Merge pull request #12 from jetesdal/master

Minor changes in Markdown

Author: ifenty

Tutorials_as_Jupyter_Notebooks/ECCO_v4_Heat_budget_closure.ipynb

@@ -5,12 +5,13 @@
    "metadata": {},
    "source": [
     "# ECCOv4 Global Heat Budget Closure\n",
-    "### Jan-Erik Tesdal$^1$$^{*}$, Ryan Abernathey$^1$ and Ian Fenty$^2$\n",
-    "**$^1$ Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY, USA**\n",
-    "\n",
-    "**$^2$ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA**\n",
-    "\n",
-    "$^{*}$*Corresponding author*: jt2796@columbia.edu\n",
+    "**Jan-Erik Tesdal<sup>1,*</sup>, Ryan Abernathey<sup>1</sup> and Ian Fenty<sup>2</sup>**\n",
+    "<br>\n",
+    "<sup>1</sup> Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY, USA\n",
+    "<br>\n",
+    "<sup>2</sup> Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA\n",
+    "<br>\n",
+    "<sup>*</sup>*Corresponding author*: jt2796@columbia.edu\n",
     "\n",
     "This section demonstrates the closure of the global heat budget in ECCOv4. The steps and Python code has been directly derived from the calculations and MATLAB code in \"*A Note on Practical Evaluation of Budgets in ECCO Version 4 Release 3\"* by Christopher G. Piecuch (https://ecco.jpl.nasa.gov/drive/files/Version4/Release3/doc/v4r3_budgets_howto.pdf)."
    ]
@@ -53,11 +54,13 @@
     "\\underbrace{\\frac{\\partial(s^*\\theta)}{\\partial t}}_{G^{\\theta}_\\textrm{total}} = \\underbrace{-\\nabla_{z^{*}} \\cdot(s^*\\theta\\,\\mathbf{v}_{res}) - \\frac{\\partial(\\theta\\,w_{res})}{\\partial z^{*}}}_{G^{\\theta}_\\textrm{advection}}\\underbrace{- s^* ({\\nabla\\cdot\\mathbf{F}_\\textrm{diff}^{\\theta}})}_{G^{\\theta}_\\textrm{diffusion}} + \\underbrace{s^* {F}_\\textrm{forc}^{\\theta}}_{G^{\\theta}_\\textrm{forcing}}\n",
     "\\end{equation}\n",
     "\n",
-    "where $z^{*} = \\frac{z - \\eta}{H + \\eta}H$ and $\\nabla_{z^{*}}$/$\\frac{\\partial}{\\partial z^{*}}$ are horizontal/vertical divergences in the $z^*$ frame. Also note that the advection is now separated into horizontal ($\\mathbf{v}_{res}$) and vertical ($w_{res}$) components, and there is a scaling factor ($s^* = 1+ \\frac{\\eta}{H}$) applied to the horizontal advection as well as the diffusion term ($G^{\\theta}_\\textrm{diffusion}$) and forcing term ($G^{\\theta}_\\textrm{forcing}$). $s^*$ is a function of $\\eta$ which is the displacement of the ocean surface from its resting position of $z=0$ (i.e., sea height anomaly). $H$ is the ocean depth. $s^{*}$ comes from the coordinate transformation from z to $z^*$ (Campin and Adcroft, 2004; Campin et al., 2004). See `ECCO_v4_Volume_budget_closure.ipynb` for a more detailed explanation of the $z^*$ coordinate system.\n",
+    "where $z^{*} = \\frac{z - \\eta}{H + \\eta}H$ and $\\nabla_{z^{*}}$/$\\frac{\\partial}{\\partial z^{*}}$ are horizontal/vertical divergences in the $z^*$ frame. Also note that the advection is now separated into horizontal ($\\mathbf{v}_{res}$) and vertical ($w_{res}$) components, and there is a scaling factor ($s^* = 1+ \\frac{\\eta}{H}$) applied to the horizontal advection as well as the diffusion term ($G^{\\theta}_\\textrm{diffusion}$) and forcing term ($G^{\\theta}_\\textrm{forcing}$). $s^*$ is a function of $\\eta$ which is the displacement of the ocean surface from its resting position of $z=0$ (i.e., sea height anomaly). $H$ is the ocean depth. $s^{*}$ comes from the coordinate transformation from z to $z^*$ (Campin and Adcroft, 2004; Campin et al., 2004). See [ECCOv4 Global Volume Budget Closure](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Volume_budget_closure.html#ECCOv4-Global-Volume-Budget-Closure) for a more detailed explanation of the $z^*$ coordinate system.\n",
     "\n",
     "Note that the velocity terms in the ECCOv4 heat budget equation ($\\mathbf{v}_{res}$ and $w_{res}$) are described as the \"residual mean\" velocities, which contain both the resolved (Eulerian) flow field, as well as the \"GM bolus\" velocity (i.e., parameterizing unresolved eddy effects):\n",
-    "$$(u_{res},v_{res},w_{res})= (u,v,w)+ (u_b,v_b,w_b),$$\n",
-    "where $(u_b,v_b,w_b)$ is the bolus velocity parameter, taking into account the correlation between velocity and thickness (also known as the eddy induced transportor the eddy advection term)."
+    "\n",
+    "$$(u_{res},v_{res},w_{res})= (u,v,w)+ (u_b,v_b,w_b)$$\n",
+    "\n",
+    "Here $(u_b,v_b,w_b)$ is the bolus velocity parameter, taking into account the correlation between velocity and thickness (also known as the eddy induced transportor the eddy advection term)."
    ]
   },
   {
@@ -84,26 +87,23 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Prepare environment and load ECCOv4 diagnostic output\n",
+    "## Prepare environment and load ECCOv4 diagnostic output"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "### Import relevant Python modules"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "/usr/local/lib/python3.6/dist-packages/pandas/compat/_optional.py:107: UserWarning: Pandas requires version '1.2.1' or newer of 'bottleneck' (version '1.2.0' currently installed).\n",
-      "  warnings.warn(msg, UserWarning)\n"
-     ]
-    }
-   ],
+   "outputs": [],
    "source": [
-    "import numpy as np\n",
+    "# import numpy as np\n",
     "import xarray as xr"
    ]
   },
@@ -2165,7 +2165,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.5"
+   "version": "3.6.9"
   }
  },
  "nbformat": 4,