Changes between Version 13 and Version 14 of doc/tec/sgs

Timestamp:

Mar 27, 2020 1:15:00 PM (5 years ago)

Author:

maronga

Comment:

--

doc/tec/sgs

@@ -46 +46 @@
  \frac{\partial{\theta_{\mathrm{v}}}}{\partial z}\right)^{-\frac{1}{2}} \right) &\text{for~}
     \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} > 0, \\
- \min\left(\boldsymbol{\varDelta}\right) &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
+ \\boldsymbol{\varDelta} &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
 \end{cases}
 \end{align*}
@@ -109 +109 @@
 Note that this parametrization of the SGS buoyancy flux differs from that used with bulk cloud microphysics (see Sect. [wiki:doc/tec/microphysics#Turbulenceclosure turbulence closure in cloud microphysics]).
 
+== [=#deardorff_sgs-dai Modified Deardorff subgrid-scale model] ==
+A modificaiton of the Deardorff scheme described above is available that provides an improved mixing length calculation for locally stable stratification, particularly if used with fine grid spacings. Details about the deficiencies of the Deardorff scheme and the following modified version are given by [#dai Dai et al. (2020)]. In the following, we only describe the modifications of the Deardorff scheme.
+
+The mixing length is calculated as
+{{{
+#!Latex
+\begin{align*}
+  & l =
+\begin{cases}
+\left(\dfrac{1}{\kappa z} +\dfrac{1}{0.76\sqrt{e}\left(\frac{g}{\theta_{\mathrm{v},0}} \frac{\partial{\theta_{\mathrm{v}}}}{\partial z}\right)^{-\frac{1}{2}} }\right)^{-1} &\text{for~}
+    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} > 0, \\
+ \boldsymbol{\varDelta} &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
+\end{cases}
+\end{align*}
+}}}
 
 == [=#dynamic_sgs Dynamic subgrid-scale model] ==
@@ -166 +181 @@
 
 == References ==
+
+* [=dai]'''Dai, Y., Basu, S., Maronga, B., de Roode, SR.''' 2020. Addressing the Grid-size Sensititivty Issue in Large-eddy Simulations of Stable Boundary Layers. Boundary-layer Meteorol., submitted. http://arxiv.org/abs/2003.09463.
+
 * [=#deardorff]'''Deardorff JW.''' 1980. Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Lay. Meteorol. 18: 495–527.