Publications associated with Ice and Fluid Dynamics


Modelling binary alloy solidification with adaptive mesh refinement

Journal of Computational Physics: X 5 (2020)

JRG Parkinson, DF Martin, AJ Wells, RF Katz

© 2019 The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time.


There is no Plan B for dealing with the climate crisis

BULLETIN OF THE ATOMIC SCIENTISTS 75 (2019) 215-221

R Pierrehumbert


Thermal Convection over Fractal Surfaces

ArXiV [physics.flu-dyn] (2019)

S Toppaladoddi, A Wells, CR Doering, JS Wettlaufer


Atmospheric circulation of tide-locked exoplanets

Annual Review of Fluid Mechanics Annual Reviews 51 (2018) 75-303

R Pierrehumbert, M Hammond

Tide-locked planets are planets in which tidal stresses from the host star have spun down the planet’s rotation to the point where its length of side-real day equals its length of year. In a nearly circular orbit, such planets have a permanent dayside and a permanent nightside, leading to extreme heating contrasts. In this article, the atmospheric circulations forced by this heating contrast are explored, with a focus on terrestrial planets; here, “terrestrial” refers to planets with a condensed solid or liquid surface at which most of the incident stellar radiation is absorbed and does not imply habitability in the Earthlike sense. The census of exoplanets contains many terrestrial planets that are very likely to be tide locked, including extremely hot close-orbit planets around Sunlike stars and habitable zone (and hotter) planets around lower-mass stars. The circulations are discussed in terms of fluid dynamical concepts arising from study of the Earth’s tropics, supplemented by general circulation model simulations. Even in the relatively simple context of dry (noncondensing) dynamics, there are a number of important unresolved issues that require further study.


Solidification of binary aqueous solutions under periodic cooling. Part 1. Dynamics of mushy-layer growth

Journal of Fluid Mechanics Cambridge University Press 870 (2019) 121-146

G-Y Ding, A Wells, J-Q Zhong

We present studies of the solidification of binary aqueous solutions that undergo time-periodic cooling from below. We develop an experiment for solidification of aqueous NH4Cl solutions, where the temperature of the cooling boundary is modulated as a simple periodic function of time with independent variations of the modulation amplitude and frequency. The thickness of the mushy layer exhibits oscillations about the background growth obtained for constant cooling. We consider the deviation given by the difference between states with modulated and fixed cooling, which increases when the modulation amplitude increases but decreases with increasing modulation frequency. At early times, the deviation amplitude is consistent with a scaling argument for growth with quasi-steady modulation. In situ measurements of the mush temperature reveal thermal waves propagating through the mushy layer, with amplitude decaying with height within the mushy layer, whilst the phase lag behind the cooling boundary increases with height. This also leads to phase lags in the variation of the mushy-layer thickness compared to the boundary cooling. There is an asymmetry of the deviation of mushy-layer thickness: during a positive modulation (where the boundary temperature increases at the start of a cycle) the peak thickness deviation has a greater magnitude than the troughs in a negative modulation mode (where the boundary temperature decreases at the start of the cycle). A numerical model is formulated to describe mushy-layer growth with constant bulk concentration and turbulent heat transport at the mush–liquid interface driven by compositional convection associated with a finite interfacial solid fraction. The model recovers key features of the experimental results at early times, including the propagation of thermal waves and oscillations in mushy-layer thickness, although tends to overpredict the mean thickness.


Solidification of binary aqueous solutions under periodic cooling. Part 2. Distribution of solid fraction

