## 2.5.1. Monin-Obukhov Similarity Theory[¶](#monin-obukhov-similarity-theory "Permalink to this headline")
--------------------------------------------------------------------------------------------------------

The surface vertical kinematic fluxes of momentum \\(\\overline{u'w'}\\) and \\(\\overline{v'w'}\\) (m2 s\-2), sensible heat \\(\\overline{\\theta 'w'}\\) (K m s \-1), and latent heat \\(\\overline{q'w'}\\) (kg kg\-1 m s\-1), where \\(u'\\), \\(v'\\), \\(w'\\), \\(\\theta '\\), and \\(q'\\) are zonal horizontal wind, meridional horizontal wind, vertical velocity, potential temperature, and specific humidity turbulent fluctuations about the mean, are defined from Monin-Obukhov similarity applied to the surface layer. This theory states that when scaled appropriately, the dimensionless mean horizontal wind speed, mean potential temperature, and mean specific humidity profile gradients depend on unique functions of \\(\\zeta =\\frac{z-d}{L}\\) ([Zeng et al. 1998](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#zengetal1998)) as

(2.5.10)[¶](#equation-5-10 "Permalink to this equation")\\\[\\frac{k\\left(z-d\\right)}{u\_{\*} } \\frac{\\partial \\left|{\\it u}\\right|}{\\partial z} =\\phi \_{m} \\left(\\zeta \\right)\\\]

(2.5.11)[¶](#equation-5-11 "Permalink to this equation")\\\[\\frac{k\\left(z-d\\right)}{\\theta \_{\*} } \\frac{\\partial \\theta }{\\partial z} =\\phi \_{h} \\left(\\zeta \\right)\\\]

(2.5.12)[¶](#equation-5-12 "Permalink to this equation")\\\[\\frac{k\\left(z-d\\right)}{q\_{\*} } \\frac{\\partial q}{\\partial z} =\\phi \_{w} \\left(\\zeta \\right)\\\]

where \\(z\\) is height in the surface layer (m), \\(d\\) is the displacement height (m), \\(L\\) is the Monin-Obukhov length scale (m) that accounts for buoyancy effects resulting from vertical density gradients (i.e., the atmospheric stability), k is the von Karman 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)), and \\(\\left|{\\it u}\\right|\\) is the atmospheric wind speed (m s\-1). \\(\\phi \_{m}\\), \\(\\phi \_{h}\\), and \\(\\phi \_{w}\\) are universal (over any surface) similarity functions of \\(\\zeta\\) that relate the constant fluxes of momentum, sensible heat, and latent heat to the mean profile gradients of \\(\\left|{\\it u}\\right|\\), \\(\\theta\\), and \\(q\\) in the surface layer. In neutral conditions, \\(\\phi \_{m} =\\phi \_{h} =\\phi \_{w} =1\\). The velocity (i.e., friction velocity) \\(u\_{\*}\\) (m s\-1), temperature \\(\\theta \_{\*}\\) (K), and moisture \\(q\_{\*}\\) (kg kg\-1) scales are

(2.5.13)[¶](#equation-5-13 "Permalink to this equation")\\\[u\_{\*}^{2} =\\sqrt{\\left(\\overline{u'w'}\\right)^{2} +\\left(\\overline{v'w'}\\right)^{2} } =\\frac{\\left|{\\it \\tau }\\right|}{\\rho \_{atm} }\\\]

(2.5.14)[¶](#equation-5-14 "Permalink to this equation")\\\[\\theta \_{\*} u\_{\*} =-\\overline{\\theta 'w'}=-\\frac{H}{\\rho \_{atm} C\_{p} }\\\]

(2.5.15)[¶](#equation-5-15 "Permalink to this equation")\\\[q\_{\*} u\_{\*} =-\\overline{q'w'}=-\\frac{E}{\\rho \_{atm} }\\\]

where \\(\\left|{\\it \\tau }\\right|\\) is the shearing stress (kg m\-1 s\-2), with zonal and meridional components \\(\\overline{u'w'}=-\\frac{\\tau \_{x} }{\\rho \_{atm} }\\) and \\(\\overline{v'w'}=-\\frac{\\tau \_{y} }{\\rho \_{atm} }\\), respectively, \\(H\\) is the sensible heat flux (W m\-2) and \\(E\\) is the water vapor flux (kg m\-2 s\-1).

