An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
An energy-conserving quasi-hydrostatic deep-atmosphere dynamical
core
Marine Tort1 , Thomas Dubos1 and Thomas Melvin2
[email protected]
www.lmd.jussieu.fr/∼mtlmd
Monday, April 7 2014
PDEs on the Sphere, NCAR, Co
1 Laboratoire
de Météorologie Dynamique, Ecole Polytechnique, France
2 Met Office, Exeter, UK
1
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Motivations : LMD-Z and Planets
Planetary atmosphere version
e.g. Mars
Atmospheric component of the
IPSL model
Common dynamical core on a longitude/latitude grid :
shallow-atmosphere hydrostatic equations (HPE),
enstrophy-conserving scheme (Sadourny, 1975a).
2
Summary
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Motivations : LMD-Z and Planets
Planetary atmosphere version
e.g. Giant gas planets, Titan
→ DEEP ATMOSPHERES
Atmospheric component of the
IPSL model
The dynamical core requires a deep-atmosphere version.
GOALS
solve the deep-atmosphere quasi-hydrostatic equations (QHE) (White and Bromley, 1995 )
AND the recently derived non-traditional shallow-atmosphere equations (NTE) with
complete Coriolis force (Tort and Dubos, 2014a),
preserve some discrete conservation properties.
3
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
An energy-conserving scheme (Lagrangian vertical coordinate case)
Curl-form and Hamiltonian formulation
How to conserve energy ?
Iterative procedure to solve hydrostasy
Application to LMD-Z
Mass coordinate : how is that different from a Lagrangian vertical coordinate ?
Idealized test cases
Summary
4
Summary
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
An energy-conserving scheme (Lagrangian vertical coordinate case)
Curl-form and Hamiltonian formulation
How to conserve energy ?
Iterative procedure to solve hydrostasy
Application to LMD-Z
Mass coordinate : how is that different from a Lagrangian vertical coordinate ?
Idealized test cases
Summary
5
Summary
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Curl-form and Hamiltonian formulation
Derivation in curl-form using a time-dependent curvilinear coordinates system
from White and Bromley (1995)’s equations (spherical coordinates and advective form),
using a general vertical coordinate η : r (λ, φ, η; t).
Momentum prognostic variables ũ = (ũ, ṽ ) (Dubos and Tort, 2014 )
shallow-atmosphere ũ = (r0 cos φ(u + Ωr0 cos φ), r0 v ),
deep-atmosphere ũ = (r cos φ(u + Ωr cos φ), rv ),
Momentum
∂t ũ +
1
(∇ × ũ) × U + ∇ (K + Φ) + θ∇π = 0
ρ̃
Mass
ρ̃ = ρr 2 cos φ∂η r ,
∂t ρ̃ + ∇ · U = 0
Entropy
Θ = ρ̃θ,
∂t Θ + ∇ · (θU) = 0
6
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Curl-form and Hamiltonian formulation
Variational interpretation
cos φ∂η r̃ 3 Θ
dλdφdη ρ̃ K (ũ, r ) + Φ(r ) + e
,
3ρ̃
ρ̃
V
equations are written in term of functional derivatives of H
Z
Hamiltonian H(ρ̃, ũ, Θ, r ) =
Functional derivatives
δH
δH
δH
=K +Φ
B
=U
B
=π
δ ρ̃ δũ δΘ
2
2
u +v
δH
= −ρ̃
+ 2Ω cos φu + r 2 cos φ∂η p + ρ̃g (r )
B
δr
r
B
HYDROSTATIC CONSTRAINT :
δH
=0
δr
7
Summary
ṽ, V
An energy-conserving scheme (Lagrangian vertical coordinate case)
ũ, U
Application to LMD-Z
Summary
q
ṽ, V
˜ ⇡, p, ✓, K,
⇢,
How to
conserve energy ?
q
r, M, W, ↵,
˜ ⇡, p, ✓,ũ,K,
⇢,
U
i
Imitate the exact Hamiltonian
formulation at discrete level
ũ, U
ũ, U
Salmon, 2004
ṽ, V
r, M, W,ṽ,↵,V
i + 1/2
ṽ, V
1. Choose the spatial gridj
ũ, U
q
j + 1/2
˜ ⇡, p, ✓, K,
⇢,
ṽ, V
k
ũ, U q
ṽ, V
˜ṽ,⇡,
⇢,
V p, ✓, K,
i, j, k k
q
r,qM, W, ↵,
i + 1/2
j + 1/2
r, M, W, ↵,r, M, W,i +
↵,1/2
vertical :j +Lorenz
grid.
1/2
j W, ↵,
r, M,
k
˜ ⇡, p, ✓, K,
⇢,
ṽ, V
r, M, W, ↵,
q
i
˜ ⇡, p, ✓, K,
⇢,
ki +
+ 1/2
1/2
i + 1/2
r, M, W, ↵,
1
j
j
˜ ⇡, p, ✓, K,
⇢,
˜ ⇡, p, ✓, K,i
⇢,
i + 1/2
q
ũ, U
˜i +⇡,1/2
⇢,
p, ✓, K,
k + 1/2
r, M, W, ↵, j + 1/2
1
i
ũ, U
horizontal
i
i j: C-grid,
i q
j
FIG
1. Grid straggering
j + 1/2
k
k + 1/2
k + 1/2
k
1
k + 1/2
j
1
1
j + 1/2
i + 1/2
k
j
i
j + 1/2
i + 1/2
kj
k + 1/2
k + 1/2
1
1
8
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
How to conserve energy ?
2. Express the discrete energy budget, take care that the terms compensate.
dH
dt
X δH
δH
δH
δH
δH
∂t ρ̃ +
∂t Θ +
∂t ũ +
∂t ṽ +
∂t r ,
δ ρ̃
δΘ
δũ
δṽ
δr
X δH δH
δH
δH
i δH
j δH
δi
+ δj
+
δi θ
+ δj θ
−
δ ρ̃
δũ
δṽ
δΘ
δũ
δṽ


