EDIPI workshop qgs tutorial - Part II.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# EDIPI Workshop qgs tutorial - Part II: Computing the LLE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook is the third one of a tutorial given during the EDIPI workshop at RMI in January 2023.\n",
    "In this tutorial, you will learn:\n",
    "\n",
    "1. How to install the [qgs](https://github.com/Climdyn/qgs) framework for low-order climate and weather modeling\n",
    "2. How to run the model\n",
    "3. How to compute the correlation properties of the underlying dynamical model\n",
    "4. **How to compute the Largest Lyapunov Exponent (LLE) in the model**\n",
    "5. How to compute the full Lyapunov Exponent spectrum\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we will be concerned with task 4 above: learning how to compute the largest [Lyapunov exponent](https://en.wikipedia.org/wiki/Lyapunov_exponent) in the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We assume that you have already been through the [Introduction notebook](https://github.com/jodemaey/EDIPI-qgs-tutorial-on-predictability/blob/main/EDIPI%20workshop%20qgs%20tutorial%20-%20Introduction.ipynb), qgs and the tutorial being installed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup of the model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use again the simple 2-layer channel QG atmosphere truncated at wavenumber 2 on a beta-plane with a simple orography (a montain and a valley), as in the Introduction notebook. But now, we are going to repeat a perturbation experiment to obtain the largest Lyapunov exponent. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we need to load the model, and integrate it to find an initial condition on the attractor, as in the Introduction notebook:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modules import"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, load some modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Importing the model's modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qgs.params.params import QgParams\n",
    "from qgs.integrators.integrator import RungeKuttaIntegrator, RungeKuttaTglsIntegrator\n",
    "from qgs.functions.tendencies import create_tendencies\n",
    "from qgs.plotting.util import std_plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and diagnostics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qgs.diagnostics.streamfunctions import MiddleAtmosphericStreamfunctionDiagnostic\n",
    "from qgs.diagnostics.variables import VariablesDiagnostic\n",
    "from qgs.diagnostics.multi import MultiDiagnostic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining the model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we define some general parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Time increment parameter\n",
    "dt = 0.1\n",
    "# Saving the model state every 5 steps\n",
    "write_steps = 5\n",
    "# transient time\n",
    "transient_time = 200000."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we create the model parameters object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_parameters = QgParams()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We set up some parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# here we define the latitude to be 50 degrees and a predefined amplitude of the meridional temperature gradient\n",
    "model_parameters.set_params({'phi0_npi': np.deg2rad(50.)/np.pi, 'hd':0.045})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and indicate that we want an atmospheric channel for the atmosphere, with Fourier modes up to wavenumber 2 in each spatial direction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_parameters.set_atmospheric_channel_fourier_modes(2, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also set some topography"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_parameters.ground_params.set_orography(0.2, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and indicate the amplitude of the meridional temperature gradient which forces the model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_parameters.atemperature_params.set_thetas(0.1, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we are done configuring the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we create the tendencies $\\boldsymbol{f}$ that will allow us to integrate the model equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%time\n",
    "f, Df = create_tendencies(model_parameters)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now integrate our model with the qgs built-in integrator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "integrator = RungeKuttaIntegrator()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We tell this integrator to use our defined model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "integrator.set_func(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now start from a small random initial condition and integrate over a transient time to obtain an initial condition on the attractors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%time\n",
    "ic = np.random.rand(model_parameters.ndim)*0.1\n",
    "integrator.integrate(0., transient_time, dt, ic=ic, write_steps=0)  # write_steps=0 will only give us the last step\n",
    "time, ic = integrator.get_trajectories()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we are ready to start our exercise on the computation of the LLE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise: Computing the LLE"