The length scale \\(L\\) is the Monin-Obukhov length defined as

(2.5.16)[¶](#equation-5-16 "Permalink to this equation")\\\[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\*} }\\\]

where \\(g\\) is the acceleration of gravity (m s\-2) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)), and \\(\\overline{\\theta \_{v,\\, atm} }=\\overline{\\theta \_{atm} }\\left(1+0.61q\_{atm} \\right)\\) is the reference virtual potential temperature. \\(L>0\\) indicates stable conditions. \\(L<0\\) indicates unstable conditions. \\(L=\\infty\\) for neutral conditions. The temperature scale \\(\\theta \_{v\*}\\) is defined as

(2.5.17)[¶](#equation-5-17 "Permalink to this equation")\\\[\\theta \_{v\*} u\_{\*} =\\left\[\\theta \_{\*} \\left(1+0.61q\_{atm} \\right)+0.61\\overline{\\theta \_{atm} }q\_{\*} \\right\]u\_{\*}\\\]

where \\(\\overline{\\theta \_{atm} }\\) is the atmospheric potential temperature.

Following [Panofsky and Dutton (1984)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#panofskydutton1984), the differential equations for \\(\\phi \_{m} \\left(\\zeta \\right)\\), \\(\\phi \_{h} \\left(\\zeta \\right)\\), and \\(\\phi \_{w} \\left(\\zeta \\right)\\) can be integrated formally without commitment to their exact forms. Integration between two arbitrary heights in the surface layer \\(z\_{2}\\) and \\(z\_{1}\\) (\\(z\_{2} >z\_{1}\\) ) with horizontal winds \\(\\left|{\\it u}\\right|\_{1}\\) and \\(\\left|{\\it u}\\right|\_{2}\\), potential temperatures \\(\\theta \_{1}\\) and \\(\\theta \_{2}\\), and specific humidities \\(q\_{1}\\) and \\(q\_{2}\\) results in

(2.5.18)[¶](#equation-5-18 "Permalink to this equation")\\\[\\left|{\\it u}\\right|\_{2} -\\left|{\\it u}\\right|\_{1} =\\frac{u\_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{2} -d}{z\_{1} -d} \\right)-\\psi \_{m} \\left(\\frac{z\_{2} -d}{L} \\right)+\\psi \_{m} \\left(\\frac{z\_{1} -d}{L} \\right)\\right\]\\\]

(2.5.19)[¶](#equation-5-19 "Permalink to this equation")\\\[\\theta \_{2} -\\theta \_{1} =\\frac{\\theta \_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{2} -d}{z\_{1} -d} \\right)-\\psi \_{h} \\left(\\frac{z\_{2} -d}{L} \\right)+\\psi \_{h} \\left(\\frac{z\_{1} -d}{L} \\right)\\right\]\\\]

(2.5.20)[¶](#equation-5-20 "Permalink to this equation")\\\[q\_{2} -q\_{1} =\\frac{q\_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{2} -d}{z\_{1} -d} \\right)-\\psi \_{w} \\left(\\frac{z\_{2} -d}{L} \\right)+\\psi \_{w} \\left(\\frac{z\_{1} -d}{L} \\right)\\right\].\\\]

The functions \\(\\psi \_{m} \\left(\\zeta \\right)\\), \\(\\psi \_{h} \\left(\\zeta \\right)\\), and \\(\\psi \_{w} \\left(\\zeta \\right)\\) are defined as

(2.5.21)[¶](#equation-5-21 "Permalink to this equation")\\\[\\psi \_{m} \\left(\\zeta \\right)=\\int \_{{z\_{0m} \\mathord{\\left/ {\\vphantom {z\_{0m} L}} \\right.} L} }^{\\zeta }\\frac{\\left\[1-\\phi \_{m} \\left(x\\right)\\right\]}{x} \\, dx\\\]

(2.5.22)[¶](#equation-5-22 "Permalink to this equation")\\\[\\psi \_{h} \\left(\\zeta \\right)=\\int \_{{z\_{0h} \\mathord{\\left/ {\\vphantom {z\_{0h} L}} \\right.} L} }^{\\zeta }\\frac{\\left\[1-\\phi \_{h} \\left(x\\right)\\right\]}{x} \\, dx\\\]

(2.5.23)[¶](#equation-5-23 "Permalink to this equation")\\\[\\psi \_{w} \\left(\\zeta \\right)=\\int \_{{z\_{0w} \\mathord{\\left/ {\\vphantom {z\_{0w} L}} \\right.} L} }^{\\zeta }\\frac{\\left\[1-\\phi \_{w} \\left(x\\right)\\right\]}{x} \\, dx\\\]

where \\(z\_{0m}\\), \\(z\_{0h}\\), and \\(z\_{0w}\\) are the roughness lengths (m) for momentum, sensible heat, and water vapor, respectively.

Defining the surface values

\\\[\\left|{\\it u}\\right|\_{1} =0{\\rm \\; at\\; }z\_{1} =z\_{0m} +d,\\\]

\\\[\\theta \_{1} =\\theta \_{s} {\\rm \\; at\\; }z\_{1} =z\_{0h} +d,{\\rm \\; and}\\\]

\\\[q\_{1} =q\_{s} {\\rm \\; at\\; }z\_{1} =z\_{0w} +d,\\\]

and the atmospheric values at \\(z\_{2} =z\_{atm,\\, x}\\)

(2.5.24)[¶](#equation-5-24 "Permalink to this equation")\\\[\\left|{\\it u}\\right|\_{2} =V\_{a} {\\rm =\\; }\\sqrt{u\_{atm}^{2} +v\_{atm}^{2} +U\_{c}^{2} } \\ge 1,\\\]

\\\[\\theta \_{2} =\\theta \_{atm} {\\rm ,\\; and}\\\]

\\\[q\_{2} =q\_{atm} {\\rm ,\\; }\\\]

the integral forms of the flux-gradient relations are

(2.5.25)[¶](#equation-5-25 "Permalink to this equation")\\\[V\_{a} =\\frac{u\_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{atm,\\, m} -d}{z\_{0m} } \\right)-\\psi \_{m} \\left(\\frac{z\_{atm,\\, m} -d}{L} \\right)+\\psi \_{m} \\left(\\frac{z\_{0m} }{L} \\right)\\right\]\\\]

(2.5.26)[¶](#equation-5-26 "Permalink to this equation")\\\[\\theta \_{atm} -\\theta \_{s} =\\frac{\\theta \_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{atm,\\, h} -d}{z\_{0h} } \\right)-\\psi \_{h} \\left(\\frac{z\_{atm,\\, h} -d}{L} \\right)+\\psi \_{h} \\left(\\frac{z\_{0h} }{L} \\right)\\right\]\\\]

(2.5.27)[¶](#equation-5-27 "Permalink to this equation")\\\[q\_{atm} -q\_{s} =\\frac{q\_{\*} }{k} \\left\[\\ln \\left(\\frac{z\_{atm,\\, w} -d}{z\_{0w} } \\right)-\\psi \_{w} \\left(\\frac{z\_{atm,\\, w} -d}{L} \\right)+\\psi \_{w} \\left(\\frac{z\_{0w} }{L} \\right)\\right\].\\\]

The constraint \\(V\_{a} \\ge 1\\) is required simply for numerical reasons to prevent \\(H\\) and \\(E\\) from becoming small with small wind speeds. The convective velocity \\(U\_{c}\\) accounts for the contribution of large eddies in the convective boundary layer to surface fluxes as follows

(2.5.28)[¶](#equation-5-28 "Permalink to this equation")\\\[\\begin{split}U\_{c} = \\left\\{ \\begin{array}{ll} 0 & \\qquad \\zeta \\ge {\\rm 0} \\quad {\\rm (stable)} \\\\ \\beta w\_{\*} & \\qquad \\zeta < 0 \\quad {\\rm (unstable)} \\end{array} \\right\\}\\end{split}\\\]

where \\(w\_{\*}\\) is the convective velocity scale

(2.5.29)[¶](#equation-5-29 "Permalink to this equation")\\\[w\_{\*} =\\left(\\frac{-gu\_{\*} \\theta \_{v\*} z\_{i} }{\\overline{\\theta \_{v,\\, atm} }} \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } ,\\\]

\\(z\_{i} =1000\\) is the convective boundary layer height (m), and \\(\\beta =1\\).

