## Faint Young Sun Snowball Earth and Hysteresis

We we now have enough basic theoretical equipment to take a first quantitative look at the Faint Young Sun problem. To allow for the greenhouse effect of the Earth's atmosphere, we take prad = 670mb, which gives the correct surface temperature with the observed current albedo a = .3. How much colder does the Earth get if we ratchet the Solar constant down to 960W/m2, as it was 4.7 billion years ago when the Earth was new? As a first estimate, we can compute the new temperature from Eq. 3.8 holding prad and the albedo fixed at their present values. This yields 261K. This is substantially colder than the present Earth. The fixed albedo assumption is unrealistic,however, since the albedo would increase for a colder and more ice-covered Earth, leading to a substantially colder temperature than we have estimated. In addition, the strength of the atmospheric greenhouse effect could have been different for the Early Earth, owing to changes in the composition of the atmosphere.

An attempt at incorporating the ice-albedo feedback can be made by using the energy balance Eq. 3.10 with the albedo parameterization given by Eq. 3.9. For this calculation, we choose constants in the albedo formula that give a somewhat more realistic Earthlike climate than

Net Flux

O Unstable equilibrium • Stable equilibrium

O Unstable equilibrium • Stable equilibrium

T

Figure 3.9: Sketch illustrating stable vs. unstable equilibrium temperatures.

those used in Figure 3.8. Specifically, we set ao = .28 to allow for the albedo of clouds and land, and To = 295 to allow a slightly bigger polar ice sheet. The position of the equilibria can be determined by drawing a graph like Fig. 3.8, or by applying a root-finding algorithm like Newton's method to Eq. 3.10. The resulting equilibria are shown as a function of Lq in Figure 3.10, with prad held fixed at 670mb. Some techniques for generating diagrams of this type are developed in Problem ??. For the modern Solar constant, and prad = 670mb, the system has a stable equilibrium at Ts = 286K, close to the observed modern surface temperature, and is partially ice covered. However, the system has a second stable equilibrium, which is a globally ice-covered Snowball state having Ts = 249K. Even today, the Earth would stay in a Snowball state if it were somehow put there. The two stable equilibria are separated by an unstable equilibrium at Ts = 270K, which defines the boundary between the set of initial conditions that go to the "modern" type state, and the set that go to a Snowball state. The attractor boundary for the modern open-ocean state is comfortably far from the present temperature, so it would not be easy to succumb to a Snowball.

Now we turn down the Solar constant, and re-do the calculation. For Lq = 960W/m2, there is only a single equilibrium point if we keep prad = 670mb. This is a stable Snowball state with Ts = 228K. Thus, if the Early Earth had the same atmospheric composition as today,leading to a greenhouse effect no stronger than the present one, the Earth would have inevitably been in a Snowball state. The open ocean state only comes into being when Lq is increased to 1330W/m2, which was not attained until the relatively recent past. This contradicts the abundant geological evidence for prevalent open water throughout several billion years of Earth's history. Even worse, if the Earth were initially in a stable snowball state four billion years ago, it would stay in that state until Lq increases to 1640W/m2, at which point the stable snowball state would disappear and the Earth would deglaciate. Since this far exceeds the present Solar constant, the Earth would be globally glaciated today. This even more obviously contradicts the data.

The currently favored resolution to the paradox of the Faint Young Sun is the supposition that the atmospheric composition of the early Earth must have resulted in a stronger greenhouse effect than the modern atmosphere produces. The prime candidate gases for mediating this change are CO2 and CH4. The radiative basis of the idea will be elaborated further in Chapter 4, and some ideas about why the atmosphere might have adjusted over time so as to maintain an equable

Figure 3.10: Hysteresis diagram obtained by varying Lq with prad/ps fixed at .67. Arrows indicate path followed by the system as Lq is first increased, then decreased. The unstable solution branch is indicated by a dashed curve.

