation results from this problem in configurations involving both premixed and nonpremixed reactants are then compared with zero-dimensional Reynolds-averaged Navier–Stokes results utilizing a new model for multicomponent reacting mixtures. The new model is shown to appropriately respect the no-mix limit and outperforms an earlier model (Morgan, 2022), particularly at early times when components are near the no-mix limit.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

The influence of temperature history and flow mixing on the vapor-phase speciation of uranium oxide nanoparticles

The transport of radioactive material following a nuclear event depends heavily on fallout characteristics (e.g., composition, shape, and size), which in turn are controlled by the conditions under which the fallout forms. To this end, we use a novel plasma flow reactor (PFR) to explore how conditions including temperature history and gas mixing influence the chemical speciation and physical characteristics of uranium oxide nanoparticles. The PFR consists of a glass tube attached to an inductively coupled plasma (ICP) torch via an adaptor piece (ring flow injector) located downstream of the plasma-generating RF coil. We increase the gas flow rates from normal ICP operating conditions to higher values both upstream at the RF coil and downstream at the ring flow injector to create four distinct temperature profiles and two distinct gas flow mixing conditions (normal and higher Ar atomic ratios). Ex situ transmission electron microscopy (TEM) analysis is used to determine the crystal structure, morphology, and size distribution of the synthesized particles as a function of temperature and mixing. The particles display a striking dependence on these conditions in terms of the resulting speciation of uranium oxide: lower temperature (higher upstream) and slower cooling (normal downstream with normal Ar atomic ratio mixing) favor the formation of α-UO 3 particles, while faster cooling (higher downstream with higher Ar atomic ratio mixing) and higher temperature (normal upstream) favor the formation of UO 2 particles. The extent of gas mixing dictates particle aggregation: better mixing encourages the particles to aggregate more extensively. Here these results demonstrate the sensitivity of uranium to initial local conditions and can be used to improve our understanding of the kinetics that drive uranium speciation and ultimately inform fallout transport models.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

Constraining Black Carbon Aging in Global Models to Reflect Timescales for Internal Mixing

The radiative effects of black carbon depend critically on its atmospheric lifetime, which is controlled by the rate at which freshly emitted combustion particles become internally mixed with other aerosol components. Global aerosol models strive to represent this process, but the timescale for aerosol mixing is not easily constrained using observations. In this study, we apply a timescale parameterization derived from particle‐resolved simulations to quantify, in a global aerosol model, the timescale for internal mixing. We show that, while highly variable, the average timescale for internal mixing is approximately 3 hr, which is much shorter than the 24‐hr aging timescale traditionally applied in bulk aerosol models. We then use the mixing timescale to constrain the aging criterion in the Modal Aerosol Module. Our analysis reveals that, to best reflect timescales for internal mixing, modal models should assume that particles transition from the hydrophobic (fresh) to the hydrophilic (aged) class once they accumulate a coating thickness equal to four monolayers of sulfuric acid, as opposed to the model's current aging criterion of eight monolayers. We show that, in remote regions like the Arctic and Antarctic, predictions of black carbon loading and its seasonal variation are particularly sensitive to the model representation of aging. By constraining aging in global models to reflect mixing timescales simulated by the particle‐resolved model, we eliminate one of the free parameters governing black carbon's long‐range transport and spatiotemporal distribution.

54 ENVIRONMENTAL SCIENCES↗