More on Exponential Growth

Understanding Exponential Growth

Often times exponential growth is plotted as a straight line on a semi-log plot. The Y-axis is logarithmic and the X-axis is linear. Here is an example:

In exponential growth, the rate of growth may well change, but the growth is still exponential!

What kinds of things grow exponentially?

Clearly exponential rates of growth are an integral part of the planning process. Different aspects of a growing population have different exponential growth rates and these need to be considered.

For instance, suppose your urban area is growing at the rate of 5% a year. How does this translate into the following:

Exponential Growth and Trend Extrapolation

Whenever schools get crowded, freeways get jammed, airline hubs get crowded, oil gets used up, there are no more available phone numbers, the federal debt goes beyond recovery, etc, etc is an indication of poor planning and trend extrapolation.

In a nutshell: there is no reason that we should ever be surprised at the rate of resource utilization. If we simply pay attention to past history, in general, its a fairly good guide for future resource use.

The difference between linear growth (constant number of units growth per year) vs exponential growth (constant percentage increase) is difficult to see initially, if the exponential growth rate is small.

We can use the Statistical Graphical Tool to help understand this.

Here is some data:

First we try a linear fit which will clearly be seen as bogus.

Then we will try an exponential fit with 10% growth rate to see its much better.

So in this case its obvious that the growth rate is exponential but in some other case such as the population data for California its not so obvious!

Global Warming: Is it Real?

What's the evidence for global warming?

What do the latest models show? Here we see the predicted temperature increases on the land surface when the data is analyzed in cumulative time slices.

Here is the equivalent data for the oceans

Later we will compare the models with some data.

Current concentration of CO2 is 360 ppm

Current concentration of CH4 is 1.8 ppm or 25x1.8 = 45 ppm CO2 equivalent. This is 12% of CO2 concentration and its growing 2.0 -- 2.5 times as fast.

The methane content of our atmosphere has more than doubled since pre-industrial times and in the last few years has shown a small decrease (no one knows why). If the sources of methane deposition stay constant then we will not have a problem. This seems unlikely however as anthropogenic soruces of methane production (e.g. rice,cows,sewage) continue to rise. Eventually (100-200 years) these sources will dominate over natural sources of methane emission.

More About Methane:

Methane Removal Mechanism (some estimates suggest the methane residence time is up to 10 years so there is no short term equilibrium control):

Atmospheric Chemistry:

CH4 + 2O2 CO2 + 2H20

  • OH + CH4 + 02 H2O + CH3O2
  • CH3O2 + NO +02 H2CO + HO2 + NO2
  • CH3O2 + HO2 CH3OOH + O2
  • H2CO + uv H2 + CO
  • CH3OOH +OH H2O + CH3O2a

    Is this apparent reduction in the rate of methane build up due to a threshold effect that is does methane build up until the atmospheric chemistry is temporarily accelerated? Was Pinatubo to blame?

    An expert's view

    Ice Core Data:

    Manifestations and Predictions of the Greenhouse Effect:

    An example of Myth Perpetuation: Deforestation of the Amazon Rain Forest greatly contributes to the Greenhouse effect:

  • Its the total amount of biomass on the planet that contributes to the Carbon Cycle . Because of agriculture, this biomass is actually increasing with time (which contributes to the methane problem).

  • Trees are temporary storage only; they fall down and rot and release Carbon eventually

  • Biggest problem is that the Amazon Rain Forest is a net producer of greenhouse gases rather than a net sink. That is, this environment produces a great deal of methane which is emitted by the forest and that is larger than the amount of CO2 which is absorbed. This point is a bit controversial still, however.

  • The Amazon Rain Forest should be preserved as a species habitat; arguments to preserve it to "reverse" the Greenhouse effect are incorrect and demonstrate that people don't know about methane production and its effects.

    Simplest Prediction is increase in global mean temperature. Can We see evidence for this in the Data?

    Role of clouds is crucial --> Net Cooling or Warming? --> There is general consensus that cloud cover will increase as the capacity for the atmosphere to retain water increases with increasing temperature.

    What does the actual data look like?

  • All the Data: Global Warming NCDC Document

    Long Term Temperature Trends:

    Potential Problems:

    Distribution of Temperature trends on the Earth. Where is it getting warmest the fastest:

  • All the historical data

  • 1953-1973:

  • 1973-1993:

    When positional data is combined with temperature data and timesliced in this way - it seems clear the Global Warming is occurring. This conclusion was reached by a National Academcy of Science panel in Sept. 1995 and is accepted by most.

  • Temperature Variations with Altitude

    What about Stratospheric Cooling?

    The temperature of the stratosphere - an upper region of the atmosphere that extends from about ten to fifty kilometers (six to thirty miles) above the surface of the Earth - will be significantly COOLED.

    The cooling comes about through the combined effect of increases in carbon dioxide and the observed depletion in stratospheric ozone, and the manner in which the two gases absorb and re-emit energy. Opposite in sign to what is expected near the ground, the change has been predicted by models and has now been OBSERVED. As such, it provides potential early evidence of greenhouse warming.

    Some Additional Internet Resources:

  • Public Perception on Global Warming
  • Nice Overview

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