asynchronous lecture 20.ipynb

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    "Asynchronous class 20: April 12, 2021\n",
    "============================\n",
    "\n",
    "Today we take our coupled acoustic/gravity wave system to a simple non-constant system: the polytropic atmosphere.\n",
    "\n",
    "# Homework\n",
    "No written homework.  Instead, read two things this week.\n",
    "1. Kataoka et al 2017, \"Tilting at wave beams: a new perspective on the St. Andrew’s Cross\"\n",
    "2. Chandrasekhar chapter 1 (\"The Benard Problem\")\n",
    "\n",
    "# Lecture\n",
    "1. Watch [simplifying linear systems with vector displacements](https://youtu.be/3LaEx0T2l4g)\n",
    "\n",
    "2. Read [Gough (2007) \"An Elementary Introduction to the JWKB approximation\"](https://ui.adsabs.harvard.edu/abs/2007AN....328..273G/abstract)."
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    "We've learned a lot about solving for waves in constant coefficient systems, where the system reduces to a matrix algebra system and we can take the determinant of the matrix to find the waves (and the eigenfunctions are simple functions that are pre-chosen using knowledge about the system).\n",
    "\n",
    "What happens if we have non-constant coefficients?  Now that beautiful machinery goes out the window and it's time for different tools.  If you're trying to solve problems analytically, your major tool is the JWKB approximation.  Please read [Gough (2007) \"An Elementary Introduction to the JWKB approximation\"](https://ui.adsabs.harvard.edu/abs/2007AN....328..273G/abstract) (and posted on Canvas) to learn more about this.\n",
    "\n",
    "Your other major tool are numerical approaches, which we will explore in our next lecture.\n",
    "\n",
    "Non-constant coefficient atmospheres: polytropes\n",
    "----------------------------------\n",
    "Before we begin solving for waves in a non-constant coefficient atmosphere, we need a nice example problem to work from.  Polytropes are a classic pen-and-paper non-constant coefficient system.  They're the solution to a system with hydrostatic balance and thermal balance, with thermal diffusion as the only physical process acting in the stable atmosphere,\n",
    "\\begin{align}\n",
    "    \\nabla P  = -\\rho g \\hat{z} \\\\\n",
    "    \\nabla\\cdot(K \\nabla T) = 0\n",
    "\\end{align}\n",
    "when $g$ and $K=\\rho\\chi$ are both constant.  This system can also be written:\n",
    "\\begin{align}\n",
    "    \\nabla \\ln T + \\nabla \\ln \\rho = -\\frac{g}{R T} \\hat{z} \\\\\n",
    "    \\nabla\\cdot(K \\nabla T) = 0\n",
    "\\end{align}\n",
    "Non-dimensionalizing the system, we have:\n",
    "\\begin{align}\n",
    "    \\nabla \\ln T + \\nabla \\ln \\rho = -\\frac{g L}{R T} \\hat{z} \\\\\n",
    "    \\nabla\\cdot(K \\nabla T) = 0\n",
    "\\end{align}\n",
    "The solutions have $\\nabla T = \\text{const}$ and $\\rho \\propto T^m$ where $m$ is the polytropic index.  Hydrostatic equilbrium in this system becomes:\n",
    "\\begin{align}\n",
    "    (m+1)\\nabla \\ln T = -\\frac{g L}{R T} \\hat{z} = -\\frac{g L}{R T_0}\\left(\\frac{T}{T_0}\\right)^{-1} \\hat{z} \\\\\n",
    "\\end{align}\n",
    "If the lengthscale is the pressure scale height, then $L=R T_0/g$ at some reference temperature $T_0$, and\n",
    "\\begin{align}\n",
    "\\nabla \\ln T = \\frac{\\nabla T}{T} = \\frac{\\nabla T}{T_0} \\left(\\frac{T}{T_0}\\right)^{-1} = -\\frac{1}{m+1}\\left(\\frac{T}{T_0}\\right)^{-1} \\hat{z}\n",
    "\\end{align}\n",
    "which simplifies to\n",
    "\\begin{align}\n",
    "    \\frac{\\nabla T}{T_0} = \\frac{-1}{m+1},\n",
    "\\end{align}\n",
    "or, integrating in $z$,\n",
    "\\begin{align}\n",
    "\\frac{T(z)}{T_0} = C - \\frac{z}{m+1},\n",
    "\\end{align}\n",
    "and if we take the reference temperature $T_0=T(z=L_z)=1$ at $z=L_z$ (the top of the atmosphere) then $C=1+L_z/(m+1)$ and\n",
    "\\begin{align}\n",
    "    T(z) = \\frac{1}{m+1}\\left(L_z + 1 + m - z\\right)\n",
    "\\end{align}\n",
    "since it comes up a lot, we'll call\n",
    "\\begin{align}\n",
    "    z_0 \\equiv Lz + 1 + m\n",
    "\\end{align}\n",
    "with\n",
    "\\begin{align}\n",
    "    T(z) = \\frac{1}{m+1}\\left(z_0 - z\\right).\n",
    "\\end{align}\n",
    "\n",
    "The density scaleheight in this system is:\n",
    "\\begin{align}\n",
    "    H_\\rho(z) = \\frac{-1}{\\nabla \\ln \\rho} = \\frac{-1}{m \\nabla \\ln T} = \\frac{1}{m}\\left(z_0 - z\\right)\n",
    "\\end{align}\n",
    "\n",
    "All of this means that:\n",
    "\\begin{align}\n",
