stomatal_cond_eqs.ipynb

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   "cell_type": "code",
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   "source": [
    "%%capture storage\n",
    "# The above redirects all output of the below commands to the variable 'storage' instead of displaying it.\n",
    "# It can be viewed by typing: 'storage()'\n",
    "# Setting up worksheet and importing equations from other worksheets\n",
    "load('temp/Worksheet_setup.sage')\n",
    "load_session('temp/leaf_enbalance_eqs.sobj')\n",
    "fun_loadvars(dict_vars) # any variables defined using var2() are saved with their attributes in dict_vars. Here we re-define them based on dict_vars from the other worksheet."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equations to compute stomatal conductance based on sizes and densities of stomata\n",
    "Equation numbers in comments refer to Lehmann & Or, 2013"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Definitions of additional variables</h2>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "F_p"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Following Lehmann_2015_Effects\n",
    "var2('B_l', 'Boundary layer thickness', meter, domain1='positive')\n",
    "var2('g_sp', 'Diffusive conductance of a stomatal pore', mole/second/meter^2, domain1='positive')\n",
    "var2('r_sp', 'Diffusive resistance of a stomatal pore', meter^2*second/mole, domain1='positive')\n",
    "var2('r_vs', 'Diffusive resistance of a stomatal vapour shell', meter^2*second/mole, domain1='positive')\n",
    "var2('r_bwmol', 'Leaf BL resistance in molar units',  meter^2*second/mole, domain1='positive', latexname = 'r_{bw,mol}')\n",
    "var2('r_end', 'End correction, representing resistance between evaporating sites and pores', meter^2*second/mole, domain1='positive')\n",
    "var2('A_p', 'Cross-sectional pore area', meter^2, domain1='positive')\n",
    "var2('d_p', 'Pore depth', meter, domain1='positive')\n",
    "var2('V_m', 'Molar volume of air', meter^3/mole, domain1='positive')\n",
    "var2('n_p', 'Pore density', 1/meter^2, domain1='positive')\n",
    "var2('r_p', 'Pore radius (for ellipsoidal pores, half the pore width)', meter, domain1='positive')\n",
    "var2('l_p', 'Pore length', meter, domain1='positive')\n",
    "var2('k_dv', 'Ratio $D_{va}/V_m$', mole/meter/second, domain1='positive', latexname='k_{dv}')\n",
    "var2('s_p', 'Spacing between stomata', meter, domain1='positive')\n",
    "var2('F_p', 'Fractional pore area (pore area per unit leaf area)', meter^2/meter^2, domain1='positive')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Molar volume of air'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "docdict[V_m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equations from Lehmann & Or, 2013"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{k_{dv}} = \\frac{{D_{va}}}{V_{m}}</script></html>"
      ],
      "text/plain": [
       "k_dv == D_va/V_m"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mole/(meter*second) == mole/(meter*second)\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g_{\\mathit{sp}} = \\frac{A_{p} {k_{dv}} n_{p}}{d_{p}}</script></html>"
      ],
      "text/plain": [
       "g_sp == A_p*k_dv*n_p/d_p"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mole/(meter^2*second) == mole/(meter^2*second)\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}A_{p} = \\pi r_{p}^{2}</script></html>"
      ],
      "text/plain": [
       "A_p == pi*r_p^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "meter^2 == meter^2\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{sp}} = \\frac{d_{p}}{A_{p} {k_{dv}} n_{p}}</script></html>"
      ],
      "text/plain": [
       "r_sp == d_p/(A_p*k_dv*n_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "meter^2*second/mole == meter^2*second/mole\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{sp}} = \\frac{d_{p}}{\\pi {k_{dv}} n_{p} r_{p}^{2}}</script></html>"
      ],
      "text/plain": [
       "r_sp == d_p/(pi*k_dv*n_p*r_p^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "meter^2*second/mole == meter^2*second/mole\n"
     ]
    }
   ],
   "source": [
    "# Eqn1\n",
    "eq_kdv = k_dv == D_va/V_m\n",
    "print units_check(eq_kdv)\n",
    "eq_gsp_A = g_sp == A_p/d_p*k_dv*n_p\n",
    "print units_check(eq_gsp_A)\n",
    "eq_Ap_rp = A_p == pi*r_p^2\n",
    "print units_check(eq_Ap_rp)\n",
    "eq_rsp_A = r_sp == 1/eq_gsp_A.rhs()\n",
    "print units_check(eq_rsp_A)\n",
    "eq_rsp_rp = r_sp == 1/eq_gsp_A.rhs().subs(eq_Ap_rp)\n",
    "print units_check(eq_rsp_rp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{vs}} = \\frac{\\frac{1}{r_{p}} - \\frac{4}{\\pi s_{p}}}{4 \\, {k_{dv}} n_{p}}</script></html>"
      ],
      "text/plain": [
       "r_vs == 1/4*(1/r_p - 4/(pi*s_p))/(k_dv*n_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2*second/mole == meter^2*second/mole"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eq. 2(c)\n",
