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

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

Bogdan Rosa, Marcin Kurowski,Zbigniew Piotrowski and Michał Ziemiański

COSMO General Meeting, 7-11 September 2009


Outline

1. Two dimensional 2D time dependent simulation of inertia-gravity waves (Skamarock and Klemp Mon. Wea. Rev. 1994) using three different approaches

• Linear numericalLinear numerical• Incompressible BoussinesqIncompressible Boussinesq• Quasi-compressible BoussinesqQuasi-compressible Boussinesq

2.2. 2D simulation of hydrostatic waves generated in stable air passing 2D simulation of hydrostatic waves generated in stable air passing over mountain. over mountain. ((Bonaventura JCP. 2000) Bonaventura JCP. 2000)


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

Initial potential temperature perturbation

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 N=0.01 s-1

max. temperature perturbation 0.01K Coriolis force included

Constant ambient flow within channel 300 km and 6000 km long

km

km

outletinlet


The Methods

220/1

)/sin()0,,(

axx

Hztzx

c

zw

xu

t

zw

xu

z

w

x

uc

t

z

ww

x

wug

zt

w

fuz

vw

x

vu

t

v

fvz

uw

x

uu

xt

u

s

2

1

wNgx

Ut

z

w

x

u

x

wUg

zt

w

fux

vU

t

v

fvx

uU

xt

u

20

0

1

Quassi-compressible Boussinesq

zw

xu

t

z

w

x

u

z

ww

x

wug

zt

w

fuz

vw

x

vu

t

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fvz

uw

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uu

xt

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0

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Incompressible Boussinesq

Linear

Initial potential temperature perturbation

smtw

smtv

smtu

/0)0(

/0)0(

/20)0(

Initail velocity

The terms responsiblefor the acoustic modes


Time evolution of flow field potential temperature and velocity (Incompressible Boussinesq)

tim

e' w

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


Continuation...

'ti

me

w

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


Convergence study for resolution

Analytical solution based onlinear approximation(Skamarock and Klemp 1994)

dx = dz = 1km

dx = dz = 0.5 km

dx = dz = 250 m

θ' (after 50min)

Numerical solution from EULAG(incompressibleBoussinesq approach)

Contour values between −0.0015K and 0.003K with a contour interval of 0.0005K


Profiles of potential temperature along 5000m height

Convergence toanalytical solution

'


Time evolution of potential temperature in long channel (6000 km)

'ti

me

tim

e

'

Time evolution of ’ (contour values between −0.0015K and 0.003K with a interval of 0.0005K)


Solution convergence (long channel)

Analytical solution based onlinear approximation(Skamarock and Klemp 1994)

dx = 20 km dz = 1km

dx = 10 kmdz = 0.5 km

dx = 5kmdz = 250 m

Numerical solution from EULAG(inocompressibleBoussinesq approach)

20dz

dx


Profiles of potential temperature along 5000m height

Convergence toanalytical solution

Analytical SolutionΔx = 5 km Δz = 0.25 kmΔx = 10 km Δz = 0.5 kmΔx = 20 km Δz = 1 km


Comparison of the results obtained from four different approaches (dx = dz = 0.25 km - short channel)

Linear analytical


Compressible Boussinesq

Linear numerical


Comparison of the results obtained from four different approaches (long channel)

Linear analytical


Compressible Boussinesq

Linear numerical


Quantitative comparison

Differences between three numerical solutions: LIN - linear, IB - incompressible Boussinesq and ELAS quassi-compressible Boussinesq

dx = dz = 1km

dx = 1kmdz = 20km


Quantitative comparison

Differences of ’ between solutions obtained using two different approaches incompressible Boussinesq and quassi-compressible Boussinesq. The contour interval is 0.00001K.


Comparison with compressible model

EULAG (Incompressible Boussinesq) Klemp and Wilhelmson (JAS, 1978)(Compressible)


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, 186–213, 2000

1000 km

25 k

m outletinlet

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)

Lxaxx

hxh

0,

/1)( 2

0

0

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

1/ UaN

kma 16


Horizontal and vertical component of velocity in a linear hydrostatic stationary lee wave test case.

horizontal

vertical

EULAG (anelastic approximation) Bonaventura (JCP. 2000) (fully elastic)

horizontal

vertical


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)

222

2/0

2/1

sin2/1cos2/1),(

xRC

xRCRCxeNhxu

p

ppRCp

In linear hydrostatic regime analytical solution has form

where )/ln(, 0 uN

g

0 is surface level potential temperature


The vertical flux of horizontal momentum for steady, inviscid mountain waves.

EULAG (2009)anelastic

The flux normalized by linear analitic solution from (Klemp and Lilly JAS. 1978)

2002

2

200 )4/(

21

)4/( huN

R

C

ghM

p

analitic

Bonaventura (JCP. 2000)

0.97 0.97

Pinty et al. (MWR. 1995)fully compressible t =11.11 [h]

t =11.11 [h]

H


Summary and conclusions Results computed using Eulag code converge to Results computed using Eulag code converge to

analitical solutions when grid resolutions increase.analitical solutions when grid resolutions increase. In considered problems we showed that anelastic In considered problems we showed that anelastic

approximation gives both qualitative and approximation gives both qualitative and quantitative agrement with with fully compressible quantitative agrement with with fully compressible models.models.

EULAG gives correct results even if EULAG gives correct results even if computational grids have significant anisotrophy.computational grids have significant anisotrophy.

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

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