Earth Systems ScienceChapter 2: SYSTEMS

1. Systems Analysis – some basic concepts / definitions

2. Daisyworld – a “heuristic” model to demonstrate the potential for negative feedbacks on a planet to stabilize the climate

3. Equilibrium vs Dynamical models

Systems Analysis: some basic concepts / definitions

1. System – a set of interrelated parts, or components

2. State of a system – a set of attributes that characterize the system (depth of water in the tub; temperature of earth)

3. Coupling – a link between 2 components

+ coupling –component 1 increases, component 2 increases- coupling - component 1 increases, component 2 decreases

4. Feedback loops – positive and negative

5. Equilibrium States – stable and unstable

6. Perturbations & Forcings

Jimmy & Rosalynn Carter

Each was inadvertently controlling the temperature of the other’s blanket

Positive feedback, unstable equilibrium

Each person controlling their own blanket temperature (negative feedback, stable equilibrium)

STABLE / UNSTABLE EQUILIBRIUM

Stable Equilibrium:If the system is perturbed by a small amount, it will return to the same equilibrium state

Unstable Equilibrium:If the system is perturbed by a small amount, it will NOT return to the same equilibrium state

Example: a thermostat

Perturbation: sudden / temporary disturbance to a systemThe disturbance is temporary, but the system might take a while to recover

a volcanic eruption injects SO2 into the atmosphere, which is washed out of the atmosphere in a few years

Impact of asteroid injects massive amount of particulates into the atmosphere

Forcinga persistent disturbance to a system

e.g. a gradual change in solar radiation over long time,the faint young sun paradox

Heuristic (dictionary.reference.com) :1. Of or relating to a usually

speculative formulation serving as a guide in the investigation or solution of a problem: “The historian discovers the past by the judicious use of such a heuristic device as the ‘ideal type’” (Karl J. Weintraub).

2. Of or constituting an educational method in which learning takes place through discoveries that result from investigations made by the student.

DAISYWORLD: A HEURISTIC MODEL

Response of average surface temperature to daisy coverage

Graph

Systems Diagram

(c) Equation:

Temp = a*daisy + b

Notice the axis

Systems Diagram explicitly including albedo

Response of daisy coverage to average surface temperature

(c) Equation:

Daisy = 100 – (T-22)2/4

Equilibrium States: Graphical Determination

1. Overlay the two graphs (this is the graphical way of setting them equal to each other).

2. The points where they meet are equilibrium points.

3. Draw a systems diagram to determine whether each one is stable or unstable.

Equilibrium States: Algebraic Determination

1. Response of Temp to daisies: Temp = a*daisy + bdaisy = (Temp – b)/a

2. Response of daisies to Temp: daisy = 100 – (Temp-22)2/4

3. Set the equal to each other:(Temp – b)/a = 100 – (Temp-22)2/4

4. Do some algebraic manipulation, you get a quadratic equationT2 – (44-4/a)T + (84-4b/a) = 0

5. solution to a quadratic (aT2 + bT + c = 0) is:T = [-b ± sqrt(b2 – 4ac) ] / 2a

6. This will give 2 solutions, corresponding to P1 and P2

External Forcing: the response of Daisy World

1. Assume that the external forcing is an increase in solar luminosity

2. The effect of temperature on daisy coverage should not change (this depends on the physiology of daisies)

3. The effect of daisy coverage on temperature should change: for the same daisy coverage, higher temperature

Notice the axis

Algebraic: Temp = a*daisy + b+T0

Response of the Equilibrium State to the Forcing

1. Use the new line for the effect of daisy coverage on temperature

2. Notice that the new equilibrium points have changed: P1, the stable point, is at a higher temperature

3. Notice that P1 is not as high a temperature as it would have been without the daisies responding

4. Feedback factor: Teq = To - Tf

Climate history of Daisyworld: solar luminosity increasing

1. As solar luminosity increases, with no feedback (or no daisies) the average temperature will increase close to linearly

2. With the daisy feedback, the temperatures on Daisyworld are kept much more stable compared to the case without feedback

3. No “intention” is required on the part of the daisies to stabilize the climate. All this is required is a negative feedback

EQUILIBRIUM MODELS

The model of Daisyworld in the text is an equilibrium model, just like the models of the bathtub and the earth’s radiation balance from lab 1. These models do not allow one to determine if it ever actually reaches equilibrium, or how long it takes to get there.

In these equilibrium models, the STATE of one variable (daisy coverage, water level in the tub, or earth’s temperature) is a function of the state of a second variable (planetary temperature, rate of water coming out of faucet, solar luminosity).

DYNAMICAL MODELS

In dynamical models, the CHANGE of one variable (daisy growth rate, water level rate of increase, rate of earth’s temperature change) is a function of the state of the second variable.