David B. Bahr

The physical basis of glacier volume‐area scaling

Ice volumes are known for only a few of the roughly 160,000 glaciers worldwide but are important components of many climate and sea level studies which require water flux estimates. A scaling analysis of the mass and momentum conservation equations shows that glacier volumes can be related by a power law to more easily observed glacier surface areas. The relationship requires four closure choices for the scaling behavior of glacier widths, slopes, side drag and mass balance. Reasonable closures predict a volume-area scaling exponent which is consistent with observations, giving a physical and practical basis for estimating ice volumes. Glacier volume is insensitive to perturbations in the mass balance scaling, but changes in average accumulation area ratios reflect significant changes in the scaling of both mass balance and ice volume.

Exploiting social networks to mitigate the obesity epidemic.

Despite significant efforts, obesity continues to be a major public health problem, and there are surprisingly few effective strategies for its prevention and treatment. We now realize that healthy diet and activity patterns are difficult to maintain in the current physical environment. Recently, it was suggested that the social environment also contributes to obesity. Therefore, using network‐based interaction models, we simulate how obesity spreads along social networks and predict the effectiveness of large‐scale weight management interventions. For a wide variety of conditions and networks, we show that individuals with similar BMIs will cluster together into groups, and if left unchecked, current social forces will drive these groups toward increasing obesity. Our simulations show that many traditional weight management interventions fail because they target overweight and obese individuals without consideration of their surrounding cluster and wider social network. The popular strategy for dieting with friends is shown to be an ineffective long‐term weight loss strategy, whereas dieting with friends of friends can be somewhat more effective by forcing a shift in cluster boundaries. Fortunately, our simulations also show that interventions targeting well‐connected and/or normal weight individuals at the edges of a cluster may quickly halt the spread of obesity. Furthermore, by changing social forces and altering the behavior of a small but random assortment of both obese and normal weight individuals, highly effective network‐driven strategies can reverse current trends and return large segments of the population to a healthier weight.

Response time of glaciers as a function of size and mass balance: 1. Theory

A simple interpretation of the traditional definitions of glacier and ice sheet response time (e.g., thickness divided by mass balance rate, h/b˙) suggests that larger glaciers respond more slowly than small glaciers to a perturbation in climate. However, with reasonable choices for mass balance behavior, a scaling analysis shows that the response time of valley glaciers decreases as a function of increasing size when other variables are held constant. In essence, this is because larger valley glaciers push further into the ablation zone, and ablation increases more rapidly than the thickness (so that h/b˙ gets smaller). Ice sheets have different mass balance regimes than valley glaciers, and as they grow larger, the ablation does not increase faster than the thickness. Therefore, as ice sheets grow in surface area, the response time increases. For both ice sheets and valley glaciers, the response time also depends on a mass balance index, which is defined as the slope of the balance curve as a function of horizontal distance along the glacier surface (rather than elevation). The response time decreases as the balance index increases, so for valley glaciers, an increase in the balance index and an increase in glacier size have opposite effects on the response time. The balance index is typically larger for smaller valley glaciers. Therefore a small glacier will typically respond faster than a large glacier, but this quicker response is because of mass balance considerations and not because of the dynamic characteristics of the glacier arising from its small size, as often assumed.

Width and length scaling of glaciers

An analysis of hundreds of mountain and valley glaciers in the former Soviet Union and the Alps shows that characteristic glacier widths scale as characteristic glacier lengths raised to an exponent of 0.6. This is in contrast to most previous analyses which implicitly or explicitly assumed scaling exponents of either 0 or 1. The exponent 0.6 implies that average glacier widths are proportional to average glacier thicknesses. Although this seems to suggest V-shaped glacier valleys, the linear width-thickness relationship is not inconsistent with parabolic valley cross-sections, because the characteristic (or average) width of a glacier depends on many other aspects of channel and glacier morphology, including variations in the channel width with distance up- and downstream.

A review of volume-area scaling of glaciers

Abstract Volume‐area power law scaling, one of a set of analytical scaling techniques based on principals of dimensional analysis, has become an increasingly important and widely used method for estimating the future response of the worlds glaciers and ice caps to environmental change. Over 60 papers since 1988 have been published in the glaciological and environmental change literature containing applications of volume‐area scaling, mostly for the purpose of estimating total global glacier and ice cap volume and modeling future contributions to sea level rise from glaciers and ice caps. The application of the theory is not entirely straightforward, however, and many of the recently published results contain analyses that are in conflict with the theory as originally described by Bahr et al. (1997). In this review we describe the general theory of scaling for glaciers in full three‐dimensional detail without simplifications, including an improved derivation of both the volume‐area scaling exponent γ and a new derivation of the multiplicative scaling parameter c. We discuss some common misconceptions of the theory, presenting examples of both appropriate and inappropriate applications. We also discuss potential future developments in power law scaling beyond its present uses, the relationship between power law scaling and other modeling approaches, and some of the advantages and limitations of scaling techniques.