=
=
δH  δi ṽ − δi ũ δH
+
−
ij
δũ
δṽ
ρ̃

δH  δi ṽ − δi ũ δH
+

ij
δṽ
δũ
ρ̃
j
i
j
+ δi
δH
i δH 
+ θ δi

δ ρ̃
δΘ

i

δH
δH
j δH 

+ θ δj
∂t r  = 0 !
+ δj
−
δ ρ̃
δΘ
δr
3. Discretize the Hamiltonian and deduce the discrete derivatives

H=
X
ρ̃
1
ρ̃ 
2
ũ
r ik
− Ωr ik cos φ
2 i
1
+
2
9
ṽ
r jk
2 j
+e
cos φδk r 3 Θ
,
3ρ̃
ρ̃

+ Φ(r k )
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Iterative procedure to solve hydrostasy
HPE
k
δk p = −ρ̃ g
QHE
k
δk p = −ρ̃g (r k )
ũ 2
i
+ ρ̃
a 2 δk r =
ρ̃κθcp p κ−1
prκ cos φ
δk r 3 =
r ik
3
cos2 φ
!ik
2 ik
+Ω r
cos φ
+ ρ̃
j
ṽ 2
3
r jk
3ρ̃κθcp p κ−1
prκ cos φ
r is the solution of a non-linear elliptic problem whose p is a byproduct.
HPE : direct obvious solution,
QHE : iterative solution (fix point or Newton’s method).
10
!jk
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
An energy-conserving scheme (Lagrangian vertical coordinate case)
Curl-form and Hamiltonian formulation
How to conserve energy ?
Iterative procedure to solve hydrostasy
Application to LMD-Z
Mass coordinate : how is that different from a Lagrangian vertical coordinate ?
Idealized test cases
Summary
11
Summary
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Mass coordinate : how is that different from a Lagrangian vertical coordinate ?
Mass budget : ∂t ρ̃ + ∂λ U + ∂φ V + ∂η (ρ̃η̇)
ρ̃ is not a prognostic variable anymore and H is expressed with respect to integrated mass
Ms instead of local mass ρ̃,
η̇ 6= 0, there is additional non-zero vertical transport in the equations.
Vertical relabeling symmetry (Dubos and Tort, 2014 )
compensation of vertical transports terms due to vertical relabeling symmetry,
we may imitate to the discrete level this relabeling to cancel vertical transport in the
discrete energy budget (Tort et al, in preparation).
12
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Idealized test cases
Like-Earth planet experiment
Newtonian relaxation
Held and Suarez, 1994
Baroclinic instability
Ullrich et al, 2013
LMD−Z
LMD−Z
20km
90N
60N
10km
30N
0
0
ENDGame
ENDGame
20km
90N
60N
10km
30N
0
120W
920
930
940
60W
950
960
0
970
60E
980
990
0
120E
60S
−30
1000 1010 1020 1030
−20
30S
−10
0
0
30N
10
30N
20
30
FIG 5. Zonally averaged zonal velocity over
1000 planetary rotations
FIG 4. Surface pressure ps at day 10 llll
lmmmmmmmmmmmmmmmmmmm
13
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Idealized test cases
re
Small like-Earth rplanet
experiment
0=
X
⌦ = X⌦e r
e
r0 =
X
H
⌦ = X⌦e
Non-traditional regime
X ∼ U/(Ωe H) ⇒ X = 15
non−traditional formulation
r0 =
20km
H
re
X
non−traditional formulation
20km
⌦ = X⌦e
10km
10km
H
0
Deep-atmosphere regime
X U/(Ωe H) e.g X = 50
0
deep−atmosphere formulation
deep−atmosphere formulation
20km
20km
10km
10km
0
60S
−20
−15
30S
−10
−5
0
0
5
30N
10
15
0
30N
20
25
30
1
FIG 6. Zonally averaged zonal velocity over
1000 planetary rotations X = 15
60S
−20
1
30S
−10
0
30N
30N
0
10
20
FIG 7. Zonally averaged zonal velocity over
1000 planetary rotations X = 50
development of a zonally averaged easterly flow in the tropics scaled by U = −2X Ω0 H cos φ
(White and Bromley, 1995, Wedi and Smolarkiewicz, 2009 )
1
14
30
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Summary
Implementation of the QHE into LMD-Z
energy-conserving,
systematic method to discretize hydrostatic systems,
ongoing : newtonian relaxation on a Titan-like planet.
Tort M. and Dubos T., 2014a Dynamically consistent shallow-atmosphere equations with a
complete Coriolis force. Q. J. R. Meteor. Soc. (in press, early view)
Tort M. and Dubos T., 2014b. Usual approximations to the equations of atmospheric motion : a
variational perspective. J. Atmos. Sci. (accepted)
Tort M. et al, 2014. Consistent shallow-water equations on the rotating sphere with complete
Coriolis force and topography. J. Fluid Mech. (accepted)
Dubos T. and Tort M., 2014. Equations of atmospheric motion in non-Eulerian vertical
coordinates : vector-invariant form and Hamiltonian formulation. Mont. Weath. Rev.
(submitted)
15
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Consequence of vertical relabeling
Energy budget induced by vertical flux (mass-based coordinate) W = ρ̃η̇
dH
dt
=
=
X δH
δH
δH
δH
δH
∂t ρ̃ +
∂t Θ +
∂t ũ +
∂t ṽ +
∂t r ,
δ ρ̃
δΘ
δũ
δṽ
δr
k X δH δH
δH
δH
i δH
j δH
−
δi
+ δj
+ δk W +
δi θ
+ δj θ
+ δk θ W
δ ρ̃
δũ
δṽ
δΘ
δũ
δṽ


