{"title":"Snowball Earth","code":"# /// script\n# requires-python = \">=3.12\"\n# dependencies = [\"marimo>=0.25.1\", \"numpy\", \"matplotlib\"]\n# ///\n\nimport marimo\n\n__generated_with = \"0.25.1\"\napp = marimo.App(width=\"medium\")\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n # Two climates under the same Sun\n\n The simplest climate model balances sunlight absorbed against heat\n radiated to space, for the planet as a whole:\n\n $$C\\,\\frac{dT}{dt} = \\frac{S}{4}\\bigl(1 - \\alpha(T)\\bigr) - \\varepsilon\\,\\sigma T^4$$\n\n $S$ is the solar constant, $\\alpha$ the fraction of sunlight reflected\n (the albedo), and $\\varepsilon\\sigma T^4$ the infrared emission, with an\n effective emissivity $\\varepsilon$ that stands in for the greenhouse\n effect. Ice reflects more sunlight than ocean and land, so the albedo\n rises as the planet cools. That one feedback, studied by Budyko and by\n Sellers in 1969, gives this model two stable climates at today's\n sunlight: the one we live in, and a frozen one.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _():\n import marimo as mo\n import numpy as np\n import matplotlib.pyplot as plt\n return mo, np, plt\n\n\n@app.cell\ndef _(np):\n SIGMA = 5.670e-8 # Stefan-Boltzmann constant, W m^-2 K^-4\n S_TODAY = 1361.0 # solar constant, W m^-2\n EMISSIVITY = 0.61 # effective, chosen so today's climate sits near 288 K\n HEAT_CAPACITY = 4.0e8 # J m^-2 K^-1, about a 100 m ocean mixed layer\n YEAR = 3.156e7\n\n def albedo(T, ice=0.7, warm=0.3):\n \"\"\"Albedo falls smoothly from an ice-covered to an ice-free planet near 265 K.\"\"\"\n return ice + (warm - ice) * (1 + np.tanh((T - 265.0) / 10.0)) / 2\n\n def net_heating(T, solar):\n \"\"\"Absorbed minus emitted flux, W m^-2, at a solar constant `solar` * S_TODAY.\"\"\"\n return solar * S_TODAY / 4 * (1 - albedo(T)) - EMISSIVITY * SIGMA * T**4\n\n def equilibria(solar):\n \"\"\"Equilibrium temperatures and whether each one is stable.\"\"\"\n T = np.linspace(150, 360, 4200)\n f = net_heating(T, solar)\n crossing = np.flatnonzero(np.sign(f[:-1]) != np.sign(f[1:]))\n roots = T[crossing] - f[crossing] * (T[1] - T[0]) / (f[crossing + 1] - f[crossing])\n stable = f[crossing + 1] < f[crossing]\n return roots, stable\n\n return HEAT_CAPACITY, S_TODAY, YEAR, albedo, equilibria, net_heating\n\n\n@app.cell\ndef _(mo):\n solar = mo.ui.slider(0.70, 1.60, step=0.01, value=1.00, label=\"Sunlight (fraction of today's)\")\n solar\n return (solar,)\n\n\n@app.cell\ndef _(S_TODAY, albedo, equilibria, mo, np, plt, solar):\n T = np.linspace(180, 340, 400)\n roots, stable = equilibria(solar.value)\n\n fig, ax = plt.subplots(figsize=(7, 3.8))\n ax.plot(T, solar.value * S_TODAY / 4 * (1 - albedo(T)), color=\"#9a6700\", lw=2, label=\"absorbed sunlight\")\n ax.plot(T, 0.61 * 5.670e-8 * T**4, color=\"#0969da\", lw=2, label=\"emitted infrared\")\n for _root, _ok in zip(roots, stable):\n ax.plot(\n _root,\n 0.61 * 5.670e-8 * _root**4,\n \"o\",\n ms=8,\n color=\"#1f2328\" if _ok else \"white\",\n markeredgecolor=\"#1f2328\",\n )\n ax.set(xlabel=\"global mean temperature (K)\", ylabel=\"flux (W m⁻²)\")\n ax.legend(frameon=False, loc=\"upper left\")\n fig.tight_layout()\n\n described = \", \".join(\n f\"{_root:.0f} K ({_root - 273.15:+.0f} °C, {'stable' if _ok else 'unstable'})\"\n for _root, _ok in zip(roots, stable)\n )\n mo.vstack([fig, mo.md(f\"Equilibria at {solar.value:.2f} × today's sunlight: {described}.