Revisiting Surface Heat-Flux and Temperature Boundary Conditions in Models of Stably Stratified Boundary-Layer Flows

Abstract

Two formulations of the surface thermal boundary condition commonly employed in numerical modelling of atmospheric stably stratified surface-layer flows are evaluated using analytical considerations and observational data from the Cabauw site in the Netherlands. The first condition is stated in terms of the surface heat flux and the second is stated in terms of the vertical potential temperature difference. The similarity relationships used to relate the flux and the difference are based on conventional log-linear expressions for vertical profiles of wind velocity and potential temperature. The heat-flux formulation results in two physically meaningful values for the friction velocity with no obvious criteria available to choose between solutions. Both solutions can be obtained numerically, which casts doubt on discarding one of the solutions as was previously suggested based on stability arguments. This solution ambiguity problem is identified as the key issue of the heat-flux condition formulation. In addition, the agreement between the temperature difference evaluated from similarity solutions and their measurement-derived counterparts from the Cabauw dataset appears to be very poor. Extra caution should be paid to the iterative procedures used in the model algorithms realizing the heat-flux condition as they could often provide only partial solutions for the friction velocity and associated temperature difference. Using temperature difference as the lower boundary condition bypasses the ambiguity problem and provides physically meaningful values of heat flux for a broader range of stability condition in terms of the flux Richardson number. However, the agreement between solutions and observations of the heat flux is again rather poor. In general, there is a great need for practicable similarity relationships capable of treating the vertical turbulent transport of momentum and heat under conditions of strong stratification in the surface layer.

Appendix

By evaluating the discriminant ${B^2}/4 - C$ in Sect. 3.2.1 one can assess whether the roots of Eq. 27 are real or complex. The condition for solutions to be real is $B^2/4 \ge C$. Invoking (29) for $B$ and $C$, this condition provides

$$\begin{aligned} Ri_B (G-F) \ge - {F^2}/4 \text{. } \end{aligned}$$

(32)

We now show that $G-F$ on the left-hand side of Eq. 32 is non-negative. First use (28) to write $G-F$ as

$$\begin{aligned} G-F = \frac{\alpha _{\theta }}{\alpha ^2} - \frac{1}{\alpha } \frac{r_{ms} \Delta _{m0}}{r_{m0} \Delta _{ms}} \text{. } \end{aligned}$$

(33)

In view of the definitions of $\beta$, $r_{m0}$, $r_{ms}$, $z_0$, and $z_s$, we obtain

$$\begin{aligned} \frac{r_{ms} \Delta _{m0}}{r_{m0} \Delta _{ms}}&= \frac{\ln \left( z_m/z_s \right) }{\ln \left( z_m/z_0 \right) } \frac{\left( z_m - z_0\right) }{\left( z_m - z_s\right) }\\&= \frac{\ln \left[ 1 - (1-\gamma )\right] }{1-\gamma }\frac{1-\mu }{\ln \left[ 1 - (1-\mu )\right] }, \end{aligned}$$

where $\gamma = z_s/z_m$ and $\mu = z_0/z_m$. The Taylor series expansion,

$$\begin{aligned} \ln (1-x) = -\left( x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} \dots \right) , \end{aligned}$$

with $1-\gamma$ and $1-\mu$ used in place of $x$, yields

$$\begin{aligned} \frac{r_{ms} \Delta _{m0}}{r_{m0} \Delta _{ms}} = \frac{ 1 + \frac{(1-\gamma )}{2} + \frac{(1-\gamma )^2}{3} + \frac{(1-\gamma )^3}{4} + \cdots }{1 + \frac{(1-\mu )}{2} + \frac{(1-\mu )^2}{3} + \frac{(1-\mu )^3}{4} + \cdots }. \end{aligned}$$

Under the natural assumption of $z_m > z_s \ge z_0$, we have $\gamma \ge \mu \ge 0$, and therefore $r_{ms} \Delta _{m0}/(r_{m0} \Delta _{ms}) \le 1$. Further assuming (as done in most model applications) $\alpha _{\theta } = \alpha$, we conclude that $G-F$ is non-negative and thus the right-hand side of Eq. 32 is non-positive. On the other hand, ${Ri}_B$ is non-negative under stable/neutral conditions for which (32) is satisfied, and thus Eq. 27 is guaranteed to have two real roots.

