1820 lines
77 KiB
ReStructuredText
1820 lines
77 KiB
ReStructuredText
.. _rst_Momentum, Sensible Heat, and Latent Heat Fluxes:
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Momentum, Sensible Heat, and Latent Heat Fluxes
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==================================================
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The zonal :math:`\tau _{x}` and meridional :math:`\tau _{y}` momentum
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fluxes (kg m\ :sup:`-1` s\ :sup:`-2`), sensible heat flux
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:math:`H` (W m\ :sup:`-2`), and water vapor flux :math:`E` (kg m\ :sup:`-2` s\ :sup:`-1`) between the atmosphere at
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reference height :math:`z_{atm,\, x}` (m) [where :math:`x` is height
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for wind (momentum) (:math:`m`), temperature (sensible heat)
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(:math:`h`), and humidity (water vapor) (:math:`w`); with zonal and
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meridional winds :math:`u_{atm}` and :math:`v_{atm}` (m
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s\ :sup:`-1`), potential temperature :math:`\theta _{atm}` (K),
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and specific humidity :math:`q_{atm}` (kg kg\ :sup:`-1`)] and the
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surface [with :math:`u_{s}` , :math:`v_{s}` , :math:`\theta _{s}` , and
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:math:`q_{s}` ] are
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.. math::
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:label: 5.1
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\tau _{x} =-\rho _{atm} \frac{\left(u_{atm} -u_{s} \right)}{r_{am} }
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.. math::
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:label: 5.2
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\tau _{y} =-\rho _{atm} \frac{\left(v_{atm} -v_{s} \right)}{r_{am} }
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.. math::
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:label: 5.3
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H=-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -\theta _{s} \right)}{r_{ah} }
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.. math::
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:label: 5.4
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E=-\rho _{atm} \frac{\left(q_{atm} -q_{s} \right)}{r_{aw} } .
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These fluxes are derived in the next section from Monin-Obukhov
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similarity theory developed for the surface layer (i.e., the nearly
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constant flux layer above the surface sublayer). In this derivation,
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:math:`u_{s}` and :math:`v_{s}` are defined to equal zero at height
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:math:`z_{0m} +d` (the apparent sink for momentum) so that
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:math:`r_{am}` is the aerodynamic resistance (s m\ :sup:`-1`) for
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momentum between the atmosphere at height :math:`z_{atm,\, m}` and the
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surface at height :math:`z_{0m} +d`. Thus, the momentum fluxes become
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.. math::
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:label: 5.5
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\tau _{x} =-\rho _{atm} \frac{u_{atm} }{r_{am} }
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.. math::
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:label: 5.6
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\tau _{y} =-\rho _{atm} \frac{v_{atm} }{r_{am} } .
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Likewise, :math:`\theta _{s}` and :math:`q_{s}` are defined at
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heights :math:`z_{0h} +d` and :math:`z_{0w} +d` (the apparent sinks for
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heat and water vapor, respectively
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:math:`r_{aw}` are the aerodynamic resistances (s m\ :sup:`-1`)
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to sensible heat and water vapor transfer between the atmosphere at
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heights :math:`z_{atm,\, h}` and :math:`z_{atm,\, w}` and the surface
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at heights :math:`z_{0h} +d` and :math:`z_{0w} +d`, respectively. The
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specific heat capacity of air :math:`C_{p}` (J kg\ :sup:`-1`
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K\ :sup:`-1`) is a constant (:numref:`Table Physical constants`). The atmospheric potential
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temperature used here is
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.. math::
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:label: 5.7
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\theta _{atm} =T_{atm} +\Gamma _{d} z_{atm,\, h}
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where :math:`T_{atm}` is the air temperature (K) at height
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:math:`z_{atm,\, h}` and :math:`\Gamma _{d} =0.0098` K
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m\ :sup:`-1` is the negative of the dry adiabatic lapse rate [this
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expression is first-order equivalent to
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:math:`\theta _{atm} =T_{atm} \left({P_{srf} \mathord{\left/ {\vphantom {P_{srf} P_{atm} }} \right. \kern-\nulldelimiterspace} P_{atm} } \right)^{{R_{da} \mathord{\left/ {\vphantom {R_{da} C_{p} }} \right. \kern-\nulldelimiterspace} C_{p} } }`
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(:ref:`Stull 1988 <Stull1988>`), where :math:`P_{srf}` is the surface pressure (Pa),
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:math:`P_{atm}` is the atmospheric pressure (Pa), and :math:`R_{da}`
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is the gas constant for dry air (J kg\ :sup:`-1` K\ :sup:`-1`) (:numref:`Table Physical constants`)]. By definition,
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:math:`\theta _{s} =T_{s}` . The density of moist air (kg m\ :sup:`-3`) is
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.. math::
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:label: 5.8
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\rho _{atm} =\frac{P_{atm} -0.378e_{atm} }{R_{da} T_{atm} }
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where the atmospheric vapor pressure :math:`e_{atm}` (Pa) is derived
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from the atmospheric specific humidity :math:`q_{atm}`
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.. math::
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:label: 5.9
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e_{atm} =\frac{q_{atm} P_{atm} }{0.622+0.378q_{atm} } .
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.. _Monin-Obukhov Similarity Theory:
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Monin-Obukhov Similarity Theory
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-----------------------------------
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The surface vertical kinematic fluxes of momentum
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:math:`\overline{u'w'}` and :math:`\overline{v'w'}` (m\ :sup:`2` s\ :sub:`-2`), sensible heat :math:`\overline{\theta 'w'}`
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(K m s :sup:`-1`), and latent heat :math:`\overline{q'w'}` (kg kg\ :sup:`-1` m s\ :sup:`-1`), where :math:`u'`, :math:`v'`,
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:math:`w'`, :math:`\theta '`, and :math:`q'` are zonal horizontal wind,
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meridional horizontal wind, vertical velocity, potential temperature,
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and specific humidity turbulent fluctuations about the mean, are defined
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from Monin-Obukhov similarity applied to the surface layer. This theory
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states that when scaled appropriately, the dimensionless mean horizontal
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wind speed, mean potential temperature, and mean specific humidity
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profile gradients depend on unique functions of
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:math:`\zeta =\frac{z-d}{L}` (:ref:`Zeng et al. 1998<Zengetal1998>`) as
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.. math::
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:label: 5.10
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\frac{k\left(z-d\right)}{u_{*} } \frac{\partial \left|{\it u}\right|}{\partial z} =\phi _{m} \left(\zeta \right)
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.. math::
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:label: 5.11
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\frac{k\left(z-d\right)}{\theta _{*} } \frac{\partial \theta }{\partial z} =\phi _{h} \left(\zeta \right)
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.. math::
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:label: 5.12
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\frac{k\left(z-d\right)}{q_{*} } \frac{\partial q}{\partial z} =\phi _{w} \left(\zeta \right)
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where :math:`z` is height in the surface layer (m), :math:`d` is the
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displacement height (m), :math:`L` is the Monin-Obukhov length scale (m)
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that accounts for buoyancy effects resulting from vertical density
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gradients (i.e., the atmospheric stability), k is the von Karman
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constant (:numref:`Table Physical constants`), and :math:`\left|{\it u}\right|` is the
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atmospheric wind speed (m s\ :sup:`-1`). :math:`\phi _{m}` ,
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:math:`\phi _{h}` , and :math:`\phi _{w}` are universal (over any
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surface) similarity functions of :math:`\zeta` that relate the constant
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fluxes of momentum, sensible heat, and latent heat to the mean profile
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gradients of :math:`\left|{\it u}\right|`, :math:`\theta` , and
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:math:`q` in the surface layer. In neutral conditions,
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:math:`\phi _{m} =\phi _{h} =\phi _{w} =1`. The velocity (i.e., friction
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velocity) :math:`u_{\*}` (m s\ :sup:`-1`), temperature
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:math:`\theta _{\*}` (K), and moisture :math:`q_{\*}` (kg kg\ :sup:`-1`) scales are
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.. math::
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:label: 5.13
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u_{*}^{2} =\sqrt{\left(\overline{u'w'}\right)^{2} +\left(\overline{v'w'}\right)^{2} } =\frac{\left|{\it \tau }\right|}{\rho _{atm} }
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.. math::
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:label: 5.14
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\theta _{*} u_{*} =-\overline{\theta 'w'}=-\frac{H}{\rho _{atm} C_{p} }
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.. math::
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:label: 5.15
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q_{*} u_{*} =-\overline{q'w'}=-\frac{E}{\rho _{atm} }
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where :math:`\left|{\it \tau }\right|` is the shearing stress (kg m\ :sup:`-1` s\ :sup:`-2`), with zonal and meridional
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components :math:`\overline{u'w'}=-\frac{\tau _{x} }{\rho _{atm} }` and
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:math:`\overline{v'w'}=-\frac{\tau _{y} }{\rho _{atm} }` , respectively,
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:math:`H` is the sensible heat flux (W m\ :sup:`-2`) and :math:`E`
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is the water vapor flux (kg m\ :sup:`-2` s\ :sup:`-1`).
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The length scale :math:`L` is the Monin-Obukhov length defined as
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.. math::
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:label: 5.16
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L=-\frac{u_{*}^{3} }{k\left(\frac{g}{\overline{\theta _{v,\, atm} }} \right)\theta '_{v} w'} =\frac{u_{*}^{2} \overline{\theta _{v,\, atm} }}{kg\theta _{v*} }
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where :math:`g` is the acceleration of gravity (m s\ :sup:`-2`)
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(:numref:`Table Physical constants`), and
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:math:`\overline{\theta _{v,\, atm} }=\overline{\theta _{atm} }\left(1+0.61q_{atm} \right)`
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is the reference virtual potential temperature. :math:`L>0` indicates
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stable conditions. :math:`L<0` indicates unstable conditions.
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:math:`L=\infty` for neutral conditions. The temperature scale
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:math:`\theta _{v*}` is defined as
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.. math::
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:label: 5.17
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\theta _{v*} u_{*} =\left[\theta _{*} \left(1+0.61q_{atm} \right)+0.61\overline{\theta _{atm} }q_{*} \right]u_{*}
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where :math:`\overline{\theta _{atm} }` is the atmospheric potential
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temperature.
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Following :ref:`Panofsky and Dutton (1984)<PanofskyDutton1984>`, the differential equations for
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:math:`\phi _{m} \left(\zeta \right)`,
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:math:`\phi _{h} \left(\zeta \right)`, and
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:math:`\phi _{w} \left(\zeta \right)` can be integrated formally without
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commitment to their exact forms. Integration between two arbitrary
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heights in the surface layer :math:`z_{2}` and :math:`z_{1}`
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(:math:`z_{2} >z_{1}` ) with horizontal winds
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:math:`\left|{\it u}\right|_{1}` and :math:`\left|{\it u}\right|_{2}` ,
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potential temperatures :math:`\theta _{1}` and :math:`\theta _{2}` ,
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and specific humidities :math:`q_{1}` and :math:`q_{2}` results in
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.. math::
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:label: 5.18
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\left|{\it u}\right|_{2} -\left|{\it u}\right|_{1} =\frac{u_{*} }{k} \left[\ln \left(\frac{z_{2} -d}{z_{1} -d} \right)-\psi _{m} \left(\frac{z_{2} -d}{L} \right)+\psi _{m} \left(\frac{z_{1} -d}{L} \right)\right]
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.. math::
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:label: 5.19
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\theta _{2} -\theta _{1} =\frac{\theta _{*} }{k} \left[\ln \left(\frac{z_{2} -d}{z_{1} -d} \right)-\psi _{h} \left(\frac{z_{2} -d}{L} \right)+\psi _{h} \left(\frac{z_{1} -d}{L} \right)\right]
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.. math::
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:label: 5.20
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q_{2} -q_{1} =\frac{q_{*} }{k} \left[\ln \left(\frac{z_{2} -d}{z_{1} -d} \right)-\psi _{w} \left(\frac{z_{2} -d}{L} \right)+\psi _{w} \left(\frac{z_{1} -d}{L} \right)\right].
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The functions :math:`\psi _{m} \left(\zeta \right)`,
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:math:`\psi _{h} \left(\zeta \right)`, and
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:math:`\psi _{w} \left(\zeta \right)` are defined as
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.. math::
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:label: 5.21
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\psi _{m} \left(\zeta \right)=\int _{{z_{0m} \mathord{\left/ {\vphantom {z_{0m} L}} \right. \kern-\nulldelimiterspace} L} }^{\zeta }\frac{\left[1-\phi _{m} \left(x\right)\right]}{x} \, dx
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.. math::
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:label: 5.22
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\psi _{h} \left(\zeta \right)=\int _{{z_{0h} \mathord{\left/ {\vphantom {z_{0h} L}} \right. \kern-\nulldelimiterspace} L} }^{\zeta }\frac{\left[1-\phi _{h} \left(x\right)\right]}{x} \, dx
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.. math::
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:label: 5.23
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\psi _{w} \left(\zeta \right)=\int _{{z_{0w} \mathord{\left/ {\vphantom {z_{0w} L}} \right. \kern-\nulldelimiterspace} L} }^{\zeta }\frac{\left[1-\phi _{w} \left(x\right)\right]}{x} \, dx
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where :math:`z_{0m}` , :math:`z_{0h}` , and :math:`z_{0w}` are the
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roughness lengths (m) for momentum, sensible heat, and water vapor,
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respectively.