    "eq_rvs_B = r_vs == (1/(4*r_p) - 1/(pi*s_p))*1/(k_dv*n_p)\n",
    "units_check(eq_rvs_B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{vs}} = \\frac{\\frac{1}{r_{p}} - \\frac{4}{\\pi s_{p}}}{4 \\, {k_{dv}} n_{p}}</script></html>"
      ],
      "text/plain": [
       "r_vs == 1/4*(1/r_p - 4/(pi*s_p))/(k_dv*n_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2*second/mole == meter^2*second/mole"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eq. 2(d)\n",
    "eq_rvs_S = r_vs == (1/(2*pi*r_p) - 1/(pi*s_p))*1/(k_dv*n_p)\n",
    "units_check(eq_rvs_B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{p} = \\frac{1}{\\sqrt{n_{p}}}</script></html>"
      ],
      "text/plain": [
       "s_p == 1/sqrt(n_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter == sqrt(n_p)/sqrt(n_p/meter^2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Below Eq. 3(b)\n",
    "assume(n_p>0)\n",
    "eq_sp_np = s_p == 1/sqrt(n_p)\n",
    "units_check(eq_sp_np)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{end}} = \\frac{1}{4 \\, {k_{dv}} n_{p} r_{p}}</script></html>"
      ],
      "text/plain": [
       "r_end == 1/4/(k_dv*n_p*r_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2*second/mole == meter^2*second/mole"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eq. 2(a)\n",
    "eq_rend_rp = r_end == 1/(4*r_p*k_dv*n_p)\n",
    "units_check(eq_rend_rp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{\\mathit{end}} = \\frac{\\log\\left(\\frac{2 \\, l_{p}}{r_{p}}\\right)}{\\pi {k_{dv}} l_{p} n_{p}}</script></html>"
      ],
      "text/plain": [
       "r_end == log(2*l_p/r_p)/(pi*k_dv*l_p*n_p)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2*second/mole == meter^2*second/mole"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eq. 2(b)\n",
    "eq_rend_PW = r_end == log(4*(l_p/2)/r_p)/(2*pi*l_p/2*k_dv*n_p)\n",
    "units_check(eq_rend_PW)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{g_{sw,mol}} = \\frac{1}{r_{\\mathit{end}} + r_{\\mathit{sp}} + r_{\\mathit{vs}}}</script></html>"
      ],
      "text/plain": [
       "g_swmol == (1/(r_end + r_sp + r_vs))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "mole/(meter^2*second) == mole/(meter^2*second)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_gswmol = g_swmol == 1/(r_end + r_sp + r_vs)\n",
    "units_check(eq_gswmol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p>It actually does not seem to make much sense to add r_end to the resistances, as it does not reflect the geometry of the inter-cellular air space! Also eq_rvs_B is the correct one to use for stomata, as eq_rvs_S is valid for droplets, not holes!</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}n_{p} = \\frac{1}{s_{p}^{2}}</script></html>"
      ],
      "text/plain": [
       "n_p == s_p^(-2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^(-2) == meter^(-2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "assume(s_p>0)\n",
    "eq_np_sp = solve(eq_sp_np, n_p)[0]\n",
    "units_check(eq_np_sp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{r_{bw,mol}} = \\frac{B_{l}}{{k_{dv}}}</script></html>"
      ],
      "text/plain": [
       "r_bwmol == B_l/k_dv"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2*second/mole == meter^2*second/mole"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eq. 5\n",
    "eq_rbwmol = r_bwmol == B_l/k_dv\n",
    "units_check(eq_rbwmol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Ratio $D_{va}/V_m$'"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "docdict[k_dv]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below, we compare our system of equations to Eq. 7a in Lehmann & Or (2015), i.e. we check if $\\frac{r_{bwmol}}{r_{tot}} = 1/(1 + s_p^2/B_l*(1/(2*r_p) + d_p/(\\pi*r_p^2)))$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, \\pi B_{l} r_{p}^{2}}{2 \\, \\pi B_{l} r_{p}^{2} + {\\left(\\pi r_{p} + 2 \\, d_{p}\\right)} s_{p}^{2}}</script></html>"
      ],
      "text/plain": [
       "2*pi*B_l*r_p^2/(2*pi*B_l*r_p^2 + (pi*r_p + 2*d_p)*s_p^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2}{\\frac{s_{p}^{2} {\\left(\\frac{1}{r_{p}} + \\frac{2 \\, d_{p}}{\\pi r_{p}^{2}}\\right)}}{B_{l}} + 2}</script></html>"
      ],
      "text/plain": [
       "2/(s_p^2*(1/r_p + 2*d_p/(pi*r_p^2))/B_l + 2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Verification of Eq. 7a\n",
    "eq1 = (r_bwmol/(2*r_end + r_sp + r_bwmol)).subs(eq_rend_rp, eq_rsp_rp, eq_rvs_B, eq_rbwmol).subs(eq_np_sp)\n",
    "eq1.simplify_full().show()\n",
    "eq7a = 1/(1 + s_p^2/B_l*(1/(2*r_p) + d_p/(pi*r_p^2)))\n",
    "eq7a.show()\n",
    "(eq1 - eq7a).simplify_full()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional equations for calculating conductances of laser perforated foils"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}A_{p} = \\pi r_{p}^{2}</script></html>"
      ],
      "text/plain": [
       "A_p == pi*r_p^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter^2 == meter^2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Pore area for circular pores\n",
