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Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Vertical atmospheric flux (time/pressure level) variability maps. ((a)–(c)) Unique variability contribution from each atmospheric layer. (a) Deeper layer containing higher-order harmonics. Variability is interpreted to be due to Mg2SiO4 cloud modulation (e.g., J. M. Vos et al. 2023; A. M. McCarthy et al. 2024, 2025; M. K. Plummer et al. 2024). (b) Transition layer containing bright spots colocated with presumed cloud tops in the lower layer. Transition layer corresponds to water absorption bands, and the bright spots may indicate localized heating or variations in water absorption associated with cloud formation (either muting of spectral features or depletion of water). (c) Upper layer demonstrating lower-order harmonics thought to be driven by bright spots in the transition layer.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Retrieved Brewster model thermal profile (brown line, with colored shading for 1σ confidence interval) for SIMP 0136. Previous retrieval result from J. M. Vos et al. (2023) is plotted as a black curve for comparison. Self-consistent model profiles from the Sonora Diamondback grid are plotted as solid colored lines. Phase-equilibrium condensation curves for plausible cloud species are plotted as colored dashed lines. The clouds pressures are indicated in bars to the right. Purple bar indicates the median cloud location for the forsterite slab and the optically thick extent of the iron deck, with gray shading indicating the 1σ range. Graduated shading shows the range over which the deck cloud optical depth drops to 1/e. Forsterite and enstatite are fairly challenging to distinguish in retrievals of the spectra, meaning high-altitude clouds could actually be enstatite or a mixture of both.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Comparison of lightcurve morphology at three distinct wavelength bins. Black error bars show NIRISS data, similarly to Figure 1; green triangles show lightcurves extracted from NIRISS data that were projected on the first two principal components. Red squares are lightcurves of an admixture model consisting of three Diamondback spectra. Observed lightcurve morphology can be explained by two components, confirming that it is caused by three or more spectral regions rotating in and out of view. The consistency between the observed and reconstructed lightcurves suggests that the bulk properties of the underlying cloud structure are correctly reproduced.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Projection of the time-varying spectrum onto the PCP. Percentage indicates the fraction of explained variance by the components. Low-contrast and high-contrast shaded curves, respectively, show the time series at the original time sampling and in 9 minute bins. Color-coding of both curves gives the rotational phase. The two components are orthogonal but do not directly correspond to physical parameters. We projected Sonora Diamondback models to see how changing fsed, T, and effective [Fe/H] impacts the spectra. Starting at the bottom right of the plot, we can infer sequential change in T and fsed as SIMP 0136 completes a rotation.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Left: temperature–pressure profiles inferred from atmospheric retrievals of regionally extracted spectra. Red dashed line with 1σ confidence interval shows retrieved profile of a time-averaged spectrum. Phase-equilibrium condensation curves for plausible cloud species are plotted as colored dashed lines. Right: altitude and fraction of patchy forsterite cloud slab and opaque iron cloud deck inferred from T–P profiles for each region of interest. Purple bar indicates the median cloud location, with gray shading indicating the 1σ range.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Top: individual explained variance for each of the principal components shown as a histogram and cumulative explained variance shown with solid lines. The number of components is limited to 20 for display purposes. Over 80% of the variance is explained by two principal components, which suggests a minimum of three spectral regions that contribute to spectral variability. Bottom: analysis of the BIC for admixtures of Diamondback models to fit the time-averaged spectrum. The minimum is reached for a linear combination of six atmospheric models, which is consistent with the PCA.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Top: measured and smoothed RV signal at 1–1.35 μm. Middle: derivative of the lightcurve at the same wavelength bin. Bottom: photometric lightcurve at this bin. The ff﹩{}^{{\prime} }﹩ framework suggests that the RV should be in antiphase with the flux derivative. This is the case for the first harmonic only, while second and third harmonics are closely in phase with the flux derivative.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Extracted spectrum from each region of interest is shown on the main plot. The inset map shows the posterior spherical harmonics map at 2 μm. Each region of interest used in this method is shown with color- and shape-coded data points on the map. The dashed white line indicates subobserver latitude. Although mathematically sound, the degeneracies involved in 2D mapping make the regional spectra highly uncertain.