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Defining the surface values
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.. math:: \left|{\it u}\right|_{1} =0{\rm \; at\; }z_{1} =z_{0m} +d,
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.. math:: \theta _{1} =\theta _{s} {\rm \; at\; }z_{1} =z_{0h} +d,{\rm \; and}
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.. math:: q_{1} =q_{s} {\rm \; at\; }z_{1} =z_{0w} +d,
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and the atmospheric values at :math:`z_{2} =z_{atm,\, x}`
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.. math::
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:label: 5.24
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\left|{\it u}\right|_{2} =V_{a} {\rm =\; }\sqrt{u_{atm}^{2} +v_{atm}^{2} +U_{c}^{2} } \ge 1,
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.. math:: \theta _{2} =\theta _{atm} {\rm ,\; and}
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.. math:: q_{2} =q_{atm} {\rm ,\; }
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the integral forms of the flux-gradient relations are
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.. math::
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:label: 5.25
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V_{a} =\frac{u_{*} }{k} \left[\ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)-\psi _{m} \left(\frac{z_{atm,\, m} -d}{L} \right)+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right]
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.. math::
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:label: 5.26
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\theta _{atm} -\theta _{s} =\frac{\theta _{*} }{k} \left[\ln \left(\frac{z_{atm,\, h} -d}{z_{0h} } \right)-\psi _{h} \left(\frac{z_{atm,\, h} -d}{L} \right)+\psi _{h} \left(\frac{z_{0h} }{L} \right)\right]
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.. math::
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:label: 5.27
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q_{atm} -q_{s} =\frac{q_{*} }{k} \left[\ln \left(\frac{z_{atm,\, w} -d}{z_{0w} } \right)-\psi _{w} \left(\frac{z_{atm,\, w} -d}{L} \right)+\psi _{w} \left(\frac{z_{0w} }{L} \right)\right].
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The constraint :math:`V_{a} \ge 1` is required simply for numerical
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reasons to prevent :math:`H` and :math:`E` from becoming small with
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small wind speeds. The convective velocity :math:`U_{c}` accounts for
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the contribution of large eddies in the convective boundary layer to
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surface fluxes as follows
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.. math::
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:label: 5.28
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U_{c} = \left\{
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\begin{array}{ll}
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0 & \qquad \zeta \ge {\rm 0} \quad {\rm (stable)} \\
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\beta w_{*} & \qquad \zeta < 0 \quad {\rm (unstable)}
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\end{array} \right\}
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where :math:`w_{*}` is the convective velocity scale
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.. math::
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:label: 5.29
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w_{*} =\left(\frac{-gu_{\*} \theta _{v*} z_{i} }{\overline{\theta _{v,\, atm} }} \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3} } ,
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:math:`z_{i} =1000` is the convective boundary layer height (m), and :math:`\beta =1`.
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The momentum flux gradient relations are (:ref:`Zeng et al. 1998 <Zengetal1998>`)
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.. math::
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:label: 5.30
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\begin{array}{llr}
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\phi _{m} \left(\zeta \right)=0.7k^{{2\mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3} } \left(-\zeta \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3} } & \qquad {\rm for\; }\zeta <-1.574 & \ {\rm \; (very\; unstable)} \\
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\phi _{m} \left(\zeta \right)=\left(1-16\zeta \right)^{-{1\mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} } & \qquad {\rm for\; -1.574}\le \zeta <0 & \ {\rm \; (unstable)} \\
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\phi _{m} \left(\zeta \right)=1+5\zeta & \qquad {\rm for\; }0\le \zeta \le 1& \ {\rm \; (stable)} \\
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\phi _{m} \left(\zeta \right)=5+\zeta & \qquad {\rm for\; }\zeta >1 & \ {\rm\; (very\; stable).}
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\end{array}
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The sensible and latent heat flux gradient relations are (:ref:`Zeng et al. 1998 <Zengetal1998>`)
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.. math::
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:label: 5.31
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\begin{array}{llr}
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\phi _{h} \left(\zeta \right)=\phi _{w} \left(\zeta \right)=0.9k^{{4\mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3} } \left(-\zeta \right)^{{-1\mathord{\left/ {\vphantom {-1 3}} \right. \kern-\nulldelimiterspace} 3} } & \qquad {\rm for\; }\zeta <-0.465 & \ {\rm \; (very\; unstable)} \\
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\phi _{h} \left(\zeta \right)=\phi _{w} \left(\zeta \right)=\left(1-16\zeta \right)^{-{1\mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} } & \qquad {\rm for\; -0.465}\le \zeta <0 & \ {\rm \; (unstable)} \\
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\phi _{h} \left(\zeta \right)=\phi _{w} \left(\zeta \right)=1+5\zeta & \qquad {\rm for\; }0\le \zeta \le 1 & \ {\rm \; (stable)} \\
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\phi _{h} \left(\zeta \right)=\phi _{w} \left(\zeta \right)=5+\zeta & \qquad {\rm for\; }\zeta >1 & \ {\rm \; (very\; stable).}
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\end{array}
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To ensure continuous functions of
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:math:`\phi _{m} \left(\zeta \right)`,
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:math:`\phi _{h} \left(\zeta \right)`, and
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:math:`\phi _{w} \left(\zeta \right)`, the simplest approach (i.e.,
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without considering any transition regimes) is to match the relations
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for very unstable and unstable conditions at :math:`\zeta _{m} =-1.574`
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for :math:`\phi _{m} \left(\zeta \right)` and
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:math:`\zeta _{h} =\zeta _{w} =-0.465` for
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:math:`\phi _{h} \left(\zeta \right)=\phi _{w} \left(\zeta \right)`
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(:ref:`Zeng et al. 1998 <Zengetal1998>`). The flux gradient relations can be integrated to
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yield wind profiles for the following conditions:
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Very unstable :math:`\left(\zeta <-1.574\right)`
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.. math::
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:label: 5.32
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V_{a} =\frac{u_{*} }{k} \left\{\left[\ln \frac{\zeta _{m} L}{z_{0m} } -\psi _{m} \left(\zeta _{m} \right)\right]+1.14\left[\left(-\zeta \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3} } -\left(-\zeta _{m} \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3} } \right]+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right\}
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Unstable :math:`\left(-1.574\le \zeta <0\right)`
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.. math::
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:label: 5.33
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V_{a} =\frac{u_{*} }{k} \left\{\left[\ln \frac{z_{atm,\, m} -d}{z_{0m} } -\psi _{m} \left(\zeta \right)\right]+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right\}
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Stable :math:`\left(0\le \zeta \le 1\right)`
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.. math::
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:label: 5.34
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V_{a} =\frac{u_{*} }{k} \left\{\left[\ln \frac{z_{atm,\, m} -d}{z_{0m} } +5\zeta \right]-5\frac{z_{0m} }{L} \right\}
|
|
|
|
Very stable :math:`\left(\zeta >1\right)`
|
|
|
|
.. math::
|
|
:label: 5.35
|
|
|
|
V_{a} =\frac{u_{*} }{k} \left\{\left[\ln \frac{L}{z_{0m} } +5\right]+\left[5\ln \zeta +\zeta -1\right]-5\frac{z_{0m} }{L} \right\}
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.36
|
|
|
|
\psi _{m} \left(\zeta \right)=2\ln \left(\frac{1+x}{2} \right)+\ln \left(\frac{1+x^{2} }{2} \right)-2\tan ^{-1} x+\frac{\pi }{2}
|
|
|
|
and
|
|
|
|
:math:`x=\left(1-16\zeta \right)^{{1\mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} }` .
|
|
|
|
The potential temperature profiles are:
|
|
|
|
Very unstable :math:`\left(\zeta <-0.465\right)`
|
|
|
|
.. math::
|
|
:label: 5.37
|
|
|
|
\theta _{atm} -\theta _{s} =\frac{\theta _{*} }{k} \left\{\left[\ln \frac{\zeta _{h} L}{z_{0h} } -\psi _{h} \left(\zeta _{h} \right)\right]+0.8\left[\left(-\zeta _{h} \right)^{{-1\mathord{\left/ {\vphantom {-1 3}} \right. \kern-\nulldelimiterspace} 3} } -\left(-\zeta \right)^{{-1\mathord{\left/ {\vphantom {-1 3}} \right. \kern-\nulldelimiterspace} 3} } \right]+\psi _{h} \left(\frac{z_{0h} }{L} \right)\right\}
|
|
|
|
Unstable :math:`\left(-0.465\le \zeta <0\right)`
|
|
|
|
.. math::
|
|
:label: 5.38
|
|
|
|
\theta _{atm} -\theta _{s} =\frac{\theta _{*} }{k} \left\{\left[\ln \frac{z_{atm,\, h} -d}{z_{0h} } -\psi _{h} \left(\zeta \right)\right]+\psi _{h} \left(\frac{z_{0h} }{L} \right)\right\}
|
|
|
|
|
|
Stable :math:`\left(0\le \zeta \le 1\right)`
|
|
|
|
.. math::
|
|
:label: 5.39
|
|
|
|
\theta _{atm} -\theta _{s} =\frac{\theta _{*} }{k} \left\{\left[\ln \frac{z_{atm,\, h} -d}{z_{0h} } +5\zeta \right]-5\frac{z_{0h} }{L} \right\}
|
|
|
|
Very stable :math:`\left(\zeta >1\right)`
|
|
|
|
.. math::
|
|
:label: 5.40
|
|
|
|
\theta _{atm} -\theta _{s} =\frac{\theta _{*} }{k} \left\{\left[\ln \frac{L}{z_{0h} } +5\right]+\left[5\ln \zeta +\zeta -1\right]-5\frac{z_{0h} }{L} \right\}.
|
|
|
|
The specific humidity profiles are:
|
|
|
|
Very unstable :math:`\left(\zeta <-0.465\right)`
|
|
|
|
.. math::
|
|
:label: 5.41
|
|
|
|
q_{atm} -q_{s} =\frac{q_{*} }{k} \left\{\left[\ln \frac{\zeta _{w} L}{z_{0w} } -\psi _{w} \left(\zeta _{w} \right)\right]+0.8\left[\left(-\zeta _{w} \right)^{{-1\mathord{\left/ {\vphantom {-1 3}} \right. \kern-\nulldelimiterspace} 3} } -\left(-\zeta \right)^{{-1\mathord{\left/ {\vphantom {-1 3}} \right. \kern-\nulldelimiterspace} 3} } \right]+\psi _{w} \left(\frac{z_{0w} }{L} \right)\right\}
|
|
|
|
Unstable :math:`\left(-0.465\le \zeta <0\right)`
|
|
|
|
.. math::
|
|
:label: 5.42
|
|
|
|
q_{atm} -q_{s} =\frac{q_{*} }{k} \left\{\left[\ln \frac{z_{atm,\, w} -d}{z_{0w} } -\psi _{w} \left(\zeta \right)\right]+\psi _{w} \left(\frac{z_{0w} }{L} \right)\right\}
|
|
|
|
Stable :math:`\left(0\le \zeta \le 1\right)`
|
|
|
|
.. math::
|
|
:label: 5.43
|
|
|
|
q_{atm} -q_{s} =\frac{q_{*} }{k} \left\{\left[\ln \frac{z_{atm,\, w} -d}{z_{0w} } +5\zeta \right]-5\frac{z_{0w} }{L} \right\}
|
|
|
|
Very stable :math:`\left(\zeta >1\right)`
|
|
|
|
.. math::
|
|
:label: 5.44
|
|
|
|
q_{atm} -q_{s} =\frac{q_{*} }{k} \left\{\left[\ln \frac{L}{z_{0w} } +5\right]+\left[5\ln \zeta +\zeta -1\right]-5\frac{z_{0w} }{L} \right\}
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.45
|
|
|
|
\psi _{h} \left(\zeta \right)=\psi _{w} \left(\zeta \right)=2\ln \left(\frac{1+x^{2} }{2} \right).
|
|
|
|
Using the definitions of :math:`u_{*}` , :math:`\theta _{*}` , and
|
|
:math:`q_{*}` , an iterative solution of these equations can be used to
|
|
calculate the surface momentum, sensible heat, and water vapor flux
|
|
using atmospheric and surface values for :math:`\left|{\it u}\right|`,
|
|
:math:`\theta` , and :math:`q` except that :math:`L` depends on
|
|
:math:`u_{*}` , :math:`\theta _{*}` , and :math:`q_{*}` . However, the
|
|
bulk Richardson number
|
|
|
|
.. math::
|
|
:label: 5.46
|
|
|
|
R_{iB} =\frac{\theta _{v,\, atm} -\theta _{v,\, s} }{\overline{\theta _{v,\, atm} }} \frac{g\left(z_{atm,\, m} -d\right)}{V_{a}^{2} }
|
|
|
|
|
|
is related to :math:`\zeta` (:ref:`Arya 2001 <Arya2001>`) as
|
|
|
|
.. math::
|
|
:label: 5.47
|
|
|
|
R_{iB} =\zeta \left[\ln \left(\frac{z_{atm,\, h} -d}{z_{0h} } \right)-\psi _{h} \left(\zeta \right)\right]\left[\ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)-\psi _{m} \left(\zeta \right)\right]^{-2} .
