#### Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows
François Gay-Balmaz 0 1 2
Darryl D. Holm 0 1 2
B Darryl D. Holm 0 1 2
0 Department of Mathematics, Imperial College , London SW7 2AZ , UK
1 Laboratoire de Météorologie Dynamique, CNRS and École Normale Supérieure de Paris , 24 Rue Lhomond, 75005 Paris , France
2 Communicated by Charles R. Doering
Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration's “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie-Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD. Mathematics Subject Classification 37H10 · 37J15 · 60H10
Stochastic geometric mechanics; Euler-Poincaré theory; Coadjoint orbits; Geophysical fluid dynamics
1 Introduction
This paper develops data-driven stochastic models of fluid dynamics, inspired by
spatiotemporal observations from satellites of the spatial paths of objects drifting near
the surface of the ocean in the National Oceanic and Atmospheric Administration’s
“Global Drifter Program”. The Lagrangian paths of these freely drifting instruments
track the ocean currents. That is, the satellite readings of their positions approximate
the motion of a fluid parcel as a curve parameterised by time.
Figure 1
(Lilly 2017)
displays the global array of surface drifter trajectories from the
National Oceanic and Atmospheric Administration’s “Global Drifter Program” (www.
aoml.noaa.gov/phod/dac). In total, more than 10,000 drifters have been deployed since
1979, representing nearly 30 million data points of positions along the Lagrangian
paths of the drifters at 6-h intervals. This large spatiotemporal data set is a major source
of information regarding ocean circulation, which in turn is an important component
of the global climate system; see, for example,
Lumpkin and Pazos (2007)
,
Griffa et al.
(2007)
and
Sykulski et al. (2016)
. An important feature of this data is that the ocean
currents show up as time-varying spatial correlations, easily recognised visually by
the concentrations of colours representing individual paths. These spatial correlations
exhibit a variety of spatial scales for the trajectories of the drifters, corresponding
to the variety of spatiotemporal scales in the evolution of the ocean currents which
transport the drifters.
Figure 2 shows a sample of the Lagrangian trajectories of drifters released in the
vicinity of Cape Cod. Here one sees concentrations of recirculating drifters following
Western boundary currents, and splitting into three main streams just off the coast
of Cape Cod. One of these main streams forms a large-scale re-circulation some
distance away from the boundary and to the East of Cape Cod. Small-scale, erratic
deviations of the main streams of the drifter paths are also visible in this figure. These
erratic trajectories will be represented stochastically in the paper, and the larger-scale
spatial correlations they follow will be modelled as spatiotemporal modulations of the
stochasticity, which follow the resolved drift currents.
Figure 3 displays the Lagrangian path taken by Drifter 36256, deployed on
September 18, 2005, and successfully recovered on February 21, 2007, at Brest, France, after
a journey of 521 days across the North Atlantic Ocean. Along this path one sees the
effects of the interactions of the drifter with a variety of space and time scales of
evolving fluid motion, strongly suggesting a need for modelling the non-stationary
statistics of Lagrangian paths
(Sykulski et al. 2016)
.
1.1 Why Introduce Stochasticity into Fluid Dynamics?
In developing parameterisations in weather and climate prediction, the term
“unresolved scales” refers to those fluid motions and thermodynamic properties which are
either not computable in a given set of numerical simulations, or are not measurable in
a given set of observations. The effects of the smaller, faster, unresolved scales on the
larger, slower, resolved variables are often modelled deterministically, in terms of the
resolved scales and their gradients. See, e.g.,
Foias et al. (2001
, 2002),
Holm (2002)
and references therein. Although this deterministic approach may be pragmatic and
even advisable, due to the limitations of simulations and measurements, it is clear from
the variety of deterministic models which have been proposed that these models are
by no means unique. For example, the plethora of turbulence models and averaging
methods, as well as the stark differences in solution behaviour as simulations of the
fluid equations are achieved at finer and finer resolutions, show that the resolved states
may be associated with many possible unresolved states. For example, phenomena
such as vortex tubes of a certain radius, as seen in simulations at one scale, or
resolution, may no longer even be present when simulated at smaller scales, achieved
with better resolution. This is not the usual situation in the development of science.
