Momentum- and Heat-Flux Parametrization at Dome C, Antarctica: A Sensitivity Study

Abstract

An extensive meteorological observational dataset at Dome C, East Antarctic Plateau, enabled estimation of the sensitivity of surface momentum and sensible heat fluxes to aerodynamic roughness length and atmospheric stability in this region. Our study reveals that (1) because of the preferential orientation of snow micro-reliefs (sastrugi), the aerodynamic roughness length $z_{0}$ varies by more than two orders of magnitude depending on the wind direction; consequently, estimating the turbulent fluxes with a realistic but constant $z_{0}$ of 1 mm leads to a mean friction velocity bias of $24\,\%$ in near-neutral conditions; (2) the dependence of the ratio of the roughness length for heat $z_{0t}$ to $z_{0}$ on the roughness Reynolds number is shown to be in reasonable agreement with previous models; (3) the wide range of atmospheric stability at Dome C makes the flux very sensitive to the choice of the stability functions; stability function models presumed to be suitable for stable conditions were evaluated and shown to generally underestimate the dimensionless vertical temperature gradient; as these models differ increasingly with increases in the stability parameter z / L, heat flux and friction velocity relative differences reached $100\,\%$ when $z/L > 1$; (4) the shallowness of the stable boundary layer is responsible for significant sensitivity to the height of the observed temperature and wind data used to estimate the fluxes. Consistent flux results were obtained with atmospheric measurements at heights up to 2 m. Our sensitivity study revealed the need to include a dynamical parametrization of roughness length over Antarctica in climate models and to develop new parametrizations of the surface fluxes in very stable conditions, accounting, for instance, for the divergence in both radiative and turbulent fluxes in the first few metres of the boundary layer.

Appendix

1.1 Appendix 1: How Does One Remove Non-stationary Data

Assuming horizontal homogeneity and negligible effects of the Coriolis force, the simplified equations for wind speed and potential temperature in the surface layer in time-varying conditions are as follows

$$\begin{aligned} \frac{\partial u}{\partial t}= & {} -\frac{\partial \overline{u^{\prime }w^{\prime }}}{\partial z}, \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial \theta }{\partial t}= & {} -\frac{\partial \overline{\theta ^{\prime }w^{\prime }}}{\partial z}. \end{aligned}$$

(13)

Further explanations concerning the notations are given in Sect. 2.3. The surface layer in which the MO theory is valid is often defined as the layer across which the turbulent fluxes do not diverge more than 10% in norm (Stull 1990), i.e.

$$\begin{aligned} \frac{|\overline{u^{\prime }w^{\prime }}_{z}-\overline{u^{\prime }w^{\prime }}_{s}|}{z}< & {} \frac{0.1 ~|\overline{u^{\prime } w^{\prime }}_{s}|}{z}, \end{aligned}$$

(14)

$$\begin{aligned} \frac{|\overline{\theta ^{\prime } w^{\prime }}_{z}-\overline{\theta ^{\prime } w^{\prime }}_{s}|}{z}< & {} \frac{0.1~ |\overline{\theta ^{\prime } w^{\prime }}_{s}|}{z}, \end{aligned}$$

(15)

where the s indices refer to surface values, and z indices refer to values at height z in the surface layer. Approximating time and spatial derivatives by bulk gradients($\partial \approx \Delta$) and combining Eq. 12 with Eq. 14 and Eq. 13 with Eq. 15 leads to

$$\begin{aligned} {\Delta } u< & {} 0.1~ \frac{|\overline{u^{\prime }w^{\prime }}_{s}| {\Delta } t}{z}, \end{aligned}$$

(16)

$$\begin{aligned} {\Delta } \theta< & {} 0.1 ~\frac{|\overline{\theta ^{\prime }w^{\prime }}_{s}| {\Delta } t}{z}. \end{aligned}$$

(17)

