Publications associated with Ice and Fluid Dynamics


Vertically resolved magma ocean–protoatmosphere evolution: H2 , H2O, CO2, CH4, CO, O2, and N2 as primary absorbers

Journal of Geophysical Research: Planets American Geophysical Union (2021)

T Lichtenberg, DJ Bower, M Hammond, R Boukrouche, P Sanan, S Tsai, RT Pierrehumbert

The earliest atmospheres of rocky planets originate from extensive volatile release during magma ocean epochs that occur during assembly of the planet. These establish the initial distribution of the major volatile elements between different chemical reservoirs that subsequently evolve via geological cycles. Current theoretical techniques are limited in exploring the anticipated range of compositional and thermal scenarios of early planetary evolution, even though these are of prime importance to aid astronomical inferences on the environmental context and geological history of extrasolar planets. Here, we present a coupled numerical framework that links an evolutionary, vertically‐resolved model of the planetary silicate mantle with a radiative‐convective model of the atmosphere. Using this method we investigate the early evolution of idealized Earth‐sized rocky planets with end‐member, clear‐sky atmospheres dominated by either H2, H2O, CO2, CH4, CO, O2, or N2. We find central metrics of early planetary evolution, such as energy gradient, sequence of mantle solidification, surface pressure, or vertical stratification of the atmosphere, to be intimately controlled by the dominant volatile and outgassing history of the planet. Thermal sequences fall into three general classes with increasing cooling timescale: CO, N2, and O2 with minimal effect, H2O, CO2, and CH4 with intermediate influence, and H2 with several orders of magnitude increase in solidification time and atmosphere vertical stratification. Our numerical experiments exemplify the capabilities of the presented modeling framework and link the interior and atmospheric evolution of rocky exoplanets with multi‐wavelength astronomical observations.


Thermal Convection over Fractal Surfaces

ArXiV [physics.flu-dyn] (2019)

S Toppaladoddi, A Wells, CR Doering, JS Wettlaufer


Thermal convection over fractal surfaces

Journal of Fluid Mechanics Cambridge University Press 907 (2020) A12

S Toppaladoddi, A Wells, CR Doering, JS Wettlaufer

We use well resolved numerical simulations with the Lattice Boltzmann Method to study Rayleigh-B´enard convection in cells with a fractal boundary in two dimensions for P r = 1 and Ra ∈ [10^7 , 10^10]. The fractal boundaries are functions characterized by power spectral densities S(k) that decay with wavenumber, k, as S(k) ∼ k^p (p < 0). The degree of roughness is quantified by the exponent p with p < −3 for smooth (differentiable) surfaces and −3 ≤ p < −1 for rough surfaces with Hausdorff dimension D_f =1/2 (p + 5). By computing the exponent β in power law fits Nu ∼ Ra^β, where Nu and Ra are the Nusselt and the Rayleigh numbers for Ra ∈ [10^8, 10^10], we observe that heat transport scaling increases with roughness over the top two decades of Ra ∈ [10^8, 10^10]. For p = −3.0, −2.0 and −1.5 we find β = 0.288 ± 0.005, 0.329 ± 0.006 and 0.352 ± 0.011, respectively. We also observe that the Reynolds number, Re, scales as Re ∼ Ra^ξ , where ξ ≈ 0.57 over Ra ∈ [10^7, 10^10], for all p used in the study. For a given value of p, the averaged Nu and Re are insensitive to the specific realization of the roughness.


Modeling sea ice

Notices of the American Mathematical Society American Mathematical Society 67 (2020) 1535-1555

K Golden, L Bennetts, E Cherkaev, I Eisenman, D Feltham, C Horvat, E Hunke, C Jones, D Perovich, P Ponte-Castaneda, C Strong, D Sulsky, A Wells


A synthesis of thermodynamic ablation at ice-ocean interfaces from theory, observations and models

Ocean Modelling Elsevier 154 (2020) 101692

A Malyarenko, A Wells, P Langhorne, N Robinson, M Williams, K Nicholls

Thermodynamic ablation of ice in contact with the ocean is an essential element of ice sheet and ocean interactions but is challenging to model and quantify. Building on earlier observations of sea ice ablation, a variety of recent theoretical, experimental and observational studies have considered ice ablation in contrasting geometries, from vertical to near-horizontal ice faces, and reveal different scaling behaviour for predicted ablation rates in different dynamical regimes. However, uncertainties remain about when the contrasting results should be applied, as existing model parameterisations do not capture all relevant regimes of ice–ocean ablation. To progress towards improved models of ice–ocean​ interaction, we synthesise current understanding into a classification of ablation types. We examine the effect of the classification on the parameterisation of turbulent fluxes from the ocean towards the ice, and identify the dominant processes next to ice interfaces of different orientation. Four ablation types are defined: melting and dissolving based on ocean temperatures, and shear-controlled and buoyancy-controlled regimes based on the dynamics of the near-ice molecular sublayer. We describe existing observational and modelling studies of sea ice, ice shelves, and glacier termini, as well as laboratory studies, to show how they fit into this classification. Two sets of observations from the Ross and Ronne Ice Shelf cavities suggest that both the buoyancy-controlled and shear-controlled regimes may be relevant under different oceanographic conditions. Overall, buoyancy-controlled dynamics are more likely when the molecular sublayer has lower Reynolds number, and shear for higher Reynolds number, although the observations suggest some variability about this trend.


