**Dynamcis of the Earth System**

Prof. Bill Menke, Instructor

Required Text: Harte, John, Consider a Spherical Cow, A Course in Environmental Problem Solving, Univeristy Science Books, 1988, ISBN 0-935702-58-X. (available at Labrynth Books).

Syllabus: Follows text, with some additions by the instructor

- Weeks 1-3: Steady State Box Models and Residence Times
- Week 1: Organizational remarks
- Week 2: Concept of reservoirs
- Model of sea level rise due to melting of Antarctic glaciers
- Model of sea level rise due to thermal expansion of oceans
- Discussion of cobbler problem
- Model of flooding of Black sea

- Week 3: Fluxes, residence times and conservation in reservoirs
- Residence time of water in atmosphere
- Model of changing atmosphere due to burning the biosphere
- Model of changing atmosphere due to burning of fossil fuel

- Week 4: Feedback and reservoir histories
- Model of radon gas in a house basement
- Model of atmospheric CO
_{2}due to constant burning of fossil fuels - Model of atmospheric CO
_{2}due to accelerating rate of burning of fossil fuels - Model of increase in radon gas in a house basement

- Week 5: Fluxes per unit area; Stocks per unit volumes; time dependence
- Uses of fluses and stocks described in terms of flux/area and stock.volume.
- Examples from hydrology
- Time dependence: - simple cases with and without feedback

- Week 6: Time dependent reservoir models
- a
- Radon in basement problem
- Discussion of how to deal with simutaneous equations

- Week 7: The Stella modeling tool.
- Weeks 8-10: Chemical Reactions and Equilibria
- Weeks 11-13: Non-steady state box models

Homework

- Due 1/25: Read "Warm-up Exercises" chapter of the textbook (pages 1-20) and be prepared to discuss them in class.
- Due 2/1: Exercises following Warm Up Exercises 1 (Cobblers) and 7 (Sulfur) in text, done in style of class handouts.
- Due 2/7: Read first half of "Steady-state box model and residence times" chapter of the textbook (p.23-44); Exercises following "3: Carbon in the biosphere" and "4: Natural SO
_{2}". - Due 2/15: Exercises following "7: The flow of atmospheric pollutants between hemispheres" and "9: Where would all the water go?"
- Due 2/22: Handout with A) Flux problem; B) Exponential function exercises.
- Due 3/01: Handout with Stella model of population and follsil fuel.
- Due 3/08: Handout with Stella model of cooling of earth.
- Due 3/15: Handout with Stella model of Trade Winds (stella model).

Some Definitions and Concepts

**Substance**: Something we're interested in.- For example, water is a substance.
- What units do we use to measure the quantity of a substance? Recall in the case of water in the ocean, we discussed whether its better to use volume or mass.
- Mass might be preferable to volume, because mass in conserved, while volume is not.

**Reservoir**: Place that holds or stores a substance- We'll use the term 'stock' (abbreviated S) to refer to the amount of substance in a reservoir.
- The problem of estimating the stock of a reservoir (i.e. the quanity of substance that it holds) is non-trivial, and often important.
- Using units of volume, the ocean has a stock (=volume) of

Volume = Area x Height, and is measured in km^{3} - Unsing units of mass, the ocean has a stock (= mass) of

Mass = Density x Area x Height, and is measured in kg

**Transport**: moving a substance between reservoirs (abbreviated T)**Flux**: rate of trasporting a substance (abbreviated F)- If the substance has units of X, then flux has units of X per unit time. For example, if the substance is mass of water with units of kilograms, then the flux has units of kilograms per second.
- Transport = Flux x Time
- T = F x time
- If a quantity is conserved, then then:
- (Flux in - Flux out) x Time = Increase in Reservoir Stock; or
- Rate of Growth in Reservoir Stock = (Flux in - Flux out)
- dS/dt = F
_{in}- F_{out}

**Flux per unit area**: rate of trasporting a substance across a surface (abbreviated f)- If flux has units of X per unit time, then flux per unit area has
units of X per unit time per unit area. For example, if the substance
is mass of water with units of kilograms, then the flux per unit area
has units of kilograms per second per meter squared: kg/m
^{2}s - Flux = (flux per unit area) x Area
- F = f x A
- Note that when the stock is volume of a material, then the flux per unit area is "velocity".

- If flux has units of X per unit time, then flux per unit area has
units of X per unit time per unit area. For example, if the substance
is mass of water with units of kilograms, then the flux per unit area
has units of kilograms per second per meter squared: kg/m
**Stock per unit volume**: amount of a stock in a unit volume (abbreviated s)- If the stock of substance has units of X , then s has units of
X per distance cubed. For example, if the substance
is mass of water with units of kilograms, then s has units of
kilograms per meter cubed: kg/m
^{3}. - Stock = (Stock per unit volume) x Volume
- S = s V
- Note that when the substance is mass, then the substance per unit volume is "density".

- If the stock of substance has units of X , then s has units of
X per distance cubed. For example, if the substance
is mass of water with units of kilograms, then s has units of
kilograms per meter cubed: kg/m
**Time-dependent reservoir models**: dS/St = F_{in}- F_{out}= F_{net}- No Feedbcak: F
_{net}(t) a proscribed function of time, t, and is not a function of stock, S. - Positive Feedback: F
_{net}increases with stock, S. - Negative Feedback: F
_{net}decreases with stock, S.

- No Feedbcak: F
**Solution**of time-dependent problems- Mathematically, we must sole the ordinary differential equation,
dS(t)/dt = F
_{net}(t), and its associated initial condition, S(t=0) = S_{0}.- Examples:
- dS/dt=C, S(t=0)=S
_{0}: S(t)=S_{0}+Ct - dS/dt=C+at, S(t=0)=S
_{0}: S(t)=S_{0}+Ct+at^{2}/2 - dS/dt=aS, S(t=0)=S
_{0}: S(t)=S_{0}exp(at) - dS/dt=-aS, S(t=0)=S
_{0}: S(t)=S_{0}exp(-at) - dS/dt=C-aS, S(t=0)=S
_{0}: S(t)=(C/a>(1-exp(-at))

- dS/dt=C, S(t=0)=S
- Once solution is found, behavior at long time can be analyzed.

- Mathematically, we must sole the ordinary differential equation,
dS(t)/dt = F
- What
**STELLA**can and cannot do- Can: facilitate building a model through its graphical interface
- Can: extrapolate time-dependent reservioir models into the future
- Can: allow coefficients, dependencies to be easily modified
- Cannot: analyze behavior
- Cannot: solve for parameters in terms of other parameters
- Cannot: analyze stead-state problems