Test Cases

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Mountain waves

2D hydrostatic and nonhydrostatic mountain waves

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


Gravity waves over  the Schaer orography
Gravity waves over the Schaer orography
a Δx Δz
1 km 200 m 100 m
10 km 2 km 100 m
100 km 20 km 100 m


Schaer test case

  • Examples after Schaer (1990) .
  • Orography z(x)=h\exp{(-x^2/a^2)}\cos^2{(\pi x/\lambda)}\,, h=400m\,, \lambda=4000m\,,a=5000m \,.
  • Stratification N=0.01s^{-1}\,, U=10ms^{-1}\,.
  • Example was defined to promote a new type of boundary following coordinates.

 
 

Density Current

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 viscosity is used with \nu=75m^2/s\,.
    - A horizontally homogeneous environment with \bar{\theta}=300~K is used.

  • Perturbation:

    - The perturbation (cold bubble) is defined 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=0.0km\,, x_r=4.0km\,, z_c=3.0km\, and z_r=2.0km\,.

  • For the translating current the s\,-position is shifted to the left.

 




Examples of the sphere

    - Model height z_T\, = 10km
    - Twenty equidistant vertical layers
    - In longitude-latitude 128 × 64 grid points are used

  • integration time: Maximal time step is 1800s


Acoustic waves

initial profile, t=1h
initial profile, t=1h
t=4h
t=4h
t=8h
t=8h
  • Profile:

    - a=6371 \text{km}\,, R=a/3 \text{km}\,
    - Basic state is isothermal T_0=300\mbox{K}\,
    - \Delta p=100 \text{Pa}\,, amplitude of the pressure perturbation
    - (\lambda_0,\phi_0)=(0^\circ,0^\circ)\,
    - Without diffusion and rotation of the earth

  • Perturbation:

    - 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}(1+\cos(\pi r/R))& r<R, \\
    0 & r>R,
  \end{cases}
g(z)=\sin\left(\frac{n_v\pi z}{z_T}\right)

    - \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 347 ms^{-1} \, (\sim \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 and 2, initial vertical profile on the equator.
Case 1, vertical profile on the equator after 48 h.
Case 1, vertical profile on the equator after 48 h.
Case 2, 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\mbox{K}\,
    - \Delta\theta=10K\,
    - Basic state for the thermo-dynamic variables is a stratified state with a       constant Brunt Väisälä frequency N\,
    - Three test cases:
        - Case 1: N=0.01 \text{s}^{-1}\,, n_v=1\,
        - Case 1: N=0.02 \text{s}^{-1}\,, n_v=1\,
        - Case 1: N=0.01 \text{s}^{-1}\,, n_v=2\,

  • Perturbation:
Case 1 and 2, initial vertical profile on the equator.
Case 1 and 2, initial vertical profile on the equator.
Case 3, vertical profile on the equator after 48 h.
Case 3, vertical profile on the equator after 48 h.

    - 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 waves 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 estimated on the hydrostatic approximation as
c_g = \frac{Nz_T}{\pi n_v}\, (z_T\, is the top of the model domain, which is constant 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:

 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 half-width of the mountain, and r\, as the distance from the center
  • Profile:

    - Center of the mountain is placed at the equator at (\lambda_0,\phi_0)=(90^\circ W, 0^\circ)\, 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

   - A wind-, density- and temperature field is constructed fitting in the hydrostatic and gradient wind balance including a wind domain which rotates around the centre.
   - First step: Construction of a radial wind field
   - The radial profile of the azimuthal windfield   V(r)=\frac{1}{2r}\zeta_0b^2(1-exp\left(-\frac{r^2}{b^2}\right))   is computed from the radial integration of a Gaussian vorticity profile:
      \zeta(r)=\zeta_0exp\left[-\left(\frac{r}{b}\right)^2\right]=\frac{1}{r}\frac{\partial}{\partial r}[rV(r)]     with     \zeta_0=2.34*10^{-3} \text{s}^{-1}\,    and     b=53.5 \text{km}\,.

   - The vertical components of vorticity is defined by:

      \zeta(r)=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}     with     u=V(r)\sin\lambda=-V(r)\frac{y}{r}     and     v=V(r)\cos\lambda=V(r)\frac{x}{r}
       \zeta(r)=V\!\,'(r)+\frac{V(r)}{r}=\frac{1}{r}\frac{\partial}{\partial r}[rV(r)]

   - The windfield is extended into the vertical with: V(r,z)=V(r)exp\left[ -\frac{z^{\alpha}}{\alpha L_Z^{\alpha}}\right]
     where  L_Z=6 \text{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{\partial p(r,z)}{\partial z}=-\rho g   and gradient wind balance   
\frac{\partial p(r,z)}{\partial r}=\rho\left(\frac{V^2(r,z)}{r}+fV(r,z)\right)  as well as the equation of states.
   - The potential temperature profile outside of the vortex   \overline{\theta}(z)=\theta_0exp\left[ \frac{N^2}{g}z\right]  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.

   - Profile of potential temperature for the total grid:

Profile of potential temperature
Profile of potential temperature











   - 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 12500
14137 0.25 15000
13193 0.29 17500
12353 0.32 20000
10894 0.38 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 95000
141 1.16 100000
0 1.18 101630



   - Following radial geometry  \theta\left(r_i,z_k\right) may be transfered into a Cartesian coordinate system  \theta\left(x,y,z\right) by interpolation.

