Added iterative runaway script
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douwe
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#My first tijme making a plot for climate modeling.
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#Hopefully, this works.
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#Note that using spreadsheets for this course is optional;
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#the only way to get credit is with Python3!
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"""The background for this material can be found in Sections 2 and 3 of Part I
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of this class. Joules are units of energy; Watts are energy flow (J/s).
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The temperature of a planet is determined by balancing energy fluxes in and out of a planet.
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Incoming solar heat is determined by L*(1-albedo)/4,
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and outgoing infrared is calculated as epsilon * sigma * T^4."""
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"""The heat capacity of the surface planet depends on the water thickness.
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Using the fact that 1 gram of water warms by 1 K with a 4.2J (1 cal) of heat,
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and the density of water, to calculate how many Joules of energy it takes
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to raise the temperature of 1 m2 of the surface, given some water depth
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in meters."""
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"""The goal is to numerically simulate how the planetary temperature of a
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naked planet would change through time as it approaches equilibrium
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(the state at which it stops changing, which we calculated before).
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The planet starts with some initial temperature.
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The “heat capacity” (units of Joules / m2 K) of the planet is set by
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a layer of water which absorbs heat and changes its temperature.
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If the layer is very thick, it takes a lot more heat (Joules) to change the temperature.
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The differential equation you are going to solve is:"""
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#d(HeatContent)/dt = L*(1-alpha)/4 - epsilon * sigma * T^4
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"where the heat content is related to the temperature by the heat capacity:"
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#T[K] = HeatContent [J/m2] / HeatCapacity [J/m2 K]
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"""The numerical method is to take time steps, extrapolating the heat content
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from one step to the next using the incoming and outgoing heat fluxes,
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same as you would balance a bank account by adding all the income and subtracting all the expenditures
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over some time interval like a month.
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The heat content "jumps" from the value at the beginning of the time step,
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to the value at the end, by following the equation:"""
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#HeatContent(t+1) = HeatContent(t) + dHeatContent/dT * TimeStep
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"""This scheme only works if the time step is short enough so that
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nothing too huge happens through the course of the time step.
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Set the model up with the following constants:"""
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timeStep = 40 # years; convert to seconds.
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time = 0 # years; this shall denote how much time has passed in total.
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T = 0 # Kelvin, according to question demands.
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waterDepth = 2000 # meters; multiply by cH2O and water density.
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L = 1350 # W m-2, same as J s-1 m-2 (Solar Constant) -> multiply by no. of seconds in each timestep
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albedo = 0.3 # alpha
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epsilon = 1
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sigma = 5.67E-8 # W m-2 K-4
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cH2O = 4200 # specific heat capacity of water (units of 4200 J kg-1 K-1).
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rhoH2O = 1000 # Density is 1000 kg m-3.
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energy_per_m2 = waterDepth * cH2O * rhoH2O #in J m-2 K-1
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#heat influx and outflux must be calculated along the way.
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import numpy
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import matplotlib.pyplot
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#Getting our lists needed with the very first variables already in:
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nSteps = input("Please use 20 or more")
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#apparently the autograder that the MOOC uses also mistakes strings for operators.
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nSteps = int(nSteps)
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time_list = [time]
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temperature_list = [T]
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HeatContent = 0 #only true when T = 0 Kelvin, this is the heat content of the Earth.
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difference = 0
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outflux = 0
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for x in range(nSteps):
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time = time + timeStep
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time_list.append(time)
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difference = (L * (1 - albedo) / 4 - (epsilon * sigma * T**4)) #W m-2
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HeatContent = HeatContent + difference * 365 * 24 * 60 * 60 * timeStep #J m-2
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T = (HeatContent / energy_per_m2) #Kelvin
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outflux = epsilon * sigma * T**4 # as per instructions
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temperature_list.append(T)
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#derive answers and end with:
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matplotlib.pyplot.plot(time_list, temperature_list)
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matplotlib.pyplot.show()
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#Give the last temperature derived by the model:
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#print(temperature_list[-1], outflux)
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#The tolerances in the grader for temperatures are 2-3 degrees C,
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#and for the heat fluxes, +- 0.1 W/m2.
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@@ -1,32 +0,0 @@
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import matplotlib.pyplot as plt
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timeStep = 10 # years
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waterDepth = 4000 # meters
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L = 1350 # Watts/m2
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albedo = 0.3
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epsilon = 1
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sigma = 5.67E-8 # W/m2 K4
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#nSteps = int(input(""))
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nSteps=200
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heat_cap=waterDepth*4.2E6 #J/Kg/deg
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time_list=[0]
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temperature_list=[400]
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heat_content=heat_cap*temperature_list[0]
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heat_in=L*(1-albedo)/4
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heat_out=0
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secs_in_yr=3.14E7
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for t in range(0,nSteps):
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time_list.append(timeStep+time_list[-1])
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heat_out=epsilon*sigma*pow(temperature_list[-1],4)
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heat_content=heat_content+(heat_in-heat_out)*timeStep*secs_in_yr
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temperature_list.append(heat_content/heat_cap)
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heat_out = epsilon * sigma * pow(temperature_list[-1], 4)
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print(temperature_list[-1])
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#print(temperature_list[-1], heat_out)
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plt.plot(time_list, temperature_list, marker='.', color="k")
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plt.xlabel("Time (years)")
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plt.ylabel("Temperature (degrees Kelvin)")
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plt.show()
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@@ -1,68 +0,0 @@
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# %%
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# %%
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import numpy as np
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import matplotlib.pyplot as plt
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# Set up constants as given:
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timeStep_years = 100 # years
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waterDepth = 4000 # meters
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L = 1350 # W/m²
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albedo = 0.3
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epsilon = 1
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sigma = 5.67e-8 # W/m² K^4
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# The incoming solar flux is constant:
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incoming_flux = L * (1 - albedo) / 4 # W/m²
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# Calculate the heat capacity (J/m² K)
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# (water density = 1000 kg/m³, specific heat = 4184 J/(kg·K))
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heat_capacity = waterDepth * 1000 * 4184
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# Read the number of time steps from stdin
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nSteps = int(input(""))
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# Set the initial conditions.
