@let@token Stochastic Variational Principles for GFD...

Stochastic Variational Principles for GFD:Modelling the unknown unknowns

Darryl D Holm,Imperial College London

LMS Workshop, 2 Nov 2015

Darryl D Holm, Imperial College London Stochastic Variational Principles for GFD LMS Workshop, 2 Nov 2015 1 / 24

So far today we have raised these “Primary Questions”

How to develop a data-driven model? Model driven error? Whichcomes first? The data or the model?

The model may give sufficient conditions to match the data, but to whatextent are these conditions at all necessary, or even unique?

M. Bocquet, A. Carassi, A. Hannart and M. Ghil [2015] Dataassimilation to compute model evidence: The attribution problem.(In preparation).

T. Palmer et al. [2009] Stochastic Parameterisation and ModelUncertainty, ECMW Report 598, cf. also recent H. Christensen et al.

How to use data assimilation to model the model error?

Our strategy: learn the models by blending data withnumerics in two data assimilation steps

Fundamental ideas and goals of this work

Develop an approach to SPDEs in which noise terms have datadriven coefficients, linked to a stochastic variational principle.Combine geometric mechanics with probabilistic analysis, in avariational hierarchy of data driven GFD approximations whichpreserve the invariance properties of the original system.Specifically, combine different rapidly developing methods(particle filters, stochastic calculus of variations, regularisationmethods for singular SPDEs) with new ideas from geometricmechanics exploiting “stochastic advection” (Lie-derivativetransport of advected quantities by a stochastic vector field,indexed by the Eulerian spatial position).Contribute to both applications and foundations of stochastic fluiddynamics and other complex nonlinear systems, to enhancenumerical methods and statistical modelling.

What is our task?

Our task: Learn from stochastic assimilation of observed data (tracers)how to produce stochastic fluid motion equations whose numerics willyield the observed statistics and variability of the data.

Numerics Observations Our Approach

Numerical simulations show this sea surface elevation

Observations of tracers in the Ocean show this.

Figure: All satellite observations of surface drifter trajectories since 1980passing around Antarctica, courtesy Eric van Sebille [2015].

What fluid equations produce this scalar turbulence?

Figure: Here are all surface drifter trajectories since 1980 to have passedbetween Eastern Australia & New Zealand, courtesy Eric van Sebille [2014].

How would these observations guide the modelling?

Claim: Stochastic terms for the motion equation can be derived viaHamilton’s variational principle, by constraining its variations to agreewith the spatial correlations seen in the tracer data.

How? Our strategy is to impose stochastic transport of advectedquantities as a constraint in Hamilton’s principle δS(u,p,q) = 0 with,

S(u,p,q) =

∫ (`(u,q)dt︸︷︷︸Physics

p , dq + Ldxt q︸︷︷︸Tracer data

Here `(u,q) is the unperturbed deterministic fluid Lagrangian, writtenas a functional of velocity vector u, tracers q and . . .

we’ll get to dxt later.Darryl D Holm, Imperial College London Stochastic Variational Principles for GFD LMS Workshop, 2 Nov 2015 9 / 24

How would these observations guide the modelling?

Claim: Stochastic terms for the motion equation can be derived viaHamilton’s variational principle, by constraining its variations to agreewith the spatial correlations seen in the tracer data.

How? Our strategy is to impose stochastic transport of advectedquantities as a constraint in Hamilton’s principle δS(u,p,q) = 0 with,

S(u,p,q) =

Here `(u,q) is the unperturbed deterministic fluid Lagrangian, writtenas a functional of velocity vector u, tracers q and . . .

we’ll get to dxt later.Darryl D Holm, Imperial College London Stochastic Variational Principles for GFD LMS Workshop, 2 Nov 2015 9 / 24

Our strategy: learn the models by blending data withnumerics in two data assimilation steps

What would we get?

What? New stochastic GFD models for climate & weather variability.New motion equations contain stochastic perturbations which multiplyboth the solution and its spatial gradient (in a certain transport way).

Remarkably, these stochastic GFD models still preserve fundamentalproperties such as Kelvin’s circulation theorem and PV conservation.

Example. Stochastic QG.Darryl D Holm, Imperial College London Stochastic Variational Principles for GFD LMS Workshop, 2 Nov 2015 11 / 24

History: RH Kraichnan [1996, PRL] scalar turbulence

In the Kraichnan model, advection of passive scalar θ is governed by

∂tθ + v · ∇θ = κ∆θ + F , ∇ · v = 0 ,

where θ(t , r) is the scalar (temperature), F (t , r) is the external source,v(t , r) is the advecting velocity, and κ is diffusivity.

Both F (t , r) and v(t , r) are independent Gaussian random functions oft and r, which are δ-correlated in time, e.g., v(t , r) =

∑k ξk (r) ◦ dWk (t).

The dWk (t) are independent 1DBrownian motions, with ∇ · ξk = 0and with bounded

∑k ξ

Tk ξk .

Typical numerical solutions showthe patchiness in θ associated withintermittency (anomalous scaling).Very non-Gaussian!

