In Pursuit of Perfect Pinnacles

By Leif Ristroph, Jinzi Mac Huang, and Michael Shelley

Michelangelo described sculptures as captives that are held inside stone—or perhaps trapped in an artist’s mind—and freed when one cuts away all of the excess material. Nature also seems to have preconceived notions of what her sculptures should look like, assigning recognizable and repeated shapes to landforms and other structures that wind and water currents chisel from rock, soil, and ice. The morphology and development of natural formations can have unexpectedly beautiful mathematical structures, as demonstrated by recent and ongoing investigations into sharply pointed pinnacles that form in soluble rock and melting ice.

Geomorphology—the discipline that explains and interprets geological forms—is often empirical due to the many complex and interactive processes that occur over widely varying length and time scales. But in some cases, researchers can identify the dominant processes and capture their physics with models. When this method is possible, employing a mathematical view of erosion, dissolution, melting, and so forth in the context of moving or free boundary problems is a powerful approach [4, 9].

Pinnacles are brashly defiant geological formations. While most structures tend to lose their edges and acquire softer features as they yield to erosion—rounded beach pebbles and the slumped slopes of older hills and mountains are prime examples—pinnacles seem to grow sharper and stand taller. Their prickly personalities are clearly visible in landscapes called stone forests (see Figure 1a). These so-called karst formations appear in soluble and porous minerals such as limestone, which dissolve over thousands of years. The most striking instances are Madagascar’s Tsingy, an apt name in the indigenous language that roughly translates to “where one cannot walk barefoot.” Ice can also melt into pinnacles, as evident when icebergs turn over and show their undersurfaces (see Figure 1b). Dissolving and melting are similar processes during which solute concentration plays much the same role as temperature.

Figure 1. Rock and ice pinnacles in nature. 1a. Limestone formations from around the world reveal sharp spikes that often form in arrays called stone forests. 1b. Melting ice displays similar features. Photographs courtesy of Steven Alvarez, Grant Dixon, Phillip Colla, and Stephen Nicol.

Sharp spikes are quite unexpected in light of the classical Stefan problem for melting, wherein the thermodynamics of solid-to-liquid phase transitions dictate the interface dynamics. This classical problem considers thermal diffusion in the two phases, heat liberation at the boundary, and the Stefan condition for boundary motion. The latter is provided by Fick’s law, which states that the local melt rate is proportional to the normal gradient of temperature in the liquid. These effects tend to suppress high curvatures. However, gravity is missing from this idealization and proves critical to pinnacle formation. In most earthly situations, melting (or dissolving) leads to density differences in the surrounding liquid that produce gravitational or buoyancy-driven flows. These flows strongly modify the temperature (concentration) field and hence the dynamics of the receding surface. Given initial conditions, this Stefan problem with gravitational convection seeks to determine the shape development.

Recent investigations into this issue illustrate the subtlety of shape dynamics problems and the value of combining mathematical modeling and analysis with laboratory experiments and numerical simulations. A review of this pursuit begins with a 2015 study that poses the question, “Do dissolving objects converge to a universal shape?” [6]. Experiments on upright cylinders of amorphous or noncrystalline solids that were immersed in initially quiescent water revealed the formation of sharp-tipped spires that approach a quasi-paraboloidal form at long times. Further experiments used hard candy as a “mock rock” and indicated that such spires sharpen indefinitely — or at least for as long as the ever-thinning shape can be accurately measured [1]. These first experiments thus suggested that the late-stage dynamics involve a quasi-paraboloidal form whose curvature grows continuously in time. Figures 2a and 2b offer some typical images and data from the aforementioned experiments.

Subsequent works then established the mathematical foundations [5, 7]. The full dynamical equations include the incompressible Navier-Stokes equations for flow, an advection-diffusion equation for the solute concentration \(c\) in the fluid, and Fick’s law \(V_n \propto n \cdot \triangledown c\) or the recession velocity of the interface with outward normal \(n\). Since the driving flows are stably attached to the surface (see Figure 2d), one can utilize boundary layer theory to derive a moving interface model. The key dynamical equation

\[V_n=-a \frac{r(s,t)^{1/3}\cos^{1/3}\theta(s,t)}{\Big[\int^s_0r(s',t)^{\frac{4}{3}}\cos^{\frac{1}{3}}\theta(s',t)ds'\Big]^{1/4}}\]

is applicable to three-dimensional axisymmetric bodies and specifies the normal velocity at any boundary point at radius \(r\) from the axis of symmetry, arc length \(s\) from the tip, and tangent angle \(\theta\) of the surface (see Figure 2e). Fluid and solid material properties determine the constant \(a\). The numerator intuitively indicates that steeper slopes dissolve more quickly due to faster flows, while the integral in the denominator captures the lower rate of dissolution due to the accumulated solute from points upstream. This reduced system takes the form of a nonlinear integro-partial differential equation that is completed by the geometric evolution equation \(\frac{\partial\theta}{\partial t}=\frac{\partial V_n}{\partial s}+V_s\frac{\partial\theta}{\partial s}\). An artificial tangential velocity of \(V_s=\int^s_0 V_n\frac{\partial\theta}{\partial s}ds'\) is convenient for separating \(s\) and \(t\) as independent variables [2, 3, 5].

