Boundary-Layer Development and Low-level Baroclinicity during High-Latitude Cold-Air Outbreaks: A Simple Model

Abstract

A new quasi-analytical mixed-layer model is formulated describing the evolution of the convective atmospheric boundary layer (ABL) during cold-air outbreaks (CAO) over polar oceans downstream of the marginal sea-ice zones. The new model is superior to previous ones since it predicts not only temperature and mixed-layer height but also the height-averaged horizontal wind components. Results of the mixed-layer model are compared with dropsonde and aircraft observations carried out during several CAOs over the Fram Strait and also with results of a 3D non-hydrostatic (NH3D) model. It is shown that the mixed-layer model reproduces well the observed ABL height, temperature, low-level baroclinicity and its influence on the ABL wind speed. The mixed-layer model underestimates the observed ABL temperature only by about 10 %, most likely due to the neglect of condensation and subsidence. The comparison of the mixed-layer and NH3D model results shows good agreement with respect to wind speed including the formation of wind-speed maxima close to the ice edge. It is concluded that baroclinicity within the ABL governs the structure of the wind field while the baroclinicity above the ABL is important in reproducing the wind speed. It is shown that the baroclinicity in the ABL is strongest close to the ice edge and slowly decays further downwind. Analytical solutions demonstrate that the $\mathrm{e}$-folding distance of this decay is the same as for the decay of the difference between the surface temperature of open water and of the mixed-layer temperature. This distance characterizing cold-air mass transformation ranges from 450 to 850 km for high-latitude CAOs.

Appendices

Appendix 1 Terms $U_\mathrm{gt}$ and $U_\mathrm{gi}$

Terms $U_\mathrm{gt}$ and $U_\mathrm{gi}$ according to Eq. 16 are

$$\begin{aligned} U_\mathrm{gt}= & {} \frac{gz_i}{2f\theta _m}\frac{\partial \theta _m}{\partial y} , \end{aligned}$$

(30)

$$\begin{aligned} U_\mathrm{gi}= & {} - \frac{g \mathrm{\Delta } \theta }{f\theta _+}\frac{\partial z_i}{\partial y}. \end{aligned}$$

(31)

In Eq. 30 we substitute $\partial \theta _m/ \partial y$ using Eq. 3 and parametrization for $w_e$ given by Eqs. 6 and 7 to obtain

$$\begin{aligned} U_\mathrm{gt}=\frac{g(1+\beta )C_H(\theta _w - \theta _m)}{2f\theta _m \mathrm{cos} \alpha }. \end{aligned}$$

(32)

In Eq. 31 we substitute $\partial z_i/ \partial y$ using Eq. 4 with parametrization for $w_e$

$$\begin{aligned} U_\mathrm{gi}=-\frac{g \beta C_H (\theta _w - \theta _m)}{f \theta _{+} \mathrm{cos} \alpha }. \end{aligned}$$

(33)

In Eqs. 32 and 33 an assumption $\phi \approx \alpha$ is used. It is straightforward to obtain the fraction $U_\mathrm{gt} / U_\mathrm{gi}$,

$$\begin{aligned} \frac{U_\mathrm{gt}}{U_\mathrm{gi}} = - \frac{(1+\beta )\theta _{+}}{2\theta _m\beta } \approx -\frac{1+\beta }{2\beta }. \end{aligned}$$

(34)

In the denominators of the terms on the right-hand side of Eqs. 32 and 33 we can substitute $\theta _m$ and $\theta _+$ by a constant reference value $\theta _0$ that is independent from y. This is justified by the fact that $\theta _m$ and $\theta _+$ change over water by less than 10 $\%$ from their initial values over the ice. This is not the case for $(\theta _w - \theta _m)$ in the nominators in Eqs. 32 and 33, which depends very strongly upon y. After that, both $U_\mathrm{gt}$ and $U_\mathrm{gi}$ become linear functions of $\theta _m$.

Now, $U_\mathrm{gt}$ and $U_\mathrm{gi}$ can be expressed as functions of $\overline{\theta _m}$. This can be done by a substitution $\theta _w - \theta _m = (\theta _w -\theta '_\mathrm{ice})(1-\overline{\theta _m})$ in Eqs. 32 and 33, where $\theta '_\mathrm{ice} = \theta _\mathrm{ice} - \gamma _{\theta }z_{i0}(1+\beta )/(1+2\beta )$. After that, we can obtain useful relations similar to Eq. 14 describing the evolution of $U_\mathrm{gt}$ and $U_\mathrm{gi}$ as functions of $\overline{y}$ and of external forcing parameters. To that aim, we express $\overline{\theta _m}$ as a function first of $U_\mathrm{gt}$ and then of $U_\mathrm{gi}$,

$$\begin{aligned} \overline{\theta _m}= & {} 1 - \frac{2 f \theta _0 \mathrm{cos} \alpha U_\mathrm{gt}}{g (1+\beta ) C_H (\theta _w - \theta '_\mathrm{ice})} \equiv 1 - \overline{U_\mathrm{gt}} , \end{aligned}$$

