On the size and shape of drumlins
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On the size and shape of drumlins A. C. Fowler · M. Spagnolo · C. D. Clark · C. R. Stokes · A. L. C. Hughes · P. Dunlop
Received: 27 August 2013 / Accepted: 9 September 2013 / Published online: 25 September 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract We provide a mechanistic explanation for observed metrics for drumlins, which represent their sizes and shapes. Our explanation is based on a concept of drumlin growth occurring through a process of instability, whereby small amplitude wave forms first grow as ice slides over a bed of deformable sediments, followed by a coarsening process, in which the wavelength as well as the relief of the drumlins continues to grow. The observations then provide inferences about the growth process itself. Keywords
Drumlins · Size and shape · Instability theory
A. C. Fowler (B) MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland e-mail: [email protected] A. C. Fowler OCIAM, Mathematical Institute, University of Oxford, Oxford, UK M. Spagnolo School of Geosciences, University of Aberdeen, Aberdeen, Scotland C. D. Clark Department of Geography, University of Sheffield, Sheffield, UK C. R. Stokes Department of Geography, University of Durham, Durham, UK A. L. C. Hughes Department of Earth Science, University of Bergen, Bergen, Norway P. Dunlop School of Environmental Sciences, University of Ulster, Coleraine, Northern Ireland
123
156
Int J Geomath (2013) 4:155–165
Mathematics Subject Classification
86A40
1 Introduction Recent work on the size and shape of British and Irish drumlins Clark et al. (2009), Spagnolo et al. (2012) has shown that both their lengths and widths have well-defined smooth distributions, which are approximately lognormal. A lognormal random variable X takes positive values x > 0, and has the probability density function
−(ln x − μ)2 exp . f X (x) = 2σ 2 σ x 2π 1 √
(1.1)
The name lognormal is associated with the fact that the random variable Y = ln X , taking values y = ln x ∈ (−∞, ∞) has the normal distribution 1 −(y − μ)2 , f Y (y) = x f X (x) = √ exp 2σ 2 σ 2π
(1.2)
thus μ and σ 2 are the mean and variance of the distribution of ln X . Estimates of the density can be made by binning a sample population into equal sized bins and plotting the results as a histogram. If unequal bins are used, then the number in each bin must be divided by the bin width in order to provide a true measure of the frequency. Thus the histograms using equal bin widths of length L and width W provided by Clark et al. (2009), and of relief Spagnolo et al. (2012), provide an estimate of the probability density. Similarly, if we plot histograms of ln X using equal bin widths of ln x, we gain an estimate of the density of ln X (but not of X ). In Figs. 1, 2 and 3, we show estimates of the densities of ln H , ln L and ln W , where also H is drumlin relief, together with Gaussian densities computed from the formula (1.2), using the unbiased estimators μ¯ and σ¯ for μ and σ defined by
Fig. 1 Distribution of drumlin relief
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