Changes between Version 2 and Version 3 of doc/tec/noncyclic

Timestamp:

Mar 12, 2013 5:08:15 PM (12 years ago)

Author:

fricke

Comment:

--

doc/tec/noncyclic

@@ -27 +27 @@
 {{{
 #!Latex
- s^{t + \Delta t}(k,j,-1) = s_{init}(k) \; .
+ s^{t + \Delta t}(k,j,-1) = s_{init}(k) \; . \quad(1)
  }}}
 t denotes the time, Δt the time step and s,,init,, the initialization profile of the scalar quantities which is constant in time.
@@ -34 +34 @@
 {{{
 #!Latex
- e^{t + \Delta t}(k,j,-1) = e^{t + \Delta t}(k,j,0) \; .
- }}}
+ e^{t + \Delta t}(k,j,-1) = e^{t + \Delta t}(k,j,0) \; . \quad(2)
+ }}} 
+To prevent gravity waves from being reflected at the inflow, a relaxation term can be added to the prognostic equations for the potential temperature θ (Davies, 1976):
+{{{
+#!Latex
+ \theta^{t+1}(d) = ... - \Delta t \cdot K(d) \cdot \left( \theta^{t}(d) - \theta_{init} \right) \; . \quad(3)
+ }}} 
+Here, d is the distance normal to the wall and θ,,init,, the initial value of the potential temperature, which corresponds to the value at the inflow boundary. 
+The damping or relaxation function K depends only on the distance d to the inflow.
+K is calculated by
+{{{
+#!Latex
+K(d) =
+\begin{cases}
+d_f \sin^2\left( \frac{\pi}{2} \frac{d_w - d}{d_w} \right) , \text{for } d < d_w \\  \qquad\quad  0  \qquad \quad \;\;\; , \text{for } d \ge d_w
+\end{cases} . \quad (4)
+ }}} 
+d,,f,, is a damping factor to control the damping intensity, and d,,w,, is the width of the relaxation region extending from the inflow. Quantities d,,f,, and d,,w,, can be set with parameters [../../app/inipar/#pt_damping_factor pt_damping_factor] and [../../app/inipar/#pt_damping_width pt_damping_width], respectively.
+Both parameters have to be set by the user and must be adjusted case-by-case, because both parameters depend on the numerical and physical conditions, so that application of universal default values is not possible. 
+So far, we have experience with gravity waves in case of cold air outbreaks, which grow in amplitude up to quite extreme values, if no damping is applied.
+In the respective simulations, we used typical values for [../../app/inipar/#pt_damping_factor pt_damping_factor] of 0.05 and for [../../app/inipar/#pt_damping_width pt_damping_width] of 25 km in order to prevent the gravity waves from growing.