Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 77 / 4

Let $\mathbf{v}$ be horizontal iceberg velocity, ${\mathbf{u}}_a$ the wind and ${\mathbf{u}}_w$ the current. Air and water drag forces act on the relative velocities, typically with magnitudes proportional to their squares. Their exposed and immersed areas and drag coefficients differ. For iceberg mass $m$, the horizontal balance can be written schematically aswhere $f$ is the Coriolis parameter and $\zeta$ the sea-surface elevation. The last term represents the large-scale pressure gradient; contact with pack ice and grounding require additional forces. Iceberg free drift neglects such contact constraints, not the water drag or Coriolis force.

In still water with a fixed wind force $\mathbf{F}$, a transparent transient model linearizes water drag as $-m\gamma \mathbf{v}$. Put $J\mathbf{v}=\hat{\mathbf{z}}\times \mathbf{v}$. ThenThe damped free drift of an iceberg isThus an initially resting berg accelerates under the wind, turns under the Coriolis force, and approaches a balance of wind, water drag and rotation through damped inertial oscillations. Its equilibrium drift is to the right of the wind in the Northern Hemisphere and to the left in the Southern Hemisphere. With quadratic drag, the drag coefficient in the equilibrium balance depends on speed; the linear formula illustrates the turning and relaxation rather than prescribing an exact percentage of wind speed.

For a moving ocean, the background pressure gradient associated with geostrophic balance must also be included. Large tabular icebergs have substantial submerged area and respond strongly to currents; smaller bergs can have a larger wind-driven velocity relative to the water. Near Antarctica, the mainly westward Antarctic Coastal Current can carry bergs along the continental margin. Export into offshore gyres and then the eastward Antarctic Circumpolar Current changes their trajectories; bathymetry, winds, fronts and evolving size all matter. Gladstone and Bigg's Weddell Sea tracking illustrates the contrast between coastal and offshore drift. The Antarctic Convergence separates colder Antarctic surface water from warmer water farther north, where an exported iceberg can encounter enhanced melting. No single wind direction determines the complete route.

Measure $z$ downward from the iceberg surface. Its mass density is ${\rho }_i(z)=300+12z$ for $0\le z\le 50$ m and $900\mathrm{kg}{\mathrm{m}}^{-3}$ below that. The mass per unit horizontal area isBy Archimedes' principle, the draft is $d=(m/A)/{\rho }_w\simeq 248.78$ m. The iceberg freeboard is therefore $300-d\simeq 51.22$ m. The specified snow-to-ice density is reached atThe more porous snow and firn are consequently above water: atmospheric warming, meltwater infiltration, compaction and refreezing can modify them. At the waterline and below, dense ice is exposed to oceanic heat transfer and wave erosion, which can undercut the berg and cause breakage. If the porous transition were submerged, water infiltration and ocean heat could act directly on that material, changing the erosion and collapse mechanisms.

The horizontal area is $A=10^6{\mathrm{m}}^2$ and the total mass from part (a) is $m=2.55\times 10^{11}\mathrm{kg}$. The iceberg melt heat budget includes warming to $0^{\circ }\mathrm{C}$ followed by fusion. With temperature $T(z)=-0.6z$ in the upper $50$ m and $-30^{\circ }\mathrm{C}$ below,The two integrals are $525000$ and $6750000\mathrm{kg}\mathrm{K}{\mathrm{m}}^{-2}$, so using the stated ice specific heat capacity,The latent heat of fusion contribution isThereforeThis produces liquid water at $0^{\circ }\mathrm{C}$; warming the meltwater further would require additional heat.

For $10^4$ such bergs per year, the rate of heat absorption north of the Antarctic Convergence isRelative to the specified oceanic polar heat flux,The bergs themselves move equatorward carrying a thermal deficit; their melting removes heat from warmer water. This is equivalent to poleward heat transport by equatorward iceberg drift, explaining the sign of the inferred poleward contribution. The estimate includes the sensible and latent terms from part (b), and does not include the separate effects of released freshwater on density stratification or ocean circulation.