By definition, G = 0 at an equilibrium point Teq. Suppose that the slope of G is well-defined near Teq - in formal mathematical language, we say that G is continuously differentiable at Teq, meaning that the derivative of G exists and is a continous function for Ts in some neighborhood of Teq. Then, if dG/dTs < 0 at Ts, it will also be negative for some finite distance to the right and left of Ts. This is the case for points a and c in the net flux curve sketched in Fig. 3.9. If the temperature is made a little warmer than Teq in this case, G(Ts) and hence ^^^ will become negative and the solution will move back toward the equilibrium. If the temperature is made a little colder than Teq, G(Ts) and hence will become positive, and the solution will again move back toward the equilibrium. In contrast, if dG/dTs > 0 near the equilibrium, as for point c in the sketch, a temperature placed near the equilibrium moves away from it, rather than towards it. Such equilibria are unstable. If the slope happens to be exactly zero at an equilibrium, one must look to higher derivatives to determine stability. These are "rare" cases, which will be encountered only for very special settings of the parameters. If the d2G/dT2 is non zero at the equilibrium, the curve takes the form of a parabola tangent to the axis at the equilibrium. If the parabola opens upwards, then the equilibrium is stable to displacements to the left of the equilibrium, but unstable to displacements to the right. If the parabola opens downwards, the equilibrium is unstable to displacements to the left but stable to displacements to the right. Similar reasoning applies to the case in which the first non-vanishing derivative is higher order, but such cases are hardly ever encountered.
Exercise 3.4.1 Draw a sketch illustrating the behavior near marginal equilibria with d2G/dT2 > 0 and d2G/dT2 < 0. Do the same for equilibria with d2G/dT2 = 0, having d3G/dT3 > 0 and d3G/dT3 < 0
It is rare that one can completely characterize the behavior of a nonlinear system, but one dimensional problems of the sort we are dealing with are exceptional. In the situation depicted in Fig. 3.9, G is positive and dT/dt is positive throughout the interval between b and c. Hence, a temperature placed anywhere in this interval will eventually approach the solution c arbitrarily closely - it will be attracted to that stable solution. Similarly, if T is initially between a and b, the solution will be attracted to the stable equilibrium a. The unstable equilibrium b forms the boundary between the basins of attraction of a and c. No matter where we start the system within the interval between a and c (and somewhat beyond, depending on the shape of the curve further out), it will wind up approaching one of the two stable equilibrium states. In mathematical terms, we are able to characterize the global behavior of this system, as opposed to just the local behavior near equilibria.
At an equilibrium point, the curve of solar absorption crosses the OLR curve, and the stability criterion is equivalent to stating that the equilibrium is stable if the slope of the solar curve is less than that of the OLR curve where the two curves intersect. Using this criterion, we see that the intermediate-temperature large ice-sheet states, labeled A and A' in Fig. 3.8, are unstable. If the temperature is made a little bit warmer then the equilibrium the climate will continue to warm until it settles into the warm state (B or B') which has a small or nonexistent ice sheet. If the temperature is made a little bit colder than the equilibrium, the system will collapse into the snowball state (Sn2 or Sn3). The unstable state thus defines the boundary separating the basin of attraction of the warm state from that of the snowball state.
Moreover, if the net flux G(T) is continous and has a continuous derivative (i.e. if the curve has no "kinks" in it), then the sequence of consecutive equilibria always alternates between stable and unstable states. For the purpose of this theorem, the rare marginal states with dG/dT = 0 should be considered "wildcards" that can substitute for either a stable or unstable state. The basic geometrical idea leading to this property is more or less evident from Figure 3.9, but a more formalized argument runs as follows: Let Ta and Tb be equilibria, so that G(Ta) = G(Tb) = 0. Suppose that the first of these is stable, so dG/dT < 0 at Ta, and also that the two solutions are consecutive, so that G(T) does not vanish for any T between Ta and Tb. Now if dG/dT < 0 at Tb, then it follows that G > 0 just to the left of Tb. The slope near Ta similarly implies that G < 0 just to the right of Ta. Since G is continuous, it would follow that G(T) = 0 somewhere between Ta and Tb. This would contradict our assumption that the two solutions are consecutive. In consequence, dG/dT > 0 at Tb. Thus, the state Tb is either stable or marginally stable, which proves our result. The proof goes through similarly if Ta is unstable. Note that we didn't actually need to make use of the condition that dG/dT be continuous everywhere: it's enough that it be continuous near the equilibria, so we can actually tolerate a few kinks in the curve.
A consequence of this result is that, if the shape of G(T) is controlled continously by some parameter like Lq, then new solutions are born in the form of a single marginal state which, upon further change of Lq splits into a stable/unstable or unstable/stable pair. The first member of the pair will be unstable if there is a pre-existing stable solution immediately on the cold side of the new one, as is the case for the Snowball states Sn in Fig. 3.8. The first member will be stable if there is a pre-existing unstable state on cold side, or a pre-existing stable state on the warm side (e.g. the state H in Fig. 3.8). What we have just encountered is a very small taste of the very large and powerful subject of bifurcation theory.
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