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

On Galerkin approximations for the QG equations

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Supplementary material for subsection on the Eady model

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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", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The basic state

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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, 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else:\n", " ax1.plot(U_g[:,i],z,label=str(N[i]),linewidth=2.,color='g')\n", " ax3.plot(Qy_g[:,i],z,label=str(N[i]),linewidth=2.,color='g') \n", "\n", " \n", "ax1.plot(U_exact,z,'--',color='k',linewidth=2.,label='Exact')\n", "ax1.set_xlabel(r'$U/\\Lambda h$')\n", "ax1.set_ylabel(r'$z/h$')\n", "ax1.text(.025,-.05, '(a)', size=30, rotation=0.)\n", "\n", "# inset\n", "def inset_vel(xl,xr,yl,yu,xil,xir,yiu,yil):\n", " inset = plt.axes([xil,xir,yiu,yil], axisbg='.975')\n", " for i in range(N.size):\n", " if i < N.size-2:\n", " plt.plot(U_g[:,i],z,linewidth=2.) \n", " elif i == N.size-2:\n", " plt.plot(U_g[:,i],z,linewidth=2.,color='b')\n", " else:\n", " plt.plot(U_g[:,i],z,linewidth=2.,color='g')\n", " \n", " plt.plot(U_exact,z,'--',color='k',linewidth=2.)\n", " plt.ylim(yl,yu) \n", " plt.xlim(xl,xr)\n", " plt.setp(inset, xticks=[], yticks=[])\n", "\n", "inset_vel(.85,1.0,-.1,0.,.19,.7,.1,.18)\n", "inset_vel(0.,.15,-1.,-.9,.19,.135,.1,.18)\n", " \n", "## PV gradient\n", "ax2.set_xlabel(r'$-U_{zz} \\times \\frac{N_0^2}{f_0^2}\\frac{h}{\\Lambda}$')\n", "ax2.plot(U_exact*0,z,'--',color='k',linewidth=2.,label='Exact')\n", "ax2.legend(loc=4,title= r'$\\mathrm{N}$')\n", "ax2.text(-31,-.05,'(b)', size=30, rotation=0.)\n", "ax2.set_xlim(-35,35)\n", "\n", "## PV gradient N=27\n", "ax3.plot(U_exact*0,z,'--',color='k',linewidth=2.,label='Exact')\n", "ax3.set_xlabel(r'$-U_{zz} \\times \\frac{N_0^2}{f_0^2}\\frac{h}{\\Lambda}$')\n", "ax3.legend(loc=4,title= r'$\\mathrm{N}$')\n", "ax3.text(-89,-.05,'(c)', size=30, rotation=0.)\n", "\n", "## inset\n", "\n", "# mark inset region in bigger axis\n", "ax1.plot([.85,1.],[-.1,-.1],'-',color=\"k\")\n", "ax1.plot([.85,.85],[-.1,0.],'-',color=\"k\")\n", "ax1.plot([.0,.15],[-.9,-.9],'-',color=\"k\")\n", "ax1.plot([.15,.15],[-.9,-1.],'-',color=\"k\")\n", "\n", "ax3.plot([-10.,10.],[-.6,-.6],'-',color=\"k\")\n", "ax3.plot([10.,10.],[-.6,-.4],'-',color=\"k\")\n", "ax3.plot([10.,-10.],[-.4,-.4],'-',color=\"k\")\n", "ax3.plot([-10.,-10.],[-.4,-.6],'-',color=\"k\")\n", "\n", "def inset_pv(xl,xr,yl,yu,xil,xir,yiu,yil):\n", " inset = plt.axes([xil,xir,yiu,yil], axisbg='.975')\n", " for i in range(N.size): \n", " if i == N.size-2:\n", " inset.plot(Qy_g[:,i],z,linewidth=2.,color='b')\n", " elif i == N.size-1:\n", " inset.plot(Qy_g[:,i],z,linewidth=2.,color='g')\n", "\n", " plt.plot(U_exact*0,z,'--',color='k',linewidth=2.)\n", " plt.ylim(yl,yu) \n", " plt.xlim(xl,xr)\n", " plt.setp(inset, xticks=[], yticks=[])\n", " \n", "inset_pv(-3,3,-.6,-.4,.685,.425,.075,.175)\n", "\n", "plt.savefig('figs/U_Qy_exact_galerk.eps')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear stability analysis for Eady problem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Divide and conquer: First we code a couple of simple functions that will be called many times. Most of the functions have similar names/variables to the variables described in Rocha, Young, and Grooms" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numba\n", "\n", "def alpha_n(kappa,n):\n", " ''' the inverse Helmholtz operator in Fourier space '''\n", " return -1./(kappa**2 + (n*pi)**2)\n", "\n", "# nopython=True means an error will be raised\n", "# if fast compilation is not possible.