Stalactites are one of the most fascinating underground phenomena: they form slender columns, thick cones or more complex, cake-like structures. Researchers have now for the first time mathematically described what determines the shape of a stalagmite that grows upwards from the bottom. According to their newly developed formula, the shape of a stalagmite depends primarily on one parameter: the so-called Damköhler number. It describes the relationship between the precipitation of the calcite, the flow rate of the water and the base area of the stalagmite. Depending on whether the value of the Damköhler number is below, equal to or above one, conical, pointed, columnar or flat stalagmites are formed. The team also discovered that the distribution of carbon isotopes in the deposited limestone also varies depending on the shape of the stalagmite. This is important for paleoclimatology.
Inside many caves, dripping water creates iconic formations: stalactites. They arise when rainwater penetrating into the subsoil absorbs carbon dioxide from the ground and becomes acidic. If this then seeps through limestone, the calcium carbonate dissolves in the water. In the atmosphere of the cave, some of the carbon dioxide gases out of the water. This changes the solubility of the dissolved calcium carbonate and the calcite precipitates. As a result, stalactites form on the cave ceiling and stalagmites gradually grow upwards from the cave floor. These calcite columns can range in height from a few centimeters to many meters and have a variety of shapes: some resemble slender columns, others more like stubby cones or layered “pies”.
A parameter determines the shape
But what determines the shape of a stalagmite? The geochemical mechanism behind the growth of stalactites was clarified around 60 years ago. There was also already a mathematical model for an ideal stalagmite. But there was a lack of a formula that describes why stalagmites sometimes become slender and columnar and sometimes thick and rather complex. A team led by Piotr Szymczak from the University of Warsaw has now found this formula. “It turns out that the great diversity of stalagmite shapes can be explained by a single simple parameter,” says Szymczak. “This is a rare case in which the beauty of nature corresponds directly to a clear mathematical law.” As the team discovered, this one parameter is the so-called Damköhler number. It describes the relationship between the rate of calcite precipitation from the drop, the flow rate of the water and the basal area of the stalagmite.
If the Damköhler number is approximately one, as is the case with concentrated, steady drops, a slender, columnar stalagmite with a rounded top forms. On the other hand, if the flow rate is high and the drops only travel a short distance to the tip of the stalagmite, the Damköhler number is below one – a cone-shaped stalagmite with a point at the top is created. The third variant is a Damköhler number above one, as occurs with slow dripping and a large distance between the stalagmite and the cave ceiling. Then the impacting drops burst on the top of the limestone column and form a thin, widely distributed film of water. As a result, the calcite falls out over a wide area and the stalagmite has a flat top, the researchers determined. “Such flat tops are usually found when the stalagmite tips are at least ten meters away from the cave ceiling,” explains the team.
The isotope values also differ
To verify their formula, Szymczak and his team next examined stalagmites from Slovenia’s Postojna Cave using X-ray tomography. This captured the shape of the stalactites and made it possible to create digital 3D models of them. “When we compared our analytical solutions with real cave samples, the agreement was remarkable,” says co-author Matej Lipar from the Research Center of the Slovenian Academy of Sciences and Arts. “This shows that even under natural, messy conditions, the underlying geometry is there.” According to the researchers, their formula provides a comprehensive mathematical description of the parameters that determine the shape of growing stalagmites – and combines observations and theory. “Our results provide a unified view of various numerical approaches to stalagmite growth and also build a bridge to models of stalactite growth,” write Szymczak and his colleagues.
The analyzes also provide important information for climate research. Past climatic conditions leave their traces in the isotope ratios of the carbon deposited in the limestone of the stalagmite. Like the annual rings of a tree, the stalactites preserve seasonal and long-term trends. That’s why stalactites are often used to reconstruct past temperatures and precipitation conditions. As Szymczak and his team have now found out, the isotopes are not uniformly distributed inside the stalactite, depending on the shape of the stalagmite. The distribution of the carbon isotope 13C from the inside to the outside in conical or columnar stalagmites is more like a parabola, but in stalactites with a flat top the ratio is relatively constant, at least in the central area. “Stalagmites are natural climate archives, but we now realize that their geometry leaves its own imprint in the isotope record,” says co-author Anthony Ladd from the University of Florida. “Knowledge of this effect now enables us to obtain even more precise information about past climate conditions.”
Source: Piotr Szymczak (University of Warsaw) et al., Proceedings of the National Academy of Sciences, doi: 10.1073/pnas.2513263122