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 of 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 order 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 σ d B t Displacement: d X ( x , t ) = w ( X ( x , t ) , t ) dt + σ ( X ( x , t ) , t ) d ˆ B t , with X ( x , 0) = x , Eulerian description of the velocity fields: U ( x , t ) = w ( x , t ) dt + σ ( x , t ) d B t . U should be solution of Navier Stokes equation derived from Newton 2nd law ⇒ σ d B t differentiable in space

Noise term Brownian motion field avatar � n 1 ˆ B n t ( x ) = √ n B t ( x i ) ϕ ν ( x − x i ) , i =1 Limiting process denoted in a formal way as: � ˆ △ IR d B t ( x ′ ) ϕ ν ( x − x ′ ) d x ′ , B t ( x ) = B t ⋆ ϕ ν ( x ) = Uncertainty: Diffusion tensor and White noise avatar on Ω � σ ( x , t ) d ˆ σ t ( x , y ) d ˆ B t = B t ( y ) d y . Ω Covariance of the turbulent component Q ( x , s , y , t ) = dt δ ( t − s ) σ ϕ ν ( x , t ) σ ϕ ν ( y , t ) , △ = a ( x , y , t ) dt δ ( t − s ) , Homogeneous diffusion provides homogeneous covariance tensor with spatially constant variance

Noise term Kraichnan smooth model � 1 d B n ζ d B t ( x i ) ψ γ κ ⋆ f ζ ( x − x i ) , √ n t ( x ) = i f ζ ( x ) = C ζ � x � ζ/ 2 0 < ζ < 2 . Incompressible fluid d ξ ζ t = P ⋆ d B ζ t . Spectral correlation defined as Q ( k ) ij = | k | − ζ − d ( δ ij − k i k j | k | 2 )( � � ψ γ κ ) 2 . Quadratic variation process for a pass-band spectral cutoff ( 1 I [ κγ ] ( k )) 2 π d / 2 t ( x ) > ij = dtC ζ d − 1 2 ) ζ − 1 ( L ζ − ℓ ζ d < ξ ζ t ( x ) , ξ ζ D ) δ ij (2 π ) d Γ( d d

Stochastic Reynolds transport theorem Volumetric rate of change Volumetric rate of change of a scalar process q ( x , t ) transported by a velocity field d X t = w ( X t , t ) dt + σ ( X t , t ) d B t � � � ∂ 2 1 d q ( x , t ) d x = d t q + ( ∇ · ( q w ) − ( a ij q ) | ∇· σ =0 + 2 ∂ x i ∂ x j V ( t ) V ( t ) i , j � ∇ · σ � 2 q ) dt + ∇ · ( q σ d B t ) d x , Example for the smooth Kraichnan model: � � [ d t q + ( ∇ · ( q w ) − 1 T d ξ ζ d q ( x , t ) d x = 2 γ ∆ q ) dt + ∇ q t ] d x , V ( t ) V ( t )

Stochastic Reynolds transport theorem Mass conservation Mass conservation constraint on the transported volume: � ∂ 2 d t ρ + ∇ · ( ρ w ) dt = 1 ( a ij ρ ) | ∇· σ =0 + � ∇ · σ � 2 ρ ) dt − ∇ · ( ρ σ d B t ) . 2( ∂ x i ∂ x j i , j For a fluid with constant density, mass preservation implies ∇ · ( σ d B t ) = 0 , ∇ · w = 0 , ∇ · ( ∇ · a ) = 0

Stochastic Reynolds transport theorem Isochoric flows and isoneutral uncertainty Mass conservation constraint: � ∂ 2 d t ρ + ∇ ρ w dt − 1 ( ρ a ij ) dt = ∇ ρσ d B t 2 ∂ x i x j i , j Kraichnan model ⇒ advection diffusion with multiplicative stochastic forcing T w dt − γ 1 d t ρ + ∇ ρ 2∆ ρ dt = ∇ ρσ d B t If the uncertainty σ d B t lies on the isodensity surfaces: σ ij = δ ij − ∂ x i ρ ( x ) ∂ x j ρ ( y ) δ ( x − y ) . �∇ ρ � 2

Stochastic Reynolds transport theorem Isochoric flows and isoneutral uncertainty � Small slope assumption ( ( ∂ x ρ ) 2 +( ∂ y ρ ) 2 << ∂ z ρ ) ⇒ diffusion tensor (and the quadratic variation) reads:   1 0 α x ( x )   , α = − ( ∂ x ρ/∂ z ρ, ∂ y ρ/∂ z ρ, 0) a ( x ) = 0 1 α y ( x ) | α ( x ) | 2 α x ( x ) α y ( x ) ∇ · σ = 0 ⇒ α constant along the depth axis ∂ z α x = ∂ z α y = 0 and ∇ · α = 0. Conserved scalar quantity q ⇒ deterministic diffusion along the density tangent plane: � ∂ q T w = 1 ∂ t + ∇ q ∂ x i ( a ij ∂ x j q ) 2 ij Gent-McWilliams ”Isoneutral” or ”Isopycnal” diffusion in large scales ocean dynamics simulations

Conservation of momentum Conservation of momentum Newton second law � d ρ w d x = F , dt V Considering stochastic conservation principle � � ρ ( w ( x , t ) dt + σ ( x , t ) d B t ) d x = F ( x , t ) d x . d V ( t ) V ( t ) highly irregular ⇒ interpreted in the sense of distribution

Conservation of momentum Conservation of momentum For every h ∈ C ∞ 0 ( R +): � � � � h ′ ( t ) h ( t ) F ( x , t ) d x dt = − σ ( x , t ) d B t d x dt + V ( t ) V ( t ) � � h ( t ) d ρ w ( t , x ) d x dt . V ( t ) Since both side of this equation must have the same structure, the forces can be written as: � � � � h ′ ( t ) h ( t ) F ( x , t ) dt = − σ ( t , x ) d B t d x + V ( t ) V ( t ) � � h ( t ) ( f ( t , x ) d x dt + θ ( t , x ) d B t ) d x . V ( t ) First terms of both equations identical and cancel out.

Conservation of momentum Conservation of momentum We have: � � � ∂ 2 1 ( a jk ρ w i ) | ∇· σ =0 + ρ w i d x = ( d ( ρ w i ) t + ( ∇ · ( ρ w i w ) − d 2 ∂ x j ∂ x k V ( t ) V ( t ) j , k � � ∇ · σ � 2 ρ w i ) dt + ∇ · ( ρ w i σ d B t )) d x , with a ij ( x , t ) = σ ik ν ( x , t ) σ kj ν ( x , t ) . k As for the forces: Body force and external forces � G = ρ ( gdt − 2 Ω × U ) d x , V Surface forces � � p ) + µ (∆ U + 1 S = Σ dt n ds = − ∇ ( pdt + d ˆ 3 ∇ ( ∇ · U ) , ∂ V V

Navier Stokes equations Stochastic Navier Stokes equations Incorporating (stochastic) mass preservation principle and the forces expression:  � � a ij ρ ∂ 2 w (( ∂ w T w ) ρ − 1 ∂ ( a ij ρ ) ∂ w   ∂ t + w ∇ − | ∇· σ =0 ) dt +    2 ∂ x i ∂ x j ∂ x j ∂ x i   i , j i , j     T ρ σ d B t = ( ρ g − 2 ρ Ω × w − ∇ p + µ (∆ w + 1   3 ∇ ( ∇ · w )) dt − w ∇  ∇ dp t − 2 ρ Ω × ( σ d B t ) + µ (∆( σ d B t ) + 1   3 ∇ ( ∇ · ( σ d B t ))) ,      �  ∂ 2 d t ρ + ( ∇ · ( ρ w ) − 1   ( a ij ρ ) | ∇· σ =0 + � ∇ · σ � 2 ρ ) dt = ∇ · ( ρ σ d B t ) .   2 ∂ x i ∂ x j i , j

Navier Stokes equations under location uncertainty Stochastic Navier Stokes equations Equating slow terms and highly oscillating terms:  � � a ij ρ ∂ 2 w ( ∂ w T w ) ρ − 1 ∂ ( a ij ρ ) ∂ w   ∂ t + w ∇ − | ∇· σ =0 =    2 ∂ x i ∂ x j ∂ x j ∂ x i   i , j i , j     ρ g − 2 ρ Ω × w − ∇ p + µ (∆ w + 1   3 ∇ ( ∇ · w )) ,    ∇ dp t = − w ∇ T ρ σ d B t − 2 ρ Ω × ( σ d B t ) + µ (∆( σ d B t )+    1    3 ∇ ( ∇ · ( σ d B t ))) ,      � ∂ 2  d t ρ + ∇ · ( ρ w ) − 1   ( a ij ρ ) | ∇· σ =0 + � ∇ · σ � 2 ρ ) dt = ∇ · ( ρ σ d B t ) ,   2 ∂ x i ∂ x j i , j

Navier Stokes equations under location uncertainty Stochastic Navier Stokes equations for the smooth Kraichnan model ∀ x ∈ Ω , t ∈ ]0 , T ]  ( ∂ w T w − γ 1   ∂ t + w ∇ 2∆ w ) ρ = ρ g − 2 ρ Ω × w − ∇ p + µ ∆ w  ∇ d ˆ p t = − ρ ( w ∇ T ) d ξ t + 2 ρ Ω × d ξ t + µ ∆ d ξ t ,    ∇ · w = 0 , with boundary and initial conditions:   w · n = 0 on ∂ Ω , t ∈ ] 0 , T ] ,  d ξ t = 0 on ∂ Ω , t ∈ ] 0 , T ] ,   w | t =0 = w o in Ω .

Navier Stokes equations under location uncertainty Stochastic Navier Stokes equations for incompressible and divergence free general turbulent model  � ∂ 2 ( ∂ w T w − 1   ( a ij w )) ρ = ρ g − 2 ρ Ω × w − ∇ p + µ ∆ w ∂ t + w ∇   2 ∂ x i x j    i , j T ) σ d ˆ ∇ d ˆ p t = − ρ ( w ∇ B t + 2 ρ Ω × σ d B t + µ ∆ σ d B t ,     ∇ · 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 = w o in Ω .