ex1.ipynb

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    "## Excercises on ice sheet modeling at GIA training school 2023\n",
    "© Torsten Albrecht, albrecht@pik-potsdam.de, PIK, 2023\n",
    "\n",
    "- based on the book [minimal glacier model](https://webspace.science.uu.nl/~oerle102/MM2011-all.pdf) by Hans Oerlemans, 2008"
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    "### 1 Minimal glacier model\n",
    "\n",
    "Cleo and Robin are on field work in North America mapping moraines, indicating there must have been a huge ice-sheet in the past, extending about $1000\\text{km}$.\n",
    "\n",
    "In order ti understand possible GIA effects they need an estimate of the past ice sheet thickness distribution.\n",
    "\n",
    "Assuming \"perfect plasticity\" the stress in the ice cannot exceed the yield stress $\\tau_0$, and basal drag must be equal to $\\tau_0$. Hence, driving stress equals yield stress\n",
    "\n",
    "$$\n",
    "\\rho_i g H \\left|\\frac{\\partial h}{\\partial x}\\right| = \\tau_0 .\n",
    "$$\n"
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    "#### Q1: What would be the maximum thickness of an (1-D) ice sheet of length $L$ resting on a flat bed? \n",
    "\n",
    "Hint: $\\tau_0 = 50 \\text{k Pa}$, $L=1000\\text{km}$ and $\\frac{\\partial h}{\\partial x} = \\frac{\\partial H}{\\partial x}$ for $\\frac{\\partial b}{\\partial x} = 0$"
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    "<div class=\"alert alert-block alert-info\">\n",
    "<details>\n",
    "  <summary><b>A1:</b> click to show solution</summary>\n",
    "    \n",
    "Let's consider the part for $dh /dx > 0$ from $x=0$ to the summit at $x=L/2$ \n",
    "\n",
    "$$\n",
    "\\frac{d(H^2)}{dx} = \\frac{2 \\tau_0}{\\rho_i g}\n",
    "$$\n",
    "\n",
    "and integrate from $0$ to $x$\n",
    "\n",
    "$$\n",
    "H(0)^2 - H(x)^2 = \\frac{2 \\tau_0}{\\rho_i g} (x)\n",
    "$$\n",
    "\n",
    "with $H(0) = H(L) = 0$ we find\n",
    "$$\n",
    "H(x) = \\sqrt{\\frac{2 \\tau_0}{\\rho_i g} (x)}, \\;0 \\leq x \\leq L/2\n",
    "$$\n",
    "\n",
    "and \n",
    "\n",
    "$$\n",
    "H(x) = \\sqrt{\\frac{2 \\tau_0}{\\rho_i g} (L-x)}, \\;L/2 \\leq x \\leq L\n",
    "$$\n",
    "\n",
    "and $H(L/2) = \\sqrt{\\frac{2 \\tau_0}{\\rho_i g} \\frac{L}{2}}$ \n",
    "\n",
    "which yields a maximum thickness of $2.4\\text{km}$ for $1000\\text{km}$.\n",
    "    \n",
    "</details>\n",
    "    </div>"
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    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = {'svg',}\n",
    "\n",
    "import matplotlib.pylab as plt\n",
    "from matplotlib import cm, colors\n",
    "import netCDF4 as nc\n",
    "import numpy as np"
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      "Hmax 2379.7 m\n"
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       "(0.0, 1000.0)"
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    "# setting parameters\n",
    "Lext = 1000e3 #m\n",
    "tau0 = 50e3 #Pa\n",
    "g = 9.81 #m/s2\n",
    "rhoi = 900 #kg/m3 \n",
    "\n",
    "H_max = (tau0*Lext/(rhoi*g))**0.5\n",
    "print('Hmax',np.around(H_max,decimals=1),'m')\n",
    "#print(2*tau0/(rhoi*g))\n",
    "\n",
    "#ice shelf geometry and spatial resolution\n",
    "dx = 1.0  # km\n",
    "xl = np.arange(0, Lext, dx*1e3)  #m\n",
    "xkm = xl * 1e-3 # xl in km\n",
    "#print(len(xl))\n",
    "\n",
    "\n",
    "Hplast_left = (2.0*tau0*xl/(rhoi*g))**0.5\n",
    "Hplast_right = (2.0*tau0*(Lext-xl)/(rhoi*g))**0.5\n",
    "\n",
    "#print(Hplast_left)\n",
    "\n",
    "################################################################\n",
    "fig = plt.figure(10,figsize=(7,4))\n",
