Stochastic fluid flow dynamics under location uncertainty E. M - - PowerPoint PPT Presentation

Stochastic fluid flow dynamics under location uncertainty

• E. M´

emin Fluminance

Introduction

Geophysical flow analysis Strong interest on the use of stochastic filters and ensemble methods for data assimilation and forecasting Particularly interesting to combine a partially known large scale evolution law with noisy data or to specify modeling uncertainties (model errors) Example: models with small scale physical forcing, turbulence model, numerical coarsening, ... coupled with fine resolution image data Difficulties Data and state variable evolution laws generally do not live at the same scales Example: in oceanography or meteorology, models at mesoscales and image data at submesoscales ⇒smoothing of the data and use

• f subgrid models

Introduction

Requirements Construct a large scale stochastic evolution model Enabling a clear interaction with finer scale time series of Eulerian data (such as images) An explicit (Eulerian) evolution law Goals Explore such an expression of Navier-Stokes equation with location uncertainties Extend this expression for simple Geophysical models and reduced

• rder models

Use such models for variational assimilation or ensemble filtering with image data

Location uncertainties

Principle Fluid particles displacement can be separated in two components: a smooth differentiable drift components w Uncertainty function uncorrelated in time but correlated in space σdBt Displacement: dX(x, t) = w(X(x, t), t)dt + σ(X(x, t), t)d ˆ Bt, with X(x, 0) = x, Eulerian description of the velocity fields: U(x, t) = w(x, t)dt + σ(x, t)dBt. U should be solution of Navier Stokes equation derived from Newton 2nd law ⇒ σdBt differentiable in space

Noise term

Brownian motion field avatar ˆ Bn

t (x) =

1 √n

n

• i=1

Bt(xi)ϕν(x − xi), Limiting process denoted in a formal way as: ˆ Bt(x)

△

= Bt ⋆ ϕν(x) =

• IRd Bt(x′)ϕν(x − x′)dx′,

Uncertainty: Diffusion tensor and White noise avatar on Ω σ(x, t)d ˆ Bt =

• Ω

σt(x, y)d ˆ Bt(y)dy. Covariance of the turbulent component Q(x, s, y, t) = dtδ(t − s)σϕν(x, t)σϕν(y, t),

△

= a(x, y, t)dtδ(t − s), Homogeneous diffusion provides homogeneous covariance tensor with spatially constant variance

Noise term

Kraichnan smooth model dBnζ

t (x) =

1 √n

• i

dBt(xi)ψγ

κ ⋆ f ζ(x − xi),

f ζ(x) = Cζxζ/2 0 < ζ < 2. Incompressible fluid dξζ

t = P ⋆ dBζ t .

Spectral correlation defined as

• Q(k)ij = |k|−ζ−d(δij − kikj

|k|2 )( ψγ

κ)2.

Quadratic variation process for a pass-band spectral cutoff ( 1 I[κγ](k)) d < ξζ

t (x), ξζ t (x) >ij = dtCζ

(2π)d d − 1 d 2πd/2 Γ( d

2 ) ζ−1(Lζ − ℓζ

D)δij

Stochastic Reynolds transport theorem

Volumetric rate of change Volumetric rate of change of a scalar process q(x, t) transported by a velocity field dXt = w(Xt, t)dt + σ(Xt, t)dBt d

• V(t)

q(x, t)dx =

• V(t)

dtq + (∇ · (qw) −

• i,j

1 2 ∂2 ∂xi∂xj (aijq)|∇·σ=0+ ∇ · σ2q)dt + ∇ · (qσdBt)dx, Example for the smooth Kraichnan model: d

• V(t)

q(x, t)dx =

• V(t)

[dtq + (∇ · (qw) − 1 2γ∆q)dt + ∇q

Tdξζ

t ]dx,

Stochastic Reynolds transport theorem

Mass conservation Mass conservation constraint on the transported volume: dtρ + ∇ · (ρw)dt = 1 2(

• i,j

∂2 ∂xi∂xj (aijρ)|∇·σ=0 + ∇ · σ2ρ)dt − ∇ · (ρσdBt). For a fluid with constant density, mass preservation implies ∇ · (σdBt) = 0, ∇ · w = 0, ∇ · (∇ · a) = 0

Stochastic Reynolds transport theorem

Isochoric flows and isoneutral uncertainty Mass conservation constraint: dtρ + ∇ρwdt − 1 2

• i,j

∂2 ∂xixj (ρaij)dt = ∇ρσdBt Kraichnan model ⇒ advection diffusion with multiplicative stochastic forcing dtρ + ∇ρ

Twdt − γ 1

2∆ρdt = ∇ρσdBt If the uncertainty σdBt lies on the isodensity surfaces: σij = δij − ∂xiρ(x)∂xjρ(y) ∇ρ2 δ(x − y).