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Long-term evolution of solar magnetic activity and optimal source-surface height derived from various magnetograms. Top: monthly averaged (gray) and 13 month smoothed (black) sunspot numbers. Middle and bottom: optimal source-surface heights estimated from observed (middle) and ADAPT (bottom) magnetograms. Red, purple, green, and blue lines denote KPVT, SOHO/MDI, GONG, and SDO/HMI, respectively. Transparent diamonds show values for each Carrington rotation; solid lines indicate 13-CR averages.
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Process of calculating the optimal value of the source-surface height (﹩{R}_{{\rm{SS}}}^{{\rm{opt}}}﹩). The top panel presents the time evolution of the three-rotation averaged open fluxes estimated from in situ observations (gray) and the PFSS extrapolations using ADAPT-GONG as input, with different source-surface radii (blue: RSS/R⊙ = 1.5; orange: RSS/R⊙ = 1.9; green: RSS/R⊙ = 2.3). The bottom panel presents the temporal evolution of the optimal source-surface height (ADAPT-GONG) computed using Equation (3).
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Time evolution of the multipolar components of the photospheric magnetic field and their relation to RSS. Top: time evolution of dipolar (red, defined in Equation (5)), quadrupolar (green, Equation (6)), and octupolar (blue, Equation (3)) field strengths. Bottom: comparison of the time evolution of dipolar field dominance fdip (red, Equation (4)) and optimal source-surface height ﹩{R}_{{\rm{ss}}}^{{\rm{opt}}}﹩ (black). In both panels, transparent diamonds show values for each Carrington rotation; solid lines represent the 13-CR running average.
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Long-term evolution of the optimal source-surface height and solar activity. Top panel: comparison between the optimal source-surface height (﹩{R}_{{\rm{SS}}}^{{\rm{opt}}}﹩, black symbols and line) and the source-surface height given by Equation (14) (red symbols and line). Transparent diamonds show values for each Carrington rotation; solid lines indicate 13-CR averages. Bottom panel: same as the top panel of Figure 2, but limited to the time range shown here.
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Relation between optimal source-surface height (﹩{R}_{{\rm{SS}}}^{{\rm{opt}}}﹩) and mean unsigned coronal radial field strength (﹩\left\langle {B}_{r,{\rm{cor}}}\right\rangle ﹩). Data from 2007 to 2023 are shown in the upper panel, and data from 2011 to 2017 in the lower panel. Each panel displays the value of the Pearson correlation coefficient (PCC).
Paper: Empirical Optimization of the Source-surface Height in the Potential Field Source Surface Extrapolation
Authors: Munehito Shoda, Kyogo Tokoro, Daikou Shiota, Shinsuke Imada
Figure caption: Scatterplots between the optimal source-surface height and characteristic parameters. The top two panels show ﹩{R}_{{\rm{ss}}}^{{\rm{opt}}}﹩ vs. the mean unsigned radial field strength at the photosphere, while the bottom panels show ﹩{R}_{{\rm{ss}}}^{{\rm{opt}}}﹩ vs. the dominance of the dipole component. The left column covers the full period (2007–2023), and the right column shows results for solar maximum (2012–2016) and minimum (2017–2021) phases.
Paper: Mapping Atmospheric Features of the Planetary-mass Brown Dwarf SIMP 0136 with JWST NIRISS
Authors: Roman Akhmetshyn, Étienne Artigau, Nicolas B. Cowan, Michael K. Plummer, Fei Wang, Ben Burningham, Björn Benneke, René Doyon, Ray Jayawardhana, David Lafrenière, Stanimir A. Metchev, Jason F. Rowe
Figure caption: Gas abundances and gravitational acceleration inferred from the atmospheric retrieval of regionally extracted spectra. Each posterior histogram is colored according to position on the object. The dashed line is the mean abundances retrieved from the time-averaged spectrum.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Growth rates γ/Ωp (upper panel) and the corresponding resonant factors ﹩{\zeta }_{b}^{+}﹩ (middle panel) and ﹩{\zeta }_{c}^{+}﹩ (bottom panel) as functions of the wavenumber ck/ωpp. The dashed gray line in the top panel represents the maximum growth rate γm = 10−4 Ωp, which is adopted as the stability threshold throughout this study.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Maximum growth rates, γm/Ωp, (upper panel), and the corresponding wavenumbers, ckm/ωpp, (lower panel), as functions of the βc∥. The dashed gray line represents the threshold of βc∥ required for the existence of the PFH instability.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Effect of the beam temperature anisotropy Ab on the instability threshold (γm/Ωp = 10−4) as a function of ∣vb/vAc∣ and βc∥.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Effect of the beam and core temperature anisotropies on the instability threshold (γm/Ωp = 10−4) as a function of beam velocity and the beam–core density ratio nb/nc.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Effect of the suprathermal Kappa populations in the beam on the instability threshold (γm/Ωp = 10−4) as a function of ∣vb/vAc∣ and βc∥.
