{ "cells": [ { "cell_type": "code", "execution_count": 694, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import division\n", "\n", "import numpy as np\n", "from numpy import pi, sqrt,cos,sin,sinc\n", "\n", "import matplotlib.pyplot as plt\n", "plt.rcParams.update({'font.size': 25, 'legend.handlelength' : 1.25})\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 695, "metadata": { "collapsed": false }, "outputs": [], "source": [ "data_path = 'outputs/beta-eady_C_kappa8_256.npz'\n", "beta_eady = np.load(data_path)" ] }, { "cell_type": "code", "execution_count": 696, "metadata": { "collapsed": true }, "outputs": [], "source": [ "kappa = beta_eady['kappa']\n", "qn = beta_eady['efunc'][1:-1] # modal q\n", "vtht = beta_eady['efunc'][0] # theta +\n", "vthb = beta_eady['efunc'][-1] # theta -\n", "N = beta_eady['Nmodes']\n", "cr = beta_eady['c'].real" ] }, { "cell_type": "code", "execution_count": 697, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# vertical coordinate\n", "z = np.linspace(-1.,0.,N+3)\n", "dz = z[2]-z[1]\n", "\n", "# horizontal coordinate\n", "x = np.linspace(0,np.pi,150)\n", "\n", "# grid\n", "X,Z = np.meshgrid(x,z)" ] }, { "cell_type": "code", "execution_count": 710, "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": 711, "metadata": { "collapsed": false }, "outputs": [], "source": [ "for i in range(N+1):\n", " if i == 0 : \n", " q = qn[0]*np.ones(z.size)\n", " psi = (qn[0] - vtht + vthb)*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] - sqrt(2)*( vtht - ((-1)**i * vthb) ) )\\\n", " *cos(i*pi*z)*alpha_n(i,kappa)" ] }, { "cell_type": "code", "execution_count": 712, "metadata": { "collapsed": true }, "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)\n", "\n", "vthtabs = np.abs(vtht)\n", "vthbabs = np.abs(vthb)\n", "vthtphase = np.arctan2(vtht.imag,vtht.real)\n", "vthbphase = np.arctan2(vthb.imag,vthb.real)" ] }, { "cell_type": "code", "execution_count": 713, "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": 714, "metadata": { "collapsed": false }, "outputs": [], "source": [ "PV = Qabs*np.cos(kappa*X + Qphase)\n", "PSI = Psiabs*np.cos(kappa*X + Psiphase)" ] }, { "cell_type": "code", "execution_count": 715, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": [ 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"KUOUfEEQBEEQBEF4yhAlXxAEQRAEQRCeMv4P3nC2aHZWk3wAAAAASUVORK5CYII=\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(12,8))\n", "plt.contour(X,Z,PSI,20,colors='k') \n", "plt.contourf(X,Z,PV,np.linspace(-6.,6.,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(1.7, -.1, r\"Eady Problem, $\\kappa = 2$, Approximation C\", size=25, rotation=0.,\\\n", "# ha=\"center\", va=\"center\",\\\n", "# bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "#plt.ylim(-.25,0)\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_2_C_N64.eps',bbox_inches = 'tight')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Computing the buoyancy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1st method finite differences" ] }, { "cell_type": "code", "execution_count": 705, "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": "code", "execution_count": 706, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# vertical coordinate\n", "z_fd = np.linspace(-1.,0.,N+3)\n", "\n", "# horizontal coordinate\n", "x= np.linspace(0,np.pi,150)\n", "\n", "# grid\n", "X,Z_fd= np.meshgrid(x,z_fd)\n", "\n", "D1,_ = D_matrices(z_fd.size)\n", "D1 = -D1/(2*dz) # correct order for this grid" ] }, { "cell_type": "code", "execution_count": 668, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Bfd = np.empty(PSI.shape)\n", "for i in range(x.size):\n", " Bfd[:,i] = np.dot(D1,PSI[:,i]) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Append