{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# On Galerkin approximations for the QG equations\n", "## Supplementary material for subsection on the $\\beta-$Eady model \n", "

\n", "### Cesar B. Rocha*, 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, " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The $\\beta-$Eady problem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Adding a non-zero background planetary vorticity gradient ($\\beta$) significantly changes the linear stability analysis for approximations B and C. The interior equations are no longer trivially satisfied, so we have to solve $(\\mathrm{N}+3)\\times(\\mathrm{N}+3)$ eigenvalue problems. Approximation A still leads to a $(\\mathrm{N}+1)\\times(\\mathrm{N}+1)$ eigenvalue problem. \n", "\n", "This notebook builds previous code developed for the Eady problem. In fact, we will be able to recover the Eady problem when $\\beta = 0$." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import division\n", "import numpy as np\n", "import scipy as sp\n", "import scipy.linalg \n", "\n", "try:\n", " import mkl\n", " np.use_fastnumpy = True\n", "except ImportError:\n", " pass\n", "\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "plt.rcParams.update({'font.size': 30, 'legend.handlelength' : 1.25})" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numba\n", "from numpy import pi,cos,sqrt,sinh,cosh\n", "\n", "def cn(kappa,n):\n", " ''' n-th mode non-dimensional\n", " Rossby wave phase speed '''\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 = cn(kappa,i)\n", " assert i[-1] == Nmax, 'Error in forming sums'\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 stability matrix" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_matrix(kappa,Nmax,method='A',beta=1.,nu=0.):\n", " ''' Linear stability matrix for\n", " Eady Problem using method A\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", " if method == 'A':\n", "\n", " # n+1 by n+1 problem\n", " M1 = np.zeros((Nmax+1,Nmax+1)) \n", " M2 = np.zeros((Nmax+1,Nmax+1))\n", " D = 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", " # beta term and horizontal diffusivity\n", " D[k,k] = beta*cn(kappa,k) + 1.e-3*(kappa**2)\n", "\n", " M = M1 + M2 + D\n", "\n", " elif method == 'B':\n", "\n", " # n+3 by n+3 problem\n", " M2 = np.zeros((Nmax+3,Nmax+3)) + 0j\n", " D = np.zeros((Nmax+3,Nmax+3)) + 0j\n", " S = np.zeros((Nmax+3,Nmax+3)) + 0j\n", " \n", " cosech_k = 1./sinh(kappa)\n", " coth_k = cosh(kappa)/sinh(kappa)\n", "\n", " taup,taum = coth_k/kappa, cosech_k/kappa\n", " sigp,sigm = -cosech_k/kappa, -coth_k/kappa\n", "\n", " sig1,sig2 = sigma12(kappa,Nmax)\n", " up,um = UGpm(Nmax)\n", "\n", " # loop over rows \n", " for kk in range(0,Nmax+3):\n", "\n", " if kk == 0:\n", " \n", " S[0,0] += 1. - taup\n", " S[0,-1] += - sigp\n", "\n", " for mm in range(1,Nmax+2):\n", " m = mm -1\n", " if m == 0:\n", " S[0,mm] = -cn(kappa,m)\n", " else:\n", " S[0,mm] = -sqrt(2)*cn(kappa,m)\n", "\n", " elif kk == Nmax+2:\n", " \n", " S[-1,0] += - taum\n", " S[-1,-1] += 0. - sigm\n", " \n", " for mm in range(1,Nmax+2):\n", " m = mm -1\n", " if m == 0:\n", " S[-1,mm] = -cn(kappa,m)\n", " else:\n", " S[-1,mm] = -sqrt(2)*((-1)**m)*cn(kappa,m)\n", "\n", " else:\n", "\n", " k = kk-1 # use k to keep same indexing as in the Eady\n", "\n", " # M2\n", " for ss in range(1,Nmax+2):\n", " \n", " s = ss -1\n", "\n", " if k == s:\n", " M2[kk,ss] += 0.5\n", "\n", " elif (k == 0) and (s%2):\n", " M2[kk,ss] += 2*sqrt(2)/( (s*pi)**2 )\n", " elif (s == 0) and (k%2):\n", " M2[kk,ss] += 2*sqrt(2)/( (k*pi)**2 )\n", " elif (k+s)%2:\n", " M2[kk,ss] += 4.