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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 Earth’s albedo (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 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 Increased Planetary Temperature Sparser Vegetation, More Desertification Biotic factors feed back to control abiotic factors Increased Planetary Albedo 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. 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 optimum min. max. T Intersection of 2 curves means the 2 effects are balanced => equilibrium points P1 & P2. T daisy coverage daisy coverage T Effects of daisy coverage on T P1 Daisy coverage Effects of T on daisy coverage P2 T Daisy coverage Feedback loops Effects of daisy coverage on T P1 Effects of T on daisy coverage P2 T Perturb daisy coverage at P1 => sys. returns to P1 (stable equil. pt.) Daisy coverage P1 A large perturb. => daisies all die from extreme T P2 T Daisy coverage Large incr. in daisy cover => very low T => decr. in daisy cov. => very high T => lifeless. P1 P2 T From P2, incr. daisy cov. => decr. T => further incr. in daisy cov. => converge to P1 daisy coverage Daisy coverage T P1 P2 unstable equilib. pt. T Gradual incr. in solar luminosity For any particular value of daisy cov., T incr. The effect of T on Daisy unchanged Daisy coverage P1 P1 P2 To Teq Tf P2 T The key variables b: Fraction of planet covered in black daisies w: Fraction covered in white daisies Tb: Temperature where the black daisies are Tw: Temperature where the white daisies are L: Solar luminosity An equation for the black daisies dαb/dt = αb ( 1 – αb – αw)β(Tb) - γαb = αb (αg β(Tb) – γ) b(T) is a function that is zero at 5C, rises to a maximum o one at 22.5C and then falls to zero again at 40C 2 ( T 22 . 5 ) A simple and convenient choice is b (T ) 1 17.52 An equation for the white daisies We use a similar equation for the white daisies: dαw/dt = αw (αg β(Tw) – γ) 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 Tb4 = T4 + q(A-Ab) Tw4=T4 + q(A-Aw) 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. The Daisyworld Equations db/dt = b(gb(Tb) - g) dw/dt=w(gb(Tw) - g) (T+273)4 = SL(1-A) A = gAg + bAb + wAw Tb4 = T4 + q(A-Ab) Tw4 = T4 + q(A-Aw) No daisies SL T 2 1/ 4 273 Black daisies only g 0 b ( b (Tb )(1 ab ) g ) b (Tb ) 1 b ( Tb 22.5) 2 b (Tb ) 1 17.52 b g 1 Tb 22.5 17.5 b 1 Gaia Hypothesis • Proposed by James Lovelock • 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 Tp 4 S L(1 p ) SB p aunun abb abb Daisyworld Model (3) • Area of daisies is modified according to the following equations das as (aun g s deathrate) 0.001 dt 4 2 gs 1 ( 22 . 5 T ) s 2 (40 5) Ts FHA ( p s ) Tp Daisyworld Model (4) • 2D CA rules: • 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 • 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 1.1 1.2 0.9 1.3 1.0 1.4 The Effects of Death Rate on Daisyworld death-rate = 0.3 death-rate = 0.1 death-rate = 0.5 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 1.1 1.2 0.9 1.3 1.0 1.4