600 550 500 450 400 350

600 550 500 450 400 350

Figure 3.11: As in Fig. 3.10, but varying prad with Lq = 960W/m2.

climate despite the brightening Sun are introduced in Chapter 8. Fig. 3.11 shows how the equilibria depend on prad, with Lq fixed at 960W/m2. Whichever greenhouse gas is the Earth's savior, if it is present in sufficient quantities to reduce prad to 500mb or less, then a warm state with an open ocean exists (the upper branch in Fig. 3.11). However, for 420mb < prad < 500mb a stable snowball state also exists, meaning that the climate that is actually selected depends on earlier history. If the planet had already fallen into a Snowball state for some reason, the early Earth would stay in a Snowball unless the greenhouse gases build up sufficiently to reduce prad below 420mb at some point.

Figures 3.10 and 3.11 illustrate an important phenomenon known as hysteresis: the state in which a system finds itself depends not just on the value of some parameter of the system, but the history of variation of that parameter. This is possible only for systems that have multiple stable states. For example, in 3.10 suppose we start with Lq = 1000W/m2, where the system is inevitably in a Snowball state with T = 230K. Let's now gradually increase Lq. When Lq reaches 1500W/m2 the system is still in a Snowball state, having T = 254K, since we have been following a stable solution branch the whole way. However, when Lq reaches 1640W/m2, the Snowball solution disappears, and the system makes a sudden transition from a Snowball state with T = 260K to the only available stable solution, which is an ice-free state having T = 301K. As Lq increases further to 2000W/m2, we follow the warm, ice-free state and the temperature rises to 316K. Now suppose we begin to gradually dim the Sun, perhaps by making the Solar system pass through a galactic dust cloud. Now, we follow the upper, stable branch as Lq decreases, so that when we find ourselves once more at Lq = 1500W/m2 the temperature is 294K and the system is in a warm, ice-free state rather than in the Snowball state we enjoyed the last time we were there. As Lq is decreased further, the warm branch disappears at Lq = 1330W/m2 and the system drops suddenly from a temperature of 277K into a Snowball state with a temperature of 246K, whereafter the Snowball branch is again followed as Lq is reduced further. The trajectory of the system as Lq is increased then decreased back to its original value takes the form of an open loop, depicted in Fig. 3.10.

The thought experiment of varying Lq in a hysteresis loop is rather fanciful, but many atmospheric processes could act to either increase or decrease the greenhouse effect over time. For the very young Earth, with Lq = 960W/m2, the planet falls into a Snowball when prad exceeds 500mb, and thereafter would not deglaciate until prad is reduced to 420mb or less (see Fig. 3.11). The boundaries of the hysteresis loop, which are the critical thresholds for entering and leaving the Snowball, depend on the solar constant. For the modern solar constant, the hysteresis loop operates between prad = 690mb and prad = 570mb. It takes less greenhouse effect to keep out of the Snowball now than it did when the Sun was fainter, but the threshold for initiating a Snowball in modern conditions is disconcertingly close to the value of prad which reproduces the present climate.

The fact that the freeze-thaw cycle can exhibit hysteresis as atmospheric composition changes is at the heart of the Snowball Earth phenomenon. An initially warm state can fall into a globally glaciated Snowball if the atmospheric composition changes in such a way as to sufficiently weaken the greenhouse effect. Once the threshold is reached, the planet can fall into a Snowball relatively quickly - in a matter of a thousand years or less - since sea ice can form quickly. However, to deglaciate the Snowball, the greenhouse effect must be increased far beyond the threshold value at which the planet originally entered the Snowball state. Atmospheric composition must change drastically in order to achieve such a great increase, and this typically takes many millions of years. When deglaciation finally occurs, it leaves the atmosphere in a hyper-warm state, which only gradually returns to normal as the atmospheric composition evolves in such a way as to reduce the greenhouse effect. As discussed in Chapter 1, there are two periods in Earth's past when geological evidence suggests that one or more Snowball freeze-thaw cycles may have occurred. The first is in the Paleoproterozoic, around 2 billion years ago. At this time, Lq « 1170W/m2, and the thresholds for initiating and deglaciating a Snowball are prad = 600mb and prad = 500mb in our simple model. For the Neoproterozoic, about 700 million years ago, Lq « 1290W/m2 and the thresholds are at prad = 650mb and prad = 540mb.

The boundaries of the hysteresis loop shift as the Solar constant increases, but there is nothing obvious in the numbers to suggest why a Snowball state should have occurred in the Paleoproterozoic and Neoproterozoic but not at other times. Hysteresis associated with ice-albedo feedback has been a feature of the Earth's climate system throughout the entire history of the planet. Hysteresis will remain a possibility until the Solar constant increases sufficiently to render the Snowball state impossible even in the absence of any greenhouse effect (i.e. with prad = 1000mb). Could a Snowball episode happen again in the future, or is that peril safely behind us? These issues require an understanding of the processes governing the evolution of Earth's atmosphere, a subject that will be taken up in Chapter 8.

Exercise 3.4.2 Assuming an ice albedo of .6, how high does L0 have to become to eliminate the possibility of a snowball state? Will this happen within the next five billion years? What if you assume there is enough greenhouse gas in the atmosphere to make prad/ps = .5?

Note: The evolution of the Solar constant over time is approximately L0(t) = L0p ■ (.7 + (t/22.975) + (t/14.563)2), where t is the age of the Sun in billions of years (t = 4.6 being the current age) and L0p is the present Solar constant. This fit is reasonably good for the first 10 billion years of Solar evolution.

The "cold start" problem is a habitability crisis that applies to waterworlds in general. If a planet falls into a Snowball state early in its history, it could take billions of years to get out if one needs to wait for the Sun to brighten. The time to get out of a Snowball could be shortened if greenhouse gases build up in the atmosphere, reducing prad. How much greenhouse gas must build up to deglaciate a snowball? How long would that take? What could cause greenhouse gases to accumulate on a Snowball planet? These important questions will be taken up in subsequent chapters.

Another general lesson to be drawn from the preceding discussion is that the state with a stable, small icecap is very fragile, in the sense that the planetary conditions must be tuned rather precisely for the state to exist at all. For example, with the present Solar constant, the stable small icecap solution first appears when prad falls below 690mb. However, the icecap shrinks to zero as prad is reduced somewhat more, to 615mb. Hence, a moderate strengthening in the greenhouse effect would, according to the simple energy balance model, eliminate the polar ice entirely and throw the Earth into an ice-free Cretaceous hothouse state. The transition to an ice-free state of this sort is continous in the parameter being varied; unlike the collapse into a snowball state or the recovery from a snowball, it does not result from a bifurcation. In light of its fragility, it is a little surprising that the Earth's present small-icecap state has persisted for the past two million years, and that similar states have occurred at several other times in the past half billion years. Does the simple energy-balance model exaggerate the fragility of the stable small-icecap state? Does some additional feedback process adjust the greenhouse effect so as to favor such a state while resisting the peril of the Snowball? These are largely unresolved questions. Attacks on the first question require comprehensive dynamical models of the general circulation, which we will not encounter in the present volume. We will take up, though not resolve, the second question in Chapter 8. It is worth noting that small-icecap states like those of the past two million years appear to be relatively uncommon in the most recent half billion years of Earth's history, for which data is good enough to render a judgement about ice cover. The typical state appears to be more like the warm relatively ice-free states of the Cretaceous, and perhaps this reflects the fragility of the small-icecap state.

The simple models used above are too crude to produce very precise hysteresis boundaries. Among the many important effects left out of the story are water vapor radiative feedbacks, cloud feedbacks, the factors governing albedo of sea ice, ocean heat transports and variations in atmospheric heat transport. The phenomena uncovered in this exposition are general, however and can be revisited across a heirarchy of models. Indeed, the re-examination of this subject provides an unending source of amusement and enlightenment to climate scientists.