*(k**2 + s**2) / ( ((k**2 - s**2)*pi)**2 )\n", "\n", "\n", " # beta term and horizontal diffusivity\n", " D[kk,kk] = beta*cn(kappa,k) - 1j*nu*(kappa)\n", " if k == 0:\n", " D[kk,0] = -beta*cn(kappa,k)\n", " D[kk,-1] = beta*cn(kappa,k)\n", " else:\n", " D[kk,0] = -sqrt(2)*beta*cn(kappa,k)\n", " D[kk,-1] = sqrt(2)*beta*( (-1)**k )*cn(kappa,k) \n", "\n", "\n", " M = S + D + M2\n", "\n", " elif method == 'C':\n", "\n", " # n+3 by n+3 problem\n", " M1 = np.zeros((Nmax+3,Nmax+3)) + 0j \n", " M2 = np.zeros((Nmax+3,Nmax+3)) + 0j\n", " D = np.zeros((Nmax+3,Nmax+3)) + 0j\n", " S = np.zeros((Nmax+3,Nmax+3)) + 0j\n", "\n", " sig1,sig2 = sigma12(kappa,Nmax)\n", " up,um = UGpm(Nmax)\n", "\n", " # loop over rows \n", " for kk in range(0,Nmax+3):\n", "\n", " if kk == 0:\n", " S[0,0] = up + sig1\n", " S[0,-1] = -sig2\n", " for mm in range(1,Nmax+2):\n", " m = mm -1\n", " if m == 0:\n", " S[0,mm] = -cn(kappa,m)\n", " else:\n", " S[0,mm] = -sqrt(2)*cn(kappa,m)\n", "\n", " elif kk == Nmax+2:\n", " S[-1,-1] = um - sig1\n", " S[-1,0] = sig2\n", " for mm in range(1,Nmax+2):\n", " m = mm -1\n", " if m == 0:\n", " S[-1,mm] = -cn(kappa,m)\n", " else:\n", " S[-1,mm] = -sqrt(2)*((-1)**m)*cn(kappa,m)\n", "\n", " else:\n", "\n", " k = kk-1 # use k to keep same indexing as in the Eady\n", "\n", " # M2\n", " for nn in range(1,Nmax+2):\n", " n=nn-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[kk,nn] += Xi(k,n,m)*U\n", "\n", "\n", " # beta term and horizontal diffusivity\n", " D[kk,kk] = beta*cn(kappa,k) - 1j*nu*(kappa)\n", " if k == 0:\n", " D[kk,0] = -beta*cn(kappa,k)\n", " D[kk,-1] = beta*cn(kappa,k)\n", " else:\n", " D[kk,0] = -sqrt(2)*beta*cn(kappa,k)\n", " D[kk,-1] = sqrt(2)*beta*( (-1)**k )*cn(kappa,k)\n", "\n", " M = S + M2 + D\n", "\n", " return M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The stability analysis for approximations A, B, C" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_analysis_galerk(kappa,Nmax,method='A',beta=1.,nu=0.,efunc=False):\n", " '''\n", " Compute the growth rate of the linear\n", " stability problem given an array of \n", " wavenumbers kappa\n", "\n", " Nmax is the number of baroclinic modes\n", " for the solution\n", "\n", "\n", " '''\n", "\n", " try:\n", " Nkappa = kappa.size\n", " c = np.zeros(Nkappa,dtype='complex128')\n", " sig = np.zeros(Nkappa)\n", " if efunc:\n", " efun = np.zeros((Nmax+2,Nkappa),dtype='complex128') \n", "\n", " for i in range(Nkappa):\n", " M = stability_matrix(kappa[i],Nmax,method,beta=beta,nu=nu)\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", "\n", " except AttributeError:\n", "\n", " M = stability_matrix(kappa,Nmax,method,beta=beta,nu=nu)\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": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "ename": "NameError", "evalue": "name 'np' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# settings\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mkappa\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0.001\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m30.