Even with this framework in hand, understanding the shape dynamics—especially the behavior of the apex—has proven challenging. Numerical solutions of a Cartesian formulation of the model demonstrated nonuniversal behavior; different initial shapes yield different terminal dynamics [7, 8]. Sufficiently slender initial shapes, such as quasi-cylindrical forms, were shown to sharpen over time via power law growth of tip curvature. Contrarily, another study presented a local analysis of the tip region that suggests the tip curvature blows up via a finite-time singularity [5]. The addition of a curvature regularization term produced numerical solutions that yielded good agreement with the shape dynamics, which were measured experimentally. Although these studies differ in quantitative details, they do concur that hydrodynamics drive the formation of an ever-sharpening needle. They also agree on the physical origin of this effect. Higher dissolution rates near the top come from thin boundary layers that entrain the fresh outer fluid, while lower rates further down the body result from the insulating effect of thicker layers, which are laden with the solute that was released from upstream points.

Recent work paints a different picture of the longtime dynamics [3]. Numerically propagating the characteristics of the previous system causes the apex to sharpen dramatically before eventually saturating to a finite value for the tip curvature as the object approaches a terminal shape. Furthermore, the authors present an elegant analysis of the model that furnishes exact solutions for boundaries that preserve their shapes as they recede. One can express these equilibrium shapes as the distribution of curvature over the tangent angle \(\kappa/\kappa_0=\sin^5\theta/(1+2\cos^2\theta)\), where the tip curvature \(\kappa_0\) parametrizes the family of such “perfect pinnacles.” Treating \(\kappa_0\) as a fitting parameter attained a convincing comparison to the longtime shape from numerical simulations. Interestingly, the finite-time singularity of previous work [5] disappears upon the consideration of higher-order terms in the analysis of the apex region.

Figure 2. Shape dynamics of pinnacles. 2a. Experiments on pillars of hard candy that dissolve in water yield needle-like shapes with sharpening tips. 2b. Simulations closely match the measured shape dynamics. 2c. Inverted and upright pinnacles form in melting ice at low and higher temperatures respectively. 2d. Boundary layer flows are induced by density differences in the fluid. 2e. A moving interface model serves as the basis of simulations and shape analysis. 2f. A “forest” or array of pinnacles emerges in experiments on dissolving blocks of porous candy. Figures 2a, 2b, 2d, 2e, and 2f courtesy of [5]; Figure 2c courtesy of [10].

These findings show the subtleties behind the original motivating question about shape universality in dissolving [6]. Some initial shapes indeed seem to converge to a member of an infinite family of terminal shapes, but it is not easy to predict which member will ultimately be selected based on the initial form. It is also not clear whether all initial shapes eventually become members of these perfect pinnacles. As for next steps, obtaining evidence for a terminal shape in experiments and implementing numerical simulations of the full dynamical equations—which may require new methods to resolve the extremely fine scales at late times—seem to be of paramount importance. Additionally (or alternatively), future studies could prescribe a member of the predicted equilibrium family as the initial form and test this shape’s preservation thereafter. One might also wish to employ stability analysis to address whether such equilibria are stable within the model, and explore shape perturbations in experiments and simulations. The latter could provide insight into the selection problem regarding which aspects of the initial form are most important in determining the final form. This knowledge would be useful in engineering applications that exploit dissolutive reshaping to manufacture ultra-fine structures [5].

Returning to natural pinnacles, researchers have both suggested that dissolutive sharpening is responsible for karst pinnacles and proposed a hypothesis for the formation of a “forest” or array of such structures [5]. An experimental test involves the dissolution of a block that was seeded with vertical pores as a simple model of porous and fissure-riddled stone. A dramatic reshaping unfolds; the pores widen as they act as conduits for downward flows and eventually merge, leaving an array of interstitial pillars that then sharpen to produce the bed-of-nails morphology in Figure 2f. Furthermore, although ice pinnacles are very much expected based on dissolution studies, they still promise to yield more surprises. A recent paper demonstrated conventional pinnacles in ice [10], but only for melting that occurs in sufficiently warm water (see Figure 2c). The very cold far-field temperatures that are typical of melting in nature yield inverted or downward-pointing pinnacles—which could explain why icebergs often form pinnacles on their underbellies—while intermediate temperatures produce more exotic shapes. These outcomes stem from liquid water’s anomalous density-temperature profile, whose maximum at four degrees Celsius drastically alters the flows that are generated.

Looking ahead, pinnacles and other natural formations will clearly continue to serve as muses that inspire further investigations. Applied and computational mathematics provide the necessary tools that allow scientists to chip away at geomorphology problems and reveal their fascinating geometries and shape dynamics.

Leif Ristroph is an applied mathematician and experimental physicist who co-directs the Applied Mathematics Laboratory (AML) at New York University’s (NYU) Courant Institute of Mathematical Sciences. Jinzi Mac Huang is an applied and computational mathematician at NYU Shanghai whose research focuses on fluid dynamics and soft matter. Michael Shelley is an applied and computational mathematician at NYU—where he co-directs the AML—and at the Simons Foundation’s Flatiron Institute, where he leads the Center for Computational Biology.