Changes between Version 12 and Version 13 of doc/tec/sgs

Timestamp:

Mar 26, 2020 9:42:20 AM (4 years ago)

Author:

gronemeier

Comment:

--

doc/tec/sgs

@@ -29 +29 @@
 \end{align*}
 }}}
-Here, ''c'',,m,, = 0.1 is a model constant and '''''Δ''' = (Δx Δy Δz)^1/3^'' with ''Δx'', ''Δy'', ''Δz'' being the grid spacings in ''x'', ''y'' and ''z'' direction, respectively. The SGS mixing length ''l'' depends on height ''z'' (distance from the wall when topography is used), '''''Δ''''', and stratification and is calculated as
+Here, ''c'',,m,, = 0.1 is a model constant and 
+{{{
+#!Latex
+\begin{align*}
+  & \boldsymbol{\varDelta} = \min\left( 1.8z, \left(\varDelta x \varDelta y \varDelta z \right)^{\frac{1}{3}} \right)
+\end{align*}
+}}}
+with ''Δx'', ''Δy'', ''Δz'' being the grid spacings in ''x'', ''y'' and ''z'' direction, respectively. The SGS mixing length ''l'' depends on height ''z'' (distance from the wall when topography is used), grid spacing, and stratification and is calculated as
 {{{
 #!Latex
@@ -35 +42 @@
   & l =
 \begin{cases}
-\min\left(1.8z,\boldsymbol{\varDelta},
+\min\left(\boldsymbol{\varDelta},
   0.76\sqrt{e}\left(\frac{g}{\theta_{\mathrm{v},0}}
  \frac{\partial{\theta_{\mathrm{v}}}}{\partial z}\right)^{-\frac{1}{2}} \right) &\text{for~}
     \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} > 0, \\
- \min\left(1.8z, \boldsymbol{\varDelta}\right) &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
+ \min\left(\boldsymbol{\varDelta}\right) &\text{for~}    \frac{\partial{\theta_{\mathrm{v}}}}{\partial z} \leq 0.
 \end{cases}
 \end{align*}