\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m200\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mNmodes\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m25\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mNmodes\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mNmodes\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined" ] } ], "source": [ "# settings\n", "kappa = np.linspace(0.001,30.,200)\n", "Nmodes = np.arange(0,25)\n", "Nmodes = np.append(Nmodes,2**np.arange(5,8))\n", "\n", "# assemble arrays\n", "c_galerkA,sig_galerkA = np.zeros((Nmodes.size,kappa.size),\\\n", " dtype='complex128'),np.zeros((Nmodes.size,kappa.size))\n", "c_galerkB,sig_galerkB = np.zeros((Nmodes.size,kappa.size),\\\n", " dtype='complex128'),np.zeros((Nmodes.size,kappa.size))\n", "c_galerkC,sig_galerkC = np.zeros((Nmodes.size,kappa.size),\\\n", " dtype='complex128'),np.zeros((Nmodes.size,kappa.size))\n", "\n", "\n", "# solve the linear problem\n", "for i in range(Nmodes.size):\n", " c_galerkA[i,:],sig_galerkA[i,:] = stability_analysis_galerk(kappa,\\\n", " Nmodes[i],method='A',beta=1.,nu=0.,efunc=False)\n", " c_galerkB[i,:],sig_galerkB[i,:] = stability_analysis_galerk(kappa,\\\n", " Nmodes[i],method='B',beta=1.,nu=0.,efunc=False) \n", " c_galerkC[i,:],sig_galerkC[i,:] = stability_analysis_galerk(kappa,\\\n", " Nmodes[i],method='C',beta=1.,nu=0.,efunc=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It's useful to save the results" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "np.savez('outputs/beta-eady_A.npz',c=c_galerkA,sig=sig_galerkA,\\\n", " kappa=kappa,Nmodes=Nmodes)\n", "np.savez('outputs/beta-eady_B.npz',c=c_galerkB,sig=sig_galerkB,\\\n", " kappa=kappa,Nmodes=Nmodes)\n", "np.savez('outputs/beta-eady_C.npz',c=c_galerkC,sig=sig_galerkC,\\\n", " kappa=kappa,Nmodes=Nmodes)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The stability analysis using finite-differences approximation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It's convenient to create simple functions to compute differences matrices" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def D2_matrix(N):\n", " ''' create an N x N second order differences matrix '''\n", " D2 = np.zeros((N,N))\n", " for i in range(N):\n", " D2[i,i] = -2.\n", " if i0: D2[i,i-1] = 1.\n", " \n", " assert np.array_equal(D2-D2.T,np.zeros((N,N))), 'Matrix not symmetric'\n", " \n", " return D2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It's easier to formulate the problem in terms of streamfunction $\\phi$. A second order finite-differences (using ghost points at the boundaries) leads to a generalized eigenvalue problem\n", "\n", "\\begin{equation}\n", "\\mathsf{A}\\,\\mathsf{\\phi} = c\\, \\mathsf{B}\\,\\mathsf{\\phi}\n", "\\end{equation}\n", "\n", "We now assemble the matrices $\\mathsf{A}$ and $\\mathsf{B}$." ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def CharneyMatrix(kappa,N,beta=1.0,nu=.0):\n", " ''' Creates matrices for Charney linear problem\n", " using second-order finite differences\n", " scheme \n", "\n", " - kappa: wavenumber \n", " - N: number of vertical levels '''\n", "\n", " dz = 1./N # vertical resolution\n", " z = np.arange(-dz/2,-1.