normal_mode_evp.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solve the normal mode eigenvalue problem that arises in GFD problems\n",
    "\n",
    "$$\n",
    "\\frac{d^2 \\phi}{dz^2} + \\frac{N^2}{c^2}\\phi = 0\n",
    "$$\n",
    "\n",
    "Use the `waves_on_a_string.py` example as a starting point..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import dedalus.public as de\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import logging\n",
    "import time\n",
    "logger = logging.getLogger(__name__)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -3.99984940e+02,  -3.99864477e+02,  -3.99623623e+02,\n",
       "        -3.99262522e+02,  -3.98781394e+02,  -3.98180527e+02,\n",
       "        -3.97460284e+02,  -3.96621097e+02,  -3.95663474e+02,\n",
       "        -3.94587990e+02,  -3.93395294e+02,  -3.92086104e+02,\n",
       "        -3.90661208e+02,  -3.89121465e+02,  -3.87467802e+02,\n",
       "        -3.85701216e+02,  -3.83822770e+02,  -3.81833597e+02,\n",
       "        -3.79734893e+02,  -3.77527924e+02,  -3.75214019e+02,\n",
       "        -3.72794571e+02,  -3.70271039e+02,  -3.67644941e+02,\n",
       "        -3.64917861e+02,  -3.62091440e+02,  -3.59167381e+02,\n",
       "        -3.56147446e+02,  -3.53033453e+02,  -3.49827279e+02,\n",
       "        -3.46530854e+02,  -3.43146165e+02,  -3.39675250e+02,\n",
       "        -3.36120200e+02,  -3.32483156e+02,  -3.28766309e+02,\n",
       "        -3.24971898e+02,  -3.21102208e+02,  -3.17159571e+02,\n",
       "        -3.13146362e+02,  -3.09064998e+02,  -3.04917937e+02,\n",
       "        -3.00707677e+02,  -2.96436754e+02,  -2.92107742e+02,\n",
       "        -2.87723248e+02,  -2.83285912e+02,  -2.78798408e+02,\n",
       "        -2.74263439e+02,  -2.69683736e+02,  -2.65062058e+02,\n",
       "        -2.60401190e+02,  -2.55703938e+02,  -2.50973132e+02,\n",
       "        -2.46211622e+02,  -2.41422275e+02,  -2.36607978e+02,\n",
       "        -2.31771629e+02,  -2.26916142e+02,  -2.22044441e+02,\n",
       "        -2.17159462e+02,  -2.12264147e+02,  -2.07361445e+02,\n",
       "        -2.02454308e+02,  -1.97545692e+02,  -1.92638555e+02,\n",
       "        -1.87735853e+02,  -1.82840538e+02,  -1.77955559e+02,\n",
       "        -1.73083858e+02,  -1.68228371e+02,  -1.63392022e+02,\n",
       "        -1.58577725e+02,  -1.53788378e+02,  -1.49026868e+02,\n",
       "        -1.44296062e+02,  -1.39598810e+02,  -1.34937942e+02,\n",
       "        -1.30316264e+02,  -1.25736561e+02,  -1.21201592e+02,\n",
       "        -1.16714088e+02,  -1.12276752e+02,  -1.07892258e+02,\n",
       "        -1.03563246e+02,  -9.92923233e+01,  -9.50820635e+01,\n",
       "        -9.09350023e+01,  -8.68536378e+01,  -8.28404285e+01,\n",
       "        -7.88977917e+01,  -7.50281024e+01,  -7.12336914e+01,\n",
       "        -6.75168445e+01,  -6.38798004e+01,  -6.03247501e+01,\n",
       "        -5.68538349e+01,  -5.34691457e+01,  -5.01727211e+01,\n",
       "        -4.69665469e+01,  -4.38525543e+01,  -4.08326191e+01,\n",
       "        -3.79085603e+01,  -3.50821394e+01,  -3.23550589e+01,\n",
       "        -2.97289614e+01,  -2.72054288e+01,  -2.47859812e+01,\n",
       "        -2.24720759e+01,  -2.02651069e+01,  -1.81664034e+01,\n",
       "        -1.61772297e+01,  -1.42987839e+01,  -1.25321976e+01,\n",
       "        -1.08785349e+01,  -9.33879193e+00,  -7.91389612e+00,\n",
       "        -6.60470579e+00,  -5.41200956e+00,  -4.33652586e+00,\n",
       "        -3.37890251e+00,  -2.53971637e+00,  -1.81947291e+00,\n",