Stochastic Reynolds transport theorem

Isochoric flows and isoneutral uncertainty Small slope assumption (

• (∂xρ)2+(∂yρ)2<< ∂zρ) ⇒ diffusion

tensor (and the quadratic variation) reads: a(x) =   1 αx(x) 1 αy(x) αx(x) αy(x) |α(x)|2   , α = −(∂xρ/∂zρ, ∂yρ/∂zρ, 0) ∇ · σ = 0 ⇒ α constant along the depth axis ∂zαx = ∂zαy = 0 and ∇· α = 0. Conserved scalar quantity q ⇒ deterministic diffusion along the density tangent plane: ∂q ∂t + ∇q

Tw = 1

2

• ij

∂xi(aij∂xjq) Gent-McWilliams ”Isoneutral” or ”Isopycnal” diffusion in large scales ocean dynamics simulations

Conservation of momentum

Conservation of momentum Newton second law d dt

• V

ρwdx = F, Considering stochastic conservation principle d

• V(t)

ρ(w(x, t)dt + σ(x, t)dBt)dx =

• V(t)

F(x, t)dx. highly irregular ⇒ interpreted in the sense of distribution

Conservation of momentum

Conservation of momentum For every h ∈ C ∞

0 (R+):

• h(t)
• V(t)

F(x, t)dxdt = −

• h′(t)
• V(t)

σ(x, t)dBtdxdt+

• h(t)d
• V(t)

ρw(t, x)dxdt. Since both side of this equation must have the same structure, the forces can be written as:

• h(t)
• V(t)

F(x, t)dt = −

• h′(t)
• V(t)

σ(t, x)dBtdx+

• h(t)
• V(t)

(f (t, x)dxdt + θ(t, x)dBt)dx. First terms of both equations identical and cancel out.

Conservation of momentum

Conservation of momentum We have: d

• V(t)

ρwidx =

• V(t)

(d(ρwi)t + (∇ · (ρwiw) −

• j,k

1 2 ∂2 ∂xj∂xk (ajkρwi)|∇·σ=0+ ∇ · σ2ρwi)dt + ∇ · (ρwiσdBt))dx, with aij(x, t) =

• k

σik

ν (x, t)σkj ν (x, t).

As for the forces: Body force and external forces G =

• V

ρ(gdt − 2Ω × U)dx, Surface forces S =

• ∂V

Σdtnds =

• V

−∇(pdt + d ˆ p) + µ(∆U + 1 3∇(∇ · U),

Navier Stokes equations

Stochastic Navier Stokes equations Incorporating (stochastic) mass preservation principle and the forces expression:                            ((∂w ∂t + w∇

Tw)ρ − 1

2

• i,j

aijρ ∂2w ∂xi∂xj −

• i,j

∂(aijρ) ∂xj |∇·σ=0 ∂w ∂xi )dt+ w∇

TρσdBt = (ρg − 2ρΩ × w − ∇p + µ(∆w + 1

3∇(∇ · w))dt− ∇dpt − 2ρΩ × (σdBt) + µ(∆(σdBt) + 1 3∇(∇ · (σdBt))), dtρ + (∇ · (ρw) − 1 2

• i,j

∂2 ∂xi∂xj (aijρ)|∇·σ=0 + ∇ · σ2ρ)dt = ∇ · (ρσdBt).

Navier Stokes equations under location uncertainty

Stochastic Navier Stokes equations Equating slow terms and highly oscillating terms:                                  (∂w ∂t + w∇

Tw)ρ − 1

2

• i,j

aijρ ∂2w ∂xi∂xj −

• i,j

∂(aijρ) ∂xj |∇·σ=0 ∂w ∂xi = ρg − 2ρΩ × w − ∇p + µ(∆w + 1 3∇(∇ · w)), ∇dpt = −w∇

TρσdBt − 2ρΩ × (σdBt) + µ(∆(σdBt)+

1 3∇(∇ · (σdBt))), dtρ + ∇ · (ρw) − 1 2

• i,j

∂2 ∂xi∂xj (aijρ)|∇·σ=0 + ∇ · σ2ρ)dt = ∇ · (ρσdBt),

Navier Stokes equations under location uncertainty

Stochastic Navier Stokes equations for the smooth Kraichnan model ∀x ∈ Ω , t ∈]0, T]        (∂w ∂t + w∇

Tw − γ 1

2∆w)ρ = ρg − 2ρΩ × w − ∇p + µ∆w ∇d ˆ pt = −ρ(w∇

T)dξt + 2ρΩ × dξt + µ∆dξt,

∇ · w = 0, with boundary and initial conditions:      w · n = 0 on ∂Ω, t ∈]0, T], dξt = 0 on ∂Ω, t ∈]0, T], w|t=0 = wo in Ω.

