By Courtenay Strong and Kenneth M. Golden
Sea ice is frozen seawater that forms on the ocean surface in the Arctic basin and around the continent of Antarctica. Sea ice packs cover millions of square kilometers of our planet’s surface and provide a habitat crucial to a diverse array of microorganisms, small crustaceans, marine birds, and mammals. Observed declines in sea ice amounting to approximately half a million square kilometers per decade are impacting global climate and ecosystems, and positive sea ice-albedo feedback is accelerating melting [2]. Sea ice has a very high albedo, meaning that it reflects most of the incoming sunlight. Declining ice coverage due to melting results in more solar energy entering the climate system, which leads to more warming and hence more melting. In fact, the September minimum of Arctic sea ice extent dropped to about 3.4 million square kilometers in 2012, which is less than half of the 1979-2000 average value of approximately 7 million square kilometers.
Since 1972, the National Aeronautics and Space Administration has been monitoring sea ice using satellites that detect the small amounts of microwave radiation emitted by the ice. The satellites detect microwave emission through clouds during both day and night, and the resulting grids at 25-km horizontal resolution provide the most spatially-complete, long-term observational record of sea ice concentrations \((0\leq c \leq 1)\) over the polar regions in both hemispheres. Unfortunately, the orbit inclination and instrument swath of the passive microwave satellites leave a “polar data gap” around the North Pole where sea ice is not observed (see Figure 1). For many years, researchers assumed that this northernmost region of the Arctic was always covered with sea ice. However, recent precipitous losses in the polar ice pack [1] call into question this assumption, which can significantly affect overall estimates of Arctic sea ice volume. By way of anecdotal evidence, the past two Decembers (2015 and 2016) have seen freakishly warm temperatures around the North Pole, with periods of almost 50 degrees Fahrenheit above average. Such dramatic changes motivate development of an objective method for estimating unobserved concentrations within the gap.
Figure 1. The left image is an example of the polar data gap (dark blue disc) on August 30, 2007, with shading outside the disc indicating concentration. The middle and right images show the data fill presented here; the color shading at right is similar to that used by the National Snow and Ice Data Center (http://nsidc.org). Image adapted from [6].
We propose [6] a partial differential equation-based model with tuned stochastic spatial heterogeneity to estimate the concentrations within a region \(\Omega\) on Earth's surface:
\[f(\theta,\phi)=\psi(\theta,\phi)+W(\theta,\phi),\]
where \(\theta\) is longitude and \(\phi\) is latitude, or \(f(\vec{r})=\psi(\vec{r})+W(\vec{r})\), where \(\vec{r} \in \Omega\). We suggest prescribing the scalar field \(\psi\) to be a solution of Laplace’s equation
\[\Delta \psi=0,\]
in spherical coordinates with boundary conditions taken from observations on the boundary \(\partial \Omega\) of the polar data gap. A unique solution for \(\psi\) exists if \(\partial \Omega\) is sufficiently smooth and the concentration is a continuous function along \(\partial \Omega\). One can numerically obtain this solution by expressing the Laplacian as a second-order finite difference operator. The stochastic term \(W\) provides realistic deviations from \(\psi\), and was tuned by collecting samples \(W_s\) of the difference between observed concentrations and \(\psi\) in three circular regions, \(C_j, j=1,2,3\), around the polar data gap,
\[W_s(\vec{r})=f_{\mathrm{obs}}(\vec{r})-\psi(\vec{r}), \qquad \vec{r} \in C_j,\;\;\; j=1,2,3,\]
where \(f_{\mathrm{obs}}\) denotes observed concentrations. Based on analysis of thousands of samples, we formulate a seasonally varying amplitude for \(W\) and introduce realistic spatial autocorrelation by convolution of spatially uncorrelated noise with a Gaussian function. Figure 1 shows an example of this model applied to the polar data gap in map view for August 30, 2007. Figure 2 below shows this same example with concentrations represented by a third vertical dimension. Tests in regions around the polar data gap reveal observation-model correlations of \(0.6\) to \(0.7\) and absolute deviations of order \(10^{-2}\) or smaller.