Journal of Fluid Mechanics Cambridge University Press 870 (2019) 147-174

G-Y Ding, A Wells, J-Q Zhong

We report an experimental study of the distributions of temperature and solid fraction of growing NH4Cl–H2O mushy layers that are subjected to periodical cooling from below, focusing on late-time dynamics where the mushy layer oscillates about an approximate steady state. Temporal evolution of the local temperature T(z, t) at various heights in the mush demonstrates that the temperature oscillations of the bottom cooling boundary propagate through the mushy layer with phase delays and substantial decay in the amplitude. As the initial concentration C0 increases, we show that the decay rate of the thermal oscillation with height also decreases, and the propagation speed of the oscillation phase increases. We interpret this as a result of the solid fraction increasing with C0, which enhances the thermal conductivity but reduces the specific heat of the mushy layer. We present a new methodology to determine the distribution of solid fraction φ(z) in mushy layers for various C0, using only measurements of the temperature T(z, t). The method is based on the phase behaviour during thermal modulation, and opens up a new approach for inferring mushy-layer properties in geophysical and engineering settings, where direct measurements are challenging. In our experiments, profiles of the solid fraction φ(z) exhibit a cliff–ramp–cliff structure with large vertical gradients of φ near the mush–liquid interface and also near the bottom boundary, but much more gradual variation in the interior of the mushy layer. Such a profile structure is more pronounced for higher initial concentration C0. For very low concentration, the solid fraction appears to be linearly dependent on the height within the mush. The volume-average of the solid fraction, and the local fluctuations in φ(z) both increase as C0 increases. We suggest that the fast increase of φ(z) near the bottom boundary is possibly due to diffusive transport of solute away from the bottom boundary and the depletion of solute content near the basal region


Mushy layer growth and convection, with application to sea ice

Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Royal Society 377 (2019)

A Wells, J Parkinson, J Hitchen

Sea ice is a reactive porous medium of ice crystals and liquid brine, which is an example of a mushy layer. The phase behaviour of sea ice controls the evolving material properties and fluid transport through the porous ice, with consequences for ice growth, brine drainage from the ice to provide buoyancy fluxes for the polar oceans, and sea-ice biogeochemistry. We review work on the growth of mushy layers and convective flows driven by density gradients in the interstitial fluid. After introducing the fundamentals of mushy-layer theory, we discuss the effective thermal properties including the impact of salt transport on mushy-layer growth. We present a simplified model for diffusively controlled growth of mushy layers with modest cooling versus the solutal freezing-point depression. For growth from a cold isothermal boundary, salt diffusion modifies mushy layer growth by around 5-20% depending on the far-field temperature and salinity. We also review work on the onset, spatial localisation and nonlinear development of convective flows in mushy layers, highlighting recent work on transient solidification and models of nonlinear convection with dissolved solid-free brine channels. Finally, future research opportunities are identified, motivated by geophysical observations of ice growth.


Climate impacts of cultured meat and beef cattle

Frontiers in Sustainable Food Systems Frontiers Media 3 (2019) 5

J Lynch, R Pierrehumbert

Improved greenhouse gas (GHG) emission efficiency of production has been proposed as one of the biggest potential advantages of cultured meat over conventional livestock production systems. Comparisons with beef are typically highlighted, as it is a highly emissions intensive food product. In this study, we present a more rigorous comparison of the potential climate impacts of cultured meat and cattle production than has previously been made. Warming impacts are evaluated using a simple climate model that simulates the different behaviors of carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O), rather than relying on carbon dioxide equivalent (CO2e) metrics. We compare the temperature impact of beef cattle and cultured meat production at all times to 1,000 years in the future, using four synthetic meat GHG footprints currently available in the literature and three different beef production systems studied in an earlier climate modeling paper. Cattle systems are associated with the production of all three GHGs above, including significant emissions of CH4, while cultured meat emissions are almost entirely CO2 from energy generation. Under continuous high global consumption, cultured meat results in less warming than cattle initially, but this gap narrows in the long term and in some cases cattle production causes far less warming, as CH4 emissions do not accumulate, unlike CO2. We then model a decline in meat consumption to more sustainable levels following high consumption, and show that although cattle systems generally result in greater peak warming than cultured meat, the warming effect declines and stabilizes under the new emission rates of cattle systems, while the CO2 based warming from cultured meat persists and accumulates even under reduced consumption, again overtaking cattle production in some scenarios. We conclude that cultured meat is not prima facie climatically superior to cattle; its relative impact instead depends on the availability of decarbonized energy generation and the specific production systems that are realized.