Navier Stokes equations under location uncertainty

Stochastic Navier Stokes equations for incompressible and divergence free general turbulent model                (∂w ∂t + w∇

Tw − 1

2

• i,j

∂2 ∂xixj (aijw))ρ = ρg − 2ρΩ × w − ∇p + µ∆w ∇d ˆ pt = −ρ(w∇

T)σd ˆ

Bt + 2ρΩ × σdBt + µ∆σdBt, ∇ · w = 0, ∇ · (∇ · a) = 0 with boundary and initial conditions:      w · n = 0 on ∂Ω, t ∈]0, T], σ = 0 on ∂Ω, t ∈]0, T], w|t=0 = wo in Ω.

Navier Stokes equations under location uncertainty

Stochastic Navier Stokes equations for divergence free turbulent model and varying density                            (∂w ∂t + w∇

Tw − 1

2

• i,j

∂2 ∂xixj (aijw))ρ = ρg − 2ρΩ × w − ∇p + µ∆w ∇d ˆ pt = −ρ(w∇

T)σd ˆ

Bt + 2ρΩ × σdBt + µ∆σdBt, ∇ · w = 0, ∇ · (∇ · a) = 0, dtρ + (∇ · (ρw) − 1 2

• i,j

∂2 ∂xi∂xj (aijρ))dt = ∇ · (ρσdBt),

Navier Stokes equations under location uncertainty

Subgrid model Energy dissipating

• Ω

w

T

i,j

∂2 ∂xi∂xj (aijw)dx = −

• Ω

∇w2

adx

Navier Stokes equations under location uncertainty

Link to Smagorinsky model Smagorinsky model ∇ · (CSS), S2 = 1

2

• ij(∂xiw j + ∂xjw i)2

Taking a = CSI ⇒

• ij ∂xi∂xjaijw = 2C

j ∂xjS∂xjw k + S∆w k + ∆Sw k

Complemented by 2C

j ∂xj(S)∂xkw j − ∆Sw k provides the

standard trace free Smagorinsky subgrid stress The complementary term may be rewritten as 2 ∂xk

• j

∂xj(S)w j

• (1)

− 2

• j

∂xj∂xk(S)w j − ∆Sw k

• (2)

(1) gradient term ⇒ compensated by a modified pressure Assuming (2) cancels we recover the Smagorinsky model ⇒ S very smooth (respect ∇ · ∇a = 0)

Navier Stokes equations under location uncertainty

Quadratic variation scaling In the condition of Kolmogorov-Ridcharson scaling uℓ ∼ ǫ1/3ℓ1/3 and τℓ ∼ ǫ−1/3ℓ2/3 ⇒ τL τℓ ∼ (L ℓ )2/3 A coarsening in time yields a space dilation. At scale L, variance tensor: aL = E(uL − EuL)2τL ∼ ǫ1/3L4/3 Assuming uncertainties are uncorrelated over the finest grid points

• f a cube of size L, the variance is :

aL = E

i∈V (L) σℓ(xi)dBt2 = ( L ℓ )3aℓ.

⇒ aL ∼ (L ℓ )3( ℓ L)4/3E(uL − EuL)2τL, ∼ (L ℓ )5/3E(uL − EuL)2τL.

Results

Simulation Navier-Stokes drift (∂w ∂t + w∇

Tw − 1

2

• i,j

∂2 ∂xixj (aijw))ρ = −∇p + µ∆w ∇ · w = 0, ∇ · (∇ · a) = 0, periodic boundary conditions Eliminating the pressure with Leray projector P computed on a divergence free wavelet basis [S. Kadri V. Perrier 2012] ∂w ∂t − ν∆w = P[1 2

• i,j

∂2 ∂xi∂xj (aijw) − w∇

Tw],

Implicit Euler scheme expressed on w(t, x) = dj,k(t) Ψdiv

j,k (x)

(I − νδt∆)wn+1 = wn − δtP[1 2

• i,j

∂2 ∂xi∂xj (aijwn) − wn∇

Twn].