   ]
  },
  {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In principle, to compute the LLE $\\lambda$, we need to compute on points $\\boldsymbol{x}$ of the attractor\n",
    "\n",
    "$$ \\lambda(\\boldsymbol{x}, \\boldsymbol{u}) = \\lim_{t\\to\\infty} \\lim_{\\|\\boldsymbol{\\delta}_0\\|\\to 0} \\frac{1}{t} \\log \\frac{\\|\\boldsymbol{\\delta}_t\\|}{\\|\\boldsymbol{\\delta}_0\\|} \\qquad\\qquad \\mathrm{(1)}$$\n",
    "\n",
    "where $\\boldsymbol{\\delta}_0 = \\|\\boldsymbol{\\delta}_0\\| \\, \\boldsymbol{u}$ with $\\boldsymbol{u}$ the unit vector of  the direction in the model *phase space* corresponding to the *maximum growth rate* of the perturbation, i.e. the most unstable direction.\n",
    "\n",
    "Computing approximately this formula for an arbitrary vector $\\boldsymbol{u}$ is easy. Just run 2 copies of the model with initial conditions $\\boldsymbol{x}$ and $\\boldsymbol{x} + \\boldsymbol{\\delta}_0$ and measure the distance between the two trajectories $\\|\\boldsymbol{\\delta}_{\\Delta t}\\|$ after a time $\\Delta t$. This is better represented by a picture:\n",
    "![image-2.png](attachment:image-2.png)\n",
    "You can then compute the logarithm of the ratio in (1) and estimate the *local* lyapunov exponent\n",
    "$$ \\lambda^\\mathrm{loc}(\\boldsymbol{x}, \\boldsymbol{u}) \\approx \\frac{1}{\\Delta t} \\log \\frac{\\|\\boldsymbol{\\delta}_{\\Delta t}\\|}{\\|\\boldsymbol{\\delta}_0\\|} \\qquad\\qquad \\mathrm{(2)}$$\n",
    "Note that what we compute here is an approximation of an approximation, because in principle the time evolution of the perturbation $\\boldsymbol{\\delta}_t$ should have been computed with the tangent linear model, while here we are using the nonlinear system (but for the largest Lyapunov exponent, this yields a good approximation). You can then repeat this perturbation experiment $N$ times with different points $\\boldsymbol{x}_i$ on the attractor, and get the largest Lyapunov exponent as\n",
    "$$ \\lambda \\approx \\frac{1}{N} \\sum_{i=1}^N \\lambda^\\mathrm{loc}(\\boldsymbol{x}_i, \\boldsymbol{u}) \\qquad\\qquad \\mathrm{(3)}$$\n",
    "\n",
    "The more experiments you perform, the best your estimate will become (however at some point you don't get much more precision by increasing $N$). Typically, you use the final point of the unperturbed trajectory of the previous experiment as the new initial point on the attractor for the next one.\n",
    "\n",
    "**Anyway, the problem is that we still don't know how to define locally the direction corresponding to the largest Lyapunov exponent !** Thankfully, to find this direction, we can use the [Osedelet theorem](http://www.scholarpedia.org/article/Oseledets_theorem), which implies that over time, $\\boldsymbol{\\delta}_t$ will align more and more in the direction of the largest Lyapunov exponent. In consequence, we can make subsequent perturbation experiments **where the direction of the new initial perturbation** $\\boldsymbol{\\delta}_0$ **is given by the direction** $\\boldsymbol{\\delta}_t / \\|\\boldsymbol{\\delta}_t\\|$ **of the final perturbation of the previous experiments.** With this process, the perturbations that we use will progressively align with the most unstable direction. Therefore, we must typically discard the first experiments because the process of this alignement of the perturbations with the direction of the of LLE takes a while to converge.\n",
    "\n",
    "The inverse of the largest Lyapunov exponent defines a timescale which is considered to be the timespan over which the system is predictible, i.e. the range over which the pertubations evolve linearly. It is sometimes called the *Lyapunov time* or the *Lyapunov horizon*. This time can be defined globally for the whole attractor - indicating the global predictibility of the system - or locally, associated to the predictability of weather regimes.\n",
    " \n",
    "In the following, we shall thus make a code which compute the local largest Lyapunov exponent at various points of the attractor, and plot this information on it. We shall also compute the average to obtain the largest Lyapunov exponent, and see if the system is chaotic.\n",
    "\n",
    "This exercise session will be as follow:\n",
    "\n",
    "1. First, a code snippet will show you how to make a perturbed and non-perturbed trajectories. You can use it to design your code at point 3.\n",
    "2. Then, before even coding anything, it is recommended that you write a [pseudo-code](https://en.wikipedia.org/wiki/Pseudocode) to reflect about the steps needed to implement the experiments described above. \n",
    "3. Finally, you will implement the algorithm and plot the results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's proceed:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. How to implement perturbation experiments in qgs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start from the initial condition on the attractor obtained precedently, and create two initial conditions: $\\boldsymbol{x}_1$ and $\\boldsymbol{x}_2 = \\boldsymbol{x}_1 + \\boldsymbol{\\delta}_0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = ic\n",
    "delta0 = np.random.rand(model_parameters.ndim)*1.e-4  # small random perturbation\n",
    "x2 = ic + delta0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we integrate the model with both initial condition over roughly 40 days (400 model timeunits):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%time\n",