Meat consumption, health and the environment

Science American Association for the Advancement of Science 361 (2018) 5324-

C Godfray, P Aveyard, T Garnett, J Hall, T Key, J Lorimer, R Pierrehumbert, P Scarborough, M Springmann, S Jebb

Both the global average per capita consumption of meat and the total amount of meat consumed are rising, driven by increasing average individual incomes and by population growth. The consumption of different types of meat and meat products has substantial effects on people’s health, and livestock production can have major negative effects on the environment. Here, we explore the evidence base for these assertions and the options policy-makers have should they wish to intervene to affect population meat consumption. We highlight where more research is required and the great importance of integrating insights from the natural and social sciences.


Salinity control of thermal evolution of late summer melt ponds on Arctic sea ice

Geophysical Research Letters American Geophysical Union 45 (2018) 8304-8313

J Kim, AJ Wells, W Moon, JP Wilkinson, T Langton, MA Granskog, B Hwang, D Rees Jones

The thermal evolution of melt ponds on Arctic sea ice was investigated through a combination of autonomous observations and two‐dimensional high‐resolution fluid dynamics simulations. We observed one relatively fresh pond and one saline pond on the same ice floe, with similar depth. The comparison of observations and simulations indicates that thermal convection dominates in relatively fresh ponds, but conductive heat transfer dominates in salt‐stratified ponds. Using a parameterized surface energy balance, we estimate that the heat flux to the ice is larger under the saline pond than the freshwater pond when averaged over the observational period. The deviation is sensitive to assumed wind, varying between 3 and 14 W/m2 for winds from 0 to 5 m/s. If this effect persists as conditions evolve through the melt season, our results suggest that this imbalance potentially has a climatologically significant impact on sea‐ice evolution.


Exploring the atmosphere of Neoproterozoic Earth: The effect of O2 on haze formation and composition

Astrophysical Journal American Astronomical Society 858 (2018) 119

S Hörst, AM Jellinek, C He, R Pierrehumbert, MA Tolbert

Previous studies of haze formation in the atmosphere of the early Earth have focused on N2/CO2/CH4 atmospheres. Here, we experimentally investigate the effect of O2 on the formation and composition of aerosols to improve our understanding of haze formation on the Neoproterozoic Earth. We obtained in situ size, particle density, and composition measurements of aerosol particles produced from N2/CO2/CH4/O2 gas mixtures subjected to FUV radiation (115–400 nm) for a range of initial CO2/CH4/O2 mixing ratios (O2 ranging from 2 ppm to 0.2%). At the lowest O2 concentration (2 ppm), the addition increased particle production for all but one gas mixture. At higher oxygen concentrations (20 ppm and greater), particles are still produced, but the addition of O2 decreases the production rate. Both the particle size and number density decrease with increasing O2, indicating that O2 affects particle nucleation and growth. The particle density increases with increasing O2. The addition of CO2 and O2 not only increases the amount of oxygen in the aerosol, but it also increases the degree of nitrogen incorporation. In particular, the addition of O2 results in the formation of nitrate-bearing molecules. The fact that the presence of oxygen-bearing molecules increases the efficiency of nitrogen fixation has implications for the role of haze as a source of molecules required for the origin and evolution of life. The composition changes also likely affect the absorption and scattering behavior of these particles but optical property measurements are required to fully understand the implications for the effect on the planetary radiative energy balance and climate.


Global or local pure-condensible atmospheres: Importance of horizontal latent heat transport

Astrophysical Journal Institute of Physics Publishing, Inc 867 (2018)

F Ding, RT Pierrehumbert


A chemical survey of exoplanets with ARIEL

Experimental Astronomy Springer 46 (2018) 135–209-

G Tinetti, P Drossart, P Eccleston, P Hartogh, A Heske, J Leconte, G Micela, M Ollivier, P Eccleston, G Pilbratt, L Puig, D Turrini, N Bowles

