Self-sculpting of melting ice by natural convection

Scott Weady

Introduction

Glaciers and icebergs exhibit fascinating morphol-ogy, including structures such as caverns, spikes,and wave-like patterns known as scallops (Figure 1).These features are a signature of the coupling be-tween flow and shape: as ice melts into water, it pro-duces and modifies flows, which non-uniformly meltthe ice surface. The surface recedes, and the processrepeats. This feedback mechanism is not limited toice alone. In fact, melting belongs to a much broaderclass of problems known as moving boundary prob-lems, which encompass a variety of geophysical pro-cesses such as dissolution, erosion, solidification, andablation [1] [2].

(a) (b)

(c) (d)

Figure 1: Overturned icebergs reveal complexmorphology including scalloping (a),(d) and spikes(b),(c).

A unique property of melting ice that separates itfrom other moving boundary problems is that wateris densest a few degrees above its freezing point, afamiliar consequence of which is that ice floats. Thisproperty, commonly referred to as the density in-version, generates exotic flow patterns distinct fromthose arising in classical convection [3] [4].

In this report, we examine how the density inver-sion influences the shape of melting ice. We considerthe effects of natural convection alone, that is, flowdriven solely by density variations. Our approach in-

volves experimental, numerical, and theoretical tech-niques that work together to form a broader pictureof the complex shape dynamics.

We first theoretically formulate the melting prob-lem and its moving boundary conditions, and thendescribe our experimental, numerical, and analyticalapproaches. We conclude with an outlook on furtherapplications and discuss the significance of movingboundary problems in the context of climate change.

Equations of motion

Moving boundary problems are typically formu-lated as partial differential equations (PDEs) in adomain whose boundary is an unspecified functionof time, meaning the boundary itself is part of thesolution. To fully describe the solution, we then needboth boundary conditions for the PDE as well as anevolution equation for the moving interface.

As a starting point, a simple model for the densityof water as a function of temperature T is

ρ(T ) = ρ∗(1− β(T − T∗)2), (1)

where T∗ ≈ 4◦C is the temperature of maximum den-sity ρ∗, and β is a thermal expansion coefficient [5].The full equations of motion with variable densityare both numerically and analytically challenging, soto simplify matters we make the Boussinesq approx-imation, which neglects all density variations exceptthose that multiply gravity. Using this approximationand a standard non-dimensionalization procedure, wearrive at the dimensionless Boussinesq equations,

∂u

∂t+ u · ∇u = Pr(−∇p+ ∆u + Raθ2z), (2a)

∇ · u = 0, (2b)

∂θ

∂t+ u · ∇θ = ∆θ. (2c)

Here u(x, t) is the fluid velocity, p(x, t) is the pres-sure, and θ(x, t) = (T (x, t) − T∗)/(T∞ − T0) is thedeviation from the temperature of maximum density,normalized by the difference in the far-field tempera-ture T∞ and the melting temperature T0 = 0◦C. Thefirst two equations, (2a) and (2b), are the famousNavier-Stokes equations, which describe the incom-pressible velocity field within the fluid, while the last

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equation (2c) describes the evolution of the temper-ature field.

The Boussinesq equations depend on two di-mensionless parameters called the Rayleigh numberRa = gβ(T∞ − T0)2L3/νK and the Prandtl numberPr = ν/K, where g is acceleration due to gravity, L isthe length scale, ν is the viscosity, and K is the ther-mal diffusivity. The Rayleigh number, which is pro-portional to L3, characterizes how turbulent the flowis, while the Prandtl number describes the separationof time scales between the temperature and velocityfields. For water, the Prandtl number is roughly con-stant with Pr = 12, while the Rayleigh number rangesfrom 106 on laboratory scales to 1012 on geophysicalscales.

In this system the temperature and velocity fieldsare coupled through the last term in Equation (2a),Raθ2z, which describes the buoyancy force. Herelarger values of Raθ2 indicate lighter fluid, whilesmaller values indicate denser fluid. Though thisterm is always positive, it can be shown that the pres-sure comes into hydrostatic balance with the far-fieldtemperature, meaning that only the relative forceRa(θ2 − θ2∞) acts on the fluid.