Variance tensor aij(x) fixed from local covariance in a local neighborhood

1 |ν|

• y∈ν(x)(w(y) − w(x))(w(y) − w(x))Tdt

Results Green-Taylor (Re 1600)

Green-Taylor vortex initial configuration isovalue subsequent time

Results Green-Taylor (Re 1600)

2 4 6 8 10 12 14 16 0.02 0.04 0.06 0.08 0.1 0.12 0.14 t TKE Taylor−Green Vortex Re=1600 Wavelet 2563 Spatial 643 Temporal 643 DSMG 643 2 4 6 8 10 12 14 16 0.002 0.004 0.006 0.008 0.01 0.012 0.014 t Dissipation rate Taylor−Green Vortex Re=1600 Wavelet 2563 Spatial 643 Temporal 643 DSMG 643

Evolutions of the dimensionless energy dissipation rate and energy as a function of the dimensionless time.

Shallow water equations under location uncertainty

Derivation of Stochastic Shallow Water Derivation of Shallow-Water with location uncertainty principle Follows the usual derivation Pressure expressed as a decomposition p(x, y, z, t) = p′(x, y, z, t) + dp(x, y, z, t), assume to follow an hydrostatic relation ∂p′ ∂z = −gρo. together with: ∂zdpt = 0. ⇒ The turbulent pressure term and the consequently the uncertainty are constant along the vertical axis.

Shallow water equations under location uncertainty

2D momentum equations Assuming vertical shear is negligible we get the 2D momentum equation:                (∂wh ∂t + wh∇

Twh − 1

2

• i,j

∂2 ∂xi∂xj (aijwh))ρ = −∇hp′ + µ∆wh, ∇hd ˆ p = −ρ(wh∇

T)(σd ˆ

Bt)h + µ∆(σdBt)h, ∇ · w = 0, ∇ · σdBt = 0, ∇ · ∇ · a = 0. Horizontal velocity component seen as averaged along depth. 3D velocity defined as:   Ux(x, t) Uy(x, t) Uz(x, t)   =   u(x, y, t)dt + (σ(x, y, t)dBt(x, y))x v(x, y, t)dt + (σ(x, y, t)dBt(x, y))y w(x, t)dt + (σ(x, y, t)dBt(x, y))z.   .

Shallow water equations under location uncertainty

Free surface evolution Considering the upper surface hu is a material surface, from stochastic transport principle, we get: dt(hu)+  ∇huwh − 1 2

• i,j

∂2 ∂xi∂xj (aijhu)   dt+∇·(hu(σdBt)h) = − w(hu) + (σdBt)z, with w(hu) the vertical velocity at point (x, y) of the surface hu. Similarly for the stationary topographic height we get:  ∇hbwh − k 2

• i,j

∂2 ∂xi∂xj (aijhb)   dt + ∇ · (hbσdBt)) = −w(hb) + (σdBt)z.

Shallow water equations under location uncertainty

hydrostatic relation integration Integration of the hydrostatic relation from a depth z up to the free surface and isobaric boundary condition p′(x, y, hu, t) = pu(t): p′(x, y, z, t) − pu(t) = gρ(hu − z), dpt(x, y, z, t) = dpt(x, y, t) for all z. and thus ∇hp′ = gρ∇hhu, The momentum equation of the horizontal mean velocity field becomes (∂wh ∂t + wh∇

Twh − 1

2

• i,j

∂2 ∂xi∂xj (aijwh))ρ = −gρ∇hu + µ∆wh, .

Shallow water equations under location uncertainty

Divergence free constraints integration Integrating now the divergence free constraint of the mean velocity along z , we have: −∇ · wh(hu − hb) = w(x, y, hu, t) − w(x, y, hb, t). Integration along z of the uncertainty divergence free constraint : ∇h · σ(x, y)h = 0, ∇h · (∇h · ah) = 0. Introducing these relations for h = hu − hb we get : dth +  ∇ · (hwh) − 1 2

• i,j

∂2 ∂xi∂xj (aijh)   dt + ∇h

T(σdBt)h = 0.

Shallow water equations under location uncertainty

Shallow water system with uncertainty The final Shallow water stochastic representation reads:                        (∂wh ∂t + wh∇

Twh − 1

2

• i,j

∂2 ∂xi∂xj (aijwh))ρ = −gρ∇hu + µ∆wh, dth + (∇ · (hwh) − 1 2

• i,j

∂2 ∂xi∂xj (aijh))dt + ∇h(σdBt)h = 0, ∇hdpt = −ρ(wh∇

T)(σdBt)h,

∇ · (∇ · ah) = 0.