       "        -1.21860600e+00,  -7.37477563e-01,  -3.76377420e-01,\n",
       "        -1.35523082e-01,  -1.50596322e-02])"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Domain\n",
    "Nz = 128\n",
    "H = 400\n",
    "#N2 = 1e-4\n",
    "\n",
    "z_basis = de.Chebyshev('z', Nz, interval=(-H, 0))\n",
    "domain = de.Domain([z_basis], np.complex128)\n",
    "\n",
    "z = z_basis.grid()\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f76e62eb160>]"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZUAAAD8CAYAAAC/1zkdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIiLL5v8DZPpx10WY+cgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f76e63ae198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def sech(z):\n",
    "    return 1/np.cosh(z)\n",
    "\n",
    "def sech_N2(beta,z):\n",
    "    N1 = 9.81/1024*(beta[1]/beta[3])\n",
    "    cff = sech((z+beta[2])/beta[3]) #+ np.exp(z/beta[0])\n",
    "    return N1*cff*cff\n",
    "    \n",
    "beta = [2000,1.0,100,50]\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(sech_N2(beta,z),z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020-01-16 13:40:12,971 problems 0/1 INFO :: Solving EVP with homogeneity tolerance of 1.000e-10\n"
     ]
    }
   ],
   "source": [
    "# Problem\n",
    "problem = de.EVP(domain, variables=['phi', 'phi_z'],eigenvalue='c')\n",
    "\n",
    "# Constant buoyancy frequency\n",
    "#problem.parameters['N2'] = N2\n",
    "\n",
    "# Add vertically varying N^2\n",
    "ncc = domain.new_field(name = 'N2')\n",
    "ncc['g'] = sech_N2(beta,z)\n",
    "problem.parameters['N2'] = ncc\n",
    "\n",
    "problem.add_equation(\"phi * N2 *c + dz(phi_z) = 0\")\n",
    "problem.add_equation(\"phi_z - dz(phi) = 0\")\n",
    "#problem.add_bc(\"left(phi) = 0\")\n",
    "problem.add_bc(\"right(phi) = 0\")\n",
    "problem.add_bc(\"left(phi) = 0\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2020-01-16 13:40:15,199 __main__ 0/1 INFO :: Elapsed solve time: 1.364400\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/dedalus/dedalus/lib/python3.5/site-packages/scipy/linalg/decomp.py:56: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  w = alpha / beta\n",
      "/home/dedalus/dedalus/lib/python3.5/site-packages/scipy/linalg/decomp.py:56: RuntimeWarning: invalid value encountered in true_divide\n",
      "  w = alpha / beta\n"
     ]
    }
   ],
   "source": [
    "# Solver\n",
    "t1 = time.time()\n",
    "\n",
    "solver = problem.build_solver()\n",
    "solver.solve(solver.pencils[0])\n",
    "t2 = time.time()\n",
    "logger.info('Elapsed solve time: %f' %(t2-t1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Filter infinite/nan eigenmodes\n",
    "finite = np.isfinite(solver.eigenvalues)\n",
    "solver.eigenvalues = solver.eigenvalues[finite]\n",
    "solver.eigenvectors = solver.eigenvectors[:, finite]\n",
    "\n",
    "# Sort eigenmodes by eigenvalue\n",
    "order = np.argsort(solver.eigenvalues)\n",
    "solver.eigenvalues = solver.eigenvalues[order]\n",
    "solver.eigenvectors = solver.eigenvectors[:, order]\n",
    "\n",
    "# Plot error vs exact eigenvalues\n",
    "mode_number = 1 + np.arange(len(solver.eigenvalues))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  1.02304758e+00,   3.62236721e-01,   2.26857209e-01,\n",
       "         1.65917942e-01,   1.30989198e-01,   1.08283512e-01,\n",
       "         9.23195049e-02,   8.04744774e-02,   7.13326223e-02,\n",
       "         6.40614210e-02,   5.81389284e-02,   5.32211286e-02,\n",
       "         4.90719829e-02,   4.55241037e-02,   4.24554644e-02,\n",