i
+
k
δH  W δk ũ
δi ṽ − δi ũ δH
−

i
ij
δũ
δṽ
ρ̃
ρ̃

j
δH  W δk ṽ
+

j
δṽ
ρ̃
k
δi ṽ − δi ũ δH
+
ij
δũ
ρ̃
i
j
+ δi
j
δH
i δH 
+ θ δi

δ ρ̃
δΘ

i
How to cancel the blue terms ? Bernoulli function B =
k
i
k
k
δH
is such as :
δ ρ̃
j
δH
1 δH
1 δH
δk B + θ δ k
− i
δk ũ − j
δk ṽ
δΘ
δũ
ρ̃
ρ̃ δṽ
X
δk βB
k
16

δH
δH
j δH 

+ δj
+ θ δj
∂t r  = 0 ?
−
δ ρ̃
δΘ
δr
=
0
=
X
k
δk β(K + Φ)
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Consequence of vertical relabeling
Tort and Dubos, 2014a
Du
−
Dt
2z
u
1
∂p
2Ω 1 +
+
v sin φ + 2Ω cos φw +
=0
r0
r0 cos φ
ρr0 cos φ ∂λ
Dv
2z
u
1 ∂p
+ 2Ω 1 +
+
u sin φ +
=0
Dt
r0
r0 cos φ
ρr0 ∂φ
Dw
1 ∂p
δNH
+ 2Ω cos φu + g +
=0
Dt
ρ ∂z
Non-traditional shallow-atmosphere angular momentum is now conserved
a cos φ (u + Ωr0 cos φ (r0 + 2z))
17
Summary
An energy-conserving scheme (Lagrangian vertical coordinate case)
Application to LMD-Z
Summary
Stability analysis of the vertical discretization
Isothermal atmosphere at rest on the f − F -plane - free pressure surface BC
extension of Thuburn et al, 2002b’s work with a rigid lid BC.
−10
20
x 10
Mass−based coordinate
Lagrangian coordinate
15
−4
x 10
3
numerical
analytical
Re(σ)=f
Im( )
10
Re(σ)
5
0
2
−5
0
1
2
Re( )
FIG 3. Numerical eigenvalues with a
Lagrangian vs mass-based coordinate
1
0
2
4
6
8
10
m
12
14
16
18
3
−4
x 10
20
Lagrangian coordinate 1/σ → ∞,
FIG 2. Analytical vs numerical frequency
spectrum
Mass-based vertical coordinate
1/σ → 15 years.
18