Thousands of exoplanets have now been discovered with a huge range of masses, sizes and orbits: from rocky Earth-like planets to large gas giants grazing the surface of their host star. However, the essential nature of these exoplanets remains largely mysterious: there is no known, discernible pattern linking the presence, size, or orbital parameters of a planet to the nature of its parent star. We have little idea whether the chemistry of a planet is linked to its formation environment, or whether the type of host star drives the physics and chemistry of the planet’s birth, and evolution. The Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) has been selected by the European Space Agency as the next mediumclass science mission, M4, to address these scientific questions. ARIEL was conceived to observe a large number (~1000) of transiting planets for statistical understanding, including gas giants, Neptunes, super-Earths and Earth-size planets around a range of host star types using transit spectroscopy in the 1.25-7.8 μm spectral range and multiple narrow-band photometry in the optical. ARIEL will focus on warm and hot planets to take advantage of their well-mixed atmospheres which should show minimal condensation and sequestration of high-Z materials compared to their colder Solar System siblings. Said warm and hot atmospheres are expected to be more representative of the planetary bulk composition. Observations of these warm/hot exoplanets, and in particular of their elemental composition (especially C, O, N, S, Si), will allow the understanding of the early stages of planetary and atmospheric formation during the nebular phase and the following few million years. ARIEL will thus provide a representative picture of the chemical nature of the exoplanets and relate this directly to the type and chemical environment of the host star. ARIEL is designed as a dedicated survey mission for combined-light spectroscopy, capable of observing a large and well-defined planet sample within its 4-year mission lifetime. Transit, eclipse and phase-curve spectroscopy methods, whereby the signal from the star and planet are differentiated using knowledge of the planetary ephemerides, allow us to measure atmospheric signals from the planet at levels of 10-100 part per million (ppm) relative to the star and, given the bright nature of targets, also allows more sophisticated techniques, such as eclipse mapping, to give a deeper insight into the nature of the atmosphere. These types of observations require a stable payload and satellite platform with broad, instantaneous wavelength coverage to detect many molecular species, probe the thermal structure, identify clouds and monitor the stellar activity. The wavelength range proposed covers all the expected major atmospheric gases from e.g. H2O, CO2, CH4 NH3, HCN, H2S through to the more exotic metallic compounds, such as TiO, VO, and condensed species. Simulations of ARIEL performance in conducting exoplanet surveys have been performed – using conservative estimates of mission performance and a full model of all significant noise sources in the measurement – using a list of potential ARIEL targets that incorporates the latest available exoplanet statistics. The conclusion at the end of the Phase A study, is that ARIEL – in line with the stated mission objectives – will be able to observe about 1000 exoplanets depending on the details of the adopted survey strategy, thus confirming the feasibility of the main science objectives.


Frazil-ice growth rate and dynamics in mixed layers and sub-ice-shelf plumes

Cryosphere European Geosciences Union 12 (2018) 25-38

D Rees Jones, A Wells

The growth of frazil or granular ice is an important mode of ice formation in the cryosphere. Recent advances have improved our understanding of the microphysical processes that control the rate of ice-crystal growth when water is cooled beneath its freezing temperature. These advances suggest that crystals grow much faster than previously thought. In this paper, we consider models of a population of ice crystals with different sizes to provide insight into the treatment of frazil ice in large-scale models. We consider the role of crystal growth alongside the other physical processes that determine the dynamics of frazil ice. We apply our model to a simple mixed layer (such as at the surface of the ocean) and to a buoyant plume under a floating ice shelf. We provide numerical calculations and scaling arguments to predict the occurrence of frazilice explosions, which we show are controlled by crystal growth, nucleation and, gravitational removal. Faster crystal growth, higher secondary nucleation and slower gravitational removal make frazil-ice explosions more likely. We identify steady-state crystal size distributions, which are largely insensitive to crystal growth rate but are affected by the relative importance of secondary nucleation to gravitational removal. Finally, we show that the fate of plumes underneath ice shelves is dramatically affected by frazil-ice dynamics. Differences in the parameterization of crystal growth and nucleation give rise to radically different predictions of basal accretion and plume dynamics; and can even impact whether a plume reaches the end of the ice shelf or intrudes at depth.