|
|
|
|
Using
|
|
:math:`\phi _{h} =\phi _{m}^{2} =\left(1-16\zeta \right)^{-{1\mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} }`
|
|
for unstable conditions and :math:`\phi _{h} =\phi _{m} =1+5\zeta` for
|
|
stable conditions to determine :math:`\psi _{m} \left(\zeta \right)` and
|
|
:math:`\psi _{h} \left(\zeta \right)`, the inverse relationship
|
|
:math:`\zeta =f\left(R_{iB} \right)` can be solved to obtain a first
|
|
guess for :math:`\zeta` and thus :math:`L` from
|
|
|
|
.. math::
|
|
:label: 5.48
|
|
|
|
\begin{array}{lcr}
|
|
\zeta =\frac{R_{iB} \ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)}{1-5\min \left(R_{iB} ,0.19\right)} & \qquad 0.01\le \zeta \le 2 & \qquad {\rm for\; }R_{iB} \ge 0 {\rm \; (neutral\; or\; stable)} \\
|
|
\zeta =R_{iB} \ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right) & \qquad -100\le \zeta \le -0.01 & \qquad {\rm for\; }R_{iB} <0 \ {\rm \; (unstable)}
|
|
\end{array}.
|
|
|
|
Upon iteration (section :numref:`Numerical Implementation`), the following is used to determine
|
|
:math:`\zeta` and thus :math:`L`
|
|
|
|
.. math::
|
|
:label: 5.49
|
|
|
|
\zeta =\frac{\left(z_{atm,\, m} -d\right)kg\theta _{v*} }{u_{*}^{2} \overline{\theta _{v,\, atm} }}
|
|
|
|
where
|
|
|
|
.. math::
|
|
|
|
\begin{array}{cr}
|
|
0.01\le \zeta \le 2 & \qquad {\rm for\; }\zeta \ge 0{\rm \; (neutral\; or\; stable)} \\
|
|
{\rm -100}\le \zeta \le {\rm -0.01} & \qquad {\rm for\; }\zeta <0{\rm \; (unstable)}
|
|
\end{array}.
|
|
|
|
The difference in virtual potential air temperature between the
|
|
reference height and the surface is
|
|
|
|
.. math::
|
|
:label: 5.50
|
|
|
|
\theta _{v,\, atm} -\theta _{v,\, s} =\left(\theta _{atm} -\theta _{s} \right)\left(1+0.61q_{atm} \right)+0.61\overline{\theta _{atm} }\left(q_{atm} -q_{s} \right).
|
|
|
|
The momentum, sensible heat, and water vapor fluxes between the surface
|
|
and the atmosphere can also be written in the form
|
|
|
|
.. math::
|
|
:label: 5.51
|
|
|
|
\tau _{x} =-\rho _{atm} \frac{\left(u_{atm} -u_{s} \right)}{r_{am} }
|
|
|
|
.. math::
|
|
:label: 5.52
|
|
|
|
\tau _{y} =-\rho _{atm} \frac{\left(v_{atm} -v_{s} \right)}{r_{am} }
|
|
|
|
.. math::
|
|
:label: 5.53
|
|
|
|
H=-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -\theta _{s} \right)}{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.54
|
|
|
|
E=-\rho _{atm} \frac{\left(q_{atm} -q_{s} \right)}{r_{aw} }
|
|
|
|
where the aerodynamic resistances (s m\ :sup:`-1`) are
|
|
|
|
.. math::
|
|
:label: 5.55
|
|
|
|
r_{am} =\frac{V_{a} }{u_{*}^{2} } =\frac{1}{k^{2} V_{a} } \left[\ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)-\psi _{m} \left(\frac{z_{atm,\, m} -d}{L} \right)+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right]^{2}
|
|
|
|
.. math::
|
|
:label: 5.56
|
|
|
|
\begin{array}{l} {r_{ah} =\frac{\theta _{atm} -\theta _{s} }{\theta _{*} u_{*} } =\frac{1}{k^{2} V_{a} } \left[\ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)-\psi _{m} \left(\frac{z_{atm,\, m} -d}{L} \right)+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right]} \\ {\qquad \left[\ln \left(\frac{z_{atm,\, h} -d}{z_{0h} } \right)-\psi _{h} \left(\frac{z_{atm,\, h} -d}{L} \right)+\psi _{h} \left(\frac{z_{0h} }{L} \right)\right]} \end{array}
|
|
|
|
.. math::
|
|
:label: 5.57
|
|
|
|
\begin{array}{l} {r_{aw} =\frac{q_{atm} -q_{s} }{q_{*} u_{*} } =\frac{1}{k^{2} V_{a} } \left[\ln \left(\frac{z_{atm,\, m} -d}{z_{0m} } \right)-\psi _{m} \left(\frac{z_{atm,\, m} -d}{L} \right)+\psi _{m} \left(\frac{z_{0m} }{L} \right)\right]} \\ {\qquad \left[\ln \left(\frac{z_{atm,\, {\it w}} -d}{z_{0w} } \right)-\psi _{w} \left(\frac{z_{atm,\, w} -d}{L} \right)+\psi _{w} \left(\frac{z_{0w} }{L} \right)\right]} \end{array}.
|
|
|
|
A 2-m height “screen” temperature is useful for comparison with
|
|
observations
|
|
|
|
.. math::
|
|
:label: 5.58
|
|
|
|
T_{2m} =\theta _{s} +\frac{\theta _{*} }{k} \left[\ln \left(\frac{2+z_{0h} }{z_{0h} } \right)-\psi _{h} \left(\frac{2+z_{0h} }{L} \right)+\psi _{h} \left(\frac{z_{0h} }{L} \right)\right]
|
|
|
|
where for convenience, “2-m” is defined as 2 m above the apparent sink
|
|
for sensible heat (:math:`z_{0h} +d`). Similarly, a 2-m height specific
|
|
humidity is defined as
|
|
|
|
.. math::
|
|
:label: 5.59
|
|
|
|
q_{2m} =q_{s} +\frac{q_{*} }{k} \left[\ln \left(\frac{2+z_{0w} }{z_{0w} } \right)-\psi _{w} \left(\frac{2+z_{0w} }{L} \right)+\psi _{w} \left(\frac{z_{0w} }{L} \right)\right].
|
|
|
|
Relative humidity is
|
|
|
|
.. math::
|
|
:label: 5.60
|
|
|
|
RH_{2m} =\min \left(100,\, \frac{q_{2m} }{q_{sat}^{T_{2m} } } \times 100\right)
|
|
|
|
where :math:`q_{sat}^{T_{2m} }` is the saturated specific humidity at
|
|
the 2-m temperature :math:`T_{2m}` (section :numref:`Saturation Vapor Pressure`).
|
|
|
|
A 10-m wind speed is calculated as (note that this is not consistent
|
|
with the 10-m wind speed calculated for the dust model as described in
|
|
Chapter :numref:`rst_Dust Model`)
|
|
|
|
.. math::
|
|
:label: 5.61
|
|
|
|
u_{10m} =\left\{\begin{array}{l} {V_{a} \qquad z_{atm,\, m} \le 10} \\ {V_{a} -\frac{u_{*} }{k} \left[\ln \left(\frac{z_{atm,\, m} -d}{10+z_{0m} } \right)-\psi _{m} \left(\frac{z_{atm,\, m} -d}{L} \right)+\psi _{m} \left(\frac{10+z_{0m} }{L} \right)\right]\qquad z_{atm,\, m} >10} \end{array}\right\}
|
|
|
|
.. _Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces:
|
|
|
|
Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces
|
|
--------------------------------------------------------------
|
|
|
|
Surfaces are considered non-vegetated for the surface flux calculations
|
|
if leaf plus stem area index :math:`L+S<0.05` (section
|
|
:numref:`Phenology and vegetation burial by snow`). By
|
|
definition, this includes bare soil and glaciers. The
|
|
solution for lakes is described in Chapter :numref:`rst_Lake Model`. For these surfaces, the
|
|
surface may be exposed to the atmosphere, snow covered, and/or surface
|
|
water covered, so that the sensible heat flux :math:`H_{g}` (W
|
|
m\ :sup:`-2`) is, with reference to :numref:`Figure Schematic diagram of sensible heat fluxes`,
|
|
|
|
.. math::
|
|
:label: 5.62
|
|
|
|
H_{g} =\left(1-f_{sno} -f_{h2osfc} \right)H_{soil} +f_{sno} H_{snow} +f_{h2osfc} H_{h2osfc}
|
|
|
|
where :math:`\left(1-f_{sno} -f_{h2osfc} \right)`, :math:`f_{sno}` , and
|
|
:math:`f_{h2osfc}` are the exposed, snow covered, and surface water
|
|
covered fractions of the grid cell. The individual fluxes based on the
|
|
temperatures of the soil :math:`T_{1}` , snow :math:`T_{snl+1}` , and
|
|
surface water :math:`T_{h2osfc}` are
|
|
|
|
.. math::
|
|
:label: 5.63
|
|
|
|
H_{soil} =-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -T_{1} \right)}{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.64
|
|
|
|
H_{sno} =-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -T_{snl+1} \right)}{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.65
|
|
|
|
H_{h2osfc} =-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -T_{h2osfc} \right)}{r_{ah} }
|
|
|
|
where :math:`\rho _{atm}` is the density of atmospheric air (kg m\ :sup:`-3`), :math:`C_{p}` is the specific heat capacity of air
|
|
(J kg\ :sup:`-1` K\ :sup:`-1`) (:numref:`Table Physical constants`),
|
|
:math:`\theta _{atm}` is the atmospheric potential temperature (K), and
|
|
:math:`r_{ah}` is the aerodynamic resistance to sensible heat transfer
|
|
(s m\ :sup:`-1`).
|
|
|
|
The water vapor flux :math:`E_{g}` (kg m\ :sup:`-2` s\ :sup:`-1`) is, with reference to
|
|
:numref:`Figure Schematic diagram of latent heat fluxes`,
|
|
|
|
.. math::
|
|
:label: 5.66
|
|
|
|
E_{g} =\left(1-f_{sno} -f_{h2osfc} \right)E_{soil} +f_{sno} E_{snow} +f_{h2osfc} E_{h2osfc}
|
|
|
|
.. math::
|
|
:label: 5.67
|
|
|
|
E_{soil} =-\frac{\rho _{atm} \left(q_{atm} -q_{soil} \right)}{r_{aw} + r_{soil}}
|
|
|
|
.. math::
|
|
:label: 5.68
|
|
|
|
E_{sno} =-\frac{\rho _{atm} \left(q_{atm} -q_{sno} \right)}{r_{aw} }
|
|
|
|
.. math::
|
|
:label: 5.69
|
|
|
|
E_{h2osfc} =-\frac{\rho _{atm} \left(q_{atm} -q_{h2osfc} \right)}{r_{aw} }
|
|
|
|
where :math:`q_{atm}` is the atmospheric specific humidity (kg kg\ :sup:`-1`), :math:`q_{soil}` , :math:`q_{sno}` ,
|
|
and :math:`q_{h2osfc}` are the specific humidities (kg kg\ :sup:`-1`) of the soil, snow, and surface water, respectively,
|
|
:math:`r_{aw}` is the aerodynamic resistance to water vapor transfer (s m\ :sup:`-1`), and :math:`r _{soi}` is the soil
|
|
resistance to water vapor transfer (s m\ :sup:`-1`). The specific humidities of the snow :math:`q_{sno}` and surface water
|
|
:math:`q_{h2osfc}` are assumed to be at the saturation specific humidity of their respective temperatures
|
|
|
|
.. math::
|
|
:label: 5.70
|
|
|
|
q_{sno} =q_{sat}^{T_{snl+1} }
|
|
|
|
.. math::
|
|
:label: 5.71
|
|
|
|
q_{h2osfc} =q_{sat}^{T_{h2osfc} }
|
|
|
|
The specific humidity of the soil surface :math:`q_{soil}` is assumed
|
|
to be proportional to the saturation specific humidity
|
|
|
|
.. math::
|
|
:label: 5.72
|
|
|
|
q_{soil} =\alpha _{soil} q_{sat}^{T_{1} }
|
|
|
|
where :math:`q_{sat}^{T_{1} }` is the saturated specific humidity at
|
|
the soil surface temperature :math:`T_{1}` (section :numref:`Saturation Vapor Pressure`). The factor
|
|
:math:`\alpha _{soil}` is a function of the surface soil water matric
|
|
potential :math:`\psi` as in :ref:`Philip (1957)<Philip1957>`
|
|
|
|
.. math::
|
|
:label: 5.73
|
|
|
|
\alpha _{soil} =\exp \left(\frac{\psi _{1} g}{1\times 10^{3} R_{wv} T_{1} } \right)
|
|
|
|
where :math:`R_{wv}` is the gas constant for water vapor (J kg\ :sup:`-1` K\ :sup:`-1`) (:numref:`Table Physical constants`), :math:`g` is the
|
|
gravitational acceleration (m s\ :sup:`-2`) (:numref:`Table Physical constants`), and
|
|
:math:`\psi _{1}` is the soil water matric potential of the top soil
|
|
layer (mm). The soil water matric potential :math:`\psi _{1}` is
|
|
|
|
.. math::
|
|
:label: 5.74
|
|
|
|
\psi _{1} =\psi _{sat,\, 1} s_{1}^{-B_{1} } \ge -1\times 10^{8}
|
|
|
|
where :math:`\psi _{sat,\, 1}` is the saturated matric potential (mm)
|
|
(section :numref:`Hydraulic Properties`),
|
|
:math:`B_{1}` is the :ref:`Clapp and Hornberger (1978) <ClappHornberger1978>`
|
|
parameter (section :numref:`Hydraulic Properties`),
|
|
and :math:`s_{1}` is the wetness of the top soil layer with respect to saturation.
