Daisyworld

What is a System? Definition: A system is a group of different components that interact with each other Example: The climate system includes the atmosphere, oceans, polar caps, clouds, vegetation…and lots of other things

How do we study systems? Identify the components Determine the nature of the interactions between components

Systems Notation = system component = positive coupling = negative coupling

Positive Coupling Atmospheric CO2 Greenhouse effect An increase in atmospheric CO2 causes a corresponding increase in the greenhouse effect, and thus in Earth’s surface temperature Conversely, a decrease in atmospheric CO2 causes a decrease in the greenhouse effect

Negative Coupling An increase in Earth’s albedo causes a (reflectivity) Earth’s surface temperature An increase in Earth’s albedo causes a corresponding decrease in the Earth’s surface temperature by reflecting more sunlight back to space Or, a decrease in albedo causes an increase in surface temperature

Conditions under which the system will remain indefinitely Equilibrium State: Conditions under which the system will remain indefinitely --If left unperturbed

An Unstable Equilibrium State

An Unstable Equilibrium State Perturbation

When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.

A Stable Equilibrium State

A Stable Equilibrium State Perturbation

When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.

Daisy World

Gaia hypothesis Earth as a single living superorganism (James Lovelock) Gaia - a new look at life on Earth, Oxford University Press, 1979.

Lovelock’s Questions James Lovelock: NASA atmospheric chemist analyzing distant Martian atmosphere. Why has temp of earth’s surface remained in narrow range for last 3.6 billion years when heat of sun has increased by 25%?

Lovelock’s Questions Why has oxygen remained near 21%? Martian atmosphere in chemical equilibrium, whereas Earth’s atmosphere in unnatural low-entropy state.

Our Earth is a Unique Planet in the Solar System Runaway greenhouse :: No water cycle to remove carbon from atmosphere Earth Harbor of Life Loss of carbon :: No lithosphere motion on Mars to release carbon Earth is unique in our solar system in its capacity to sustain highly diversified life from Guy Brasseur (NCAR)

Lovelock´s answers Earth can’t be understood without considering role of life Abiotic factors (physical, geological and chemical) determine biological possibilities Biotic factors feed back to control abiotic factors Increased Planetary Temperature Increased Planetary Albedo Sparser Vegetation, More Desertification Reduced Temperature

Gaia Hypothesis Organisms have a significant influence on their environment Species of organisms that affect environment in a way to optimize their fitness leave more of the same – compare with natural selection. Life and environment evolve as a single system – not only the species evolve, but the environment that favors the dominant species is sustained

Daisy world White daisies Black daisies Available fertile land

About Daisyworld… Daisyworld: a mythical planet with dark soil, white daisies, and a sun shining on it. The dark soil have low albedo – they absorb solar energy, warming the planet. The white daisies have high albedo – they reflect solar energy, cooling the planet. There’s a simple flash animation of daisyworld concept out there too. http://library.thinkquest.org/C003763/flash/gaia1.htm

The number of daisies affects temperature The number of daisies influences temperature of Daisyworld. More white daisies means a cooler planet.

Temperature affects the number of daisies At 25° C many daisies cover the planet. Daisies can’t survive below 5° C or above 40° C.

White Daisy Response to Increasing Solar Luminosity Relative solar luminosity

Daisies can live between a min.T & a max. T daisy coverage T daisy coverage T Daisy coverage min. max. optimum

Effects of daisy coverage on T Intersection of 2 curves means the 2 effects are balanced => equilibrium points P1 & P2. T daisy coverage T daisy coverage P1 Effects of daisy coverage on T P2 T Daisy coverage Effects of T on daisy coverage ENSC 425/625 Chapter 3UNBC

Feedback loops P1 P2 T ENSC 425/625 Chapter 3UNBC Daisy coverage Effects of T on daisy coverage P1 Effects of daisy coverage on T P2 ENSC 425/625 Chapter 3UNBC

Perturb daisy coverage at P1 => sys. returns to P1 (stable equil. pt.) A large perturb. => daisies all die from extreme T

Large incr. in daisy cover => very low T => decr. in daisy cov. => very high T => lifeless. P1 T Daisy coverage P2

From P2, incr. daisy cov. => decr. T => further incr. in daisy cov. => converge to P1 P1 T Daisy coverage P2 T daisy coverage unstable equilib. pt. ENSC 425/625 Chapter 3UNBC

Gradual incr. in solar luminosity For any particular value of daisy cov., T incr. T Daisy coverage P1 P2 The effect of T on Daisy unchanged P1 P2 Teq To Tf ENSC 425/625 Chapter 3UNBC

The key variables Later, we’ll see that we also need T, the “effective temperature”, but that isn’t obvious until we get a bit further on in the modelling.

