Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 77 / 3

The mechanisms affect surface energy input, heat supply from below, and the transport and thickness of sea ice. They reinforce one another: atmospheric warming and ocean heat thin the cover; circulation can export old ice; thinner ice melts earlier, increasing open-water absorption and strengthening ice-albedo feedback. Heat below the cold halocline matters only to the extent that mixing delivers it upward. Consequently the seven effects cannot be assigned independent percentages from the information given; their importance varies by region and year.

In a forward-looking answer from the paper's 2009 context, sustained warming would be expected to produce continued Arctic sea ice decline, a longer summer open-water season, less multi-year sea ice, and potentially nearly ice-free late summers within coming decades. Winter darkness permits new first-year sea ice to form, so loss of summer ice does not imply immediate year-round disappearance. Winter ice would tend to become thinner, more seasonal, and slower to recover after enhanced ocean heating. A precise date requires a forcing scenario and model, and internal variability causes year-to-year departures. Wang and Overland's 2009 projection is a contemporary example of such a conditional summer outlook.

Warmer air increases surface melt during the warm season, lengthens the melt season, and reduces the accumulated freezing degree-days available for winter growth. Enhanced downward heat transfer can therefore thin the cover even without extra ocean heat. It provides a broad thermodynamic influence on Arctic sea ice decline, while regional winds and cloud conditions determine part of the local response.

Warmer Atlantic water increases the reservoir of heat beneath the Arctic surface layer. However, stable density stratification across the cold halocline often prevents that heat from directly reaching the ice. The effect on sea ice depends on upward mixing and ocean heat flux, with greater direct access in some Atlantic-influenced regions; a temperature increase at depth alone is insufficient to quantify basal melting.

Erosion or displacement of the cold halocline weakens the barrier separating sea ice from warmer water below. It can therefore increase the upward ocean heat flux without requiring a large new warming of that deeper water. Wind-driven mixing, changes in shelf-water supply, and freshwater redistribution can alter the halocline. This mechanism couples the Atlantic heat reservoir to atmospheric and circulation changes.

Pacific inflow carries seasonally warm water through the Bering Strait into the western Arctic. Much of this heat lies higher in the water column than the warm Atlantic layer, making it more accessible to the ice in the Pacific sector. Its path, timing, temperature and volume transport affect basal melting and delayed freeze-up. Shimada and collaborators' 2006 observations associate warm Pacific summer water with regional ice loss; they do not imply that it explains all Arctic loss.

The Arctic Oscillation changes pressure patterns and winds, redistributing sea ice and altering export through Fram Strait. Removing thick old ice and replacing it with thinner new ice preconditions greater summer loss; winds also open leads and change ocean circulation. This is an important dynamic contributor and source of variability, rather than a universal one-to-one relation between the oscillation index and total summer ice extent.

Long-term thinning lowers the latent heat inventory that must be supplied before a patch of sea ice disappears. Thin ice is also more mobile and vulnerable to mechanical breakup. A smaller fraction of multi-year sea ice therefore increases susceptibility to a warm or windy summer. Thin ice grows relatively rapidly in winter because it conducts heat more readily, giving a stabilizing response that partly opposes continued thinning without cancelling sustained warming.

Open water absorbs more sunlight than bright sea ice, and melt ponds also lower the surface albedo. More absorbed heat promotes further melt and can be stored in the upper ocean, delaying autumn freezing. This ice-albedo feedback amplifies an initially forced loss during the sunlit season. Perovich and collaborators' 2008 measurements identify solar heating of the upper ocean as the main heat source for enhanced bottom melting in their observed Beaufort Sea case. Feedback is especially important locally after open water develops; it does not supply solar heat during polar winter.

Invert the Lebedev ice-growth law to obtain the original accumulated freezing degree-days:The average cold over $275$ days is therefore approximately $12.56\mathrm{K}$ below the reference. Assuming the warmed days remain below the freezing-degree-day threshold, a $4\mathrm{K}$ increase removes $4\times 275=1100$ degree-days. HenceThe estimated end-of-winter thickness falls by about $0.30$ m, or $20\%$. This applies the same empirical growth law to the altered temperature forcing; it does not add changes in snow, winds or ocean heat.

The extra ocean heat flux supplies energy per unit areaIn a fixed heat-removal estimate, this offsets the latent heat of fusion that would otherwise form thickness $\Delta H=\Delta Q/({\rho }_i{L}_f)$. Using a standard ice mass density ${\rho }_i\simeq 917\mathrm{kg}{\mathrm{m}}^{-3}$, an additional material assumption,For any other assumed ice density, retain $\Delta H=1.4256\times 10^8/(336000{\rho }_i)$ metres. This is the latent-heat equivalent of the added energy, not an exact coupled growth calculation. Thinner ice conducts heat more readily, so the atmospheric heat-removal budget can respond; a full sea ice thermodynamics model, such as the single-category sea-ice thermodynamic model, includes that compensation.