|
|
The surface wetness :math:`s_{1}` is a function of the liquid water and ice content
|
|
|
|
.. math::
|
|
:label: 5.75
|
|
|
|
s_{1} =\frac{1}{\Delta z_{1} \theta _{sat,\, 1} } \left[\frac{w_{liq,\, 1} }{\rho _{liq} } +\frac{w_{ice,\, 1} }{\rho _{ice} } \right]\qquad 0.01\le s_{1} \le 1.0
|
|
|
|
where :math:`\Delta z_{1}` is the thickness of the top soil layer (m),
|
|
:math:`\rho _{liq}` and :math:`\rho _{ice}` are the density of liquid
|
|
water and ice (kg m\ :sup:`-3`) (:numref:`Table Physical constants`), :math:`w_{liq,\, 1}`
|
|
and :math:`w_{ice,\, 1}` are the mass of liquid water and ice of the
|
|
top soil layer (kg m\ :sup:`-2`) (Chapter :numref:`rst_Hydrology`), and
|
|
:math:`\theta _{sat,\, 1}` is the saturated volumetric water content
|
|
(i.e., porosity) of the top soil layer (mm\ :sup:`3` mm\ :sup:`-3`) (section :numref:`Hydraulic Properties`). If
|
|
:math:`q_{sat}^{T_{1} } >q_{atm}` and :math:`q_{atm} >q_{soil}` , then
|
|
:math:`q_{soil} =q_{atm}` and :math:`\frac{dq_{soil} }{dT} =0`. This
|
|
prevents large increases (decreases) in :math:`q_{soil}` for small
|
|
increases (decreases) in soil moisture in very dry soils.
|
|
|
|
The resistance to water vapor transfer occurring within the soil matrix
|
|
:math:`r_{soil}` (s m\ :sup:`-1`) is
|
|
|
|
.. math::
|
|
:label: 5.76
|
|
|
|
r_{soil} = \frac{DSL}{D_{v} \tau}
|
|
|
|
where :math:`DSL` is the thickness of the dry surface layer (m), :math:`D_{v}`
|
|
is the molecular diffusivity of water vapor in air (m\ :sup:`2` s\ :sup:`-2`)
|
|
and :math:`\tau` (*unitless*) describes the tortuosity of the vapor flow paths through
|
|
the soil matrix (:ref:`Swenson and Lawrence 2014 <SwensonLawrence2014>`).
|
|
|
|
The thickness of the dry surface layer is given by
|
|
|
|
.. math::
|
|
:label: 5.77
|
|
|
|
DSL =
|
|
\begin{array}{lr}
|
|
D_{max} \ \frac{\left( \theta_{init} - \theta_{1}\right)}
|
|
{\left(\theta_{init} - \theta_{air}\right)} & \qquad \theta_{1} < \theta_{init} \\
|
|
0 & \qquad \theta_{1} \ge \theta_{init}
|
|
\end{array}
|
|
|
|
where :math:`D_{max}` is a parameter specifying the length scale
|
|
of the maximum DSL thickness (default value = 15 mm),
|
|
:math:`\theta_{init}` (mm\ :sup:`3` mm\ :sup:`-3`) is the moisture value
|
|
at which the DSL initiates, :math:`\theta_{1}` (mm\ :sup:`3` mm\ :sup:`-3`)
|
|
is the moisture value of the top model soil layer, and
|
|
:math:`\theta_{air}` (mm\ :sup:`3` mm\ :sup:`-3`) is the 'air dry' soil
|
|
moisture value (:ref:`Dingman 2002 <Dingman2002>`):
|
|
|
|
.. math::
|
|
:label: 5.78
|
|
|
|
\theta_{air} = \Phi \left( \frac{\Psi_{sat}}{\Psi_{air}} \right)^{\frac{1}{B_{1}}} \ .
|
|
|
|
where :math:`\Phi` is the porosity (mm\ :sup:`3` mm\ :sup:`-3`),
|
|
:math:`\Psi_{sat}` is the saturated soil matric potential (mm),
|
|
:math:`\Psi_{air} = 10^{7}` mm is the air dry matric potential, and
|
|
:math:`B_{1}` is a function of soil texture (section
|
|
:numref:`Hydraulic Properties`).
|
|
|
|
The soil tortuosity is
|
|
|
|
.. math::
|
|
:label: 5.79
|
|
|
|
\tau = \Phi^{2}_{air}\left(\frac{\Phi_{air}}{\Phi}\right)^{\frac{3}{B_{1}}}
|
|
|
|
where :math:`\Phi_{air}` (mm\ :sup:`3` mm\ :sup:`-3`) is the air filled pore space
|
|
|
|
.. math::
|
|
:label: 5.80
|
|
|
|
\Phi_{air} = \Phi - \theta_{air} \ .
|
|
|
|
:math:`D_{v}` depends on temperature
|
|
|
|
.. math::
|
|
:label: 5.81
|
|
|
|
D_{v} = 2.12 \times 10^{-5} \left(\frac{T_{1}}{T_{f}}\right)^{1.75} \ .
|
|
|
|
where :math:`T_{1}` (K) is the temperature of the top soil layer and
|
|
:math:`T_{f}` (K) is the freezing temperature of water
|
|
(:numref:`Table Physical Constants`).
|
|
|
|
The roughness lengths used to calculate :math:`r_{am}` ,
|
|
:math:`r_{ah}` , and :math:`r_{aw}` are :math:`z_{0m} =z_{0m,\, g}` ,
|
|
:math:`z_{0h} =z_{0h,\, g}` , and :math:`z_{0w} =z_{0w,\, g}` . The
|
|
displacement height :math:`d=0`. The momentum roughness length is
|
|
:math:`z_{0m,\, g} =0.01` for soil, glaciers, and
|
|
:math:`z_{0m,\, g} =0.0024` for snow-covered surfaces
|
|
(:math:`f_{sno} >0`). In general, :math:`z_{0m}` is different from
|
|
:math:`z_{0h}` because the transfer of momentum is affected by pressure
|
|
fluctuations in the turbulent waves behind the roughness elements, while
|
|
for heat and water vapor transfer no such dynamical mechanism exists.
|
|
Rather, heat and water vapor must be transferred by molecular diffusion
|
|
across the interfacial sublayer. The following relation from
|
|
:ref:`Zilitinkevich (1970) <Zilitinkevich1970>` is adopted by
|
|
:ref:`Zeng and Dickinson 1998 <ZengDickinson1998>`
|
|
|
|
.. math::
|
|
:label: 5.82
|
|
|
|
z_{0h,\, g} =z_{0w,\, g} =z_{0m,\, g} e^{-a\left({u_{*} z_{0m,\, g} \mathord{\left/ {\vphantom {u_{*} z_{0m,\, g} \upsilon }} \right. \kern-\nulldelimiterspace} \upsilon } \right)^{0.45} }
|
|
|
|
where the quantity
|
|
:math:`{u_{\*} z_{0m,\, g} \mathord{\left/ {\vphantom {u_{*} z_{0m,\, g} \upsilon }} \right. \kern-\nulldelimiterspace} \upsilon }`
|
|
is the roughness Reynolds number (and may be interpreted as the Reynolds number of the smallest turbulent eddy in the flow) with the kinematic
|
|
viscosity of air :math:`\upsilon =1.5\times 10^{-5}` m\ :sup:`2` s\ :sup:`-1` and :math:`a=0.13`.
|
|
|
|
The numerical solution for the fluxes of momentum, sensible heat, and
|
|
water vapor flux from non-vegetated surfaces proceeds as follows:
|
|
|
|
#. An initial guess for the wind speed :math:`V_{a}` is obtained from
|
|
:eq:`5.24` assuming an initial convective velocity :math:`U_{c} =0` m
|
|
s\ :sup:`-1` for stable conditions
|
|
(:math:`\theta _{v,\, atm} -\theta _{v,\, s} \ge 0` as evaluated from
|
|
:eq:`5.50` ) and :math:`U_{c} =0.5` for unstable conditions
|
|
(:math:`\theta _{v,\, atm} -\theta _{v,\, s} <0`).
|
|
|
|
#. An initial guess for the Monin-Obukhov length :math:`L` is obtained
|
|
from the bulk Richardson number using :eq:`5.46` and :eq:`5.48`.
|
|
|
|
#. The following system of equations is iterated three times:
|
|
|
|
#. Friction velocity :math:`u_{*}` (:eq:`5.32`, :eq:`5.33`, :eq:`5.34`, :eq:`5.35`)
|
|
|
|
#. Potential temperature scale :math:`\theta _{*}` (:eq:`5.37` , :eq:`5.38`, :eq:`5.39`, :eq:`5.40`)
|
|
|
|
#. Humidity scale :math:`q_{*}` (:eq:`5.41`, :eq:`5.42`, :eq:`5.43`, :eq:`5.44`)
|
|
|
|
#. Roughness lengths for sensible :math:`z_{0h,\, g}` and latent heat
|
|
:math:`z_{0w,\, g}` (:eq:`5.82` )
|
|
|
|
#. Virtual potential temperature scale :math:`\theta _{v*}` ( :eq:`5.17`)
|
|
|
|
#. Wind speed including the convective velocity, :math:`V_{a}` ( :eq:`5.24`)
|
|
|
|
#. Monin-Obukhov length :math:`L` (:eq:`5.49`)
|
|
|
|
#. Aerodynamic resistances :math:`r_{am}` , :math:`r_{ah}` , and
|
|
:math:`r_{aw}` (:eq:`5.55`, :eq:`5.56`, :eq:`5.57`)
|
|
|
|
#. Momentum fluxes :math:`\tau _{x}` , :math:`\tau _{y}` (:eq:`5.5`, :eq:`5.6`)
|
|
|
|
#. Sensible heat flux :math:`H_{g}` (:eq:`5.62`)
|
|
|
|
#. Water vapor flux :math:`E_{g}` (:eq:`5.66`)
|
|
|
|
#. 2-m height air temperature :math:`T_{2m}` and specific humidity
|
|
:math:`q_{2m}` (:eq:`5.58` , :eq:`5.59`)
|
|
|
|
The partial derivatives of the soil surface fluxes with respect to
|
|
ground temperature, which are needed for the soil temperature calculations (section
|
|
:numref:`Numerical Solution Temperature`) and to update the soil surface fluxes
|
|
(section :numref:`Update of Ground Sensible and Latent Heat Fluxes`), are
|
|
|
|
.. math::
|
|
:label: 5.83
|
|
|
|
\frac{\partial H_{g} }{\partial T_{g} } =\frac{\rho _{atm} C_{p} }{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.84
|
|
|
|
\frac{\partial E_{g} }{\partial T_{g} } =\frac{\beta _{soi} \rho _{atm} }{r_{aw} } \frac{dq_{g} }{dT_{g} }
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.85
|
|
|
|
\frac{dq_{g} }{dT_{g} } =\left(1-f_{sno} -f_{h2osfc} \right)\alpha _{soil} \frac{dq_{sat}^{T_{soil} } }{dT_{soil} } +f_{sno} \frac{dq_{sat}^{T_{sno} } }{dT_{sno} } +f_{h2osfc} \frac{dq_{sat}^{T_{h2osfc} } }{dT_{h2osfc} } .
|
|
|
|
The partial derivatives
|
|
:math:`\frac{\partial r_{ah} }{\partial T_{g} }` and
|
|
:math:`\frac{\partial r_{aw} }{\partial T_{g} }` , which cannot be
|
|
determined analytically, are ignored for
|
|
:math:`\frac{\partial H_{g} }{\partial T_{g} }` and
|
|
:math:`\frac{\partial E_{g} }{\partial T_{g} }` .
|
|
|
|
.. _Sensible and Latent Heat Fluxes and Temperature for Vegetated Surfaces:
|
|
|
|
Sensible and Latent Heat Fluxes and Temperature for Vegetated Surfaces
|
|
--------------------------------------------------------------------------
|
|
|
|
In the case of a vegetated surface, the sensible heat :math:`H` and
|
|
water vapor flux :math:`E` are partitioned into vegetation and ground
|
|
fluxes that depend on vegetation :math:`T_{v}` and ground
|
|
:math:`T_{g}` temperatures in addition to surface temperature
|
|
:math:`T_{s}` and specific humidity :math:`q_{s}` . Because of the
|
|
coupling between vegetation temperature and fluxes, Newton-Raphson
|
|
iteration is used to solve for the vegetation temperature and the
|
|
sensible heat and water vapor fluxes from vegetation simultaneously
|
|
using the ground temperature from the previous time step. In section
|
|
:numref:`Theory`, the equations used in the iteration scheme are derived. Details
|
|
on the numerical scheme are provided in section :numref:`Numerical Implementation`.
|
|
|
|
.. _Theory:
|
|
|
|
Theory
|
|
^^^^^^^^^^^^
|
|
|
|
The air within the canopy is assumed to have negligible capacity to
|
|
store heat so that the sensible heat flux :math:`H` between the surface
|
|
at height :math:`z_{0h} +d` and the atmosphere at height
|
|
:math:`z_{atm,\, h}` must be balanced by the sum of the sensible heat
|
|
from the vegetation :math:`H_{v}` and the ground :math:`H_{g}`
|
|
|
|
.. math::
|
|
:label: 5.86
|
|
|
|
H=H_{v} +H_{g}
|
|
|
|
where, with reference to :numref:`Figure Schematic diagram of sensible heat fluxes`,
|
|
|
|
.. math::
|
|
:label: 5.87
|
|
|
|
H=-\rho _{atm} C_{p} \frac{\left(\theta _{atm} -T_{s} \right)}{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.88
|
|
|
|
H_{v} =-\rho _{atm} C_{p} \left(T_{s} -T_{v} \right)\frac{\left(L+S\right)}{r_{b} }
|
|
|
|
.. math::
|
|
:label: 5.89
|
|
|
|
H_{g} =\left(1-f_{sno} -f_{h2osfc} \right)H_{soil} +f_{sno} H_{snow} +f_{h2osfc} H_{h2osfc} \ ,
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.90
|
|
|
|
H_{soil} =-\rho _{atm} C_{p} \frac{\left(T_{s} -T_{1} \right)}{r_{ah} ^{{'} } }
|
|
|
|
.. math::
|
|
:label: 5.91
|
|
|
|
H_{sno} =-\rho _{atm} C_{p} \frac{\left(T_{s} -T_{snl+1} \right)}{r_{ah} ^{{'} } }
|
|
|
|
.. math::
|
|
:label: 5.92
|
|
|
|
H_{h2osfc} =-\rho _{atm} C_{p} \frac{\left(T_{s} -T_{h2osfc} \right)}{r_{ah} ^{{'} } }
|
|
|
|
where :math:`\rho _{atm}` is the density of atmospheric air (kg m\ :sup:`-3`), :math:`C_{p}` is the specific heat capacity of air
|
|
(J kg\ :sup:`-1` K\ :sup:`-1`) (:numref:`Table Physical constants`),
|
|
:math:`\theta _{atm}` is the atmospheric potential temperature (K), and
|
|
:math:`r_{ah}` is the aerodynamic resistance to sensible heat transfer
|
|
(s m\ :sup:`-1`).
|
|
|
|
Here, :math:`T_{s}` is the surface temperature at height
|
|
:math:`z_{0h} +d`, also referred to as the canopy air temperature.
|
|
:math:`L` and :math:`S` are the exposed leaf and stem area indices
|
|
(section :numref:`Phenology and vegetation burial by snow`), :math:`r_{b}` is the leaf boundary layer resistance (s
|
|
m\ :sup:`-1`), and :math:`r_{ah} ^{{'} }` is the aerodynamic
|
|
resistance (s m\ :sup:`-1`) to heat transfer between the ground at
|
|
height :math:`z_{0h} ^{{'} }` and the canopy air at height
|
|
:math:`z_{0h} +d`.
|
|
|
|
.. _Figure Schematic diagram of sensible heat fluxes:
|
|
|
|
.. figure:: image1.png
|
|
|
|
Figure Schematic diagram of sensible heat fluxes for (a)
|
|
non-vegetated surfaces and (b) vegetated surfaces.
|
|
|
|
.. _Figure Schematic diagram of latent heat fluxes:
|
|
|
|
.. figure:: image2.png
|
|
|
|
Figure Schematic diagram of water vapor fluxes for (a)
|
|
non-vegetated surfaces and (b) vegetated surfaces.
|
|
|
|
Equations :eq:`5.86` - :eq:`5.89` can be solved for the canopy air
|
|
temperature :math:`T_{s}`
|
|
|
|
.. math::
|
|
:label: 5.93
|
|
|
|
T_{s} =\frac{c_{a}^{h} \theta _{atm} +c_{g}^{h} T_{g} +c_{v}^{h} T_{v} }{c_{a}^{h} +c_{g}^{h} +c_{v}^{h} }
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.94
|
|
|
|
c_{a}^{h} =\frac{1}{r_{ah} }
|
|
|
|
.. math::
|
|
:label: 5.95
|
|
|
|
c_{g}^{h} =\frac{1}{r_{ah} ^{{'} } }
|
|
|
|
.. math::
|
|
:label: 5.96
|
|
|
|
c_{v}^{h} =\frac{\left(L+S\right)}{r_{b} }
|
|
|
|
are the sensible heat conductances from the canopy air to the
|
|
atmosphere, the ground to canopy air, and leaf surface to canopy air,
|
|
respectively (m s\ :sup:`-1`).
|
|
|
|
When the expression for :math:`T_{s}` is substituted into equation :eq:`5.88`,
|
|
the sensible heat flux from vegetation :math:`H_{v}` is a function of
|
|
:math:`\theta _{atm}` , :math:`T_{g}` , and :math:`T_{v}`
|
|
|
|
.. math::
|
|
:label: 5.97
|
|
|
|
H_{v} = -\rho _{atm} C_{p} \left[c_{a}^{h} \theta _{atm} +c_{g}^{h} T_{g} -\left(c_{a}^{h} +c_{g}^{h} \right)T_{v} \right]\frac{c_{v}^{h} }{c_{a}^{h} +c_{v}^{h} +c_{g}^{h} } .
|
|
|
|
Similarly, the expression for :math:`T_{s}` can be substituted into
|
|
equation to obtain the sensible heat flux from ground :math:`H_{g}`
|
|
|
|
.. math::
|
|
:label: 5.98
|
|
|
|
H_{g} = -\rho _{atm} C_{p} \left[c_{a}^{h} \theta _{atm} +c_{v}^{h} T_{v} -\left(c_{a}^{h} +c_{v}^{h} \right)T_{g} \right]\frac{c_{g}^{h} }{c_{a}^{h} +c_{v}^{h} +c_{g}^{h} } .
|
|
|
|
The air within the canopy is assumed to have negligible capacity to
|
|
store water vapor so that the water vapor flux :math:`E` between the
|
|
surface at height :math:`z_{0w} +d` and the atmosphere at height
|
|
:math:`z_{atm,\, w}` must be balanced by the sum of the water vapor
|
|
flux from the vegetation :math:`E_{v}` and the ground :math:`E_{g}`
|
|
|
|
.. math::
|
|
:label: 5.99
|
|
|
|
E = E_{v} +E_{g}
|
|
|
|
where, with reference to :numref:`Figure Schematic diagram of latent heat fluxes`,
|
|
|
|
.. math::
|
|
:label: 5.100
|
|
|
|
E = -\rho _{atm} \frac{\left(q_{atm} -q_{s} \right)}{r_{aw} }
|
|
|
|
.. math::
|
|
:label: 5.101
|
|
|
|
E_{v} = -\rho _{atm} \frac{\left(q_{s} -q_{sat}^{T_{v} } \right)}{r_{total} }
|
|
|
|
.. math::
|
|
:label: 5.102
|
|
|
|
E_{g} = \left(1-f_{sno} -f_{h2osfc} \right)E_{soil} +f_{sno} E_{snow} +f_{h2osfc} E_{h2osfc} \ ,
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.103
|
|
|
|
E_{soil} = -\rho _{atm} \frac{\left(q_{s} -q_{soil} \right)}{r_{aw} ^{{'} } +r_{soil} }
|
|
|
|
.. math::
|
|
:label: 5.104
|
|
|
|
E_{sno} = -\rho _{atm} \frac{\left(q_{s} -q_{sno} \right)}{r_{aw} ^{{'} } +r_{soil} }
|
|
|
|
.. math::
|
|
:label: 5.105
|
|
|
|
E_{h2osfc} = -\rho _{atm} \frac{\left(q_{s} -q_{h2osfc} \right)}{r_{aw} ^{{'} } +r_{soil} }
|
|
|
|
where :math:`q_{atm}` is the atmospheric specific humidity (kg kg\ :sup:`-1`), :math:`r_{aw}` is the aerodynamic resistance to
|
|
water vapor transfer (s m\ :sup:`-1`), :math:`q_{sat}^{T_{v} }`
|
|
(kg kg\ :sup:`-1`) is the saturation water vapor specific humidity
|
|
at the vegetation temperature (section :numref:`Saturation Vapor Pressure`), :math:`q_{g}` ,
|
|
:math:`q_{sno}` , and :math:`q_{h2osfc}` are the specific humidities
|
|
of the soil, snow, and surface water (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`),
|
|
:math:`r_{aw} ^{{'} }` is the aerodynamic resistance (s
|
|
m\ :sup:`-1`) to water vapor transfer between the ground at height
|
|
:math:`z_{0w} ^{{'} }` and the canopy air at height :math:`z_{0w} +d`,
|
|
and :math:`r_{soil}` (:eq:`5.76`) is a resistance to diffusion through the soil
|
|
(s m\ :sup:`-1`). :math:`r_{total}` is the total resistance to
|
|
water vapor transfer from the canopy to the canopy air and includes
|
|
contributions from leaf boundary layer and sunlit and shaded stomatal
|
|
resistances :math:`r_{b}` , :math:`r_{s}^{sun}` , and
|
|
:math:`r_{s}^{sha}` (:numref:`Figure Schematic diagram of latent heat fluxes`).
|
|
The water vapor flux from vegetation
|
|
is the sum of water vapor flux from wetted leaf and stem area
|
|
:math:`E_{v}^{w}` (evaporation of water intercepted by the canopy) and
|
|
transpiration from dry leaf surfaces :math:`E_{v}^{t}`
|
|
|
|
.. math::
|
|
:label: 5.106
|
|
|
|
E_{v} =E_{v}^{w} +E_{v}^{t} .
|
|
|
|
Equations :eq:`5.99` - :eq:`5.102` can be solved for the canopy specific humidity
|
|
:math:`q_{s}`
|
|
|
|
.. math::
|
|
:label: 5.107
|
|
|
|
q_{s} =\frac{c_{a}^{w} q_{atm} +c_{g}^{w} q_{g} +c_{v}^{w} q_{sat}^{T_{v} } }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} }
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.108
|
|
|
|
c_{a}^{w} =\frac{1}{r_{aw} }
|
|
|
|
.. math::
|
|
:label: 5.109
|
|
|
|
c_{v}^{w} =\frac{\left(L+S\right)}{r_{b} } r''
|
|
|
|
.. math::
|
|
:label: 5.110
|
|
|
|
c_{g}^{w} =\frac{1}{r_{aw} ^{{'} } +r_{soil} }
|
|
|
|
are the water vapor conductances from the canopy air to the atmosphere,
|
|
the leaf to canopy air, and ground to canopy air, respectively. The term
|
|
:math:`r''` is determined from contributions by wet leaves and
|
|
transpiration and limited by available water and potential evaporation
|
|
as
|
|
|
|
.. math::
|
|
:label: 5.111
|
|
|
|
r'' = \left\{
|
|
\begin{array}{lr}
|
|
\min \left(f_{wet} +r_{dry} ^{{'} {'} } ,\, \frac{E_{v}^{w,\, pot} r_{dry} ^{{'} {'} } +\frac{W_{can} }{\Delta t} }{E_{v}^{w,\, pot} } \right) & \qquad E_{v}^{w,\, pot} >0,\, \beta _{t} >0 \\
|
|
\min \left(f_{wet} ,\, \frac{E_{v}^{w,\, pot} r_{dry} ^{{'} {'} } +\frac{W_{can} }{\Delta t} }{E_{v}^{w,\, pot} } \right) & \qquad E_{v}^{w,\, pot} >0,\, \beta _{t} \le 0 \\
|
|
1 & \qquad E_{v}^{w,\, pot} \le 0
|
|
\end{array}\right\}
|
|
|
|
where :math:`f_{wet}` is the fraction of leaves and stems that are wet
|
|
(section :numref:`Canopy Water`), :math:`W_{can}` is canopy water (kg m\ :sup:`-2`)
|
|
(section :numref:`Canopy Water`), :math:`\Delta t` is the time step (s), and
|
|
:math:`\beta _{t}` is a soil moisture function limiting transpiration
|
|
(Chapter :numref:`rst_Stomatal Resistance and Photosynthesis`). The potential
|
|
evaporation from wet foliage per unit wetted area is
|
|
|
|
.. math::
|
|
:label: 5.112
|
|
|
|
E_{v}^{w,\, pot} =-\frac{\rho _{atm} \left(q_{s} -q_{sat}^{T_{v} } \right)}{r_{b} } .