Usually, the advent of better, more accurate, measurements lead to new effects and
new laws governing them. Instead, in the case of weather and climate prediction, the
fundamental deterministic equations and the laws of thermodynamics are completely
known.
No new deterministic laws of fluid motion should be expected. However, there
can be new statistical approaches to the physical and mathematical descriptions of
weather and climate. For example, the recognition of stochastic vector fields as the
basic paradigm in fluid dynamics has long been the province of turbulence modelling
(Monin and Yaglom 1971)
. However, recently the use of stochastic vector fields has
also been recognised in estimating statistical model uncertainty in numerical weather
prediction
(Franzke et al. 2015)
. From this viewpoint, the uncertainty and variability
of the predictions are crucial aspects of the solution. In the statistical science of
numerical weather and climate prediction, stochastic methods offer systematic approaches
towards quantitative estimates of uncertainties due to model error and inaccuracy
of data assimilation, as well as improved estimates of long-term climate variability,
including estimates of the probability of extreme events. Following
Berner et al. (2013)
we agree that “stochasticity must be incorporated at a very basic level within the design
of physical process parameterizations and improvements to the dynamical core.” The
purpose of this paper is to offer new approaches at this basic level. However, it is
beyond the scope of the present paper to analyse the large dataset of drifter
trajectories that has inspired our investigation. The work is in progress to fully analyse the
drifter dataset and will be discussed elsewhere. Here we will discuss both 2D and 3D
stochastic fluid dynamics models.
1.2 How to Do It?
How does one use stochasticity to improve the physical and mathematical basis for
designing statistical model uncertainty schemes? Recently, this question was answered
by using Hamilton’s principle to derive a new class of mathematical models of
stochastic transport in fluid dynamics
(Holm 2015)
. In this class of models, the effects of the
small, fast, unresolved fine scales of motion on the coarser ones are modelled by
introducing stochastic uncertainty into the transport velocity of fluid parcels in the
dynamics at the resolvable coarse scale. This stochastic transport velocity
decomposition encompasses both Newtonian and variational perspectives of mechanics and
also leads to proper Kelvin circulation dynamics. For a rigorous derivation of the
same decomposition into mean and fluctuating velocities using multi-time
homogenisation methods, see
Cotter et al. (2017
b). Applying this decomposition to create
stochastic fluid dynamics preserves the fundamental mathematical properties of their
deterministic counterparts
(Crisan et al. 2017)
. It also enables new approaches to
sub-grid scale parameterization, expressed both in terms of fluctuation distributions,
and spatial/temporal correlations. As such, it introduces stochastic corrections that are
amenable to statistical inference from high-resolution data (either observed, or
numerical). Moreover, this new class of models forms the ideal approach for the development
of a novel data assimilation technology based on particle filters
(Beskos et al. 2017)
.
Filtering and ensemble techniques require de facto a stochastic representation of the
dynamics. This randomization is most often achieved through random perturbations
of the initial conditions. However, this approach tends to yield insufficient
spreading of the ensemble and produce a poor representation of the error dynamics
(Berner
et al. 2013)
. The stochastic transport class of models establishes the much-needed
randomization via its rigorous derivation at the fundamental level, rather than via ad
hoc empiricism.
In this paper, we will introduce two different stochastic extensions of
Holm (2015)
in
applying the geometric mechanics framework to estimate the contribution of stochastic
transport to statistical uncertainty (error) in fluid dynamics models.
Stochastic transport means that the Lagrangian fluid parcel motion has a stochastic
component. In this context, we ask the following question in the context of numerical
weather and climate prediction: What fundamental properties of the deterministic fluid
equations would persist in a stochastic vector field representation of continuum fluid
motions? First, even if the fluid parcel velocity were stochastic, the fluid continuum
motion would still be describable as a spatially smooth but now temporally stochastic
flow, gt , depending on time t . For the sake of brevity and simplicity of the presentation,
we will take the domain of flow D to lie in R2 or R3, and we will neglect considerations
of boundary conditions in the examples discussed in this paper. Even in the presence
of stochasticity, the stochastic path of the fluid parcel which is initially at position X in
the domain of flow can still be represented by the formula for the Lagrange-to-Euler
flow map, xt = gt (X ), so that g0 at time t = 0 is the identity map, g0(X ) = X .
