Symbol | Name | Description | Definition | Default value | Units |
$\\alpha_l$ | alpha_l | Leaf albedo, fraction of shortwave radiation reflected by the leaf. | | - | 1 |
$\\epsilon_l$ | epsilon_l | Longwave emmissivity of the leaf surface. | | 1.0 | 1 |
$\\rho_{al}$ | rho_al | Density of air at the leaf surface. | | - | kg m$^{-3}$ |
$a_s$ | a_s | Fraction of one-sided leaf area covered by stomata.\n",
"\n",
" (1 if stomata are on one side only, 2 if they are on both sides).\n",
" | | - | 1 |
$a_{sh}$ | a_sh | Fraction of projected area exchanging sensible heat with the air. | | 2.0 | 1 |
$C_{wl}$ | C_wl | Concentration of water in the leaf air space. | | - | mol m$^{-3}$ |
$E_l$ | E_l | Latent heat flux from leaf. | | - | J s$^{-1}$ m$^{-2}$ |
$E_{l,mol}$ | E_lmol | Transpiration rate in molar units. | | - | mol s$^{-1}$ m$^{-2}$ |
$g_{bw,mol}$ | g_bwmol | Boundary layer conductance to water vapour. | | - | mol s$^{-1}$ m$^{-2}$ |
$g_{bw}$ | g_bw | Boundary layer conductance to water vapour. | | - | m s$^{-1}$ |
$g_{sw,mol}$ | g_swmol | Stomatal conductance to water vapour. | | - | mol s$^{-1}$ m$^{-2}$ |
$g_{sw}$ | g_sw | Stomatal conductance to water vapour. | | - | m s$^{-1}$ |
$g_{tw,mol}$ | g_twmol | Total leaf layer conductance to water vapour. | | - | mol s$^{-1}$ m$^{-2}$ |
$g_{tw}$ | g_tw | Total leaf conductance to water vapour. | | - | m s$^{-1}$ |
$h_c$ | h_c | Average 1-sided convective heat transfer coefficient. | | - | J K$^{-1}$ s$^{-1}$ m$^{-2}$ |
$H_l$ | H_l | Sensible heat flux from leaf. | | - | J s$^{-1}$ m$^{-2}$ |
$L_A$ | L_A | Leaf area. | | - | m$^{2}$ |
$L_l$ | L_l | Leaf width as characteristic length scale for convection. | | - | m |
$N_{Gr_L}$ | Gr | Grashof number. | | - | 1 |
$P_{wl}$ | P_wl | Water vapour pressure inside the leaf. | | - | Pa |
$r_{bw}$ | r_bw | Boundary layer resistance to water vapour, inverse of $g_{bw}$. | | - | s m$^{-1}$ |
$R_{la}$ | R_la | Longwave radiation absorbed by leaf. | | - | W m$^{-2}$ |
$R_{ld}$ | R_ld | Downwards emitted/reflected global radiation from leaf. | | - | W m$^{-2}$ |
$R_{ll}$ | R_ll | Longwave radiation away from leaf. | | - | W m$^{-2}$ |
$R_{lu}$ | R_lu | Upwards emitted/reflected global radiation from leaf. | | - | W m$^{-2}$ |
$r_{sw}$ | r_sw | Stomatal resistance to water vapour, inverse of $g_{sw}$. | | - | s m$^{-1}$ |
$r_{tw}$ | r_tw | Total leaf resistance to water vapour, $r_{bv} + r_{sv}$. | | - | s m$^{-1}$ |
$T_l$ | T_l | Leaf temperature. | | - | K |
$T_w$ | T_w | Radiative temperature of objects surrounding the leaf. | | - | K |
"
],
"text/plain": [
"[('Symbol', 'Name', 'Description', 'Definition', 'Default value', 'Units'),\n",
" ('$\\\\alpha_l$',\n",
" 'alpha_l',\n",
" 'Leaf albedo, fraction of shortwave radiation reflected by the leaf.',\n",
" '',\n",
" '-',\n",
" '1'),\n",
" ('$\\\\epsilon_l$',\n",
" 'epsilon_l',\n",
" 'Longwave emmissivity of the leaf surface.',\n",
" '',\n",
" '1.0',\n",
" '1'),\n",
" ('$\\\\rho_{al}$',\n",
" 'rho_al',\n",
" 'Density of air at the leaf surface.',\n",
" '',\n",
" '-',\n",
" 'kg m$^{-3}$'),\n",
" ('$a_s$',\n",
" 'a_s',\n",
" 'Fraction of one-sided leaf area covered by stomata.\\n\\n (1 if stomata are on one side only, 2 if they are on both sides).\\n ',\n",
" '',\n",
" '-',\n",
" '1'),\n",
" ('$a_{sh}$',\n",