    "    \\nabla \\ln T &= -(z_0-z)^{-1} \\\\\n",
    "    \\nabla \\ln \\rho &= -m (z_0-z)^{-1}\n",
    "\\end{align}\n",
    "\n",
    "We'll use this non-dimensionalization for now.  Here, we also have $R = 1$ and $g=1$.\n"
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    "\\renewcommand{\\vec}{\\boldsymbol}\n",
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    "# Participation\n",
    "\n",
    "Once you've read Gough, and the details above about polytropes, let's play a bit with polytropic atmospheres.\n",
    "\n",
    "$\\renewcommand{\\vec}{\\boldsymbol}$\n",
    "$\\renewcommand{\\del}{\\vec{\\nabla}}$\n",
    "\n",
    "1. Given that the entropy for an ideal gas is\n",
    "\\begin{align}\n",
    "\\frac{s}{c_P} = \\frac{1}{\\gamma}\\ln T - \\frac{\\gamma-1}{\\gamma}\\ln \\rho,\n",
    "\\end{align}\n",
    "show that the entropy gradient is zero if \n",
    "\\begin{align}\n",
    "m = m_{ad} = \\frac{1}{\\gamma-1}.\n",
    "\\end{align}\n",
    "This is called an adiabatic polytrope, and $m_{ad}$ is the adiabatic index.\n",
    "\n",
    "2. Show that the sense of the entropy gradient (positive or negative) is determined by whether $m$ is greater than or less than $m_{ad}$ (assume $\\nabla ln T <0$, if needed, as it is in many physical systems).\n",
    "\n",
    "3. Using this expression for the Brunt-Vaisalla frequency:\n",
    "\\begin{align}\n",
    "    N^2 = -\\vec{g}\\cdot\\vec{\\nabla}\\left(\\frac{s}{c_P}\\right)\n",
    "\\end{align}\n",
    "and your knowledge of Boussinesq gravity waves, argue why $\\nabla s > 0$ leads to stable gravity waves while $\\nabla s < 0$ leads to convective instability.  Assume $\\nabla s = (\\partial s/\\partial z) \\vec{\\hat{z}}$ and $\\vec{g} = -g \\vec{\\hat{z}}$.\n",
    "\n",
    "A negative entropy gradient is the stratified-atmosphere analogue of an unstable (negative) temperature gradient in Boussinesq fluids.  In the same way that a negative temperature gradient can lead to convective instability, a negative entropy gradient can lead to convective instability in a stratified atmosphere.  And in the same way that temperature is the bouyancy variable for Boussinesq fluids (hot things go up, cold things fall down), entropy is the generalization for stratifed atmospheres (high entropy goes up, low entropy falls down).  In atmospheric sciences, entropy is often written in the form of a (possibly confusing) \"potential temperature\", but you can see that the two are related, and in my opinion entropy is a bit simpler to think about.\n"
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    "Now we're going to plot some frequency diagrams for a polytrope.  Recall that $c^2 = \\gamma R T$, and here we're taking that $R=1$, $\\vec{g}=-1\\vec{\\hat{z}}$, $T_0 = 1$, so\n",
    "\\begin{align}\n",
    "        c^2 = \\frac{\\gamma}{m+1}\\left(z_0 - z\\right)\n",
    "\\end{align}\n",
    "   In a plane-parallel atmosphere, like we have here, the $\\omega_+^2$ and $\\omega_-^2$ branches are simpler than in Gough:\n",
    "\\begin{align}\n",
    "        \\omega_L^2 = k^2 c^2 \\\\\n",
    "        \\omega_+^2 = \\omega_L^2 + \\omega_c^2 \\\\\n",
    "        \\omega_-^2 = \\frac{N^2 \\omega_L^2}{\\omega_L^2 + \\omega_c^2}\n",
    "\\end{align}\n",
    "where here $\\omega_L^2$ is the squared Lamb frequency.\n",
    "\n",
    "4.  Plot the following for a polytrope with $m=2$ and $\\gamma=5/3$ and $L_z=5$:\n",
    "   - Brunt-Vaisalla frequency $N^2 = -\\vec{g}\\cdot\\vec{\\nabla} s/c_P$\n",
    "   - The acoustic cutoff frequency $\\omega_c^2$ (eqn 6 of Gough; to get dH/dz, take d/dz of $H_\\rho(z)$ above and get a simple solution)\n",
    "   - The $\\omega_+^2$ and $\\omega_-^2$ branches for $k = [0.5, 1, 2]$.\n",
    "   - Illustrate where in the frequency diagram gravity waves propagate, where acoustic waves propagate, and where both are evanescent (see Gough)\n",
    "   \n",
    "   I'd suggest making a different diagram for each of the 3 $k$ values, and I suggest using a logrithmic y-scale.  Try using filling and shading (see `fill_between` in matplotlib) to illustrate the propagation regions.\n",
    "\n",
    "  What can you say about the propagation of acoustic and gravity waves in this atmosphere?\n",
    "\n",
    "\n",
    "\n",
    "**To Turn in**:\n",
    "Send Ben your answers to 1-4 via Canvas, uploading to Asynchronous Lecture 20.  You can do that in this ipynb and upload as an ipynb.  You can also export to PDF if you prefer.  Please send by noon, Tuesday April 13."
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