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Applied Mathematics:
Polar Marine Physics and Modelling
Vernon Squire
Vast regions of the Arctic and Antarctic seas are covered with a thin yet heterogeneous veneer of sea-ice that grows in situ, but is rearranged relentlessly by winds, waves and ocean currents. Away from the margins the sea-ice appears quasi-continuous but it is actually interlaced by pressure ridges and leads that are formed by spatially- and temporally-varying compressive and tensile stresses in the ice sheet. As a result, ocean waves propagating beneath sea-ice encounter a variety of irregularities that arise because of the dynamic nature of the ice cover over large physical scales. Zones of thinner, thicker, rougher or ridged ice, changes of material property, and abrupt transitions into and from open water, for example, each have their own distinctive scattering kernel that modifies the incoming wave energy spectrum as it progresses further into the ice cover. Nearer the ice edge the ice cover takes on a different appearance. Here it behaves more like a granular material composed of many interacting separate ice floes present as a mixture of sizes at different concentrations. This region, known as the marginal ice zone or MIZ, is the segment of the seasonal sea-ice zone that is affected significantly by open ocean processes. As a result its properties are dominated by the presence of ocean waves.
Several experiments have produced data that are not easily explained, both within the ice interior and in the MIZ. In particular, features that relate to the scattering of a sea or swell as it progresses into and through pack ice, e.g. the evolution of wave spectrum's directional structure, are not described well, and neither is the transmission and reflexion of waves at the many types of irregularity observed in sea-ice. A long term theoretical project to understand these processes focuses on both the MIZ and the ice interior. The overarching aim of the research is to fully characterize the various changes to ocean waves that occur as they transit through sea-ice. This is being done at the level of individual features, e.g. pressure ridges and cracks, and for the ice cover as a whole, where the spatial randomness of scatterers is included. Large amplitude waves are also being investigated, where nonlinear plate models are being used to represent the properties of the ice floes.
Recent publications can be seen here
A related problem that has been studied in the recent past concerns vehicles travelling across sea-ice or a frozen lake. The work also applies to landing aircraft. Serious accidents have occurred when trucks and trains have broken through the ice and sunk to the bottom. As a load moves across the ice, it pushes a wave in front of it and drags a wave behind it. The deflexion experienced by the ice and the properties of the waves depend crucially on the speed at which the load is travelling, indeed a critical speed exists where the deflexion is maximum as the generated waves cannot escape from the moving load. This problem has been studied in both a theoretical setting and experimentally, where experiments have been done on a frozen lake in Norway and on the sea-ice of McMurdo Sound, Antarctica, using strain gauges to monitor passing vehicles and overflying Hercules aircraft.
The book:
Squire, V. A., Hosking, R. J., Kerr, A. D. and Langhorne, P. J. Moving Loads on Ice Plates. Solid Mechanics and its Applications, Vol. 45, G. M. L. Gladwell (Series editor), Kluwer Academic Publishers (1996) 230 pp.
describes our current state of knowledge.
The work described above has been supported by Marsden grants administered by the Royal Society of New Zealand, and by the Foundation for Research, Science and Technology. Some of the illustrated images are available at NASA visible earth.
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Anomalous transport and diffusion
Boris Baeumer in collaboration with the
Fractional Calculus Project.
In classical diffusion, particles spread in a normal
bell-shaped pattern whose width grows like the square root of time while it
might drift along at a mean velocity. Anomalous diffusion occurs when the
growth rate or the shape of the particle distribution is different than the
classical model predicts. Anomalous diffusion is observed in many physical
situations - particularly power-law leading tails and power-law breakthrough
curves - motivating the development of new mathematical and physical models.
An example of a fractal medium
Power-law (also called heavy) tailing occurs whenever
there is a power-law probability of catastrophic events, and therefore the new
models have promising applications in hydrology (transport of potentially toxic
particles in groundwater or turbulent streams), chemical engineering (for
example, build-up on electrodes), physics (for example, rays going through the
atmosphere), economics (stock-market), biology (invasion of species),
meteorology (rainfall patterns, flood events), etc.
One way to develop physically
meaningful models for anomalous diffusion is to derive the limiting
distribution of an ensemble of particles following a specified stochastic
process. Continuous time random walks, where each random particle jump occurs
after a random waiting time, have been very successful.
Power-law leading tails have been
modelled nicely using a particle jump distribution with diverging (or scaling)
second moments, applying the full version of the Central Limit Theorem (the
limits are called operator stable laws). The limit is obtained by rescaling
after centering. The
result is a Levy motion with drift that has a fractional partial differential
equation as governing equation. The process is super-diffusive and the sample
paths of particles are fractals with the fractal dimension
coinciding with the order of the spatial derivative.
Power-law breakthrough curves can be modelled using
a waiting or “sticking” time process between particle jumps. We succeeded in
giving an explicit solution formula for the scaling limit if the sticking time
process has diverging second moments, using a surrogate time process. The governing
PDE is fractional in time. The same model equations have also been applied to
chaotic dynamics and finance.
Infinite variance jumps
are associated with fractional derivatives in space, while infinite
variance waiting times lead to fractional derivatives in time.
Financial data (e.g., stock market prices) is known to be heavy tailed with a
random number of price jumps per market day. The waiting time between jumps and
the size of the subsequent jumps appears statistically coupled. The underlying
condition for the above to work is the assumption that we have independent
incremental jumps and independent incremental waiting times. If the size of the
particle jump depends on the waiting time between jumps, the limiting particle
distribution is governed by a fractional differential equation involving coupled
space-time fractional derivative operators.
Expected concentration in fracture flow
Fractal flow
MADE II-data at Columbus air force Base in Mississippi
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