-dz/2.,-dz) # level array\n", " u = 1. + z\n", "\n", " Ut,Ub = 1.,0. # top and bottom vel.\n", " \n", " #\n", " # first and second elements of the eigenvector\n", " # are ghost points\n", "\n", " D2 = D2_matrix(N+2)/(dz**2)\n", " A = np.zeros((N+2,N+2))\n", " B = np.zeros((N+2,N+2))\n", " \n", " # assemble matrices\n", " for i in range(1,N+1):\n", " A[i,:] = u[i-1]*D2[i,:]\n", " A[i,i] += beta - u[i-1]*(kappa**2)\n", "\n", " B[i,:] = D2[i,:]\n", " B[i,i] -= kappa**2\n", "\n", " # add submesoscale horizontal diffusivity (nu)\n", " A[1:N+1,1:N+1] += D2[1:N+1,1:N+1]*nu*(kappa**2) - nu*(kappa**4)*np.eye(N,N)\n", "\n", " # top\n", " A[0,0],A[0,1] = Ut/dz -.5, -(Ut/dz +.5)\n", " B[0,0],B[0,1] = 1./dz, -1./dz\n", "\n", " # bottom\n", " A[-1,-2],A[-1,-1] = Ub/dz -.5, -(Ub/dz+.5)\n", " B[-1,-2],B[-1,-1] = 1./dz,-1./dz\n", "\n", " return A,B" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now solve the stability problem" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_analysis_num(kappa,N,beta=1.,nu=0.,nmodes=1):\n", " '''\n", " Compute the growth rate of the linear\n", " stability problem given an array of \n", " wavenumbers kappa\n", "\n", " N is the of vertical levels '''\n", "\n", " try:\n", " Nkappa = kappa.size\n", " c = np.zeros(Nkappa,dtype='complex128')\n", " c2 = np.zeros(Nkappa,dtype='complex128')\n", " sig = np.zeros(Nkappa)\n", " sig2 = np.zeros(Nkappa)\n", " efun = np.zeros((N+2,Nkappa)) + 0j\n", " efun2 = np.zeros((N+2,Nkappa)) + 0j\n", "\n", " for i in range(Nkappa):\n", " M1,M2 = CharneyMatrix(kappa[i],N,beta,nu)\n", " cm,em = sp.linalg.eig(M1,M2)\n", " isort = cm.imag.argsort()\n", " c[i],sig[i] = cm[isort[-1]],kappa[i]*cm.imag[isort[-1]]\n", " efun[:,i] = em[:,isort[-1]]\n", " if nmodes == 2:\n", " c2[i],sig2[i] = cm[isort[-2]],kappa[i]*cm.imag[isort[-2]]\n", " efun2[:,i] = em[:,isort[-2]]\n", "\n", " except AttributeError:\n", "\n", " M1,M2 = CharneyMatrix(kappa,N,beta,nu)\n", " cm,em = sp.linalg.eig(M1,M2)\n", " isort = cm.imag.argsort() \n", " c,sig = cm[isort[-1]],kappa*cm.imag[isort[-1]]\n", " efun = em[:,isort[-1]]\n", "\n", " if nmodes == 2:\n", " c2,sig2 = cm[isort[-2]],kappa[i]*cm.imag[isort[-2]]\n", " efun2 = em[:,isort[-2]]\n", "\n", " if nmodes == 2:\n", " return c,efun,sig,c2,efun2,sig2\n", " else:\n", " return c,efun,sig" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# parameters\n", "N = 1000 \n", "kappa = np.linspace(0.001,10.,100)\n", "beta = 1.\n", "\n", "c_num,e_num,sig_num = stability_analysis_num(kappa,N,beta=beta,nu=0,nmodes=1)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "np.savez('outputs/beta-eady_num.npz',kappa=kappa,c=c_num,sig=sig_num,e=e_num,N=N)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we plot the results" ] }, { "cell_type": "code", "execution_count": 6, "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})\n", "\n", "def labels(field='sig',fs=50):\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", " if field == 'sig':\n", " plt.plot(lcharneyNum['kappa'],lcharneyNum['sig'],color=\"k\",\\\n", " linewidth=4.,label='Finite Differences')\n", " elif field == 'ps':\n", " plt.plot(lcharneyNum['kappa'],lcharneyNum['c'].real,color=\"k\",\\\n", " linewidth=4.,label='Finite