\n", "def sigma12(kappa,Nmax):\n", " ''' evaluate sigma1 and sigma2\n", " sums for used in methods B and C \n", " Nmax is the number of baroclinic modes '''\n", " \n", " i = np.arange(Nmax+1)\n", " c = alpha_n(kappa,i)\n", " sigma1 = c[0] + 2.*c[1:].sum() \n", " sigma2 = c[0] + 2.*(( (-1.)**i[1:] )*c[1:]).sum()\n", " \n", " return sigma1,sigma2\n", "\n", "@numba.jit(nopython=True)\n", "def UGpm(Nmax):\n", " ''' evaluate series for base velocity\n", " at the boundaries \n", " Nmax is the number of baroclinic modes'''\n", "\n", " up,um = .5,.5\n", " for i in range(1,Nmax+1,2):\n", " up += 4./( (i*pi)**2 )\n", " um += -4./( (i*pi)**2 )\n", "\n", " return up,um\n", "\n", "@numba.jit(nopython=True)\n", "def Xi(k,m,n):\n", " ''' the interaction coefficient \n", " Xi for constant stratification '''\n", "\n", " if ((k==0) and (m==n)): x = 1.\n", " elif ((m==0) and (k==n)): x = 1.\n", " elif ((n==0) and (k==m)): x = 1. \n", " elif ((k == m+n) or (k == m-n) or (k == n-m)): \n", " x = sqrt(2)/2.\n", " else: x = 0.\n", "\n", " return x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The linear stability analysis reduces to a simple eigenvalue problem in the form\n", "\n", "\\begin{equation}\n", "\\mathsf{M} \\, \\phi = c \\phi\\, ,\n", "\\end{equation}\n", "where the matrix $\\mathsf{M}$ varies from approximation to approximation. In the Eady problem, the matrix $\\mathsf{M}$ is $(\\mathrm{N}+1)\\times(\\mathrm{N}+1)$, wheres it is $2\\times 2$ in approximations B (exact in this case) and C. Thus, for the Eady problem, we only need to compute the eigenvalues numerically in approximation A. First, we create a function to assemble the matrix $\\mathsf{M}$ given the number of baroclinic modes $\\mathrm{N}$ and the wavenumber $\\kappa$. See Appendix C.\n", "\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from numpy import cos,sin,pi,sqrt\n", "import scipy as sp\n", "try:\n", " np.use_fastnumpy = True\n", "except:\n", " pass\n", " \n", "import scipy.linalg\n", "import line_profiler\n", "\n", "def stability_matrix(kappa,Nmax):\n", " ''' Linear stability matrix for\n", " Eady Problem using method A\n", "\n", " --------------------------------------- \n", " M phi = c phi, where c\n", " are the eigenvalues and phi\n", " the eigenvector \n", " \n", " M is a Nmax+1 by Nmax+1 matrix\n", " \n", " Nmax is the number of baroclinic modes\n", " for the solution\n", " -------------------------------------- '''\n", "\n", "\n", " M1 = np.zeros((Nmax+1,Nmax+1)) \n", " M2 = np.zeros((Nmax+1,Nmax+1))\n", "\n", " # loop over rows \n", " for k in range(Nmax+1):\n", " \n", " # M1 \n", " for m in range(Nmax+1):\n", " for n in range(1,Nmax+1,2):\n", " gamma = -2.*sqrt(2)/( kappa**2 + (m*pi)**2 ) \n", " M1[k,m] += Xi(k,n,m)*gamma\n", "\n", " # M2\n", " for n in range(Nmax+1):\n", " for m in np.append(0,range(1,Nmax+1,2)):\n", " if m == 0:\n", " U = 1/2.\n", " else: \n", " U = 2.*sqrt(2)/( (m*pi)**2 )\n", " M2[k,n] += Xi(k,n,m)*U\n", "\n", " M = M1 + M2\n", "\n", " return M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Testing efficient of the code" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%reload_ext line_profiler" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "*** Profile printout saved to text file u'lp_results.txt'. \n" ] } ], "source": [ "%lprun -s -f stability_matrix -T lp_results.txt stability_matrix(kappa=2.,Nmax=128)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We then create a simple function to perform the stability analysis, using either method A or C. For method A it calls **stability_matrix**. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_analysis(kappa,Nmax,method='C',efunc=False):\n", " '''\n", " Compute the growth rate of the linear\n", " stability analysis for the Eady problem \n", "\n", " -------------------------------------------- \n", " kappa is the array of wavenumbers\n", " Nmax is the number of baroclinic modes\n", " for the solution \n", " ---------------------------------------------\n", "\n", " '''\n", " \n", " if efunc == True and method == 'C':\n", " raise 'eigenfunction works only with method A'\n", " \n", " try:\n", " Nkappa = kappa.size\n", " c = np.zeros(Nkappa,dtype='complex128')\n", " sig = np.zeros(Nkappa)\n", " \n", " if method == 'C':\n", " \n", " for i in range(Nkappa):\n", " sig1,sig2 = sigma12(kappa[i],Nmax)\n", " up,um = UGpm(Nmax)\n", " uh = (up+um)/2.\n", " D = uh**2 - up*um +\\\n", " sig1*(up-um) + sig1**2 - sig2**2\n", " if D<0: cm = uh + 1j*np.sqrt(-D)\n", " else: cm = uh + np.sqrt(D) \n", " c[i],sig[i] = cm,kappa[i]*cm.imag\n", "\n", " elif method == 'A':\n", " for i in range(Nkappa):\n", " M = stability_matrix(kappa[i],Nmax)\n", " cm,em = np.linalg.eig(M)\n", " imax = cm.imag.argmax()\n", " c[i],sig[i] = cm[imax],kappa[i]*cm.imag[imax]\n", " if efunc:\n", " efun[:,i] = em[:,imax]\n", " except AttributeError:\n", " \n", " if method == 'C':\n", " sig1,sig2 = sigma12(kappa,Nmax)\n", " up,um = UGpm(Nmax)\n", " uh = (up+um)/2.\n", "\n", " D = uh**2 - up*um +\\\n", " sig1*(up-um) + sig1**2 - sig2**2\n", "\n", " if D<0: cm = uh + 1j*np.sqrt(-D)\n", " else: cm = uh + np.sqrt(D)\n", "\n", " c,sig = cm,kappa*cm.imag\n", "\n", " elif method == 'A': \n", " M = stability_matrix(kappa,Nmax)\n", " cm,em = np.linalg.eig(M)\n", " imax = cm.imag.argmax()\n", " c,sig = cm[imax],kappa*cm.imag[imax]\n", " if efunc:\n", " efun = em[:,imax]\n", "\n", " if efunc: return c,efun,sig\n", " else : return c,sig" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now loop compute the eigenvalue for different wanumber $\\kappa$ over an array of number of baroclinic modes $\\mathrm{N}$ to study the converge of different approximations. (It may take a while to run for a large number of baroclinic modes using approximation A.)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = np.linspace(0.001,10.,100)\n", "#kappa = 8\n", "\n", "# set single number or array of number of baroclinic modes\n", "Nmodes = np.arange(0,25)\n", "#Nmodes = np.arange(0,10)\n", "Nmodes = np.append(Nmodes,2**np.arange(5,8))\n", "#Nmodes = np.array([16,32,64])\n", "\n", "c_galerkA,sig_galerkA = np.zeros((Nmodes.size,kappa.size),\\\n", " dtype='complex128'),np.zeros((Nmodes.size,kappa.size))\n", "\n", "c_galerkC,sig_galerkC = np.zeros((Nmodes.size,kappa.size),\\\n", " dtype='complex128'),np.zeros((Nmodes.size,kappa.size))\n", "\n", "# solve the linear problem\n", "for i in range(Nmodes.size):\n", " \n", " c_galerkC[i,:],sig_galerkC[i,:] = stability_analysis(kappa,\\\n", " Nmodes[i],method='C')\n", " \n", " #if Nmodes = 0 , A is stable\n", " if Nmodes[i] == 0:\n", " c_galerkA[i,:],sig_galerkA[i,:] = 0.,0.5\n", " else: \n", " c_galerkA[i,:],sig_galerkA[i,:] = stability_analysis(kappa,\\\n", " Nmodes[i],method='A',efunc=False)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [], "source": [ "np.savez('outputs/eady_A.npz',c=c_galerkA,sig=sig_galerkA,\\\n", " kappa=kappa,Nmodes=Nmodes)\n", "np.savez('outputs/eady_C.npz',c=c_galerkC,sig=sig_galerkC,\\\n", " kappa=kappa,Nmodes=Nmodes)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compute and save single mode with eigenfunction to plot wave structure" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = 8\n", "Nmodes = 64\n", "c_galerkA,efunc_galerk_A,sig_galerkA = stability_analysis(kappa,\\\n", " Nmodes,method='A',efunc=True)\n", "np.savez('outputs/eady_A_k8_64_efunc.npz',c=c_galerkA,sig=sig_galerkA,\\\n", " efunc=efunc_galerk_A,kappa=kappa,Nmodes=Nmodes)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Method B is exact." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = np.linspace(0.001,10.,100)\n", "kappa_2 = kappa/2. + 0j\n", "c_exact = .5 + np.sqrt( (kappa_2 - np.tanh(kappa_2))\\\n", " *(kappa_2 - 1./np.tanh(kappa_2)) )/kappa\n", "sig_exact = kappa*c_exact.imag\n", "\n", "np.savez('outputs/eady_exact.npz',c=c_exact,sig=sig_exact,\\\n", " kappa=kappa)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we plot the results to study the convergence of different approximations. " ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#import seaborn as sns\n", "#sns.set(style=\"darkgrid\")\n", "#sns.set_context(\"paper\", font_scale=5., rc={\"lines.linewidth\": 1.5})\n", "#sns.set_style(\"darkgrid\",{'grid_color':.95})" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def labels(field='sig',fs=50):\n", " ''' set labels in figures '''\n", " plt.xlabel(r'$\\kappa \\times\\, L_d$',fontsize=fs)\n", " if field == 'sig': \n", " plt.ylabel(r'$\\sigma \\times \\, \\frac{\\Lambda H}{L}$',fontsize=fs)\n", " elif field == 'ps':\n", " plt.ylabel(r'$c_r / \\Lambda H$',fontsize=fs)\n", "\n", "def plt_exact(field='sig'):\n", " ''' plot the exact solution '''\n", " if field == 'sig':\n", " plt.plot(eady_exact['kappa'],eady_exact['sig'],color=\"k\",\\\n", " linewidth=4.,label='B, Exact')\n", " elif field == 'ps':\n", " plt.plot(eady_exact['kappa'],eady_exact['c'].real,color=\"k\",\\\n", " linewidth=4.,label='B, Exact')" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "path = 'outputs/'\n", "eadyA = np.load(path+'eady_A.npz')\n", "eadyC = np.load(path+'eady_C.npz')\n", "eady_exact = np.load(path+'eady_exact.npz')\n", "Nmodes = eadyA['Nmodes']" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "lw,aph = 4.,.75\n", "color1,color2 = \"#1e90ff\",\"#ff6347\"" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA/MAAALsCAYAAACvC+dZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xm4JFV9//H3lxmWYZV9kWFzQVlEREFABRcEF1xJXEBF\n", "xZ8aI4kkMZofKsYtLlGjJlHiyk9wV9yDSxwUBEVQkUUMqywKCDrAAAIz398fVZepW3OX7nuru7r6\n", 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plt.plot(eadyA['kappa'],eadyA['sig'][i,:],\\\n", " color=color1,linewidth=lw,label='A')\n", " plt.plot(eadyC['kappa'],eadyC['sig'][i,:],\\\n", " color=color2,linewidth=lw,label='C')\n", "\n", " labels() \n", " plt.yticks(np.arange(0.,.5,.1))\n", " plt.tick_params(axis='both', which='major', labelsize=40) \n", " plt.legend(loc=1, title = r'$\\mathrm{N} = $' + str(Nmodes[i]))\n", " plt.ylim(0.,0.4)\n", " plt.xlim(0,10)\n", " plt.text(8.65, .02, \"Eady Problem\", size=30, rotation=0.,\\\n", " ha=\"center\", va=\"center\",\\\n", " bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "\n", " figtit = 'figs/eady_sig_galerk_ABC_'+str(Nmodes[i])+'.eps'\n", " plt.savefig(figtit,format='eps') \n", " \n", " # close some of