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About the same time that the Russian metrologist Mikhail I. Budyko was working out his icealbedo model of climate, William D. Sellers of the University of Arizona was developing a similar model. Sellers' model is more complicated than the Budyko model but the basic structure and purpose is the same. In particular, Sellers' model explicitly involves interzonal transfers of energy through air and water flow.
Sellers constructed his model to determine the average sealevel temperature in 10° latitude belts. It is based upon an energy balance for the atmosphere.
The units of Sellers' analysis are latitude belts. The area of the belt is denoted as A_{0} and the length of the north boundary is l_{0} and that of the south boundary is l_{0}. (These are literally north and south rather than poleward and equatorward.) For the north pole cap belt l_{0}=0 and for the south pole cap belt l_{1}=0.
The energy flows for a belt are of four types:
and α_{s} is the albedo.
The outgoing radiation is given by a modified form of the StefanBoltzmann formula; i.e.,
where σ is the StefanBoltzmann constant (1.356x10^{12} (ly/sec)/(K°)^{4}), m is an atmosphere attenuation coefficient (0.5), and γ is an empirical constant (1.9x10^{15}).
The variables c_{0} and c_{1} denote the fluxes of water vapor across the northern and southern perimeters, respectively. The energy flow for for a material flow of c is that material flow times L, the latent heat of condensation per unit mass.
C_{0} and C_{1} denote the fluxes of sensible heat of air flow across the northern and southern perimeters, respectively.
F_{0} and F_{1} denote the fluxes of sensible heat of water flow across the northern and southern perimeters, respectively.
It is presumed that in the northern hemisphere the heat flows are into the belt at the southern boundary and out of the belt at the northern boundary. The total heat transported across a latitude circle per unit perimeter length is given by
The energy balance for a latitude belt is then
where P_{1}=Lc_{1}+C_{1}+F_{1} and P_{0}=Lc_{0}+C_{0}+F_{0}.
Albedo is given as the following function of surface temperature T_{g} in degree Kelvin
where b is an empirical coefficient which is a function of latitude.
The surface temperature is determined by the sealevel temperature of the belt and the altitude Z; i.e.,
The key variable in the model is the temperature difference ΔT between successive latitude levels with special provision for the polar caps. ΔT for a latitude belt in principle should be the difference between the sealevel temperature at its northern boundary and its southern boundary. The temperature in the belt is used for the northern boundary temperature and the latitude belt temperature to the immediate south is used for the southern boundary temperature. This temperature gradient drives the meridional wind. Because of southtonorth coordinate system of the model the meridional wind velocity v is defined as
ΔT will be negative in the northern hemisphere and thus generate a positive southtonorth wind. In the southern hemisphere ΔT is positive and will generate a negative wind velocity. The exact definition of ΔT is not given in the paper but it appears to be a weighted average of the absolute values of ΔT with the weights being the values of the southern perimeters of the latitudes belts. The reason for the asymmetric defition of v is not explained, although it is apparently a matter of the empirical equator being to the north of the actual equator. The coefficient a is a meridional exchange coefficient which varies with latitude.
The model utilizes a variable q defined as the saturation specific humidity at sealevel for a latitude belt. It is computed at each latitude circle from the equation
where e is the saturation pressure and p is atmospheric pressure, both defined at sealevel.
With the variables v and q he computes the next flux of water vapor through a latitude band from the formula
where K_{w} is the eddy diffusivity of water vapor in air, Δy is the width of the latitude belt (1.11x10^{8} cm), Δp is the pressure depth of the troposhere, and g is the gravitational acceleration which Sellers takes to be 1000 cm/sec².
The saturation vapor pressure e and the difference in saturation specific humidity Δq are computed from a version of the ClausiusClapeyron equation; i.e.,
where e_{0} is the sealevel saturation vapor pressure, and R_{d} is the gas constant (6.8579x10^{2} cal/gmK°).
The flux of sensible heat C out of a belt boundary is given by the equation
where K_{h} is the eddy thermal diffusivity of air and c_{p} is the specific heat of air at constant pressure.
The flux of sensible heat by ocean currents is given by
where K_{0} is the eddy thermal diffusivity of ocean currents, Δz is the ocean depth and l' is the oceancovered length of the latitude circle (whose length is l_{1}).
The various diffusivity coefficents are functions of latitude and Sellers gives estimates of their values.
(To be continued.)
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