Differences')\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "data_path = 'outputs/beta-eady_'\n", "data_pathA, data_pathB,data_pathC = \"linear_charney_A.npz\",\"linear_charney_B.npz\",\"linear_charney_C.npz\"\n", "data_pathNum = \"linear_charney_num.npz\"\n", "\n", "lcharneyA = np.load(data_path+'A'+'.npz')\n", "lcharneyB = np.load(data_path+'B'+'.npz')\n", "lcharneyC = np.load(data_path+'C'+'.npz')\n", "lcharneyNum = np.load(data_path+'num'+'.npz')\n", "\n", "# plot settings\n", "lw,aph = 4.,.75\n", "Nmodes = lcharneyC['Nmodes']\n", "color1,color2,color3 = \"#1e90ff\",\"#ff6347\",\"#32cd32\"" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA/MAAALsCAYAAACvC+dZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xe4JFW19/HvYoZhYJhADjpEJQgodAEiCIwYABFQLkiU\n", "YLiKIqhXrsoLYsaAqCiKXPR6uYKKyEUREFAYEEWRakGGpJKzxBnCABPW+0dV03X26RP6dFVXdffv\n", "8zz9nLN3V1WvPrNn5qyuvfY2d0dEREREREREescyZQcgIiIiIiIiIu1RMi8iIiIiIiLSY5TMi4iI\n", "iIiIiPQYJfMiIiIiIiIiPUbJvIiIiIiIiEiPUTIvIiIiIiIi0mMmlx3AIDCzycD2wHrAWsAC4D7g\n", "Wnd/vMTQREREREREpAeZ9pkvjpmtAJwAHAGs3uKQRcAlwPHuPq+bsYXMbBZwK7BG8NQb3P2qEkIS\n", 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"metadata": {}, "output_type": "display_data" } ], "source": [ "# individual modes\n", "for i in range(Nmodes.size):\n", " fig=plt.figure(figsize=(15,11))\n", " plt_exact('sig')\n", " plt.plot(lcharneyA['kappa'],lcharneyA['sig'][i,:],\\\n", " color=color1,linewidth=lw,label='A')\n", " plt.plot(lcharneyB['kappa'],lcharneyB['sig'][i,:],\\\n", " color=color3,linewidth=lw,label='B')\n", " plt.plot(lcharneyC['kappa'],lcharneyC['sig'][i,:],\\\n", " color=color2,linewidth=lw,label='C')\n", "\n", " plt.grid()\n", " labels() \n", " plt.yticks(np.arange(0.,.5,.1))\n", " plt.legend(loc=1, title = r'$\\mathrm{N} = $' + str(Nmodes[i]))\n", " plt.ylim(0.,0.4)\n", " plt.xlim(0,10)\n", " \n", " # annotate order of the figure (for publication)\n", " pm = True\n", " nm = Nmodes[i]\n", "\n", " plt.tick_params(axis='both', which='major', labelsize=40)\n", " \n", " plt.text(8.25, .02, r\"$\\beta-$Eady Problem\", size=30, rotation=0.,\\\n", " ha=\"center\", va=\"center\",\\\n", " bbox = dict(boxstyle=\"round\",ec='k',fc='w'))\n", "\n", " figtit = 'figs/beta-eady_sig_galerk_ABC_'+str(Nmodes[i])+'.eps'\n", " plt.savefig(figtit,format='eps') \n", "\n", " if i < Nmodes.size-1:\n", " if i%2: \n", " plt.close()\n", " else:\n", " plt.text(.1,.05,'Green')\n", " plt.text(.1,.03,'modes')\n", " plt.text(1.,.32,'Modified Eady modes')\n", " plt.text(5.,.1,'Charney modes')\n", " figtit = 'figs/beta-eady_sig_galerk_ABC_'+str(Nmodes[i])+'.eps'\n", " plt.savefig(figtit,format='eps', bbox_inches = 'tight')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Convergence rate" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#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\n", "\n", "def absolute_error(exact,approx):\n", " ''' compute absolute error '''\n", " return np.abs(exact-approx)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# the fasted growing mode\n", "imax = lcharneyNum['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": "markdown", "metadata": {}, "source": [ "Now compute the error" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [], "source": [ "errorA = np.zeros((Nmodes.size,kappas.size))\n", "errorB = 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(lcharneyNum['sig'][inds[k]],\\\n", " lcharneyA['sig'][n,inds[k]])\n", " errorB[n,k] = absolute_error(lcharneyNum['sig'][inds[k]],\\\n", " lcharneyB['sig'][n,inds[k]]) \n", " errorC[n,k] = absolute_error(lcharneyNum['sig'][inds[k]],\\\n", " lcharneyC['sig'][n,inds[k]]) " ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAqsAAAIpCAYAAACFY3EiAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXd4W9X5xz+vvOPsHRJQEkIWWegywg7bbMJqCy3TlNmK\n", "2V9LoWWUUdpSRFtaqChQyg6EjdgJe+SKhJmdKITs5SROYsf2+f1xjixZkrdsy/b5PI8eSfeee+9x\n", "Ilvf+573/b6ilMJisVgsFovFYslEPG09AYvFYrFYLBaLpTasWLVYLBaLxWKxZCxWrFosFovFYrFY\n", "MhYrVi0Wi8VisVgsGYsVqxmOiOSJyI0iMldEdojIKhF5XERGtPXcLBaLxWKxWFoasW4AmYuI5AFv\n", "AQcCK4D3gWHAvsBWYIpSKtx2M7RYLBaLxWJpWWxkNbP5DVqofgSMVEr9RCk1GbgG6Ao8JiL2/9Bi\n", "sVgsFkuHxUZWMxQRyQHWAN0Bn1JqTsL+2cAE4FSl1PNtMEWLxWKxWCyWFsdG5TKXA4EewKJEoWqY\n", "Zp5Par0pWSwWi8VisbQuVqxmLpPMc205qdHtE1phLhaLxWKxWCxtghWrmctu5nl5Lft/MM/eVpiL\n", "xWKxWCwWS5tgxWrm0tU8b6tlf6l57tYKc7FYLBaLxWJpE6xYtVgsFovFYrFkLFasGkQkS0TGich5\n", 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line error at k ~8 (max growth rate)\n", "plt.loglog(Nmodes,errorA[:,-1],'--',color=color1,linewidth=lw)\n", "plt.loglog(Nmodes,errorB[:,-1],'--',color=color3,linewidth=lw)\n", "plt.loglog(Nmodes,errorC[:,-1],'--',color=color2,linewidth=lw)\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}$\",fontsize=30)\n", "plt.ylabel(r\"Absolute error\",fontsize=30)\n", "plt.xlim(0,350)\n", "plt.text(100, .5e-10, r\"$\\beta-$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_beta_eady.eps',format='eps')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "As sanity a test for our code, we compare the function stability curve from **stability_analysis** with $\\mathrm{N}=0$ with the simplest model. First, we construct the basic state, which should be a steady solution to the non-linear equations for approximation B. In particular we choose a solution independent of $x$. A possiblity, consistent with Eady's basic state, is\n", "

\n", "\\begin{equation}\n", "\\Sigma = -y\\left(z+1\\right) \\qquad \\text{and} \\qquad Q_0 = \\Phi_0 = 0\\,.\n", "\\end{equation}\n", "Hence\n", "\n", "\\begin{equation}\n", "T_0 = \\int_{-1}^{0} \\Sigma \\,d z = -\\frac{y}{2}\\, .\n", "\\end{equation}\n", "

\n", "Notice that\n", "\\begin{equation}\n", "\\Delta T_0 = 0 \\longrightarrow \\Theta^{+} = \\Theta^{-} = \\Theta\\, .\n", "\\end{equation}\n", "Matching the shear at the boundaries we obtain $\\Theta = -y$.\n", "

\n", "The linearized interior equation is \n", "\n", "\\begin{equation}\n", "\\partial_t q_0 -\\partial_y \\,T_0\\,\\partial_x\\,q_0 + \\hat{\\beta}\\, \\partial_x \\left(\\phi_0 + \\tau_0\\right) = 0\\, .\n", "\\end{equation}\n", "

\n", "The boundary conditions are\n", "\\begin{equation}\n", "\\partial_t \\vartheta^{\\pm} - \\partial_y\\,\\Theta\\,\\partial_x\\,\\phi_0 - \\Sigma_y^{\\pm} \\partial_x \\vartheta^{\\pm} = 0\\, .\n", "\\end{equation}\n", "

\n", "Fourier transforming, and using the inversion relationships \n", "

\n", "\n", "\\begin{equation}\n", "\\Delta \\phi_0 = q_0 \\qquad \\text{and} \\qquad \\Delta \\tau_0 = -\\left(\\vartheta^{+}-\\vartheta^{-}\\right)\n", "\\end{equation}\n", "