    "eq_Ap = A_p == pi*r_p^2\n",
    "units_check(eq_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_{p} = \\sqrt{\\frac{A_{p}}{\\pi}}</script></html>"
      ],
      "text/plain": [
       "r_p == sqrt(A_p/pi)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter == sqrt(A_p*meter^2/pi)/sqrt(A_p/pi)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To deduce pore radius from pore area, assuming circular pore:\n",
    "eq_rp_Ap = solve(eq_Ap_rp, r_p)[0]\n",
    "units_check(eq_rp_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# For regular pore distribution we can derive the pore density from porosity and pore area\n",
    "eq_np = n_p == F_p/A_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{g_{sw}} = -\\frac{{\\left({P_{wa}} - {P_{wl}}\\right)} {R_{mol}} T_{a} T_{l} {g_{sw,mol}}}{P_{a} {P_{wl}} T_{a} - P_{a} {P_{wa}} T_{l}}</script></html>"
      ],
      "text/plain": [
       "g_sw == -(P_wa - P_wl)*R_mol*T_a*T_l*g_swmol/(P_a*P_wl*T_a - P_a*P_wa*T_l)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter/second == meter/second"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Conversion from mol/m2/s to m/s, i.e. from g_swmol to g_sw\n",
    "eq_gsw_gswmol = solve(eq_gtwmol_gtw(g_tw = g_sw, g_twmol = g_swmol), g_sw)[0]\n",
    "units_check(eq_gsw_gswmol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{g_{sw}} = \\frac{{R_{mol}} T_{a} {g_{sw,mol}}}{P_{a}}</script></html>"
      ],
      "text/plain": [
       "g_sw == R_mol*T_a*g_swmol/P_a"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "meter/second == meter/second"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Simplification, neglecting effect of leaf-air temperature difference\n",
    "eq_gsw_gswmol_iso = eq_gsw_gswmol(T_l = T_a).simplify_full()\n",
    "units_check(eq_gsw_gswmol_iso)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Saving session so that it can be loaded using load_session().\n",
    "save_session('temp/stomatal_cond_eqs.sobj')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Table of symbols</h2>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class=\"notruncate\">\n",
       "<table  class=\"table_form\">\n",
       "<tbody>\n",
       "<tr>\n",
       "<th>Variable</th>\n",
       "<th>Description (value)</th>\n",
       "<th>Units</th>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">A_{p}</script></td>\n",
       "<td>Cross-sectional pore area</td>\n",
       "<td> m<script type=\"math/tex\">^{2}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">a_{s}</script></td>\n",
       "<td>Fraction of one-sided leaf area covered by stomata (1 if stomata are on one side only, 2 if they are on both sides)</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{a_{sh}}</script></td>\n",
       "<td>Fraction of projected area exchanging sensible heat with the air (2)</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">\\alpha_{a}</script></td>\n",
       "<td>Thermal diffusivity of dry air</td>\n",
       "<td> m<script type=\"math/tex\">^{2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">B_{l}</script></td>\n",
       "<td>Boundary layer thickness</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{c_{pa}}</script></td>\n",
       "<td>Specific heat of dry air (1010) </td>\n",
       "<td>J  K<script type=\"math/tex\">^{-1}</script>  kg<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{C_{wa}}</script></td>\n",
       "<td>Concentration of water in the free air </td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-3}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{C_{wl}}</script></td>\n",
       "<td>Concentration of water in the leaf air space </td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-3}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">d_{p}</script></td>\n",
       "<td>Pore depth</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{D_{va}}</script></td>\n",
       "<td>Binary diffusion coefficient of water vapour in air</td>\n",
       "<td> m<script type=\"math/tex\">^{2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">E_{l}</script></td>\n",
       "<td>Latent heat flux from leaf</td>\n",
       "<td>J  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{E_{l,mol}}</script></td>\n",
       "<td>Transpiration rate in molar units</td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">\\epsilon_{l}</script></td>\n",
       "<td>Longwave emmissivity of the leaf surface (1.0)</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">F_{p}</script></td>\n",
       "<td>Fractional pore area (pore area per unit leaf area)</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">g</script></td>\n",
       "<td>Gravitational acceleration (9.81)</td>\n",
       "<td>m  s<script type=\"math/tex\">^{-2}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{g_{bw}}</script></td>\n",
       "<td>Boundary layer conductance to water vapour </td>\n",