Wave-mean flow interactions in the atmospheric circulation of tidally locked planets

Astrophysical Journal IOP Publishing 869 (2018)

M Hammond, R Pierrehumbert

We use a linear shallow-water model to investigate the global circulation of the atmospheres of tidally locked planets. Simulations, observations, and simple models show that if these planets are sufficiently rapidly rotating, their atmospheres have an eastward equatorial jet and a hot-spot east of the substellar point. We linearize the shallow-water model about this eastward flow and its associated geostrophic height perturbation. The forced solutions of this system show that the shear flow explains the form of the global circulation, particularly the hot-spot shift and the positions of the cold standing waves on the night-side. We suggest that the eastward hot-spot shift in observations and 3D simulations of these atmospheres is caused by the zonal flow Doppler-shifting the stationary wave response eastwards, summed with the geostrophic height perturbation from the flow itself. This differs from other studies which explained the hot-spot shift as pure advection of heat from air flowing eastward from the substellar point, or as equatorial waves travelling eastwards. We compare our solutions to simulations in our climate model Exo-FMS and show that they matched the position of the eastward-shifted hot-spot, and the global wind pattern. We discuss how planetary properties affect the global circulation, and how they change observables such as the hot-spot shift or day-night contrast. We conclude that the wave-mean flow interaction be tween the stationary planetary waves and the equatorial jet is a vital part of the equilibrium circulation on tidally locked planets.


Ice-shelf damming in the glacial Arctic Ocean: dynamical regimes of a basin-covering kilometre-thick ice shelf

Cryosphere European Geosciences Union 11 (2017) 1745-1765

J Nilsson, M Jakobsson, N Kirchner, C Borstad, G Björk, R Pierrehumbert, C Stranne

Recent geological and geophysical data suggest that a 1km thick ice shelf extended over the glacial Arctic Ocean during Marine Isotope Stage 6, about 140000 years ago. Here, we theoretically analyse the development and equilibrium features of such an ice shelf, using scaling analyses and a one-dimensional ice-sheet–ice-shelf model. We find that the dynamically most consistent scenario is an ice shelf with a nearly uniform thickness that covers the entire Arctic Ocean. Further, the ice shelf has two regions with distinctly different dynamics: a vast interior region covering the central Arctic Ocean and an exit region towards the Fram Strait. In the interior region, which is effectively dammed by the Fram Strait constriction, there are strong back stresses and the mean ice-shelf thickness is controlled primarily by the horizontally integrated mass balance. A narrow transition zone is found near the continental grounding line, in which the ice-shelf thickness decreases offshore and approaches the mean basin thickness. If the surface accumulation and mass flow from the continental ice masses are sufficiently large, the ice-shelf thickness grows to the point where the ice shelf grounds on the Lomonosov Ridge. As this occurs, the back stress increases in the Amerasian Basin and the ice-shelf thickness becomes larger there than in the Eurasian Basin towards the Fram Strait. Using a one-dimensional ice-dynamic model, the stability of equilibrium ice-shelf configurations without and with grounding on the Lomonosov Ridge are examined. We find that the grounded ice-shelf configuration should be stable if the two Lomonosov Ridge grounding lines are located on the opposites sides of the ridge crest, implying that the downstream grounding line is located on a downward sloping bed. This result shares similarities with the classical result on marine ice-sheet stability of Weertman, but due to interactions between the Amerasian and Eurasian ice-shelf segments the mass flux at the downstream grounding line decreases rather than increases with ice thickness.