    "ax1 = fig.add_subplot(1,1,1)\n",
    "\n",
    "# bottom\n",
    "ax1.plot(xkm, np.zeros_like(xl), ls='dotted', color='C9')\n",
    "\n",
    "ax1.plot(xkm[xl<Lext/2.0], Hplast_left[xl<Lext/2.0], color='C0')\n",
    "ax1.plot(xkm[xl>=Lext/2.0], Hplast_right[xl>=Lext/2.0], color='C0')\n",
    "\n",
    "# ocean water\n",
    "#ax.plot(x, lan, color='C1')\n",
    "#ax1.fill_between(x,-1e3,lan,color='C0')\n",
    "\n",
    "ax1.set_xlabel('distance from grounding line, km')\n",
    "ax1.set_ylabel('elevation, m')\n",
    "ax1.set_xlim(0,Lext*1e-3)\n"
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   "source": [
    "#### Q1*: What would be the maximum surface elevation and subsidence, if the ice sheet load was in equilibrium with the solid Earth underneath?\n",
    "\n",
    "$$\n",
    "H = \\left(1-\\frac{b}{h}\\right) h = \\left(1-\\frac{\\rho_i}{\\rho_m - \\rho_i}\\right) h  = \\left(1-\\zeta\\right) h\n",
    "$$\n",
    "\n",
    "\n",
    "Hint: Assume a mantle density of $\\rho_m = 3300 \\text{kg m}^{-3}$"
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   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "<details>\n",
    "  <summary><b>A1*:</b> click to show solution</summary>\n",
    "\n",
    "$$\n",
    "H(x) = \\sqrt{\\frac{2 \\tau_0 (1-\\zeta)}{\\rho_i g} (x)}, \\;0 \\leq x \\leq L/2\n",
    "$$\n",
    " \n",
    " \n",
    "$$\n",
    "h(x) = \\sqrt{\\frac{2 \\tau_0}{\\rho_i g (1-\\zeta)} (x)}, \\;0 \\leq x \\leq L/2\n",
    "$$\n",
    "\n",
    "The ice sheet becomes thicker with up to 3.0km at the summit, while surface elevation is lower with about 1.9km.\n",
    "    \n",
    "</details>\n",
    "    </div>"
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      "0.375\n",
      "hmax 1881.3 m\n",
      "Hmax 3010.2 m\n",
      "bmax -1128.8 m\n"
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      "text/plain": [
       "<Figure size 504x288 with 1 Axes>"
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   ],
   "source": [
    "rhom = 3300.0\n",
    "zeta = rhoi/(rhom-rhoi)\n",
    "print(zeta)\n",
    "\n",
    "Hiso_left = (2.0*tau0*xl/(rhoi*g*(1.0-zeta)))**0.5\n",
    "Hiso_right = (2.0*tau0*(Lext-xl)/(rhoi*g*(1.0-zeta)))**0.5\n",
    "\n",
    "hiso_left = (2.0*tau0*xl*(1.0-zeta)/(rhoi*g))**0.5\n",
    "hiso_right = (2.0*tau0*(Lext-xl)*(1.0-zeta)/(rhoi*g))**0.5\n",
    "\n",
    "Hiso_max = (tau0*Lext/(rhoi*g*(1.0-zeta)))**0.5\n",
    "hiso_max = (tau0*Lext*(1.0-zeta)/(rhoi*g))**0.5\n",
    "\n",
    "print('hmax',np.around(hiso_max,decimals=1),'m')\n",
    "print('Hmax',np.around(Hiso_max,decimals=1),'m')\n",
    "print('bmax',np.around(hiso_max-Hiso_max,decimals=1),'m')\n",
    "\n",
    "################################################################\n",
    "fig = plt.figure(10,figsize=(7,4))\n",
    "ax1 = fig.add_subplot(1,1,1)\n",
    "\n",
    "# bottom\n",
    "ax1.plot(xkm, np.zeros_like(xl), ls='dotted', color='C9')\n",
    "\n",
    "Hplast = np.r_[Hplast_left[xl<Lext/2.0], Hplast_right[xl>=Lext/2.0]]\n",
    "ax1.plot(xkm, Hplast, color='C0',ls='dashed',label='flat-bed ice sheet surface')\n",
    "\n",
    "hiso = np.r_[hiso_left[xl<Lext/2.0], hiso_right[xl>=Lext/2.0]]\n",
    "ax1.plot(xkm, hiso, color='C1',label='isostatic ice sheet surface')\n",
    "\n",
    "biso = np.r_[hiso_left[xl<Lext/2.0]-Hiso_left[xl<Lext/2.0], hiso_right[xl>=Lext/2.0]-Hiso_right[xl>=Lext/2.0]]\n",
    "ax1.plot(xkm, biso, color='C2',label='isostatic ice sheet bed')\n",
    "\n",
    "ax1.set_xlabel('distance from grounding line, km')\n",
    "ax1.set_ylabel('elevation, m')\n",
    "ax1.set_xlim(0,Lext*1e-3)\n",
    "ax1.legend()"
   ]
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