\n", "We obtain the eigenvalue problem\n", "

\n", "\\begin{equation}\n", "\\begin{bmatrix}\n", "1 - \\frac{\\coth{\\kappa}}{\\kappa} \\qquad \\frac{1}{\\kappa^2} \\qquad\\:\\: \\frac{\\text{csech}\\,\\kappa}{\\kappa}\\\\\n", "\\:\\:\\:\\:\\:\\frac{\\hat{\\beta}}{\\kappa^2} \\qquad \\frac{1}{2}-\\frac{\\hat{\\beta}}{\\kappa^2} \\qquad -\\frac{\\hat{\\beta}}{\\kappa^2}\\\\\n", "-\\frac{\\text{csech}\\,\\kappa}{\\kappa} \\qquad \\frac{1}{\\kappa^2} \\qquad \\frac{\\coth{\\kappa}}{\\kappa}\n", "\\end{bmatrix}\n", "\\begin{bmatrix}\n", "\\hat{\\vartheta}^{+}\\\\\n", "\\hat{{q}_0}\\\\\n", "\\hat{\\vartheta}^{-}\n", "\\end{bmatrix}\n", "=\n", "c\\, \n", "\\begin{bmatrix}\n", "\\hat{\\vartheta}^{+}\\\\\n", "\\hat{{q}_0}\\\\\n", "\\hat{\\vartheta}^{-}\n", "\\end{bmatrix}\n", "\\end{equation}\n", "\n", "\n", "

\n", "This matrix is much easier to code than the general case, and therefore provide a simple sanity check on our previous code." ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def matrixB_simplest(kappa,beta=1.):\n", " M = np.zeros((3,3))\n", " \n", " cosech_k = 1./sinh(kappa)\n", " coth_k = cosh(kappa)/sinh(kappa)\n", "\n", " M[0,0],M[0,1],M[0,2] = 1.-coth_k/kappa,1./kappa**2,cosech_k/kappa\n", " M[1,0],M[1,1],M[1,2] = beta/kappa**2,-beta/kappa**2+0.5,-beta/kappa**2\n", " M[2,0],M[2,1],M[2,2] = -cosech_k/kappa,1./kappa**2,coth_k/kappa\n", "\n", " return M \n", "\n", "def stability_simplest(kappa,beta=1.,debug=False):\n", " \n", " try:\n", " Nk = kappa.size\n", " except AttributeError:\n", " Nk = 1\n", " \n", " if Nk>1:\n", " c,sig = np.zeros(Nk)+0j, np.zeros(Nk)\n", " e = np.zeros((3,Nk),dtype='complex128')\n", " \n", " for i in range(Nk):\n", " M = matrixB_simplest(kappa[i],beta)\n", " \n", " cm,em = np.linalg.eig(M)\n", " \n", " assert ((3/2. - 1/kappa[i]**2) - (cm.real.sum()))<1.e-14,\\\n", " \"Sum of eigenvalue does not match the trace of the matrix\"\n", " \n", " isort = cm.imag.argsort()\n", " c[i],sig[i] = cm[isort[-1]],kappa[i]*cm.imag[isort[-1]]\n", " e[:,i] = em[:,isort[-1]] \n", " \n", " if debug: \n", " if i == 0: \n", " print \"The eigenvalues are\"\n", " print \" \" \n", " print cm\n", " \n", " \n", " else: \n", " M = matrixB_simplest(kappa,beta)\n", " cm,em = np.linalg.eig(M)\n", " \n", " assert ((3/2. - 1/kappa**2) - (cm.real.sum()))<1.e-14,\\\n", " \"Sum of eigenvalue does not match the trace of the matrix\"\n", " \n", " isort = cm.imag.argsort()\n", " c,sig = cm[isort[-1]],kappa*cm.imag[isort[-1]]\n", " e = em[:,isort[-1]]\n", " \n", " return c,e,sig" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = np.linspace(0.001,10,100)\n", "c_simplest,e_simples,sig_simplest = stability_simplest(kappa,beta=1.,debug=False)\n", "c_galerkB0,sig_galerkB0 = stability_analysis_galerk(kappa,0,method='B',\\\n", " beta=1.,nu=0.0,efunc=False) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We compare the two stability diagrams to make sure our code is not bogus" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAqYAAAIXCAYAAABO7ZxPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4FOXawOHfpPcEQiAkEEIAB6SLCEpRRBEUjEiTJtWG\n", "oFg+LAfhCIigcvCoiPQq6gFBQCxY0JCISBWkDCVAICGQBJKQnk32+2M3IWU3Pexm89zX5TXJzOzM\n", "M+8M7pN33qLo9XqEEEIIIYSwNDtLByCEEEIIIQRIYiqEEEIIIayEJKZCCCGEEMIqSGIqhBBCCCGs\n", "giSmQgghhBDCKkhiKoQQQgghrIKDpQOoDqqq1gFmAgOBhkAc8D0wS9O0qCo4/j3AbiBK07SmJez3\n", "JDAZaAtkAoeA/2iatr2yMQghhBBC2BqbqzE1JqV/AC8A3sBhwBkYDxxWVbVtJY/vAqwAFMDsILCq\n", "qs4DVgN3AKeAK8C9wFZVVadXJgYhhBBCCFtkc4kpsAxQgR1AoKZpdwGBGJJEH+BLVVUrc90zjcc3\n", 