{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Convert Specific Humidity (q) to Water Vapor Pressure (e)\n",
    "* Wenchang Yang (wenchang@prnceton.edu)\n",
    "* Department of Geosciences, Princeton University"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T14:26:07.738686Z",
     "start_time": "2018-10-25T14:26:07.573042Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Thu Oct 25 10:26:07 EDT 2018\r\n"
     ]
    }
   ],
   "source": [
    "!date"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By definition, the specific humidity is:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ q = \\frac{\\rho_v}{\\rho_d + \\rho_v} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\rho_v$ and $\\rho_d$ are the density of the water vapor and dry air, respectively. So we have"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T14:05:49.026403Z",
     "start_time": "2018-10-25T14:05:49.022579Z"
    }
   },
   "source": [
    "$$ \\frac{\\rho_v}{\\rho_d} = \\frac{q}{1-q} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to ideal gas law:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$e = \\rho_v R_v T$$\n",
    "\n",
    "$$p_d = \\rho_d R_d T$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $e$ is the water vapor and $p_d$ is the dry air pressure. So we have"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\frac{e}{p_d} = \\frac{\\rho_v}{\\rho_d}\\cdot \\frac{R_v}{R_d} = \\frac{q}{1 - q}\\cdot \\frac{M_d}{M_v} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $M_d$ and $M_v$ are the molar mass for dry air and water vapor, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the total air pressure is the sum of the dry air pressure and the water vapor pressure, i.e."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ p_a = p_d + e $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we have"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\frac{e}{p_a} = \\frac{ \\frac{q}{1 - q}\\cdot \\frac{M_d}{M_v} }{  \\frac{q}{1 - q}\\cdot \\frac{M_d}{M_v} + 1 } $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ e = p_a \\cdot \\frac{ 1 }{ 1 + \\frac{1}{ \\frac{q}{1 - q}\\cdot \\frac{M_d}{M_v} }  } $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we assume $q<<1$ and $e << p_d$, we have the following approximation (though not necessary if we use a computer to calculate $e$):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T14:12:04.835436Z",
     "start_time": "2018-10-25T14:12:04.832116Z"
    }
   },
   "source": [
    "$$ e \\approx p_a \\cdot q \\cdot \\frac{M_d}{M_v} $$"
   ]
  }
 ],
 "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": "3.6.6"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}