updated some of the text

LinearStabilityAnalysis/LinearStabilityAnalysis.ipynb

@@ -32,6 +32,7 @@
     "import scipy\n",
     "from scipy.sparse.linalg import eigs\n",
     "import scipy.sparse as sps\n",
+    "from scipy.optimize import minimize_scalar\n",
     "import xarray as xr\n",
     "import matplotlib\n",
     "import matplotlib.pyplot as plt\n",
@@ -65,7 +66,11 @@
    "source": [
     "## Parameters of the problem\n",
     "\n",
-    "Below we set parameters relating to the domain size (including grid spacing) and the flow itself. We also create a matrix representation of the $f\\zeta$ and $\\frac{d^2}{dx^2}$ operators."
+    "Below we set parameters relating to the domain size (including grid spacing) and the flow itself. We also create a matrix representation of the $f\\zeta$ and $\\frac{d^2}{dx^2}$ operators.\n",
+    "\n",
+    "When running this notebook I reccomend increasing the grid-spacing (`dx`) to something larger ($\\sim$`2e3` should still give ok results). The value of `1e2` was used here for producing publication quality figures and is overkill even for that.\n",
+    "\n",
+    "If you want to play around with different vorticity profiles this can be done by redefining `zeta` in the below code to return the absolute vorticity profile you fancy."
    ]
   },
   {
@@ -81,18 +86,21 @@
     "\n",
     "N2 = 2.5e-5  # Buoyancy frequency (squared) in s^{-2}\n",
     "f = 1.01e-5  # Planetary vorticity in s^{-1}\n",
+    "A_r = 4e-4  # Viscosity in m^{2} s^{-1}\n",
+    "\n",
     "x_mid = 40e3  # Mid point of the jet in m\n",
     "delta_b = 30e3  # Width of the jet in m\n",
     "V_0 = 0.87  # Velocity of the jet in m/s\n",
-    "A_r = 4e-4\n",
+    "\n",
     "\n",
     "def zeta(x):\n",
     "    \"\"\" returns the absolute vorticity at x in s^{-1}\n",
     "    \"\"\"\n",
     "    rel_vort = - 2 * V_0 / delta_b * np.tanh((x - x_mid) / delta_b) / np.square(np.cosh((x - x_mid) / delta_b))\n",
     "    return rel_vort + f\n",
     "\n",
-    "x_vorticity_minima = 0.5 * np.log(2 + np.sqrt(3)) * delta_b + x_mid  # Gives x at the vorticity minima\n",
+    "\n",
+    "x_vorticity_minima = minimize_scalar(zeta, bounds=[0, Lx], method='bounded', options={'xatol': dx / 10}).x  # Gives x at the vorticity minima\n",
     "\n",
     "# Create a matrix representation of the curvature operator\n",
     "diagonals = np.array([1, -2, 1]) / np.square(dx)\n",
@@ -227,7 +235,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Viscualise the output"
+    "## Visualise the output"
    ]
   },
   {
@@ -312,7 +320,7 @@
     "ax.set_xticks(np.linspace(0, 400, 9))\n",
     "\n",
     "ax2 = ax.twinx()\n",
-    "ax2.plot(x * 1e-3, zeta(ds['lon']), 'k', lw=1.5)\n",
+    "ax2.plot(x * 1e-3, zeta(ds['lon']) / f, 'k', lw=1.5)\n",
     "ax2.set_ylabel('$\\zeta / f$')\n",
     "ax2.axhline(0, c='k', ls=':')\n",
     "\n",
@@ -328,7 +336,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Plot $\\hat{\\psi}(x)$ for different wavelengths. This is not contained in the paper. Note this also takes a long time to run."
+    "Plot $\\hat{\\psi}(x)$ for different wavelengths. This is not contained in the paper. Note this may take a while to run."
    ]
   },
   {