Taking ${\Delta } t=30$ min (period of data averaging before storage), $\overline{u^{\prime }w^{\prime }}_{s}\approx 0.015$ m$^{2}$ s$^{-2}$, $\overline{\theta ^{\prime }w^{\prime }}_{s}\approx 0.01$ m K s$^{-1}$ (typical values for Dome C), $z=0.9$ m for the temperature measurement height and $z=2.35$ m for the wind measurement height in the surface layer, we obtain ${\Delta } \theta < 2$ K and ${\Delta } u < 1.1$ m s$^{-1}$. In other words, the stationary conditions to apply the MO theory are guaranteed if the variations in temperature and wind speed over a 30-min period are lower than the aforementioned thresholds. To ensure this condition was met, we removed all the 30-min data for which the differences in temperature or wind speed from the previous half-hour were above the thresholds ${\Delta } \theta$ and ${\Delta } u$ respectively.

1.2 Appendix 2: How Does an Error in $\theta _{*}$ Affect the Estimation of $z_{0t}$ Using MO Similarity Equations?

Equation 3 applied to measurements at height $z_{t}$ and the ground can be written as,

$$\begin{aligned} \theta (z_{t})-\theta _{s}=\frac{\theta _{*}}{\kappa }\left( \ln \left( \frac{z_{t}}{z_{0t}}\right) -\psi _{h}\left( \frac{z_{t}}{L}\right) \right) , \end{aligned}$$

(18)

and writing $\theta (z_{t})-\theta _{s}=\Delta \theta$, this directly leads to an expression for $z_{0t}$:

$$\begin{aligned} \ln (z_{0t})=\ln (z_{t})-\psi _{h}\left( \frac{z_{t}}{L}\right) +\frac{\kappa {\Delta } \theta }{\theta _{*}}. \end{aligned}$$

(19)

The dependence of $\ln (z_{0t})$ on the measured $\theta _{*}$ by a sonic thermo-anemometer is thus,

$$\begin{aligned} \frac{\partial \ln (z_{0t})}{\partial \theta _{*}}= \frac{\kappa {\Delta } \theta }{\theta _{*} ^{2}}- \frac{\partial \phi _{h}}{\partial \theta _{*}}, \end{aligned}$$

(20)

and in near-neutral conditions, $\partial \phi _{h}/\partial \theta _{*}$ is negligible, and thus an error $\partial \theta _{*}$ on $\theta _{*}$ results in an error $\partial \ln (z_{0t})$ proportional to $\theta _{*} ^{-2}$. Given that $\theta _{*}$ is very small when the ABL stratification tends to neutrality, the error made on $z_{0t}$ is significant.

1.3 Appendix 3: Dimensionless Gradients for Stable Conditions

We recall here the relations for the dimensionless gradients taken from the literature and compared in the present study; $\zeta$ denotes the stability parameter z / L.

• Holtslag and De Bruin (1988):

  $$\begin{aligned} \phi _{m}(\zeta )=\phi _{h}(\zeta )=1+0.7\zeta +0.75\zeta (6-0.35\zeta )\exp (-0.35\zeta ). \end{aligned}$$

  (21)

• Grachev et al. (2007):

  $$\begin{aligned} \phi _{m}(\zeta )= & {} 1+\frac{6.5\zeta (1+\zeta )^{1/3}}{1.3+\zeta }, \end{aligned}$$

  (22)
  $$\begin{aligned} \phi _{h}(\zeta )= & {} 1+\frac{5\zeta +5\zeta ^{2}}{1+3\zeta +\zeta ^{2}}. \end{aligned}$$

  (23)

• King and Anderson (1994)

  $$\begin{aligned} \phi _{m}(\zeta )= & {} 1+5.7\zeta \quad \text {and} \quad \phi _{m}(\zeta )<12, \end{aligned}$$

  (24)
  $$\begin{aligned} \phi _{h}(\zeta )= & {} 0.95+4.99\zeta \quad \text {and} \quad \phi _{h}(\zeta )<12. \end{aligned}$$

  (25)

• Lettau (1979)

  $$\begin{aligned} \phi _{m}(\zeta )= & {} (1+4.5\zeta )^{3/4}, \end{aligned}$$

  (26)
  $$\begin{aligned} \phi _{h}(\zeta )= & {} (1+4.5\zeta )^{3/2}. \end{aligned}$$

  (27)