       "<td>m  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{g_{bw,mol}}</script></td>\n",
       "<td>Boundary layer conductance to water vapour </td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">g_{\\mathit{sp}}</script></td>\n",
       "<td>Diffusive conductance of a stomatal pore</td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{g_{sw}}</script></td>\n",
       "<td>Stomatal conductance to water vapour</td>\n",
       "<td>m  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{g_{sw,mol}}</script></td>\n",
       "<td>Stomatal conductance to water vapour</td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{g_{tw}}</script></td>\n",
       "<td>Total leaf conductance to water vapour</td>\n",
       "<td>m  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{g_{tw,mol}}</script></td>\n",
       "<td>Total leaf layer conductance to water vapour</td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">h_{c}</script></td>\n",
       "<td>Average 1-sided convective transfer coefficient</td>\n",
       "<td>J  K<script type=\"math/tex\">^{-1}</script>  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">H_{l}</script></td>\n",
       "<td>Sensible heat flux from leaf</td>\n",
       "<td>J  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">k_{a}</script></td>\n",
       "<td>Thermal conductivity of dry air</td>\n",
       "<td>J  K<script type=\"math/tex\">^{-1}</script>  m<script type=\"math/tex\">^{-1}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{k_{dv}}</script></td>\n",
       "<td>Ratio <script type=\"math/tex\">D_{va}/V_m</script></td>\n",
       "<td>mol  m<script type=\"math/tex\">^{-1}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">L_{l}</script></td>\n",
       "<td>Characteristic length scale for convection (size of leaf)</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">l_{p}</script></td>\n",
       "<td>Pore length</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">\\lambda_{E}</script></td>\n",
       "<td>Latent heat of evaporation (2.45e6)</td>\n",
       "<td>J  kg<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">M_{N_{2}}</script></td>\n",
       "<td>Molar mass of nitrogen (0.028)</td>\n",
       "<td>kg  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">M_{O_{2}}</script></td>\n",
       "<td>Molar mass of oxygen (0.032)</td>\n",
       "<td>kg  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">M_{w}</script></td>\n",
       "<td>Molar mass of water (0.018)</td>\n",
       "<td>kg  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{N_{Gr_L}}</script></td>\n",
       "<td>Grashof number</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{N_{Le}}</script></td>\n",
       "<td>Lewis number</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{N_{Nu_L}}</script></td>\n",
       "<td>Nusselt number</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">n_{p}</script></td>\n",
       "<td>Pore density</td>\n",
       "<td> m<script type=\"math/tex\">^{-2}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{N_{Re_c}}</script></td>\n",
       "<td>Critical Reynolds number for the onset of turbulence</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{N_{Re_L}}</script></td>\n",
       "<td>Reynolds number</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{N_{Sh_L}}</script></td>\n",
       "<td>Sherwood number</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">\\nu_{a}</script></td>\n",
       "<td>Kinematic viscosity of dry air</td>\n",
       "<td> m<script type=\"math/tex\">^{2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">P_{a}</script></td>\n",
       "<td>Air pressure</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{P_{N2}}</script></td>\n",
       "<td>Partial pressure of nitrogen in the atmosphere</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{P_{O2}}</script></td>\n",
       "<td>Partial pressure of oxygen in the atmosphere</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{P_{wa}}</script></td>\n",
       "<td>Vapour pressure in the atmosphere</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{P_{was}}</script></td>\n",
       "<td>Saturation vapour pressure at air temperature</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{P_{wl}}</script></td>\n",
       "<td>Vapour pressure inside the leaf</td>\n",
       "<td>Pa </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{N_{Pr}}</script></td>\n",
       "<td>Prandtl number (0.71)</td>\n",
       "<td>1 </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{r_{bw}}</script></td>\n",
       "<td>Boundary layer resistance to water vapour, inverse of <script type=\"math/tex\">g_{bw}</script></td>\n",
       "<td>s  m<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{r_{bw,mol}}</script></td>\n",
       "<td>Leaf BL resistance in molar units</td>\n",
       "<td>s  m<script type=\"math/tex\">^{2}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">r_{\\mathit{end}}</script></td>\n",
       "<td>End correction, representing resistance between evaporating sites and pores</td>\n",