How to derive stochastic GFD motion equations?

How? Our strategy is to impose stochastic transport of advectedquantities (a la Kraichnan) as a constraint in Hamilton’s principle,

0 = δS(u,p,q) = δ

Here `(u,q) is the unperturbed deterministic fluid Lagrangian, writtenas a functional of velocity vector field u, and . . .

Ldxt is the transport operator (Lie derivative) for any advected quantityq ∈ V by a stochastic vector field, dxt , at fixed Eulerian position y as

dxt (y) = u(y , t) dt︸︷︷︸Flow

ξk (y) ◦ dWk (t)︸︷︷︸Noise

The stochastic vector field dxt (y) contains cylindrical Stratonovichnoise whose spatial correlations are given by ξk (y) (a la Kraichnan).

How to derive stochastic GFD motion equations?

How? Our strategy is to impose stochastic transport of advectedquantities (a la Kraichnan) as a constraint in Hamilton’s principle,

0 = δS(u,p,q) = δ

Here `(u,q) is the unperturbed deterministic fluid Lagrangian, writtenas a functional of velocity vector field u, and . . .

Ldxt is the transport operator (Lie derivative) for any advected quantityq ∈ V by a stochastic vector field, dxt , at fixed Eulerian position y as

dxt (y) = u(y , t) dt︸︷︷︸Flow

ξk (y) ◦ dWk (t)︸︷︷︸Noise

The stochastic vector field dxt (y) contains cylindrical Stratonovichnoise whose spatial correlations are given by ξk (y) (a la Kraichnan).

Example: How to make deterministic QG stochastic?

Deterministic QG is basically PV transport:

∂Q∂t

= −u ·∇Q ,

where PV (Potential Vorticity) is defined

Q := ∆ψ −Fψ + f ,

in terms of the stream function ψ of the geostrophicfluid velocity, u := ∇⊥ψ = (− ∂ψ

∂x2, ∂ψ∂x1

Stochastic QG is basically stochastic PV transport

In Stochastic QG, Hamilton’s principle =⇒ stochastic PV transport,

dQ = −dx t · ∇Q ,

dx t := ∇⊥dΨ = u dt +∑

ξk(x) ◦ dWk(t) .

For viscous such eqns, see: Z Brzezniak, F Flandoli and M Maurelli [ 2014]Existence and uniqueness for stochastic 2D Euler flows with boundedvorticity, arXiv:1401.5938v1 [math.PR] 23 Jan 2014

Two further properties of Stochastic QG

1.) Ito form. The Ito form of stochastic QG is,

dQ =−(

u dt +∑

ξk(x) dWk(t))· ∇Q

div(∑

ξTk ξk

)grad Q dt ⇐= (Ito drift) .

Ito drift is how the noise contributes to the mean flow velocity.

2.) PV enstrophies. Both the Stratonovich and Ito stochastic QG

equations preserve the infinite family of potential enstrophies,

Φ(Q) ,

for any smooth function Φ. Proof: dCΦ = −∫R2 dx t · ∇Φ(Q) = 0.

Two further properties of Stochastic QG

1.) Ito form. The Ito form of stochastic QG is,

dQ =−(

u dt +∑

ξk(x) dWk(t))· ∇Q

div(∑

ξTk ξk

)grad Q dt ⇐= (Ito drift) .

Ito drift is how the noise contributes to the mean flow velocity.

2.) PV enstrophies. Both the Stratonovich and Ito stochastic QG

equations preserve the infinite family of potential enstrophies,

Φ(Q) ,

for any smooth function Φ. Proof: dCΦ = −∫R2 dx t · ∇Φ(Q) = 0.

Summary Why? How? What? QG Example.

Why? We needed to learn from stochastic assimilation of tracer datahow to produce stochastic fluid motion equations which wouldreproduce the observed statistics and variability of the data.We asked, How would observed stochastic tracer transport data guideus in modifying the deterministic GFD motion equations?

How? Stochastic terms for the motion equation were derived viaHamilton’s variational principle, by constraining its variations to agreewith the spatial correlations seen in tracer transport data.

What? New methods of finding GFD models for climate and weather.New motion equations contained stochastic perturbations multiplyingboth the solution and its spatial gradient (in a certain transport way).Remarkably, these stochastic GFD models preserved fundamentalproperties such as Kelvin’s circulation theorem and PV conservation.

Conclusion: This is just the beginning of the work!

1 The fundamental mathematical structure of fluids is preserved by(1) injecting stochasticity via Hamilton’s principle, using(2) a stochastic transport constraint for advected quantities.

2 E.g., deterministic transport became stochastic transport for QG.3 And, for QG, stochastic transport still preserved PV enstrophies.4 The theory applies to all fluid models derived from Hamilton’s

principle. (The spatial correlations∑

k ξTk ξk derive from data.)

