## The general damping of climate variability by the ocean

n>t only does the ocean provide a moderating influence on the march of the seasons, but it also can provide a moderating influence on the variability of climate on other timescales too. We talk more about the mechanisms that give rise to climate variability in the next chapter, but for now let us just suppose that the climate system excluding the ocean is able to vary on multiple timescales, from days to years. Then, just as the ocean is able to damp the seasonal variability, the ocean damps variability on all these timescales. However, the ocean does not damp the variations equally on all timescales; rather, because on longer timescales the ocean itself can heat up or cool down in response to climate variations, the damping effects are larger on shorter timescales. We give a brief mathematical treatment of this argument in the next section, and a more complete treatment in appendix A of this chapter.

### Mathematical treatment of damping

The surface temperature of the ocean and the land are maintained by a balance between heating and cooling. The heating occurs both by solar radiation and by downward longwave radiation from the atmosphere and is proximately independent of the temperature of the surface itself. The cooling, on the other hand, increases with the temperatureâ€”a hot object cools down faster than a warm one. If for simplicity we suppose that the cooling rate varies linearly with temperature, then we can model the surface temperature by the equation cdr S XT. (5.1)

Here, S is the heating source, T is the temperature, and t, the time. The parameter C is the heat capacity of the system, and X is a constant that determines how fast the body cools when it is hot. Obviously, this equation is too simple to realistically describe how the surface temperature varies (it ignores lateral variations, for one thing), but it illustrates the point we wish to make.

The equation says that the heat capacity times the rate of the temperature increase (the left-hand side) is equal to the heating (S) minus the cooling (XT). If we set S = 0 for the moment, then a solution of this equation is T = T0 exp(-Xt/C), where T0 is the initial temperature.

If S is a constant, then the full solution is

This equation tells us that if there is a perturbation to the system, the perturbation will decay away on the timescale C/X. With a mixed-layer depth of 100 m and X = 15 Wm-2 K-1 (which is suggested by observations for air-sea interactions), we obtain a timescale of a little less than a year. That is to say, the ocean mixed layer can absorb or give out heat on the timescale of about a year. Variability on timescales significantly longer than this is not greatly damped by the presence of an ocean mixed layer because on these timescales the mixed layer itself heats up and cools down and so provides no damping to the system. However, on timescales much shorter than this, the mixed layer absorbs heat from a warm atmosphere, or alternatively gives up heat to a cold atmosphere, thus damping the variability that the atmosphere otherwise might have. The land surface has a much smaller heat capacity, so that the timescale C/m is much smaller for land than it is for the ocean. There is thus a much smaller damping effect over land than over the ocean.

The situation is not quite as straightforward as this argument suggests. A complicating factor is that the entirety of the ocean mixed layer does not respond to fast variations in the atmosphere. Thus, for example, only the top few meters of water may respond to diurnal variations in temperature, and such variations are therefore damped less than one might expect. Nevertheless, the overall effect is clear: The heat capacity of the ocean mixed layer damps variations on timescales up to and including the annual variations. Interested readers can find a more complete description of this effect in appendix A of this chapter.

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