Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 69 / 3

A polynya is a persistent or recurrent area of open water or relatively thin ice within an otherwise ice-covered region. Persistence of the opening does not mean absence of ice formation: a latent-heat polynya can continuously make ice and export it. In contrast, a sensible-heat polynya remains open because ocean heat inhibits freezing. Both processes may occur at the same site.

At an Antarctic coast, cold air descending from the elevated continent produces strong katabatic winds. Their offshore component drives sea ice away from a fixed coastline, fast-ice edge or grounded-iceberg barrier. Open water exposed to the cold atmosphere loses heat rapidly; frazil ice forms, aggregates into pancake ice or consolidated new ice, and is exported offshore. The opening continues as long as export balances production. Headlands, the shelter provided by fast ice and the direction of the prevailing winds determine the location and size. Terra Nova Bay is a characteristic coastal example. Such latent-heat polynyas are ice factories rather than permanently warm holes in the ice pack.

At an ice shelf front the fixed boundary is floating shelf ice instead of land. Offshore winds blowing across the shelf can similarly remove newly formed sea ice, so an ice-front polynya need not have a different thermodynamic mechanism from a coastal one; the Ross Ice Shelf front provides an important wind-driven example. Shelf-front geometry, ice tongues, grounded icebergs and changes in calving alter the shelter and the supply of drifting pack ice. Some ice-front openings also receive significant sensible heat from currents, tidal mixing or upwelling, and heat carried out of an ice-shelf cavity can affect freezing. A sensible-heat polynya requires a sufficiently warm ocean supply, whereas the latent-heat mechanism requires continuing removal of the ice that does form. Consequently a strict identification of all coastal openings with one mechanism and all ice-shelf openings with the other would be incorrect.

The oceanic roles are substantial. Rapid heat loss and freezing produce brine rejection, raising shelf-water salinity and mass density. The resulting convection and circulation can carry dense shelf water into ice-shelf cavities; melting beneath an ice shelf modifies it into colder, fresher ice-shelf water. Dense shelf and cavity-modified waters can subsequently descend the continental slope, mix with adjacent waters and contribute to Antarctic bottom water. The outcome depends on bathymetry and the density after mixing: not all rejected brine reaches the abyss.

These localized regions can therefore have a disproportionate effect on deep-ocean ventilation, oxygen supply, carbon transport and the large-scale ocean circulation. They are also intense sites of ocean-to-atmosphere heat and moisture transfer. Seasonal sunlight arriving in open water supports high biological production where nutrients are available, with consequences for marine food webs and carbon cycling. Antarctic polynyas connect wind-driven ice export, concentrated ice production and dense-water formation; ice-shelf fronts add a floating boundary and possible cavity-related heat and freshwater exchanges.

Off northeast Greenland, the Northeast Water Polynya is organized by the interaction of local currents and winds with coastal geometry and relatively fixed ice barriers. A regional northward coastal current encounters the Norske Øer shelf-ice barrier; divergence and shelter on its lee side help create the southern opening. This local northward current should not be replaced in the explanation by assuming uniform southward flow along the entire Greenland coast. In summer, weak winds, the current and the barrier can maintain the opening, while solar heating and the seasonal heat budget allow it to expand northward without continuous new-ice formation.

Winter conditions differ. Strong northerly winds can remove newly formed sea ice from the northern region around Ob Bank, with the Ob Bank ice barrier and Kronprins Christian Land controlling shelter and divergence. Large air–sea temperature differences then sustain rapid freezing and export. Thus the recurring geographical opening has a changing balance of dynamical export and seasonal thermodynamics; long-lived does not mean identical formation conditions throughout the year. Ice barriers themselves can also change between years.

The North Water Polynya, or Pikialasorsuaq, occupies the Smith Sound and northern Baffin Bay region between Greenland and the Canadian Arctic. An ice arch across the narrow upstream channel restricts the arrival of thick drifting Arctic ice. Northerly winds and predominantly southward transport through the channel remove newly formed ice into Baffin Bay. The combination of blocked upstream ice supply and continuing downstream export maintains low ice concentration despite strong winter cooling. The location and stability of the arch therefore matter as well as the local wind. Warm Atlantic-origin water carried along the Greenland side can provide a sensible-heat contribution, with tidal and other mixing helping deliver heat upward; the entire opening should not be attributed either exclusively to warm-water convection or exclusively to wind.

Both regions permit much greater winter ocean-to-atmosphere heat transfer than nearby thick ice, and both can concentrate ice formation and brine rejection. The production of saline shelf water affects local density stratification and exchanges with surrounding waters; it is not automatically equivalent to deep or abyssal water formation. In spring and summer the early availability of light and open water promotes biological production and supports rich marine food webs. These persistent or recurrent openings also provide habitat and access of particular importance to Arctic animals and coastal communities. The key Arctic mechanisms are ice-barrier control of supply, current- and wind-driven divergence, and a seasonally varying contribution from ocean heat.

In the Pease polynya model, take a straight fixed coast at $x=0$ and an offshore polynya edge at $x=L(t)$. Let consolidated pack ice move offshore at speed $U>0$, let new ice be produced throughout the opening at equivalent thickness rate $F>0$, and let the collected ice at the edge have a prescribed thickness $H$. All fluxes below are per unit length of coastline. The new-ice volume production is $FL$, while the consolidated-ice export relative to the moving edge is $H(U-\dot{L})$. Neglect storage of drifting frazil ice inside the opening, so conservation givesThis is the simplest Pease closure: newly formed ice is collected at the edge rapidly enough that production can be equated to the instantaneous edge flux. A model retaining frazil storage would instead include the local balance ${\partial }_{t}h+{\partial }_x(uh)=F$, with frazil thickness $h$ and transport velocity $u$, together with an edge condition; it should not silently be identified with the instantaneous-collection approximation.