       "<td>s  m<script type=\"math/tex\">^{2}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{R_{ll}}</script></td>\n",
       "<td>Longwave radiation away from leaf</td>\n",
       "<td>J  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{R_{mol}}</script></td>\n",
       "<td>Molar gas constant (8.314472)</td>\n",
       "<td>J  K<script type=\"math/tex\">^{-1}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">r_{p}</script></td>\n",
       "<td>Pore radius (for ellipsoidal pores, half the pore width)</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">R_{s}</script></td>\n",
       "<td>Solar shortwave flux</td>\n",
       "<td>J  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">r_{\\mathit{sp}}</script></td>\n",
       "<td>Diffusive resistance of a stomatal pore</td>\n",
       "<td>s  m<script type=\"math/tex\">^{2}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{r_{sw}}</script></td>\n",
       "<td>Stomatal resistance to water vapour, inverse of <script type=\"math/tex\">g_{sw}</script></td>\n",
       "<td>s  m<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">{r_{tw}}</script></td>\n",
       "<td>Total leaf resistance to water vapour, <script type=\"math/tex\">r_{bv} + r_{sv}</script></td>\n",
       "<td>s  m<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">r_{\\mathit{vs}}</script></td>\n",
       "<td>Diffusive resistance of a stomatal vapour shell</td>\n",
       "<td>s  m<script type=\"math/tex\">^{2}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">\\rho_{a}</script></td>\n",
       "<td>Density of dry air</td>\n",
       "<td>kg  m<script type=\"math/tex\">^{-3}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">\\rho_{\\mathit{al}}</script></td>\n",
       "<td>Density of air at the leaf surface</td>\n",
       "<td>kg  m<script type=\"math/tex\">^{-3}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">s_{p}</script></td>\n",
       "<td>Spacing between stomata</td>\n",
       "<td>m </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">{\\sigma}</script></td>\n",
       "<td>Stefan-Boltzmann constant (5.67e-8)</td>\n",
       "<td>J  K<script type=\"math/tex\">^{-4}</script>  m<script type=\"math/tex\">^{-2}</script>  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">T_{a}</script></td>\n",
       "<td>Air temperature</td>\n",
       "<td>K </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">T_{l}</script></td>\n",
       "<td>Leaf temperature</td>\n",
       "<td>K </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">T_{w}</script></td>\n",
       "<td>Radiative temperature of objects surrounding the leaf</td>\n",
       "<td>K </td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">V_{m}</script></td>\n",
       "<td>Molar volume of air</td>\n",
       "<td> m<script type=\"math/tex\">^{3}</script>  mol<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">v_{w}</script></td>\n",
       "<td>Wind velocity</td>\n",
       "<td>m  s<script type=\"math/tex\">^{-1}</script> </td>\n",
       "</tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  Variable    Description (value)                                                                                                   Units\n",
       "+-----------+---------------------------------------------------------------------------------------------------------------------+----------------------------------+\n",
       "  A_p         Cross-sectional pore area                                                                                              m$^{2}$\n",
       "  a_s         Fraction of one-sided leaf area covered by stomata (1 if stomata are on one side only, 2 if they are on both sides)   1\n",
       "  a_sh        Fraction of projected area exchanging sensible heat with the air (2)                                                  1\n",
       "  alpha_a     Thermal diffusivity of dry air                                                                                         m$^{2}$  s$^{-1}$\n",
       "  B_l         Boundary layer thickness                                                                                              m\n",
       "  c_pa        Specific heat of dry air (1010)                                                                                       J  K$^{-1}$  kg$^{-1}$\n",
       "  C_wa        Concentration of water in the free air                                                                                mol  m$^{-3}$\n",
       "  C_wl        Concentration of water in the leaf air space                                                                          mol  m$^{-3}$\n",
       "  d_p         Pore depth                                                                                                            m\n",
       "  D_va        Binary diffusion coefficient of water vapour in air                                                                    m$^{2}$  s$^{-1}$\n",