The Stefan condition

Classically, melting boundaries are described bythe Stefan condition, which says the melt rate is pro-portional to the temperature gradient along the sur-face,

vn = St∂θ

∂n.

Here vn is the (dimensionless) normal velocity of theinterface and St = cp(T∞−T0)/L is the Stefan num-ber, where cp is the heat capacity and L is the latentheat of fusion. The Stefan number can roughly becharacterized as the energy required to melt a unitmass of ice. In addition to the Stefan condition, ice isgoverned by the Gibbs-Thomson effect, which causesregions of high curvature to melt faster. To leadingorder, this can be incorporated into the boundarycondition by modifying the normal velocity as

vn = vn(1 + γκ), (3)

where κ is the surface curvature and γ is a materialconstant. With the Boussinesq equations (2a)–(2c)and the Stefan condition (3), we have a complete an-alytic description of the moving boundary problem.

Experiments and Simulation

Our study of melting ice is driven by table-top ex-periments and numerical simulation which motivate

(a)

1 cm

(b)

Figure 2: Close-up images of ice during experimentsat (a) T∞ = 4◦C and (b) T∞ = 6◦C. For T∞ = 4◦Cwe find the ice sharpens from below, while at T∞ =6◦C it develops scallops.

mathematical models. Experiments strip away math-ematical complications, providing a direct avenue tothe full moving boundary problem, while simulationsallow us to perform systematic parameter studies. Inthis section, we review our experimental and numer-ical frameworks and provide an overview of some re-sults. Throughout our study, we are primarily in-terested in characterizing the effects of the far-fieldtemperature T∞, which determines the level of influ-ence the density inversion has on the flow.

Table-top geophysics

As a laboratory model of an iceberg, we consider acylinder of ice, approximately 20cm tall, submergedin a tank of water maintained at temperature T∞. Tocontrol T∞, we work in a cold room, typically used forbiological research, which has operating temperaturesin the range 0− 10◦C.

Our experiments reveal three primary classes of dy-namics. At higher temperatures, T∞ > 8◦C, we findthe ice sharpens from the top, analogous to what hasrecently been discovered during dissolution [2]. Tounderstand this analogy, note that in this regime thedimensionless ice temperature θ0 = (T0 − T∗)/(T∞ −T0) approaches zero and the dimensionless far-fieldtemperature θ∞ = (T∞ − T∗)/(T∞ − T0) approachesone. In terms of the buoyancy force Raθ2, this meanscold fluid near the ice surface is less buoyant, or heav-ier, than warm fluid away from the surface, just assaturated water near a dissolving body is heavier thanfresh water away from it. In both these cases, the dif-ference in buoyancy forces at the surface and the far-

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(a) (b)

Figure 3: Temperature fields from a 2D phase-fieldsimulation with (a) T∞ = 4◦C and (b) T∞ = 6◦C.The yellow region denotes the ice where T = 0◦C, andthe red curve denotes the phase boundary φ = 1/2.

field causes the fluid near the surface to sink, whichsharpens the body from the top.

At low temperatures, T0 < T∞ < T∗, we discoverthe ice also sharpens, but this time from the bot-tom (Figure 2a). The mechanism here is similar tothe previous case. In this temperature range, the di-mensionless melting temperature θ0 is now larger inmagnitude than the dimensionless far-field tempera-ture θ∞, meaning the buoyancy force Raθ2 is strongernear the surface than in the far-field. This causeslighter fluid near the surface to rise, opposite to be-fore, which sharpens the ice from the bottom. Thisinverse sharpening is a typical characteristic of ice-bergs, both in the form of large pinnacles (Figure1b), as well as smaller ridges (Figure 1c).

In between these regimes, when T∗ < T∞ < 8◦C,something very different occurs. Instead of strictlyupward or downward flow, we find the flow near theice consists of adjacent regions of rising and sinkingfluid. This shear flow drives an instability, whichcarves wave-like patterns into the ice surface (Fig-ure 2b), similar to scallops found on icebergs. Suchpatterns are known to form under imposed flow, butour experiments demonstrate they can be created bybuoyancy forces alone.

