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

On Galerkin approximations for the QG equations

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Supplementary material for subsection on the $\\beta-$Eady model

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Wave structure for Charney mode

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\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" ] }, { "cell_type": "code", "execution_count": 23, "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": [ "### A function to compute difference matrices" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def D_matrices(N):\n", " ''' create an N x N difference matrices \n", " \n", " D1 :: first-order, centered difference\n", " D2 :: second-order, centered difference \n", " \n", " '''\n", " D2 = np.zeros((N,N))\n", " D1 = np.zeros((N,N))\n", " for i in range(N):\n", " D2[i,i] = -2.\n", " if i0: \n", " D2[i,i-1],D1[i,i-1] = 1.,1.\n", " return D1,D2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Load data" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": true }, "outputs": [], "source": [ "data_path = 'linear_charney_num_kappa_8.npz'\n", "charney = np.load(data_path)\n", "\n", "kappa = charney['kappa']\n", "phi_max = charney['e_num'][1:-1] # do no consider ghost points\n", "N = charney['N']\n", "\n", "# the critical level\n", "zc = charney['c_num'].real - 1. # recall the domain has depth 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### set up domain" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": true }, "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": [ "### compute wavestructure\n", "We solved the problem for the streamfunction $\\phi$ vertical structure. The streamfunction is them \n", "\n", "\\begin{equation}\n", "\\psi = |\\phi(z)|\\cos{(i k x +P_{\\psi}(z))}\n", "\\end{equation}\n", "\n", "The associated PV is\n", "\n", "\\begin{equation}\n", "q = \\partial^2_{xx}\\psi + \\partial^2_{zz}\\psi = (-k^2\\,|\\phi(z)| + \\partial^2_{zz}\\,\\,|\\phi(z)| - |\\phi(z)|(\\partial_zP_{\\psi}(z))^2)) \\cos{(i k x +P_{\\psi}(z)}) -(2\\partial_z|\\phi(z)|\\partial_z P_{\\psi}(z)) \\sin{(i k x +P_{\\psi}(z)})\\,\n", "\\end{equation}\n", "\n", "\n", "The phase is\n", "\\begin{equation}\n", "P_{\\psi}(z) = \\text{tan$^{-1}$}\\frac{\\text{Im}(\\hat{\\psi})}{\\text{Re}(\\hat{\\psi})}\\,.\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# wave structure in xz-plane\n", "phi_max_abs = np.abs(phi_max)\n", "phi_max_phase = np.arctan2(phi_max.imag,phi_max.real)\n", "\n", "phase = np.repeat(phi_max_phase,x.size).reshape(z.size,x.size)\n", "mag = np.repeat(phi_max_abs,x.size).reshape(z.size,x.size)\n", "\n", "# wave structure\n", "PSI = mag*np.cos( kappa*X + phase )\n", "\n", "phi = charney['e_num'][:]\n", "phi_abs = np.abs(phi)\n", "phi_phase = np.arctan2(phi.imag,phi.real)\n", "\n", "D1,D2 = D_matrices(N+2)\n", "D1,D2 = np.matrix(D1),np.matrix(D2)\n", " \n", "phi_abs_prime = np.array(D1*np.matrix(phi_abs).T)[1:-1]/(2*dz)\n", "phi_abs_dprime = np.array(D2*np.matrix(phi_abs).T)[1:-1]/(dz**2)\n", "phi_phase_prime = np.array(D1*np.matrix(phi_phase).T)[1:-1]/(2*dz)\n", "phi_phase_dprime = np.array(D2*np.matrix(phi_phase).T)[1:-1]/(dz**2)\n", "\n", "mag_prime = np.repeat(phi_abs_prime,x.size).reshape(z.size,x.size)\n", "mag_dprime = np.repeat(phi_abs_dprime,x.size).reshape(z.size,x.size)\n", "\n", "phase_prime = np.repeat(phi_phase_prime,x.size).reshape(z.size,x.size)\n", "phase_dprime = np.repeat(phi_phase_dprime,x.size).reshape(z.size,x.size)\n", "\n", "cost = np.cos( kappa*X + phase)\n", "sint = np.sin( kappa*X + phase)\n", "\n", "PV = (-(kappa**2)*mag + mag_dprime - mag*(phase_prime**2) )*cost \\\n", " - (2.*mag_prime*phase_prime + mag*phase_dprime)*sint\n" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAzQAAAJ/CAYAAACumpZsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XeYVOXZx/HvUpYiIAIiKlWFGwNG0AAGwQYaxV6wF1Di\n", "K/beojEao0GjUYMxFgSNXdQQxQ6KAgKKGgtyC6L0JksRlrYw7x8zu5ydnd2durPD/j7XtZfnPHOe\n", "89yzzjJzz9PyQqEQIiIiIiIiuahWtgMQERERERFJlhIaERERERHJWUpoREREREQkZymhERERERGR\n", "nKWERkREREREcpYSGhERERERyVl1sh1ATWFm+cApwEBgf2BnYCOwEPgUeNbd38twDPsBFwB9gXZA\n", "PWAxMBsYDbzo7qsyGYOIiIiISDrlaR+azDOz7sCzQOdKLh0HDHb3BWluvwHwMDCokkuXAxe6+5h0\n", "ti8iIiIikilKaDLMzA4A3gUaBYq3AksJ95A0i6qyEOjl7ovS1H79SPt9AsUhYA2wFmgF1I6qNtjd\n", "n0pH+yIiIiIimaQ5NBlkZs2BV9mWzGwG7gRaufvu7t4C6AK8Eqi2O/C6maXr/81DlE5mPgR6uvtO\n", 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"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", "\n", "plt.ylabel('$z/H$')\n", "plt.xlabel(r'$x/L_d$')\n", "\n", "plt.savefig('figs/wave-structure_pv_psi_kappa_8_num.eps')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }