Title: 2.5. Momentum, Sensible Heat, and Latent Heat Fluxes — ctsm CTSM master documentation

URL Source: https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html

Markdown Content:
The zonal \\(\\tau \_{x}\\) and meridional \\(\\tau \_{y}\\) momentum fluxes (kg m\-1 s\-2), sensible heat flux \\(H\\) (W m\-2), and water vapor flux \\(E\\) (kg m\-2 s\-1) between the atmosphere at reference height \\(z\_{atm,\\, x}\\) (m) \[where \\(x\\) is height for wind (momentum) (\\(m\\)), temperature (sensible heat) (\\(h\\)), and humidity (water vapor) (\\(w\\)); with zonal and meridional winds \\(u\_{atm}\\) and \\(v\_{atm}\\) (m s\-1), potential temperature \\(\\theta \_{atm}\\) (K), and specific humidity \\(q\_{atm}\\) (kg kg\-1)\] and the surface \[with \\(u\_{s}\\), \\(v\_{s}\\), \\(\\theta \_{s}\\), and \\(q\_{s}\\) \] are

(2.5.1)[¶](#equation-5-1 "Permalink to this equation")\\\[\\tau \_{x} =-\\rho \_{atm} \\frac{\\left(u\_{atm} -u\_{s} \\right)}{r\_{am} }\\\]

(2.5.2)[¶](#equation-5-2 "Permalink to this equation")\\\[\\tau \_{y} =-\\rho \_{atm} \\frac{\\left(v\_{atm} -v\_{s} \\right)}{r\_{am} }\\\]

(2.5.3)[¶](#equation-5-3 "Permalink to this equation")\\\[H=-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -\\theta \_{s} \\right)}{r\_{ah} }\\\]

(2.5.4)[¶](#equation-5-4 "Permalink to this equation")\\\[E=-\\rho \_{atm} \\frac{\\left(q\_{atm} -q\_{s} \\right)}{r\_{aw} } .\\\]

These fluxes are derived in the next section from Monin-Obukhov similarity theory developed for the surface layer (i.e., the nearly constant flux layer above the surface sublayer). In this derivation, \\(u\_{s}\\) and \\(v\_{s}\\) are defined to equal zero at height \\(z\_{0m} +d\\) (the apparent sink for momentum) so that \\(r\_{am}\\) is the aerodynamic resistance (s m\-1) for momentum between the atmosphere at height \\(z\_{atm,\\, m}\\) and the surface at height \\(z\_{0m} +d\\). Thus, the momentum fluxes become

(2.5.5)[¶](#equation-5-5 "Permalink to this equation")\\\[\\tau \_{x} =-\\rho \_{atm} \\frac{u\_{atm} }{r\_{am} }\\\]

(2.5.6)[¶](#equation-5-6 "Permalink to this equation")\\\[\\tau \_{y} =-\\rho \_{atm} \\frac{v\_{atm} }{r\_{am} } .\\\]

Likewise, \\(\\theta \_{s}\\) and \\(q\_{s}\\) are defined at heights \\(z\_{0h} +d\\) and \\(z\_{0w} +d\\) (the apparent sinks for heat and water vapor, respectively \\(r\_{aw}\\) are the aerodynamic resistances (s m\-1) to sensible heat and water vapor transfer between the atmosphere at heights \\(z\_{atm,\\, h}\\) and \\(z\_{atm,\\, w}\\) and the surface at heights \\(z\_{0h} +d\\) and \\(z\_{0w} +d\\), respectively. The specific heat capacity of air \\(C\_{p}\\) (J kg\-1 K\-1) is a constant ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). The atmospheric potential temperature used here is

(2.5.7)[¶](#equation-5-7 "Permalink to this equation")\\\[\\theta \_{atm} =T\_{atm} +\\Gamma \_{d} z\_{atm,\\, h}\\\]

where \\(T\_{atm}\\) is the air temperature (K) at height \\(z\_{atm,\\, h}\\) and \\(\\Gamma \_{d} =0.0098\\) K m\-1 is the negative of the dry adiabatic lapse rate \[this expression is first-order equivalent to \\(\\theta \_{atm} =T\_{atm} \\left({P\_{srf} \\mathord{\\left/ {\\vphantom {P\_{srf} P\_{atm} }} \\right.} P\_{atm} } \\right)^{{R\_{da} \\mathord{\\left/ {\\vphantom {R\_{da} C\_{p} }} \\right.} C\_{p} } }\\) ([Stull 1988](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#stull1988)), where \\(P\_{srf}\\) is the surface pressure (Pa), \\(P\_{atm}\\) is the atmospheric pressure (Pa), and \\(R\_{da}\\) is the gas constant for dry air (J kg\-1 K\-1) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants))\]. By definition, \\(\\theta \_{s} =T\_{s}\\). The density of moist air (kg m\-3) is

(2.5.8)[¶](#equation-5-8 "Permalink to this equation")\\\[\\rho \_{atm} =\\frac{P\_{atm} -0.378e\_{atm} }{R\_{da} T\_{atm} }\\\]

where the atmospheric vapor pressure \\(e\_{atm}\\) (Pa) is derived from the atmospheric specific humidity \\(q\_{atm}\\)

(2.5.9)[¶](#equation-5-9 "Permalink to this equation")\\\[e\_{atm} =\\frac{q\_{atm} P\_{atm} }{0.622+0.378q\_{atm} } .\\\]