\")])\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n Where the curves cross, the planet neither warms nor cools. A filled\n point is stable: nudge the temperature and it returns. The open point is\n unstable and separates the two basins. Lower the sunlight until the warm\n crossing disappears; the planet then has nowhere to go but the frozen\n state.\n\n ## The hysteresis loop\n\n Follow the equilibria as the Sun slowly dims and then brightens again.\n A warm planet stays warm until the warm branch ends, then drops to a\n snowball. Once frozen it stays frozen until the cold branch ends, at a\n much brighter Sun. This is the Snowball Earth puzzle: how the planet\n froze over during the Neoproterozoic, and how it ever thawed. The\n accepted answer is volcanic carbon dioxide accumulating for millions of\n years under the ice, which this model represents only through\n $\\varepsilon$.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _(equilibria, np, plt, solar):\n fractions = np.linspace(0.7, 1.6, 361)\n points = [(s, r, ok) for s in fractions for r, ok in zip(*equilibria(s))]\n s_all, r_all, ok_all = map(np.array, zip(*points))\n warm_end = min(s for s, r, ok in points if ok and r > 260)\n cold_end = max(s for s, r, ok in points if ok and r < 260)\n\n fig2, ax2 = plt.subplots(figsize=(7, 3.8))\n ax2.scatter(s_all[ok_all], r_all[ok_all], s=3, color=\"#1f2328\", label=\"stable\")\n ax2.scatter(s_all[~ok_all], r_all[~ok_all], s=3, color=\"#8c959f\", label=\"unstable\")\n ax2.annotate(\"\", xy=(warm_end, 225), xytext=(warm_end, 270), arrowprops={\"arrowstyle\": \"->\", \"color\": \"#0969da\"})\n ax2.annotate(\"\", xy=(cold_end, 300), xytext=(cold_end, 255), arrowprops={\"arrowstyle\": \"->\", \"color\": \"#cf222e\"})\n ax2.axvline(solar.value, color=\"#9a6700\", lw=0.8)\n ax2.set(xlabel=\"sunlight (fraction of today's)\", ylabel=\"equilibrium temperature (K)\")\n ax2.legend(frameon=False, loc=\"upper left\")\n fig2.tight_layout()\n fig2\n return cold_end, warm_end\n\n\n@app.cell\ndef _(cold_end, mo, warm_end):\n mo.md(\n f\"\"\"\n In this model the warm climate ends at **{warm_end:.2f}** times today's\n sunlight and the frozen one only at **{cold_end:.2f}** times. Between\n the two, which climate the planet has depends on where it came from.\n The Sun was about 70 percent as bright four billion years ago, when\n geology shows liquid water: the faint young Sun paradox. A stronger\n greenhouse effect then is the usual resolution. Lower the emissivity\n in your fork to try it.\n \"\"\"\n )\n return\n\n\n@app.cell\ndef _(mo):\n start_T = mo.ui.slider(200, 320, step=1, value=262, label=\"Starting temperature (K)\")\n start_T\n return (start_T,)\n\n\n@app.cell\ndef _(HEAT_CAPACITY, YEAR, net_heating, np, plt, solar, start_T):\n dt = 0.05 * YEAR\n path = [float(start_T.value)]\n for _ in range(1200):\n path.append(path[-1] + dt * net_heating(path[-1], solar.value) / HEAT_CAPACITY)\n years = np.arange(len(path)) * dt / YEAR\n\n fig3, ax3 = plt.subplots(figsize=(7, 3))\n ax3.plot(years, path, color=\"#0969da\", lw=2)\n ax3.set(xlabel=\"years\", ylabel=\"temperature (K)\")\n ax3.set_title(f\"From {start_T.value} K at {solar.value:.2f} × today's sunlight: {path[-1]:.0f} K after {years[-1]:.0f} years\")\n fig3.tight_layout()\n fig3\n return\n\n\n@app.cell(hide_code=True)\ndef _(mo):\n mo.md(\n r\"\"\"\n ---\n **Model and limits.** Zero-dimensional energy balance: one temperature\n for the whole planet, no latitude, no ocean circulation, no carbon\n cycle. Albedo 0.7 for an ice-covered and 0.3 for an ice-free planet,\n joined smoothly around 265 K; effective emissivity 0.61; heat capacity\n of a 100 m ocean layer, which sets only the time scale. The thresholds\n depend on these choices, the existence of two branches does not.\n **References.** M. I. Budyko, *Tellus* 21 (1969) 611;\n W. D. Sellers, *J. Appl. Meteor.* 8 (1969) 392;\n P. F. Hoffman et al., *Science* 281 (1998) 1342.\n Fork it to add latitude bands, as Budyko and Sellers did.\n \"\"\"\n )\n return\n\n\nif __name__ == \"__main__\":\n app.run()\n","revision":"2026-10-13T02:22:20.737Z","appConfig":{"width":"medium"},"firstPaint":{"notebook":{"cells":[{"code":"mo.md(\n r\"\"\"\n # Two climates under the same Sun\n\n The simplest climate model balances sunlight absorbed against heat\n radiated to space, for the planet as a whole:\n\n $$C\\,\\frac{dT}{dt} = \\frac{S}{4}\\bigl(1 - \\alpha(T)\\bigr) - \\varepsilon\\,\\sigma T^4$$\n\n $S$ is the solar constant, $\\alpha$ the fraction of sunlight reflected\n (the albedo), and $\\varepsilon\\sigma T^4$ the infrared emission, with an\n effective emissivity $\\varepsilon$ that stands in for the greenhouse\n effect. Ice reflects more sunlight than ocean and land, so the albedo\n rises as the planet cools. That one feedback, studied by Budyko and by\n Sellers in 1969, gives this model two stable climates at today's\n sunlight: the one we live in, and a frozen one.\n \"\"\"\n)","code_hash":"a57bf038ae837b41d023b92f7db9df8e","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"Hbol","name":"_"},{"code":"import marimo as mo\nimport numpy as np\nimport matplotlib.pyplot as plt","code_hash":"ef0032c4370d821ec8e83d0954968f66","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"MJUe","name":"_"},{"code":"SIGMA = 5.670e-8 # Stefan-Boltzmann constant, W m^-2 K^-4\nS_TODAY = 1361.0 # solar constant, W m^-2\nEMISSIVITY = 0.61 # effective, chosen so today's climate sits near 288 K\nHEAT_CAPACITY = 4.0e8 # J m^-2 K^-1, about a 100 m ocean mixed layer\nYEAR = 3.156e7\n\ndef albedo(T, ice=0.7, warm=0.3):\n \"\"\"Albedo falls smoothly from an ice-covered to an ice-free planet near 265 K.\"\"\"\n return ice + (warm - ice) * (1 + np.tanh((T - 265.0) / 10.0)) / 2\n\ndef net_heating(T, solar):\n \"\"\"Absorbed minus emitted flux, W m^-2, at a solar constant `solar` * S_TODAY.\"\"\"\n return solar * S_TODAY / 4 * (1 - albedo(T)) - EMISSIVITY * SIGMA * T**4\n\ndef equilibria(solar):\n \"\"\"Equilibrium temperatures and whether each one is stable.\"\"\"\n T = np.linspace(150, 360, 4200)\n f = net_heating(T, solar)\n crossing = np.flatnonzero(np.sign(f[:-1]) != np.sign(f[1:]))\n roots = T[crossing] - f[crossing] * (T[1] - T[0]) / (f[crossing + 1] - f[crossing])\n stable = f[crossing + 1] < f[crossing]\n return roots, stable","code_hash":"1afdf6d26382e26b6780cda6390c4f74","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"vblA","name":"_"},{"code":"solar = mo.ui.slider(0.70, 1.60, step=0.01, value=1.00, label=\"Sunlight (fraction of today's)\")\nsolar","code_hash":"dedfc6c0e9038f36be347527103609ef","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"bkHC","name":"_"},{"code":"T = np.linspace(180, 340, 400)\nroots, stable = equilibria(solar.value)\n\nfig, ax = plt.subplots(figsize=(7, 3.8))\nax.plot(T, solar.value * S_TODAY / 4 * (1 - albedo(T)), color=\"#9a6700\", lw=2, label=\"absorbed sunlight\")\nax.plot(T, 0.61 * 5.670e-8 * T**4, color=\"#0969da\", lw=2, label=\"emitted infrared\")\nfor _root, _ok in zip(roots, stable):\n ax.plot(\n _root,\n 0.61 * 5.670e-8 * _root**4,\n \"o\",\n ms=8,\n color=\"#1f2328\" if _ok else \"white\",\n markeredgecolor=\"#1f2328\",\n )\nax.set(xlabel=\"global mean temperature (K)\", ylabel=\"flux (W m⁻²)\")\nax.legend(frameon=False, loc=\"upper left\")\nfig.tight_layout()\n\ndescribed = \", \".join(\n f\"{_root:.0f} K ({_root - 273.15:+.0f} °C, {'stable' if _ok else 'unstable'})\"\n for _root, _ok in zip(roots, stable)\n)\nmo.vstack([fig, mo.md(f\"Equilibria at {solar.value:.2f} × today's sunlight: {described}.\")])","code_hash":"827188de3e375a091cea154687f0d18a","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"lEQa","name":"_"},{"code":"mo.md(\n r\"\"\"\n Where the curves cross, the planet neither warms nor cools. A filled\n point is stable: nudge the temperature and it returns. The open point is\n unstable and separates the two basins. Lower the sunlight until the warm\n crossing disappears; the planet then has nowhere to go but the frozen\n state.\n\n ## The hysteresis loop\n\n Follow the equilibria as the Sun slowly dims and then brightens again.\n A warm planet stays warm until the warm branch ends, then drops to a\n snowball. Once frozen it stays frozen until the cold branch ends, at a\n much brighter Sun. This is the Snowball Earth puzzle: how the planet\n froze over during the Neoproterozoic, and how it ever thawed. The\n accepted answer is volcanic carbon dioxide accumulating for millions of\n years under the ice, which this model represents only through\n $\\varepsilon$.\n \"\"\"\n)","code_hash":"e23b50d56049165b6d965fa9d5bc1255","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"PKri","name":"_"},{"code":"fractions = np.linspace(0.7, 1.6, 361)\npoints = [(s, r, ok) for s in fractions for r, ok in zip(*equilibria(s))]\ns_all, r_all, ok_all = map(np.array, zip(*points))\nwarm_end = min(s for s, r, ok in points if ok and r > 260)\ncold_end = max(s for s, r, ok in points if ok and r < 260)\n\nfig2, ax2 = plt.subplots(figsize=(7, 3.8))\nax2.scatter(s_all[ok_all], r_all[ok_all], s=3, color=\"#1f2328\", label=\"stable\")\nax2.scatter(s_all[~ok_all], r_all[~ok_all], s=3, color=\"#8c959f\", label=\"unstable\")\nax2.annotate(\"\", xy=(warm_end, 225), xytext=(warm_end, 270), arrowprops={\"arrowstyle\": \"->\", \"color\": \"#0969da\"})\nax2.annotate(\"\", xy=(cold_end, 300), xytext=(cold_end, 255), arrowprops={\"arrowstyle\": \"->\", \"color\": \"#cf222e\"})\nax2.axvline(solar.value, color=\"#9a6700\", lw=0.8)\nax2.set(xlabel=\"sunlight (fraction of today's)\", ylabel=\"equilibrium temperature (K)\")\nax2.legend(frameon=False, loc=\"upper left\")\nfig2.tight_layout()\nfig2","code_hash":"b104c724b79902ec56197579a2aa8f9b","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"Xref","name":"_"},{"code":"mo.md(\n f\"\"\"\n In this model the warm climate ends at **{warm_end:.2f}** times today's\n sunlight and the frozen one only at **{cold_end:.2f}** times. Between\n the two, which climate the planet has depends on where it came from.\n The Sun was about 70 percent as bright four billion years ago, when\n geology shows liquid water: the faint young Sun paradox. A stronger\n greenhouse effect then is the usual resolution. Lower the emissivity\n in your fork to try it.\n \"\"\"\n)","code_hash":"b64da7df77bd09e0eb65ae7edd9c9f96","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"SFPL","name":"_"},{"code":"start_T = mo.ui.slider(200, 320, step=1, value=262, label=\"Starting temperature (K)\")\nstart_T","code_hash":"12bd171a9792c40ce76fee576d393d1b","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"BYtC","name":"_"},{"code":"dt = 0.05 * YEAR\npath = [float(start_T.value)]\nfor _ in range(1200):\n path.append(path[-1] + dt * net_heating(path[-1], solar.value) / HEAT_CAPACITY)\nyears = np.arange(len(path)) * dt / YEAR\n\nfig3, ax3 = plt.subplots(figsize=(7, 3))\nax3.plot(years, path, color=\"#0969da\", lw=2)\nax3.set(xlabel=\"years\", ylabel=\"temperature (K)\")\nax3.set_title(f\"From {start_T.value} K at {solar.value:.2f} × today's sunlight: {path[-1]:.0f} K after {years[-1]:.0f} years\")\nfig3.tight_layout()\nfig3","code_hash":"98bfe36eabaa8f210d4c7da202776198","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":false},"id":"RGSE","name":"_"},{"code":"mo.md(\n r\"\"\"\n ---\n **Model and limits.** Zero-dimensional energy balance: one temperature\n for the whole planet, no latitude, no ocean circulation, no carbon\n cycle. Albedo 0.7 for an ice-covered and 0.3 for an ice-free planet,\n joined smoothly around 265 K; effective emissivity 0.61; heat capacity\n of a 100 m ocean layer, which sets only the time scale. The thresholds\n depend on these choices, the existence of two branches does not.\n **References.** M. I. Budyko, *Tellus* 21 (1969) 611;\n W. D. Sellers, *J. Appl. Meteor.* 8 (1969) 392;\n P. F. Hoffman et al., *Science* 281 (1998) 1342.\n Fork it to add latitude bands, as Budyko and Sellers did.\n \"\"\"\n)","code_hash":"d195309a9020e3f6d5d6cb8c8107efd7","config":{"column":null,"disabled":false,"expand_output":false,"hide_code":true},"id":"Kclp","name":"_"}],"metadata":{"marimo_version":"0.25.1"},"version":"1"},"session":{"cells":[{"code_hash":"a57bf038ae837b41d023b92f7db9df8e","console":[],"id":"Hbol","outputs":[{"data":{"text/markdown":"

Two climates under the same Sun

\nThe simplest climate model balances sunlight absorbed against heat\nradiated to space, for the planet as a whole:\n||[C\\,\\frac{dT}{dt} = \\frac{S}{4}\\bigl(1 - \\alpha(T)\\bigr) - \\varepsilon\\,\\sigma T^4||]||(S||) is the solar constant, ||(\\alpha||) the fraction of sunlight reflected\n(the albedo), and ||(\\varepsilon\\sigma T^4||) the infrared emission, with an\neffective emissivity ||(\\varepsilon||) that stands in for the greenhouse\neffect. Ice reflects more sunlight than ocean and land, so the albedo\nrises as the planet cools. That one feedback, studied by Budyko and by\nSellers in 1969, gives this model two stable climates at today's\nsunlight: the one we live in, and a frozen one."},"type":"data"}]},{"code_hash":"ef0032c4370d821ec8e83d0954968f66","console":[],"id":"MJUe","outputs":[{"data":{"text/plain":""},"type":"data"}]},{"code_hash":"1afdf6d26382e26b6780cda6390c4f74","console":[],"id":"vblA","outputs":[{"data":{"text/plain":""},"type":"data"}]},{"code_hash":"dedfc6c0e9038f36be347527103609ef","console":[],"id":"bkHC","outputs":[{"data":{"text/html":""},"type":"data"}]},{"code_hash":"827188de3e375a091cea154687f0d18a","console":[],"id":"lEQa","outputs":[{"data":{"text/html":"
Equilibria at 1.00 × today's sunlight: 233 K (-40 °C, stable), 265 K (-8 °C, unstable), 288 K (+14 °C, stable).
"},"type":"data"}]},{"code_hash":"e23b50d56049165b6d965fa9d5bc1255","console":[],"id":"PKri","outputs":[{"data":{"text/markdown":"Where the curves cross, the planet neither warms nor cools. A filled\npoint is stable: nudge the temperature and it returns. The open point is\nunstable and separates the two basins. Lower the sunlight until the warm\ncrossing disappears; the planet then has nowhere to go but the frozen\nstate.\n

The hysteresis loop

\nFollow the equilibria as the Sun slowly dims and then brightens again.\nA warm planet stays warm until the warm branch ends, then drops to a\nsnowball. Once frozen it stays frozen until the cold branch ends, at a\nmuch brighter Sun. This is the Snowball Earth puzzle: how the planet\nfroze over during the Neoproterozoic, and how it ever thawed. The\naccepted answer is volcanic carbon dioxide accumulating for millions of\nyears under the ice, which this model represents only through\n||(\\varepsilon||)."},"type":"data"}]},{"code_hash":"b104c724b79902ec56197579a2aa8f9b","console":[],"id":"Xref","outputs":[{"data":{"application/vnd.marimo+mimebundle":"{\"image/png\":\"https://p6tmzgq3m2oibnwv7skm4v2cuc.notebooks.sciencehubusercontent.com/_outputs/eyJuYW1lc3BhY2VJZCI6Im5zLXdicjZya2tkOWZqMG1neGUiLCJwcm9qZWN0SWQiOiJwcm9qLWF2Nnh2dnByZW1rOGVoNjQiLCJub3RlYm9va0lkIjoibmItcXd5dDA0Y3I0cGU2cWc3eCIsImhhc2giOiJmN2Y3OTA3YmZiYTU3MGRjOGRlMWJkM2M3ZTM1YzVlMyJ9.0.lm_sZP2FR45w7HdxL12qQ93D5Wsek0PPxx6pwciOOr0\",\"__metadata__\":{\"image/png\":{\"width\":690,\"height\":370}}}"},"type":"data"}]},{"code_hash":"b64da7df77bd09e0eb65ae7edd9c9f96","console":[],"id":"SFPL","outputs":[{"data":{"text/markdown":"In this model the warm climate ends at 0.89 times today's\nsunlight and the frozen one only at 1.25 times. Between\nthe two, which climate the planet has depends on where it came from.\nThe Sun was about 70 percent as bright four billion years ago, when\ngeology shows liquid water: the faint young Sun paradox. A stronger\ngreenhouse effect then is the usual resolution. Lower the emissivity\nin your fork to try it."},"type":"data"}]},{"code_hash":"12bd171a9792c40ce76fee576d393d1b","console":[],"id":"BYtC","outputs":[{"data":{"text/html":""},"type":"data"}]},{"code_hash":"98bfe36eabaa8f210d4c7da202776198","console":[],"id":"RGSE","outputs":[{"data":{"application/vnd.marimo+mimebundle":"{\"image/png\":\"https://p6tmzgq3m2oibnwv7skm4v2cuc.notebooks.sciencehubusercontent.com/_outputs/eyJuYW1lc3BhY2VJZCI6Im5zLXdicjZya2tkOWZqMG1neGUiLCJwcm9qZWN0SWQiOiJwcm9qLWF2Nnh2dnByZW1rOGVoNjQiLCJub3RlYm9va0lkIjoibmItcXd5dDA0Y3I0cGU2cWc3eCIsImhhc2giOiI1ZmYxMmFlZTRhZWNmYzMwYzlmMTVkMDNmMGQyMjRlYiJ9.0.iOPQpn4uOxPUjwurVme38zkS-AFEqOYqkeeYzw4Bl-c\",\"__metadata__\":{\"image/png\":{\"width\":690,\"height\":290}}}"},"type":"data"}]},{"code_hash":"d195309a9020e3f6d5d6cb8c8107efd7","console":[],"id":"Kclp","outputs":[{"data":{"text/markdown":"
\nModel and limits. Zero-dimensional energy balance: one temperature\nfor the whole planet, no latitude, no ocean circulation, no carbon\ncycle. Albedo 0.7 for an ice-covered and 0.3 for an ice-free planet,\njoined smoothly around 265 K; effective emissivity 0.61; heat capacity\nof a 100 m ocean layer, which sets only the time scale. The thresholds\ndepend on these choices, the existence of two branches does not.\nReferences. M. I. Budyko, Tellus 21 (1969) 611;\nW. D. Sellers, J. Appl. Meteor. 8 (1969) 392;\nP. F. Hoffman et al., Science 281 (1998) 1342.\nFork it to add latitude bands, as Budyko and Sellers did."},"type":"data"}]}],"metadata":{"marimo_version":"0.25.1","script_metadata_hash":null},"version":"1"},"appConfig":{"sql_output":"auto","width":"medium"},"layout":null,"renderedOn":"browser","formulas":{"||[C\\,\\frac{dT}{dt} = \\frac{S}{4}\\bigl(1 - \\alpha(T)\\bigr) - \\varepsilon\\,\\sigma T^4||]":"CdtdT​=4S​(1−α(T))−εσT4","||(S||)":"S","||(\\alpha||)":"α","||(\\varepsilon\\sigma T^4||)":"εσT4","||(\\varepsilon||)":"ε"}}}