The Phase-curve Signature of Condensible Water-rich Atmospheres on Slowly Rotating Tidally Locked Exoplanets

ASTROPHYSICAL JOURNAL LETTERS 901 (2020) ARTN L33

F Ding, RT Pierrehumbert


The Equatorial Jet Speed on Tidally Locked Planets. I. Terrestrial Planets

ASTROPHYSICAL JOURNAL 901 (2020) ARTN 78

M Hammond, S-M Tsai, RT Pierrehumbert


Simplified 3D GCM modelling of the irradiated brown dwarf WD 0137−349B

Monthly Notices of the Royal Astronomical Society Oxford University Press 496 (2020) 4674-4687

GKH Lee, SL Casewell, KL Chubb, M Hammond, X Tan, S-M Tsai, R Pierrehumbert

White dwarf–brown dwarf short-period binaries (Porb ≲ 2 h) are some of the most extreme irradiated atmospheric environments known. These systems offer an opportunity to explore theoretical and modelling efforts of irradiated atmospheres different to typical hot Jupiter systems. We aim to investigate the three-dimensional (3D) atmospheric structural and dynamical properties of the brown dwarf WD 0137−349B. We use the 3D global circulation model (GCM) Exo-Flexible Modelling System (FMS) with a dual-band grey radiative transfer scheme to model the atmosphere of WD 0137−349B. The results of the GCM model are post-processed using the 3D Monte Carlo radiative transfer model CMCRT. Our results suggest inefficient day–night energy transport and a large day–night temperature contrast for WD 0137−349B. Multiple flow patterns are present, shifting energy asymmetrically eastward or westward depending on their zonal direction and latitude. Regions of overturning are produced on the western terminator. We are able to reproduce the start of the system near-infrared (IR) emission excess at ≳1.95 μm as observed by the Gemini Near-Infrared Spectrograph (GNIRS) instrument. Our model overpredicts the IR phase curve fluxes by factors of ≈1–3, but generally fits the shape of the phase curves well. Chemical kinetic modelling using VULCAN suggests a highly ionized region at high altitudes can form on the dayside of the brown dwarf. We present a first attempt at simulating the atmosphere of a short-period white dwarf–brown dwarf binary in a 3D setting. Further studies into the radiative and photochemical heating from the ultraviolet irradiation are required to more accurately capture the energy balance inside the brown dwarf atmosphere. Cloud formation may also play an important role in shaping the emission spectra of the brown dwarf.


The dynamics of a subglacial salt wedge

Journal of Fluid Mechanics Cambridge University Press 895 (2020) A20

EA Wilson, AJ Wells, IJ Hewitt, C Cenedese

Marine-terminating glaciers, such as those along the coastline of Greenland, often release meltwater into the ocean in the form of subglacial discharge plumes. Though these plumes can dramatically alter the mass loss along the front of a glacier, the conditions surrounding their genesis remain poorly constrained. In particular, little is known about the geometry of subglacial outlets and the extent to which seawater may intrude into them. Here, the latter is addressed by exploring the dynamics of an arrested salt wedge – a steady-state, two-layer flow system where salty water partially intrudes a channel carrying fresh water. Building on existing theory, we formulate a model that predicts the length of a non-entraining salt wedge as a function of the Froude number, the slope of the channel and coefficients for interfacial and wall drag. In conjunction, a series of laboratory experiments were conducted to observe a salt wedge within a rectangular channel. For experiments conducted with laminar flow (Reynolds number <i>Re < 800</i>), good agreement with theoretical predictions are obtained when the drag coefficients are modelled as being inversely proportional to <i>Re</i>. However, for fully turbulent flows on geophysical scales, these drag coefficients are expected to asymptote toward finite values. Adopting reasonable drag coefficient estimates for this flow regime, our theoretical model suggests that typical subglacial channels may permit seawater intrusions of the order of several kilometres. While crude, these results indicate that the ocean has a strong tendency to penetrate subglacial channels and potentially undercut the face of marine-terminating glaciers.


Thermodynamic and energetic limits on continental silicate weathering strongly impact the climate and habitability of wet, rocky worlds

Astrophysical Journal American Astronomical Society 896 (2020) 115

R Graham, R Pierrehumbert

The “liquid water habitable zone” (HZ) concept is predicated on the ability of the silicate weathering feedback to stabilize climate across a wide range of instellations. However, representations of silicate weathering used in current estimates of the effective outer edge of the HZ do not account for the thermodynamic limit on concentration of weathering products in runoff set by clay precipitation, nor for the energetic limit on precipitation set by planetary instellation. We find that when the thermodynamic limit is included in an idealized coupled climate/weathering model, steady-state planetary climate loses sensitivity to silicate dissolution kinetics, becoming sensitive to temperature primarily through the effect of temperature on runoff and to pCO2 through an effect on solute concentration mediated by pH. This increases sensitivity to land fraction, CO2 outgassing, and geological factors such as soil age and lithology, all of which are found to have a profound effect on the position of the effective outer edge of the HZ. The interplay between runoff sensitivity and the energetic limit on precipitation leads to novel warm states in the outer reaches of the HZ, owing to the decoupling of temperature and precipitation. We discuss strategies for detecting the signature of silicate weathering feedback through exoplanet observations in light of insights derived from the revised picture of weathering.


