Semi-Lagrangian Approach to4D-Discrete Linear Equations
of Atmospheric Dynamicswith Arbitrary Stratification
and Orography
Rein Room , Marko Zirk
Tartu University, Estonia
[email protected], [email protected]
1
1 Introduction
A method for numerical solu-tion of non-hydrostatic linearequations of atmospheric dynam-ics for horizontally homogeneousbut otherwise arbitrary refer-ence state and arbitrary orog-raphy is introduced.
1
The developed solution is
4D-discrete (x,y,z,t),spectral,
semi-implicit,semi-Lagrangian(SISL) scheme
for both stationary and nonsta-tionary cases
2
Motivation
Originally this algorithm was de-veloped for testing and qualitycheck of nonhydrostatic adia-batic kernels of SISL-based NWPmodels (HIRLAM, in particu-lar)
3
Testing NH HIRLAM
260220
T
600
0
1000
800
400
200
4020
U
U, m/sT, Kp,hPa
Reference temperature and wind
K/km
K/km
γ=4.5
γ=8.0
0
200
400
600
800
10000 20 40 60 80 100 120 140
p, h
Pa
X, km
HIRLAM, Vz: D(Vz)=0.05m/s,ax=3km, h=100m, MLEV=100,dx=.55km,dt=30s,600 steps
260220
T
600
0
1000
800
400
200
4020
U
U, m/sT, Kp,hPa
Reference temperature and wind
K/km
K/km
γ=4.5
γ=8.0
0
200
400
600
800
10000 20 40 60 80 100 120 140
p, h
Pa
X, km
Vz: D(Vz)=0.05m/s; U,T-HIRLAM,h=100m,ax= 3km,dx=.55km,MLEV=200,dz=100m
4
The actual domain of applica-tion is much wider:
Investigation of specific details of oro-graphic flows for complex wind andtemperature stratification.
Investigation of non-stationary devel-opment and buoyant instability.
Investigation of the impact of discretiza-tion to the solution quality.
5
2 Model description
Initial continuous equations: Lin-ear, NH pressure-coordinate equa-tions with filtered internal sound-waves(Miller-Pearce-White model)
6
Linearization with respect toT (p) , U(p), ps(x, y)
7
Discretization:
3D staggering with constant hor-izontal grid-step ∆x = ∆y andvariable vertical step ∆pk
Two time level,semi-implicit,semi-Lagrangian time scheme
8
Solution in the form of discreteFourier series
3D (x,y,t) presentation of dis-crete solution:
Ψnijk =
∑
qrsΨs
qrkei(ηx
q i+ηyr j−ds
qrkn),
where
Ψ = {T ′, u, v, ω, ϕ}
9
Discrete spectral equations forspectral amplitudes arrivefrom which a
one-dimensional wave equation
follows for ω-velocity.
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The wave equation is solved forboundary conditions :
Free-slip condition on the sur-face
Radiative boundary conditionon the top
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A special feature of the waveequation:
As normal mode intrinsic fre-quency ν depends on height, thewave-equation coefficients are func-tions of both ν and ∆ν/∆p
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In stationary case height-dependentν presents an ordinary thing;
In nonstationary case it involvesa dispersion equation which isa nonlinear first order differen-tial equation with respect to ν
13
Special: Solution of wave equa-tion is designed as a cumulativeproduct of decrease factors
ωk =k∏
j=1cj, |cj| < 1,
which results in an effective nu-merical algorithm, where solu-tion ωk is alwas an exponent func-tion of a complex argument.
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SOLUTION EXAMPLES
I.
Stationary 2D orographic flowover 1D mountain ridge
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Vertical velocity waves
Homogeneous stratification, U = 10 m/s, T = 280 KMountain ridge: ax = 2 km, h = 200 m
0
5
10
15
20
25
30
0 50 100 150 200 250
Z,
km
X, km
Vz: u = 10 m/s, T = 265 K, h = 0.2 km, ax = 2 km, zlev = 300, xlev = 1024, ∆z = 0.1 km, ∆x = 0.4 km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
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Vertical velocity waves
Refraction and reflection on tropopause.U = 12 m/s, γ = 6.5 K/km;
0
12
UT
Z, km30
K/kmγ= 6.5
U, m/s105
T, K160 220 280
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180 200
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
17
Vertical velocity waves
Refraction and reflection on tropopause.U = 12 m/s, γ = 8.5 K/km;
0
12
UT
Z, km30
U, m/s105
T, K160 220 280
K/kmγ= 8.5
0
5
10
15
20
25
30
0 50 100 150 200 250
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
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Vertical velocity waves
Refraction and reflection on tropopause in the case oflinear wind shear in the troposphere.U = 12-15 m/s, γ = 6.5 K/km;
0
12
UT
Z, km30
K/kmγ= 6.5
U, m/sT, K160 220 280 6 12
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
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Vertical velocity waves
Refraction and reflection on tropopause in the case oflinear wind shear in troposphere.U = 12-24 m/s, γ = 6.5 K/km;
Z, km
0
12
30
UT
U, m/sT, K160 220 280 10 20
K/kmγ=6.5
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800 900
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
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Vertical velocity waves
Hyperbolic wind, U = 10-30 m/s, γ = 6.5 K/km
30
Z, km
0
12
30
UT
U, m/sT, K160 220 280 10 20
K/kmγ=6.5
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 800
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
21
Vertical velocity waves
Hyperbolic wind, U = 10-30 m/s, γ = 6.5 K/km;
30
Z, km
0
12
30
UT
U, m/sT, K160 220 280 10 20
K/kmγ=6.5
0
5
10
15
20
25
30
20 40 60 80 100 120 140 160 180 200
Z, km
X , km
Green - positive, red - negative velocity; ∆Vz = 0.1 m/s
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SOLUTION EXAMPLES
II.
Baroclinic instability oflong waves
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E-folding time of normal modes in case ofconstant wind shear dU/dz = 2 m/s/km
Z = 0.0: τ1= 28 h, ∆τ= 6h
1.00.1
.01 X
1.00.1
.01.001
Y
24
48
72
96
τ ,h
X = Hkx
Y = Hky
Z = Hkz=0
H = 10 km- scale height
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As previous Fig., except Z = 1 (Lz ∼ 10 km)
Z = 1.0: τ1= 64 h, ∆τ= 6h
1.00.1
.01 X1.0
0.1.01
Y
60
80
100
120
τ ,h
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3 CONCLUSIONS
Though the numerical scheme was initially developed for testpurpose, its actual application area is wider:
Investigation of specific details of orographic flows forcomplex wind and temperature stratification:
Impact of tropopause, discontinuity of the Brunt-Vaisala fre-quency, wind shear (including directional shear), boundarylayer
Investigation of non-stationary development of lineardisturbances, including buoyant instability study
Investigation of the impact of discretization to numer-ical solution quality:
Vertical discretization (variable ∆z)
Accessible time step size and numerical stability issues
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