An Efficient Non-iterative Bulk Parametrization of Surface Fluxes for Stable Atmospheric Conditions Over Polar Sea-Ice

Abstract

In climate and weather prediction models the near-surface turbulent fluxes of heat and momentum and related transfer coefficients are usually parametrized on the basis of Monin–Obukhov similarity theory (MOST). To avoid iteration, required for the numerical solution of the MOST equations, many models apply parametrizations of the transfer coefficients based on an approach relating these coefficients to the bulk Richardson number $Ri_{b}$. However, the parametrizations that are presently used in most climate models are valid only for weaker stability and larger surface roughnesses than those documented during the Surface Heat Budget of the Arctic Ocean campaign (SHEBA). The latter delivered a well-accepted set of turbulence data in the stable surface layer over polar sea-ice. Using stability functions based on the SHEBA data, we solve the MOST equations applying a new semi-analytic approach that results in transfer coefficients as a function of $Ri_{b}$ and roughness lengths for momentum and heat. It is shown that the new coefficients reproduce the coefficients obtained by the numerical iterative method with a good accuracy in the most relevant range of stability and roughness lengths. For small $Ri_{b}$, the new bulk transfer coefficients are similar to the traditional coefficients, but for large $Ri_{b}$ they are much smaller than currently used coefficients. Finally, a possible adjustment of the latter and the implementation of the new proposed parametrizations in models are discussed.

Appendix: Fluxes of Latent Heat in the Surface Layer

In general, the set of bulk MOST equations includes in addition to the equations for momentum and heat a bulk equation for the flux of specific humidity E. The latter equation can be written (see, e.g., Garratt 1992) as

$$\begin{aligned} E = -\rho \, \lambda \,C_{q}\,|\mathbf {v}(z)|(q - q_0) \end{aligned}$$

(41)

with the transfer coefficient for latent heat $C_q$. Here, $\lambda$ is the latent heat of evaporation and q is the specific humidity. Index 0 refers to the surface value as before.

The coefficient $C_{q}$ can be written similar to Eq. 4a and 4b as

$$\begin{aligned} C_q = C_{qn} \,f_q\,, \end{aligned}$$

(42)

where

$$\begin{aligned} C_{qn} = \frac{\kappa ^2}{\ln \epsilon \,\ln \epsilon _q}\,, \end{aligned}$$

(43)

with the normalized transfer coefficients for latent heat $f_q$, given as

$$\begin{aligned} f_q = \bigg [ 1 - \frac{\psi _m (\zeta )-\psi _m (\zeta /\epsilon )}{\ln \epsilon }\bigg ]^{-1}\,\bigg [ 1 - \frac{\psi _q (\zeta )-\psi _q (\zeta /\epsilon _q)}{\ln \epsilon _q}\bigg ]^{-1}, \end{aligned}$$

(44)

and where $f_q$ is normalized as $f_q = 1$ for neutral stability. Here, $\psi _q(\zeta )$ is the MOST stability correction function for latent heat.

Following, e.g., G2007 and Grachev and Fairall (1997), we applied the Reynolds analogy for humidity and heat using an approximation $C_q =C_h$ in the MOST framework. The flux H is then defined according to Eq. 2 as a combination of the heat flux $F= - \rho \,c_p\,C_h\,|\mathbf {v}(z)|\,[\theta - \theta _0]$ and specific humidity flux (see Eq. 41), if the virtual potential temperature $\theta _{v}=\theta +0.61{\varTheta }q$ is introduced. The bulk Richardson number is defined then as a non-dimensional combination of wind speed, potential temperature and specific humidity according to Eq. 12 in Sect. 2.

The Reynolds analogy is not always a correct assumption but for polar ocean conditions it is a reasonable first approximation.