4.10 Surface Exchange Formulations

The surface exchange of heat, moisture and momentum between the atmosphere and land, ocean or ice surfaces are treated with a bulk exchange formulation. We present a description of each surface exchange separately. Although the functional forms of the exchange relations are identical, we present the descriptions of these components as developed and represented in the various subroutines in CAM 3.0. The differences in the exchange expressions are predominantly in the definition of roughness lengths and exchange coefficients. The description of surface exchange over ocean follows from Bryan et al. [29], and the surface exchange over sea ice is discussed in chapter 6. Over lakes, exchanges are computed by a lake model embedded in the land surface model described in the following section.

4.10.1 Land

In CAM 3.0, the NCAR Land Surface Model (LSM) [22] has been replaced by the Community Land Model CLM2 [23]. This new model includes components treating hydrological and biogeochemical processes, dynamic vegetation, and biogeophysics. Because of the increased complexity of this new model and since a complete description is available online, users of CAM 3.0 interested in CLM should consult this documentation at
http://www.cgd.ucar.edu/tss/clm/. A discussion is provided here only of the component of CLM which controls surface exchange processes.

Land surface fluxes of momentum, sensible heat, and latent heat are calculated from Monin-Obukhov similarity theory applied to the surface (i.e. constant flux) layer. The zonal $\tau_x$ and meridional $\tau_y$ momentum fluxes (kg m${}^{-1}$s${}^{-2}$), sensible heat $H$ (W m${}^{-2}$) and water vapor $E$ (kg m${}^{-2}$s${}^{-1}$) fluxes between the surface and the lowest model level $z_1$ are:

$$\tau_x = - \rho_1 \overline {(u'w')} = - \rho_1 u_*^2 (u_1 /V_a ) = \rho_1 \frac{{u_s - u_1 }}{{r_{am} }}$$ (4.406)

$$\tau_y = - \rho_1 \overline {(v'w')} = - \rho_1 u_*^2 (v_1 /V_a ) = \rho_1 \frac{{v_s - v_1 }}{{r_{am} }}$$ (4.407)

$$H = \phantom{-}\rho_1 c_p (\overline {w'\theta '} ) = - \rho_1 c_p u_* \theta_* = \rho_1 c_p \frac{{\theta_{s} - \theta_1 }} {{r_{ah} }}$$ (4.408)

$$E = \phantom{-}\rho_1 (\overline {w'q'} ) = - \rho_1 u_* q_* = \rho_1 \frac{{q_{s} - q_1 }}{{r_{aw} }}$$ (4.409)

$$r_{am} = V_a /u_*^2$$ (4.410)

$$r_{ah} = (\theta_1 - \theta_s )/u_* \theta_*$$ (4.411)

$$r_{aw} = (q_1 - q_s )/u_* q_*$$ (4.412)

where $\rho_1$, $u_1$, $v_1$, $\theta_1$ and $q_1$ are the density (kg m$^{-3}$), zonal wind (m s$^{-1}$), meridional wind (m s$^{-1}$), air potential temperature (K), and specific humidity (kg kg$^{-1}$) at the lowest model level. By definition, the surface winds $u_s$ and $v_s$ equal zero. The symbol $\theta_1$ represents temperature, and $q_1$ is specific humidity at surface. The terms $r_{am}$, $r_{ah}$, and $r_{aw}$ are the aerodynamic resistances (s m$^{-1}$) for momentum, sensible heat, and water vapor between the lowest model level at height $z_1$ and the surface at height $z_{0m}+d$ [$z_{0h}+d$]. Here $z_{0m}$ [$z_{0h}$] is the roughness length (m) for momentum [scalar] fluxes, and $d$ is the displacement height (m).

For the vegetated fraction of the grid, $\theta_s = T_{af}$ and $q_s = q_{af}$, where $T_{af}$ and $q_{af}$ are the air temperature and specific humidity within canopy space. For the non-vegetated fraction, $\theta_s = T_g$ and $q_s = q_g$, where $T_g$ and $q_g$ are the air temperature and specific humidity at ground surface. These terms are described by Dai et al. [45].

4.10.1.1 Roughness lengths and zero-plane displacement

The aerodynamic roughness $z_{0m}$ is used for wind, while the thermal roughness $z_{0h}$ is used for heat and water vapor. In general, $z_{0m}$ is different from $z_{0h}$, 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 ultimately be transferred by molecular diffusion across the interfacial sublayer. Over bare soil and snow cover, the simple relation from Zilitinkevich [201] can be used [197]:

$$\ln \frac{{z_{0m} }} {{z_{0h} }} = a\left( {\frac{{u_* z_{0m} }} {\nu }} \right)^{0.45}$$ (4.413)

$$a = 0.13$$ (4.414)

$$\nu = 1.5 \times 10^{ - 5} {\text{m}}^2 {\text{s}}^{-1}$$ (4.415)

Over canopy, the application of energy balance

$$R_n - H - L_v E = 0$$ (4.416)

(where $R_n$ is the net radiation absorbed by the canopy) is equivalent to the use of different $z_{0m}$ versus $z_{0h}$ over bare soil, and hence thermal roughness is not needed over canopy [198].