In this intermediate range, fine-tuning the far-fieldtemperature T∞ can be challenging, so to gain furtherinsight we turn to numerical simulation.

Numerical methods

Numerical methods for moving boundary problemsare generally classified as interface-tracking or fixed-grid methods. In interface-tracking methods, themelting body is explicitly parametrized and then re-lated to the fluid, typically by deforming the compu-tational grid [6] or using an immersed boundary typemethod [7]. These methods can resolve fine-scale fea-tures on the interface, but are often computationally

intensive. Fixed grid methods, on the other hand,use a thermodynamic model for phase change, whichtypically appears as a source term in the temperatureequation. The PDEs can then be solved on a stan-dard Cartesian grid, making it easy to incorporatemelting dynamics into existing fluid solvers.

One of the most popular fixed-grid methods is thephase-field method [8] [9], which we use in this study.Starting from the Stefan condition (3), one can de-rive a nonlinear diffusion equation for a phase pa-rameter φ, which couples to the velocity and temper-ature equations (2a) and (2c). In this framework, φranges smoothly between the solid (φ = 0) and liquid(φ = 1) phases, leading to higher regularity in the so-lution and therefore better overall accuracy comparedto other fixed grid methods.

Snapshots of the temperature field from our phase-field simulations at T∞ = 4◦C and T∞ = 6◦C areshown in Figure 3. The ice is represented by thesolid yellow region, outlined by the phase boundaryφ = 1/2 in red. At T∞ = 4◦C, the ice sharpens fromthe bottom, while at T∞ = 6◦C it forms scallops onthe surface. This behavior is in agreement with theexperimental results from Figure 2, with similar tipcurvatures and scallop wavelengths.

To summarize, our experiments and simulations re-veal three distinct shape regimes: sharpening frombelow (T0 < T∞ < T∗), sharpening from above(T∞ > 8◦C), and scalloping (T∗ < T∞ < 8◦). Theseobservations suggest several modeling approaches,which we describe in the next section.

Modeling

In this section we propose two analytical modelsto quantify our experimental and numerical observa-tions. The first model, based on boundary layer the-ory, characterizes the sharpening behavior, while thesecond, based on linear stability analysis, describesthe scalloping patterns.

Boundary layer theory

Boundary layer theory is a useful tool for analyzingmoving boundary problems coupled to flows [10] [11].First, we assume the flow is two-dimensional and in-troduce a body fitting coordinate system (x, y), wherex = 0 corresponds to the tip of the ice. In this coor-dinate system, the ice surface can be parametrized bythe angle α(x) made between the tangent line to thesurface and the horizontal axis. Next, assuming theflow is steady, unidirectional, and localized near theice surface, it can be shown that the Boussinesq equa-

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tions (2a)–(2c) asymptotically reduce to the boundarylayer equations,

uux + vuy = Pr[uyy + Ra(θ2 − θ2∞) sinα(x)

],

uθx + vθy = θyy,

ux + vy = 0.

(4)

In this formulation u and v are the fluid velocity tan-gent and normal to the surface, respectively, and θ isthe temperature, non-dimensionalized as before.

Introducing a similarity variable η(x, y), which de-pends on the Rayleigh number, the Prandtl num-ber, and the tangent angle α(x), the boundary layerequations (4) can be transformed into two cou-pled ordinary differential equations (ODEs) for asimilarity-stream function f(η) and similarity tem-perature h(η),

f ′′′ + 3ff ′′ − 2f ′2 + (h2 − θ2∞) = 0,

h′′ + 3Prfh′ = 0.

From these ODEs, the normal velocity of the ice-water interface can be computed explicitly by eval-uating ∂θ/∂y = h′(η)∂η/∂y at the surface y = 0 andusing the Stefan condition (3),

vn(x) = St(3Ra

4Pr

)1/4 h′(0) sin1/3 α(x)( ∫ x0

sin1/3 α(x′)dx′)1/4 .