Shallow water equations under location uncertainty

Shallow water system with uncertainty Location uncertainty provides general framework to derive deterministic/stochastic large scale geophysical models Example Shallow Water system for the free surface conditional expectation (∂wh ∂t + wh∇

Twh − 1

2

• (i,j)h

∂xi∂xj(aijwh))ρ = −gρ∇¯ hu + µ∆wh, ∂¯ h ∂t + ∇ · (¯ hwh) − 1 2

• (i,j)h

∂xi∂xj(aij ¯ h) = 0, ∇ · (∇ · ah) = 0, ∇ · σh = 0.

Reduced order modeling under uncertainty

Reduced order modeling Decomposition of the flow as u(x, t) = u +

m

• i=1

bi(t)φi(x), Optimal representation subspace for u(x, ti) maximizing

• (u, ψ)
• 2

2,

subject to (ψ, ψ) = 1, Solution, φ, eigenvalue problem

• Ω

K(x, x′)φk(x′)dx′ = φk(x)λk, with the spatial autocorrelation – or two points correlation – tensor K(x, x′) = (u(x, t)u(x′, t)) = 1 M

M

• i=1

u(x, ti)u(x′, ti).

Navier Stokes equations under uncertainty

Reduced order modeling K linear, positive semi-definite and self-adjoint. Positive real eigenvalues and by the Mercer theorem, can be represented as the (uniformly convergent) series: K(x, x′) =

+∞

• j=0

λjφj(x)φ

T

j (x′).

Galerkin projection of Navier Stokes under uncertainty: dbk dt + ik +

m

• i=1

likbi +

s

• i=1

m

• j=i

bicijkbj = 0 ∀k = 1, . . . , m , Anisotropic diffusion enters now into the linear and constant coefficients (resp. diffusion on the resolved modes and the mean field) ⇒ Useful for reduced order dynamics from numerical simulation data

Navier Stokes equations under uncertainty

Reduced order modeling Representation of the variance tensor on the residual: a(x) = (∆t)2 T

f

• i=1

[u(x, ti)−¯ u−

m

• j=1

bj(ti)φj(x))][(u(x, ti)−¯ u−

m

• j=1

bj(ti)φj(x)]

T

Representation on complement space basis {φi, i = m + 1, . . . , f } E

• σ(x)dBt(σ(x′)dBt)

T

=

f

• k=m+1

λkφk(x)φ

T

k(x′).

The uncertainty term can then be written in a spectral form as σ(x)dBt =

f

• j=m+1

λ1/2

j

φjdβj(t), ⇒ injected in reduced model simulation dXt =

m

• i=1

bi(t)φidt +

f

• j=m+1

λ1/2

j

φjdβj(t).

Wake flow Re 3900

Test with 2 temporal modes: a1

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 50 100 150 200 250 300 350 a

tU D

init data forecast assimilation

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 50 100 150 200 250 300 350 a

tU D

init+unc data forecast assimilation

Wake flow Re 3900

Test with 2 temporal modes: a2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 50 100 150 200 250 300 350 a

tU D

init data forecast assimilation

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 50 100 150 200 250 300 350 a

tU D

init+unc data forecast assimilation

Wake flow Re 3900

Test with several modes

5 10 15 0.3 0.5209 0.7144 1 e rror(log) Time n = 2 5 10 15 0.3 0.4751 0.7144 1 e rror(log) Time n = 4 5 10 15 0.3 0.4397 0.7144 1 e rror(log) Time n = 6 5 10 15 0.3 0.4126 0.7144 1 error(log) Time n = 8 5 10 15 0.3 0.3882 0.7144 1 error(log) Time n = 10 5 10 15 0.3 0.3656 0.7144 1 error(log) Time n = 12 5 10 15 0.3 0.3472 0.7144 1 error(log) Time n = 14 5 10 15 0.3 0.3298 0.7144 1 error(log) Time n = 16

Conclusion

Derivation of a stochastic expression of Navier-Stokes Identification of the drift evolution equation Subgrid term related to the variance of the random turbulent term Identification of this variance through image data ? No model for the variance or covariance evolution

• E. M´
• emin. Fluid flow dynamics under location uncertainty. Geophysical

& Astrophysical Fluid Dynamics, 2014.

Navier Stokes equations under uncertainty

Perspectives Derivation of geophysical models under uncertainty Ertel’s theorem ? Ocean atmosphere models ? Data assimilation from small scales observations