|
|
|
|
The term :math:`r_{dry} ^{{'} {'} }` is
|
|
|
|
.. math::
|
|
:label: 5.113
|
|
|
|
r_{dry} ^{{'} {'} } =\frac{f_{dry} r_{b} }{L} \left(\frac{L^{sun} }{r_{b} +r_{s}^{sun} } +\frac{L^{sha} }{r_{b} +r_{s}^{sha} } \right)
|
|
|
|
where :math:`f_{dry}` is the fraction of leaves that are dry (section
|
|
:numref:`Canopy Water`), :math:`L^{sun}` and :math:`L^{sha}` are the sunlit and shaded
|
|
leaf area indices (section :numref:`Solar Fluxes`), and :math:`r_{s}^{sun}` and
|
|
:math:`r_{s}^{sha}` are the sunlit and shaded stomatal resistances (s
|
|
m\ :sup:`-1`) (Chapter :numref:`rst_Stomatal Resistance and Photosynthesis`).
|
|
|
|
When the expression for :math:`q_{s}` is substituted into equation :eq:`5.101`,
|
|
the water vapor flux from vegetation :math:`E_{v}` is a function of
|
|
:math:`q_{atm}` , :math:`q_{g}` , and :math:`q_{sat}^{T_{v} }`
|
|
|
|
.. math::
|
|
:label: 5.114
|
|
|
|
E_{v} =-\rho _{atm} \left[c_{a}^{w} q_{atm} +c_{g}^{w} q_{g} -\left(c_{a}^{w} +c_{g}^{w} \right)q_{sat}^{T_{v} } \right]\frac{c_{v}^{w} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} } .
|
|
|
|
Similarly, the expression for :math:`q_{s}` can be substituted into
|
|
:eq:`5.84` to obtain the water vapor flux from the ground beneath the
|
|
canopy :math:`E_{g}`
|
|
|
|
.. math::
|
|
:label: 5.115
|
|
|
|
E_{g} =-\rho _{atm} \left[c_{a}^{w} q_{atm} +c_{v}^{w} q_{sat}^{T_{v} } -\left(c_{a}^{w} +c_{v}^{w} \right)q_{g} \right]\frac{c_{g}^{w} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} } .
|
|
|
|
The aerodynamic resistances to heat (moisture) transfer between the
|
|
ground at height :math:`z_{0h} ^{{'} }` (:math:`z_{0w} ^{{'} }` ) and
|
|
the canopy air at height :math:`z_{0h} +d` (:math:`z_{0w} +d`) are
|
|
|
|
.. math::
|
|
:label: 5.116
|
|
|
|
r_{ah} ^{{'} } =r_{aw} ^{{'} } =\frac{1}{C_{s} U_{av} }
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.117
|
|
|
|
U_{av} =V_{a} \sqrt{\frac{1}{r_{am} V_{a} } } =u_{*}
|
|
|
|
is the magnitude of the wind velocity incident on the leaves
|
|
(equivalent here to friction velocity) (m s\ :sup:`-1`) and
|
|
:math:`C_{s}` is the turbulent transfer coefficient between the
|
|
underlying soil and the canopy air. :math:`C_{s}` is obtained by
|
|
interpolation between values for dense canopy and bare soil
|
|
(:ref:`Zeng et al. 2005 <Zengetal2005>`)
|
|
|
|
.. math::
|
|
:label: 5.118
|
|
|
|
C_{s} =C_{s,\, bare} W+C_{s,\, dense} (1-W)
|
|
|
|
where the weight :math:`W` is
|
|
|
|
.. math::
|
|
:label: 5.119
|
|
|
|
W=e^{-\left(L+S\right)} .
|
|
|
|
The dense canopy turbulent transfer coefficient
|
|
(:ref:`Dickinson et al. 1993 <Dickinsonetal1993>`) is
|
|
|
|
.. math::
|
|
:label: 5.120)
|
|
|
|
C_{s,\, dense} =0.004 \ .
|
|
|
|
The bare soil turbulent transfer coefficient is
|
|
|
|
.. math::
|
|
:label: 5.121
|
|
|
|
C_{s,\, bare} =\frac{k}{a} \left(\frac{z_{0m,\, g} U_{av} }{\upsilon } \right)^{-0.45}
|
|
|
|
where the kinematic viscosity of air
|
|
:math:`\upsilon =1.5\times 10^{-5}` m\ :sup:`2` s\ :sup:`-1` and :math:`a=0.13`.
|
|
|
|
The leaf boundary layer resistance :math:`r_{b}` is
|
|
|
|
.. math::
|
|
:label: 5.122
|
|
|
|
r_{b} =\frac{1}{C_{v} } \left({U_{av} \mathord{\left/ {\vphantom {U_{av} d_{leaf} }} \right. \kern-\nulldelimiterspace} d_{leaf} } \right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} }
|
|
|
|
where :math:`C_{v} =0.01` m\ s\ :sup:`-1/2` is the turbulent
|
|
transfer coefficient between the canopy surface and canopy air, and
|
|
:math:`d_{leaf}` is the characteristic dimension of the leaves in the
|
|
direction of wind flow (:numref:`Table Plant functional type aerodynamic parameters`).
|
|
|
|
The partial derivatives of the fluxes from the soil beneath the canopy
|
|
with respect to ground temperature, which are needed for the soil
|
|
temperature calculations (section :numref:`Numerical Solution Temperature`)
|
|
and to update the soil surface fluxes (section
|
|
:numref:`Update of Ground Sensible and Latent Heat Fluxes`), are
|
|
|
|
.. math::
|
|
:label: 5.123
|
|
|
|
\frac{\partial H_{g} }{\partial T_{g} } = \frac{\rho _{atm} C_{p} }{r'_{ah} } \frac{c_{a}^{h} +c_{v}^{h} }{c_{a}^{h} +c_{v}^{h} +c_{g}^{h} }
|
|
|
|
.. math::
|
|
:label: 5.124
|
|
|
|
\frac{\partial E_{g} }{\partial T_{g} } = \frac{\rho _{atm} }{r'_{aw} +r_{soil} } \frac{c_{a}^{w} +c_{v}^{w} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} } \frac{dq_{g} }{dT_{g} } .
|
|
|
|
The partial derivatives
|
|
:math:`\frac{\partial r'_{ah} }{\partial T_{g} }` and
|
|
:math:`\frac{\partial r'_{aw} }{\partial T_{g} }` , which cannot be
|
|
determined analytically, are ignored for
|
|
:math:`\frac{\partial H_{g} }{\partial T_{g} }` and
|
|
:math:`\frac{\partial E_{g} }{\partial T_{g} }` .
|
|
|
|
The roughness lengths used to calculate :math:`r_{am}` ,
|
|
:math:`r_{ah}` , and :math:`r_{aw}` from :eq:`5.55`, :eq:`5.56`, and :eq:`5.57` are
|
|
:math:`z_{0m} =z_{0m,\, v}` , :math:`z_{0h} =z_{0h,\, v}` , and
|
|
:math:`z_{0w} =z_{0w,\, v}` . The vegetation displacement height
|
|
:math:`d` and the roughness lengths are a function of plant height and
|
|
adjusted for canopy density following :ref:`Zeng and Wang (2007) <ZengWang2007>`
|
|
|
|
.. math::
|
|
:label: 5.125
|
|
|
|
z_{0m,\, v} = z_{0h,\, v} =z_{0w,\, v} =\exp \left[V\ln \left(z_{top} R_{z0m} \right)+\left(1-V\right)\ln \left(z_{0m,\, g} \right)\right]
|
|
|
|
.. math::
|
|
:label: 5.126
|
|
|
|
d = z_{top} R_{d} V
|
|
|
|
where :math:`z_{top}` is canopy top height (m)
|
|
(:numref:`Table Plant functional type canopy top and bottom heights`),
|
|
:math:`R_{z0m}` and :math:`R_{d}` are the ratio of momentum roughness
|
|
length and displacement height to canopy top height, respectively
|
|
(:numref:`Table Plant functional type aerodynamic parameters`), and :math:`z_{0m,\, g}`
|
|
is the ground momentum roughness length (m) (section
|
|
:numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`). The
|
|
fractional weight :math:`V` is determined from
|
|
|
|
.. math::
|
|
:label: 5.127
|
|
|
|
V = \frac{1-\exp \left\{-\beta \min \left[L+S,\, \left(L+S\right)_{cr} \right]\right\}}{1-\exp \left[-\beta \left(L+S\right)_{cr} \right]}
|
|
|
|
where :math:`\beta =1` and :math:`\left(L+S\right)_{cr} = 2`
|
|
(m\ :sup:`2` m\ :sup:`-2`) is a critical value of exposed leaf
|
|
plus stem area for which :math:`z_{0m}` reaches its maximum.
|
|
|
|
.. _Table Plant functional type aerodynamic parameters:
|
|
|
|
.. table:: Plant functional type aerodynamic parameters
|
|
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Plant functional type | :math:`R_{z0m}` | :math:`R_{d}` | :math:`d_{leaf}` (m) |
|
|
+==================================+====================+==================+=========================+
|
|
| NET Temperate | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| NET Boreal | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| NDT Boreal | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BET Tropical | 0.075 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BET temperate | 0.075 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BDT tropical | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BDT temperate | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BDT boreal | 0.055 | 0.67 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BES temperate | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BDS temperate | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| BDS boreal | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| C\ :sub:`3` arctic grass | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| C\ :sub:`3` grass | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| C\ :sub:`4` grass | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Crop R | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Crop I | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Corn R | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Corn I | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Temp Cereal R | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Temp Cereal I | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Winter Cereal R | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Winter Cereal I | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Soybean R | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
| Soybean I | 0.120 | 0.68 | 0.04 |
|
|
+----------------------------------+--------------------+------------------+-------------------------+
|
|
|
|
.. _Numerical Implementation:
|
|
|
|
Numerical Implementation
|
|
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
|
|
|
|
Canopy energy conservation gives
|
|
|
|
.. math::
|
|
:label: 5.128
|
|
|
|
-\overrightarrow{S}_{v} +\overrightarrow{L}_{v} \left(T_{v} \right)+H_{v} \left(T_{v} \right)+\lambda E_{v} \left(T_{v} \right)=0
|
|
|
|
where :math:`\overrightarrow{S}_{v}` is the solar radiation absorbed by
|
|
the vegetation (section :numref:`Solar Fluxes`), :math:`\overrightarrow{L}_{v}` is the net
|
|
longwave radiation absorbed by vegetation (section :numref:`Longwave Fluxes`), and
|
|
:math:`H_{v}` and :math:`\lambda E_{v}` are the sensible and latent
|
|
heat fluxes from vegetation, respectively. The term :math:`\lambda` is
|
|
taken to be the latent heat of vaporization :math:`\lambda _{vap}`
|
|
(:numref:`Table Physical constants`).
|
|
|
|
:math:`\overrightarrow{L}_{v}` , :math:`H_{v}` , and
|
|
:math:`\lambda E_{v}` depend on the vegetation temperature
|
|
:math:`T_{v}` . The Newton-Raphson method for finding roots of
|
|
non-linear systems of equations can be applied to iteratively solve for
|
|
:math:`T_{v}` as
|
|
|
|
.. math::
|
|
:label: 5.129
|
|
|
|
\Delta T_{v} =\frac{\overrightarrow{S}_{v} -\overrightarrow{L}_{v} -H_{v} -\lambda E_{v} }{\frac{\partial \overrightarrow{L}_{v} }{\partial T_{v} } +\frac{\partial H_{v} }{\partial T_{v} } +\frac{\partial \lambda E_{v} }{\partial T_{v} } }
|
|
|
|
where :math:`\Delta T_{v} =T_{v}^{n+1} -T_{v}^{n}` and the subscript
|
|
“n” indicates the iteration.
|
|
|
|
The partial derivatives are
|
|
|
|
.. math::
|
|
:label: 5.130
|
|
|
|
\frac{\partial \overrightarrow{L}_{v} }{\partial T_{v} } =4\varepsilon _{v} \sigma \left[2-\varepsilon _{v} \left(1-\varepsilon _{g} \right)\right]T_{v}^{3}
|
|
|
|
.. math::
|
|
:label: 5.131
|
|
|
|
\frac{\partial H_{v} }{\partial T_{v} } =\rho _{atm} C_{p} \left(c_{a}^{h} +c_{g}^{h} \right)\frac{c_{v}^{h} }{c_{a}^{h} +c_{v}^{h} +c_{g}^{h} }
|
|
|
|
.. math::
|
|
:label: 5.132
|
|
|
|
\frac{\partial \lambda E_{v} }{\partial T_{v} } =\lambda \rho _{atm} \left(c_{a}^{w} +c_{g}^{w} \right)\frac{c_{v}^{w} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} } \frac{dq_{sat}^{T_{v} } }{dT_{v} } .