       "  E_l         Latent heat flux from leaf                                                                                            J  m$^{-2}$  s$^{-1}$\n",
       "  E_lmol      Transpiration rate in molar units                                                                                     mol  m$^{-2}$  s$^{-1}$\n",
       "  epsilon_l   Longwave emmissivity of the leaf surface (1.0)                                                                        1\n",
       "  F_p         Fractional pore area (pore area per unit leaf area)                                                                   1\n",
       "  g           Gravitational acceleration (9.81)                                                                                     m  s$^{-2}$\n",
       "  g_bw        Boundary layer conductance to water vapour                                                                            m  s$^{-1}$\n",
       "  g_bwmol     Boundary layer conductance to water vapour                                                                            mol  m$^{-2}$  s$^{-1}$\n",
       "  g_sp        Diffusive conductance of a stomatal pore                                                                              mol  m$^{-2}$  s$^{-1}$\n",
       "  g_sw        Stomatal conductance to water vapour                                                                                  m  s$^{-1}$\n",
       "  g_swmol     Stomatal conductance to water vapour                                                                                  mol  m$^{-2}$  s$^{-1}$\n",
       "  g_tw        Total leaf conductance to water vapour                                                                                m  s$^{-1}$\n",
       "  g_twmol     Total leaf layer conductance to water vapour                                                                          mol  m$^{-2}$  s$^{-1}$\n",
       "  h_c         Average 1-sided convective transfer coefficient                                                                       J  K$^{-1}$  m$^{-2}$  s$^{-1}$\n",
       "  H_l         Sensible heat flux from leaf                                                                                          J  m$^{-2}$  s$^{-1}$\n",
       "  k_a         Thermal conductivity of dry air                                                                                       J  K$^{-1}$  m$^{-1}$  s$^{-1}$\n",
       "  k_dv        Ratio $D_{va}/V_m$                                                                                                    mol  m$^{-1}$  s$^{-1}$\n",
       "  L_l         Characteristic length scale for convection (size of leaf)                                                             m\n",
       "  l_p         Pore length                                                                                                           m\n",
       "  lambda_E    Latent heat of evaporation (2.45e6)                                                                                   J  kg$^{-1}$\n",
       "  M_N2        Molar mass of nitrogen (0.028)                                                                                        kg  mol$^{-1}$\n",
       "  M_O2        Molar mass of oxygen (0.032)                                                                                          kg  mol$^{-1}$\n",
       "  M_w         Molar mass of water (0.018)                                                                                           kg  mol$^{-1}$\n",
       "  Gr          Grashof number                                                                                                        1\n",
       "  Le          Lewis number                                                                                                          1\n",
       "  Nu          Nusselt number                                                                                                        1\n",
       "  n_p         Pore density                                                                                                           m$^{-2}$\n",
       "  Re_c        Critical Reynolds number for the onset of turbulence                                                                  1\n",
       "  Re          Reynolds number                                                                                                       1\n",
       "  Sh          Sherwood number                                                                                                       1\n",
       "  nu_a        Kinematic viscosity of dry air                                                                                         m$^{2}$  s$^{-1}$\n",
       "  P_a         Air pressure                                                                                                          Pa\n",
       "  P_N2        Partial pressure of nitrogen in the atmosphere                                                                        Pa\n",
