The Use of Simple Models to Investigate Climate 
Michael A. Kelly* and David A. Randall*
Department of Atmospheric Science, Colorado State University, Fort Collins,
Colorado 80523, USA
Our goal has been to develop a simple model which can represent the
tropical atmosphere of Earth. We use the model to develop ideas
about the interactions of various physical processes, which can then be tested
against observations and the results of more realistic models. Simple
"toy" models are attractive, because their results are easy to
understand. If the researcher determines that the
assumptions used to derive the model are wrong, it is usually a simple
task to revise the model.
Background
Our model is designed to simulate a zonal circulation along the Equator, which is called the Walker circulation. The Walker
circulation has rising motion in the western Pacific
and sinking motion in the eastern Pacific. Bjerknes (1969) named this circulation in honor of Sir Gilbert Walker.
Bjerknes posulated that when the cold tongue of sea surface temperatures on the equator is well developed, the cool, dry air just
above the ocean cannot ascend to join the Hadley circulation (a tropical, meridional circulation). Instead, the air is heated and
moistened as it moves westward, until convects over the Warm Pool (WP). If there were no mass exchange with adjacent latitudes, a simple
circulation would develop in which the flow is easterly at low altitudes and westerly at upper levels. For a more detailed
description, see Kelly (1998).
Bjerknes, J., 1969: Atmospheric teleconnections from the equatorial Pacific. Mon. Wea. Rev. 97, 163-172.
Kelly, M.A., 1998: A Simple Model of Ocean-Atmosphere Interactions in the Tropical Climate System. Ph.D. Dissertation, Colorado State
University, 205pp.
Model Features
- Separate boxes for the rising and descending branches of the Walker circulation and for the ocean.
- A simplified but physically based radiative-transfer parameterization
- Momentum budgets for the atmosphere and ocean
- Radiatively active clouds in the Warm Pool and Cold Pool.
- Cloud fraction/optical depths for both regions are calculated.
Figure 1: Schematic of the two-box atmosphere model. Heavy arrows denote the direction of the winds; thin arrows depict the vertical
boundaries of the Cold-Pool free troposhere. The trade-wind inversion (TWI) separates the boundary layer from the free troposphere.
The quantities L1 and L2 represent the widths of the WP and CP, respectively.
Stand-alone results from the WP model
Model simulations which include cloud radiative effects and lateral heat/moisture transports
give realistic solutions. A runaway greenhouse (Ingersoll, 1969) results for simulations in which cloud radiative effects and lateral
heat/moisture transports are ignored. If cloud radiative effects are simulated but lateral energy/moisture transports are neglected,
the model reaches a very warm, very dry equilibrium. These results indicate the requirement for lateral energy/moisture transports
and radiatively active clouds in order for the solution to resemble the present-day climate. Figure 2 illustrates the sensitivity of
the model to specified cloud fraction and the "autoconversion" cloud timescale. As the timescale increases, so does the thickness of
the cloud.
Figure 2: Tropical mean sea surface temperature from our simple, semi-analytical model of the tropical Warm Pool as a function of specified cloud fraction and specified
"autoconversion" timescale. See Kelly, Randall, and Stephens (1999) for details.
We have also discovered that the tropopause height and temperature depend on
cloud thickness and cloud fraction. As the cloud amount increases, the cloud emits at
much colder temperatures than the surface, which causes the upper troposphere to cool. It is
well known that the tropical temperature profile convectively adjusts toward a moist-adiabatic lapse
rate. As temperature in the upper troposphere decreases, the height of the tropopause therefore
increases as the atmosphere adjusts toward a moist adiabat.
Figure 3: A contour plot of zT as a function of sea surface temperature, TS, and
precipitable water, W with a) cloud fraction = 0, b) cloud fraction = 0.4 with an autoconversion
cloud timescale of 1000 s. Values are not plotted for TS-W combinations in which the relative
humidity exceeds 100% or the lapse rate becomes superadiabatic at the tropopause. See Kelly, Randall, and Stephens (1999) for details.
Ingersoll, A.P., 1969: The runaway greenhous: A history of water on Venus. J. Atmos. Sci.
26, 1191-1198.
Kelly, M.A., D.A. Randall and G.L. Stephens, 1999: A simple radiative-convective model with a hydrological cycle and interactive clouds. Quart. J. Roy. Met. Soc.
125, 837-869.
Current work
Our current research has focused on the connection between the amount and vertical distribution of water vapor in the Cold Pool and the
intensity of the Walker circulation. As the Cold Pool moistens, the radiative cooling rate in the free troposphere increases. To a
good approximation, radiative cooling in the subsiding branches of the tropical circulation is balanced by subsidence warming. If the
width of the Cold Pool remains constant, an increase of the radiative cooling rate implies a stronger Walker circulation. In order to
preserve energy/moisture balance, the atmosphere must adjust so as to reduce the east-west precipitable water and/or sea surface temperature
difference. We have found that the model solution strongly depends on the altitude at which WP convection detrains. As this altitude
increases, the WP transports increasingly drier air to the CP. This drying changes the radiative cooling rate in the CP, and so the
atmosphere must adjust.
Kelly, M.A., and D.A. Randall, 1999: The Effects of the Vertical Distribution of Water Vapor on the Strength of the Walker
Circulation. Submitted to J. Climate
Michael A. Kelly
Dept of Atmospheric Science
Colorado State University
(970)491-5237
mike@atmos.colostate.edu