Stochastic Filtering for Rotating Shallow Water EquationsStochastic Filtering for Rotating Shallow Water EquationsOana LangOana Lang

Supervised by D. Crisan, P. J. van Leeuwen, R. PotthastSupervised by D. Crisan, P. J. van Leeuwen, R. PotthastImperial College London and University of ReadingImperial College London and University of Reading

IntroductionIntroduction

We investigate a stochastic filtering problem consisting of a signal thatmodels the motion of an incompressible fluid below a free surface when thevertical length scale is much smaller than the horizontal one. The evolution ofthe two-dimensional rotating system is represented by an infinite dimensionalstochastic PDE and observed via a finite dimensional observation process. Thedeterministic part of the SPDE consists of a classical shallow waterequation and a new type of noise, namely the one introduced in [2].

Atmospheric CirculationAtmospheric Circulation

Source: http://www.ux1.eiu.edu/ cfjps/1400/circulation.html

New Type of NoiseNew Type of Noise

Following [2] we introduce the noise:

∞∑i=1

[ξi , ω] ◦ dB it ,

where ω is the vorticity vector, ξi are zero divergence vector fields, B it are

independent Brownian motions, and [·, ·] is the Jacobi-Lie bracket.

Why is this better than other types of noise?Why is this better than other types of noise?The hierarchy of model equations that govern climate dynamics is designed toprovide the most accurate approximation for the original equations near ageostrophically balanced state (quasigeostrophic balance in this case). One ofthe necessary conditions required in this hierarchy is the existence of aconservation law. By choosing this type of noise, added to the deterministicpart, certain physical properties of the solution of the resultingstochastic PDE are conserved.

MotivationMotivation

Climate change is one of the most challenging problems that we arecurrently facing, fundamental research in this area being, therefore, of crucialimportance. Large-scale circulatory motions of ocean and atmosphere involvecomplex geophysical phenomena that have a determinant influence on climatedynamics. Although one will never be able to formalise reality in its entirecomplexity, the introduction of stochasticity into ideal fluid dynamicsis a realistic tool for modelling the sub-grid scale processes that cannototherwise be resolved.

Rotating Shallow Water Equations (RSW) and Quasigeostrophy (QG)Rotating Shallow Water Equations (RSW) and Quasigeostrophy (QG)

The dynamics of a rotating shallow water system on a two dimensional domainis given, in nondimensional form, by

εdu

dt+ f z× u +∇p = 0 and

∂h

∂t+∇ · hu = 0,

where: ε is the Rossby number, u is the horizontal fluid velocity vector, f is theCoriolis parameter, z is a unit vector pointing away from the center of theEarth, η is the total depth, h is the thickness of the fluid, p := h

εF and F isthe Froude number.

Figure : A single layer shallow water system. η is the total

depth, ηb is the height of the bottom surface, h := η − ηb, and

H is the mean depth.

These two equationsare classically derivedfrom the momentum andmass continuity equationswhich fully characterisethe motion of afluid. Hydrostatic balanceand constant density areassumed by convention.

A useful model wheninvestigating large-scale geophysical fluid dynamics is the quasigeostrophicapproximation which can be obtained from an asymptotic expansion ofRSW. In potential vorticity form, QG has the following expression:

∂q

∂t+ u · ∇q = 0,

where q = ∆ψ −Fψ + f is the potential vorticity, ψ being the streamfunction.

Alternatively, RSW can be derived from a variational principle. In thissituation the corresponding model for the QG motion (in its curl form) is

∂q

∂t+ u · ∇q + q∇ · u = 0,

with q = ε(∆ψ −Fψ) + f .

Figure from G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics [4]

Filtering for Stochastic RSWFiltering for Stochastic RSW

Our research is focused on the analysis of filtering problems where the signal satisfies the following SPDE:

d

(ω

h

)+

[u,ω

h

]dt +

∞∑i=1

[ξi ,ω

h

]◦ dB i

t = 0 and dh +∇ · (hu)dt +∞∑i=1

(∇ · (ξih)) ◦ dB it = 0,

where ω = z · curl (εu + R(x)), R(x) is the vector potential for the zero divergence rotation rate about the vertical direction, and the other elements are as above,with q = ω/h The qualitative study of this stochastic system concerns issues like existence, uniqueness, and smoothness properties of the solution. The evolution ofthe signal will be conditioned by observations of an underlying true system, which are pointwise measurements corresponding to the velocity field. According to[1], the resulting conditional distribution (the posterior distribution) satisfies the Kushner-Stratonovitch equation. Unfortunately, in most of the cases thecorresponding solution cannot be computed explicitly. For this reason, the quantitative analysis will be done using Ensemble Methods (Particle Filters).

ConclusionConclusion

A proper incorporation of stochastics into fluid dynamics could contribute to theunresolved processes issue. Although this is a single layer model, hence it does not completelyreflect the complex stratification of the real atmosphere, it allows for important geophysicalphenomena such as gravity and Rossby waves, eddy formation and geophysical turbulence.

ReferencesReferences

[1] A. Bain, D. Crisan, Fundamentals of Stochastic Filtering, 2009[2] D. Holm , Variational Principles for Stochastic Fluid Dynamics, 2015[3] D. Holm et. al., Geometric Mechanics and Symmetry, 2009[4] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2005[5] G. Nakamura, R. Potthast, Inverse Modeling - An introduction to thetheory and methods of inverse problems and data assimilation, 2015[6] P. J. van Leeuwen, Aspects of Particle Filtering in High-DimensionalSpaces, 2015