Anomalous convective flows carve pinnacles and scallops in melting ice

Scott Weady1, Joshua Tong1,2, Alexandra Zidovska2 & Leif Ristroph1∗1Applied Math Lab, Courant Institute, New York University, New York NY 10012 USA

2Department of Physics, New York University, New York NY 10003 USA(Dated: January 28, 2022)

We report on the shape dynamics of ice suspended in cold fresh water and subject to the naturalconvective flows generated during melting. Experiments reveal shape motifs for increasing far-fieldtemperature: Sharp pinnacles directed downward at low temperatures, scalloped waves for inter-mediate temperatures between 5 and 7◦C, and upward pointing pinnacles at higher temperatures.Phase-field simulations reproduce these morphologies, which are closely linked to the anomalousdensity-temperature profile of liquid water. Boundary layer flows yield pinnacles that sharpen withaccelerating growth of tip curvature while scallops emerge from a Kelvin-Helmholtz-like instabilitycaused by counterflowing currents that roll up to form vortex arrays. By linking the molecular-scaleeffects underlying water’s density anomaly to the macro-scale flows that imprint the surface, theseresults show that the morphology of melted ice is a sensitive indicator of ambient temperature.

The shape of a landform or landscape holds clues toits history and the environmental conditions under whichit developed. However, interpreting geological morpholo-gies is challenging due to the complex multi-scale andinteractive processes involved, such as erosion and de-position, dissolution and solidification, and melting andfreezing [1–6]. The latter yield examples across scales, in-cluding rippled icicles, pinnacle shaped icebergs, texturedice caves, and larger icescapes [7–12]. Understanding howto interpret such forms and the physical mechanisms be-hind them is all the more important due to the increasingmelt rate of the Earth’s ice reserves [13, 14].

Melting is an example of a Stefan problem, which clas-sically seeks to determine interface motion induced bya phase transition [15]. Here the solid-liquid interfacerecedes due to temperature gradients normal to the sur-face, and the energy released during phase change in turnmodifies the temperature field in the fluid. In many situa-tions, these temperature changes cause density variationsthat drive gravitational convective flows, which also feedback on the interface motion [16, 17]. This convectiveStefan problem has recently been studied in the relatedcontext of solids dissolving into liquids, where the effectsof flows due to solutal convection can be seen in fine-scalesurface features and overall forms [18–20].

Convective flows are well studied in heat transfer prob-lems involving fixed boundaries [21, 22], but the effectsof shape-flow coupling are less understood. The problemis uniquely complex for melting ice due to the unusualeffect of temperature on liquid water’s density, which dis-plays a maximum at about 4◦C. This so-called densityanomaly, while ultimately a molecular-scale effect thatleads to relative density differences on the order of 0.01%[23], nonetheless strongly affects convective flows, heattransfer characteristics, and hydrodynamic instabilitiesacross a wide range of scales [24–32].

Here we show that the density anomaly and conse-quent flows are imprinted onto the shape of melting ice.We consider the highly simplified context of ice sub-

merged within fresh water while subject to the convec-tive flows generated during melting. Reporting first onexperiments, we manufacture clear ice using a directionalfreezing method [33, 34], and immerse it in water of fixedfar-field temperature T∞ ∈ [2, 10]◦C. We focus on cylin-drical initial forms that are sized, supported, and ori-ented vertically to allow for observation of the long-timeshape dynamics. The ice is rigidly supported underwa-ter on a plastic base that is located either at the top orbottom of the ice depending on the ambient water tem-perature. Using a large tank in a cold room facility, thefar-field water temperature is controlled and systemati-cally varied to assess its impact on shape development,as captured by time-lapse photography. Prior to meltingin water, nearly uniform internal temperature of 0◦C is

(a) (b) (c)4 C 5.6 C 8 C

1 cm

FIG. 1. Representative morphologies formed by melting icein laboratory experiments. (a) For sufficiently cold ambi-ent temperatures T∞ . 5◦C, the ice tapers from below toform an inverted pinnacle. (b) For intermediate temperatures5◦C . T∞ . 7◦C, scalloped patterns form on the surface. (c)Upright pinnacles form for warmer temperatures T∞ & 7◦C.Photographs capture the late-stage ice removed from waterand under diffuse lighting.

arX

iv:2

111.

0993

7v2

[ph

ysic

s.fl

u-dy

n] 2

7 Ja

n 20

22

2

FIG. 2. Flow velocity (arrows) and temperature (color) fields from phase-field simulations for (a) T∞ = 4◦C, (b) 5.6◦C, and(c) 8◦C at early (left) and late (right) times. Curves show the flow speed (yellow) and density (pink) profiles at early times.Melting in cold water is associated with upward flow and downwards tapering of the ice as in (a), whereas warmer temperaturesinvolve downwards flow and tapering at the top as in (c). Intermediate temperatures yield shear flows involving rising fluidnear the surface and sinking outer flows, driving an instability that later patterns the surface.

achieved by leaving the ice in room temperature air for atleast 30 minutes, which is long compared to the timescaleof thermal diffusion. Experimental details are availableas Supplemental Material.

We first present some motivating observations from ex-periments, which reveal three distinct morphologies thatarise for specific intervals of the far-field temperature.Representative photographs are shown in Fig. 1. Forsufficiently low temperatures T∞ . 5◦C, the ice becomestapered at its lower end to form an inverted pinnaclewith its apex pointing downward, as shown in panel(a). For such conditions, the base is at the top of theice, which avoids interference with the upward boundarylayer flows to be discussed below. At higher tempera-tures T∞ & 7◦C, a similarly shaped but upright pinnacleforms, shown in (c), where the base is at the bottom ofthe ice. For intermediate temperatures 5◦C . T∞ . 7◦C,intricate wavy and scalloped features pattern the ice sur-face, as shown in (b).

The sensitive dependence of shape on temperatureevokes water’s anomalous density-temperature profile,whose peak at T∗ ≈ 4◦C suggests distinct scenarios cate-gorized by the far-field temperature T∞. For sufficientlywarm temperatures, the cold liquid near the surface isuniformly denser than that in the far-field and is expectedto sink. For cold temperatures, however, the densityanomaly upends intuition: Cold liquid near the surfaceis less dense and will rise. Intermediate temperatures aremore subtle, since the coldest fluid near the surface is lessdense than that in the far field, while fluid slightly fur-ther away must be at or near T∗ and is thus more dense.The resulting flows are not easily inferred.

To more clearly interpret the observed morphologies,we formulate and implement simulations of the shape dy-namics coupled to the natural convective flows. Here weuse the phase-field model [35, 36], which has proven suc-cessful for moving boundary problems with natural con-vection [37–39]. In this model, material is implicitly rep-resented by a continuous phase parameter φ(x, t), which

takes values φ = 0 in the solid phase and φ = 1 in theliquid, with the interface defined as the level set φ = 1/2.This phase parameter is then used to describe energycontributions from phase change and to approximate theno-slip boundary condition on the ice, while admittingnumerical discretization on a Cartesian grid. Introducingthe quadratic equation of state ρ(T ) = ρ∗[1−β(T −T∗)2]as a basic model for the density anomaly [24, 26], thefluid motion is described by the Navier-Stokes equationin the Boussinesq approximation

Du

Dt= Pr

(−∇p+ ∆u + Raθ2z

)− η(1− φ)2u, (1)

∇ · u = 0, (2)

where u(x, t) is the velocity, p(x, t) is the pressure, andθ(x, t) = (T (x, t)− T∗)/(T∞ − T0) is the dimensionlesstemperature. Parameters include the Rayleigh numberRa = gβ(T∞ − T0)2H3/νκT , which compares buoyantand viscous forces, and the Prandtl number Pr = ν/κT ,which assesses viscous and thermal diffusivities. Here gis acceleration due to gravity, H is the initial height ofthe ice, ν is the fluid viscosity, and κT is the thermal dif-fusivity. The last term in (1) is a Brinkman penalizationforce that models the ice as a porous medium, in whichthe velocity vanishes for large resistivity η � 1 [40].

