Numerical simulations of inertia-gravity waves and hydrostatic mountain waves using EULAG model

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Numerical simulations of inertia-gravity waves
and hydrostatic mountain waves using EULAG model
Bogdan Rosa, Marcin Kurowski, Zbigniew Piotrowski
and Michal Ziemianski
COSMO General Meeting, 7-11 September 2009
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Outline

• Two dimensional 2D time dependent simulation of
  inertia-gravity waves (Skamarock and Klemp Mon.
  Wea. Rev. 1994) using three different approaches
• Linear numerical
• Incompressible Boussinesq
• Quasi-compressible Boussinesq
• 2D simulation of hydrostatic waves generated in
  stable air passing over mountain. (Bonaventura
  JCP. 2000)

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Two dimensional time dependent simulation of
inertia-gravity waves
Skamarock W. C. and Klemp J. B. Efficiency and
accuracy of Klemp-Wilhelmson time-splitting
technique. Mon. Wea. Rev. 122 2623-2630, 1994

• Setup overview
• domain size 300x10 km
• resolution 1x1km,
• 0.5x0.5 km, 0.25x0.25 km
• rigid free-slip b.c.
• periodic lateral boundaries
• constant horizontal flow
• 20m/s at inlet
• no subgrid mixing
• hydrostatic balance
• stable stratification N0.01 s-1
• max. temperature
• perturbation 0.01K
• Coriolis force included

Constant ambient flow within channel 300 km and
6000 km long
Initial potential temperature perturbation
outlet
inlet
km
km
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The Methods
Quassi-compressible Boussinesq
Incompressible Boussinesq
Linear
The terms responsible for the acoustic modes
Initial potential temperature perturbation
Initail velocity
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Time evolution of flow field potential
temperature and velocity (Incompressible
Boussinesq)
time
Time evolution of ? (contour values between
-0.0015K and 0.003K with a interval of 0.0005K)
and vertical velocity (contour values between
-0.0025m/s and 0.002m/s with a interval of
0.0005m/s). Grid resolution dx dz 1km.
Channel size is 300km 10km
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Continuation...
time
Time evolution of ? (contour values between
-0.0015K and 0.003K with a interval of 0.0005K)
and vertical velocity (contour values between
-0.0025m/s and 0.002m/s with a interval of
0.0005m/s). Grid resolution dx dz 1km.
Channel size is 300km 10km
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Convergence study for resolution
?' (after 50min)
Analytical solution based on linear
approximation (Skamarock and Klemp 1994)
dx dz 1km
Numerical solution from EULAG (incompressible Bou
ssinesq approach)
dx dz 0.5 km
dx dz 250 m
Contour values between -0.0015K and 0.003K with a
contour interval of 0.0005K
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Profiles of potential temperature along 5000m
height
Convergence to analytical solution
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Time evolution of potential temperature in long
channel (6000 km)
time
time
Time evolution of ? (contour values between
-0.0015K and 0.003K with a interval of 0.0005K)
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Solution convergence (long channel)
Analytical solution based on linear
approximation (Skamarock and Klemp 1994)
dx 20 km dz 1km
Numerical solution from EULAG (inocompressible Bo
ussinesq approach)
dx 10 km dz 0.5 km
dx 5km dz 250 m
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Profiles of potential temperature along 5000m
height
Analytical Solution ?x 5 km ?z 0.25 km ?x
10 km ?z 0.5 km ?x 20 km ?z 1 km
Convergence to analytical solution
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Comparison of the results obtained from four
different approaches (dx dz 0.25 km - short
channel)
Linear analytical
Incompressible Boussinesq
Linear numerical
Compressible Boussinesq
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Comparison of the results obtained from four
different approaches (long channel)
Linear analytical
Incompressible Boussinesq
Linear numerical
Compressible Boussinesq
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Quantitative comparison
Differences between three numerical solutions
LIN - linear, IB - incompressible Boussinesq and
ELAS quassi-compressible Boussinesq
dx dz 1km
dx 1km dz 20km
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Quantitative comparison
Differences of ? between solutions obtained
using two different approaches incompressible
Boussinesq and quassi-compressible Boussinesq.
The contour interval is 0.00001K.
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Comparison with compressible model
Klemp and Wilhelmson (JAS, 1978) (Compressible)
EULAG (Incompressible Boussinesq)
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2D simulation of hydrostatic waves generated in
stable air passing over mountain.
Bonaventura L. A Semi-implicit Semi-Lagrangian
Scheme Using the Height Coordinate for a
Nonhydrostatic and Fully Elastic Model of
Atmospheric Flows JCP. 158, 186213, 2000

• Profile of the two-dimensional mountain defines
  the symmetrical Agnesi formula.

1000 km
inlet
25 km
outlet
1 m

• Initial horizontal velocity U 32 m/s
• Grid resolution ?x 3km, ?z 250 m
• Terrain following coordinates have been used
• Problem belongs to linear hydrostatic regime
• Profiles of vertical and horizontal sponge zones
  from Pinty et al. (MWR 1995)

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Horizontal and vertical component of velocity in
a linear hydrostatic stationary lee wave test
case.
EULAG (anelastic approximation)
Bonaventura (JCP. 2000) (fully elastic)
horizontal
horizontal
vertical
vertical
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Horizontal component of velocity - comparison of
numerical solution based on anelastic
approximation (solid line) with linear
analitical solution (dashed line) form Klemp and
Lilly (JAS. 1978)
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The vertical flux of horizontal momentum for
steady, inviscid mountain waves.
Bonaventura (JCP. 2000)
Pinty et al. (MWR. 1995) fully compressible
EULAG (2009) anelastic
t 11.11 h
0.97
0.97
t 11.11 h
H
The flux normalized by linear analitic solution
from (Klemp and Lilly JAS. 1978)
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Summary and conclusions

• Results computed using Eulag code converge to
  analitical solutions when grid resolutions
  increase.
• In considered problems we showed that anelastic
  approximation gives both qualitative and
  quantitative agrement with with fully
  compressible models.
• EULAG gives correct results even if
  computational grids have significant anisotrophy.