An equation for the black daisies ( 1 – αb – αw) β(Tb) - γαb dαb/dt = = αb (αg β(Tb) – γ) b(T) is a function that is zero at 5C, rises to a maximum of one at 22.5C and then falls to zero again at 40C A simple and convenient choice is

An equation for the white daisies We use a similar equation for the white daisies: dαw/dt = αw (αg β(Tw) – γ) Another reason for using a different growth function and death rate later on is to check that the result doesn’t depend on using the same for both. But we have only two equations for four unknowns, so we have to think about what else is going on that we haven’t included so far. We don’t have to use the same b(T) and g but it keeps things simple. We can use different ones later if we want to.

Heat Flow Because different regions of Daisyworld are at different temperatures, there will be heat flow. We include this in the model using the equations Note that if q=0 the whole planet is at the same temperature, i.e., the heat flow is very rapid indeed. As q increases, so do the temperature differences. Don’t worry about the 4th powers; they’re only there to make the calculations easier and don’t make any real difference. T is now properly defined. Note that if q=0, then heat flow is so rapid that the whole planet is at the same temperature.

The Daisyworld Equations

No daisies On a dead planet, as the solar luminosity steadily increases, so does the temperature.

Black daisies only This is not a full derivation because we haven’t got L into it yet, but you can already see that the nonlinearity of the equations means the value of the local temperature may not be uniquely determined by L.

Gaia Hypothesis Proposed by James Lovelock Definition of Gaia: Developed in 1960s First published in 1975 Definition of Gaia: a complex entity involving the Earth's biosphere, atmosphere, oceans, and soil; the totality constituting a feedback or cybernetic system which seeks an optimal physical and chemical environment for life on this planet. (Lovelock)

Daisyworld Model Daisyworld is a hypothetical planet orbiting a sun that increases in intensity The planet is inhabited by 2 species Black daisies White daisies Original Daisyworld model consisted of a system of differential equations This project uses these equations to build a 2D cellular automata representation of Daisyworld

Daisyworld Model (2) Temperature of Daisyworld is based on the assumption that the planet is in radiative equilibrium (i.e. energy emitted = energy absorbed) Albedo of the planet is computed based on the albedos of each type of daisy and the area covered by them -SB= 5.669e-8W/degK^4 -S= 917W/m^2

Daisyworld Model (3) Area of daisies is modified according to the following equations

Daisyworld Model (4) 2D CA rules: If da/dt > 0 If da/dt <= 0 If neighbors with no daisies < spreading threshold Bare neighbors grow daisy with probability: p = c*da/dt Else if neighbors with no daisies >= spreading threshold Start new patch of daisies If da/dt <= 0 Daisies die with probability p = -da/dt

Example of Daisy Crowding Spreading-threshold = 6 => Start new patch of daisies => Don’t start new patch

Parameter Settings Two different temperature models Death-rate: 0.3 Automatic linear increase of solar luminosity Manual adjustment of solar luminosity Death-rate: 0.3 Albedo of white daisies: 0.75 Albedo of black daisies: 0.25 Albedo of bare land: 0.50 Spreading threshold: 8 Optimal daisy growth temperature: 22.5 C

Spatial Daisyworld vs. Mathematical Daisyworld Area Occupied by Daisies (Mathematical Model) (Spatial Model)

Spatial Daisyworld vs. Mathematical Daisyworld (2) Temperature of Daisyworld (Mathematical Model) (Spatial Model)

Effects of Solar Luminosity on Daisyworld 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

The Effects of Death Rate on Daisyworld

Daisyworld with Four Species of Daisies Area covered by daisies Temperature of Daisyworld

Effects of Solar Luminosity on Daisyworld with Four Species 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4