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

Gravity waves over a Witch of agnesi with length of 10km.

Examples after Gallus and Klemp (2000)

Orography $z(x)=\frac{H}{(1+(\frac{x}{a^{2}})}\,$, $h =400\,m\,$,

a variable.

Stratification $N = 0.01s^{-1}\,$, $U = 10ms^{-1}$.

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

Gravity waves over the Schaer orography.

Examples after Schaer (1999).

Orography:
$z(x)= h\,exp(\frac{-x^{2}}{a^{2}})cos^{2} (\frac{\pi\,x}{\lambda}) $,
$h\,=\,400\,m\,$,
$\lambda = 4000\,m\,$,
$ a = 5000\,m $.

Stratification $N = 0.01s^{-1}\,$, $U = 10ms^{-1}$.

Example was defined to promote a new type of boundary following coordinates.

Density Durrent

The density current test case is documented in Straka et al. (1993).

Geometry:

Computational domain extends in horizontal direction from -25.6 to 25.6 km and in vertical direction from 0 to 6.4 km.

Integration time: t= 1800s

Profile:

A fixed physical viscosiry is used with $v=75\,m^{2}s^{-1}$.

A horizontally homogeneous environment with $ \bar{\theta} = 300\,K $ is used.

Pertubation:

The perturbation (cold bubble) is defines by

$
\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
$L=[(x-x_c)x_r^{-1}]^2+[(z-z_c)z_r^{-1}]^2)^{1/2}$
and $x_{c}=0.0\,km$, $x_{r}=4.0\,km$, $z_{c}=3.0\,km$, $z_{r}=2.0\,km$.

For the translating current the $S$-position is shifted to the left.

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

Examples taken from Tomita and Satoh (2004)

Geometry:

Model height $z_{T}= 10\,km$

Twenty equidistant vertical layers

In longitude-latitude 128 x 64 grid points are used

integration time: Maximal time step is 1800s

Acoustic waves

initial profile, t=1h

t=4h

t=8h

Profile:

$ a = 6371\,km\,$, $ R=\frac{a}{3\,km}$

Basic state is isothermal $T_{0}= 100\,Pa\,$ amplitude of the pressure perturbation

$ \Delta p = 100\, Pa\,$ amplitude of the pressure perturbation

$(\lambda_{0},\phi_{0}) = (0^{\circ},0^{\circ})$

Without diffusion and rotation of the earth

Pertubation:

A perturbation $p’$ is superimposed on the basic pressure

$p’ = \Delta p f(\lambda,\phi)g(z)$
$ f(\lambda , \phi) = \begin{cases} \frac{1}{2}cos(\pi \frac{r}{R}) & r < R \\
0 & r > R \end{cases}$
$g(z)= sin (\frac{n_{v}\,\phi\,z}{z_{T}})$

$\Delta p$ is the amplitude of the pressure perturbation

$R$ is a constant distance

$n_{v}$ stands for the vertical mode

$r$ is the distance along a great circle from $(\lambda_{0},\phi_{0})$ to $(\lambda,\phi)$ with

$ r = a\,cos^{-1}\,[sin\,\phi_{0}\,sin\,\phi\,+cos\,\phi_{0}\,cos\,\phi\,cos\,(\lambda-\lambda_{0})]$

$f(\lambda,\phi)$ and $g(z)$ are horizontal and vertical distribution functions

Pressure perturbation horizontally propagates as a concentric circle with the acoustic wave speed (theoretical value of acoustic wave speed in this case is $347ms^{-1}(\tilde\,\sqrt{\gamma\,R_{d}\,T_{0}})$

Perturbation field has a structure similar to the Lamb wave; its amplitude exponentially decays in the upward direction

Gravity waves

Case 1 and 2, initial vertical profile on the equator.

Case 1, vertical profile on the equator after 48 h.

Case 2, vertical profile on the equator after 48 h.

Profile:

Basic state is isothermal $T=300\,K$

$\Delta \theta = 10\,K$

basic state the thermodynamic variables is a stratified state with a constant Brunt-Väisälä frequency $N$

Three test cases:

Case 1: $N=0.01s^{-1}, n_{v}=1$

Case 2: $N=0.02s^{-1}, n_{v}=1$

Case 3: $N=0.01s^{-1}, n_{v}=2$

Case 1 and 2, initial vertical profile on the equator.

Case 3, vertical profile on the equator after 48h.

Perturbation:

A perturbation $ \theta ‘$ is superimposed on the basic potential temperature

$\theta’ = \Delta\theta f(\lambda,\phi)\,g(z)$

$\Delta \theta $ is the amplitude of the potential temperature perturbation

Gravity wavey propagate as a concentric circle similar to the acoustic wave experiment

Theoretically: phase speed of the gravity wave with vertical mode $n_{v}$ in the stratified layer with the constant $N$ is estimates on the hydrostatic approximation as

$c_{g}=\frac{N\,z_{T}}{\pi\,n_{v}}$

\(z_{T}[latex] it the top of the model domain, which is content in the entire globe

Mountain waves

Orography:

Previous: gravity waves generated by the initial potential temperature perturbation

Now: gravity waves generated by the orography

Terrain following coordinates

Shape of mountain is given as an ideal bell shape:
[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 halt-width of the mountain and $r$ as the distance from the center.

Profil:

Center of the mountain is placed at the equator at $(\lambda_{0},\phi_{09}) = (90^\circ\text{C},0^\circ\text{C})$ and half-width is set to 1250 km

To avoid the reflection of gravity waves at the top boundary, the Rayleigh damping for the velocity and the Newtonian cooling to the basic state for the temperature are imposed

Linear anelastic equations for atmospheric vortices

Examples after Hodyss and Nolan

Construction of a test example:

A wind-, density and temperature field is constructed fitting in the hydrostatic and gradient wind balance including a wind domain which rotates around the center.

First step: Construction of a radial wind field

The radial profile of the azimuthal windfield
$V(r)= \frac{1}{2r}\,\zeta_{0}\,b^{2}\,(1-exp(-\frac{r^{2}}{b^{2}}))$
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$.

The vertical components of vorticity is defiend by:
$\zeta(r)=\frac{\delta v}{\delta x}-\frac{\delta u}{\delta y}$
with $u= V(r) sin \lambda = -V(r) \frac{y}{r}$ and $u= V(r) cos \lambda = V(r) \frac{x}{r}$
-> $\zeta(r) = V’(r)+\frac{V(r)}{r} =\frac{1}{r}\frac{\delta}{\delta r}[rV(r)]$

The windfield is extended into the vertical with:
$V(r,z)=V(r)exp\, [-\frac{z^{a}}{\alpha\,L_{z}^{\alpha}}]$
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.

Second step: Evaluation of the thermodynamic variables: density pressure, temperature

This has to fit the hydrostatic $\frac{\delta p(r,z)}{\delta z}=-\rho\,g$ and gradient wind balance $\frac{\delta p(r,z)}{\delta r}= \rho\,(\frac{V^{2}(r,z)}{r}+f\,V(r,z))$ as well as the equation of states.

Profile of potential temperature for the total grid.

The potential temperature profile outside of the vortex
$\theta(z)=\theta_0 \,exp[\frac{N^2}{g}z]$ 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 z-direction for every grid cell from the edge of the grid to the centre.

A more realistic profile:

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

Following radial geometry $\theta (r_{i},z_{k})$ may be transferred into a Cartesian coordinate system $\theta (x,y,z)$ by interpolation.

$ \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}}}$

Next step is to add a potential perturbation to the potential temperature as followed:
$\theta_{i}(r,\lambda,z)= A\,cos\,(3\lambda)\,exp\,[-(\frac{(r-r_{b})^{2}}{\sigma _{r}^{2}})+\frac{(z-z_{b})^{2}}{\sigma _{z}^{4}}]$
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 3-km 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:

$ v \le 40\,ms^{-1}$

Vertical gravity wave reflection from the top of the domain is suppressed by a Rayleigh damping layer.

advective timestep: 20 sec

internal diffusivity for momentum: $20\,m^{2}s^{-1}$ and for temperture: $60\,m^{2}s{-1} $

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}$) at 300sec$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 300sec	$Horizontal cross section of vertical velocity ($ms ^{− 1}$) at 0.5hours$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 0.5hours
$Horizontal cross section of vertical velocity ($ms ^{− 1}$) at 1 hour$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 1 hour	$Horizontal cross section of vertical velocity ($ms ^{− 1}$) at 1.5 hour$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 1.5 hour	$Horizontal cross section of vertical velocity ($ms ^{− 1}$) at 2 hours$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 2 hours

Horizontal cross section of vertical velocity $(ms^{-1})$ with/without potential perturbation at 2 hours,$z=8.2\,km, 180×180\,km$ $Horizontal cross section of vertical velocity ($ms^{− 1})$ with the potential perturbation at 2hours$ Horizontal cross section of vertical velocity ([latex]ms^{− 1})[/latex] with the potential perturbation at 2hours $Horizontal cross section of vertical velocity ($ms^{− 1})$ without the potential perturbation at 2hours$ Horizontal cross section of vertical velocity ([latex]ms^{− 1})[/latex] without the potential perturbation at 2hours Valley wind systems
Examples after Schmidli (2008)
part of the Terrain-induced Rotor Experiment
Grid:
across valley (x-direction): 120 km with spacing of 1 km
along valley (y-direction): 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 cross-valley direction $ P_{x}=1\,km $
plateau half width in along-valley 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)]$

with $ \theta _{s}=289\,K, \Gamma= 3.2\,K\,km^{-1}, \Delta \theta = 5\,K, \beta= 0.002\,m^{-1}$

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 wind speed v across the valley wind speed v across the valley wind vektor across the valley wind speed w across the valley
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 (x-direction): 120 km with spacing of 1 km
along valley (y-direction): 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.

$Horizontal cross section of vertical velocity (\(ms ^{− 1}\)) at 300sec$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 300sec	$Horizontal cross section of vertical velocity (\(ms ^{− 1}\)) at 0.5hours$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 0.5hours
$Horizontal cross section of vertical velocity (\(ms ^{− 1}\)) at 1 hour$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 1 hour	$Horizontal cross section of vertical velocity (\(ms ^{− 1}\)) at 1.5 hour$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 1.5 hour	$Horizontal cross section of vertical velocity (\(ms ^{− 1}\)) at 2 hours$ Horizontal cross section of vertical velocity ([latex]ms ^{− 1}[/latex]) at 2 hours

a	\(\delta x\)	\(\delta z\)
1 km	200 m	100 m
10 km	2 km	100 m
100 km	20 km	100 m

Mountain Waves

Density Durrent

Examples of the sphere

Mountain waves

Linear anelastic equations for atmospheric vortices

Valley wind systems

Valley wind systems II