6.4 Ice to Atmosphere Flux Exchange

Atmospheric states and downwelling fluxes, along with surface states and properties, are used to compute atmosphere-ice shortwave and longwave fluxes, stress, sensible and latent heat fluxes. Surface states are temperature $T_{s}$ and albedos $\alpha_{vsdr}$, $\alpha_{vsdf}$, $\alpha_{nidr}$, $\alpha_{nidf}$ (see section 6.3), while surface properties are longwave emissivity $\varepsilon$ and aerodynamic roughness $z_i$ (note that these properties in general vary with ice thickness, but are here assumed constant). Additionally, certain flux temperature derivatives required for the ice temperature calculation are computed, as well as a reference diagnostic surface air temperature.

The following formulas are for the absorbed shortwave fluxes and upwelling longwave flux:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...\varepsilon\sigma_{sb} T_s^4+(1-\varepsilon)F_{LWDN} \cr\crcr}}$$ (6.13)

for $T_{s}$ in Kelvin and $\sigma_{sb}$ denotes the Stefan-Boltzmann constant. The downwelling shortwave flux and albedos distinguish between visible ( $vs, \lambda < 0.7\mu m$), near-infrared ( $ni, \lambda > 0.7\mu m$), direct ($dr$) and diffuse ($df$) radiation for each category. Note that the upwelling longwave flux has a reflected component from the downwelling longwave whenever $\varepsilon < 1$.

For stress components $\tau_{ax}$ and $\tau_{ay}$ and sensible and latent heat fluxes the following bulk formulas are used [29]:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...r_{e} u^\ast \left( q_a - {\overline q}^{*} \right). \cr\crcr}}$$ (6.14)

The quantities from the lowest layer of the atmosphere include wind components $u_a$ and $v_a$, the density of air $\rho_a$, the potential temperature $\theta_a$, and the specific humidity $q_a$. The surface saturation specific humidity is

$${\overline q}^{*} = (q_1/ \rho_a) e^{-q_2/T_{s}}$$ (6.15)

where the values of $q_1$ and $q_2$ were kindly supplied by Xubin Zeng of the University of Arizona. The specific heat of the air in the lowest layer is evaluated from

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...\overline q}^{*}) \cr C_{pvir} = & (C_{pwv}/C_p) - 1 \cr\crcr}}$$ (6.16)

where specific heat of dry air and water vapor are $C_p$ and $C_{pwv}$, respectively. Values for the exchange coefficients for momentum, sensible and latent heat $r_{m,h,e}$ and the friction velocity $u^\ast$ require further consideration.

The bulk formulas are based on Monin-Obukhov similarity theory. Among boundary layer scalings, this is the most well tested [99]. It is based on the assumption that in the surface layer (typically the lowest tenth of the atmospheric boundary layer), but away from the surface roughness elements, only the distance from the boundary and the surface kinematic fluxes are important in the turbulent exchange. The fundamental turbulence scales that are formed from these quantities are the friction velocity $u^\ast$, the temperature and moisture fluctuations $\theta^\ast$ and $q^\ast$ respectively, and the Monin-Obukhov length scale $L$:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...\overline q}^{*})) \cr L = & u^{\ast 3} / (\kappa F) \cr\crcr}}$$ (6.17)

with

$$V_{mag} = \max(1.0,\sqrt{u_a^2 + v_a^2}),$$ (6.18)

to prevent zero or small fluxes under quiescent wind conditions, $\kappa$ is von Karman's constant (0.4), and $F$ is the buoyancy flux, defined as:

$$F = \frac{u^\ast} {g} \left[ \frac{\theta^\ast}{\theta_{v}} + \frac{q^\ast}{z_v^{-1}+q_a} \right]$$ (6.19)

with g the gravitational acceleration and the virtual potential temperature $\theta_{v} = \theta_a(1+z_vq_a)$ where $z_v=\rho_{wv}/\rho_a - 1$.

Similarity theory holds that the vertical gradients of mean horizontal wind, potential temperature and specific humidity are universal functions of stability parameter $\zeta = z / L$, where $z$ is height above the surface ($\zeta$ is positive for a stable surface layer and negative for an unstable surface layer). These universal similarity functions are determined from observations in the atmospheric boundary layer [73] though no single form is widely accepted. Integrals of the vertical gradient relations result in the familiar logarithmic mean profiles, from which the exchange coefficients can be defined, where $\zeta = z_a / L$:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...\chi_h(\zeta)\right]\right\}^{-1}\cr r_{e} = & r_{h} \cr\crcr}}$$ (6.20)

with the neutral coefficient

$$r_0 = \frac{\kappa}{ln(z_{ref}/z_i)}.$$ (6.21)

The flux profile functions (integrals of the similarity functions mentioned above) for momentum $m$ and heat/moisture $h$ are:

$$\chi_m(\zeta) = \chi_h(\zeta) = -5\zeta$$ (6.22)

for stable conditions ($\zeta > 0$). For unstable conditions ($\zeta < 0$):

$$\chi_m(\zeta) = \ln \{(1+X(2+X))(1+X^2)/8\} - 2 \tan^{-1}(X) + 0.5\pi$$ (6.23)

$$\chi_h(\zeta) = 2 \ln\{(1+X^2)/2\}$$ (6.24)
with

$$[-1.0em]$$

$$[-2.0em] X = \left\{ \max((1-16\zeta)^{1/2}),1\right\}^{1/2}.$$ (6.25)

The stability parameter $\zeta$ is a function of the turbulent scales and thus the fluxes, so an iterative solution is necessary. The coefficients are initialized with their neutral value $r_0$, from which the turbulent scales, stability, and then flux profile functions can be evaluated. This order is repeated for five iterations to ensure convergence to an acceptable solution.

The surface temperature derivatives required by the ice temperature calculation are evaluated as:

$$\frac{dF_{LWUP}}{dT_{s}} = -4\varepsilon\sigma_{sb} T_{s}^3$$ (6.26)

$$\frac{dF_{SH}}{dT_{s}} = - \rho_a c_a r_{h} u^\ast$$ (6.27)

$$\frac{dF_{LH}}{dT_{s}} = - \rho_a L_s r_{e} u^\ast \frac{d{\overline q}^{*}(T_{s})}{dT_{s}}$$ (6.28)

where the small temperature dependencies of $c_a$, the exchange coefficients $r_{h}$ and $r_{e}$ and velocity scale $u^\ast$ are ignored.

For diagnostic purposes, an air temperature ($T_{REF}$) at the reference height of $z_{2m}=2m$ is computed, making use of the stability and momentum/sensible heat exchange coefficients. Defining $b_m = \kappa / r_{m}$, and $b_h = \kappa / r_{h}$, we have:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ... \cr \ln_h = & \ln\{(1+z_{2m}/z_a)(e^{b_m-b_h}-1)\}. \cr\crcr}}$$ (6.29)

For stable conditions ($\zeta > 0$)

$$f_{int} = (\ln_m-(z_{2m}/z_a)(b_m-b_h))/b_h$$ (6.30)

$$\zeta < 0$$

$$[-2.0em] f_{int} = (\ln_m-\ln_h)/b_h$$ (6.31)

where $f_{int}$ is bounded by 0 and 1. The resulting reference temperature is:

$$T_{ref} = T_{s} + (T_a - T_{s})f_{int}.$$ (6.32)