boundary values" ] }, { "cell_type": "code", "execution_count": 707, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Bfd[0,:] = vthbabs*cos(kappa*x + vthbphase)\n", "Bfd[-1,:] = vthtabs*cos(kappa*x + vthtphase)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 708, "metadata": { "collapsed": false }, "outputs": [], "source": [ "ct = np.linspace(-.7,.7,20)" ] }, { "cell_type": "code", "execution_count": 709, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ 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"code", "execution_count": 672, "metadata": { "collapsed": false }, "outputs": [], "source": [ "z2 = z_fd[1:-1]\n", "\n", "for i in range(N+1):\n", " if i == 0 : \n", " b = np.zeros(z2.size) + 0.j\n", " else: \n", " b+= -i*pi*sqrt(2)*(qn[i] - sqrt(2)*vtht + sqrt(2)*((-1)**i * vthb))\\\n", " *sin(i*pi*z2)*alpha_n(i,kappa)" ] }, { "cell_type": "code", "execution_count": 673, "metadata": { "collapsed": false }, "outputs": [], "source": [ "b = np.append(vthb,b)\n", "b = np.append(b,vtht)\n", "z2 = np.append(-1,z2)\n", "z2 = np.append(z2,0)\n", "babs = np.abs(b)\n", "bphase = np.arctan2(b.imag,b.real)\n", "\n", "# grid\n", "X2,Z2 = np.meshgrid(x,z2)" ] }, { "cell_type": "code", "execution_count": 674, "metadata": { "collapsed": false }, "outputs": [], "source": [ "babs = babs.repeat(x.size).reshape(z.size,x.size)\n", "bphase = bphase.repeat(x.size).reshape(z.size,x.size)" ] }, { "cell_type": "code", "execution_count": 688, "metadata": { "collapsed": false }, "outputs": [], "source": [ "B2 = babs*np.cos(kappa*X + bphase)" ] }, { "cell_type": "code", "execution_count": 689, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAxUAAAIeCAYAAADNrSJ9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XeYbFWVsPF3CUhWUTAhygUEBCQqyQCImAARxIw4KmAC\n", "MXyGMYA566iYRh0xKxJUBBEJEkWyBBEQEARFQATJOsD6/ti7te253be7z6k6Fd7f89ynqrpO7b1u\n", "d1Wds3aMzESSJEmS5us+XQcgSZIkabiZVEiSJElqxKRCkiRJUiMmFZIkSZIaMamQJEmS1IhJhSRJ\n", "kqRGTCqqiHhORPwsIq6LiLsi4oqI+HxErNZC2ctHxH4RcX5E3BoRN0fE6RGxb0Qs3kb8kiRJUldi\n", "3PepiIgAvgq8vP5o4hcS9fY24AWZedQ8y18AHAssmKb8s4HtMvPm+ZQvSZIkdc2eCvhP/pVQHARs\n", "ADwYeDbwe2A54KCIePRcC46I+wI/oSQUdwB7A48AVgPeA9wLbFLrlSRJkobSWPdURMRDgMuBZYAj\n", "MvPZU55/FHAhsCxwSGY+f47l7w18ltI78fzMPHTK828GPl4f7pCZP53Xf0SSJEnq0Lj3VOxOSSgS\n", "eOfUJzPzKuDL9eEuEbHSHMt/Tb09d2pCUX0GuH7KsZIkSdJQGfekYsd6+7vMvGCaYw6pt/cBdpht\n", "wXWC92OmlPFvMvNu4Mf14VMiYunZli9JkiQNinFPKjaqt6fPcMw5lLkPABvPoeyJY3MR5U88txSw\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", "plt.ylabel('$z/H$')\n", "plt.xlabel(r'$x/L_d$')\n", "plt.savefig('figs/wave-structure_b_psi_kappa_2_C_N128.eps',bbox_inches = 'tight')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Trying to remove gibbs oscillations: Laczos $\\sigma$ factor " ] }, { "cell_type": "code", "execution_count": 677, "metadata": { "collapsed": false }, "outputs": [], "source": [ "z3 = z_fd[1:-1]\n", "\n", "for i in range(N+1):\n", " if i == 0 : \n", " b3 = np.zeros(z3.size) + 0.j\n", " else: \n", " b3+= -i*pi*sqrt(2)*sinc(i/((N+1)))*(qn[i] - sqrt(2)*vtht + sqrt(2)*((-1)**i * vthb))\\\n", " *sin(i*pi*z3)*alpha_n(i,kappa)" ] }, { "cell_type": "code", "execution_count": 678, "metadata": { "collapsed": true }, "outputs": [], "source": [ "b3 = np.append(vthb,b3)\n", "b3 = np.append(b3,vtht)\n", "z3 = np.append(-1,z3)\n", "z3 = np.append(z3,0)\n", "babs = np.abs(b3)\n", "bphase = np.arctan2(b3.imag,b3.real)\n", "\n", "# grid\n", "X3,Z3 = np.meshgrid(x,z3)" ] }, { "cell_type": "code", "execution_count": 679, "metadata": { "collapsed": false }, "outputs": [], "source": [ "babs = babs.repeat(x.size).reshape(z3.size,x.size)\n", "bphase= bphase.repeat(x.size).reshape(z3.size,x.size)" ] }, { "cell_type": "code", "execution_count": 