Following thermodynamic derivations [37], the temper-ature and phase fields satisfy the evolution equations

Dθ

Dt= ∆θ − 1

St

df

dφ

∂φ

∂t, (3)

∂φ

∂t= m∆φ+

m(θ − θ0)

δ2df

dφ− m

4δ2dg

dφ. (4)

The last term in Eq. (3) captures energy contributionsfrom phase change, the magnitude of which is controlledby the Stefan number St = cp(T∞ − T0)/L, with cpthe heat capacity and L the latent heat of fusion. InEq. (4), m is a regularization parameter, δ is an ef-fective interface thickness, θ0 = (T0 − T∗)/(T∞ − T0) is

3

FIG. 3. Dynamics of pinnacle formation. Panels (a)-(d) show interfaces for T∞ = 4◦C and T∞ = 8◦C extracted over time(dark to light blue) in experiments and simulations. The corresponding tip curvatures are plotted in (e), which exhibit rapid

sharpening to micro-scales. At early times, all data follow the scaling law κ0(t) = κ0(0)(1 − t/ts)−4/5, as shown in (f) by the

linear behavior 1 − t/ts of the transformed quantity [κ0(t/ts)]−5/4 = [κ0(t/ts)/κ0(0)]−5/4.

the dimensionless melting temperature, and the functionsf(φ) = φ3(10 − 15φ + 5φ2) and g(φ) = φ2(1 − φ)2 arepotentials that ensure no phase-change occurs away fromthe interface. In the limit δ → 0 and η →∞, the system(1)-(4) recovers the Navier-Stokes equations with no-slipboundary conditions on the ice and the Stefan conditionfor the interface velocity Vn = St ∂θ/∂n, where the tem-perature is assumed to be T0 throughout the solid [37].

For comparison with the cylindrical geometries in ex-periments, we solve the system of equations (1)-(4) onan axisymmetric domain sufficiently wide such that thefar-field temperature remains constant to within 0.1◦C.Estimated from the experimental parameters, we takePr = 12 and Ra/T 2

∞ = 2.5 × 106 (◦C)−2. The selectedStefan number St/T∞ = 0.05 (◦C)−1 is larger than theexperimental value St/T∞ = 0.012 (◦C)−1, which reducessimulation run time while having negligible effect on theshape dynamics. Upon initialization, the temperaturein the solid phase is set to T0 while the temperature inthe liquid is T∞. See the Supplemental Material for im-plementation details, including spatial and temporal dis-cretization [41].

Figure 2 shows the interfaces at early and late timesfor three representative values of T∞. Also shown arethe flow velocity and temperature fields as well as curvesrepresenting the density and velocity profiles along a hor-izontal transect at early times, the latter substantiatingour inferences based on the density anomaly. For thecase T∞ = 4◦C of (a), an upward boundary layer flowpersists for all times. This flow is fed by warmer outer

fluid that is continuously entrained from the sides andbottom, offering a mechanism for the enhanced melt ratethat tapers the ice from below. For T∞ = 8◦C, shownin (c), the situation is similar but inverted: A downwardboundary layer flow tapers the top of the ice to form anupright pinnacle. For the intermediate case T∞ = 5.6◦Cof (b), early times are marked by a thin region of upwardflow near the surface surrounded by a broader region ofdownward flow. This shear flow eventually destabilizesand forms recirculating vortices that entrain the warmerouter fluid and carve scallop-shaped indentations. Acrossall cases, stagnation points of the flow are associated withsharp features of the surface, including the apexes of thepinnacles and cusped crests of the scalloped waves.

Further analysis of the experiments and simulationsprovides insight into the mathematical structure of theshape dynamics. Considering first the pinnacles observedfor sufficiently low or high T∞, we show in Fig. 3(a)-(d)comparisons of the shape progression as measured in ex-periments and computed in simulations for T∞ = 4◦Cand 8◦C. The strong agreement across all times serves asa cross-validation of the simulations and experiments anddemonstrates the robustness of the pinnacle form. Thesepinnacles are reminiscent of those recently observed forbodies dissolving in natural convective flows [6, 17–19],for which a boundary layer theory analysis predicts thatthe pinnacle apex sharpens via a power law growth of cur-vature: κ0(t) = κ0(0)(1 − t/ts)−4/5 [19, 22]. Here κ0(0)is the initial tip curvature and ts is the blow-up time forthe predicted singular dynamics, which were shown to

4

FIG. 4. Dynamics of scallop formation. Panels (a) and (b) show extracted interfaces over time for T∞ = 5.6◦C in experimentand simulation, respectively. Panel (c) plots the cusp-to-cusp wavelength of the scallops versus their vertical location or height,

confirming the scaling λ ∼ h−3/4 predicted by an analysis of the viscous Kelvin-Helmholtz instability. Additional simulations(filled squares) for a taller body and thus higher Ra confirm the trend over wider ranges of the variables.

accurately describe the initial stages of sharpening [19].

To test this law, we extract the interfaces over timefrom experiments and simulations and evaluate the apexcurvature by fitting and differentiating a fourth-orderpolynomial for the tip height as a function of radius.Strikingly, the tip curvature κ0(t), shown in Fig. 3(e),exhibits steep and seemingly unbounded growth. The ra-dius of curvature reaches values smaller than 100 microns,as fine as a human hair and approaching the resolutions ofthe experiments and simulations. In panel (f) we plot therescaled quantity [κ0(t/ts)]

−5/4 = [κ0(t/ts)/κ0(0)]−5/4,where ts is treated as a fitting parameter based on thepredicted power law. Remarkably, all data collapse tothe predicted linear form 1− t/ts (dashed line) for earlytimes, indicating a mechanism shared with dissolution forthe formation of ultra-sharp structures. The curvaturecontinues to grow at later times but falls off the singularpace, an effect also observed for dissolution pinnacles andwhich is the subject of recent studies [19, 42].

Experiments and simulations are also in agreement forintermediate temperatures, yielding scallops of compara-ble scales as seen in Figs. 4(a) and (b) for T∞ = 5.6◦C.Some disparities are expected as the simulations are ax-isymmetric while the patterns are three-dimensional inexperiments, with the images of (a) representing cross-sectional views. Nonetheless, the wave-like structurescommon to both are indicative of a hydrodynamic insta-bility. Shear flows of the form observed in Fig. 2(b) areknown to undergo the Kelvin-Helmholtz instability, theclassic analysis of which involves counter-flowing layers ofinviscid fluid [43, 44]. Such flows are unstable, with thesmallest wavelengths growing at the fastest rates. Viscos-

ity, however, suppresses high frequency modes, yieldinga most unstable wavelength λ ∼ Re−1/2, where Re isthe Reynolds number [45, 46]. While Re is somewhatpoorly defined since the flows accelerate along the sur-face, scaling arguments predict Re ∼ Ra1/2 for Ra � 1[47], with experimental evidence suggesting an exponentslightly below 1/2 [48, 49]. Observing that Ra ∼ h3,where h is the vertical distance from the bottom of theice, we predict the wavelength decreases up the surfaceof the ice as λ ∼ h−3/4.

This scaling law can be used to test the hypothesis thatthe maximally unstable wavelength of the shear flow isimprinted on the ice in the form of scalloped waves. Wedetermine the locations of the wave crests at early timesand identify the wavelength λ with differences betweensuccessive crests and the height h with their midpoint.Figure 4(c) shows results from experiments (open circles)and simulations (open squares) at T∞ = 5.6◦C. Here, h isnormalized by the total height and λ by its value at h = 1,and the location of h = 0 is treated as a fitting parame-ter due to the ambiguous location of the lowest scallop.The uppermost scallop, whose longer wavelength seemsto arise from the top boundary conditions, is excludedfrom this analysis. Over the few wavelengths present,the data indeed follow the −3/4 scaling law. Additionalsimulations of taller ice (filled squares), details of whichare given in the Supplemental Material, yield more wave-lengths and correspondingly more convincing agreement.

These findings show that the shape of ice is a sen-sitive indicator of the ambient temperature at which itmelted. Sharply-pointed pinnacles directed downwardsfor T∞ . 5◦C and upwards for T∞ & 7◦C are formed by

5

persistent and unidirectional boundary layer flows thatrise in the former case and sink in the latter. This lat-ter case parallels the upright pinnacles carved by down-ward flows observed during dissolution [6, 17–20], whichis expected by the analogous mathematical descriptionsof thermal and solutal convection. It remains for futurestudies to determine if the shape dynamics are quantita-tively different for melting and dissolving due to the dif-ferences in their microscale physics. The scalloped wavesobserved here for 5◦C . T∞ . 7◦C have their origin inbidirectional flows due to the buoyant rise of cold waternear the surface and the sinking of warmer water furtheroutward in the boundary layer. The resulting shear flowsundergo a Kelvin-Helmholtz instability and roll up intovortices that carve pits in the surface.

Pinnacles are commonly observed on icebergs andhave been qualitatively attributed to buoyancy-drivenflows [11, 50, 51]. However, they have not previouslybeen reproduced in laboratory experiments, nor vali-dated through fluid dynamical models or simulations.Scalloped patterns on icebergs, ice shelves, and bore holesare generally attributed to instabilities due to externally-driven flows [7, 52, 53]. In contrast, the mechanismrevealed here is rooted in the intrinsic flows generatedby water’s anomalous density characteristics, and scal-lops formed in this way can be distinguished by theirincreasing wavelength with depth. While our resultspertain strictly to fresh water, the identified shape mo-tifs may persist in saltwater solutions up to the criti-cal concentration above which the anomaly is lost anddensity decreases monotonically with temperature [25].Future studies that vary both far-field temperature andsalinity should assess how the long-time shape dynamicsis impacted by the associated double-diffusive processes[25, 54].

We are grateful for support from an NSF graduate fel-lowship to S.W., an NYU WiPhy fellowship to J.T., andNSF grants PHY-1554880 to A.Z. and CBET-1805506and DMS-1646339 to L.R.

∗ ristroph@cims.nyu.edu[1] J. M. Huang, M. N. J. Moore, and L. Ristroph, Journal

of Fluid Mechanics 765, R3 (2015).[2] W. W. Mullins and R. F. Sekerka, Journal of Applied

Physics 34, 323 (1963).[3] J. S. Wettlaufer, M. G. Worster, and H. E. Huppert,

Journal of Fluid Mechanics 344, 291–316 (1997).[4] L. Ristroph, M. N. J. Moore, S. Childress, M. J. Shelley,

and J. Zhang, Proceedings of the National Academy ofSciences 109, 19606 (2012).

[5] J. N. Hewett and M. Sellier, Journal of Fluids and Struc-tures 70, 295 (2017).

[6] E. Nakouzi, R. E. Goldstein, and O. Steinbock, Lang-muir : the ACS journal of surfaces and colloids 31 (2015),10.1021/la503562z.

[7] R. Curl, Trans. Cave Res. Group 7 , 121 (1966).[8] S. Kobayashi, Contributions from the Institute of Low

Temperature Science A29, 1 (1980).[9] J. A. Neufeld, R. E. Goldstein, and M. G. Worster, Jour-

nal of Fluid Mechanics 647, 287–308 (2010).[10] C. Camporeale and L. Ridolfi, Journal of Fluid Mechanics

694, 225–251 (2012).[11] Y. A. Romanov, N. A. Romanova, and P. Romanov,

Antarctic Science 24, 77–87 (2012).[12] S. Filhol and M. Sturm, Journal of Geophysical Research:

Earth Surface 120, 1645 (2015).[13] J. Chen, C. Wilson, and B. Tapley, Science 313, 1958

(2006).[14] J. Chen, C. Wilson, D. Blankenship, and B. Tapley,

Nature Geoscience 2, 859 (2009).[15] L. I. Rubenstein, The Stefan Problem (American Math-

ematical Society, 1971).[16] L. Ristroph, Journal of Fluid Mechanics 838, 1–4 (2018).[17] S. S. Pegler and M. S. Davies Wykes, Journal of Fluid

Mechanics 915, A86 (2021).[18] M. S. Davies Wykes, J. M. Huang, G. A. Hajjar, and

L. Ristroph, Phys. Rev. Fluids 3, 043801 (2018).[19] J. M. Huang, J. Tong, M. Shelley, and L. Ristroph, Pro-

ceedings of the National Academy of Sciences 117, 23339(2020).

[20] S. S. Pegler and M. S. Davies Wykes, Journal of FluidMechanics 900, A35 (2020).