The momentum flux gradient relations are ([Zeng et al. 1998](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#zengetal1998))

(2.5.30)[¶](#equation-5-30 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{llr} \\phi \_{m} \\left(\\zeta \\right)=0.7k^{{2\\mathord{\\left/ {\\vphantom {2 3}} \\right.} 3} } \\left(-\\zeta \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } & \\qquad {\\rm for\\; }\\zeta <-1.574 & \\ {\\rm \\; (very\\; unstable)} \\\\ \\phi \_{m} \\left(\\zeta \\right)=\\left(1-16\\zeta \\right)^{-{1\\mathord{\\left/ {\\vphantom {1 4}} \\right.} 4} } & \\qquad {\\rm for\\; -1.574}\\le \\zeta <0 & \\ {\\rm \\; (unstable)} \\\\ \\phi \_{m} \\left(\\zeta \\right)=1+5\\zeta & \\qquad {\\rm for\\; }0\\le \\zeta \\le 1& \\ {\\rm \\; (stable)} \\\\ \\phi \_{m} \\left(\\zeta \\right)=5+\\zeta & \\qquad {\\rm for\\; }\\zeta >1 & \\ {\\rm\\; (very\\; stable).} \\end{array}\\end{split}\\\]

The sensible and latent heat flux gradient relations are ([Zeng et al. 1998](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#zengetal1998))

(2.5.31)[¶](#equation-5-31 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{llr} \\phi \_{h} \\left(\\zeta \\right)=\\phi \_{w} \\left(\\zeta \\right)=0.9k^{{4\\mathord{\\left/ {\\vphantom {4 3}} \\right.} 3} } \\left(-\\zeta \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 3}} \\right.} 3} } & \\qquad {\\rm for\\; }\\zeta <-0.465 & \\ {\\rm \\; (very\\; unstable)} \\\\ \\phi \_{h} \\left(\\zeta \\right)=\\phi \_{w} \\left(\\zeta \\right)=\\left(1-16\\zeta \\right)^{-{1\\mathord{\\left/ {\\vphantom {1 2}} \\right.} 2} } & \\qquad {\\rm for\\; -0.465}\\le \\zeta <0 & \\ {\\rm \\; (unstable)} \\\\ \\phi \_{h} \\left(\\zeta \\right)=\\phi \_{w} \\left(\\zeta \\right)=1+5\\zeta & \\qquad {\\rm for\\; }0\\le \\zeta \\le 1 & \\ {\\rm \\; (stable)} \\\\ \\phi \_{h} \\left(\\zeta \\right)=\\phi \_{w} \\left(\\zeta \\right)=5+\\zeta & \\qquad {\\rm for\\; }\\zeta >1 & \\ {\\rm \\; (very\\; stable).} \\end{array}\\end{split}\\\]

To ensure continuous functions of \\(\\phi \_{m} \\left(\\zeta \\right)\\), \\(\\phi \_{h} \\left(\\zeta \\right)\\), and \\(\\phi \_{w} \\left(\\zeta \\right)\\), the simplest approach (i.e., without considering any transition regimes) is to match the relations for very unstable and unstable conditions at \\(\\zeta \_{m} =-1.574\\) for \\(\\phi \_{m} \\left(\\zeta \\right)\\) and \\(\\zeta \_{h} =\\zeta \_{w} =-0.465\\) for \\(\\phi \_{h} \\left(\\zeta \\right)=\\phi \_{w} \\left(\\zeta \\right)\\) ([Zeng et al. 1998](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#zengetal1998)). The flux gradient relations can be integrated to yield wind profiles for the following conditions:

Very unstable \\(\\left(\\zeta <-1.574\\right)\\)

(2.5.32)[¶](#equation-5-32 "Permalink to this equation")\\\[V\_{a} =\\frac{u\_{\*} }{k} \\left\\{\\left\[\\ln \\frac{\\zeta \_{m} L}{z\_{0m} } -\\psi \_{m} \\left(\\zeta \_{m} \\right)\\right\]+1.14\\left\[\\left(-\\zeta \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } -\\left(-\\zeta \_{m} \\right)^{{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} } \\right\]+\\psi \_{m} \\left(\\frac{z\_{0m} }{L} \\right)\\right\\}\\\]

Unstable \\(\\left(-1.574\\le \\zeta <0\\right)\\)

(2.5.33)[¶](#equation-5-33 "Permalink to this equation")\\\[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\\}\\\]

Stable \\(\\left(0\\le \\zeta \\le 1\\right)\\)

(2.5.34)[¶](#equation-5-34 "Permalink to this equation")\\\[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 \\(\\left(\\zeta >1\\right)\\)

(2.5.35)[¶](#equation-5-35 "Permalink to this equation")\\\[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

(2.5.36)[¶](#equation-5-36 "Permalink to this equation")\\\[\\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

\\(x=\\left(1-16\\zeta \\right)^{{1\\mathord{\\left/ {\\vphantom {1 4}} \\right.} 4} }\\) .

The potential temperature profiles are:

Very unstable \\(\\left(\\zeta <-0.465\\right)\\)

(2.5.37)[¶](#equation-5-37 "Permalink to this equation")\\\[\\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\[\\left(-\\zeta \_{h} \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 3}} \\right.} 3} } -\\left(-\\zeta \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 3}} \\right.} 3} } \\right\]+\\psi \_{h} \\left(\\frac{z\_{0h} }{L} \\right)\\right\\}\\\]

Unstable \\(\\left(-0.465\\le \\zeta <0\\right)\\)

(2.5.38)[¶](#equation-5-38 "Permalink to this equation")\\\[\\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 \\(\\left(0\\le \\zeta \\le 1\\right)\\)

(2.5.39)[¶](#equation-5-39 "Permalink to this equation")\\\[\\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 \\(\\left(\\zeta >1\\right)\\)

(2.5.40)[¶](#equation-5-40 "Permalink to this equation")\\\[\\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 \\(\\left(\\zeta <-0.465\\right)\\)

(2.5.41)[¶](#equation-5-41 "Permalink to this equation")\\\[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\[\\left(-\\zeta \_{w} \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 3}} \\right.} 3} } -\\left(-\\zeta \\right)^{{-1\\mathord{\\left/ {\\vphantom {-1 3}} \\right.} 3} } \\right\]+\\psi \_{w} \\left(\\frac{z\_{0w} }{L} \\right)\\right\\}\\\]

Unstable \\(\\left(-0.465\\le \\zeta <0\\right)\\)

(2.5.42)[¶](#equation-5-42 "Permalink to this equation")\\\[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 \\(\\left(0\\le \\zeta \\le 1\\right)\\)

(2.5.43)[¶](#equation-5-43 "Permalink to this equation")\\\[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 \\(\\left(\\zeta >1\\right)\\)

(2.5.44)[¶](#equation-5-44 "Permalink to this equation")\\\[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

(2.5.45)[¶](#equation-5-45 "Permalink to this equation")\\\[\\psi \_{h} \\left(\\zeta \\right)=\\psi \_{w} \\left(\\zeta \\right)=2\\ln \\left(\\frac{1+x^{2} }{2} \\right).\\\]

Using the definitions of \\(u\_{\*}\\), \\(\\theta \_{\*}\\), and \\(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 \\(\\left|{\\it u}\\right|\\), \\(\\theta\\), and \\(q\\) except that \\(L\\) depends on \\(u\_{\*}\\), \\(\\theta \_{\*}\\), and \\(q\_{\*}\\). However, the bulk Richardson number