" 'a_sh',\n",
" 'Fraction of projected area exchanging sensible heat with the air.',\n",
" '',\n",
" '2.0',\n",
" '1'),\n",
" ('$C_{wl}$',\n",
" 'C_wl',\n",
" 'Concentration of water in the leaf air space.',\n",
" '',\n",
" '-',\n",
" 'mol m$^{-3}$'),\n",
" ('$E_l$',\n",
" 'E_l',\n",
" 'Latent heat flux from leaf.',\n",
" '',\n",
" '-',\n",
" 'J s$^{-1}$ m$^{-2}$'),\n",
" ('$E_{l,mol}$',\n",
" 'E_lmol',\n",
" 'Transpiration rate in molar units.',\n",
" '',\n",
" '-',\n",
" 'mol s$^{-1}$ m$^{-2}$'),\n",
" ('$g_{bw,mol}$',\n",
" 'g_bwmol',\n",
" 'Boundary layer conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'mol s$^{-1}$ m$^{-2}$'),\n",
" ('$g_{bw}$',\n",
" 'g_bw',\n",
" 'Boundary layer conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'm s$^{-1}$'),\n",
" ('$g_{sw,mol}$',\n",
" 'g_swmol',\n",
" 'Stomatal conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'mol s$^{-1}$ m$^{-2}$'),\n",
" ('$g_{sw}$',\n",
" 'g_sw',\n",
" 'Stomatal conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'm s$^{-1}$'),\n",
" ('$g_{tw,mol}$',\n",
" 'g_twmol',\n",
" 'Total leaf layer conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'mol s$^{-1}$ m$^{-2}$'),\n",
" ('$g_{tw}$',\n",
" 'g_tw',\n",
" 'Total leaf conductance to water vapour.',\n",
" '',\n",
" '-',\n",
" 'm s$^{-1}$'),\n",
" ('$h_c$',\n",
" 'h_c',\n",
" 'Average 1-sided convective heat transfer coefficient.',\n",
" '',\n",
" '-',\n",
" 'J K$^{-1}$ s$^{-1}$ m$^{-2}$'),\n",
" ('$H_l$',\n",
" 'H_l',\n",
" 'Sensible heat flux from leaf.',\n",
" '',\n",
" '-',\n",
" 'J s$^{-1}$ m$^{-2}$'),\n",
" ('$L_A$', 'L_A', 'Leaf area.', '', '-', 'm$^{2}$'),\n",
" ('$L_l$',\n",
" 'L_l',\n",
" 'Leaf width as characteristic length scale for convection.',\n",
" '',\n",
" '-',\n",
" 'm'),\n",
" ('$N_{Gr_L}$', 'Gr', 'Grashof number.', '', '-', '1'),\n",
" ('$P_{wl}$', 'P_wl', 'Water vapour pressure inside the leaf.', '', '-', 'Pa'),\n",
" ('$r_{bw}$',\n",
" 'r_bw',\n",
" 'Boundary layer resistance to water vapour, inverse of $g_{bw}$.',\n",
" '',\n",
" '-',\n",
" 's m$^{-1}$'),\n",
" ('$R_{la}$',\n",
" 'R_la',\n",
" 'Longwave radiation absorbed by leaf.',\n",
" '',\n",
" '-',\n",
" 'W m$^{-2}$'),\n",
" ('$R_{ld}$',\n",
" 'R_ld',\n",
" 'Downwards emitted/reflected global radiation from leaf.',\n",
" '',\n",
" '-',\n",
" 'W m$^{-2}$'),\n",
" ('$R_{ll}$',\n",
" 'R_ll',\n",
" 'Longwave radiation away from leaf.',\n",
" '',\n",
" '-',\n",
" 'W m$^{-2}$'),\n",
" ('$R_{lu}$',\n",
" 'R_lu',\n",
" 'Upwards emitted/reflected global radiation from leaf.',\n",
" '',\n",
" '-',\n",
" 'W m$^{-2}$'),\n",
" ('$r_{sw}$',\n",
" 'r_sw',\n",
" 'Stomatal resistance to water vapour, inverse of $g_{sw}$.',\n",
" '',\n",
" '-',\n",
" 's m$^{-1}$'),\n",
" ('$r_{tw}$',\n",
" 'r_tw',\n",
" 'Total leaf resistance to water vapour, $r_{bv} + r_{sv}$.',\n",
" '',\n",
" '-',\n",
" 's m$^{-1}$'),\n",
" ('$T_l$', 'T_l', 'Leaf temperature.', '', '-', 'K'),\n",
" ('$T_w$',\n",
" 'T_w',\n",
" 'Radiative temperature of objects surrounding the leaf.',\n",
" '',\n",
" '-',\n",
" 'K')]"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import essm.variables.leaf.energy_water as leaf_energy\n",
"vars = ['leaf_energy.' + name for name in leaf_energy.__all__]\n",
