San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Simultaneous Determination of the Latitudinal Temperature Profile and Latitudinal Heat Flux

In meteorological studies of the transfer of heat energy from the equatorial regions toward the polar regions the problem is characterized as one of establishing the flow of heat energy from regions of a heat surplus to regions of a heat deficit. The regions of heat surplus are those in which the absorption of radiant energy from the Sun exceeds the thermal radiation. The heat deficit regions where more energy is radiated away into space than comes in from the sun. This characterization of the problem implies that there are some regions that are intrinsically heat surplus areas and others that are heat deficit areas. This gets the causalities mixed up. The temperature profile is established simultaneous with the heat flows. One does not determine the other; they are simultaneously and mutually determined. The models examined below are ones in which this simultaneous determination is made explicit.

## The Simple Energy Balance Model

Let φ be latitude and F(φ) be the incoming solar radiation energy (insolation) at φ. The albedo (reflectance) α could be a function of latitude and/or temperature T. The temperature T is generally a function of latitude, T(φ).

The energy radiated from the surface is equal to εσT4, where σ is the Stefan-Boltzmann constant and ε is an emissivity factor representing the possibility that the surface is not a perfect blackbody radiator.

A balance of incoming and outgoing radiative energy at the surface requires that

#### (1−α(φ))F(φ) = εσT4(φ) and hence T(φ) = [(1−α(φ))F(φ)/(εσ)]1/4

The computation of the insolation F(φ) as a function of latitude is complicated for a planet whose axis of rotation is not perpendicular to the plane of its orbit, but for a planet with a zero angle of inclination the value is simply μcos(φ) where μ is the solar constant. For a constant albedo then

#### T(φ) = [(1−α)μcos(φ)/(εσ)]1/4which is of the form T(φ) = βcos1/4(φ)

which has the shape shown below.

## A Model in Which the Heat Flux is Driven by the Latitudinal Temperature Gradient

In materials heat diffuses through molecular interaction from areas of high temperature to areas of low temperature. In a bar of material of cross-sectional area A the amount of heat-flow S at a position x is proportional to the temperature gradient but in the opposite direction; i.e., it is given by

#### S = −k(∂T/∂x)A

where k is a constant, the coefficient of heat transmission. This type of heat transmission equation could apply to the atmosphere through other processes of diffusion such as eddy diffusion.

To work out the equation that must be satisfied by the temperature gradient and the heat fluxes consider a ring-wall of atmosphere between latitude φ−Δφ/2 and latitude φ+Δφ/2. Let a be the radius of the Earth and for convenience let that radius extend to the midpoint of the atmosphere which is of depth H. The ring-wall then has the following dimensions;

• a perimeter of L = 2πacos(φ)
• a width of W = Δφ
• a height of H

These dimensions then result in:

• a top area of LW = 2πacos(φ)Δφ
• a side area of LH = 2πacos(φ)H
• a volume V = LWH = 2πacos(φ)ΔφH

It is desirable to use a linear transmission coefficient k which would be such that the heat flux through an area A is

#### −k(∂T/∂(aφ)A = −(k/a)(∂T/∂φ)A

The heat flux through the wall side at φ−Δφ/2 is then

#### −k2πacos(φ−Δφ/2)H(∂T/∂(aφ) = −k2πcos(φ−Δφ/2)H(∂T/∂(φ) whereas that at φ+Δφ/2 is −k2πacos(φ+Δφ/2)H(∂T/∂(aφ)

The net gain in heat within the volume of the ring-wall is then

#### −2πkH[cos(φ+Δφ/2)(∂T/∂φ)−cos(φ−Δφ/2)(∂T/∂φ)

The expression within the bracket may be replaced by

#### −(∂/∂φ)[cos(φ)(∂T/∂φ)]Δφ

When the radiation flows are included the net gain in heat to the ring-wall is

#### LW(μcos(φ)−εσT4) −2πkH(∂/∂φ)[cos(φ)(∂T/∂φ)]Δφ

Now consider the volume of the ring-wall. Let cv and ρ be the heat capacity and the mass density of the atmosphere. The volume of the ring-wall is (2πacos(φ))(aΔφ)H. The rate of increase of heat energy in the ring-wall is

#### cvρLWH(∂T/∂t)

This net rate of increase of heat energy in the ring-wall has to equal the net flux of heat energy through the surface of the ring-wall. There is a common factor of 2πΔφ which can be factored leaving as the requirement for the heat balance

#### cvρaHcos(φ)(∂T/∂t) = (a²cos(φ)(μcos(φ)−εσT4) −kH(∂/∂φ)[cos(φ)(∂T/∂φ)]

where ∂T/∂t is the derivative of temperature T with respect to time t.

Dividing through by the coefficient of ∂T/∂t gives

#### ∂T/∂t = (1/(cvρ))[(a/H)(μcos(φ)−εσT4) −k(1/cos(φ))(∂/∂φ)[cos(φ)(∂T/∂φ)]

At equilibrium where ∂T/∂t=0

#### (a/H)(μcos(φ)−εσT4) = k(1/cos(φ))(∂/∂φ)[cos(φ)(∂T/∂φ) or, equivalently (∂/∂φ)[cos(φ)(∂T/∂φ) = (a/Hk)cos(φ)(μcos(φ)−εσT4)

In principle this equation can be solved subject to the boundary conditions that ∂T/∂φ=0 at φ=0 and φ=π/2.

(To be continued.)