(2.5.46)[¶](#equation-5-46 "Permalink to this equation")\\\[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 \\(\\zeta\\) ([Arya 2001](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#arya2001)) as

(2.5.47)[¶](#equation-5-47 "Permalink to this equation")\\\[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 \\(\\phi \_{h} =\\phi \_{m}^{2} =\\left(1-16\\zeta \\right)^{-{1\\mathord{\\left/ {\\vphantom {1 2}} \\right.} 2} }\\) for unstable conditions and \\(\\phi \_{h} =\\phi \_{m} =1+5\\zeta\\) for stable conditions to determine \\(\\psi \_{m} \\left(\\zeta \\right)\\) and \\(\\psi \_{h} \\left(\\zeta \\right)\\), the inverse relationship \\(\\zeta =f\\left(R\_{iB} \\right)\\) can be solved to obtain a first guess for \\(\\zeta\\) and thus \\(L\\) from

(2.5.48)[¶](#equation-5-48 "Permalink to this equation")\\\[\\begin{split}\\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}.\\end{split}\\\]

Upon iteration (section [2.5.3.2](#numerical-implementation)), the following is used to determine \\(\\zeta\\) and thus \\(L\\)

(2.5.49)[¶](#equation-5-49 "Permalink to this equation")\\\[\\zeta =\\frac{\\left(z\_{atm,\\, m} -d\\right)kg\\theta \_{v\*} }{u\_{\*}^{2} \\overline{\\theta \_{v,\\, atm} }}\\\]

where

\\\[\\begin{split}\\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}.\\end{split}\\\]

The difference in virtual potential air temperature between the reference height and the surface is

(2.5.50)[¶](#equation-5-50 "Permalink to this equation")\\\[\\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

(2.5.51)[¶](#equation-5-51 "Permalink to this equation")\\\[\\tau \_{x} =-\\rho \_{atm} \\frac{\\left(u\_{atm} -u\_{s} \\right)}{r\_{am} }\\\]

(2.5.52)[¶](#equation-5-52 "Permalink to this equation")\\\[\\tau \_{y} =-\\rho \_{atm} \\frac{\\left(v\_{atm} -v\_{s} \\right)}{r\_{am} }\\\]

(2.5.53)[¶](#equation-5-53 "Permalink to this equation")\\\[H=-\\rho \_{atm} C\_{p} \\frac{\\left(\\theta \_{atm} -\\theta \_{s} \\right)}{r\_{ah} }\\\]

(2.5.54)[¶](#equation-5-54 "Permalink to this equation")\\\[E=-\\rho \_{atm} \\frac{\\left(q\_{atm} -q\_{s} \\right)}{r\_{aw} }\\\]

where the aerodynamic resistances (s m\-1) are

(2.5.55)[¶](#equation-5-55 "Permalink to this equation")\\\[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}\\\]

(2.5.56)[¶](#equation-5-56 "Permalink to this equation")\\\[\\begin{split}\\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}\\end{split}\\\]

(2.5.57)[¶](#equation-5-57 "Permalink to this equation")\\\[\\begin{split}\\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}.\\end{split}\\\]

A 2-m height “screen” temperature is useful for comparison with observations

(2.5.58)[¶](#equation-5-58 "Permalink to this equation")\\\[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 (\\(z\_{0h} +d\\)). Similarly, a 2-m height specific humidity is defined as

(2.5.59)[¶](#equation-5-59 "Permalink to this equation")\\\[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

(2.5.60)[¶](#equation-5-60 "Permalink to this equation")\\\[RH\_{2m} =\\min \\left(100,\\, \\frac{q\_{2m} }{q\_{sat}^{T\_{2m} } } \\times 100\\right)\\\]

where \\(q\_{sat}^{T\_{2m} }\\) is the saturated specific humidity at the 2-m temperature \\(T\_{2m}\\) (section [2.5.5](#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 [2.30](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Dust/CLM50_Tech_Note_Dust.html#rst-dust-model))

(2.5.61)[¶](#equation-5-61 "Permalink to this equation")\\\[\\begin{split}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\\}\\end{split}\\\]