"generate_metadata_table([eval(name) for name in vars])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Equations for leaf energy and water balance\n",
"General equations based on the above variables can be imported from `essm.equations.physics.thermodynamics`:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"eq_Rs_enbal | Calculate R_s from energy balance.\n",
"\n",
" (Eq. 1 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $R_s = E_l + H_l + R_{ll}$ |
eq_Rll | R_ll as function of T_l and T_w.\n",
"\n",
" (Eq. 2 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $R_{ll} = a_{sh} \\epsilon_l \\sigma \\left(T_l^{4} - T_w^{4}\\right)$ |
eq_Hl | H_l as function of T_l.\n",
"\n",
" (Eq. 3 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $H_l = a_{sh} h_c \\left(- T_a + T_l\\right)$ |
eq_El | E_l as function of E_lmol.\n",
"\n",
" (Eq. 4 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $E_l = E_{l,mol} M_w \\lambda_E$ |
eq_Elmol | E_lmol as functino of g_tw and C_wl.\n",
"\n",
" (Eq. 5 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $E_{l,mol} = g_{tw} \\left(- C_{wa} + C_{wl}\\right)$ |
eq_gtw | g_tw from g_sw and g_bw.\n",
"\n",
" (Eq. 6 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $g_{tw} = \\frac{1}{\\frac{1}{g_{sw}} + \\frac{1}{g_{bw}}}$ |
eq_gbw_hc | g_bw as function of h_c.\n",
"\n",
" (Eq. B2 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $g_{bw} = \\frac{a_s h_c}{N_{Le}^{\\frac{2}{3}} c_{pa} \\rho_a}$ |
eq_Cwl | C_wl as function of P_wl and T_l.\n",
"\n",
" (Eq. B4 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $C_{wl} = \\frac{P_{wl}}{R_{mol} T_l}$ |
eq_Pwl | Clausius-Clapeyron P_wl as function of T_l.\n",
"\n",
" (Eq. B3 in :cite:`hartmann_global_1994`)\n",
" | $P_{wl} = p_1 e^{- \\frac{M_w \\lambda_E \\left(- \\frac{1}{p_2} + \\frac{1}{T_l}\\right)}{R_{mol}}}$ |
eq_Elmol_conv | E_lmol as function of g_twmol and P_wl.\n",
"\n",
" (Eq. B6 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $E_{l,mol} = \\frac{g_{tw,mol} \\left(- P_{wa} + P_{wl}\\right)}{P_a}$ |
eq_gtwmol_gtw | g_twmol as a function of g_tw.\n",
"\n",
" It uses eq_Elmol, eq_Cwl and eq_Elmol_conv.\n",
" | $g_{tw,mol} = \\frac{g_{tw} \\left(P_a P_{wa} T_l - P_a P_{wl} T_a\\right)}{R_{mol} T_a T_l \\left(P_{wa} - P_{wl}\\right)}$ |
eq_gtwmol_gtw_iso | g_twmol as a function of g_tw at isothermal conditions. | $g_{tw,mol} = \\frac{P_a g_{tw}}{R_{mol} T_a}$ |
eq_hc | h_c as a function of Nu and L_l.\n",
"\n",
" (Eq. B10 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $h_c = \\frac{N_{Nu_L} k_a}{L_l}$ |
eq_Re | Re as a function of v_w and L_l.\n",
"\n",
" (Eq. B11 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $N_{Re_L} = \\frac{L_l v_w}{\\nu_a}$ |
eq_Gr | Gr as function of air density within and outside of leaf.\n",
"\n",
" (Eq. B12 in :cite:`schymanski_leaf-scale_2017`)\n",
" | $N_{Gr_L} = \\frac{L_l^{3} g \\left(\\rho_a - \\rho_{al}\\right)}{\\nu_a^{2} \\rho_{al}}$ |
"
],
"text/plain": [
"[('eq_Rs_enbal',\n",
" 'Calculate R_s from energy balance.\\n\\n (Eq. 1 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$R_s = E_l + H_l + R_{ll}$'),\n",
" ('eq_Rll',\n",
" 'R_ll as function of T_l and T_w.\\n\\n (Eq. 2 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$R_{ll} = a_{sh} \\\\epsilon_l \\\\sigma \\\\left(T_l^{4} - T_w^{4}\\\\right)$'),\n",
" ('eq_Hl',\n",