Paper: Cumulative Proton Right-handed Instabilities in the Young Solar Wind Observed by PSP
Authors: Shaaban M. Shaaban, M. Lazar, H. Fichtner, R. A. López, S. Poedts
Figure caption: Variation of the maximum growth rate γm/Ωp as a function of the core temperature anisotropy Ac.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Dispersions of dust density normalized by the midplane gas density (first row; Equation (8)) and gas velocities (normalized by Πcs; second through fourth rows; Equation (9)) as a function of z for the snapshots shown in Figure 1. Radial, azimuthal, and vertical components of the gas velocity dispersions are shown in order from the second to fourth rows. In order from left to right, the panels correspond to t/T = 1, 1.5, 2.0, and 3.0. In each panel, darker to brighter colors represent lower to higher Π values: 0.01, 0.02, 0.05, and 0.1. As all runs have nearly identical Hp/(ΠH) at a given time, we use the shaded region to mark ±Hp at a given time. While the density and the velocity dispersions for Π = 0.05 and 0.1 peak near the midplane as early as t/T = 1, those for Π = 0.01 and 0.02 peak near ±Hp and only become prominent by t/T = 1.5. After sedimentation balances turbulent diffusion (t/T = 3.0), the density and velocity profiles look similar across different Π values.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Evolution of dust density (ρp) during the sedimentation phase. From left to right, the columns correspond to Π = 0.01, 0.02, 0.05, and 0.1, respectively. Each row corresponds to snapshots at different times: t/T = 1, 1.5, 2.0, and 3.0 from top to bottom. The vertical domain is zoomed in to −1.0 ≤ z/(ΠH) ≤ 1.0, while the full vertical extent is 8ΠH. The simulations presented here have the same value of Z/Π. Dust voids emerge earlier at higher Π, appearing at the midplane around t/T ≈ 1 for Π = 0.1 (top right) and away from the midplane around t/T ≈ 2 for Π = 0.01 (third row, first column). These voids develop into filaments as time goes on.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Similar to Figure 8 but for different domain sizes (Lx and Lz) at Z = 0.012 and Π = 0.05 with NΠH = 256. The left and right columns correspond to Lz = 8ΠH and 16ΠH, respectively. In order from top to bottom, the rows correspond to density and to radial, azimuthal, and vertical velocity dispersions. Different colors denote different Lx values, with solid and dashed curves representing the dust and the gas, respectively. The gray shaded regions indicate the saturated state, spanning from t/T = 10 to 80 for Lz = 8ΠH and from t/T = 10 to 70 for Lz = 16ΠH. While runs with Lx = 8ΠH show a sharp increase in ﹩{\sigma }_{{\rho }_{p}}(z=0)﹩ during the final 20–30 orbits depending on Lz, the dispersions do not significantly depend on domain size during the saturated state.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: The vertical gradient of radial velocity (upper panels; Equation (10)) and the radial Richardson number (lower panels; Equation (18)) as a function of z across different Π values during the sedimentation phase. In order from left to right, panels correspond to Π = 0.01, 0.02, 0.05, and 0.1. For each Π, we include snapshots before the dust layer becomes turbulent. We mark ±Hp at a given time using dotted lines in each panel, with the dotted lines matching the color of the corresponding profile at that time. The magnitude of the vertical gradient of radial velocity vanishes at the midplane but increases with height up to ≈±Hp in all snapshots and Π values. The Richardson number is above Rir,crit (0.25; horizontal dashed lines in the lower panels) within the dust layer making it stable to KHI.