Department of Meteorology & ClimatologyDepartment of Meteorology & ClimatologyFMFI UK BratislavaFMFI UK Bratislava

Davies CouplingDavies Couplingin a in a ShallowShallow--WaterWater ModelModel

MatMatúš Múš MARTARTÍNIÍNI

�� ArakawaArakawa, A., 1984. Boundary conditions in limited, A., 1984. Boundary conditions in limited--area model. Dep. of area model. Dep. of Atmospheric Sciences. University of California, Los Angeles: 28pAtmospheric Sciences. University of California, Los Angeles: 28pp.p.

�� DaviesDavies, H.C., 1976. A lateral boundary formulation for multi, H.C., 1976. A lateral boundary formulation for multi--level level prediction models. Q. J. Roy. Met. Soc., Vol. 102, 405prediction models. Q. J. Roy. Met. Soc., Vol. 102, 405--418.418.

�� McDonaldMcDonald, A., 1997. Lateral boundary conditions for operational , A., 1997. Lateral boundary conditions for operational regional forecast models; a review. Irish Meteorogical Service, regional forecast models; a review. Irish Meteorogical Service, Dublin: Dublin: 25 pp.25 pp.

�� MesingerMesinger, F., Arakawa A., 1976. Numerical Methods Used in , F., Arakawa A., 1976. Numerical Methods Used in Atmospherical Models. Vol. 1, WMO/ICSU Joint Organizing Atmospherical Models. Vol. 1, WMO/ICSU Joint Organizing Committee, GARP Publication Series No. 17, 53Committee, GARP Publication Series No. 17, 53--54.54.

�� PhillipsPhillips, N. A., 1990. Dispersion processes in large, N. A., 1990. Dispersion processes in large--scale weather scale weather prediction. WMO prediction. WMO -- No. 700, Sixth IMO Lecture: 1No. 700, Sixth IMO Lecture: 1--23.23.

�� TermoniaTermonia, P., 2002. The specific LAM coupling problem seen as a , P., 2002. The specific LAM coupling problem seen as a filter. Kransjka Gora: 25 pp.filter. Kransjka Gora: 25 pp.

MotivationMotivation

�� global model with variable resolutionglobal model with variable resolutionARPEGE ARPEGE 22 22 –– 270 km270 km

�� low resolution driving model low resolution driving model with nested high resolution LAMwith nested high resolution LAMDWD/GME DWD/LM 60 km 7 kmDWD/GME DWD/LM 60 km 7 km

�� combination of both methodscombination of both methodsARPEGE ALADIN/LACE ALADIN/SLOKARPEGE ALADIN/LACE ALADIN/SLOK

25 km25 km 12 km 12 km 7 km7 km

High resolution NWP techniques:High resolution NWP techniques:

WHY WHY NESTEDNESTED MODELSMODELS IMPROVE IMPROVE WEATHER WEATHER -- FORECASTFORECAST

�� the surface is more accurately characterized the surface is more accurately characterized (orography, roughness, type of soil, (orography, roughness, type of soil, vegetation, vegetation, albedoalbedo …)…)

�� more realistic parametrizations might be more realistic parametrizations might be used, eventually some of the physical used, eventually some of the physical processes can be fully resolved in LAMprocesses can be fully resolved in LAM

�� own assimilation system own assimilation system �� better initial better initial conditions (early phases of integration)conditions (early phases of integration)

ShallowShallow--water water equationsequations

xu

Hth

,xh

gtu

∂∂−=

∂∂

∂∂−=

∂∂

•• 1D system (Coriolis acceleration not considered)1D system (Coriolis acceleration not considered)•• linearization around resting backgroundlinearization around resting background

const)H(g, =

•• forwardforward--backward schemebackward scheme•• centered finite differencescentered finite differences

DISCRETIZATION}

Davies relaxation schemeDavies relaxation scheme

),hK(x)(h DM−−

continuous formulationcontinuous formulation inin shallowshallowshallow---water systemwater systemwater system

xu

Hth

xh

gtu

∂∂−=

∂∂

∂∂−=

∂∂

discretediscrete formulationformulation -- general formalism:general formalism:

DMXX)(1X

LAM

++ ⋅+⋅−=+ ���

���=

h

uX

),uK(x)(u DM−−0K(x) >

PROPERTIES OF PROPERTIES OF DAVIES RELAXATION SCHEMEDAVIES RELAXATION SCHEME

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

Input of the wave from the driving modelInput of the wave from the driving model

u

j

Difference betweenDifference betweennumerical and analytical solutionnumerical and analytical solution

(no relaxation)(no relaxation) 88--point relaxation zonepoint relaxation zone

-0.1

-0.05

0

0.05

0.1

0 20 40 60 80

-0.1

-0.05

0

0.05

0.1

0 20 40 60 80

Outcome of the wave, which is Outcome of the wave, which is not represented in driving modelnot represented in driving model

-1

-0.5

0

0.5

1

0 20 40 60 80-1

-0.5

0

0.5

1

0 20 40 60 80

-1

-0.5

0

0.5

1

0 20 40 60 80

-1

-0.5

0

0.5

1

0 20 40 60 80-1

-0.5

0

0.5

1

0 20 40 60 80-1

-0.5

0

0.5

1

0 20 40 60 80

t15t ∆=0t = t20t ∆=

t25t ∆= t30t ∆= t50t ∆=

88--point relaxation zone point relaxation zone

-0.1

0.2

0.5

0.8

1.1

0 20 40 60 80

-0.1

0.2

0.5

0.8

1.1

0 20 40 60 80

-0.1

0.2

0.5

0.8

1.1

0 20 40 60 80

-0.1

0.2

0.5

0.8

1.1

0 20 40 60 80

88 72

8

8

8 72 72

72

t20t ∆= t20t ∆= t25t ∆=

t30t ∆= t40t ∆=

analytical solution

Minimalization of the reflectionMinimalization of the reflection

•• weight functionweight function•• width of the relaxation zonewidth of the relaxation zone

�� the velocity of the wave the velocity of the wave (4 different (4 different velocities satisfying CFL stability criterion)velocities satisfying CFL stability criterion)

(simulation of dispersive system)(simulation of dispersive system)�� wavewave--length length x}40x,20x,{10 ∆∆∆

Choosing the weight functionChoosing the weight function

20

25

30

35

40

3 4 5 6 7 8 9 10

15

20

25

30

35

40

3 4 5 6 7 8 9 10

r [%] r [%]

convex-concave (ALADIN)

cosine

quadratic

quarticquartic

tan hyperbolictan hyperbolic

linearlinear

number of points in relaxation zonenumber of points in relaxation zone

testing criterion testing criterion -- critical reflection coefficient critical reflection coefficient rr

(more accurate representation of surface) (more accurate representation of surface)

-0.5

-0.2

0.1

0.4

0.7

1

0 10 20 30 40-0.5

-0.2

0.1

0.4

0.7

1

0 10 20 30 40-0.5

-0.2

0.1

0.4

0.7

1

0 10 20 30 40

-0.5

-0.2

0.1

0.4

0.7

1

0 20 40-0.5

-0.2

0.1

0.4

0.7

1

0 20 40-0.5

-0.2

0.1

0.4

0.7

1

0 20 40

DM

8 8 8 323232

LAM

H0.81H =LAM DM

)c0.9(c =LAM DM

DMDM--driving modeldriving model LAMLAM--limited area modellimited area model