Response time of glaciers as a function of size and mass balance: 2. Numerical experiments

Numerical experiments were made using a time-dependent nonlinear finite element model of glacier flow to seek confirmation of theoretical predictions made in the companion to this paper [Bahr et al., this issue]. The theoretical analysis indicates that under conditions of equal gradients in mass balance along the glacier surface ( ∂b˙/∂x), other variables being held constant, response time to mass balance perturbations will decrease as glacier size increases. This behavior is confirmed by the numerical model experiments, which are performed on a suite of 23 radially symmetric ice caps with sizes ranging over ∼4 orders of magnitude. The shape and mass balance profiles of the ice caps are defined so that ∂b˙/∂x is constant over all ice caps, and for uniform perturbations in mass balance, the modeled response times are faster for larger ice caps. The model results also confirm a volume/area scaling relationship established in the theoretical discussion and confirm the equivalence between two measures of response time, one arising from the present theory and the other a conventional expression of response time. The apparent discrepancy between the present results and preexisting theory is discussed, and it is shown that the results are compatible.

Theoretical limitations to englacial velocity calculations

To study the dynamics of ice sheets and glaciers, velocities at the bed of a glacier must be measured directly or calculated using data gathered from boreholes and surface surveys. Boreholes to the bed are expensive and time-consuming to drill, so the determination of basal velocity is almost exclusively by numerical inversion of velocities observed at the surface. For non-linearly viscous glaciers, a perturbation analysis demonstrates that inversions for englacial velocities will magnify measurement errors at an exponential rate with depth. The rate at which calculation errors grow is proportional to a Lyapunov exponent, a measure of “information loss” which is shown to be a simple linear function of spatial frequency with a coefficient depending on Glen’s flow-law exponent, n. The coefficient decreases with increasing non-linearity, demonstrating that inversions with non-linearly viscous ice have smaller calculation errors than inversions with linearly viscous ice. In both the linear and nonlinear cases, the Lyapunov exponent (and rate of error growth) increases with decreasing wavelength, which limits velocity calculations at the bed to wavelengths on the order of one ice thickness or greater. This limitation is theoretical and cannot be countered by more accurate survey data or special numerical techniques.

Snow patch and glacier size distributions

A simple theoretical model demonstrates that some amount of randomness in snow and ice mass balance is sufficient to reproduce empirically observed power law and exponential distributions of snow patch and glacier sizes. No other assumptions about the Underlying topography or snow accumulation and ablation processes are necessary to extract this important spatial property. The inclusion of additional geometrical and physical processes can alter the specific scaling constants of the size distributions, but the fundamental behavior remains unchanged. Specifically, for snow patch and glacier sizes less than some correlation length the size distribution is a decreasing power law, and for sizes larger than the correlation length the distribution decreases rapidly as an exponential. The solution is based on a mapping to a relatively well explored class of problems in percolation theory.

Spatial variability in the flow of a valley glacier: Deformation of a large array of boreholes

Measurements of the deformation of a dense array of boreholes in Worthington Glacier, Alaska, show that the glacier moves with generally bed-parallel motion. Strain in the 200 m deep valley glacier is constant near the surface but follows a nonlinear vertical profile below a depth of about 120 m. By a depth of 180 m, the octahedral strain rate reaches 0.35 yr -1 . The three-dimensional velocity field shows spatial complexity with significant deviations from plane strain, despite relatively simple valley geometry in the vicinity of the 6 × 10 6 m 3 study volume. No evidence was found for time-varying deformation or movement along localized shear planes. Observations were made by repeatedly measuring the long-axis geometry of 31 closely spaced boreholes over a 70 day period, and three additional holes after 1 full year of deformation. The holes were spaced 15 to 30 m apart. Installation and measurement of such a large number of boreholes required the development of a semiautomated hot water drilling system that creates straight and vertical boreholes with uniform walls. The equipment and procedures enables borehole profiles to be measured without the use of hole casing. Inclinometry measurements collected in the holes were processed, analyzed for error, and visualized as a fully three-dimensional data set. The new methods offer unique insight into small-scale spatial and temporal variations in the pattern of flow in a valley glacier.

Theory of lattice Boltzmann simulations of glacier flow

A lattice Boltzmann technique for modeling Navier-Stokes fluid flow is extended to allow steady-state simulations of glaciers and other slow-flowing solids. The technique is based on a statistical mechanical representation of flowing ice as a set of particles (populations) which translate and collide on a face-centered cubic lattice. The average trajectories of the populations give the velocities of the ice at any point in the glacier. The method has considerable advantages over other techniques, including its ability to handle complex realistic geometries without additional complications to the code. Examples are presented for two-dimensional simulations.