Analytical Model Coupling Ekman and Surface Layer Structure in Atmospheric Boundary Layer Flows

Abstract

We introduce an analytical model that describes the vertical structure of Ekman boundary layer flows coupled to the Monin-Obukhov Similarity Theory (MOST) surface layer representation, which is valid for conventionally neutral (CNBL) and stable (SBL) atmospheric conditions. The model is based on a self-similar profile of horizontal stress for both CNBL and SBL flows that merges the classic 3/2 power law profile with a MOST-consistent stress profile in the surface layer. The velocity profiles are then obtained from the Ekman momentum balance equation. The same stress model is used to derive a new self-consistent Geostrophic Drag Law (GDL). We determine the ABL height (h) using an equilibrium boundary layer height model and parameterize the surface heat flux for quasi-steady SBL flows as a function of a prescribed surface temperature cooling rate. The ABL height and GDL equations can then be solved together to obtain the friction velocity $(u_*)$ and the cross-isobaric angle ($\alpha _0$) as a function of known input parameters such as the Geostrophic wind speed and surface roughness $(z_0)$. We show that the model predictions agree well with simulation data from the literature and newly generated Large Eddy Simulations (LES). These results indicate that the proposed model provides an efficient and relatively accurate self-consistent approach for predicting the mean wind velocity distribution in CNBL and SBL flows.

Appendix A: Summary of ABL model and iterative solution method

In this Appendix section, we summarize the iterative method to solve the coupled equations for the proposed ABL model discussed in Sect. 2.

• 1. Input values: $N_\infty ,f_c,G,z_0,C_r$ (assuming $f_c>0$, northern hemisphere).

• 2. Iterative calculation of $h,u_*,\alpha _0$:

  • a. Assume initial values for $h^0,u_*^0$ and set $h^n=h^0,u_*^n=u_*^0$.

  • b. Compute $u_*^{n+1}$ using Eq. (35):

    $$\begin{aligned} u_*^{n+1}&=\frac{\kappa G}{\sqrt{[\ln \left( Ro^n\right) -A^n]^2+(B^n)^2}}, \,\, \textrm{where} \\ A^n&=-\ln c_m {\hat{h}}^n-\kappa \Biggl [(5\mu ^n+0.3\mu _N)(c_m{\hat{h}}^n-{\hat{\xi }}_0^n)\nonumber \\&\quad +g^\prime (c_m{\hat{h}}^n)\left( 1-c_m\right) ^{3/2} -g(c_m{\hat{h}}^n)\frac{3}{2{\hat{h}}^n}\sqrt{1-c_m}\Biggr ],\nonumber \\ B^n&=\frac{3\kappa }{2{\hat{h}}^n}, \,\,\, \textrm{and} \nonumber \\ g(c_m{\hat{h}}^n)&=c_g\left[ 1-e^{- c_m/{ \Gamma }}\right] ,\ g^\prime (c_m{\hat{h}}^n)=\frac{c_g}{{ \Gamma }{\hat{h}}^n}e^{-c_m/{ \Gamma } }, \ c_g=1.43, \ { \Gamma }={ 0.83},\nonumber \\ \mu ^n&=\frac{g\, (-C_r)}{u_*^n f_c^2 \, \Theta _0} {\hat{h}}^n, \ \,\, {\hat{\xi }}_0=\frac{z_0 f_c}{u_*^n}, \,\,\ {\hat{h}}^n=\frac{h^n f_c}{u_*^n}, \,\, c_m=0.20.\nonumber \end{aligned}$$

    (47)

  • c. Compute $h^{n+1}$ using $u_*^{n+1}$ from Eq. (47) in Eq. (30):

    $$\begin{aligned} h^{n+1}&=\frac{u_*^{n+1}}{f_c}\left[ \frac{1}{C_{TN}^2} +\frac{\mu _N}{C_{CN}^2}+\frac{1}{C_{NS}^2}\frac{(g/\theta _0)(-C_r)h^n }{ {u_*^{n+1}}^2 f_c}\right] ^{-1/2},\\ C_{TN}&=0.5, \ C_{CN}=1.6, \ C_{NS}=0.78, \ \mu _N=N_\infty /f_c. \end{aligned}$$

  • d. Iterate till convergence to get final values of $u_*,h$.

• 3. Then evaluate:

  $$\begin{aligned} {\hat{h}}&=\frac{h f_c }{u_*}, \ {\hat{\xi }}_0=\frac{z_0 f_c }{u_*},\ \mu =\frac{g \,(-C_r)}{u_* f_c^2\,\Theta _0}{\hat{h}}, \ Ro=\frac{u_*}{z_0 f_c}, \end{aligned}$$

  as well as the converged values of A and B.

• 4. Evaluate $U_g,V_g$ using the GDL equations (Eqs. 5, 6, 33, 34):

  $$\begin{aligned} U_g =\frac{u_*}{\kappa } \left( \ln \left( Ro\right) -A \right) , \ \,\, V_g = - \frac{u_*}{\kappa } B, \end{aligned}$$

  and obtain $\alpha _0=\tan ^{-1}\left( \dfrac{V_g}{U_g}\right)$.

• 5. Evaluate U(z) and V(z) with ${\hat{\xi }} = z f_c/u_*$:

  $$\begin{aligned} U(z)&= {\left\{ \begin{array}{ll} u_*\left( -g^\prime (\hat{\xi })\left[ 1-\dfrac{\hat{\xi }}{\hat{h}}\right] ^{3/2}+g(\hat{\xi })\dfrac{3}{2\hat{h}} \sqrt{1-\dfrac{\hat{\xi }}{\hat{h}}} \right) + U_g &{}, \ \hat{\xi }\ge \hat{\xi }_m\\ &{}\\ u_*\left( \dfrac{1}{\kappa }\ln \dfrac{\hat{\xi }}{\hat{\xi }_0} +(5\mu +0.3\mu _N)(\hat{\xi }-\hat{\xi }_0)\right) &{},\ \hat{\xi }\le \hat{\xi }_m \end{array}\right. }.\\ V(z)&= u_* \left( \dfrac{g(\hat{\xi })g^\prime (\hat{\xi })}{\sqrt{1-g(\hat{\xi })^2}} \left[ 1-\dfrac{\hat{\xi }}{\hat{h}}\right] ^{3/2} +\dfrac{3}{2\hat{h}}\sqrt{1-g(\hat{\xi })^2} \left[ 1-\dfrac{\hat{\xi }}{\hat{h}}\right] ^{1/2}\right) + V_g . \end{aligned}$$