﻿ The Sellers' Energy-Balance Global Climate Model

San José State University

San José State University

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 The Sellers' Energy-Balance Global Climate Model

About the same time that the Russian meteorologist Mikhail I. Budyko was working out his ice-albedo 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 sea-level 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 A0 and the length of the north boundary is l0 and that of the south boundary is l0. (These are literally north and south rather than poleward and equatorward.) For the North Pole cap belt l0=0 and for the South Pole cap belt l1=0.

The energy flows for a belt are of four types:

• Radiation energy flows, Qs incoming, and Is outgoing. Q and I are per unit area. The net radiation is denoted as Rs where

#### Rs = Qs(1-αs) − Is

and αs is the albedo.

The outgoing radiation is given by a modified form of the Stefan-Boltzmann formula; i.e.,

#### Is = σT04[1 - m tanh(γT06)]

where σ is the Stefan-Boltzmann 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).

• Water vapor flow.

The variables c0 and c1 denote the fluxes of water vapor across the northern and southern perimeters, respectively. The energy flow for a material flow of c is that material flow times L, the latent heat of condensation per unit mass.

• Air flow.

C0 and C1 denote the fluxes of sensible heat of air flow across the northern and southern perimeters, respectively.

• Ocean current flow.

F0 and F1 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

#### P = (Lc + C + F)

The energy balance for a latitude belt is then

#### RsA0 + P1l1 − P0l0 = 0

where P1=Lc1+C1+F1 and P0=Lc0+C0+F0.

Albedo is given as the following function of surface temperature Tg in degree Kelvin

• αs = b − 0.009Tg if Tg < 283.16 K
= b − 2.548     if Tg ≥ 283.16 K

where b is an empirical coefficient which is a function of latitude.

The surface temperature is determined by the sea-level temperature of the belt and the altitude Z; i.e.,

## The Determination of the Material Fluxes

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 sea-level 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 south-to-north coordinate system of the model the meridional wind velocity v is defined as

#### v = −a(ΔT+|ΔT|) for latitudes north of 5N and v = −a(ΔT−|ΔT|) for latitudes south of 5N

ΔT will be negative in the northern hemisphere and thus generate a positive south-to-north 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 definition 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 sea-level for a latitude belt. It is computed at each latitude circle from the equation

#### q = 0.622e/p

where e is the saturation pressure and p is atmospheric pressure, both defined at sea-level.

With the variables v and q he computes the next flux of water vapor through a latitude band from the formula

#### c = [vq − Kw(Δq/Δy)](Δp/g)

where Kw is the eddy diffusivity of water vapor in air, Δy is the width of the latitude belt (1.11x108 cm), Δp is the pressure depth of the troposphere, 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 Clausius-Clapeyron equation; i.e.,

#### e = e0[1 − 0.5(0.622)LΔT/(RdT0²) Δq = ((0.622)²LeΔT)/(pRdT0²)

where e0 is the sea-level saturation vapor pressure, and Rd is the gas constant (6.8579x10-2 cal/gm-K°).

The flux of sensible heat C out of a belt boundary is given by the equation

#### C = [vT0 − Kh(ΔT/Δy)](cpΔp/g)

where Kh is the eddy thermal diffusivity of air and cp is the specific heat of air at constant pressure.

The flux of sensible heat by ocean currents is given by

#### F = −K0Δz(l'/l1)(ΔT/Δy)

where K0 is the eddy thermal diffusivity of ocean currents, Δz is the ocean depth and l' is the ocean-covered length of the latitude circle (whose length is l1).

The various diffusivity coefficients are functions of latitude and Sellers gives estimates of their values.

(To be continued.)

Source:

• William D. Sellers, "A Global Climatic Model Based on the Energy Balance of the Earth-Atmosphere System," Journal of Applied Meteorology, vol. 8 (June 1969), pp. 392-400.