    "integrator.integrate(0., 400., dt, ic=x1, write_steps=5)\n",
    "time1, trajectory1 = integrator.get_trajectories()\n",
    "integrator.integrate(0., 400., dt, ic=x2, write_steps=5)\n",
    "time2, trajectory2 = integrator.get_trajectories()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we plot the result. First, one component of both trajectories as a function of the lead time $t$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "var = 0\n",
    "plt.figure(figsize=(10, 8))\n",
    "\n",
    "plt.plot(model_parameters.dimensional_time*time1, trajectory1[var], label=\"unperturbed\")\n",
    "plt.plot(model_parameters.dimensional_time*time2, trajectory2[var], label=\"perturbed\")\n",
    "\n",
    "plt.legend()\n",
    "plt.xlabel('time (days)')\n",
    "plt.ylabel('$'+model_parameters.latex_var_string[var]+'$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can also plot the distance $\\|\\boldsymbol{\\delta}_t\\|$ between the two trajectories as a function of the lead time $t$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10, 8))\n",
    "\n",
    "plt.plot(model_parameters.dimensional_time*time1, np.linalg.norm(trajectory1 - trajectory2, axis=0))\n",
    "\n",
    "plt.xlabel('time (days)')\n",
    "plt.ylabel(r'$\\|\\mathbf{\\delta}_t\\|$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Write a pseudo-code for the computation of the local and global largest Lyapunov exponent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can type your pseudo-code here:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Implement the algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the pseudo-code written in Section 2 and the code snippet in Section 1 above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the local Lyapunov exponent as a function of the state on the attractor. You can for instance uncomment and modify the code below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# varx = 2\n",
    "# vary = 1\n",
    "# plt.figure(figsize=(10, 8))\n",
    "\n",
    "# sc = plt.scatter(experiment_trajectory[varx], experiment_trajectory[vary], c=largest_lyapunov_exponent, vmin=-0.5, vmax=0.5, cmap=plt.get_cmap('viridis'), marker='o', s=10.)\n",
    "# plt.colorbar(sc)\n",
    "\n",
    "# plt.xlabel('$'+model_parameters.latex_var_string[varx]+'$')\n",
    "# plt.ylabel('$'+model_parameters.latex_var_string[vary]+'$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the global largest Lyapunov exponent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estimate the average predictability of the system (in days)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. Plot the spatial fields for the corresponding the trajectory vectors and LLE vectors side by side for various time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For that, your algoritm to compute the LLE must store the trajectory and the LLE vector data. You can plot them in parallel to show the system state field and also the field correponding to the most unstable direction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hint: use the diagnostic classes detailed in the introduction to easily do that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5. Compare with the Mean Square Error time evolution of the model "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeat $N$ perturbations experiments as in point **1.** and compute the averaged Mean Square Error of the perturbation as a function of the lead time $t$:\n",
    "$$\\mathrm{MSE} \\,(t) = \\frac{1}{N} \\sum_{n=1}^N  \\| \\boldsymbol{x}_{2}(t) - \\boldsymbol{x}_1(t)\\|^2_n $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the result as a function of the lead time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fit this curve over the first 25 days:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and compute its exponential growth and compare it to the averaged LLE computed beforehand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6. Compare with the average variance of an ensemble of perturbed trajectories"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create an ensemble of **$M=15$ random perturbations** $\\boldsymbol{\\delta}_{0, m}$, $m \\in \\{1,\\ldots,15\\}$ of an initial condition $\\boldsymbol{x}$ on the attractor and compute the time evolution of the trajectories starting at $\\boldsymbol{x}_m = \\boldsymbol{x} + \\boldsymbol{\\delta}_{0,m}$. Repeat this over $N$ experiments. Compute and average the variance of the ensemble\n",
    "$$\\frac{1}{N} \\sum_{n=1}^N  \\frac{1}{M-1}\\sum_{m=1}^M \\| \\boldsymbol{x}_{m}(t) - \\bar{\\boldsymbol{x}}(t)\\|^2_n$$\n",
    "as a function of the lead time $t$, and where $\\bar{\\boldsymbol{x}}(t) = \\left(\\left. \\sum_{m=1}^M \\boldsymbol{x}_m(t) \\right)\\right/ M$ is the ensemble mean. Basically, the formula above is the averaged ensemble variance at any time $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the result as a function of the lead time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fit this curve over the first 25 days:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and compute its exponential growth and compare it to the averaged LLE computed beforehand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The End."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Optional: Redo this part with a more chaotic atmosphere...."
   ]
  }
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