Taylor expanding this expression near the tip x = 0and using a curve-shortening equation for the tangentangle α(x), we can derive a power law for the tipcurvature,

κ(t) = κ0

(1− t

ts

)−4/5.

Notably, the negative exponent −4/5 on the righthand side of this equation indicates a finite-time sin-gularity at ts, which is a signature of the sharpeningseen in the two regimes T0 < T∞ < T∗ and T∞ > 8◦C.In reality, such singularities do not exist, and theGibbs-Thomson effect limits their growth.

In the scalloping regime T∗ < T∞ < 8◦C, it seemsthe boundary layer solution is no longer valid, so adifferent approach is necessary.

Linear stability analysis

Wave formation is generally an indicator of linearinstability. To analyze these instabilities, we typi-cally look for a steady state solution and considersmall perturbations about that state. As a basicmodel for flow near the ice surface, we consider a

Figure 4: Steady state velocity profiles from lin-ear stability analysis. For 5◦C < T∞ < 6.5◦C thesteady state exhibits both upward and downward flowin agreement with observations.

fluid, governed by the Boussinesq equations (2a) –(2c), confined between two vertical walls. We as-sume one wall is kept at the ice temperature θ0 andthe other at the far-field temperature θ∞. Assumingthe vertical derivatives are small, there is a simplesteady-state in this geometry, which we denote byu(0) = (0, v(0)) and θ(0). The vertical velocity profilev(0) is shown in Figure 4, demonstrating the bidi-rectional flow observed in experiments and simula-tions. Introducing the stream function ψ such thatu = (ψz,−ψx) and perturbing the Boussinesq equa-

tions about the steady state ψ = ψ(0) + εψeikz+σt

and θ = θ(0) + εθeikz+σt, we get a generalized eigen-value problem for the perturbation profiles ψ and θ,and their growth rate σ,

σ(D2 − k2)ψ − ikv(0)xx ψ + ikv(0)(D2 − k2)ψ

= Pr[(D2 − k2)2ψ − 2RaD(θ(0)θ)

],

σθ + ikθ(0)x ψ + ikv(0)θ = (D2 − k2)θ,(5)

where D = d/dx. This eigenvalue problem, called theOrr-Sommerfeld equation, can provide insight intothe wavelength of instability, its growth rate, and theway such properties vary with the far-field tempera-ture. In particular, for all wavenumbers k that havean eigenvalue σ∗ with positive real part, the pertur-bations will grow and the steady state u(0) and θ(0)

becomes unstable.

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Figure 5: Region of unstable wavelengths k versusthe Rayleigh number for T∞ = 6◦C. The shadingindicates the growth rate of instability Re(σ∗).

To solve the Orr-Sommerfeld equation (5), we ex-

pand the perturbation profiles ψ and θ in a finiteseries of orthogonal basis functions, such as Cheby-shev polynomials, and solve the corresponding eigen-value problem for the system of coefficients. Figure5 shows the computed region of unstable wavenum-bers k versus the Rayleigh number for T∞ = 6◦C.For the experimental conditions where we estimateRa ≈ 2 × 107, this analysis predicts a scallop wave-length in the range 2–5cm, in rough agreement withthe scallop from the experiment shown in Figure 2b.

Outlook

The class of moving boundary problems is im-mense, and accelerating climate change is makingthem more relevant every day. The oceans are warm-ing and polar ice is melting faster; the sea level isrising and coastal regions are eroding. Melting iceis just one part of this global process, and we hopeto expand the tools we developed here to further un-derstand other moving boundary problems related toclimate change.

Throughout this study we assumed the ice was heldfixed at the surface. In reality, as icebergs melt andchange shape they can become gravitationally unsta-ble and capsize. Polar scientists have noticed this ishappening more often, and attribute the higher fre-quency to warmer waters. By generalizing our modelto let the ice float freely, we can analyze the couplingbetween temperature, shape, and stability, as well asits impact on overall melt rates.

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