For constant $U,F,H$, the stable equilibrium and relaxation time areIn particular, an initially closed opening has $L(t)=L_{\infty }(1-e^{-t/\tau })$. It initially opens at speed $U$, then slows as its larger area produces more ice. It reaches $63\%$ of its ultimate width after $\tau$ and $95\%$ after $-\tau \log 0.05\simeq 3\tau$; the exact equilibrium is approached asymptotically, not reached at a finite opening time.

The production rate is determined by the energy budget. With $Q_{\mathrm{surf}}$ positive for net upward surface heat loss and $Q_o$ the heat supplied from the underlying ocean, water at its local freezing temperature givesThe surface loss includes sensible and latent turbulent transfers and net radiation. For example, the sensible component is approximately ${\rho }_a{c}_{p,a}{C}_{H}V(T_f-T_a)$, where $V$ is wind speed and $C_H$ an exchange coefficient. Ice motion is commonly parameterized as a small fraction of the offshore wind speed, but currents and the direction relative to the coast also matter. Thus colder air and stronger net cooling increase $F$ and narrow the opening; faster export increases $U$ and widens it; thicker collected ice increases both $L_{\infty }$ and $\tau$. Stronger winds can increase both export and heat loss, so the width's dependence on wind can be much weaker than the formula with artificially fixed $F$ would suggest.

The assumptions are steady spatially uniform forcing; a one-dimensional straight coast; fixed collection thickness and drift speed; no alongshore import of pack ice; water already at freezing rather than an appreciable sensible-heat reservoir; and negligible residence time of newly formed ice. Real coastlines, time-dependent winds, ocean currents, atmospheric adjustment over warm open water and variable ice accumulation can all violate these assumptions. The model describes a latent-heat polynya, not an opening maintained principally by a warm deep-ocean heat supply.

If air temperature rises while wind, $H$ and the other external parameters are unchanged, the smaller air–sea temperature difference reduces net heat loss and hence $F$. For $0<F^{\prime }<F$,The opening becomes wider and takes longer to attain the same fraction of its new equilibrium width. Its initial opening speed remains $U$; the longer equilibration follows from weaker ice production, rather than slower initial export. If warming eliminates net freezing, $F=0$ gives $L=L(0)+Ut$ in the simple kinematic closure, with no finite freezing-balanced width. A negative $F$ represents melt and is outside the positive-production Pease model.

For the shelf-water calculation, a $30$-day month at $0.05\mathrm{m}\mathrm{da}{\mathrm{y}}^{-1}$ produces $h_i=1.5\mathrm{m}$ of ice. Assume a well-mixed $D=100\mathrm{m}$ shelf column, no compensating advection or precipitation, and fixed ice and water densities ${\rho }_i=917$ and ${\rho }_w=1025\mathrm{kg}{\mathrm{m}}^{-3}$. The frozen fraction of its initial liquid mass isLet the liquid initially have salinity $S_0$, and newly formed ice retain $rS$ with $r=0.3$ and $S$ the instantaneous ambient value. For remaining liquid mass $M$, freezing removes salt with the departing ice, so conservation of salt with salt-retaining ice givesThereforeAn initial shelf-water salinity is needed for an absolute answer. Taking $S_0=34$ psu gives $\boxed{\Delta S\simeq 0.32\mathrm{psu}}$. At the accuracy of an equal-density thickness calculation, $f\simeq 1.5/100$ and $\Delta S\simeq 0.7S_{0}f$, giving about $\boxed{0.36\mathrm{psu}}$ for $S_0=34$ psu. These estimates describe the same modest increase. Treating ice as completely fresh would overestimate brine rejection by omitting the factor $0.7$.

The resulting water is denser and can leave the shelf and intrude beneath fresher surface water. It is therefore a physically plausible contribution to the cold halocline. Nevertheless, this one-month column budget yields only a few tenths of a psu, and the salinity contrast across the Arctic halocline can be several psu. If water from a shelf occupying one-third of the area were spread through an equally deep layer over the whole ocean, the average increase would be only about $0.1$ psu. This illustration is not a volume budget for the actual deep Arctic: shelf area fraction is not ocean water-volume fraction, and the exported water can be concentrated within a relatively thin halocline layer.

Consequently this calculation supports a shelf contribution but does not establish its dominance. Repeated seasonal ice production, enhanced production in polynyas, shelf-water export, freshwater dilution, and the thickness and residence time of the receiving layer would all be needed to test dominance. The assumption that the shelf initially becomes ice-free limits the first month's salt production; it does not bound an entire winter's cumulative production if new ice is continually removed.

Another pathway is winter cooling and limited convection of Atlantic-origin water as it enters the Arctic and encounters the ice-covered ocean. Cooling and local freezing can make it near-freezing water without requiring that it first be formed throughout the Siberian shelf. A subsequent fresh surface cap limits further convection; the cooled water is advected into the basin beneath that cap, producing a cold, salinity-stratified lower halocline. Pacific-origin waters also contribute to halocline structure in the western Arctic. Both formation and maintenance depend on circulation and freshwater supply, not simply on the presence of freezing somewhere. The strong stable density stratification then limits upward transfer of heat from the deeper Atlantic layer to sea ice.