the figures\n", " if i%2:\n", " plt.close()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may also be interested in computing the relative error, e.g. at the most unstable mode" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def absolute_error(exact,approx):\n", " ''' compute relative error '''\n", " return np.abs(exact-approx)\n", "#relative_error(exact,approx)\n", "def find_nearest(x, x0):\n", " \"find closest to x0 in array value\"\n", " idx = np.abs(x - x0).argmin()\n", " return x[idx],idx" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# the fasted growing mode\n", "imax = eady_exact['sig'].argmax()\n", "\n", "# calculate error at some wanumbers\n", "kappas = np.array([.5,kappa[imax],4.,8.])\n", "inds = np.zeros(kappas.size)\n", "\n", "for i in range(kappas.size):\n", " kappas[i],inds[i] = find_nearest(kappa, kappas[i])" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [], "source": [ "errorA = np.zeros((Nmodes.size,kappas.size))\n", "errorC = np.zeros((Nmodes.size,kappas.size))\n", "\n", "for n in range(Nmodes.size):\n", " for k in range(inds.size):\n", " errorA[n,k] = absolute_error(eady_exact['sig'][inds[k]],\\\n", " eadyA['sig'][n,inds[k]]) \n", " errorC[n,k] = absolute_error(eady_exact['sig'][inds[k]],\\\n", " eadyC['sig'][n,inds[k]])" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA4oAAAKYCAYAAAA15AbcAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XmcXFWd///XJwvZ94UtUAECgbApCCqgLCK0yOo2zjgq\n", "SDuOfr8z9zs6+htnc5mfzuZXp5zFcaYdUEdHUWSRpVnECCI7siWBECA3JCGks5IFkpCc7x/nVO6t\n", "qlud6u6qru39fDzq0d23bt06Sbo79a5zzudjzjlERERERERECkY0egAiIiIiIiLSXBQURURERERE\n", "pIiCooiIiIiIiBRRUBQREREREZEiCopNzszGmNlfmdnTZvaama0xsx+a2bxGj01ERERERNqTqepp\n", "8zKzMcCdwOnAauAe4DDgVGArcJZz7tHGjVBERERERNqRZhSb2+fxIfE3wFHOud91zr0F+AwwEfiB\n", "menfUEREREREakozik3KzEYDa4HJwEnOucdL7n8MOAF4j3Pu+gYMUURERERE2pRmo5rX6cAU4LnS\n", "kBj8NHy8ePiGJCIiIiIinUBBsXm9IXystAexcPyEYRiLiIiIiIh0EAXF5nVo+Liywv2rwsfcMIxF\n", 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rate)\n", "plt.loglog(Nmodes,errorA[:,1],color=color1,label='A',linewidth=lw)\n", "plt.loglog(Nmodes,errorC[:,1],color=color2,label='C',linewidth=lw)\n", "# dashed line error at k ~8 (max growth rate)\n", "plt.loglog(Nmodes,errorA[:,-1],'--',color=color1,linewidth=lw)\n", "plt.loglog(Nmodes,errorC[:,-1],'--',color=color2,linewidth=lw)\n", "plt.legend(loc=3)\n", "plt.yticks(np.array([1.e-10,1.e-8,1.e-6,1.e-4,1.e-2,1.e0]))\n", "plt.tick_params(axis='both', which='major', labelsize=30) \n", "plt.text(150,4.e-9,r'$\\mathrm{N}^{-4}$')\n", "plt.text(150,5.e-5,r'$\\mathrm{N}^{-2}$')\n", "plt.text(150,1.e-2,r'$\\mathrm{N}^{-1}$')\n", "plt.xlabel(r\"$\\mathrm{N}$\",fontsize=30)\n", "plt.ylabel(r\"Absolute error\",fontsize=30)\n", "plt.xlim(0,350)\n", "plt.text(100, .5e-10, r\"Eady Problem\", size=25, rotation=0.,\\\n", " ha=\"center\", va=\"center\",\\\n", " bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "plt.savefig('figs/Error_eady.eps',format='eps')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plot eady growth rate for approximation A with extended $\\kappa$\n", "see the bogus critical high-wavenumber instabilities in approximation A" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "eady30 = np.load('outputs/eady_A_30.npz')\n", "plt.figure(figsize=(10,8))\n", "plt.plot(eady30['kappa'],eady30['sig'][-1,:])\n", "plt.xlabel(r'$\\kappa$')\n", "plt.ylabel(r'$k\\,c_i$')\n", "plt.savefig('figs/eady_stability_A_ext_kappa')" ] }, { "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 }