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With these definitions, the solution can be written

The character of the response depends on the period of the forcing relative to the characteristic response time of the system. This determines both the amplitude of the fluctuation and the phase shift relative to the forcing. For wr << 1 we have A = 0 and |A| = Si/a. For wr >> 1 we have A = n/2 and |A| = S1/(pcpHw). Note that in this case the temperature fluctuation becomes weak in inverse proportion to the frequency of the solar forcing fluctuation. These are special cases of the limits discussed previously, but we now have the further advantage of an explicit formula showing how the phase and amplitude of the seasonal cycle vary between the two extreme cases.

So far, we have not specified the flux which is to be used for the heat loss term F(T) in Eq. 7.11, or for the damping coefficient b in the linearized form of the equation. One candidate for this flux is the top-of-atmosphere infrared heat loss. The other is the combined turbulent and infrared heat exchange between the planetary surface and the atmosphere, discussed in Chapter 6. There are two circumstances in which the top-of-atmosphere flux (the OLR) is the appropriate one to use. If the time scale under consideration is long enough that the surface budget can come into equilibrium, then the net solar flux absorbed at the surface is equal to the net turbulent and infrared flux passing from the surface into the atmosphere. In this case, we may consider the energy budget of the surface-atmosphere system as a whole, whence the OLR gives the heat loss from the system. The thermal inertia is provided by the atmosphere, and one uses he atmosphere's equivalent mixed layer depth in the mixed layer model equations. Alternately, if the response time of the atmosphere is short enough compared to the time scale under consideration, the energy budget of the atmosphere comes into equilibrium. In this case, the net energy exchange between the surface and the atmosphere must equal the OLR, since otherwise the atmosphere would warm up or cool down until equilibrium is achieved. Hence, one can use the OLR for the heat loss term in the surface energy budget, obviating the need to know the detailed physics behind the surface to atmosphere energy transfer. In this case, the thermal inertia is provided by the heat capacity of the mixed layer ocean.

In either case, one can compute OLR(T) using a radiation model and some assumption linking the temperature and humidity profile to surface temperature, or one can use one of the linear or polynomial fits to the OLR curve discussed in Section 4. For example, with a linear fit to the OLR curve for a terrestrial atmosphere with 300ppmv CO2 and 50% relative humidity, b is about 2(W/m2)/K in the range 250K to 310K. The corresponding relaxation time t is 1200 days for a 50 meter mixed layer, or 60 days for the 2.4m mixed layer which is equivalent to the thermal inertia of the Earth's atmosphere. In consequence, the seasonal cycle is expected to be strongly attenuated on the ocean- covered parts of the Earth (apart from coastal effects). The atmosphere alone does not have enough thermal inertia to damp out the seasonal cycle, but it does have enough thermal inertia to keep the atmospheric temperature roughly constant in the course of the diurnal cycle. Colder temperatures tend to make the relaxation time longer. For example, in an Earthlike atmosphere with 300ppm CO2, the relaxation time roughly doubles at 160K. As noted earlier, Titan has a very long relaxation time owing to its thick atmosphere and low temperature; now we can make the statement more precise. Ignoring the greenhouse effect and setting b = 4<rT3, T = 90K we find a relaxation time of 20 Earth years, based on the equivalent 24m mixed layer depth of Titan's atmosphere. Since Titan's year (which is the same as Saturn's year) is about 30 Earth years, the seasonal cycle on Titan is expected to be considerably damped, though not so much so as the seasonal cycle over the Earth's oceans. The weak greenhouse effect from methane in Titan's atmosphere would somewhat enhance the damping. In contrast, a similar calculation for the thin atmosphere of present Mars gives a relaxation time of only .8 Earth days, based on T = 200K. Since a Mars day is approximately the same as an Earth day, the thermal inertia of the Martian atmosphere at present has relatively little damping effect on the diurnal cycle.

The thermal relaxation process is different if the time scale under consideration is short compared to the response time of the atmosphere, but long compared to the response time of the surface. In this case, the atmospheric temperature remains approximately constant while the surface temperature fluctuates. This is the way the diurnal cycle works on Earth over ice or land. The relaxation time of surface temperature is then determined using the turbulent and radiative surface-atmosphere flux formulae discussed in Chapter 6, rather than the OLR. The situation of present Mars is not like this, since the atmosphere has little thermal inertia. There, the diurnal cycle affects the entire depth of the atmosphere, and the diurnal response is approximately governed by the OLR and the thermal inertia of the surface, much as for the Earth's seasonal cycle.

In the general case, where neither the atmosphere nor the ocean are in equilibrium, one must write a separate mixed layer model for each of these two components, coupling them through the surface exchange flux formulae, and allowing the atmosphere to lose energy through it's top via OLR. The exploration of this case will be left to the reader.

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