Aofor T To

qualitatively reproduces the shape of the albedo curve which is found in detailed calculations. In particular, the slope of albedo vs temperature is large when the temperature is low and the planet is nearly ice-covered, because there is more area near the Equator, where ice melts first. Conversely, the slope reduces to zero as the temperature threshold for an ice-free planet is approached, because there is little area near the poles where the last ice survives; moreover, the poles receive relatively little sunlight in the course of the year, so the albedo there contributes less to the global mean than does the albedo at lower latitudes. Note that this description assumes an Earthlike planet, which on average is warmest near the Equator. As will be discussed in Chapter 7, other orbital configurations could lead to the poles being warmer, and this would call for a different shape of albedo curve.

Ice albedo feedback of a similar sort could arise on a planet with land, through snow accumulation and glacier formation on the continents. The albedo could have a similar temperature dependence, in that glaciers are unlikely to survive where temperatures are very much above freezing, but can accumulate readily near places that are below freezing - provided there is enough precipitation. It is the latter requirement that makes land-based snow/ice albedo feedback much more complicated than the oceanic case. Precipitation is determined by complex atmospheric circulation patterns that are not solely determined by local temperature. A region with no precipitation will not form glaciers no matter how cold it is made. The present state of Mars provides a good example: its small polar glaciers do not advance to the Equator, even though the daily average equatorial temperature is well below freezing. Still, for a planet like Earth with a widespread ocean to act as a source for precipitation, it may be reasonable to assume that most continental areas will eventually become ice covered if they are located at sufficiently cold latitudes. In fairness, we should point out that even the formation of sea ice is considerably more complex than we have made it out to be, particularly since it is affected by the mixing of deep unfrozen water with surface waters which are trying to freeze.

Earth is the only known planet that has an evident ice/snow albedo feedback, but it is reasonable to inquire as to whether a planet without Earth's water-dominated climate could behave analogously. Snow is always "white" more or less regardless of the substance it is made of, since its reflectivity is due to the refractive index discontinuity between snow crystals and the ambient gas or vacuum. Therefore, a snow-albedo feedback could operate with substances other than water (e.g. nitrogen or methane). Titan presents an exotic possibility, in that its surface is bathed in a rain of tarry hydrocarbon sludge, raising the speculative possibility of "dark glacier" albedo feedbacks. Sea ice forming on Earth's ocean gets its high albedo from trapped air bubbles, which act like snowflakes in reverse. The same could happen for ices of other substances, but sea-ice albedo feedback is likely to require a water ocean. The reason is that water, alone among likely planetary materials, floats when it freezes. Ice forming on, say, a carbon dioxide or methane ocean would sink as soon as it formed, preventing it from having much effect on surface albedo.

Returning attention to an Earthlike waterworld, we write down the energy budget

This determines Ts as before, with the important difference that the Solar absorption on the left hand side is now a function of Ts instead of being a constant. Analogously to Fig. 3.5, the equilibrium surface temperature can be found by plotting the absorbed Solar radiation and the OLR vs. Ts on the same graph. This is done in Fig. 3.8, for four different choices of Lq. In this plot, we have taken OLR = aT4, which assumes no greenhouse effect 4. In contrast with the fixed-albedo case, the ice-albedo feedback allows the climate system to have multiple equilibria: there can be more than one climate compatible with a given Solar constant, and additional information is required to determine which state the planet actually settles into. The nature of the equilibria depends on Lq. When Lq is sufficiently small (as in the case Lq = 1516W/m2 in Fig. 3.8) there is only one solution, which is a very cold globally ice-covered Snowball state, marked Sn\ on the graph. Note that the Solar constant that produces a unique Snowball state exceeds the present Solar constant at Earth's orbit. Thus, were it not for the greenhouse effect, Earth would be in such a state, and would have been for its entire history. When L_odot is sufficiently large (as in the case Lq = 2865W/m2 in Fig. 3.8) there is again a unique solution, which is a very hot globally ice-free state, marked H on the graph. However, for a wide range of intermediate Lq, there are three solutions: a Snowball state (Sn2), a partially ice covered state with a relatively large ice sheet (e.g. A), and a warmer state (e.g. B) which may have a small ice sheet or be ice free, depending on the precise value of Lq. In the intermediate range of Solar constant, the warmest state is suggestive of the present or Pleistocene climate when there is a small ice-cap, and suggestive of Cretaceous-type hothouse climates when it is ice-free. In either case, the frigid Snowball state is available as an alternate possibility.

As the parameter Lq is increased smoothly from low values, the temperature of the the Snowball state increases smoothly but at some point an additional solution discontinuously comes into being at a temperature far from the previous equilibrium, and splits into a pair as Lq is further increased. As Lq is increased further, at some point, the intermediate temperature state merges with the snowball state, and disappears. This sort of behavior, in which the behavior of a system changes discontinuously as some control parameter is continuously varied, is an example of a bifurcation.

Finding the equilibria tells only part of the story. A system placed exactly at an equilbrium point will stay there forever, but what if it is made a little warmer than the equilibrium? Will it heat up yet more, perhaps aided by melting of ice, and ultimately wander far from the equilibrium? Or will it cool down and move back toward the equilibrium? Similar questions apply if the state is made initially slightly cooler than an equilibrium. This leads us to the question of stability. In order to address stability, we must first write down an equation describing the time evolution of the system. To this end, we suppose that the mean energy storage per unit area of the planet's surface can be written as a function of the mean temperature; let's call this function E(Ts). Changes in the energy storage could represent the energy required to heat up or cool down a layer of water of

4Of course, this is an unrealistic assumption, since a waterworld would inevitably have at least water vapor — a good greenhouse gas — in its atmosphere

Figure 3.8: Graphical determination of the possible equilibrium states of a planet whose albedo depends on temperature in accordance with Eq. 3.9. The OLR is computed assuming the atmosphere has no greenhouse effect, and the albedo parameters are ao = .1, a = .6, T = 260K and To = 290K. The Solar constant for the various solar absorption curves is indicated in the legend.

Figure 3.8: Graphical determination of the possible equilibrium states of a planet whose albedo depends on temperature in accordance with Eq. 3.9. The OLR is computed assuming the atmosphere has no greenhouse effect, and the albedo parameters are ao = .1, a = .6, T = 260K and To = 290K. The Solar constant for the various solar absorption curves is indicated in the legend.

some characteristic depth, and could also include the energy needed to melt ice, or released by the freezing of sea water. For our purposes, all we need to know is that E is a monotonically increasing function of Ts. The energy balance for a time-varying system can then be written

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