Water World

Abstract: A model is described for an Earth like planet “Water World”  with a surface 100%  covered in water able to self-regulate it’s temperature as incident solar radiation increases by a third. The model assumes that convective clouds increase while normalized greenhouse effects decrease as the sun brightens over 4 billion years. Global average temperatures rise by just 5 degreesC. while cloud cover increases by 13% and normalized greenhouse effects fall by 0.15 over the planet’s history.

Imagine a world 100% covered in water with an atmosphere similar to that on Earth but with no other greenhouse gas present except water vapor. The climate is then driven just  by the thermodynamics of water evaporation and solar forcing. Due to solar radiation the atmosphere is never in a state of thermodynamic equilibrium as energy and temperature gradients are always present. For simplicity, the axis of rotation is taken as perpendicular to the orbit plane so there are no seasons. Let’s call this imaginary world “Water World”. In all other respects conditions on Water World are exactly the same as on Earth.  Can such a water covered planet self regulate its temperature as the sun’s output gradually increases? The motivation for proposing such a Water World follows  Daisy World [1] proposed by James Lovelock to justify Gaia theory. When the planet’s sun is 4 billion years younger its output is 33% less than it is today, so  under clear skies with an albedo for water of 0.1,  the  average incident solar energy would be ~ 274 watts/m2. The solar constant slowly increases over the following 4 billion years resulting in a current average  value of 342 watts/m2 equivalent to that on Earth today.

The only greenhouse gas present on Water World is water vapor and  its concentration  is determined by thermodynamic balances in the atmosphere. Evaporation transfers latent heat from the surface to the atmosphere, enhancing H2O greenhouse effect and condensing to form low level clouds which increase the planet’s albedo. Further heating in the daytime can  trigger thunderstorms which transfer heat directly to the top of the troposphere and rain out humidity at the end of the day. The 12 hours of darkness then allow cooling from the surface with a reduced greenhouse effect. The average global temperature is then given by the total Outgoing Long Wave radiation (OLR) at the top of the atmosphere through steffan-bolltzman’s law = e*sigmaTeff**4.

Figure 1: Schematic of Water World

The distribution of solar energy with latitude is shown in figure 2, and the hourly dependence of incident energy for the equator is shown in figure 3. The tropics are defined as the area with latitude +- 30 degrees which receive the highest intensity of solar radiation and drive the humidity and cloud production. Convection currents transfer energy  north and south to higher latitudes driving winds and currents. However, rather than try to model all these complex processes we will just simply propose a  macro-model for the greenhouse effect from water vapor and the albedo for clouds which depend only on solar radiance.

Figure 2: Distribution of average solar energy versus latitdude

Figure 3: Solar forcing in the tropics

The model for our “Water World” assumes 2 basic  dependences. For  low solar forcing epochs evaporation from  the surface mostly increases greenhouse warming  and favors high altitude clouds. In the tropics stronger evaporation during the day causes low convection clouds and eventually thunderstorms. The increase in low level clouds increases the planet’s albedo thus attenuating incident solar energy. In parallel, more saturated tropical air will tend to increase the lapse rate acting to dampen the greenhouse effect. The final global average temperature is some (complex) balance between these effects. To try to keep things as simple as possible, I will just assume that these two  drivers are simple linear functions of incident solar forcing:

It will be just assumed that there is a simple relationships between low clouds and the net greenhouse effect upon incident solar energy. Defining x = S0/342 as the normalized solar flux on  Water World relative to that incident on Earth today.

1. Low Cloud Cover is assumed to be driven by solar heating: CC = 0.4*x. Intense heating (in the tropics) causes more convection clouds, boundary level clouds etc. The albedo for convection clouds is taken as 0.5 so the planet albedo varies as 0.1+0.2*x. This value is chosen so that the planet albedo today is 0.3 (about the same as that on Earth).

2. The net total normalised greenhouse effect g  is assumed to depend inversely on x.  Water evaporation and  high clouds at low x yields  a high g value which decreases as higher forcing drives evaporation leading to a lower lapse rate and more direct latent heat loss to the upper atmosphere. Today g is 0.3 and the proposal is that g depends inversely on increasing x so   g= 0.3/x. Therefore this implies that 4 billion years ago  g was 0.45.

Then the global Energy balance is simply:

(0.9-0.2x)S0 = SU(1.0-0.3/x),  where S0(now) is 342 watts/m2

–>   SU = ((0.9-0.2x)x*342)/(1-0.3/X)

–>   Tsurf(x) = T(now)*4th root(SU(x)/SU(now))

This is very easy to calculate and the results are shown in figure 5.

Fig 5:Water World as compared to fixed g & a

As expected there is a clear dampening effect of the model on increases in surface temperature, as compared to assuming constant values of today’s values of albedo and greenhouse effect. The fact that we know liquid water was present 4 billion years ago effectively rules out constant values. Others have argued that on Earth a greatly enhanced CO2 greenhouse effect is responsible for warming in the early history of the Earth. One problem with this is CO2 forcing increases only logarithmically so to offset reduced solar forcing of 100watts/m2 would appear to require an impossible 10**8 increase in CO2 concentrations. We can easily see what the model implies for warming due to a doubling of CO2 above current levels. Figure 6. shows the details for a CO2 doubling forcing of 5.3Ln(2) = 3.6 watts/m2.

Fig 6: Temperature increase in response to doubling of CO2 (3.6 watts/m2)

The temperature increase is just 0.2 degrees for the model showing the built in negative feedback. Linear feedback parameter would work out at about -14 watt m2/degK ! which is grossly different to the positive feedbacks used by current climate models for Earth, which all assume  positive feedbacks for water vapor and clouds.

Discussion: In this model the average temperature changes by just 5 degrees over 4 billion years. The parameters of the model were  not tuned but were simply  estimated and the calculation was done just once. The objective is to demonstrate that a planet covered in water can self regulate its temperature in response to large increases in radiative forcing. We know that the Earth has maintained its oceans for the last 4 billion years during which time the sun’s output has risen by a third.  It is almost impossible that CO2 alone could have been responsible for this negative feedback, so it is proposed that the main driver was from water. This model for a water covered planet is a gross oversimplification of complex processes of cloud formation, evaporation and advection, but the evidence suggests that they do act to minimise temperature changes to external forcing.

References:

1. Lovelock, J. E. (1983b), Daisy world—A cybernetic proof of the Gaia hypothesis, CoEvol. Q., Summer, 66 – 72.

2.  Willis Eschenbach-  Its not about the feedbacks

3. Hsien-Wang Ou, Possible Bounds on the Earth’s Surface Temperature, Journal of Climate, Vol 14, 2976, 2000.

History:

• Updated 11am 21/8 to make albedo = 0.3 for x=1 as on Earth.
• Updated 12pm 22/8 to show CO2 doubling effect.