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Geophysics applies physics and mathematics to the structure, composition, and evolution of planets, including Earth. It connects measurements with mechanisms: plate tectonics explains moving crust, the geodynamo explains a planetary magnetic field, and albedo connects reflected sunlight to a planetary surface energy balance. Even a small model can reveal a useful organizing principle, such as the competition between ice-ocean albedo feedback and stabilizing thermal radiation.

The solar constant measures the incident power per area normal to the sunlight. A spherical Earth of radius $R$ intercepts sunlight over the disc $\pi R^2$, but radiates over its surface $4\pi R^2$. For uniform surface albedo $a$ in the global energy-budget approximation, absorbed power is $(1-a)S\pi R^2$. Treat the surface as a blackbody, assume efficient heat redistribution and neglect internal heating. The Stefan–Boltzmann law and the planetary surface energy balance then giveFor a nonuniform surface the same power balance instead defines an effective planetary equilibrium temperature, $T_E^4=⟨T^4⟩$; its fourth root is not generally the arithmetic mean of the local temperatures. Also, area-averaging the surface albedo is an approximation to the properly insolation-weighted averaging. These are the simplifications implicit in the naked-planet calculation.

Take $R=6.371\times 10^6\mathrm{m}$, so $A_E=4\pi R^2=5.10\times 10^{14}{\mathrm{m}}^2$. The changed area is $A_i=10^{12}{\mathrm{m}}^2$, givingThe initial planetary equilibrium temperature is $T_0=254.8\mathrm{K}$. ThusThe differential estimate $\Delta T\simeq -T_0\Delta a/[4(1-a_0)]$ gives the same result. This is the global equilibrium response of the specified model, not the immediate local Arctic warming or the full atmospheric response.

The exceptional 2007 Arctic sea ice decline is best understood as a weather event acting on an already vulnerable ice cover, with several reinforcing pathways. Warmer air increases downward long-wave heating, lengthens the melt season and delays autumn freezing. Changes in snowfall and the development of melt ponds also change the surface energy balance. Air temperature alone, however, does not identify the source of the observed bottom melt: heat must reach the underside through the ocean.

Warmer Atlantic-origin water supplies a larger subsurface ocean heat content after entering through Fram Strait and the Barents Sea. Its influence on sea ice depends on vertical exchange, because a cold, relatively fresh halocline normally separates it from the ice. Erosion of that halocline, through altered freshwater distributions, wind-driven mixing or changes in circulation, weakens stable density stratification and can increase upward ocean heat flux. A warmer Atlantic layer without this exchange need not produce immediate extra bottom melt. Conversely, warmer Pacific inflow through Bering Strait introduces heat into the shallower western Arctic; advection through the Chukchi Sea provides another route to ice retreat and ocean heat storage.

The Arctic Oscillation changes winds, ice motion and export. Positive phases during earlier years helped remove old, thick ice through Fram Strait and left a younger, thinner pack. Summer 2007 also had a pronounced pressure contrast between the Beaufort region and the Eurasian Arctic, associated with the Arctic dipole anomaly. Its winds promoted ice displacement and warm-air transport. The summer circulation should not be described merely as a particular Arctic Oscillation index: the spatial pattern and the preconditioning from earlier years both matter.

Long-term thinning reduces the latent heat of fusion needed to remove the remaining ice. Replacement of multi-year sea ice by first-year sea ice therefore makes a given summer heat input more effective, while mobile thin ice is more readily redistributed or exported. These dynamical changes can expose open water early enough for substantial solar absorption. Water and melt ponds have lower surface albedo than snow-covered ice, so absorbed short-wave energy rises, increasing upper-ocean heat storage and bottom melt. This ice-ocean albedo feedback also delays refreezing. In the Beaufort Sea in 2007, enhanced solar heating of the newly exposed upper ocean was an especially important explanation of extreme bottom melt. Thus thinning and wind-driven opening can initiate retreat, while solar heating amplifies it; changes in Atlantic or Pacific heat supply can further reinforce the process where circulation permits.

A further positive climate feedback involves terrestrial carbon. Loss of the insulating ice cover permits greater autumn and winter transfer of stored ocean heat to the atmosphere. The resulting regional warming can increase thaw of coastal and inland permafrost. Microbial decomposition of previously frozen organic material then releases carbon dioxide, and oxygen-poor wet environments also release methane. These greenhouse gases strengthen the greenhouse effect, increasing global warming and further thaw. The permafrost carbon feedback provides the causal chain from summer ice retreat to an additional global influence. An Arctic methane feedback involving subsea sources is also possible, but the amount actually reaching the atmosphere depends on oxidation and transport; an abrupt, wholesale release is not a necessary consequence of summer ice loss.

For the bottom-melt estimate, use ${\rho }_i=917\mathrm{kg}{\mathrm{m}}^{-3}$ and the given $L_f=336000\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. Assume the ice at the melting interface is already at its local freezing temperature; there is then no separate sensible-heating term. Per square metre the latent heat requirement isJuly through September has $92$ days, so the mean ocean heat flux to the ice isUsing ${\rho }_i\simeq 900\mathrm{kg}{\mathrm{m}}^{-3}$ instead gives about $76\mathrm{W}{\mathrm{m}}^{-2}$, within the intended accuracy. Around the date of this examination, the estimated net anthropogenic radiative forcing since preindustrial times was about $1.6\mathrm{W}{\mathrm{m}}^{-2}$; doubling carbon dioxide gives about $3.7\mathrm{W}{\mathrm{m}}^{-2}$. The calculated bottom flux is tens of times larger. There is no energy-budget contradiction: it is a local, summer ocean-to-ice heat transfer, supplied by absorbed sunshine, stored heat and transport, whereas radiative forcing is a global perturbation to the radiation budget. Redistribution and local absorption can produce much larger regional fluxes than the global forcing.

For an already ice-free $100\mathrm{m}$ water column, take ${\rho }_w=1025\mathrm{kg}{\mathrm{m}}^{-3}$ and liquid-water specific heat capacity $c_w\simeq 4000\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Assuming all the same energy remains in that uniformly mixed column,If the water initially lies near the seawater freezing point, about $-1.8{}^{\circ }\mathrm{C}$, its final temperature is approximately $\boxed{-0.3{}^{\circ }\mathrm{C}}$. An absolute final temperature requires that additional initial condition; the energy calculation itself determines only the temperature increase. The given ice specific heat capacity is not the appropriate value for heating liquid seawater.

The upward arrows at the surface are reflected short-wave flux $\alpha F_r$, emitted long-wave flux $ϵ\sigma T_s^4$ from the Stefan–Boltzmann law, and, when positive, conductive heat arriving from the planetary ice shell. The downward arrow is the incident solar flux $F_r$. The planetary surface energy balance is thereforeThe relevant planetary albedo is the Bond albedo. In a steady one-dimensional shell, Fourier's law and $k(T)=a/T$ make the upward conductive fluxwhere $z$ points downward and $T(h)=T_m$. Substitution gives the requested implicit equation