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"plt.plot(beady30['kappa'],beady30['sig'][-1,:])\n", "plt.xlabel(r'$\\kappa$')\n", "plt.ylabel(r'$k\\,c_i$')\n", "plt.savefig('figs/beta-eady_stability_A_ext_kappa')" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from sympy import *\n", "from sympy import init_printing\n", "init_printing() " ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = symbols('kappa')" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAP4AAAAwBAMAAADDWtqmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZkRU\n", "MokUreuSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFYUlEQVRYCc1YXWhcRRT+9t7dzf4nKUqgot6i\n", "0j41CzFQ6kMWwQepkjXWiIhJBE2gqFkSkooviVpU/GsUW1TUrggqReg+FKm+dCuoT9V9MOBTumKb\n", 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\\frac{2 \\kappa^{5}}{945} + \\mathcal{O}\\left(\\kappa^{6}\\right)$$" ], "text/plain": [ " 3 5 \n", "1 κ κ 2⋅κ ⎛ 6⎞\n", "─ + ─ - ── + ──── + O⎝κ ⎠\n", "κ 3 45 945 " ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(coth(kappa),kappa,0)" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAR0AAAAwBAMAAADeAbMUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZkRU\n", "MokUreuSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGCElEQVRYCc1YX2gcRRj/bvdy//bukhRUapFO\n", "teibOaiBqmDuoYpSJWdN0qqQnNVSlWrOlkTxoWlVsNJWLTUFi9B7LEXMIVXqU1bBNyX34oMIuROt\n", "UcFrEqklTdr1m9md3dndubtJCdaB3Hzz/X7fN7/MzuzMLMANlezB3QpxmVceKCvQ1oDyDAwoZDGs\n", "fxRYa0EZhimFNMb3RIG1NpQnFdIYFQXS2lC04wp5jJ1jRIEWpKwLOpq0f9zTaNRtTHutJCM91Hgc\n", "3RGCP7RouXSeGav6uW9RkT5lWVaRc2XzWS/ATBn0fsJJoP3tmqrG5iFVPU8BxNysNdM1XSO+Ap0f\n", 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"PArUlhSepwePJDruc1hwc/YdGG7K+kpZ+x4csgeCvm/EA4P3PqJqmxT+fzWBld1unvPWZNWOouMh\n", "Hhiav5+zc87Vz1tfoDd+NXEPREQbHnMy+tSsw21gf+ZZ9KGlD26r+oiuHi8mUfFsat3mb3otbYO9\n", "IKnHKDv+XkhfBYZETO2k4xQq+iEgYVklAGrdDdllEIluHi+ko+7ZaGkYJi+ROnt3+sEJE5aAIfic\n", "f/Zj2GIfAiK/HXCs3QBH6AYVJnqRmUOejVaq6GsKjamc0ODmhYq2AAx5BC9YhLvdmh58Y6xFrfcA\n", "RqtyohsB+zwTraRsEjDGoz6e28DnxRD8LDhtul5u+PXga3i0KifyAHwNeSZan9GWfdP2+QEWZscC\n", "U40RIgWGaJdRD06TQGF6ztGZTi0sZ+6UE724dMGzQc/ThuymrS2U4ZjAdMzZSWCIfh2PUvUQTlVE\n", "iY4XSVuPcb0JUYj8TrA7CG3IbtqaRWAXQynDK9njDLmruR7kPsb1JPPt9Xi5uSW9aS/he1z2wI4Q\n", "inyOepo8L8w6knPGZwA0OZF3La2lN+3D2Gs5SN8C0G8yBKfFjBmEmYrnAC7hxYXOn3gRQE4MRQoO\n", "+U0bV1F4fPAh9pcZ8hHCREhim1QFyh0htp5vQc/JiaFIwSG/aePbZZdAss13cMXkGIL7oOR7ONWD\n", "6+WEPX8yBYjn5MRQZsERk960o3UtvL7OQfoKMCRpyr6HUz29kHrb1nN/4+JLeImXEYXuQ2aTm/bF\n", "IXyNBIoxOIBznCLa3HgYZh8C9OE9xP4ksNGyluTEQNr/dfNfMzDfJd6IAkcAAAAASUVORK5CYII=\n" ], "text/latex": [ "$$\\frac{1}{\\kappa} - \\frac{\\kappa}{6} + \\frac{7 \\kappa^{3}}{360} - \\frac{31 \\kappa^{5}}{15120} + \\mathcal{O}\\left(\\kappa^{6}\\right)$$" ], "text/plain": [ " 3 5 \n", "1 κ 7⋅κ 31⋅κ ⎛ 6⎞\n", "─ - ─ + ──── - ───── + O⎝κ ⎠\n", "κ 6 360 15120 " ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(1/sinh(kappa),kappa,0)" ] }, { "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 }