5 The theory includes stochastic versions of the usual GFDEuler-Boussinesq equations, primitive equations, etc.

6 There’s so much more to do, e.g., in analysing and applying thesenew stochastic GFD equations!

7 Until recently, the existence and uniqueness for our example ofstochastic 2D QG flows were still open! Now they are solved

8 Recently, we have shown finite time existence, uniqueness andregularity of 3D stochastic Euler equations derived this way!

3D stochastic Euler equations are derived this way

The Ito vorticity equation for incompressible stochastic Euler flow is

dω + [u, ω] dt − 12

[ξi(x), [ξi(x), ω]

]︸︷︷︸Double Lie bracket

dt +∑

[ξk (x), ω]dWk (t) = 0 ,

with Lie bracket [u, ω] = u · ∇ω − ω · ∇u =: Buω with B†u = −Bu,

and ω := curl u. Equivalently,

dω + (Bu +∑

B†ξi

Bξi )ω dt +∑

Bξkω dWk (t) = 0 .

The operator formed by the sum

Bu +∑

B†ξi

is hypo-coercive [Villani, 2012], which greatly facilitates the analysis ofthese eqns, in showing finite time existence, uniqueness and regularity!

Research Project: Colin Cotter, Dan Crisan, D Holm

Cotter Crisan Holm ProjectThe issues at the heart of this project relate to Stochastic PartialDifferential Equations (SPDEs), Stochastic Variational Principles(SVPs) and Numerical Modelling, in connection with ComplexNonlinear Systems, Applied Analysis, Numerical Analysis, StochasticGeophysical Fluid Dynamics (SGFD) and Turbulence.

The new stochastic methodology fits together

Objectives of the new stochastic methodology

• Create new parameterisation approaches in SGFD formathematics of climate change and weather variability

• Quantify variability in SGFD models due to stochastic transport, bydetermining the Fokker-Planck evolution of the probability density

• Apply to quantify variability for each member of the new SGFDhierarchy, first for the lowest level approximation, later for more realisticSGFD models at higher orders in the GFD asymptotic expansion

• Use PV preservation and the dissipative double-bracket operatorsin the Ito interpretation of these SGFD models as input for obtainingfinite-horizon parameterising manifolds [Chekroun & Liu, 2015]

Machine learning & the new stochastic methodology

• In our Input Step, we could use methods of belief propagation,message passing, statistical inference and particle image velocimetry;all of which are standard in tracking fluid particle paths in turbulence.

• Can we regard our data assimilation and data-driven modellingsteps as a form of machine learning? How is our approach differentfrom usual machine learning problems?

• Our approach imposes a probabilistic structure on the “outsidemodel”, and propagates it from the tracer equations to the motionequation via Hamilton’s principle and then to a numerical solver.

• How to compare our approach with standard methods in machinelearning, which focus on a stochastic representation of the uncertaintyin individual terms within the model, and then draw from that ensembleand propagate the results through a numerical solver?

References

rspa.royalsocietypublishing.org

ResearchCite this article: Holm DD. 2015 Variationalprinciples for stochastic fluid dynamics. Proc.R. Soc. A 471: 20140963.http://dx.doi.org/10.1098/rspa.2014.0963

Received: 15 December 2014Accepted: 24 February 2015

Subject Areas:mathematical physics, applied mathematics,fluid mechanics

Keywords:geometric mechanics, stochastic fluid models,cylindrical stochastic processes, multiscalefluid dynamics, symmetry reducedvariational principles

Author for correspondence:Darryl D. Holme-mail: [email protected]

Variational principles forstochastic fluid dynamicsDarryl D. Holm

Department of Mathematics, Imperial College, London, UK

This paper derives stochastic partial differentialequations (SPDEs) for fluid dynamics from astochastic variational principle (SVP). The paperproceeds by taking variations in the SVP to derivestochastic Stratonovich fluid equations; writingtheir Itô representation; and then investigatingthe properties of these stochastic fluid modelsin comparison with each other, and with thecorresponding deterministic fluid models. Thecirculation properties of the stochastic Stratonovichfluid equations are found to closely mimic thoseof the deterministic ideal fluid models. As withdeterministic ideal flows, motion along the stochasticStratonovich paths also preserves the helicity ofthe vortex field lines in incompressible stochasticflows. However, these Stratonovich properties are notapparent in the equivalent Itô representation, becausethey are disguised by the quadratic covariation driftterm arising in the Stratonovich to Itô transformation.This term is a geometric generalization of thequadratic covariation drift term already found forscalar densities in Stratonovich’s famous 1966 paper.The paper also derives motion equations for twoexamples of stochastic geophysical fluid dynamics;namely, the Euler–Boussinesq and quasi-geostropicapproximations.

1. IntroductionIn this paper, we propose an approach for includingstochastic processes as cylindrical noise in systems ofevolutionary partial differential equations (PDEs) thatderive from variational principles which are invariantunder a Lie group action. Such dynamical systemsare called Euler–Poincaré equations [1,2]. The mainobjective of the paper is the inclusion of stochasticprocesses in ideal fluid dynamics, in which case thevariational principle is invariant under the Lie group ofsmooth invertible maps acting to relabel the reference

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