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Stochastic subgrid tensor for geophysical flow modeling
Valentin Resseguier, Etienne Mémin, Bertrand Chapron
To cite this version:
Valentin Resseguier, Etienne Mémin, Bertrand Chapron. Stochastic subgrid tensor for geophysical flow modeling. 8th European Postgraduate Fluid Dynamics Conference, Jul 2016, Varsovie, Poland.
�hal-01377741�
STOCHASTIC SUBGRID TENSOR FOR GEOPHYSICAL MODELING
Valentin Resseguier, Etienne Mémin, Bertrand Chapron
Conservations
(mass, linear momentum, …)
Navier-Stokes
Boussinesq
SQG under Moderate Uncertainty
QG MU
Uncertainty
D Dt
Systematic derivation of random models with the
new
• Usual SQG relationship and transport under uncertainty of buoyancy
• Good illustration of our stochastic transport
• Simulations done with homogeneous small-scale velocity Assumption:
Velocity decomposition: with
From stochastic calculus theory, we proofed:
QG assumption under strong uncertainty:
• kills the interior dynamics
• creates horizontal velocity divergence
• provides a diagnostic relation
Transport under location uncertainty
Conclusion
• Quantification of the dynamics errors:
- Diagnostic for numerical simulations (mesh refinements, …)
- Ensemble data assimilation and forecasts
• Rigorous identification of sudgrid dynamics effects
• Taking into account likely small-scale dynamics (stochastic backscatter)
• Forecast of likely distinct scenarios
Motivations
Advection
Inhomogeneous and anisotropic
diffusion
Balanced
energy exchanges
D⇥
Dt = @
t⇥ + w
?· r ⇥ + B ˙ · r ⇥ r ·
✓ 1
2 a r ⇥
◆
= 0
Multiplicative random
forcing Drift
correction
v = w + B˙
8<
:
w B˙
large-scale velocity
uncorrelated in time, divergence-free, correlated in space, Gaussian,
inhomogeneous and anisotropic
y(m)
x(m)
t = 0 d ay s
0 2 4 6 8
x 105 0
5
10 x 105
−1
−0.5 0 0.5 x 101 −3
Initial condition
10−5 10−4
100 102 104
|ˆf(κ)|2
κ(rad.m−1) Initial spectrum ofwandσdBt
-5/3
x ( m)
y(m)
t = 17 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
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ!
r a d . m−1"
t = 17 d ay s
Reference: HR deterministic simulation (5122)
One realization: better small-scale representation
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 of simulations
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
x ( m)
y(m)
t = 17 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
10−5 10−4
10−6 10−4 10−2
|ˆ b(κ)|2
κ!
r a d . m−1"
t = 17 d ay s
LR stochastic simulation (1282)
x ( m)
y(m)
t = 17 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
10−5 10−4
10−6 10−4 10−2
|ˆb(κ)|2
κ!
r a d . m−1"
t = 17 d ay s
LR deterministic simulation (1282) Spectrum of
w|t=0 and B˙
x ( m)
y(m)
Me an e r r or
0 2 4 6 8
x 105 0
2 4 6 8 10x 105
0.5 1 1.5 2 x 10−4
x ( m)
y(m)
O u r e st i mat i on
0 2 4 6 8
x 105 0
2 4 6 8
x 105
0.5 1 1.5 2 x 10−4
x ( m)
y(m)
Ran d om I C
0 2 4 6 8
x 105 0
2 4 6 8
x 105
0.5 1 1.5 2 x 10−4
10−5 10−4
10−8 10−6 10−4
|ˆe(κ)|2
κ!
r a d . m−1"
S p e c t r u m of t h e e r r or s an d i t s e st i mat i on at t = 13 d ay s
Our error Our estimation
Error with random IC Estimation random IC
Errors estimation
x ( m)
y(m)
S t an d ar d d e v i at i on
0 5 10
x 105 0
2 4 6 8
x 105
x ( m)
y(m)
S k e w n e ss
0 5 10
x 105 0
2 4 6 8
x 105
x ( m)
y(m)
l og( K u r t osi s-3)
0 5 10
x 105 0
2 4 6 8
x 105 x ( m)
y(m)
Me an at t = 17 d ay s
0 5 10
x 105 0
2 4 6 8
x 105
−2
−1 0 1 2
−4
−2 0 2
−1
−0.5 0 0.5 x 101 −3
0 0.5 1 x 101.5 −4
Point-wise moments:
variability, extreme events, … Description
• Random transport applicable to any dynamics
• Better representation of small scales
• Extreme events
• Good errors estimation in location and amplitude
• Likely scenarios
code available online: link from Fluminance website - V. Resseguier
Description
r · u / r
?· u
Geostrophic balance
Horizontal Diffusion
f ⇥ u = 1
⇢b rp0 + a
2 u
Modified geostrophic balance
Testbed data:
very high-resolution model outputs
Gula, J., Molemaker, J., and McWilliams J.
"Gulf Stream dynamics along the southeastern US seaboard.”
Journal of Physical Oceanography 45.3 (2015): 690-715.
x(m)
y(m)
Te mp e r at u r e
4 5 6 7 8
x 105 0
1 2 3 4 5 6
x 105
15 20 25
x(m)
y(m)
Te mp e r at u r e
0 2 4 6 8
x 105 0
2 4 6 8 10
x 105
15 20 25
Original Filtered
Results
x(m)
y(m)
D i v e r ge n c e
4 6 8
x 105 0
2 4 6
x 105
−1 0 1 x 10−5
x(m)
y(m)
E st i mat i on
4 6 8
x 105 0
2 4 6
x 105
−2
−1 0 1 2
x 10−12
Conclusion
• Frontolysis / frontogenesis on warm / cold side of fronts
• Estimation of horizontal divergence or diffusive coefficient and subgrid variance
• Strong uncertainty assumption verified a posteriori for the Gulf Stream
during wintertime at mesoscales
SQG under Strong Uncertainty
w⇤ = w 1
2r · a a = T
