8 Flux Calculations

The Coupler is required to compute certain fluxes and other quantities related to fluxes, such computing the ocean's surface albedo. These calculations are described below.

8.1 Atmosphere/Ocean Fluxes

8.1.1 General Expressions

The fluxes across the interface are calculated from bulk formulae and general expressions are

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$$\begin{eqnarray*} \vec \tau & = & \rho_A  u^{*2}   \Delta \vec U    \vert \Del... ...  - \epsilon  \sigma    T^4     +    \alpha^L    L\downarrow, \end{eqnarray*}$$

where the turbulent velocity scales are given by

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$$\begin{eqnarray*} u^* & = &   CD^{1/2}   \vert \Delta \vec U \vert \\ Q^* & = &... ... = &   CH  \vert \Delta \vec U \vert  (\Delta \theta)  u^{*-1}, \end{eqnarray*}$$

where $\rho_A$ is atmospheric surface density, $Cp_A$ is the specific heat, $\sigma = 5.67 \times 10^{-8}$W/m$^2$/K$^4$ is the Stefan-Boltzmann constant, $\epsilon$ is the emissivity of the interface, and $\alpha^L$ is the surface albedo for incident longwave radiation, $L\downarrow$. In (8.2) the differences $\Delta \vec U$, $\Delta q$ and $\Delta \theta$ are defined at each interface in accord with the convention of fluxes being positive down. The reflected downward incident longwave radiation is simply accounted for by assuming an emissivity, $\epsilon = 1$, and the water surface albedo for incident longwave radiation, $\alpha^L = 0.0$.

The transfer coefficients in (8.2), shifted to a height, $Z$, and considering the appropriate stability parameter, $\zeta$, are :

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$$\begin{eqnarray*} CD & = & \kappa ^2 \left[ ln \left( {Z \over Z^o} \right) - \p... ...\left[ ln \left( {Z \over Z^h} \right) - \psi _s \right] ^{-1}, \end{eqnarray*}$$

where $\kappa = 0.4$ is von Karman's constant and the integrated flux profiles, $\psi_m$ for momentum and $\psi_s$ for scalars, are functions of the stability parameter, $\zeta$. These functions as used in the coupler are:

$$\begin{eqnarray*} \psi_m(\zeta)   &=&   \psi_s(\zeta)  =  -5\zeta               ... ...                \zeta < 0 \\ X    &=&   ( 1 - 16 \zeta )^{1/4} \end{eqnarray*}$$

Above the atmospheric interfaces $i=1, 2$ and $3$ the stability parameter

$$\zeta  =  {\kappa g Z_A \over u^{*2} } \Big( {\theta^* \over \theta_v} + {Q^* \over (Z_v^{-1} + q_A) } \Big)    ,$$

where virtual potential temperature is computed as $\theta_v = \theta_A (1 + Z_v q_A)$, $q_A$ and $\theta_A$ are the lowest level atmospheric humidity, and potential temperature, respectively, and $Z_v = (\varrho(water) / \varrho(air)) - 1 = 0.606$.

In addition to surface fluxes, the atmospheric model requires effective surface albedos for both direct $\alpha (dir)$, and diffuse, $\alpha (dif)$, radiation at each wavelength. They are used in a single call to the computationally demanding atmospheric radiation routines. This call gives downward atmospheric albedo for diffuse radiation, $\alpha_a$($dif$). If direct and diffuse albedos, $\alpha^m$($dir$) and $\alpha^m$($dif$) for $m = 1, M$ different surfaces below an atmospheric grid cell are known, the multiple scattering coefficients, $R^m = \left( 1 - \alpha_a (dif) \alpha^m(dif) \right)^{-1}$ are computed, and with $f^m$ the fractional coverage of each surface type, then the albedos are given by

$$\begin{eqnarray*} \alpha (dif) &=& \left( 1 - F_2 \right) \big/ \left( 1 - F_2... ... (dir) &=& F_1 \left( 1 - \alpha_a (dif) \alpha (dif) \right) \end{eqnarray*}$$

$$F_1 = \sum\limits^M_{m=1} f^m R^m \alpha^m (dir)$$

$$F_2 = \sum\limits^M_{m=1} f^m R^m \left( 1 - \alpha^m (dif) \right)$$

Use of these albedos ensures that the solar radiation exiting the bottom of the atmosphere is identical to the sum of $M$ radiation calculations, each using an $\alpha^m (dif)$ and $\alpha^m (dir)$. At present, however, the albedo calculations are greatly simplified by assuming $\alpha_a(dif) = 0$, such that $R^m = 1$. This albedo then does not need to be passed from the atmosphere to the coupler, and the coupler simply computes the effective albedos to be passed to the atmosphere as

$$\begin{eqnarray*} \alpha (dif) &=& \sum\limits^M_{m=1} f^m \alpha^m (dif) \\ \alpha (dir) &=& \sum\limits^M_{m=1} f^m \alpha^m (dir) \end{eqnarray*}$$

In general the partition of radiation among the $M$ surfaces is a function of $R^m$, $\alpha^m (dif)$, and $\alpha^m (dir)$, and hence of wavelength. Hence, correct partitioning would need to be performed for each wavelength band within the radiation transfer code of the atmospheric model. This procedure is greatly simplified by partitioning the net solar radiation within the Coupler.

8.1.2 Specific Expressions

At the atmosphere-ocean interface the near-surface air is also assumed to be saturated,

$$q = 0.98    \rho_A^{-1}    C_5   \exp (C_6/T),$$

where the sea surface temperature is $T = SST$, $C_5 = 640380.0$ kg/m$^3$ and $C_6 = -5107.4$ K, and the factor 0.98 accounts for the salinity of the ocean. The differences (8.2) become

$$\begin{eqnarray*} \Delta \vec U &=& \vec U_A - \vec U \\ \Delta q &=& q_A - q \\ \Delta \theta &=& \theta_A - T, \end{eqnarray*}$$

where the surface current, $\vec U$, is presently assumed to be negligible.