Frazil-ice growth rate and dynamics in mixed layers and sub-ice-shelf plumes

Cryosphere Discussions European Geosciences Union 12 (2017) 25-38

D Rees Jones, AJ Wells

The growth of frazil or granular ice is an important mode of ice formation in the cryosphere. Recent advances have improved our understanding of the microphysical processes that control the rate of ice-crystal growth when water is cooled beneath its freezing temperature. These advances suggest that crystals grow much faster than previously thought. In this paper, we consider models of a population of ice crystals with different sizes to provide insight into the treatment of frazil ice in large-scale models. We consider the role of crystal growth alongside the other physical processes that determine the dynamics of frazil ice. We apply our model to a simple mixed layer (such as at the surface of the ocean) and to a buoyant plume under a floating ice shelf. We provide numerical calculations and scaling arguments to predict the occurrence of frazil-ice explosions, which we show are controlled by crystal growth, nucleation and, gravitational removal. Faster crystal growth, higher secondary nucleation and slower gravitational removal make frazil-ice explosions more likely. We identify steady-state crystal size distributions, which are largely insensitive to crystal growth rate but are affected by the relative importance of secondary nucleation to gravitational removal. Finally, we show that the fate of plumes underneath ice shelves is dramatically affected by frazil-ice dynamics. Differences in the parameterization of crystal growth and nucleation give rise to radically different predictions of basal accretion and plume dynamics; and can even impact whether a plume reaches the end of the ice shelf or intrudes at depth.


Observational evidence against strongly stabilizing tropical cloud feedbacks

Geophysical Research Letters American Geophysical Union 44 (2017) 1503-1510

IN Williams, R Pierrehumbert

We present a method to attribute cloud radiative feedbacks to convective processes, using sub-cloud layer buoyancy as a diagnostic of stable and deep convective regimes. Applying this approach to tropical remote-sensing measurements over years 2000-2016 shows that an inferred negative short-term cloud feedback from deep convection was nearly offset by a positive cloud feedback from stable regimes. The net cloud feedback was within statistical uncertainty of the NCAR Community Atmosphere Model (CAM5) with historical forcings, with discrepancies in the partitioning of the cloud feedback into convective regimes. Compensation between high-cloud responses to tropics-wide warming in stable and unstable regimes resulted in smaller net changes in high-cloud fraction with warming. In addition, deep convection and associated high clouds set in at warmer temperatures in response to warming, as a consequence of nearly invariant sub-cloud buoyancy. This invariance further constrained the magnitude of cloud radiative feedbacks, and is consistent with climate model projections.


Linking the climate and thermal phase curve of 55 Cancri e

Astrophysical Journal American Astronomical Society (2017)

M Hammond, R Pierrehumbert

The thermal phase curve of 55 Cancri e is the first measurement of the temperature distribution of a tidally locked Super-Earth, but raises a number of puzzling questions about the planet's climate. The phase curve has a high amplitude and peak offset, suggesting that it has a significant eastward hot-spot shift as well as a large day-night temperature contrast. We use a general circulation model to model potential climates, and investigate the relation between bulk atmospheric composition and the magnitude of these seemingly contradictory features. We confirm theoretical models of tidally locked circulation are consistent with our numerical model of 55 Cnc e, and rule out certain atmospheric compositions based on their thermodynamic properties. Our best-fitting atmosphere has a significant hot- spot shift and day-night contrast, although these are not as large as the observed phase curve. We discuss possible physical processes which could explain the observations, and show that night-side cloud formation from species such as SiO from a day-side magma ocean could potentially increase the phase curve amplitude and explain the observations. We conclude that the observations could be explained by an optically thick atmosphere with a low mean molecular weight, a surface pressure of several bar and a strong eastward circulation, with night-side cloud formation a possible explanation for the difference between our model and the observations.


Mountain glaciers as paleoclimate proxies

Annual Review of Earth and Planetary Sciences Annual Reviews 49 (2017) 649-680

AN Mackintosh, BM Anderson, R Pierrehumbert

Glaciers offer the potential to reconstruct past climate over timescales from decades to millennia. They are found on nearly every continent, and at the Last Glacial Maximum, glaciers were larger in all regions on Earth. The physics of glacier-climate interaction is relatively well understood, and glacier models can be used to reconstruct past climate from geological evidence of past glacier extent. This can lead to significant insights regarding past, present and future climate. For example, glacier modelling has demonstrated that the near ubiquitous global pattern of glacier retreat during the last few centuries resulted from a global-scale climate warming of ~1°C, consistent with instrumental data and climate proxy records. Climate reconstructions from glaciers also demonstrated that the tropics were colder at the Last Glacial Maximum than was originally inferred from sea surface temperature reconstructions. Future efforts to reconstruct climate from glaciers may provide new constraints on climate sensitivity to CO2 forcing, polar amplification of climate change, and more.

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