       "  P_O2        Partial pressure of oxygen in the atmosphere                                                                          Pa\n",
       "  P_wa        Vapour pressure in the atmosphere                                                                                     Pa\n",
       "  P_was       Saturation vapour pressure at air temperature                                                                         Pa\n",
       "  P_wl        Vapour pressure inside the leaf                                                                                       Pa\n",
       "  Pr          Prandtl number (0.71)                                                                                                 1\n",
       "  r_bw        Boundary layer resistance to water vapour, inverse of $g_{bw}$                                                        s  m$^{-1}$\n",
       "  r_bwmol     Leaf BL resistance in molar units                                                                                     s  m$^{2}$  mol$^{-1}$\n",
       "  r_end       End correction, representing resistance between evaporating sites and pores                                           s  m$^{2}$  mol$^{-1}$\n",
       "  R_ll        Longwave radiation away from leaf                                                                                     J  m$^{-2}$  s$^{-1}$\n",
       "  R_mol       Molar gas constant (8.314472)                                                                                         J  K$^{-1}$  mol$^{-1}$\n",
       "  r_p         Pore radius (for ellipsoidal pores, half the pore width)                                                              m\n",
       "  R_s         Solar shortwave flux                                                                                                  J  m$^{-2}$  s$^{-1}$\n",
       "  r_sp        Diffusive resistance of a stomatal pore                                                                               s  m$^{2}$  mol$^{-1}$\n",
       "  r_sw        Stomatal resistance to water vapour, inverse of $g_{sw}$                                                              s  m$^{-1}$\n",
       "  r_tw        Total leaf resistance to water vapour, $r_{bv} + r_{sv}$                                                              s  m$^{-1}$\n",
       "  r_vs        Diffusive resistance of a stomatal vapour shell                                                                       s  m$^{2}$  mol$^{-1}$\n",
       "  rho_a       Density of dry air                                                                                                    kg  m$^{-3}$\n",
       "  rho_al      Density of air at the leaf surface                                                                                    kg  m$^{-3}$\n",
       "  s_p         Spacing between stomata                                                                                               m\n",
       "  sigm        Stefan-Boltzmann constant (5.67e-8)                                                                                   J  K$^{-4}$  m$^{-2}$  s$^{-1}$\n",
       "  T_a         Air temperature                                                                                                       K\n",
       "  T_l         Leaf temperature                                                                                                      K\n",
       "  T_w         Radiative temperature of objects surrounding the leaf                                                                 K\n",
       "  V_m         Molar volume of air                                                                                                    m$^{3}$  mol$^{-1}$\n",
       "  v_w         Wind velocity                                                                                                         m  s$^{-1}$"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Creating dictionary to substitute names of units with shorter forms\n",
    "var('m s J Pa K kg mol')\n",
    "subsdict = {meter: m, second: s, joule: J, pascal: Pa, kelvin: K, kilogram: kg, mole: mol}\n",
    "var('N_Re_L N_Re_c N_Le N_Nu_L N_Gr_L N_Sh_L')\n",
    "dict_varnew = {Re: N_Re_L, Re_c: N_Re_c, Le: N_Le, Nu: N_Nu_L, Gr: N_Gr_L, Sh: N_Sh_L}\n",
    "dict_varold = {v: k for k, v in dict_varnew.iteritems()}\n",
    "variables = sorted([str(variable.subs(dict_varnew)) for variable in udict.keys()],key=str.lower)\n",
    "tableheader = [('Variable', 'Description (value)', 'Units')]\n",
    "tabledata = [('Variable', 'Description (value)', 'Units')]\n",
    "for variable1 in variables:\n",
    "    variable2 = eval(variable1).subs(dict_varold)\n",
    "    variable = str(variable2)\n",
    "    tabledata.append((eval(variable),docdict[eval(variable)],fun_units_formatted(variable)))\n",
    "\n",
    "table(tabledata, header_row=True)"
   ]
  }
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