The concept of thermal inertia is well illustrated by consideration of heat storage in the mixed layer of an ocean. Consider a layer of incompressible fluid with density p and specific heat cp, which is well-mixed by turbulence to a depth H. The assumption of well-mixedness implies that any heating or cooling applied to the surface is distributed instantaneously throughout the depth of the mixed layer, whose temperature thus remains uniform. Let S(t) be the solar flux heating the mixed layer, and assume that the cooling of the mixed layer (by infrared radiation or other means) can be written as a function of temperature, which we shall call F(T). For example, if the atmosphere above the ocean has no greenhouse effect and carries no heat away from the surface by turbulent transport, the cooling is the radiative cooling F(T) = <rT4. The energy balance equation for the mixed layer is then dtpcpHT = S(t) - F(T) (7.11)

Exercise 7.4.1 Consider a planet with a 50m deep mixed layer water ocean (cp = 4218 J/kg, p = 1000kg/m3). Suppose that the atmosphere for some reason has no effect whatsoever on the surface energy budget. (Why would this situation be hard to arrange, even for a pure N2 atmosphere?) Hence F(T) = aT4. Suppose that the temperature of the polar ocean is 300K when the sun sets and the long polar night begins. Find a solution to Eq. 7.11 for this situation, and use it to determine how long it takes for the ocean to fall to the freezing point (about 271K for salt water)?

We may define a thermal inertia coefficient y = pcpH for the mixed layer ocean. If an amount of energy AE is added to or removed from a column of the ocean having a cross section of one square meter, the corresponding temperature change is AE/y. For a 50m mixed layer water ocean, y = 2.1 • 108 J/(m2K), so that an energy flux of 100W/m2 out of the surface would lead to a cooling rate of 100/y = 4.74 • 10-7K/s = .04K/day. Clearly, a rather shallow layer of well-mixed water can buffer a considerable surface flux imbalance. The Earth's ocean is several kilometers deep, but it is only the upper few tens of meters that are well mixed on short time scales; 50m is in fact a reasonable approximation to the overall mixed layer depth of Earth's ocean, though there are geographical variations. Most other liquids would do about as well as water at storing heat. It is primarily the mixing depth that determines the thermal buffering effect of a planet's ocean.

Atmospheres also have thermal inertia, which can be considered in a fashion analogous to a mixed layer. The entire mass of the troposphere is well-mixed, and this generally makes up most of the mass of an atmosphere. For a well-mixed atmosphere the temperature profile can be tied to the temperature at any convenient fixed level (usually the ground), and we need to determine how much energy it takes to change this index temeprature by one degree K. This is where the notion of moist or dry static energy, introduced in Chapter 2, comes into its own. For simplicity, let's consider a noncondensing atmosphere, for which case the dry static energy (per unit mass) cpT + gz is independent of height within the troposphere. Hence, by evaluating its value at z = 0, the dry static energy per unit mass can be written cpTsa, where Tsa is the near-surface air temperature. The mass per unit area of the atmosphere is ps/g, so the energy required to change the surface air temprature by 1K while keeping the dry static energy well-mixed in the vertical is simply p = cpps/g. When the atmosphere has a condensible component, one needs to take into account latent heat storage as well. Exploration of that aspect of the problem will be relegated to the Workbook section of this chapter. The following discussion is most valid for noncondensing atmospheres, but will nonetheless apply approximately to condensible atmospheres undergoing fluctuations in which the latent heat storage doesn't change too much.

It is convenient to express p in terms of the depth of a water mixed layer ocean, Heq which would have the same thermal inertia coefficient. For the Earth atmosphere, Heq = 2.4m, which is insignificant in comparison to the mixed layer depth of the ocean. Hence, one expects the Earth's atmosphere to come into equilibrium much more quickly than the ocean. The current 6mb CO2 atmosphere of Mars has Heq = .03m, while the massive atmosphere of Venus has Heq in excess of 155m. Neither Mars nor Venus has an ocean to buffer the seasonal cycle, but the Venus atmosphere alone can be expected to have a considerable moderating effect, whereas present Mars should behave more or less as if each point of the globe is in instantaneous equilibrium. Early Mars (circa 4 billion years ago) may have had a 2 bar CO2 atmosphere, which would translate into a 10m equivalent mixed layer. This is considerably greater than that of Earth's atmosphere, but still not enough to have much moderating effect, in view of the fact that Mars' year is about twice as long as Earth's. Titan has a mostly N2 atmosphere with a surface pressure only slightly in excess of Earth, but its weak gravitational acceleration of 1.35m/s2 means that this pressure translates into a much greater mass of atmosphere per square meter of planetary surface. Thus,the Titan atmosphere has an equivalent mixed layer depth of about 24m. Given the low temperature of Titan, and consequent low rate of energy loss by infrared emission, this value is expected to yield a very considerable buffering effect on Titan's seasonal cycle, regardless of whether there is a liquid ocean at the surface. For example, based on a typical surface temperature of 90K, blackbody emission would cool the planet only at a rate of about 1K per 300 Earth days, if insolation were completely shut off.

Exercise 7.4.2 The specific heat of liquid Methane is 3450 J/K. How deep would a well-mixed methane ocean on Titan have to be for it to have thermal inertia comparable to Titan's atmosphere?

We shall now consider some simple solutions to the mixed layer model, keeping in mind that this model applies to atmospheres as well as oceans, with a suitable choice of the equivalent mixed layer depth. At this point we assume that pcpH is constant, though models with a time-varying mixed layer depth are possible. Without any loss of generality we may write the insolation and temperature in the form

where So and To are the time means of S and T and the deviations have zero time mean. Now, suppose that T' << To for whatever reason; this need not require that S' << So, since the temperature fluctuations might be small by virtue of a slow response time of the system. Because the temperature fluctuations are small, the surface cooling can be expanded about To and approximated by a linear function:

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