      1.)  \overline{r}=\sqrt{x^2+y^2}   

      2.)  \overline{i}=r_{\overline{i}}\leq \overline{r}\leq r_{\overline{i}+1}

      \overline{\theta}(r)=\frac{\left[\theta_{\overline{i}}\cdot(r_{\overline{i}+1}-\overline{r})+\theta_{\overline{i}+1}\cdot(\overline{r}-r_{\overline{i}})\right]}
    {r_{\overline{i}+1}-r_{\overline{i}}}

   - Next step is to add a potential perturbation to the potential temperature as followed:

      \theta_i(r,\lambda,z)=Acos(3\lambda)exp\left[-\left(\frac{(r-r_b)^2}{\sigma_r^2}+\frac{(z-z_b)^4}{\sigma_z^4}\right)\right]

     where r_b=50\text{km}\,, z_b=6\text{km}\,, \sigma_r=15\text{km}\,, \sigma_z=3\text{km}\,, and A=0.5\text{K}\,.

  • Configuration of model domain:

   - 400\times400\text{km} horizontal grid points with 3\text{-km}\, spacing
   - z=0 \text{to} 20\text{km}\,, spacing of 200\text{m}\,
   - necessity of periodic outer boundary conditions in both directions
   - necessity of constant timestep of 10\text{sec}\,

  • Profile:

   - v\le40\text{ms}^{-1}\,
   - Vertical gravity wave reflection from the top of the domain is supressed by a Rayleigh damping layer.
   - advective time step: 20\text{s}\,
   - internal diffusivity for momentum: 20 \text{m}^2 \text{s}^{-1}\,, and for temperature: 60 \text{m}^2 \text{s}^{-1}\,

  • Simulation with ASAM:

   - Horizontal cross section of vertical velocity (\text{ms}^{-1})\,  ,  z=8,2\text{km}\,  ,  1200\times1200\text{km}

Horizontal cross section of vertical velocity (ms − 1) at 300sec
Horizontal cross section of vertical velocity (ms − 1) at 300sec
Horizontal cross section of vertical velocity (ms − 1) at 0.5hours
Horizontal cross section of vertical velocity (ms − 1) at 0.5hours
Horizontal cross section of vertical velocity (ms − 1) at 1hours
Horizontal cross section of vertical velocity (ms − 1) at 1hours
Horizontal cross section of vertical velocity (ms − 1) at 1.5hours
Horizontal cross section of vertical velocity (ms − 1) at 1.5hours
Horizontal cross section of vertical velocity (ms − 1) at 2hours
Horizontal cross section of vertical velocity (ms − 1) at 2hours














   - Horizontal cross section of vertical velocity (\text{ms}^{-1})\, with/without potential perturbation at 2\text{hours}\,  ,  z=8,2\text{km}\,  ,  180\times180\text{km}

Horizontal cross section of vertical velocity (ms − 1) with the potential perturbation at 2hours
Horizontal cross section of vertical velocity (ms − 1) 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 (ms − 1) 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 km with spacing of 1 km
   - vertical direction: 6.2 km with spacing of 20 m near 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 (1800 LT)

  • Valley:

    z = h(x,y) = hphx(x)hy(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\, \text{km}
    - valley floor half width V_x=0.5\, \text{km}
    - sloping sidewall width S_x=9\,\text{km}
    - plateau half width in cross-valley direction P_x=1\,\text{km}
    - plateau half width in along-valley direction P_y=100\,\text{km}
    - S_y=9\,\text{km}
    - X2 = Vx + Sx
    - X3 = Vx + Sx + Px
    - X4 = Vx + 2Sx + Px
    - Y2 = Py + Sy

  • Profile:

   - all simulations are started from an atmosphere in rest
   - potential temperature: \theta(z)=\theta_s + \Gamma z + \Delta \theta \left [ 1 - \exp (-\beta z) \right ]
   - with \theta_s=280\,\text{K}, \Gamma =3.2\,\text{K km}^{-1}, \Delta \theta=5\,\text{K}, \beta = 0.002\,\text{m}^{-1}
   - suface pressure: p_s=1000\,\text{hPa}
   - constant relative humidity of 40 %
   - constant stratification of N \approx 0.011\,\text{s}^{-1}
   - sensible heat flux for the uncoupled simulations: Q(t) = Q0sin(ωt) with Q_0=200\,\text{Wm}^{-2}, \omega = \frac{2 \pi}{24\,\text{h}} and the time t donates hours since sunrise

Valley wind systems II

wind speed v along the valley
wind speed v along the valley
wind speed v across the valley
wind speed v across the valley
wind vektor  across the valley
wind vektor \vec{v} across the valley
wind speed w across the valley
wind speed w across the valley
  • Examples after Schmidli (2008)
  • Construction of a testexample for the uncoupled simulations


   - 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 spacing of 100 m

  • Second step: construction of the startprofiles after Schmidli (2008)
  • Profile:

   - all simulations are started from an atmosphere in rest
   - potential temperature: \theta(z)=\theta_s + \Gamma z + \Delta \theta \left [ 1 - \exp (-\beta z) \right ]
   - with \theta_s=280\,\text{K}, \Gamma =3.2\,\text{K km}^{-1}, \Delta \theta=5\,\text{K}, \beta = 0.002\,\text{m}^{-1}
   - constant relative humidity of 40 %
   - testcase like the uncoupled simulations, but only with heating process

  • Results:

   - maximum of the alongvalley wind speed: v = 5.77m/s
   - vertical wind speeds across the valley: w = 0.5m/s upward and w = 0.15m/s downward

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