Material Flows

Conveyor transit time= ______ Does not allow for skipping grades or being held back. That is, it has no distortion

Unrealistic distortion ????

Diffusion and 1-D random walk The ending position for any walk is random but the expected average ending position of many walks is zero.

This is a 1-D diffusion model driven by a diurnal cycle. We can think of it as simulating either an exchange of gas or energy downward into the soil. If the diurnal cycle is solar energy then the energy content of each soil layer is directly linked to its temperature. The diffusion flow between adjacent layers is assumed to have a 1 hour transport time. (flowAB=(A-B)/1.0 ) Notice: 1) That the amplitude of fluctuations decreases as you go deeper into the soil (from A to L) 2) the peak in temperature lags the peak in solar forcing by mor and more as you go deeper into the soil. 3) the time behavior has not yet reached a steady state (constant average).

When performing a model sensitivity experiment it is important to spin up the model to a steady state and then make a change. In the above case we took 500 hours to spin up the soil energy diffusion model driven by a diurnal cycle. At 500 year the amplitude of the diurnal solar forcing is cut in half resulting in not only smaller oscillations in soil temperature but also cooler mean soil temperatures.

2-D Diffusion and Random walk

Data suggests that the NH concentrations grew from 290 ppt in 1979 to 490 ppt in 1990, and the SH concentrations grew from 260 ppt in 1979 to 460 ppt in 1990.

2-Box CFC model

Atmospheric lifetime of CFC-12 used in the 2007 IPCC report

CFC emissions of 43.8 ppt/yr and inter-hemispheric transport time of 1.45 years gives a good fit to observations. This suggests that it takes about 1.45 years for NH air to exchange with SH air.

4-box diffusion model

You can see that: A initially has 100 and B,C,D have zero. Equilibrium is 25 in each box so mass is conserved. Because of symmetry, B and C behave exactly the same.

If A and C both start at 100 then symmetry allows us to conclude that A and C will follow each other as do B and D. In fact this is identical to 2-Box model since there is never any transport between A and C nor B and D. (see next page)

In this example A and D are set to 100 and C and B are set to zero. Again symmetry arguments suggest that A and D follow each other and C and B follow each other. Since A and D have two paths to lose their material (or energy) the approach to equilibrium is twice as fast compared to either of the last two examples. i.e. the gap closes to 37% of it’s initial value in 5 yrs instead of 10 yrs.

When A=C=D=100 to start and B=0 we get a mirror image of what we had when A=100 and B=C=D=0 initially.

You can see that: A initially has 100 and B,C,D have zero. Equilibrium is 25 in each box so mass is conserved. Because of symmetry, B and C behave exactly the same.

The behavior of D is interesting in that it exhibits S-shaped growth. What’s going on to give this behavior??

Because both C and B start at zero the transport into D is zero but grows steadily as C and B fill up to 25. Eventually the rate at which D accumulates Matter peaks and then decays to back zero.

Another perspective Flow= [TM] [Content] Change in content= [TM] [content] This is just like exponential decay or growth except with a matrix for transport and vector whose components are the content of each box (remember that dX/dt=aX is exponential growth)

Use an online calculator to find the eigenvalues and eigenvectors of the transport matrix.

The Matrix with eigenvectors as columns, its inverse, and the transport matrix eigenvalues can be used to find an exact analytical solution to the 4 box model. D=25(1-2e -.1t +1e -.2t )

The moral of the story: Apparent S-shaped growth may be the result of competing negative exponential functions with different time delays D=25(1-2e -.1t +1e -.2t )

How about a 1-D 5-Box model? A B C D E A B C E What are the initial concentrations of each box? Be able to sketch in the shape of D on this graph. What is the final equilibrium concentration of each region? Explain why B behaves as it does. Explain the apparent s-shaped behavior of E.

How about a 1-D 5-Box model? A B C D E A B C E What are the initial concentrations of each box? {100, 0, 0, 0, 0}

How about a 1-D 5-Box model? A B C D E A B C E Be able to sketch in the shape of D on this graph. D fits between E and C E D C

How about a 1-D 5-Box model? A B C D E A B C E What is the final equilibrium concentration of each region? 20 (100/5) the material is divided evenly between all boxes at equilibrium.

How about a 1-D 5-Box model? A B C D E A B C E Explain why B behaves as it does. At first material rushes into B from A with very little loss to C. At the peak of the B curve it is losing as much to C as it gains from A. After the peak it loses more to C than A gives it and it slowly approaches equilibrium.

How about a 1-D 5-Box model? A B C D E A B C E Explain the apparent s-shaped behavior of E. E is so far removed from the material source that it does not get any material at first. Finally after about 1 year it begins to gain material from D and then increases rapidly to equilibrium for the next two years and then more slowly as it gets closer to equilibrium.

How about a 1-D 5-Box model? A B C D E A B C E A final note: The behavior of this five box model can be described by using different combinations of four exponential decay functions and a constant. (5 eigenvalues) Also it should be stressed that the boxes could represent the spatial components of a system or completely different components of a system that exchanges material with each other.