The roughness $z_{0m}$ is proportional to canopy height, and is also affected by fractional vegetation cover, leaf area index, and leaf shapes. The roughness is derived from the simple relationship $z_{0m} = 0.07 h_c$, where $h_c$ is the canopy height. Similarly, the zero-plane displacement height $d$ is proportional to canopy height, and is also affected by fractional vegetation cover, leaf area index, and leaf shapes. The simple relationship $d/h_c = 2/3$ is used to obtain the height.

4.10.1.2 Monin-Obukhov similarity theory

(1) Turbulence scaling parameters

A length scale (the Monin-Obukhov length) $L$ is defined by

$$L = \frac{{\theta_v u_*^2 }} {{kg\theta_{v*}}}$$ (4.417)

where $k$ is the von Kàrman constant, and $g$ is the gravitational acceleration. $L > 0$ indicates stable conditions, $L < 0$ indicates unstable conditions, and $L = \infty$ applies to neutral conditions. The virtual potential temperature $\theta_v$ is defined by

$$\theta_v = \theta_1 (1 + 0.61q_1 ) = T_a \left( {\frac{{p_s }} {{p_l }}} \right)^{R/c_p } (1 + 0.61q_1 )$$ (4.418)

where $T_1$ and $q_1$ are the air temperature and specific humidity at height $z_1$ respectively, $\theta_1$ is the atmospheric potential temperature, $p_l$ is the atmospheric pressure, and $p_s$ is the surface pressure. The surface friction velocity $u_*$ is defined by

$$u_*^2 = [\overline {u'w'}^2 + \overline {v'w'}^2 ]^{1/2}$$ (4.419)

The temperature scale $\theta_*$ and $\theta_{ * v}$ and a humidity scale $q_*$ are defined by

$$\theta_* = - \overline {w'\theta '} /u_*$$ (4.420)

$$q_* = - \overline {w'q'} /u_*$$ (4.421)

$$\theta_{v * } = - \overline {w'\theta '_v } /u_*$$

$$\approx - (\overline {w'\theta '} + 0.61\overline \theta \overline {w'q'} )/u_*$$ (4.422)

$$= \theta_* + 0.61\overline \theta q_*$$

(where the mean temperature $\overline \theta$ serves as a reference temperature in this linearized form of $\theta_v$ ).

The stability parameter is defined as

$$\varsigma = \frac{{z_1 - d}}{L}\quad ,$$ (4.423)

with the restriction that $- 100 \leqslant \varsigma \leqslant 2$. The scalar wind speed is defined as

$$V_a^2 = u_1^2 + v_1^2 + U_c^2$$ (4.424)

$$U_c = \left\{ \begin{array}{ll} 0.1\>{\text{ms}}^{-1} & {\text{, if }... ...\text{, if }}\varsigma < {\text{0 (unstable)}} . \hfill \ \end{array} \right.$$ (4.425)

Here $w_*$ is the convective velocity scale, $z_i$ is the convective boundary layer height, and $\beta$ = 1. The value of $z_i$ is taken as 1000 m

(2) Flux-gradient relations [198]

The flux-gradient relations are given by:

$$\frac{{k(z_1 - d)}} {{\theta_* }}\frac{{\partial \theta }} {{\partial z}} = \phi_h (\varsigma )$$ (4.426)

$$\frac{{k(z_1 - d)}} {{q_* }}\frac{{\partial q}} {{\partial z}} = \phi_q (\varsigma )$$ (4.427)

$$\phi_h = \phi_q$$ (4.428)

$$\phi_m (\varsigma ) = \left\{\begin{array}{ll} (1 - 16\varsigma )^{ - 1/4} & \mbox{fo... ...arsigma < 0 \ 1 + 5\varsigma & \mbox{for} 0 < \varsigma < 1 \end{array}\right.$$ (4.429)

$$\phi_h (\varsigma ) = \left\{\begin{array}{ll} (1 - 16\varsigma )^{ - 1/2} & \mbox{fo... ...arsigma < 0 \ 1 + 5\varsigma & \mbox{for} 0 < \varsigma < 1 \end{array}\right.$$ (4.430)

Under very unstable conditions, the flux-gradient relations are taken from Kader and Yaglom [81]:

$$\phi_m = 0.7k^{2/3} ( - \varsigma )^{1/3}$$ (4.431)

$$\phi_h = 0.9k^{4/3} ( - \varsigma )^{ - 1/3}$$ (4.432)

To ensure the functions $\phi_m (\varsigma )$ and $\phi_h (\varsigma )$ are continuous, the simplest approach (i.e., without considering any transition regions) is to match the above equations at $\varsigma_m = - 1.574$ for $\phi_m (\varsigma )$ and $\varsigma_h = - 0.465$ for $\phi_h (\varsigma )$ .