Demonstrating GWP*: a means of reporting warming-equivalent emissions that captures the contrasting impacts of short- and long-lived climate pollutants

Environmental Research Letters IOP Publishing 15 (2020) 044023

JM Lynch, M Cain, RT Pierrehumbert, M Allen

The atmospheric lifetime and radiative impacts of different climate pollutants can both differ markedly, so metrics that equate emissions using a single scaling factor, such as the 100-year Global Warming Potential (GWP100), can be misleading. An alternative approach is to report emissions as 'warming-equivalents' that result in similar warming impacts without requiring a like-for-like weighting per emission. GWP*, an alternative application of GWPs where the CO2-equivalence of short-lived climate pollutant (SLCP) emissions is predominantly determined by changes in their emission rate, provides a straightforward means of generating warming-equivalent emissions. In this letter we illustrate the contrasting climate impacts resulting from emissions of methane, a short-lived greenhouse gas, and CO2, and compare GWP100 and GWP* CO2-equivalents for a number of simple emissions scenarios. We demonstrate that GWP* provides a useful indication of warming, while conventional application of GWP100 falls short in many scenarios and particularly when methane emissions are stable or declining, with important implications for how we consider 'zero emission' or 'climate neutral' targets for sectors emitting different compositions of gases. We then illustrate how GWP* can provide an improved means of assessing alternative mitigation strategies. GWP* allows warming-equivalent emissions to be calculated directly from CO2-equivalent emissions reported using GWP100, consistent with the "Paris Rulebook" agreed by the UNFCCC. It provides a direct link between emissions and anticipated warming impacts, supporting stocktakes of progress towards a long-term temperature goal and compatible with cumulative emissions budgets.


Ice, fire, or fizzle: The climate footprint of Earth's supercontinental cycles

Geochemistry, Geophysics, Geosystems American Geophysical Union 21 (2020) e2019GC008464

M Jellinek, A Lenardic, R Pierrehumbert

Supercontinent assembly and breakup can influence the rate and global extent to which insulated and relatively warm subcontinental mantle is mixed globally, potentially introducing lateral oceanic‐continental mantle temperature variations that regulate volcanic and weathering controls on Earth's long‐term carbon cycle for a few hundred million years. We propose that the relatively warm and unchanging climate of the Nuna supercontinental epoch (1.8–1.3 Ga) is characteristic of thorough mantle thermal mixing. By contrast, the extreme cooling‐warming climate variability of the Neoproterozoic Rodinia episode (1–0.63 Ga) and the more modest but similar climate change during the Mesozoic Pangea cycle (0.3–0.05 Ga) are characteristic features of the effects of subcontinental mantle thermal isolation with differing longevity. A tectonically modulated carbon cycle model coupled to a one‐dimensional energy balance climate model predicts the qualitative form of Mesozoic climate evolution expressed in tropical sea‐surface temperature and ice sheet proxy data. Applied to the Neoproterozoic, this supercontinental control can drive Earth into, as well as out of, a continuous or intermittently panglacial climate, consistent with aspects of proxy data for the Cryogenian‐Ediacaran period. The timing and magnitude of this cooling‐warming climate variability depends, however, on the detailed character of mantle thermal mixing, which is incompletely constrained. We show also that the predominant modes of chemical weathering and a tectonically paced abiotic methane production at mid‐ocean ridges can modulate the intensity of this climate change. For the Nuna epoch, the model predicts a relatively warm and ice‐free climate related to mantle dynamics potentially consistent with the intense anorogenic magmatism of this period.


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 Informa UK Limited 75 (2019) 215-221

R Pierrehumbert

© 2019, © 2019 Bulletin of the Atomic Scientists. To halt global warming, the emission of carbon dioxide into the atmosphere by human activities such as fossil fuel burning, cement production, and deforestation needs to be brought all the way to zero. The longer it takes to do so, the hotter the world will get. Lack of progress towards decarbonization has created justifiable panic about the climate crisis. This has led to an intensified interest in technological climate interventions that involve increasing the reflection of sunlight to space by injecting substances into the stratosphere which lead to the formation of highly reflective particles. When first suggested, such albedo modification schemes were introduced as a “Plan B,” in case the world economy fails to decarbonize, and this scenario has dominated much of the public perception of albedo modification as a savior waiting in the wings to protect the world against massive climate change arising from a failure to decarbonize. But because of the mismatch between the millennial persistence time of carbon dioxide and the sub-decadal persistence of stratospheric particles, albedo modification can never safely play more than a very minor role in the portfolio of solutions. There is simply no substitute for decarbonization.


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 Hitchen, J Parkinson

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.


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.


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

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