Contents:
 1. Mountain Waves
 2. Density Current
 3. Examples of the sphere
 4. Linear anelastic equations for atmospheric vortices
 5. Valley wind system
 6. Valley wind system II
Mountain Waves
2D hydrostatic and nonhydrostatic mountain waves
a  \(\delta x\)  \(\delta z\) 
1 km  200 m  100 m 
10 km  2 km  100 m 
100 km  20 km  100 m 
Schaer test case
\(z(x)= h\,exp(\frac{x^{2}}{a^{2}})cos^{2} (\frac{\pi\,x}{\lambda}) \),
\(h\,=\,400\,m\,\),
\(\lambda = 4000\,m\,\),
\( a = 5000\,m \).
Density Durrent
The density current test case is documented in Straka et al. (1993).
Geometry:
\(
\Delta T=\begin{cases}
0.0^\circ C & \text{if}\quad L>1.0, \\
15.0^\circ C \,[\cos(\pi L)+1.0]\, /2. & \text{if}\quad L\le 1.0
\end{cases}
\)
where
\(L=[(xx_c)x_r^{1}]^2+[(zz_c)z_r^{1}]^2)^{1/2}\)
and \(x_{c}=0.0\,km\), \(x_{r}=4.0\,km\), \(z_{c}=3.0\,km\), \(z_{r}=2.0\,km\).
Density current at the beginning and after 300 s, 600 s and 900 s, starting from rest  Translating density current at the beginning and after 300 s, 600 s and 900 s, horizontal wind 20 m/s 

Examples of the sphere
Geometry:
integration time: Maximal time step is 1800s
Acoustic waves
 Pertubation:
 \(p’ = \Delta p f(\lambda,\phi)g(z)\)
 \( f(\lambda , \phi) = \begin{cases} \frac{1}{2}cos(\pi \frac{r}{R}) & r < R \\
0 & r > R \end{cases}\)  \(g(z)= sin (\frac{n_{v}\,\phi\,z}{z_{T}})\)
Gravity waves
Profile:
 Case 1: \(N=0.01s^{1}, n_{v}=1\)
 Case 2: \(N=0.02s^{1}, n_{v}=1\)
 Case 3: \(N=0.01s^{1}, n_{v}=2\)
Perturbation:
 \(\theta’ = \Delta\theta f(\lambda,\phi)\,g(z)\)
Mountain waves
Orography:
[latex] h(\lambda,\phi)=\frac{{h_{0}}{1+(r/d)^{2}}\)
with \(h_{0}\) as the height at the center of the mountain, \(d\) as the haltwidth of the mountain and \(r\) as the distance from the center.
Profil:
Linear anelastic equations for atmospheric vortices
Construction of a test example:
\(V(r)= \frac{1}{2r}\,\zeta_{0}\,b^{2}\,(1exp(\frac{r^{2}}{b^{2}}))\)
is computed from the radial integration of a Gaussian vorticity profile:
\(\zeta(r)=\zeta_{0}\,exp[(\frac{r}{b})^{2}]=\frac{1}{r}\frac{\delta}{\delta r}[rV(r)]\) with
\( \zeta_{0}= 2.34 \cdot 10^{3}\,s^{1}\) and \( b = 53.4\,km\).
\(\zeta(r)=\frac{\delta v}{\delta x}\frac{\delta u}{\delta y}\)
with \(u= V(r) sin \lambda = V(r) \frac{y}{r}\) and \(u= V(r) cos \lambda = V(r) \frac{x}{r}\)
> \(\zeta(r) = V’(r)+\frac{V(r)}{r} =\frac{1}{r}\frac{\delta}{\delta r}[rV(r)]\)
\(V(r,z)=V(r)exp\, [\frac{z^{a}}{\alpha\,L_{z}^{\alpha}}]\)
where \(L_{z}=6\,km\) indicates the depth of the barotropic part of the vortex and \(\alpha = 2.5\) is the vertical decay rate of the wind in the baroclinic region.
\(\theta(z)=\theta_0 \,exp[\frac{N^2}{g}z]\) is given by the buoyancy frequency \( N=10^{2}\text{s}^{2}\,\) which represents boundary conditions for a sufficient large radius. This potential temperature is used to calculate the density and pressure profile. These profiles may be computed by integration in r and zdirection for every grid cell from the edge of the grid to the centre.
height [m] 
density [kg/m^3] 
pressure [Pa] 
23867  0.05  3000 
22047  0.07  4000 
20658  0.08  5000 
19546  0.1  6000 
17826  0.14  8000 
16535  0.18  10000 
15227  0.22  15000 
14137  0.25  20000 
13193  0.29  25000 
9647  0.44  30000 
8553  0.49  35000 
7573  0.55  40000 
6682  0.6  45000 
5870  0.66  50000 
5123  0.71  55000 
4427  0.76  60000 
3779  0.81  65000 
3171  0.87  70000 
2599  0.92  75000 
2058  0.97  80000 
1545  1.02  85000 
1057  1.07  90000 
590  1.12  950000 
141  1.16  100000 
0  1.18  10163 
 \( \bar{r}=\sqrt{x^{2}+y^{2}}\)
 \( \bar{i}=r_{\bar{i}} \le \bar{r} \le r_{\bar{i}+1}\)
 \( \bar{\theta} (r)=\frac{[\theta _{\bar{i}+1}\, (r_{\bar{i}+1}\bar{r})+\theta_{\bar{i}+1}\, (\bar{r}r_{\bar{i}})]}{r_{\bar{i}+1}r_{\bar{i}}}\)
\(\theta_{i}(r,\lambda,z)= A\,cos\,(3\lambda)\,exp\,[(\frac{(rr_{b})^{2}}{\sigma _{r}^{2}})+\frac{(zz_{b})^{2}}{\sigma _{z}^{4}}]\)
where \(r_{b}=50\,km, z_{b}=6\,km, \sigma _{r}=15\,km, \sigma _{z}=3\,km\) and \(A=0.5\,K\).
Configuration of model domain:

 400 x 400 km horizontal grid points with 3km spacing

 z= 0 to 20 km, spacing pf 200m

 necessity of periodic outer boundary conditions in both directions

 necessity of constant time steps of 10 sec
Profile:
Simulation with ASAM:

 Horizontal cross section of vertical velocity \((ms^{1}), z=8.2\,km, 1200×1200\,km\)

 Horizontal cross section of vertical velocity \((ms^{1})\) with/without potential perturbation at 2 hours,\(z=8.2\,km, 180×180\,km\)
Valley wind systems
 Horizontal cross section of vertical velocity \((ms^{1})\) with/without potential perturbation at 2 hours,\(z=8.2\,km, 180×180\,km\)

 Examples after Schmidli (2008)

 part of the Terraininduced Rotor Experiment

 Grid:

 across valley (xdirection): 120 km with spacing of 1 km

 along valley (ydirection): 400 m with a spacing of 1 km

 vertical direction: 6.2 km with spacing of 20 m newr the ground up to 200 m above 2 km

 lateral boundary conditions are periodic

 as top boundary conditions a Rayleigh sponge was specified

 Coriolis force is turned off for all simulations

 models were integrated for 12 hours from sunrise (0600 local time(LT)) to sunset (1800LT)

 Valley:

 \( z=h(x,y)=h_{p}h_{x}(x)h_{y}(y) \)
where
\(h_x(x) = \left \{ \begin{array}{ll} 0 & x \le V_x \\ \frac{1}{2}\frac{1}{2} \cos \left ( \pi \frac{x – V_x}{S_x} \right ) & V_x < x < X_2 \\ 1 & X_2 \le x \le X_3 \\ \frac{1}{2} + \frac{1}{2} \cos \left ( \pi \frac{x – X_3}{S_x} \right ) & X_3 < x < X_4 \\ 0 & x \ge X_4 \end{array} \right .\)
and
\(h_y(y) = \left \{ \begin{array}{ll} 1 &  y  \le P_y \\ \frac{1}{2} + \frac{1}{2} \cos \left ( \pi \frac{ y  – P_y}{S_y} \right ) & P_y <  y  < Y_2 \\ 0 &  y  \ge Y_2 \end{array} \right .\)
with

 valley depth \( h_{p}=1.5\,km \)

 valley floor half width \( V_{x}=0.5\,km \)

 sloping sidewall width \( S_{x}=9\,km \)

 plateau hald width in crossvalley direction \( P_{x}=1\,km \)

 plateau half width in alongvalley direction \( P_{y}=100\,km \)
 \( S_{y}=9\,km \)
 \( X_{2}= V_{x} + S_{x} \)
 \( X_{3}= V_{x} + S_{x} + P_{x} \)
 \( X_{4}= V_{x} + 2S_{x} + P_{x} \)
 \( Y_{2}= P_{y} + S_{y} \)
Profile:

 all simulations are started from an atmosphere in rest

 potential temperature \( \theta (z)= \theta _{s}+\Gamma z + \Delta \theta [1 exp\,(\beta z)]\)

 surface pressure: \( p_{s}=1000\,hPa\)

 constant relative humidity of 40 %

 constant stratification of \(N \approx 0.011s^{1} \)

 sensible heat flux for the uncoupled simulations: \( Q(t)=Q_{0}\,sin(\omega \,t) \) with \( Q_{0} = 200\,Wm^{2} , \omega = \frac{2\pi}{24\,h}\) and the time t donates hours since sunrise.
Valley wind systems II
 sensible heat flux for the uncoupled simulations: \( Q(t)=Q_{0}\,sin(\omega \,t) \) with \( Q_{0} = 200\,Wm^{2} , \omega = \frac{2\pi}{24\,h}\) and the time t donates hours since sunrise.

 Examples after Schmidli (2008)

 Construction of a testaxample for the uncoupled simulations

 First step: construction of the test valley after Schmidli (2008)
 Grid:

 across valley (xdirection): 120 km with spacing of 1 km

 along valley (ydirection): 400 km with spacing of 1 km

 vertical direction: 10 km with spacng of 100 m
 Profile:

 all simulations are started from an atmosphere in rest

 potential temperature: \(\theta (z) = \theta _{s} +\Gamma z + \Delta \theta [1 exp\,(\beta z)]\)

 with

 \(\theta _{s} = 280\,K, \Gamma = 3.2\,K\,km^{1}, \Delta \theta= 5\,K \)

 and \( \beta = 0.002m^{1}\)

 constant relative humidity of 40%

 testcase like the uncoupled simulations, but only with heating process
 Results:

 maximum of the along valley wind speed: \(v = 5.77ms^{1}\)

 vertical wind speeds across the valley: \(w = 0.5ms^{1} \) and \( w = 0.15ms^{1}\) downward.