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

On Galerkin approximations for the QG equations

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

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

Winter 2015

*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", "Our code in this notebook build on the on previously developed for the Eady problem. In fact, we will be able to recover the Eady problem when $\\beta = 0$." ] }, { "cell_type": "code", "execution_count": 1, "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': 25, 'legend.handlelength' : 1.25})" ] }, { "cell_type": "code", "execution_count": 63, "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 = N\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": 77, "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": 78, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_analysis(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": 64, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# settings\n", "kappa = np.linspace(0.001,10.,100)\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(kappa,\\\n", " Nmodes[i],method='A',beta=1.,nu=0.,efunc=False)\n", " c_galerkB[i,:],sig_galerkB[i,:] = stability_analysis(kappa,\\\n", " Nmodes[i],method='B',beta=1.,nu=0.,efunc=False) \n", " c_galerkC[i,:],sig_galerkC[i,:] = stability_analysis(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": 69, "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": 5, "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 skew-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": 59, "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": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stability_analysis(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": 8, "metadata": { "collapsed": false }, "outputs": [ { "ename": "KeyboardInterrupt", "evalue": "", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mKeyboardInterrupt\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 4\u001b[0m \u001b[0mbeta\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.\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;32m----> 6\u001b[0;31m \u001b[0mc_num\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0me_num\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msig_num\u001b[0m \u001b[0;34m=\u001b[0m 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\u001b[0;34m(\u001b[0m\u001b[0malphar\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0m_I\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0malphai\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mbeta\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 60\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0minfo\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mKeyboardInterrupt\u001b[0m: " ] } ], "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(kappa,N,beta=beta,nu=0,nmodes=1)" ] }, { "cell_type": "code", "execution_count": null, "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": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "plt.rcParams.update({'font.size': 25, 'legend.handlelength' : 1.25})\n", "import numpy as np\n", "import sys\n", "\n", "import seaborn as sns\n", "sns.set(style=\"darkgrid\")\n", "sns.set_context(\"paper\", font_scale=3., rc={\"lines.linewidth\": 1.5})\n", "sns.set_style(\"darkgrid\",{'grid_color':.95})\n", "\n", "def labels(field='sig',fs=25):\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_num'],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", "\n", "def relative_error(exact,approx):\n", " ''' compute relative error '''\n", " return np.abs(exact-approx)/np.abs(exact)" ] }, { "cell_type": "code", "execution_count": 10, "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": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ 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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", " labels() \n", " plt.legend(loc=1, title = r'$\\mathrm{N} = $' + str(Nmodes[i]))\n", " plt.ylim(0.,0.4)\n", "\n", " # annotate order of the figure (for publication)\n", " pm = True\n", " nm = Nmodes[i]\n", "\n", " plt.text(8.65, .02, r\"$\\beta-$Eady Problem\", size=25, 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')" ] }, { "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 the the general case, and therefore provide a simple sanity check on our previous code." ] }, { "cell_type": "code", "execution_count": 95, "metadata": { "collapsed": true }, "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.):\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", " cm,em = np.linalg.eig(M)\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", " else: \n", " M = matrixB_simplest(kappa,beta)\n", " cm,em = np.linalg.eig(M)\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": 96, "metadata": { "collapsed": false }, "outputs": [], "source": [ "kappa = np.linspace(0.001,10,100)\n", "c_simplest,e_simples,sig_simples = stability_simplest(kappa,beta=1.)\n", "c_galerkB0,sig_galerkB0 = stability_analysis(kappa,0,method='B',beta=1.,nu=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": 94, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ 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"plt.plot(kappa,sig,label=r'matrixB$\\_$simplest code')\n", "plt.plot(kappa,sig_galerkB0,'ro',label=r'stability$\\_$analysis code')\n", "plt.xlabel(r'$\\kappa \\times\\, L_d$') \n", "plt.ylabel(r'$\\sigma \\times \\, \\frac{\\Lambda H}{L}$')\n", "plt.legend(loc=1, title = r'')\n", "plt.savefig('figs/comparisons_methodB_N_0')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "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 }