|
|
|
|
The partial derivatives
|
|
:math:`\frac{\partial r_{ah} }{\partial T_{v} }` and
|
|
:math:`\frac{\partial r_{aw} }{\partial T_{v} }` , which cannot be
|
|
determined analytically, are ignored for
|
|
:math:`\frac{\partial H_{v} }{\partial T_{v} }` and
|
|
:math:`\frac{\partial \lambda E_{v} }{\partial T_{v} }` . However, if
|
|
:math:`\zeta` changes sign more than four times during the temperature
|
|
iteration, :math:`\zeta =-0.01`. This helps prevent “flip-flopping”
|
|
between stable and unstable conditions. The total water vapor flux
|
|
:math:`E_{v}` , transpiration flux :math:`E_{v}^{t}` , and sensible heat
|
|
flux :math:`H_{v}` are updated for changes in leaf temperature as
|
|
|
|
.. math::
|
|
:label: 5.133
|
|
|
|
E_{v} =-\rho _{atm} \left[c_{a}^{w} q_{atm} +c_{g}^{w} q_{g} -\left(c_{a}^{w} +c_{g}^{w} \right)\left(q_{sat}^{T_{v} } +\frac{dq_{sat}^{T_{v} } }{dT_{v} } \Delta T_{v} \right)\right]\frac{c_{v}^{w} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} }
|
|
|
|
.. math::
|
|
:label: 5.134
|
|
|
|
E_{v}^{t} =-r_{dry} ^{{'} {'} } \rho _{atm} \left[c_{a}^{w} q_{atm} +c_{g}^{w} q_{g} -\left(c_{a}^{w} +c_{g}^{w} \right)\left(q_{sat}^{T_{v} } +\frac{dq_{sat}^{T_{v} } }{dT_{v} } \Delta T_{v} \right)\right]\frac{c_{v}^{h} }{c_{a}^{w} +c_{v}^{w} +c_{g}^{w} }
|
|
|
|
.. math::
|
|
:label: 5.135
|
|
|
|
H_{v} =-\rho _{atm} C_{p} \left[c_{a}^{h} \theta _{atm} +c_{g}^{h} T_{g} -\left(c_{a}^{h} +c_{g}^{h} \right)\left(T_{v} +\Delta T_{v} \right)\right]\frac{c_{v}^{h} }{c_{a}^{h} +c_{v}^{h} +c_{g}^{h} } .
|
|
|
|
The numerical solution for vegetation temperature and the fluxes of
|
|
momentum, sensible heat, and water vapor flux from vegetated surfaces
|
|
proceeds as follows:
|
|
|
|
#. Initial values for canopy air temperature and specific humidity are
|
|
obtained from
|
|
|
|
.. math::
|
|
:label: 5.136
|
|
|
|
T_{s} =\frac{T_{g} +\theta _{atm} }{2}
|
|
|
|
.. math::
|
|
:label: 5.137
|
|
|
|
q_{s} =\frac{q_{g} +q_{atm} }{2} .
|
|
|
|
#. An initial guess for the wind speed :math:`V_{a}` is obtained from
|
|
:eq:`5.24` assuming an initial convective velocity :math:`U_{c} =0` m
|
|
s\ :sup:`-1` for stable conditions
|
|
(:math:`\theta _{v,\, atm} -\theta _{v,\, s} \ge 0` as evaluated from
|
|
:eq:`5.50` ) and :math:`U_{c} =0.5` for unstable conditions
|
|
(:math:`\theta _{v,\, atm} -\theta _{v,\, s} <0`).
|
|
|
|
#. An initial guess for the Monin-Obukhov length :math:`L` is obtained
|
|
from the bulk Richardson number using equation and :eq:`5.46` and :eq:`5.48`.
|
|
|
|
#. Iteration proceeds on the following system of equations:
|
|
|
|
#. Friction velocity :math:`u_{*}` (:eq:`5.32`, :eq:`5.33`, :eq:`5.34`, :eq:`5.35`)
|
|
|
|
#. Ratio :math:`\frac{\theta _{*} }{\theta _{atm} -\theta _{s} }`
|
|
(:eq:`5.37` , :eq:`5.38`, :eq:`5.39`, :eq:`5.40`)
|
|
|
|
#. Ratio :math:`\frac{q_{*} }{q_{atm} -q_{s} }` (:eq:`5.41`, :eq:`5.42`, :eq:`5.43`, :eq:`5.44`)
|
|
|
|
#. Aerodynamic resistances :math:`r_{am}` , :math:`r_{ah}` , and
|
|
:math:`r_{aw}` (:eq:`5.55`, :eq:`5.56`, :eq:`5.57`)
|
|
|
|
#. Magnitude of the wind velocity incident on the leaves :math:`U_{av}`
|
|
(:eq:`5.117` )
|
|
|
|
#. Leaf boundary layer resistance :math:`r_{b}` (:eq:`5.136` )
|
|
|
|
#. Aerodynamic resistances :math:`r_{ah} ^{{'} }` and
|
|
:math:`r_{aw} ^{{'} }` (:eq:`5.116` )
|
|
|
|
#. Sunlit and shaded stomatal resistances :math:`r_{s}^{sun}` and
|
|
:math:`r_{s}^{sha}` (Chapter :numref:`rst_Stomatal Resistance and Photosynthesis`)
|
|
|
|
#. Sensible heat conductances :math:`c_{a}^{h}` , :math:`c_{g}^{h}` ,
|
|
and :math:`c_{v}^{h}` (:eq:`5.94`, :eq:`5.95`, :eq:`5.96`)
|
|
|
|
#. Latent heat conductances :math:`c_{a}^{w}` , :math:`c_{v}^{w}` , and
|
|
:math:`c_{g}^{w}` (:eq:`5.108`, :eq:`5.109`, :eq:`5.110`)
|
|
|
|
#. Sensible heat flux from vegetation :math:`H_{v}` (:eq:`5.97` )
|
|
|
|
#. Latent heat flux from vegetation :math:`\lambda E_{v}` (:eq:`5.101` )
|
|
|
|
#. If the latent heat flux has changed sign from the latent heat flux
|
|
computed at the previous iteration
|
|
(:math:`\lambda E_{v} ^{n+1} \times \lambda E_{v} ^{n} <0`), the
|
|
latent heat flux is constrained to be 10% of the computed value. The
|
|
difference between the constrained and computed value
|
|
(:math:`\Delta _{1} =0.1\lambda E_{v} ^{n+1} -\lambda E_{v} ^{n+1}` )
|
|
is added to the sensible heat flux later.
|
|
|
|
#. Change in vegetation temperature :math:`\Delta T_{v}` (:eq:`5.129` ) and
|
|
update the vegetation temperature as
|
|
:math:`T_{v}^{n+1} =T_{v}^{n} +\Delta T_{v}` . :math:`T_{v}` is
|
|
constrained to change by no more than 1ºK in one iteration. If this
|
|
limit is exceeded, the energy error is
|
|
|
|
.. math::
|
|
:label: 5.138
|
|
|
|
\Delta _{2} =\overrightarrow{S}_{v} -\overrightarrow{L}_{v} -\frac{\partial \overrightarrow{L}_{v} }{\partial T_{v} } \Delta T_{v} -H_{v} -\frac{\partial H_{v} }{\partial T_{v} } \Delta T_{v} -\lambda E_{v} -\frac{\partial \lambda E_{v} }{\partial T_{v} } \Delta T_{v}
|
|
|
|
where :math:`\Delta T_{v} =1{\rm \; or\; }-1`. The error
|
|
:math:`\Delta _{2}` is added to the sensible heat flux later.
|
|
|
|
#. Water vapor flux :math:`E_{v}` (:eq:`5.133` )
|
|
|
|
#. Transpiration :math:`E_{v}^{t}` (:eq:`5.134` if :math:`\beta_{t} >0`,
|
|
otherwise :math:`E_{v}^{t} =0`)
|
|
|
|
#. The water vapor flux :math:`E_{v}` is constrained to be less than or
|
|
equal to the sum of transpiration :math:`E_{v}^{t}` and the water
|
|
available from wetted leaves and stems
|
|
:math:`{W_{can} \mathord{\left/ {\vphantom {W_{can} \Delta t}} \right. \kern-\nulldelimiterspace} \Delta t}` .
|
|
The energy error due to this constraint is
|
|
|
|
.. math::
|
|
:label: 5.139
|
|
|
|
\Delta _{3} =\max \left(0,\, E_{v} -E_{v}^{t} -\frac{W_{can} }{\Delta t} \right).
|
|
|
|
The error :math:`\lambda \Delta _{3}` is added to the sensible heat
|
|
flux later.
|
|
|
|
#. Sensible heat flux :math:`H_{v}` (:eq:`5.135` ). The three energy error
|
|
terms, :math:`\Delta _{1}` , :math:`\Delta _{2}` , and
|
|
:math:`\lambda \Delta _{3}` are also added to the sensible heat
|
|
flux.
|
|
|
|
#. The saturated vapor pressure :math:`e_{i}` (Chapter
|
|
:numref:`rst_Stomatal Resistance and Photosynthesis`), saturated
|
|
specific humidity :math:`q_{sat}^{T_{v} }` and its derivative
|
|
:math:`\frac{dq_{sat}^{T_{v} } }{dT_{v} }` at the leaf surface
|
|
(section :numref:`Saturation Vapor Pressure`), are re-evaluated based on
|
|
the new :math:`T_{v}` .
|
|
|
|
#. Canopy air temperature :math:`T_{s}` (:eq:`5.93` )
|
|
|
|
#. Canopy air specific humidity :math:`q_{s}` (:eq:`5.107` )
|
|
|
|
#. Temperature difference :math:`\theta _{atm} -\theta _{s}`
|
|
|
|
#. Specific humidity difference :math:`q_{atm} -q_{s}`
|
|
|
|
#. Potential temperature scale
|
|
:math:`\theta _{*} =\frac{\theta _{*} }{\theta _{atm} -\theta _{s} } \left(\theta _{atm} -\theta _{s} \right)`
|
|
where :math:`\frac{\theta _{*} }{\theta _{atm} -\theta _{s} }` was
|
|
calculated earlier in the iteration
|
|
|
|
#. Humidity scale
|
|
:math:`q_{*} =\frac{q_{*} }{q_{atm} -q_{s} } \left(q_{atm} -q_{s} \right)`
|
|
where :math:`\frac{q_{*} }{q_{atm} -q_{s} }` was calculated earlier
|
|
in the iteration
|
|
|
|
#. Virtual potential temperature scale :math:`\theta _{v*}` (:eq:`5.17` )
|
|
|
|
#. Wind speed including the convective velocity, :math:`V_{a}` (:eq:`5.24` )
|
|
|
|
#. Monin-Obukhov length :math:`L` (:eq:`5.49` )
|
|
|
|
#. The iteration is stopped after two or more steps if
|
|
:math:`\tilde{\Delta }T_{v} <0.01` and
|
|
:math:`\left|\lambda E_{v}^{n+1} -\lambda E_{v}^{n} \right|<0.1`
|
|
where
|
|
:math:`\tilde{\Delta }T_{v} =\max \left(\left|T_{v}^{n+1} -T_{v}^{n} \right|,\, \left|T_{v}^{n} -T_{v}^{n-1} \right|\right)`,
|
|
or after forty iterations have been carried out.
