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   "source": [
    "# Exercise 2, melting the snowball earth\n",
    "\n",
    "> Start after finishing excercise 1 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the last lectures we implemented a energy balance model. The energy balance equation is give by\n",
    "\n",
    "\\begin{gather}\n",
    "\\color{brown}{C \\frac{dT}{dt}}\n",
    "\\; \\color{black}{=} \\; \\color{orange}{\\frac{(1 - α)S}{4}}\n",
    "\\; \\color{black}{-} \\; \\color{blue}{(A - BT)}\n",
    "\\; \\color{black}{+} \\; \\color{grey}{a \\ln \\left( \\frac{[\\text{CO}₂]}{[\\text{CO}₂]_{\\text{PI}}} \\right)},\n",
    "\\end{gather}\n",
    "\n",
    "Recall that in the last lecture we implemented a simplied ice albedo feedback by allowing the albedo to depend on temperature:\n",
    "\n",
    "$$\\alpha(T) = \\begin{cases}\n",
    "\\alpha_{i} & \\text{if }\\;\\; T \\leq -10\\text{°C} &\\text{(completely frozen)}\\\\\n",
    "\\alpha_{i} + (\\alpha_{0}-\\alpha_{i})\\frac{T + 10}{20} & \\text{if }\\;\\; -10\\text{°C} \\leq T \\leq 10\\text{°C} &\\text{(partially frozen)}\\\\\n",
    "\\alpha_{0} &\\text{if }\\;\\; T \\geq 10\\text{°C} &\\text{(no ice)}\n",
    "\\end{cases}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One thing that we did not adress in the last lecture was the impact of CO$_2$ increase. We simply set:\n",
    "\n",
    "$\\ln \\left( \\frac{ [\\text{CO}₂]_{\\text{PI}} }{[\\text{CO}₂]_{\\text{PI}}} \\right) = \\ln(1) = 0$\n",
    "\n",
    "We then evaluated how an increasing solar constant changes the equalibirum temperature on earth. In this excercise you keep $S$ at $1368 W/m^2$ and instead increase the $CO_2$ concentration.\n",
    "\n",
    "Replot the bifurcations diagramm for $CO_2$ increase instead of solar radiation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import xarray as xr\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "\n",
    "from IPython.display import HTML\n",
    "from IPython.display import display\n",
    "\n",
    "from energy_balance_model import ebm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def CO2_change(t): \n",
    "    return 280 + co2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that `co2` is a global variable here. In general all variables that are assigned in a function call are private (only accessible within the function). However, if this variable is not defined within the function, python looks for a global variable that is called `co2` (in your complete code). Therefore, remember to increase `co2` for the following task.\n",
    "\n",
    "Start by running the model for with different C02 concentrations and plot the equlibrium temperature. \n",
    "\n",
    "Hint: You can copy nearly all code from the Lecture *Snowball earth*"
   ]
  },
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   "execution_count": null,
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   "source": []
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   "execution_count": null,
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize = (8,6))\n",
    "plt.plot(co2vec[0:len(co2vec)//2], tvec[0:len(co2vec)//2], color = \"blue\", label = \"cool branch\", alpha = 0.3)\n",
    "plt.plot(co2vec[len(co2vec)//2:], tvec[len(co2vec)//2:], color = \"red\", label = \"warm branch\", alpha = 0.3)\n",
    "plt.axvline(1368, color = \"yellow\", lw = 5, alpha = 0.2, label = \"Pre-industiral / present insolation\")\n",
    "\n",
    "plt.plot(420, 14, marker=\"o\", label=\"Our preindustrial climate\", color=\"orange\", markersize=8)\n",
    "plt.plot(420, -38, marker=\"o\", label=\"Alternate preindustrial climate\", color=\"lightblue\", markersize=8)\n",
    "plt.plot(280, -48, marker=\"o\", label=\"neoproterozoic (700 Mya)\", color=\"lightgrey\", markersize=8)\n",
    "\n",
    "plt.xlabel(\"CO$_2$ concentration [ppm]\")\n",
    "plt.ylabel(\"Global temperature T [°C]\")\n",
    "\n",
    "plt.legend()\n",
    "plt.grid()"
   ]
  }
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