Under very stable conditions (i.e., $\varsigma > 1$ ), the relations are taken from Holtslag et al. [75]:

$$\phi_m = \phi_h = 5 + \varsigma$$ (4.433)

(3) Integral forms of the flux-gradient relations

Integration of the wind profile yields:

$$V_a = \frac{{u_* }} {k}f_M (\varsigma )$$ (4.434)

$$f_M (\varsigma ) = \left\{ {\left[ {\ln \left( {\frac{{\varsigma_m L}} {{z_{0m} }}... ...} \right] + 1.14[( - \varsigma )^{1/3} - ( - \varsigma_m )^{1/3} ]} \right\} , \varsigma < \varsigma_m = - 1.574$$ (4.434a)

$$f_M (\varsigma ) = \left[ {\ln \left( {\frac{{z_1 - d}} {{z_{0m} }}} \right) - \psi_m (\varsigma ) + \psi_m \left( {\frac{{z_{0m} }} {L}} \right)} \right] , \varsigma_m < \varsigma < 0$$ (4.434b)

$$f_M (\varsigma ) = \left[ {\ln \left( {\frac{{z_1 - d}} {{z_{0m} }}} \right) + 5\varsigma } \right] , < \varsigma < 1$$ 0 (4.434c)

$$f_M (\varsigma ) = \left\{ {\left[ {\ln \left( {\frac{L} {{z_{0m} }}} \right) + 5} \right] + [5\ln (\varsigma ) + \varsigma - 1]} \right\} , \varsigma > 1$$ (4.434d)

Integration of the potential temperature profile yields:

$$\theta_1 - \theta_s = \frac{{\theta_* }} {k}f_T (\varsigma )$$ (4.435)

$$f_T (\varsigma ) = \left\{ {\left[ {\ln \left( {\frac{{\varsigma_h L}} {{z_{0h} }}... ...ght] + 0.8[( - \varsigma_h )^{ - 1/3} - ( - \varsigma )^{ - 1/3} ]} \right\} , \varsigma < \varsigma_h = - 0.465$$ (4.435a)

$$f_T (\varsigma ) = \left[ {\ln \left( {\frac{{z_1 - d}} {{z_{0h} }}} \right) - \psi_h (\varsigma ) + \psi_h \left( {\frac{{z_{0h} }} {L}} \right)} \right] , \varsigma_h < \varsigma < 0$$ (4.435b)

$$f_T (\varsigma ) = \left[ {\ln \left( {\frac{{z_1 - d}} {{z_{0h} }}} \right) + 5\varsigma } \right] , < \varsigma < 1$$ 0 (4.435c)

$$f_T (\varsigma ) = \left\{ {\left[ {\ln \left( {\frac{L} {{z_{0h} }}} \right) + 5} \right] + [5\ln (\varsigma ) + \varsigma - 1]} \right\} , \varsigma > 1$$ (4.435d)

The expressions for the specific humidity profiles are the same as those for potential temperature except that ( $\theta_1 - \theta_s$ ), $\theta_*$ and $z_{0h}$ are replaced by ($q_1 - q_s$ ), $q_*$ and $z_{0q}$ respectively. The stability functions for $\varsigma < 0$ are

$$\psi_m = 2\ln\left( {\frac{{1 + \chi }}{2}} \right) + \ln\left( {\frac{{1 + \chi^2 }}{2}} \right) - 2\tan^{ - 1} \chi + \frac{\pi }{2}$$ (4.436)

$$\psi_h = \psi_q = 2\ln\left( {\frac{{1 + \chi^2 }}{2}} \right)$$ (4.437)
where

$$\chi = (1 - 16\varsigma )^{1/4}$$ (4.438)

Note that the CLM code contains extra terms involving $z_{0m} /\varsigma$, $z_{0h} /\varsigma$, and $z_{0q} /\varsigma$ for completeness. These terms are very small most of the time and hence are omitted in Eqs. 4.434 and 4.435.

In addition to the momentum, sensible heat, and latent heat fluxes, land surface albedos and upward longwave radiation are needed for the atmospheric radiation calculations. Surface albedos depend on the solar zenith angle, the amount of leaf and stem material present, their optical properties, and the optical properties of snow and soil. The upward longwave radiation is the difference between the incident and absorbed fluxes. These and other aspects of the land surface fluxes have been described by Dai et al. [45].