[21] D. J. Tritton, Physical Fluid Dynamics (Springer Nether-lands, 1977).

[22] H. Schlichting and K. Gersten, Boundary layer theory(Springer, Berlin, Heidelberg, 2017).

[23] A. Nilsson and L. G. M. Pettersson, Nature Communi-cations 6 (2015), 10.1038/ncomms9998.

[24] G. Veronis, Astrophysical Journal 137, 641 (1963).[25] J. M. Higgins and B. Gebhart, Journal of Heat Transfer

105, 767 (1983).[26] S. Toppaladoddi and J. S. Wettlaufer, Phys. Rev. Fluids

3, 043501 (2018).[27] M. S. Bendell and B. Gebhart, International Journal of

Heat and Mass Transfer 19, 1081 (1976).[28] T. Saitoh, Applied Scientific Research 32 (1976),

10.1007/BF00385849.[29] B. Gebhart and T. Wang, Chemical Engineering Com-

munications 13, 197 (1982).[30] D. White, R. Viskanta, and W. Leidenfrost, Experiments

in Fluids 4, 171 (1986).[31] L.-A. Couston and M. Siegert, Science Advances 7,

eabc3972 (2021).[32] Q. Wang, P. Reiter, D. Lohse, and O. Shishkina, Phys.

Rev. Fluids 6, 063502 (2021).[33] A. E. Carte, Proceedings of the Physical Society (1958-

1967) 77, 757 (1961).[34] N. Maeno, Physics of Snow and Ice: proceedings 1, 207

(1967).[35] S.-L. Wang, R. Sekerka, A. Wheeler, B. Murray,

S. Coriell, R. Braun, and G. McFadden, Physica D: Non-linear Phenomena 69, 189 (1993).

[36] C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma,and X. Tong, Journal of Computational Physics 154, 468(1999).

[37] B. Favier, J. Purseed, and L. Duchemin, Journal of FluidMechanics 858, 437–473 (2019).

[38] L.-A. Couston, E. Hester, B. Favier, J. R. Taylor, P. R.Holland, and A. Jenkins, Journal of Fluid Mechanics

6

911, A44 (2021).[39] E. W. Hester, C. D. McConnochie, C. Cenedese, L.-A.

Couston, and G. Vasil, Phys. Rev. Fluids 6, 023802(2021).

[40] P. Angot, C.-H. Bruneau, and P. Fabrie, NumerischeMathematik 81, 497 (1999).

[41] D. L. Brown, R. Cortez, and M. L. Minion, Journal ofComputational Physics 168, 464 (2001).

[42] J. M. Huang and N. J. Moore, “Morphological at-tractors in natural convective dissolution,” (2021),arXiv:2109.02212.

[43] S. Chandrasekhar, Hydrodynamic and HydromagneticStability (Dover, 1961).

[44] P. G. Drazin and W. H. Reid, Hydrodynamic Stability ,2nd ed., Cambridge Mathematical Library (CambridgeUniversity Press, 2004).

[45] D. W. Moore, Studies in Applied Mathematics 58, 119(1978).

[46] M. R. Dhanak, Journal of Fluid Mechanics 269, 265–281(1994).

[47] S. Grossmann and D. Lohse, Journal of Fluid Mechanics407, 27–56 (2000).

[48] X.-L. Qiu and P. Tong, Phys. Rev. E 64, 036304 (2001).[49] S. Grossmann and D. Lohse, Phys. Rev. E 66, 016305

(2002).[50] T. Moon, D. A. Sutherland, D. Carroll, D. Felikson,

L. Kehrl, and F. Straneo, Nature Geoscience 11 (2018),10.1038/s41561-017-0018-z.

[51] G. Brostrom, A. Melsom, M. Sayed, and I. Kubat, Jour-nal of Geophysical Research 99, 3337 (2009).

[52] P. Claudin, O. Duran, and B. Andreotti, Journal of FluidMechanics 832, R2 (2017).

[53] M. Bushuk, D. M. Holland, T. P. Stanton, A. Stern,and C. Gray, Journal of Fluid Mechanics 873, 942–976(2019).

[54] E. G. Josberger and S. Martin, Journal of Fluid Mechan-ics 111, 439–473 (1981).