Next we must determine whether these roots are physically meaningful. First we note that the dimensionless friction velocity must be greater than zero. Combining the definitions of $\beta$, $r_{m0}$, $r_{ms}$, $z_0$, and $z_s$ with (1), (5), and (7), we obtain

$$\begin{aligned} {\hat{u}_*} = \frac{r_{m0}}{r_{m0} - \Psi _{u0}} \text{. } \end{aligned}$$

(34)

Noting that $\Psi _{u0}$ is negative (see Eq. 3), we conclude that ${\hat{u}_*}$ must also be less than unity. Thus, the roots of Eq. 27 must satisfy $0 < {\hat{u}_*} < 1$.

We also need to determine ranges of variability of $B$ and $C$ (Eq. 29). Taking into account that $G-F$ is non-negative and $\alpha _{\theta } = \alpha$, the difference

$$\begin{aligned} F - 2G = \frac{1}{\alpha } \left( \frac{r_{ms} \Delta _{m0}}{r_{m0} \Delta _{ms}} - 2 \right) \end{aligned}$$

(35)

is negative since $r_{ms} \Delta _{m0}/(r_{m0} \Delta _{ms}) \le 1$ (see above), it becomes clear that $B$ is always negative. Considering the numerator of $C$,

$$\begin{aligned} G - Ri_B = \frac{1}{\alpha } - Ri_B, \end{aligned}$$

(36)

we find that

$$\begin{aligned} C \ge 0&\quad \text {if}\ {Ri}_B \le 1/\alpha , \end{aligned}$$

(37a)

$$\begin{aligned} C < 0&\quad \text {if}\ {Ri}_B > 1/\alpha . \end{aligned}$$

(37b)

Now consider

$$\begin{aligned} x_{1} = -\frac{B}{2} + \sqrt{\frac{B^2}{4} - C}, x_{2} = -\frac{B}{2} - \sqrt{\frac{B^2}{4} - C} \text{. } \end{aligned}$$

(38)

Because the condition $B^2/4 \ge C$ is globally satisfied and $B$ is non-negative, $x_1 >0$ regardless of whether $C$ is positive or negative. On the other hand, $x_2 < 0$ when $C < 0$, and $x_2 > 0$ when $C > 0$.

However, as previously noted, we need ${\hat{u}_*} < 1$, so the following condition:

$$\begin{aligned} x_{1} =-\frac{B}{2} + \sqrt{\frac{B^2}{4} - C} < 1 \end{aligned}$$

must be satisfied for $x_1$. According to (28) and (29), this would require ${Ri}_B < 0$, so $x_1$ is physically irrelevant. Conversely, with $C > 0$ and ${Ri}_B < 1/\alpha$, see (37), we come to $0 < x_2 < 1$, so in this case $x_2$ is a physically relevant root.

Collecting results, we see that there is no physical solution for ${\hat{u}_*}$ when ${Ri}_B > 1/\alpha$. When ${Ri}_B \le 1/\alpha$, Eq. 27 provides only one physical root given by $-B/2 - \sqrt{B^2/4 - C}$. Thus, $1/\alpha =0.2$ is the maximum value of ${Ri}_B$ for which a physically relevant ${\hat{u}_*}$ exists within the framework of the adopted assumptions. This solution behaviour is illustrated in Fig. 8.

Fig. 8

Roots of Eq. 27 as functions of ${Ri}_B$. The blue line indicates the $x_1$ root (note that its value at ${Ri}_B=0$ is equal to $-B$ ), and the red line indicates the $x_2$ root