{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
Using non-dimensional variables, the basic state for the Eady problem is
\n", "\\begin{equation}\n", "U(z) = z + 1 \\, ,\\qquad \\text{and} \\qquad U'(z) = 1 \\, .\n", "\\end{equation}\n", "\n", "The base state expansion in standard modes is\n", "\n", "\\begin{equation}\n", "U^G(z) = \\sum_{n=0}^{\\mathrm{N}} U_n \\mathsf{p}_n = \\frac{1}{2} + 2\\sqrt{2} \\sum_{n = 1, n~ \\text{odd}}^{\\mathrm{N}} \\frac{\\mathsf{p}_n}{(n\\pi)^2}\\, .\n", "\\end{equation}\n", "\n", "Its associated potential vorticity gradient is\n", "\n", "\\begin{equation}\n", "-[U^G(z)]'' = 2\\sqrt{2} \\sum_{n = 1, n~ \\text{odd}}^{\\mathrm{N}} \\mathsf{p}_n\\, = 2\\,\\frac{\\sin[(N+1)\\pi z]}{\\sin \\pi z} .\n", "\\end{equation}\n", "\n", "Figure 1 in Rocha, Young and Grooms indents to convey two important points: \n", "\n", "1. While standard modes have zero slope at the boundaries, a modest number of terms provides a reasonable approximation to the exact mean velocity.\n", "2. The potential vorticity associated with this base state is a Dirichlet kernel (a sum of cosines), which is essentially a finite approximation for the two delta functions at the boundaries." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To produce Figure 1, we first create a simple function that evaluates the series" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import division\n", "from numpy import cos, pi\n", "\n", "def eady_base_series(N):\n", " ''' Galerkin approximatin to Eady's \n", " basic state U = z + 1 \n", " N is the number of baroclinic modes '''\n", " Ug, Qy = 0.5, 0.\n", " for n in range(1,N+1,2):\n", " Ug += 4.*cos(n*pi*z)/(n*pi)**2\n", " Qy += 4.*cos(n*pi*z)\n", " return Ug,Qy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We then call this function for various $\\mathrm{N}$, and plot it using matplotlib" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "plt.rcParams.update({'font.size': 40, 'legend.handlelength' : 1.25})\n", "\n", "z = np.linspace(-1.,0,300) \n", "U_exact = z + 1.\n", "\n", "N = np.array([1,3,5,7,9,15,27,49])\n", "\n", "U_g,Qy_g = np.zeros((z.size,N.size)),np.zeros((z.size,N.size))\n", "\n", "for i in range(N.size):\n", " U_g[:,i],Qy_g[:,i] = eady_base_series(N[i])" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABw8AAAOACAYAAADM4zxZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8XFX9//HXJ2vbdN8oLVAKtEApUNYCslwWBREEBMEF\n", "BVEURNy3r/4kBEH5Kn5lEfyCKMoiwhdBXFgE5Mom+94W2tKF0kL30qZp9s/vj3OnmaaTZJLMTNLk\n", "/Xw87uPO5N459yRpM2fu53w+x9wdEREREREREREREREREZGinu6AiIiIiIiIiIiIiIiIiPQOCh6K\n", 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