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Time-averaged power spectrum of the specific kinetic energy (﹩\overline{{\rm{KE}}}﹩; Equation (20)) over the saturated state as a function of radial wavenumber kx. All panels display the same power spectra but with different scaling on the x- and y-axes. The left two panels share the same x-axis (kxΠH/(2π)), while in the lower left panel, ﹩\overline{{\rm{KE}}}﹩ is scaled by Π2. The upper panels share the same y-axis (not scaled by Π2), but in the top right panel, kx is expressed in terms of the gas scale height H. The power at any given kxΠH increases with Π (upper left) and scales with Π2 (lower left). Most of the kinetic energy is concentrated near kxΠH/(2π) ∼ 1, corresponding to a wavelength of ΠH (lower left). Given that both length and velocity scales are proportional to ΠH, the SI plays a dominant role in the dynamics during the saturated state.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Similar to Figure 4 but showing only vertical velocities of gas (top) and dust (bottom) for simulations with varying radial domain sizes at fixed Π = 0.05 and Lz/(ΠH) = 8. Blue, red, and green curves correspond to Lx/(ΠH) = 2, 4 (fiducial), and 8, respectively. All of the runs have time-averaged Hp of ≈0.24ΠH, which is indicated by the gray vertical band. In the gas, the Lx/(ΠH) = 8 run shows zero vertical velocity at the midplane, while the smaller-domain runs reach zero velocity away from the midplane. All runs exhibit nonzero dust vertical velocity outside of ≈±2Hp, with the radial domain size having a negligible effect on the dust velocity.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Similar to Figure 12 but for the vertical gradient of azimuthal velocity (upper row; Equation (11)) and the azimuthal Richardson number (lower row, Equation (19)). Unlike Rir, Riϕ falls below the threshold (Riϕ,crit; the horizontal line in the lower panels) within the dust layer (marked by vertical lines) as dust settles toward the midplane across all simulations presented here, making it unstable to SymI.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Snapshot of ρp at t/T = 50 for the simulations shown in Figure A3. From top to bottom, the panels correspond to Lx/(ΠH) = 2, 4, and 8, respectively. Only the central vertical region (Lz/(ΠH) ∈ [−1, 1]) is displayed, although the full vertical extent is 8ΠH. White dashed curves show the mean vertical position of dust as a function of x. As Lx increases, more wavelengths of the undulating dust layer are captured, with Lx ≤ 4ΠH appearing to contain only a single mode or less.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Snapshots of the dust density (ρp; upper panels) and radial gas velocity (ux; lower panels) at t/T = 2. In order from left to right, the panels correspond to Π = 0.01, 0.02, 0.05, and 0.1, respectively. For Π = 0.05 and 0.1, the dust filaments are nearly fully developed, and the midplane is already turbulent. By contrast, for Π = 0.01 and 0.02, the midplane is still under development, while the radial velocity reaches its largest magnitude near the top of the dust layer, where large radial structures (as large as ∼ΠH) develop.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Similar to Figure 10 but for different domain sizes (Lx and Lz) at Z = 0.012 and Π = 0.05 with NΠH = 256. Upper and lower panels correspond to Lz = 8ΠH and 16ΠH, respectively. Different colors denote different Lx values. The title of each panel indicates the time period over which the distributions are averaged. The comparison shows convergence of the dust density distributions across various domain sizes, except for the maximum densities, which increase with Lx.
Paper: Bridging Unstratified and Stratified Simulations of the Streaming Instability for τs = 0.1 Grains
Authors: Jeonghoon Lim, 정훈 임, Stanley A. Baronett, Jacob B. Simon, Chao-Chin Yang, 朝欽 楊, Debanjan Sengupta, Orkan M. Umurhan, Wladimir Lyra
Figure caption: Time-averaged vertical profiles of Rir (upper panel; Equation (18)) and Riϕ (lower panel; Equation (19)) over the saturated state. Their critical values are denoted by horizontal lines in each panel. The gray shaded region marks ±Hp. Regardless of Π, the Richardson numbers remain well above their corresponding critical values (0.25 for Rir and 1.0 for Riϕ).