The roughness length for momentum, $Z^o$ in meters, is a function of the atmospheric wind at 10 meters height, $U_{10}$:

$$Z^o = 10  \exp - \left[ \kappa \left( {C_1 \over U_{10}} + C_2 + C_3 U_{10} \right) ^{-1/2} \right],$$

where $C_1 = 0.0027$ m/s, $C_2 = 0.000142,$ and $C_3 = 0.0000764$ m$^{-1}$s. The corresponding drag coefficient at 10m height and neutral stability is

$$C_{10}^N   =   C_1   U_{10}^{-1}   +   C_2   +   C_3   U_{10}   .$$

The roughness length for heat, $Z^h$, is a function of stability, and for evaporation, $Z^e$, is a different constant:

$$\begin{eqnarray*} Z^h &=& 2.2 \times 10^{-9} m     \zeta > 0 \\ &=& 4.9 \times 10^{-5} m     \zeta \leq 0 \\ Z^e &=& 9.5 \times 10^{-5} m. \end{eqnarray*}$$

Since $\zeta$ is itself a function of the turbulent scales (8.2), and hence the fluxes, an iterative procedure is generally required to solve (8.1). First, $\zeta$ is set incrementally greater than zero when the air-sea temperature difference suggests stable stratification, otherwise it is set to zero. In either case, $\psi_m = \psi_s = 0$, and the initial transfer coefficients are then found from the roughness lengths at this $\zeta$ and $U_{10} = U_A$. As with sea-ice, these coefficients are used to approximate the initial flux scales (8.2) and the first iteration begins with an updated $\zeta$ and calculations of $\psi_m$ and $\psi_s$. The wind speed, $U_A$, is then shifted to its equivalent neutral value at 10m height :

$$U_{10} = U_A   \Big( 1 + { \sqrt{C_{10}^N} \over \kappa} ln( {Z_A \over 10} - \psi_m(\zeta))   \Big)^{-1}   .$$

This wind speed is used to update the transfer coefficients and hence the flux scales. The second and final iteration begins with another update of $\zeta$. The final flux scales then give the fluxes calculated by (8.1).

8.2 Surface Albedo and Net Absorbed Solar Radiation

For the ice, land, and ocean components there are four surface albedos that are used by the atmosphere component to compute four corresponding components of downward shortwave. Subsequently, the four albedos, together with the four downward shortwave fields, are used to compute net absorbed shortwave flux from the atmosphere to the surface components:

$$SW_{net} = \sum\limits^4_{m=1} SW_{down}^m \left( 1 - \alpha^m \right)   ,$$

where $m = 1,..., 4$ corresponds to near-infrared/diffuse, visible/diffuse, near-infrared/direct, and visible/direct shortwave components.

The ice and land components each compute their own surface albedos, and subsequently, give the downward shortwave fields, compute their own net absorbed shortwave radiation. For the ocean component, it is the Coupler that computes the ocean surface albedo and subsequently computes net absorbed shortwave radiation. The Coupler then sends the net absorbed shortwave field to the ocean component.

8.2.1 Land Surface Albedos

The land surface albedos are computed by the land component and passed on to the atmosphere component. These albedos are not altered by the Coupler in any way. Note that the atmosphere component may be an active model computing downward shortwave once per hour, or it may be a data model feeding the Coupler a daily average downward shortwave. It is the user's responsibility to ensure that the albedos the land model sends are appropriate considering what type of downward shortwave fields atmosphere component is providing.

8.2.2 Ice Surface Albedo

The ice surface albedos are computed by the ice component and passed through the Coupler and on to the atmosphere component. These albedos are "60 degree reference albedos" that have no diurnal cycle. Based on an input namelist variable, "flx_albav" (see the namelist section of the Coupler User's Guide), the Coupler will either pass these albedos on to the atmosphere component unaltered (in which case they are considered "daily average" albedos), or impose a diurnal cycle on the albedos (in which case they are considered "instantaneous" albedos). When the Coupler does add a diurnal cycle to the ice albedo, this consists of merely setting the albedos to 1.0 on the dark side of the earth.

8.2.3 Ocean Surface Albedos

Unlike the ice and land components, the Coupler computes the ocean component's surface albedo. There are two ways the Coupler can compute the ocean albedo: with a diurnal cycle ("instantaneous") or without a diurnal cycle ("daily average"). An input namelist variable (the "flx_albav" namelist variable) selects which option the Coupler implements.

If the albedos are computed as daily average albedos, then all four ocean albedos are set to 0.06 everywhere, regardless of time of day or time of year.

If the albedos are computed as instantaneous albedos, then all four ocean will be set to 1.0 on the dark side of the earth, and where the solar angle is greater than zero, the albedos are set to a value which has both an annual and a diurnal cycle. Ocean albedo distinguishes between direct and diffuse radiation. The direct albedo is solar zenith angle dependent, while the diffuse is not. There is no spectral dependence of the albedo, nor dependence on surface wind speed. The expressions for both direct and diffuse albedo are taken from Briegleb et al. (1986), based on fits to observations of ocean albedo, good to within $\pm 0.3\%$. The albedo expressions are valid for open ocean, and do not include the effects of suspended hydrosols in near-surface waters.

For complete details see Briegleb, B.P., P.Minnis, V.Ramanathan, and E.Harrison, 1986. Comparison of regional clear-sky albedos inferred from satellite observations and model comparisons. Journal of Climate and Applied Meteorology, Vol. 25, pp.214-226.