4.10.2 Ocean

The bulk formulas used to determine the turbulent fluxes of momentum (stress), water (evaporation, or latent heat), and sensible heat into the atmosphere over ocean surfaces are

$$( \boldsymbol {\tau}, E, H) = \rho_A \left\vert\Delta {\boldsymb... ...right\vert(C_D \Delta {\boldsymbol {v}}, C_E \Delta q, C_p C_H \Delta\theta),$$ (4.439)

where $\rho_A$ is atmospheric surface density and $C_p$ is the specific heat. Since CAM 3.0 does not allow for motion of the ocean surface, the velocity difference between surface and atmosphere is $\Delta {\boldsymbol {v}}= {\boldsymbol {v}}_A$, the velocity of the lowest model level. The potential temperature difference is $\Delta\theta = \theta_A - T_s$, where $T_s$ is the surface temperature. The specific humidity difference is $\Delta q = q_A - q_s(T_s)$, where $q_s(T_s)$ is the saturation specific humidity at the sea-surface temperature.

In (4.439), the transfer coefficients between the ocean surface and the atmosphere are computed at a height $Z_A$ and are functions of the stability, $\zeta$:

$$C_{(D,E,H)} = \kappa^2 {\left[\ln\left(\frac{Z_A}{Z_{0m}}\right) ... ...} {\left[\ln\left(\frac{Z_A}{Z_{0(m,e,h)}}\right) - \psi_{(m,s,s)}\right]}^{-1}$$ (4.440)

where $\kappa = 0.4$ is von Kármán's constant and $Z_{0(m,e,h)}$ is the roughness length for momentum, evaporation, or heat, respectively. The integrated flux profiles, $\psi_m$ for momentum and $\psi_s$ for scalars, under stable conditions ($\zeta > 0$) are

$$\psi_m(\zeta) = \psi_s(\zeta) = -5 \zeta.$$ (4.441)

For unstable conditions ($\zeta < 0$), the flux profiles are

$$\psi_m(\zeta) = 2 \ln[0.5(1 + X)] + \ln[0.5(1 + X^2 )]$$

$$- 2 \tan^{-1} X + 0.5 \pi,$$ (4.442)

$$\psi_s(\zeta) = 2 \ln[0.5(1 + X^2 )],$$ (4.443)

$$X = (1 - 16 \zeta)^{1/4} .$$ (4.444)

The stability parameter used in (4.441)-(4.444) is

$$\zeta = \frac{\kappa g Z_A}{u^{*2}}\left(\frac{\theta^*}{\theta_v} + \frac{Q^*}{(\epsilon^{-1} + q_A)}\right),$$ (4.445)

where the virtual potential temperature is $\theta_v = \theta_A(1 + \epsilon q_A)$; $q_A$ and $\theta_A$ are the lowest level atmospheric humidity and potential temperature, respectively; and $\epsilon = 0.606$. The turbulent velocity scales in (4.445) are

$$u^* = C_D^{1/2} \vert\Delta {\boldsymbol {v}}\vert,$$

$$(Q^*,\theta^*) = C_{(E,H)}\frac{\vert\Delta {\boldsymbol {v}}\vert}{u^*} (\Delta q,\Delta\theta).$$ (4.446)

Over oceans, $Z_{0e} = 9.5 \times 10^{-5}$ m under all conditions and $Z_{0h} = 2.2 \times 10^{-9}$ m for $\zeta > 0$, $Z_{0h} = 4.9 \times 10^{-5}$ m for $\zeta \le 0$, which are given in Large and Pond [101]. The momentum roughness length depends on the wind speed evaluated at 10 m as

$$Z_{om} = 10 \exp\left[-\kappa{\left(\frac{c_4}{U_{10}} + c_5 + c_6 U_{10}\right)}^{-1}\right] ,$$

$$U_{10} = U_A {\left[1 + \frac{\sqrt{C_{10}^N}}{\kappa}\ln\left(\frac{Z_A}{10} - \psi_m\right)\right]}^{-1} ,$$ (4.447)

where $c_4 = 0.0027$  m s${}^{-1}$, $c_5 = 0.000142$, $c_6 = 0.0000764$ m${}^{-1}$ s, and the required drag coefficient at 10-m height and neutral stability is $C^{N}_{10} = c_4 U^{-1}_{10} + c_5 + c_6 U_{10}$ as given by Large et al. [100].

The transfer coefficients in (4.439) and (4.440) depend on the stability following (4.441)-(4.444), which itself depends on the surface fluxes (4.445) and (4.446). The transfer coefficients also depend on the momentum roughness, which itself varies with the surface fluxes over oceans (4.447). The above system of equations is solved by iteration.

4.10.3 Sea Ice

The fluxes between the atmosphere and sea ice are described in detail in chapter 6.