Figure 2. The example in Figure 1 with concentration indicated by shading and surface elevation. Panels show the polar data gap (left), the solution to Laplace’s equation within the gap (middle), and the solution with realistic spatial heterogeneity added (right). Image adapted from [6].
Our formulation of the data fill was motivated by our prior work [5] using Laplace’s equation to approximate sea ice concentrations within the marginal ice zone (MIZ), the region where sea ice concentrations transition from dense pack ice \((c\ge 0.8)\) to open ocean \((c\le 0.15)\) (see Figure 3a). The MIZ is important from both climatic and ecological perspectives, and is characterized by strong ocean-ice interactions where waves penetrate the sea ice pack. By adapting medical imaging techniques for measuring non-convex shapes and volumes in the human body [3], we define the width of the MIZ as the arc length of a streamline through the solution to Laplace’s equation (see Figure 3b). Spatially averaging the widths reveals a dramatic 39% widening of the MIZ over the satellite record [7]. Figure 3c illustrates an example of the Laplace method applied to measuring the thickness of a rodent cerebral cortex.
Figure 3. 3a. For September 29, 2010, pack ice is shaded gray, the marginal ice zone is shown in white, sparse ice and open ocean are shaded blue, land is shaded black, and islands over which concentrations were interpolated are outlined in black. Image adapted from [7]. 3b. The solution ψ to Laplace’s equation within the marginal ice zone (MIZ) is shaded, and the black curves are examples of streamlines through ψ whose arc lengths define MIZ width. Image adapted from [7]. 3c. Colored curves are examples of streamlines of a solution to Laplace’s equation on a cross-section of the cerebral cortex of a rodent brain, the arc lengths of which are used to objectively measure cortical thickness. Image adapted from [4].
In the formulation for filling the polar data gap, a least-squares linear function could replace the solution to Laplace’s equation, but at the expense of observation-model agreement along \(\partial \Omega\). One could think of the function \(\psi\) more generally as the solution to a Poisson equation, or a more general elliptic equation incorporating a local conductivity or diffusivity \(D(\vec{r})\),
\[\nabla\cdot(D\nabla\psi)=0,\]
thereby accommodating local extrema precluded by Laplace’s equation. By further increasing the complexity and computational expense, the polar data gap could also be filled by sophisticated numerical models of sea ice evolution, which incorporate dynamics and thermodynamics. In any event, developing methods to objectively fill this critical data gap is a worthy mathematical challenge and will impact our understanding of Earth’s rapidly-changing climate.
References
[1] Cavalieri, D.J., & Parkinson, C.L. (2012). Arctic sea ice variability and trends, 1979-2010. The Cryosphere, 6, 881-889.
[2] Intergovernmental Panel on Climate Change. (2013). Summary for Policymakers. In Climate Change 2013: The Physical Science Basis (pp. 1-30). New York, NY: Cambridge University Press.
[3] Jones, S.E., Buchbinder, B.R., & Aharon, I. (2000). Three-dimensional mapping of cortical thickness using Laplace’s equation. Hum. Brain Mapp., 11, 12-32.
[4] Lee, J., Kim, S.H., Oguz, I., & Styner, M. (2016). Enhanced cortical thickness measurements for rodent brains via Lagrangian-based RK4 streamline computation. Proc. SPIE Intl. Soc. Opt. Eng., 9784, 97840B.
[5] Strong, C. (2012). Atmospheric influence on Arctic marginal ice zone position and width in the Atlantic sector, February-April 1979-2010. Climate Dynamics, 39, 3091-3102.
[6] Strong, C., & Golden, K.M. (2016). Filling the polar data gap in sea ice concentration fields using partial differential equations. Remote Sensing, 8(6), 442-451.
[7] Strong, C., & Rigor, I.G. (2013). Arctic marginal ice zone trending wider in summer and narrower in winter. Geophys. Res. Lett., 40(18), 4864-4868.
Courtenay Strong is an associate professor of atmospheric sciences at the University of Utah. A substantial component of his research focuses on modeling and analysis of the cryosphere, which includes sea ice and snow. Kenneth M. Golden is a distinguished professor of mathematics and an adjunct professor of bioengineering at the University of Utah. His research is focused on developing mathematics of composite materials and statistical physics to model sea ice structures and processes.