Changes between Version 2 and Version 3 of doc/tec/1dmodel

Timestamp:

Jul 1, 2016 10:03:48 AM (8 years ago)

Author:

Giersch

Comment:

--

doc/tec/1dmodel

@@ +1 @@
+= 1-D model for precursor runs =
 
-== 1d-model ==
+The initial profiles of the horizontal wind components in PALM can be prescribed by the user by piecewise linear gradients or by directly
+using observational data. Alternatively, a 1-D model can be employed to calculate stationary boundary-layer wind profiles. This is particularly useful in neutral stratification, where inertial oscillations can persist for several days in case that non-balanced profiles are used for initialization. By employing the embedded computationally inexpensive 1-D model with a Reynolds-average based turbulence parametrization, these oscillations can be significantly damped. A stationary state of the wind profiles can thus be provided much faster in the 3-D model. The arrays of the 3-D variables are then initialized with the (stationary) solution of the 1-D model. These variables are ''u'',,i,, where ''i ∈ {1, 2}'', ''e'', ''K'',,h,,, ''K'',,m,, and, with MOST applied between the surface and the first vertical grid level, also ''L'', ''u'',,∗,, as well as
+{{{
+#!Latex
+$\overline{u_i^{\prime\prime} u_3^{\prime\prime}}$
+}}} 
+(where ''i ∈ {1, 2}'').
 
-A description of the 1d-model will follow.
+The 1-D model assumes the profiles of ''θ'' and ''q'',,v,,, as prescribed by the user, to be constant in time. The model solves the prognostic equations for ''u'',,i,, and ''e'':
+{{{
+#!Latex
+\begin{align*}
+  & \frac{\partial u_i}{\partial t} = -\varepsilon_{i3j}f_3 u_j +
+  \varepsilon_{i3j}f_3 {u_{\mathrm{g},j}} - \frac{\partial
+    \overline{u_i^{\prime\prime}u_3^{\prime\prime}}}{\partial x_3}
+\end{align*}
+}}}
+and
+{{{
+#!Latex
+\begin{align*}
+  & \frac{\partial e}{\partial t} = - \frac{\partial
+    \overline{u^{\prime\prime}w^{\prime\prime}}}{\partial z} -
+  \frac{\partial \overline{v^{\prime\prime}w^{\prime\prime}}}{\partial
+    z} - \frac{g}{\theta} \frac{\partial
+    \overline{w^{\prime\prime}\theta^{\prime\prime}}}{\partial z} -
+  \frac{\partial \overline{w^{\prime\prime}e^{\prime\prime}}}{\partial
+    z} - \epsilon\;.
+\end{align*}
+}}}
+The dissipation rate is parametrized by
+{{{
+#!Latex
+\begin{align*}
+  & \epsilon = 0.064 \frac{e^{\frac{3}{2}}}{l}
+\end{align*}
+}}}
+after [#detering1985 Detering and Etling (1985)]. The mixing length is calculated after [#blackadar1997 Blackadar (1997)] as
+{{{
+#!Latex
+\begin{align*}
+  & l = \frac{\kappa z}{1 + \frac{\kappa
+      z}{l_{\text{Bl}}}}\,\;\text{with}\,\;l_{\text{Bl}} = 2.7\times
+      10^{-4}\sqrt{u_{\mathrm{g}}^2 + v_{\mathrm{g}}^2}\;.
+\end{align*}
+}}}
+The turbulent fluxes are calculated using a 1st-order closure:
+{{{
+#!Latex
+\begin{align*}
+  & \overline{u_i^{\prime\prime}u_3^{\prime\prime}} = - K_\mathrm{m}
+  \frac{\partial u_i}{\partial
+    x_3}\;,\;\overline{w^{\prime\prime}\theta^{\prime\prime}} = -
+  K_\mathrm{h} \frac{\partial \theta}{\partial z}
+  \;,\,\overline{w^{\prime\prime}e^{\prime\prime}} = - K_\mathrm{m}
+  \frac{\partial e}{\partial z}\;,
+\end{align*}
+}}}
+where ''K'',,m,, and ''K'',,h,, are calculated as
+{{{
+#!Latex
+\begin{align*}
+ K_\mathrm{m} = c_\mathrm{m}\;\sqrt{e}\
+  \begin{cases}         l & \text{for~}  \textit{Ri} \geq 0\\
+    l_{\text{Bl}} & \text{for~}  \textit{Ri} < 0\\
+  \end{cases}\;,\\
+  &K_\mathrm{h} = \frac{\Phi_\mathrm{h}}{\Phi_\mathrm{m}} K_\mathrm{m}
+\end{align*}
+}}}
+with the similarity functions ''Φ'',,h,, and ''Φ'',,m,, (see Eqs. in Sect [wiki:/doc/tec/bc boundary conditions]), using the gradient Richardson number:
+{{{
+#!Latex
+\begin{align*}
+  & \textit{Ri} =
+  \frac{\frac{g}{\theta_\mathrm{v}}\frac{\partial\theta}{\partial
+      z}}{\left[\left(\frac{\partial u}{\partial z}\right)^2 +
+      \left(\frac{\partial v}{\partial z}\right)^2 \right]} \cdot
+  \begin{cases}
+    1\,\;&\text{for~}\,\;\textit{Ri} \geq 0\;,\\
+    (1 - 16 \cdot
+    \textit{Ri})^{\frac{1}{4}}\,\;&\text{for~}\,\;\textit{Ri} < 0\;.
+  \end{cases}
+\end{align*}
+}}}
+Note that the distinction of cases in the Eq. above is done with the value of ''Ri'' from the previous time step.
+
+Moreover, a Rayleigh damping can be switched on to speed up the damping of inertial oscillations. The 1-D model is discretized in space using finite differences. Discretization in time is achieved using the 3rd-order Runge--Kutta time-stepping scheme ([#williamson1980 Williamson, 1980]). Dirichlet boundary conditions are used at the top and bottom boundaries of the model, except for ''e'', for which Neumann conditions are set at the surface (see also Sect. [wiki:/doc/tec/bc boundary conditions]).
+
+
+
+== References ==
+* [=#detering1985] '''Detering HW, Etling D.''' 1985. Application of the ''E''-''ε'' turbulence model to the atmospheric boundary layer. Bound.-Lay. Meteorol. 33: 113–133.
+
+* [=#blackadar1997] '''Blackadar AK.''' 1997. Turbulence and Diffusion in the Atmosphere. Springer. Berlin. Heidelberg. NewYork. 185 pp.
+
+* [=#williamson1980]'''Williamson JH.''' 1980. Low-storage Runge–Kutta schemes. J. Comput. Phys. 35: 48–56.