|
|
|
|
#. Momentum fluxes :math:`\tau _{x}` , :math:`\tau _{y}` (:eq:`5.5`, :eq:`5.6`)
|
|
|
|
#. Sensible heat flux from ground :math:`H_{g}` (:eq:`5.89` )
|
|
|
|
#. Water vapor flux from ground :math:`E_{g}` (:eq:`5.102` )
|
|
|
|
#. 2-m height air temperature :math:`T_{2m}` , specific humidity
|
|
:math:`q_{2m}` , relative humidity :math:`RH_{2m}` \ (:eq:`5.58` , :eq:`5.59`, :eq:`5.60`)
|
|
|
|
.. _Update of Ground Sensible and Latent Heat Fluxes:
|
|
|
|
Update of Ground Sensible and Latent Heat Fluxes
|
|
----------------------------------------------------
|
|
|
|
The sensible and water vapor heat fluxes derived above for bare soil and
|
|
soil beneath canopy are based on the ground surface temperature from the
|
|
previous time step :math:`T_{g}^{n}` and are used as the surface
|
|
forcing for the solution of the soil temperature equations (section
|
|
:numref:`Numerical Solution Temperature`). This solution yields a new ground
|
|
surface temperature :math:`T_{g}^{n+1}` . The ground sensible and water
|
|
vapor fluxes are then updated for :math:`T_{g}^{n+1}` as
|
|
|
|
.. math::
|
|
:label: 5.140
|
|
|
|
H'_{g} =H_{g} +\left(T_{g}^{n+1} -T_{g}^{n} \right)\frac{\partial H_{g} }{\partial T_{g} }
|
|
|
|
.. math::
|
|
:label: 5.141
|
|
|
|
E'_{g} =E_{g} +\left(T_{g}^{n+1} -T_{g}^{n} \right)\frac{\partial E_{g} }{\partial T_{g} }
|
|
|
|
where :math:`H_{g}` and :math:`E_{g}` are the sensible heat and water
|
|
vapor fluxes derived from equations and for non-vegetated surfaces and
|
|
equations and for vegetated surfaces using :math:`T_{g}^{n}` . One
|
|
further adjustment is made to :math:`H'_{g}` and :math:`E'_{g}` . If
|
|
the soil moisture in the top snow/soil layer is not sufficient to
|
|
support the updated ground evaporation, i.e., if :math:`E'_{g} > 0` and
|
|
:math:`f_{evap} < 1` where
|
|
|
|
.. math::
|
|
:label: 5.142
|
|
|
|
f_{evap} =\frac{{\left(w_{ice,\; snl+1} +w_{liq,\, snl+1} \right)\mathord{\left/ {\vphantom {\left(w_{ice,\; snl+1} +w_{liq,\, snl+1} \right) \Delta t}} \right. \kern-\nulldelimiterspace} \Delta t} }{\sum _{j=1}^{npft}\left(E'_{g} \right)_{j} \left(wt\right)_{j} } \le 1,
|
|
|
|
an adjustment is made to reduce the ground evaporation accordingly as
|
|
|
|
.. math::
|
|
:label: 5.143
|
|
|
|
E''_{g} =f_{evap} E'_{g} .
|
|
|
|
The term
|
|
:math:`\sum _{j=1}^{npft}\left(E'_{g} \right)_{j} \left(wt\right)_{j}`
|
|
is the sum of :math:`E'_{g}` over all evaporating PFTs where
|
|
:math:`\left(E'_{g} \right)_{j}` is the ground evaporation from the
|
|
:math:`j^{th}` PFT on the column, :math:`\left(wt\right)_{j}` is the
|
|
relative area of the :math:`j^{th}` PFT with respect to the column, and
|
|
:math:`npft` is the number of PFTs on the column.
|
|
:math:`w_{ice,\, snl+1}` and :math:`w_{liq,\, snl+1}` are the ice and
|
|
liquid water contents (kg m\ :sup:`-2`) of the top snow/soil layer
|
|
(Chapter :numref:`rst_Hydrology`). Any resulting energy deficit is assigned
|
|
to sensible heat
|
|
as
|
|
|
|
.. math::
|
|
:label: 5.144
|
|
|
|
H''_{g} =H_{g} +\lambda \left(E'_{g} -E''_{g} \right).
|
|
|
|
The ground water vapor flux :math:`E''_{g}` is partitioned into evaporation
|
|
of liquid water from snow/soil :math:`q_{seva}` (kg\ m\ :sup:`-2` s\ :sup:`-1`),
|
|
sublimation from snow/soil ice :math:`q_{subl}` (kg m\ :sup:`-2` s\ :sup:`-1`),
|
|
liquid dew on snow/soil :math:`q_{sdew}` (kg m\ :sup:`-2` s\ :sup:`-1`), or
|
|
frost on snow/soil :math:`q_{frost}` (kg m\ :sup:`-2` s\ :sup:`-1`) as
|
|
|
|
.. math::
|
|
:label: 5.145
|
|
|
|
q_{seva} =\max \left(E''_{sno} \frac{w_{liq,\, snl+1} }{w_{ice,\; snl+1} +w_{liq,\, snl+1} } ,0\right)\qquad E''_{sno} \ge 0,\, w_{ice,\; snl+1} +w_{liq,\, snl+1} >0
|
|
|
|
.. math::
|
|
:label: 5.146
|
|
|
|
q_{subl} =E''_{sno} -q_{seva} \qquad E''_{sno} \ge 0
|
|
|
|
.. math::
|
|
:label: 5.147
|
|
|
|
q_{sdew} =\left|E''_{sno} \right|\qquad E''_{sno} <0{\rm \; and\; }T_{g} \ge T_{f}
|
|
|
|
.. math::
|
|
:label: 5.148
|
|
|
|
q_{frost} =\left|E''_{sno} \right|\qquad E''_{sno} <0{\rm \; and\; }T_{g} <T_{f} .
|
|
|
|
The loss or gain in snow mass due to :math:`q_{seva}` ,
|
|
:math:`q_{subl}` , :math:`q_{sdew}` , and :math:`q_{frost}` on a snow
|
|
surface are accounted for during the snow hydrology calculations
|
|
(Chapter :numref:`rst_Snow Hydrology`). The loss of soil and surface water due to
|
|
:math:`q_{seva}` is accounted for in the calculation of infiltration
|
|
(section :numref:`Infiltration`), while losses or gains due to :math:`q_{subl}` ,
|
|
:math:`q_{sdew}` , and :math:`q_{frost}` on a soil surface are
|
|
accounted for following the sub-surface drainage calculations (section
|
|
:numref:`Lateral Sub-surface Runoff`).
|
|
|
|
The ground heat flux :math:`G` is calculated as
|
|
|
|
.. math::
|
|
:label: 5.149
|
|
|
|
G=\overrightarrow{S}_{g} -\overrightarrow{L}_{g} -H_{g} -\lambda E_{g}
|
|
|
|
where :math:`\overrightarrow{S}_{g}` is the solar radiation absorbed by
|
|
the ground (section :numref:`Solar Fluxes`), :math:`\overrightarrow{L}_{g}` is the net
|
|
longwave radiation absorbed by the ground (section :numref:`Longwave Fluxes`)
|
|
|
|
.. math::
|
|
:label: 5.150
|
|
|
|
\vec{L}_{g} =L_{g} \uparrow -\delta _{veg} \varepsilon _{g} L_{v} \, \downarrow -\left(1-\delta _{veg} \right)\varepsilon _{g} L_{atm} \, \downarrow +4\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{3} \left(T_{g}^{n+1} -T_{g}^{n} \right),
|
|
|
|
where
|
|
|
|
.. math::
|
|
:label: 5.151
|
|
|
|
L_{g} \uparrow =\varepsilon _{g} \sigma \left[\left(1-f_{sno} -f_{h2osfc} \right)\left(T_{1}^{n} \right)^{4} +f_{sno} \left(T_{sno}^{n} \right)^{4} +f_{h2osfc} \left(T_{h2osfc}^{n} \right)^{4} \right]
|
|
|
|
and :math:`H_{g}` and :math:`\lambda E_{g}` are the sensible and
|
|
latent heat fluxes after the adjustments described above.
|
|
|
|
When converting ground water vapor flux to an energy flux, the term
|
|
:math:`\lambda` is arbitrarily assumed to be
|
|
|
|
.. math::
|
|
:label: 5.152
|
|
|
|
\lambda =\left\{\begin{array}{l} {\lambda _{sub} \qquad {\rm if\; }w_{liq,\, snl+1} =0{\rm \; and\; }w_{ice,\, snl+1} >0} \\ {\lambda _{vap} \qquad {\rm otherwise}} \end{array}\right\}
|
|
|
|
where :math:`\lambda _{sub}` and :math:`\lambda _{vap}` are the latent
|
|
heat of sublimation and vaporization, respectively (J
|
|
(kg\ :sup:`-1`) (:numref:`Table Physical constants`). When converting vegetation water vapor
|
|
flux to an energy flux, :math:`\lambda _{vap}` is used.
|
|
|
|
The system balances energy as
|
|
|
|
.. math::
|
|
:label: 5.153
|
|
|
|
\overrightarrow{S}_{g} +\overrightarrow{S}_{v} +L_{atm} \, \downarrow -L\, \uparrow -H_{v} -H_{g} -\lambda _{vap} E_{v} -\lambda E_{g} -G=0.
|
|
|
|
.. _Saturation Vapor Pressure:
|
|
|
|
Saturation Vapor Pressure
|
|
-----------------------------
|
|
|
|
Saturation vapor pressure :math:`e_{sat}^{T}` (Pa) and its derivative
|
|
:math:`\frac{de_{sat}^{T} }{dT}` , as a function of temperature
|
|
:math:`T` (ºC), are calculated from the eighth-order polynomial fits of
|
|
:ref:`Flatau et al. (1992) <Flatauetal1992>`
|
|
|
|
.. math::
|
|
:label: 5.154
|
|
|
|
e_{sat}^{T} =100\left[a_{0} +a_{1} T+\cdots +a_{n} T^{n} \right]
|
|
|
|
.. math::
|
|
:label: 5.155
|
|
|
|
\frac{de_{sat}^{T} }{dT} =100\left[b_{0} +b_{1} T+\cdots +b_{n} T^{n} \right]
|
|
|
|
where the coefficients for ice are valid for
|
|
:math:`-75\, ^{\circ } {\rm C}\le T<0\, ^{\circ } {\rm C}` and the
|
|
coefficients for water are valid for
|
|
:math:`0\, ^{\circ } {\rm C}\le T\le 100\, ^{\circ } {\rm C}`
|
|
(:numref:`Table Coefficients for saturation vapor pressure` and
|
|
:numref:`Table Coefficients for derivative of esat`).
|
|
The saturated water vapor specific humidity :math:`q_{sat}^{T}` and its derivative
|
|
:math:`\frac{dq_{sat}^{T} }{dT}` are
|
|
|
|
.. math::
|
|
:label: 5.156
|
|
|
|
q_{sat}^{T} =\frac{0.622e_{sat}^{T} }{P_{atm} -0.378e_{sat}^{T} }
|
|
|
|
.. math::
|
|
:label: 5.157
|
|
|
|
\frac{dq_{sat}^{T} }{dT} =\frac{0.622P_{atm} }{\left(P_{atm} -0.378e_{sat}^{T} \right)^{2} } \frac{de_{sat}^{T} }{dT} .
|
|
|
|
.. _Table Coefficients for saturation vapor pressure:
|
|
|
|
.. table:: Coefficients for :math:`e_{sat}^{T}`
|
|
|
|
+------------------+------------------------------------------+----------------------------------------+
|
|
| | water | ice |
|
|
+==================+==========================================+========================================+
|
|
| :math:`a_{0}` | 6.11213476 | 6.11123516 |
|
|
+------------------+------------------------------------------+----------------------------------------+
|
|
| :math:`a_{1}` | 4.44007856 :math:`\times 10^{-1}` | 5.03109514\ :math:`\times 10^{-1}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{2}` | 1.43064234 :math:`\times 10^{-2}` | 1.88369801\ :math:`\times 10^{-2}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{3}` | 2.64461437 :math:`\times 10^{-4}` | 4.20547422\ :math:`\times 10^{-4}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{4}` | 3.05903558 :math:`\times 10^{-6}` | 6.14396778\ :math:`\times 10^{-6}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{5}` | 1.96237241 :math:`\times 10^{-8}` | 6.02780717\ :math:`\times 10^{-8}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{6}` | 8.92344772 :math:`\times 10^{-11}` | 3.87940929\ :math:`\times 10^{-10}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{7}` | -3.73208410 :math:`\times 10^{-13}` | 1.49436277\ :math:`\times 10^{-12}` |
|
|
+------------------+-------------------------------------------+---------------------------------------+
|
|
| :math:`a_{8}` | 2.09339997 :math:`\times 10^{-16}` | 2.62655803\ :math:`\times 10^{-15}` |
|
|
+------------------+------------------------------------------+----------------------------------------+
|
|
|
|
.. _Table Coefficients for derivative of esat:
|
|
|
|
.. table:: Coefficients for :math:`\frac{de_{sat}^{T} }{dT}`
|
|
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| | water | ice |
|
|
+==================+========================================+========================================+
|
|
| :math:`b_{0}` | 4.44017302\ :math:`\times 10^{-1}` | 5.03277922\ :math:`\times 10^{-1}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{1}` | 2.86064092\ :math:`\times 10^{-2}` | 3.77289173\ :math:`\times 10^{-2}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{2}` | 7.94683137\ :math:`\times 10^{-4}` | 1.26801703\ :math:`\times 10^{-3}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{3}` | 1.21211669\ :math:`\times 10^{-5}` | 2.49468427\ :math:`\times 10^{-5}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{4}` | 1.03354611\ :math:`\times 10^{-7}` | 3.13703411\ :math:`\times 10^{-7}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{5}` | 4.04125005\ :math:`\times 10^{-10}` | 2.57180651\ :math:`\times 10^{-9}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{6}` | -7.88037859 :math:`\times 10^{-13}` | 1.33268878\ :math:`\times 10^{-11}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{7}` | -1.14596802 :math:`\times 10^{-14}` | 3.94116744\ :math:`\times 10^{-14}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
| :math:`b_{8}` | 3.81294516\ :math:`\times 10^{-17}` | 4.98070196\ :math:`\times 10^{-17}` |
|
|
+------------------+----------------------------------------+----------------------------------------+
|
|
|