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Randomized fluid dynamics based on subgrid transport
Valentin Resseguier, Etienne Mémin, Bertrand Chapron
To cite this version:
Valentin Resseguier, Etienne Mémin, Bertrand Chapron. Randomized fluid dynamics based on subgrid transport. Workshop on Stochastic Weather Generators, May 2016, Vannes, France. �hal-01377747�
Randomized fluid
dynamics based on subgrid transport
Valentin Resseguier, Etienne Mémin, Bertrand Chapron
Motivations
• Rigorously identified sudgrid dynamics effects
• Injecting likely small-scale dynamics
• Predicting possible distinct scenarios
• Quantification of modeling errors:
Diagnose to design numerical simulations (mesh refinements, …) Data assimilation: ensemble forecasts
2
• Randomized dynamics
• SQG under Moderate Uncertainty
Contents
Randomized dynamics
4
Random equations
• Random initial conditions
• Arbitrary Gaussian forcing
• Averaging, homogenization
• Adding white random velocity
Underdispersive
Adding energy + wrong phase
Previous talk
v = w + B ˙
Advection of tracer
D ⇥
Dt = 0
Θ
Advection
Diffusion
Advection of tracer Θ
Drift
correction
Multiplicative random
forcing
Balanced energy exchanges
@
t⇥ + w
?· r ⇥ + B ˙ · r ⇥ = r ·
✓ 1
2 a r ⇥
◆
Drift correction
8
Drift correction
w? = w 1
2 (r · a)T
Uncertainty
Derived random models
Conservations (mass, linear momentum, …)
D Dt
Navier-Stokes
Boussinesq
QG MU
SQG MU
SQG SU
10
SQG under Moderate Uncertainty
SQG MU
Code available online
Reference flow:
deterministic SQG
512 x 512
12
One realization
x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s
One realization
Our model Larger noise
Lower noise
x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s
14
x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s
Ensemble
x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s
Ensemble
16
Conclusion
Conclusion
• Random transport applicable to any dynamics
• Better small scales
• Estimate position and amplitude of errors
• Extreme events
• Likely scenarios
• under Strong Uncertainty:
Simple 2D description of frontolysis/frontogenesis
18
Code SQG MU:
link from Fluminance website - V. Resseguier
Thank you for your attention
Likely SQG scenarios
tracked by SQG MU
20
pdf of the 1st PCA coefficient along time
20 30 40 50 60 70 80
Time (day) -4
-2 0 2 4
1st PCAcoefficient
×10-4
0 2000 4000 6000 8000 10000
x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s x ( m)
y(m)
t= 17 d ay s
0 2 4 6 8 10
x 105 0
2 4 6 8
x 105
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ! r a d . m−1"
t= 17 d ay s
Ensemble
y(m)
x(m) t = 0 d ay s
0 2 4 6 8
x 105 0
2 4 6 8
x 105
−1
−0.5 0 0.5 x 101 −3
y(m)
x(m)
t = 70 d ay s
0 5 10
x 105 0
2 4 6 8
x 105
−1
−0.5 0 0.5 x 101 −3
512x512
Chaotic
transition y(m
)
x(m)
t = 70 d ay s
0 2 4 6 8
x 105 0
2 4 6 8
x 105
−1
−0.5 0 0.5 x 101 −3
128x128
512x512
t=70 days
t=0
SQG under Strong Uncertainty
SQG SU
22
Mesoscale divergence
Horizontal Diffusion Geostrophic balance
f ⇥ u = 1
⇢
br p
0+ a
2 u
r · u / r ? · u
Filtering of model outputs:
Gula, Jonathan, M. Jeroen Molemaker, and James C. McWilliams
"Gulf Stream dynamics along the southeastern US seaboard.”
Journal of Physical Oceanography 45.3 (2015): 690-715.
24
Spatial test
10−4 10−2
10−1 100 101
|ˆf1(κ)|/|ˆf2(κ)|
κ!
r a d . m−1"
Me an sp e c t r u m r at i o 10−4
10−4 10−2 100 102
|ˆf(κ)|2
κ!
r a d . m−1"
Nor mal i z e d me an sp e c t r u m of t h e i r r ot at i on al v e l o c i ty an d of i t s e st i mat i on
10−4 0
0.5 1 1.5
θ
κ!
r a d . s−1"
Me an p h ase sh i f t
Spectral test
26