" 'H_l as function of T_l.\\n\\n (Eq. 3 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$H_l = a_{sh} h_c \\\\left(- T_a + T_l\\\\right)$'),\n",
" ('eq_El',\n",
" 'E_l as function of E_lmol.\\n\\n (Eq. 4 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$E_l = E_{l,mol} M_w \\\\lambda_E$'),\n",
" ('eq_Elmol',\n",
" 'E_lmol as functino of g_tw and C_wl.\\n\\n (Eq. 5 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$E_{l,mol} = g_{tw} \\\\left(- C_{wa} + C_{wl}\\\\right)$'),\n",
" ('eq_gtw',\n",
" 'g_tw from g_sw and g_bw.\\n\\n (Eq. 6 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$g_{tw} = \\\\frac{1}{\\\\frac{1}{g_{sw}} + \\\\frac{1}{g_{bw}}}$'),\n",
" ('eq_gbw_hc',\n",
" 'g_bw as function of h_c.\\n\\n (Eq. B2 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$g_{bw} = \\\\frac{a_s h_c}{N_{Le}^{\\\\frac{2}{3}} c_{pa} \\\\rho_a}$'),\n",
" ('eq_Cwl',\n",
" 'C_wl as function of P_wl and T_l.\\n\\n (Eq. B4 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$C_{wl} = \\\\frac{P_{wl}}{R_{mol} T_l}$'),\n",
" ('eq_Pwl',\n",
" 'Clausius-Clapeyron P_wl as function of T_l.\\n\\n (Eq. B3 in :cite:`hartmann_global_1994`)\\n ',\n",
" '$P_{wl} = p_1 e^{- \\\\frac{M_w \\\\lambda_E \\\\left(- \\\\frac{1}{p_2} + \\\\frac{1}{T_l}\\\\right)}{R_{mol}}}$'),\n",
" ('eq_Elmol_conv',\n",
" 'E_lmol as function of g_twmol and P_wl.\\n\\n (Eq. B6 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$E_{l,mol} = \\\\frac{g_{tw,mol} \\\\left(- P_{wa} + P_{wl}\\\\right)}{P_a}$'),\n",
" ('eq_gtwmol_gtw',\n",
" 'g_twmol as a function of g_tw.\\n\\n It uses eq_Elmol, eq_Cwl and eq_Elmol_conv.\\n ',\n",
" '$g_{tw,mol} = \\\\frac{g_{tw} \\\\left(P_a P_{wa} T_l - P_a P_{wl} T_a\\\\right)}{R_{mol} T_a T_l \\\\left(P_{wa} - P_{wl}\\\\right)}$'),\n",
" ('eq_gtwmol_gtw_iso',\n",
" 'g_twmol as a function of g_tw at isothermal conditions.',\n",
" '$g_{tw,mol} = \\\\frac{P_a g_{tw}}{R_{mol} T_a}$'),\n",
" ('eq_hc',\n",
" 'h_c as a function of Nu and L_l.\\n\\n (Eq. B10 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$h_c = \\\\frac{N_{Nu_L} k_a}{L_l}$'),\n",
" ('eq_Re',\n",
" 'Re as a function of v_w and L_l.\\n\\n (Eq. B11 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$N_{Re_L} = \\\\frac{L_l v_w}{\\\\nu_a}$'),\n",
" ('eq_Gr',\n",
" 'Gr as function of air density within and outside of leaf.\\n\\n (Eq. B12 in :cite:`schymanski_leaf-scale_2017`)\\n ',\n",
" '$N_{Gr_L} = \\\\frac{L_l^{3} g \\\\left(\\\\rho_a - \\\\rho_{al}\\\\right)}{\\\\nu_a^{2} \\\\rho_{al}}$')]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import essm.equations.leaf.energy_water as leaf_energy\n",
"modstr = 'leaf_energy.'\n",
"eqs = [name for name in leaf_energy.__all__]\n",
"table = ListTable()\n",
"#table.append(('Name', 'Description', 'Equation'))\n",
"for name in eqs:\n",
" table.append((name, eval(modstr+name).__doc__, latex('$'+latex(eval(modstr+name))+'$')))\n",
"table"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Variables for leaf radiative balance\n",
"Variables related to the model by can be imported from `essm.variables.leaf.radiation`:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/home/stan/Programs/essm/essm/variables/_core.py:89: UserWarning: \"essm.variables.leaf.energy_water:alpha_l\" will be overridden by \"essm.variables.leaf.radiation: