In these equations, t is time, Tm and Td are the temperature anomalies of the mixed layer and deep ocean, respectively, F is the anomalous radiative forcing caused by greenhouse warming, and Cm and Cd are the heat capacities of the mixed layer and deep ocean, respectively. The parameters and X2 are exchange coefficients that determine the rate at which heat is transferred from the upper ocean to the atmosphere and from the upper ocean to the deep ocean, respectively. Although an exact solution of the above equations is often possible (depending on the form of F), it is more informative, and more general, to look at approximate solutions, and that is how we will proceed. The main assumption we make is that the heat capacity of the deep ocean is far greater than that of the mixed layer (Cd & Cm), which is a good assumption considering that the depth of the mixed layer is typically # 100 m, whereas the depth of the ocean itself is on average about 4,000 m.
Given the big disparity in heat capacities, there will be two timescales to the problem: a short timescale over which the mixed layer comes into a quasi-equilibrium, and a much longer timescale over which the full ocean equilibrates. In the short timescale, the deep ocean does not respond and its temperature stays at the initial temperature, namely zero (because all temperatures are measured relative to the initial temperature). Equation 7.2a becomes
If we turn on the forcing and then hold it constant, the solution of equation 7.3 is found to be
m where A = Ax + A2. There are two conclusions to be drawn at this stage:
1. The system evolves toward a quasi-equilibrium on a short timescale of ts = Cm/A. Observations and experiments with comprehensive climate models suggest that this timescale is on the order of a few years to a decade.
2. The quasi-equilibrium temperature reached on this short timescale is given by T2 = 0 and
Let us now consider timescales much longer than Cm/A. We suppose that the mixed layer is in a quasi-equilibrium and that equation 7.2 may be approximated by
The mixed-layer temperature is thus given by
and substituting this in equation 7.6b gives
The system now evolves on the long timescale tl = Cd A* (A1A2), which, given the large value of Cd, may be measured in centuries. The final equilibrium reached has the temperature
which is higher than the temperature given by equation 7.5.
Let us end with few cautionary notes and general remarks. First, and to summarize, there is a fast evolution to the temperature FI(X1 + X2), followed by a much slower evolution to the final temperature FIX 1. However, the real ocean does not consist of just two boxes; the real ocean is immensely complicated. There is indeed a big separation between the timescale on which the mixed layer responds and that on which the full ocean equilibrates, but there are a number of intermediate timescales on which other aspects of the oceans respond, like the gyre and the thermocline. So our treatment is a simplification. Nevertheless, it does contain an essential truth, and that is that it will take a long time for the ocean to equilibrate fully. Even if we were to curtail our emissions of greenhouse gases into the atmosphere and the levels were to stop increasing, the temperature would slowly keep on increasing, possibly for hundreds of years, until the true equilibrium were reached.
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