(35)

$$\begin{aligned} \overline{\theta _m}= & {} 1 + \frac{f \theta _0 \mathrm{cos} \alpha U_\mathrm{gi} }{g \beta C_H (\theta _w - \theta '_\mathrm{ice})} \equiv 1 + \overline{U_\mathrm{gi}}. \end{aligned}$$

(36)

Then, we use Eqs 35–36 in Eq. 14 and obtain

$$\begin{aligned} \mathrm{ln}( \overline{U_\mathrm{gt}}) - \overline{U_\mathrm{gt}} + 1= & {} -C_1 \overline{y} +C_2 , \end{aligned}$$

(37)

$$\begin{aligned} \mathrm{ln}(- \overline{U_\mathrm{gi}}) + \overline{U_\mathrm{gi}} + 1= & {} -C_1 \overline{y} +C_2. \end{aligned}$$

(38)

Appendix 2 Geostrophic Wind in the NH3D Model

Here, the calculation of the ABL-mean x-component $u_\mathrm{gm}$ of the geostrophic wind vector and the baroclinic terms $U_\mathrm{gt}$, $U_\mathrm{gi}$ and $U_{g+}$ is addressed. To obtain them from the NH3D model variables we consider the equation for the v-component of wind vector used in the NH3D model. It is given by

$$\begin{aligned} \frac{d v}{d t} = - \frac{\partial \phi '}{\partial y} + \frac{\partial \phi '}{\partial \sigma } \frac{\sigma }{p_*} \frac{\partial p_*}{\partial y} + fU_g - f u , \end{aligned}$$

(39)

where we neglect diffusion as we are interested now only in the u-component of the geostrophic wind. In Eq. 39 $\phi ' = \phi - \phi _{ref}$ is the deviation of geopotential from its reference-state value $\phi _{ref}$; $\sigma = (p-p_{top})/p_*$ is the terrain-following vertical coordinate where $p_* = p_{surf} - p_{top}$ is the pressure difference between the surface and the model-top values.

The first three terms on the right-hand side of Eq. 39 represent the horizontal pressure gradient force. In particular, the first two terms are associated with baroclinicity. The third term is the constant in space and time barotropic forcing.

After averaging Eq. 39 over the ABL height one easily obtains the correspondence between the baroclinic terms of the geostrophic wind in the mixed-layer and the NH3D models

$$\begin{aligned} u_\mathrm{gm} - U_g = U_{g+} + U_{g_i} + U_\mathrm{gt} = \frac{1}{fz_i} \int ^{z_i}_0 - \frac{\partial \phi '}{\partial y} + \frac{\partial \phi '}{\partial \sigma } \frac{\sigma }{p_*} \frac{\partial p_*}{\partial y} dz. \end{aligned}$$

(40)

Note, that Eq. 40 differs from Eq. 16 as the latter is based on several assumptions of the mixed-layer model.

The baroclinic part of the x-component of the geostrophic wind vector right above the inversion height at $z = z_{i+}$ is

$$\begin{aligned} U_{g+} = - \frac{\partial \phi '}{\partial y}\big |_{z=z_{i+}} + \frac{\partial \phi '}{\partial \sigma } \frac{\sigma }{p_*} \frac{\partial p_*}{\partial y}\big |_{z=z_{i+}}. \end{aligned}$$

(41)

It is not possible to define $z_{i+}$ in the NH3D model exactly as in the mixed-layer model since there it is based on the assumption of an idealized structure of the ABL and its capping inversion. So, we evaluate the terms on the right-hand side of Eq. 41 at height $z = (1 + \delta ) z_i$ with $\delta =0.1$. We found that a variation of $\delta$ by ± 50% changed the results presented in Sect. 5 only marginally.

It is also impossible to evaluate $U_\mathrm{gi}$ directly from the results of the NH3D model because one cannot give an exact location and value of the horizontal gradient of geopotential produced by the sloping inversion at the ABL top. However, it is sufficient to evaluate the sum $U_\mathrm{gt} + U_\mathrm{gi}$ since we expect $U_\mathrm{gi}$ to be a fraction of $U_\mathrm{gt}$ according to Eq. 22 so that $U_\mathrm{gt}/U_\mathrm{gi} \approx - (1+\beta )/2\beta$. The sum $U_\mathrm{gt} + U_\mathrm{gi}$ is evaluated by subtracting the value of $U_{g+}$ diagnosed using Eq. 41 from the value of $U_{g+} + U_\mathrm{gi} + U_\mathrm{gt}$ diagnosed using Eq. 40.