{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "

On Galerkin approximations for the QG equations

\n", "

Supplementary material for subsection on the Eady model

\n", "

\n", "Cesar B. Rocha*\n", ", William R. Young, and Ian Grooms \n", "

\n", "Winter 2015 \n", "

\n", "*Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr. MC 0213, La Jolla, CA/USA, \n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from __future__ import division\n", "\n", "import numpy as np\n", "from numpy import pi, sqrt,cos\n", "\n", "import matplotlib.pyplot as plt\n", "plt.rcParams.update({'font.size': 25, 'legend.handlelength' : 1.25})\n", "%matplotlib inline\n", "\n", "import seaborn as sns\n", "#sns.set(style=\"darkgrid\")\n", "sns.set_context(\"paper\", font_scale=5, rc={\"lines.linewidth\": 1.5})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Loading data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We computed and saved the eigenvalue and eigenfunction using Approximation A with $\\mathrm{N}=64$ using the linear_eady notebook. We now load the data and plot the wavestructure in the $(x,z)$ plane." ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [], "source": [ "data_path = 'outputs/eady_A_k8_64_efunc.npz'\n", "eady = np.load(data_path)" ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "collapsed": true }, "outputs": [], "source": [ "kappa = eady['kappa']\n", "qn = eady['efunc'] \n", "N = eady['Nmodes']\n", "cr = eady['c'].real" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Seting up the domain grid in $(x,z)$ space" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# vertical coordinate\n", "dz = 1./N # vertical resolution\n", "z = np.arange(-dz/2,-1.-dz/2.,-dz) # level array \n", "\n", "# horizontal coordinate\n", "x = np.linspace(0,np.pi,100)\n", "\n", "# grid\n", "X,Z = np.meshgrid(x,z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The eigenfunctions are the modal coefficients $\\,\\breve{q}_n$. We use the standard modes $\\mathsf{p}_0 = 1$ and $\\mathsf{p}_n = \\sqrt{2}\\,\\cos(n \\pi z)$, $n>1$ to compute the eigenstructure in physical space:\n", "\n", "\\begin{equation}\n", "\\hat{q}(z) = \\sum_{n=0}^{\\mathrm{N}} \\breve{q}_n\\, \\mathsf{p}_n \\,\\,\\,.\n", "\\end{equation}\n", "\n", "In approximation A, the modal streamfunction $\\breve{\\psi}_n$ is related to the PV by\n", "\n", "\\begin{equation}\n", "\\breve{\\psi}_n = \\alpha_n\\, \\breve{q}_n = - (\\kappa^2 + (n \\pi)^2)^{-1}\\,\\,\\breve{q}_n\\,\\,.\n", "\\end{equation}\n", "\n", "The streamfunction $\\hat{\\psi}$ is then\n", "\n", "\\begin{equation}\n", "\\hat{q}(z) = \\sum_{n=0}^{\\mathrm{N}} \\breve{q}_n\\, \\mathsf{p}_n \\,.\n", "\\end{equation}\n", "\n", "\\begin{equation}\n", "\\hat{\\psi}(z) = \\sum_{n=0}^{\\mathrm{N}} \\breve{\\psi}_n\\, \\mathsf{p}_n \\,.\n", "\\end{equation}\n" ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def alpha_n(n,kappa):\n", " \"\"\" Compute the inverse Helmholtz operator in\n", " Fourier space \"\"\"\n", " return -1./( kappa**2 + (n*pi)**2 )" ] }, { "cell_type": "code", "execution_count": 109, "metadata": { "collapsed": false }, "outputs": [], "source": [ "for i in range(N):\n", " if i == 0 : \n", " q = qn[0]*np.ones(z.size)\n", " psi = qn[0]*np.ones(z.size)*alpha_n(0,kappa)\n", " else: \n", " q+= sqrt(2)*qn[i]*cos(i*pi*z)\n", " psi+= sqrt(2)*qn[i]*cos(i*pi*z)*alpha_n(i,kappa)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that $q$ and $\\psi$ are complex. The wavestructure is then\n", "\n", "\\begin{equation}\n", "\\psi(x,z) = |\\psi(z)| \\exp({ik\\,x + P_{\\psi}(z)})\\,, \\qquad \\text{and}\\qquad q(x,z) = |q(z)|\\, \\exp({ik\\,x + P_{q}(z)}) \n", "\\end{equation}\n", "\n", "where vertical bars denote the magnitude and the phases are\n", "\n", "\\begin{equation}\n", "P_{\\psi}(z) = \\text{tan$^{-1}$}\\frac{\\text{Im}(\\hat{\\psi})}{\\text{Re}(\\hat{\\psi})}\\,, \\qquad \\text{and}\\qquad P_{q}(z) = \\text{tan$^{-1}$}\\frac{\\text{Im}(\\hat{q})}{\\text{Re}(\\hat{q})}\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [], "source": [ "qabs = np.abs(q)\n", "qphase = np.arctan2(q.imag,q.real)\n", "psiabs = np.abs(psi)\n", "psiphase = np.arctan2(psi.imag,psi.real)" ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [], "source": [ "qabs = qabs.repeat(x.size).reshape(z.size,x.size)\n", "qphase = qphase.repeat(x.size).reshape(z.size,x.size)\n", "psiabs = psiabs.repeat(x.size).reshape(z.size,x.size)\n", "psiphase = psiphase.repeat(x.size).reshape(z.size,x.size)" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": true }, "outputs": [], "source": [ "PV = qabs*np.cos(kappa*X + qphase)\n", "PSI = psiabs*np.cos(kappa*X + psiphase)" ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 103, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAwIAAAJHCAYAAAAjXuHTAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXd4FFXbh+9N2WTTeyEhhECYUEOAABJCL9IEVFQEewHF\n", "3gs2FPW1N2y8omIBBH1BpSO99w5DLwnpCel1d74/JrvZ3ZlNI1E/nfu6uMhOPbO/nXPOc87zPEcn\n", "SRIaGhoaGhoaGhoaGv8unP7qAmhoaGhoaGhoaGho/PlohoCGhoaGhoaGhobGvxDNENDQ0NDQ0NDQ\n", 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"0P8DW/AQgU8GCbIAAAAASUVORK5CYII=\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(11,8))\n", "plt.contour(X,Z,PSI,colors='k') \n", "plt.contourf(X,Z,PV,np.linspace(-2.,2.,9),cmap='RdBu_r',extend='both') \n", "#plt.text(-0.325,zc,r' $z_c \\rightarrow$',fontsize=25)\n", "cb = plt.colorbar(extend='both',shrink=.9)\n", "cb.ax.text(.0,1.075,'PV',rotation=0,fontsize=25)\n", "plt.text(1.725, -.9, r\"Eady Problem, $\\kappa = 8$, Approximation A\", size=25, rotation=0.,\\\n", " ha=\"center\", va=\"center\",\\\n", " bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "plt.ylim(-1.,0)\n", "plt.xticks([0.,pi/4,pi/2,3*pi/4,pi],[r'$0$',r'$\\pi/4$',r'$\\pi/2$',\\\n", " r'$3\\,\\pi/4$',r'$\\pi$'])\n", "plt.ylabel('$z/H$')\n", "plt.xlabel(r'$x/L_d$')\n", "#plt.savefig('figs/wave-structure_pv_psi_kappa_8_A_N64.eps')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], 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