Reducing Parametrization Errors for Polar Surface Turbulent Fluxes Using Machine Learning

Abstract

Turbulent exchanges between sea ice and the atmosphere are known to influence the melting rate of sea ice, the development of atmospheric circulation anomalies and, potentially, teleconnections between polar and non-polar regions. Large model errors remain in the parametrization of turbulent heat fluxes over sea ice in climate models, resulting in significant uncertainties in projections of future climate. Fluxes are typically calculated using bulk formulae, based on Monin-Obukhov similarity theory, which have shown particular limitations in polar regions. Parametrizations developed specifically for polar conditions (e.g. representing form drag from ridges or melt ponds on sea ice) rely on sparse observations and thus may not be universally applicable. In this study, new data-driven parametrizations have been developed for surface turbulent fluxes of momentum, sensible heat and latent heat in the Arctic. Machine learning has already been used outside the polar regions to provide accurate and computationally inexpensive estimates of surface turbulent fluxes. To investigate the feasibility of this approach in the Arctic, we have fitted neural-network models to a reference dataset (SHEBA). Predictive performance has been tested using data from other observational campaigns. For momentum and sensible heat, performance of the neural networks is found to be comparable to, and in some cases substantially better than, that of a state-of-the-art bulk formulation. These results offer an efficient alternative to the traditional bulk approach in cases where the latter fails, and can serve to inform further physically based developments.

A: Polar-Specific Bulk Parametrizations

1.1 A.1 Stability Correction of Grachev et al. (2007)

Grachev et al. (2007) assumed the form of the vertical profiles to be:

$$\begin{aligned} a(z) = p_1 (\ln z)^2 + p_2 \ln (z) + p_3, \end{aligned}$$

(8)

where a can be the wind speed, potential temperature, specific humidity, or any other variable, and $p_1$, $p_2$, $p_3$ are constants to be determined. By taking the derivative of eqn (8), the vertical gradients can be obtained through a fit to observations. Using data from the SHEBA tower, Grachev et al. (2007) proposed the “SHEBA profile functions”:

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

(9)

$$\begin{aligned} \phi _H&= 1 + \dfrac{5 \zeta + 5 \zeta ^2}{1 + 3 \zeta + \zeta ^2}, \end{aligned}$$

(10)

where $\zeta$ is the Monin-Obukhov stability parameter and $\phi _M$,$\phi _H$ are the non-dimensional stability profile functions for momentum and sensible/latent heat respectively.

1.2 A.2 Aerodynamic Roughness Model of Andreas et al. (2010b)

Based on the SHEBA winter data, Andreas et al. (2010b) concluded that the aerodynamic roughness $z_0$ did not significantly depend on the atmospheric stability. They proposed the following unified parametrization:

$$\begin{aligned} z_0 = 0.135 \dfrac{\nu }{u_\star } + B \tanh ^3 (13 u_\star ), \end{aligned}$$

(11)

where $\nu$ is the kinematic viscosity of air in m²s^-1 and $B = 2.3 \times 10^{-4}$. The first term on the right models aerodynamically smooth regimes, while the second treats aerodynamically rough regimes as well as the transition from smooth to rough flows.

1.3 A.3 Scalar Roughness Model of Andreas (1987)

Andreas (1987) proposed modelling the ratio of the scalar and aerodynamic roughness lengths as a function of the roughness Reynolds number:

$$\begin{aligned} \ln \dfrac{z_s}{z_0} = b_{0,s} + b_{1,s} \ln R_\star + b_{2,s} (\ln R_\star )^2, \end{aligned}$$

(12)

where $z_s$ is the scalar roughness for temperature ($s=T$) or humidity ($s=Q$) and $R_\star = \frac{u_\star z_0}{\nu }$ is the roughness Reynolds number. Polynomial coefficients $b_{i,s}$ are tabulated for smooth ($R_\star \le 0.135$), transitional ($0.135< R_\star < 2.5$) and rough ($2.5 \le R_\star < 1000$) surfaces in Andreas (1987).

1.4 A.4 Form Drag Parametrization

The form drag parametrization used in this study assumes that the drag coefficient $C_D$ in eqn (1) (the bulk exchange coefficient for momentum) can be partitioned as:

$$\begin{aligned} C_{D} = C_{D,w} (1-C_i) + C_{D,i} C_i + C_{D,f}, \end{aligned}$$

(13)

where $C_{D,w}$ and $C_{D,i}$ denote the skin drag coefficients over water and ice respectively, $C_i$ is the sea ice concentration, and $C_{D,f}$ is the form drag contribution. We obtain $C_{D,f}$ by applying a stability correction to its neutral counterpart $C_{DN,f}$:

$$\begin{aligned} C_{DN,f} = C_{DN,f,w} (1-C_i) + C_{DN,f,i} C_i, \end{aligned}$$

(14)

where $C_{DN,f,k}$ are form-induced drag coefficients over water ($k=w$) and ice ($k=i$). Following Lüpkes et al. (2012) and Lüpkes and Gryanik (2015), we use:

$$\begin{aligned} C_{DN,f,k} = \dfrac{c_e}{2} \left[ \dfrac{\ln \dfrac{h_{fc}}{z_{0,k}}}{\ln \dfrac{10}{z_{0,k}}} \right] ^2 \dfrac{h_{fc}}{D} (1-C_i)^\beta C_i, \end{aligned}$$

(15)

where $c_e = 0.4$ is the effective resistance coefficient, $h_{fc} = 0.41$ m is the ice floe freeboard, $z_{0,k}$ is the skin drag aerodynamic roughness over water/ice, $D = 8$ m is the average diameter of leads or melt ponds and $\beta = 1$ is an optional exponent.