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# (Initial temperature is chosen in Kelvins; 0 K is allowed though not physical.)
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initial_temperature = 0.0 # Kelvin
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heatContent = initial_temperature * heat_capacity # in J/m²
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# Prepare lists for time and temperature (for plotting, if desired)
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time_list = [0]
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temperature_list = [initial_temperature]
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# Convert the time step to seconds.
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dt = timeStep_years * 365 * 24 * 3600
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# Integration loop: update heat content and temperature each step.
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for i in range(nSteps):
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# Current temperature in Kelvin:
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T = heatContent / heat_capacity
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# Differential equation: dHeat/dt = incoming_flux - ε·σ·(T)⁴
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dH_dt = incoming_flux - epsilon * sigma * (T**4)
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# Update the heat content over the time step:
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heatContent = heatContent + dH_dt * dt
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# Append new time and temperature to lists:
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time_list.append((i+1) * timeStep_years)
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temperature_list.append(heatContent / heat_capacity)
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# Compute the final temperature (in Kelvin) and the corresponding outgoing heat flux.
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final_temperature = heatContent / heat_capacity
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final_heat_flux = epsilon * sigma * (final_temperature**4)
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# Output only the final temperature and heat flux.
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print(final_temperature, final_heat_flux)
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# Plotting code
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plt.plot(time_list, temperature_list)
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plt.show()
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# %%
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80
iterative_runaway_ice_albedo_feedback.py
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80
iterative_runaway_ice_albedo_feedback.py
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import numpy as np
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import matplotlib.pyplot as plt
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### Properties ###
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steps = 100
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# Data
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T_data = np.array([265, 255, 245, 235, 225, 215])
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il_data = np.array([75, 60, 45, 30, 15, 0])
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albedo_data = np.array([0.15, 0.25, 0.35, 0.45, 0.55, 0.65])
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# Ice latitude
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m1, b1 = np.polyfit(T_data, il_data, 1)
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def calc_ice_latitude(T):
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return m1 * T + b1
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# Planetary albedo
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m2, b2 = np.polyfit(T_data, albedo_data, 1)
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def calc_albedo(T):
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return np.clip(m2 * T + b2, 0, 1)
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### Constants and Data ###
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epsilon = 1
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sigma = 5.67E-8 # W/m2 K4
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### Energy balance equation ###
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def calc_temperature(albedo, L):
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return pow((L * (1 - albedo)) / (4 * epsilon * sigma), 1/4)
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### Iteration step ###
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def iterate(L, initial_albedo, steps):
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T = calc_temperature(initial_albedo, L)
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for i in range(steps):
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albedo = calc_albedo(T)
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new_T = calc_temperature(albedo, L)
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T = new_T
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return T, albedo
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### Loop ###
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l_range_bounds = [ 1200, 1600 ]
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current_albedo = 0.15
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temp_down = []
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l_range_down = []
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L = l_range_bounds[1]
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while L > l_range_bounds[0]-1:
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T, current_albedo = iterate(L, current_albedo, steps)
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temp_down.append(T)
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L = L - 10
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l_range_down.append(L)
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current_albedo = 0.65
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temp_up = []
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l_range_up = []
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L = l_range_bounds[0]
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while L < l_range_bounds[1]+1:
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T, current_albedo = iterate(L, current_albedo, steps)
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temp_up.append(T)
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L = L + 10
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l_range_up.append(L)
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### Input ###
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# L, albedo, nIters = input("").split()
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# L, albedo, nIters = [ float(L), float(albedo), int(nIters) ]
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# T, albedo = iterate(L, albedo, nIters)
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# print(T, " ", albedo)
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### Plot hysteresis ###
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plt.plot(l_range_down, temp_down, label="Cooling")
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plt.plot(l_range_up, temp_up, label="Warming")
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plt.xlabel("Solar Constant (L)")
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plt.ylabel("Equilibrium Temperature (K)")
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plt.legend()
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plt.show()
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@@ -3,10 +3,10 @@ import matplotlib.pyplot
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# Inputs
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timeDuration = 2000 # years
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timeStep = 10 # years
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waterDepth = 4000 # meters
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initialTemp = 400
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timeDuration = 2000 # years
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timeStep = 10 # years
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waterDepth = 4000 # meters
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initialTemp = 400 # K
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# Constants
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secondToYear = 60*60*24*365
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@@ -41,7 +41,9 @@ for i in range(nSteps):
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# Save timestep for plotting
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time.append(time[-1] + timeStep)
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print(T[-1])
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print("final temperature: ", T[-1])
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print("Initial heat out: ", heatOut[0])
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print("Final heat out: ", heatOut[-1])
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matplotlib.pyplot.plot(time, T)
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