Changes between Version 11 and Version 12 of doc/tec/microphysics

Timestamp:

Jun 12, 2016 2:11:30 PM (9 years ago)

Author:

Giersch

Comment:

--

doc/tec/microphysics

@@ -324 +324 @@
 == Turbulence closure ==
 
+Using bulk cloud microphysics, PALM predicts liquid water temperature ''θ'',,l,, and total water content ''q'' instead of ''θ''
+and ''q'',,v,,. Consequently, some terms in the Eq. for 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}{\theta_{\mathrm{v}}}^{\prime\prime}}$
+}}}
+of Sect. [wiki:/doc/tec/sgs turbulence closure] are unknown. We thus follow [#cuijpers1993 Cuijpers and Duynkerke (1993)] and calculate the SGS buoyancy flux from the known SGS fluxes
+{{{
+#!Latex 
+$\overline{w^{\prime\prime}{\theta_{\mathrm{l}}}^{\prime\prime}}$ 
+}}}
+and 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}{q}^{\prime\prime}}$.
+}}}
+In unsaturated air (''q'',,c,, = 0) the Eq. for 
+{{{
+#!Latex
+$\overline{w^{\prime\prime}
+    {\theta_{\mathrm{v}}}^{\prime\prime}}$
+}}}
+of Sect. [wiki:/doc/tec/sgs turbulence closure] is then replaced by
+{{{
+#!Latex
+\begin{align*}
+  & \overline{w^{\prime\prime}
+    {\theta_{\mathrm{v}}}^{\prime\prime}}=K_1\,\cdot\,\overline{w^{\prime\prime}
+    {\theta_\mathrm{l}}^{\prime\prime}} +
+  K_2\,\cdot\,\overline{w^{\prime\prime} {q}^{\prime\prime}},
+\end{align*}
+}}}
+with
+{{{
+#!Latex
+\begin{align*}
+  &  K_1 = 1+\left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right)\,\cdot\,q,\\
+  & K_2 =
+  \left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right)\,\cdot\,\theta_\mathrm{l},
+\end{align*}
+}}}
+and in saturated air (''q'',,c,, > 0) by
+{{{
+#!Latex
+\begin{align*}
+  &
+  K_1 =\frac{1 - q + \frac{R_\mathrm{v}}{R_\mathrm{d}} (q-q_\mathrm{l}) \cdot \left(1 + \frac{L_\mathrm{V}}{R_\mathrm{v} T} \right)}{1 + \frac{L_\mathrm{V}^2}{R_\mathrm{v} c_p T^2} (q-q_\mathrm{l})},\\
+  & K_2 = \left(\frac{L_\mathrm{V}}{c_p T} K_1 - 1 \right)
+  \cdot \theta.
+\end{align*}
+}}}
+
 == Recent applications ==
 
@@ -356 +408 @@
 * [=#rogers1993]'''Rogers RR, Baumgardner D, Ethier SA, Carter DA, Ecklund WL.''' 1993. Comparison of raindrop size distributions measured by radar wind profiler and by airplane. J. Appl. Meteorol. 32: 694–699.
 
-
+* [=#cuijpers1993]'''Cuijpers JWM, Duynkerke PG.''' 1993. Large eddy simulation of trade wind cumulus clouds. J. Atmos. Sci. 50: 3894–3908.