       "         3.97749966e-02,   3.74133508e-02,   3.53167824e-02,\n",
       "         3.34429931e-02,   3.17582362e-02,   3.02352585e-02,\n",
       "         2.88518103e-02,   2.75895404e-02,   2.64331742e-02,\n",
       "         2.53699282e-02,   2.43889592e-02,   2.34808409e-02,\n",
       "         2.26368992e-02,   2.18500511e-02,   2.11095392e-02,\n",
       "         2.04080712e-02,   1.97251671e-02,   1.90686397e-02,\n",
       "         1.84044260e-02,   1.77618801e-02,   1.70966980e-02,\n",
       "         1.64701658e-02,   1.58193994e-02,   1.52084063e-02,\n",
       "         1.45740371e-02,   1.39938562e-02,   1.33934266e-02,\n",
       "         1.28376273e-02,   1.22649273e-02,   1.17481491e-02,\n",
       "         1.12157846e-02,   1.07266754e-02,   1.02227440e-02,\n",
       "         9.77632120e-03,   9.31100513e-03,   8.89390229e-03,\n",
       "         8.45156287e-03,   8.07901380e-03,   7.66965599e-03,\n",
       "         7.32808462e-03,   6.93340336e-03,   6.63638989e-03,\n",
       "         6.26889200e-03,   6.00333854e-03,   5.64486812e-03,\n",
       "         5.41656706e-03,   5.08709764e-03,   4.88217438e-03,\n",
       "         4.57138846e-03,   4.37443495e-03,   4.11457273e-03,\n",
       "         3.91536735e-03,   3.69974809e-03,   3.47186618e-03,\n",
       "         3.32146119e-03,   3.09286332e-03,   2.96642964e-03,\n",
       "         2.75970577e-03,   2.60312454e-03,   2.47318772e-03,\n",
       "         2.29096529e-03,   2.19703967e-03,   2.03967082e-03,\n",
       "         1.89024064e-03,   1.81365706e-03,   1.67503918e-03,\n",
       "         1.55980915e-03,   1.49242123e-03,   1.37053136e-03,\n",
       "         1.25498707e-03,   1.19441780e-03,   1.11869993e-03,\n",
       "         1.02151955e-03,   9.43785537e-04,   9.04418907e-04,\n",
       "         8.33418333e-04,   7.58099345e-04,   6.89144659e-04,\n",
       "         6.30295765e-04,   5.98947770e-04,   5.60310128e-04,\n",
       "         5.10063180e-04,   4.63209183e-04,   4.20683541e-04,\n",
       "         3.83510703e-04,   3.63009483e-04,   3.41080695e-04,\n",
       "         3.10482785e-04,   2.81178332e-04,   2.54363254e-04,\n",
       "         2.29784702e-04,   2.07422293e-04,   1.87001949e-04,\n",
       "         1.68377023e-04,   1.51331397e-04,   1.35734625e-04,\n",
       "         1.21481074e-04,   1.08449968e-04,   9.65213737e-05,\n",
       "         8.55032052e-05,   7.53264541e-05,   6.59073908e-05,\n",
       "         5.72839362e-05,   4.92837164e-05,   4.18221950e-05,\n",
       "         3.47244250e-05,   2.81440729e-05,   2.19261310e-05,\n",
       "         1.61164935e-05,   1.03196299e-05,   5.25374307e-06])"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/np.sqrt(np.abs(solver.eigenvalues))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f76e61ccc18>]"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYMAAAD8CAYAAACVZ8iyAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nZjgMzMwMh4GZmeEwMDMzHAZmZgb8f+UAFoU0zxafAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f76e6281710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def norm(V):\n",
    "    if np.max(V)>np.abs(np.min(phi)):\n",
    "        return V/np.max(V)\n",
    "    else:\n",
    "        return V/np.min(V)\n",
    "\n",
    "# This is the method for extracting the eigenvalue in physical space    \n",
    "mode=0\n",
    "solver.set_state(mode)\n",
    "phi = solver.state['phi']['g'].real\n",
    "\n",
    "plt.figure()\n",
    "\n",
    "plt.plot(norm(phi),z,'.')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}