680, "metadata": { "collapsed": false }, "outputs": [], "source": [ "B3 = babs*np.cos(kappa*X3 + bphase)" ] }, { "cell_type": "code", "execution_count": 681, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAxUAAAIeCAYAAADNrSJ9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8VPW9//H3B1AIiyQhIewhrAIiq6C4K7T1Vlur3mpv\n", "q2ZCCKtbrV287rVatXUDlTWJS2u9VW9btbau4IqKCMheCGEnhEAgbMry/f2R4ZrmRwLJyZwzy+vZ\n", "Rx4zkznz/XyINMx7zvl+v+acEwAAAADUV6OgGwAAAAAQ2wgVAAAAADwhVAAAAADwhFABAAAAwBNC\n", 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"D1[-1,-1],D1[-1,-2] = -1./dz,1/dz" ] }, { "cell_type": "code", "execution_count": 417, "metadata": { "collapsed": false }, "outputs": [], "source": [ "psiabs = np.abs(psi)\n", "psiphase = np.arctan2(psi.imag,psi.real)\n", "psiabs_prime_fd = np.array( D1*np.matrix(psiabs).T )\n", "psiphase_prime_fd = np.array( D1*np.matrix(psiphase).T )" ] }, { "cell_type": "code", "execution_count": 418, "metadata": { "collapsed": true }, "outputs": [], "source": [ "vthtabs = np.abs(vtht)\n", "vthbabs = np.abs(vthb)\n", "vthtphase = np.arctan2(vtht.imag,vtht.real)\n", "vthbphase = np.arctan2(vthb.imag,vthb.real)" ] }, { "cell_type": "code", "execution_count": 419, "metadata": { "collapsed": false }, "outputs": [], "source": [ "vthtabs = vthtabs.repeat(x.size).reshape(1,x.size)\n", "vthbabs = vthbabs.repeat(x.size).reshape(1,x.size)\n", "\n", "vthtphase = vthtphase.repeat(x.size).reshape(1,x.size)\n", "vthbphase = vthbphase.repeat(x.size).reshape(1,x.size)\n" ] }, { "cell_type": "code", "execution_count": 420, "metadata": { "collapsed": false }, "outputs": [], "source": [ "psiabs = psiabs.repeat(x.size).reshape(z.size,x.size)\n", "psiabs_prime_fd = psiabs_prime_fd.repeat(x.size).reshape(z.size,x.size)\n", "psiphase_prime_fd = psiphase_prime_fd.repeat(x.size).reshape(z.size,x.size)" ] }, { "cell_type": "code", "execution_count": 421, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Bfd = psiabs_prime_fd*cos(kappa*X + psiphase) - psiabs*psiphase_prime_fd*sin(kappa*X + psiphase)" ] }, { "cell_type": "code", "execution_count": 422, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Bfd = np.concatenate((vthbabs*cos(kappa*x+vthbphase),Bfd),axis=0)" ] }, { "cell_type": "code", "execution_count": 424, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# vertical coordinate\n", "dz = .01 # vertical resolution\n", "z = np.linspace(-1.+dz,-dz,100)\n", "\n", "# horizontal coordinate\n", "x = np.linspace(0,np.pi,100)\n", "\n", "z = np.append(-1.,z)\n", "\n", "# grid\n", "X,Z = np.meshgrid(x,z)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 368, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAxUAAAIeCAYAAADNrSJ9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXeYFEXawH/vwLJLWpYMJpQkAmYEDCgImE5BBQPgCfop\n", "pxcU9fQMGDiMd+fdqXd6giIYMKGoKJhIKkkUMQcEEUmCkjPsvt8f1TPM7s5smNDVM1u/5+lnZre7\n", "q97u6a6qt+oNoqo4HA6Hw+FwOBwOR6KEbAvgcDgcDofD4XA4MhunVDgcDofD4XA4HI6kcEqFw+Fw\n", "OBwOh8PhSAqnVDgcDofD4XA4HI6kcEqFw+FwOBwOh8PhSAqnVDgcDofD4XA4HI6kcEqFh4icLSJv\n", "isjPIrJDRJaIyH9FpGUKyq4rIreJyGcisllENojIPBG5WkSqp0J+h8PhcDgcDofDFlLV81SIiACP\n", "AZd4/wrfEPE+twAXqOqUBMs/CHgXOChO+R8DvVV1QyLlOxwOh8PhcDgctnErFXATexWK54HDgSZA\n", "H+AHoA7wvIi0qWzBIlIDmIRRKLYBfwT2A1oCdwBFwNFevQ6Hw+FwOBwOR0ZSpVcqRKQpsBioBbyu\n", 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-.1, r\"Eady Problem, $\\kappa = 2$, Approximation C\", size=25, rotation=0.,\\\n", " ha=\"center\", va=\"center\",\\\n", " bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "#plt.ylim(-.25,0)\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_b_psi_kappa_2_C_N128_fd.eps',bbox_inches = 'tight')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "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 }