Since the stochasticity is Markovian, the flow map still corresponds to a stochastic
time-dependent curve gt on the group of compositions of smooth invertible maps, i.e.,
the diffeomorphisms, acting on flow domain D, see
Crisan et al. (2017)
.
Thus, following
Kraichnan (1994)
,
Mikulevicius and Rozovskii (2004)
, and
Holm
(2015)
, we may begin by assuming that the stochastic paths xt = gt (X ) solve a
Lagrangian stochastic differential equation (SDE) with prescribed spatially dependent
function ξt = ξ(gt (X ))
dgt (X ) = ut (gt (X ))dt + ξ(gt (X )) ◦ dW (t ), with g0(X ) = X ∈ D,
(1.1)
where gt : D → D is a spatially smooth invertible map depending on time, t ∈ R.
The corresponding Eulerian stochastic velocity decomposition is given in terms of
cylindrical noise, introduced in
Schaumlöffel (1988)
, as
dgt gt−1(x ) = ut (x ) dt + ξ(x ) ◦ dW (t ), with g0(X ) = X ∈ D.
(1.2)
In this approach, the particle-and-field duality of the Lagrangian and Eulerian
descriptions of continuum fluid motion is still available, even through the fluid parcel
motions are stochastic. This means the symmetry of the Eulerian description under
relabelling of Lagrangian fluid parcels persists, even when the Lagrangian fluid paths
are stochastic, so that the Lagrange-to-Euler fluid map is also stochastic. This
relabelling symmetry implies the Kelvin circulation theorems for fluid dynamics models
(Holm et al. 1998)
. Of course, these remarks generalise to any number of dimensions.
1.3 What Does This Paper Do?
Given the stochasticity in the flow map, how does one derive the corresponding
Eulerian equations of continuum motion? The paper lays out a geometric framework for
deriving stochastic Eulerian motion equations for geophysical fluid dynamics (GFD)
using the method of symmetry reduction for a modified Hamilton’s principle for fluids
with advected quantities.
Although the paper is based on previous developments in
Kraichnan (1994)
,
Mikulevicius and Rozovskii (2004)
, and
Holm (2015)
, it introduces two new approaches for
incorporating non-stationary statistics due to flow dependence, as seen in the NOAA
drifter data shown in Figs. 1, 2 and 3, and analysed in
Sykulski et al. (2016)
.
In particular, the paper allows for flow dependence of the eigenvectors ξt of the
spatial correlations in the stochastic process in (1.2). To obtain this flow dependence,
we postulate two different approaches for allowing the eigenvectors for the stochastic
process to evolve along with the advected quantities. These two models are each
modifications of the approach in
Holm (2015)
. The two models provide alternative
systematic avenues for forecasting with evolving time-dependent statistics, such as
those seen in the NOAA drifter data analysed in
Sykulski et al. (2016)
, rather than using
spatially dependent but steady statistics. While steady statistics may be appropriate
for climate science, analysis of flows at shorter time scales in forecasting weather
variability, for example, may require flow-dependent, evolving statistics. Our purpose
here is to present a systematic framework for modelling non-stationary statistics in
stochastic fluid flows.
1.4 Plan of the Paper
In Sect. 2, we quickly review the approach of
Holm (2015)
as it applies to the Euler
equations of a perfect fluid. The methodology of this approach has two primary
features: (1) stochastic variational principles; and (2) stochastic Hamiltonian
formulations. These two primary features will allow us to introduce two new stochastic
extensions of Geophysical Fluid Dynamics (GFD), one with advection by the drift
velocity of the eigenvectors ξt discussed in Sect. 3, and the other with eigenvectors ξt
depending on advected fluid quantities discussed in Sect. 4. Both Stratonovich and Itô
forms of the equations are provided. We conclude by comparing the three stochastic
models in terms of their Kelvin circulation theorems in Sect. 5.
2 Model 1: Review of Stochastic Variational Principles for Fluids
Following the geometric approach of
Arnold (1966)
, we consider the group G of
volume preserving diffeomorphisms of the fluid domain D, as the configuration manifold
for incompressible fluids. Curves gt ∈ G in this group describe Lagrangian
trajectories xt = gt (X ) of the fluid motion. To simplify our discussion, we will take the
domain D to lie in R2 or R3 and neglect considerations of boundary conditions. Our
developments extend easily to the case where D is a manifold with smooth boundary.
The Lagrangian of the incompressible fluid is defined on the tangent bundle T G of
the group G and is given by the kinetic energy, i.e.,
L(g, v) =
1
D 2
|v(X )|2d nX,
1
D 2
(u) =
|u(x )|2dn x ,
for n = 2, 3. By a change of variables, we note that L is right invariant: L(gh, vh) =
L(g, v), for all h in G. We can thus write L(g, v) = (vg−1), where : g → R. That
is, we write
as the reduced Lagrangian defined on the Lie algebra g of G, given by the space of
divergence free vector fields, denoted u := vg−1 ∈ g.
Hamilton’s principle δ 0T L(g, g˙)dt = 0 yields the Euler equations in Lagrangian
description, i.e., geodesics on G, whereas the variational principle induced on , the
Euler–Poincaré principle, yields the Euler equations in their standard spatial
description.
(2.1)
In particular, the stochastic variational principle in
Holm (2015)
is obtained by
selecting a space V of tensor fields on D, denoted q(t, x ), on which the group G acts
linearly by the pull-back operation
q ∈ V → g∗q ∈ V ,
(2.2)
which is the natural transport operation of tensor fields by the fluid motion. The
associated Lie algebra action of divergence free vector fields u ∈ g is given by the Lie
derivative as
d
q ∈ V → £uq := dε ε=0 gε∗q ∈ V ,
where gε is the flow of u. We fix a space V ∗ of tensor fields in nondegenerate duality
with V relative to the L2 pairing
p, q V =
Remark 2.1 (Determining the correlation eigenvectors ξi (x )) How may the
eigenvectors ξi (x ) be found in practice? One approach for determining them would be based
on running an ensemble of computer simulations of the flow started with observed
initial velocity data measured in the NOAA data base at each grid point X on a
computational mesh erected in the flow domain. This computational mesh would
be significantly coarser that the resolution of the data. An ensemble of these
computer runs would produce a distribution of simulated end points Xr = U r dt on the
coarse mesh, where r labels the ensemble member. The distribution of differences
Xr of the simulations from the observed data around each coarse grid point would
be modelled (up to some specified tolerance) as a local stochastic process given by
d X = Nj=1 ξ j (X ) ◦ dW j (t ), where {ξ j (X )} is a set of orthogonal functions whose
squared amplitudes are given by λ j (kinetic energy) for the spectrum of the
correlation tensor C (X, Y ) = ξ({X }) ⊗ ξ T ({Y }) where vectors {X } and {Y } represent all of
the points on the coarse grid. Finally, the choice of tolerance determines the choice
of N . See
Cotter et al. (2017
a) for an application of this stochastic data
assimilation approach, in which the “observed data”, or “truth”, is given by a highly resolved
numerical simulation. In practice, the noise correlation eigenvectors must be inferred
from observed data, on a case by case basis which will depend on the quality and
completeness of the data. However, the inference of velocity-velocity correlation tensors
is standard practice in fluid turbulence experiments, cf.
Berkooz et al. (1993)
.
Given N time-independent divergence free vector fields ξi (x ), i = 1, . . . , N , the
stochastic variational principle in
Holm (2015)
is formally written as
δ
0
T
(u)dt + p, dq + £dxt q V = 0,
(2.3)
with respect to variations δu, δq, δp and where dxt is defined as
dxt := u(t, x )dt +
ξi (x ) ◦ dWi (t ),
in which the vector fields ξi (x ) represent spatial correlations of the stochasticity and
dWi (t ) are independent Brownian motions, introduced in the Stratonovich sense. The
stationarity conditions are computed by fixing a space g∗ in nondegenerate duality
with g, relative to a pairing m, u g, u ∈ g, m ∈ g∗. The variations of (2.3) in δu, δp
and δq yield, respectively, the conditions
δ
δu = p
q,
q ∈ g∗, δδu ∈ g∗, and £uT p ∈ V ∗ are defined as
for q ∈ V , p ∈ V ∗, and u, δu ∈ g. The conditions (2.5) imply the following stochastic
equation:
δ δ
d δu + add∗xt δu = 0,
where adu∗ : g∗ → g∗ denotes the coadjoint operator defined by adu∗ m, v g =
m, [u, v] g, with [u, v] = v · ∇u − u · ∇v, and dxt is given in (2.4).
The notations used in (2.7) are general enough to make this equation valid for any
Lie group G and Lagrangian : g → R. See
Arnaudon et al. (2017
a) for a parallel
treatment of Model 1 for the rigid body and the group S O(3), as well as for the heavy
top, which involves advected quantities arising from symmetry breaking from S O(3)
to S O(2).
Upon choosing for g∗ the space of divergence free vector fields on D, i.e., g∗ = g,
and the duality pairing
m, u g =
m(x )·u(x ) dn x ,
the coadjoint operator is adu∗ m = P(u · ∇m + ∇uT · m), where P is the Hodge
projection onto divergence free vector fields. With the Lagrangian (2.1), the stochastic
Euler equation (2.7) becomes, in 3D,
D
N
i=1
du + P(u · ∇u)dt +
P(curl u × ξi ) ◦ dWi (t ) = 0.
(2.8)
Equation (2.8) can be written equivalently in vorticity form as
where ω = curl u is the vorticity and the stochastic vector field dxt is given in Eq. (2.4).
In 2D we identify the space of divergence free vector fields with the space of
functions on D modulo constants, the stream functions ψ . As explained in
Marsden
and Weinstein (1983)
, the dual space g∗ is identified with functions on D with zero
integrals, the absolute vorticities , via the duality pairing
, ψ g =
The absolute vorticity is related to the total fluid momentum m as = curl m · z.
For instance, for the Euler equation, the absolute vorticity coincide with the vorticity
ω = curl u · z, whereas for the rotating Euler equation, we have = curl u · z + f =
ω + f , where f is Earth’s frequency rotation.
In 2D, the stochastic Euler equation (2.7) becomes
dω + {ω, ψ }dt +
{ω, ψi } ◦ dWi (t ) = 0,
(2.10)
(2.9)
δ
0
T
1 This is a variational principle on the Pontryagin bundle T G ⊕ T ∗G → G, defined as the vector bundle
over G with vector fibre at g ∈ G given by Tg G ⊕ Tg∗G.
where, for two functions f, g on D, the function { f, g} is the Jacobian defined by
{ f, g} := ∂x1 f ∂x2 g − ∂x2 f ∂x1 g, with x = (x1, x2). In (2.10), ψ (t, x ) is the stream
function of the fluid velocity u(t, x ), ω(t, x ) = − ψ (t, x ) is its vorticity, and the
functions ψi (x ) are the stream functions of the divergence free vector fields ξi (x ). The
deterministic Euler equations are recovered in (2.8) and (2.10) when ξi = 0, for all
= 1, . . . , N .
We shall now make two crucial observations about this approach, that will allow us
later to extend the approach further to develop the other two stochastic models treated
in the paper.
Observation 1 Stochastic variational principles. Knowing that the deterministic
Euler equations in the Lagrangian fluid description arise from the Hamilton principle
δ
0
T
L(g, g˙)dt = 0,
(2.11)
for the right invariant Lagrangian L : T G → R given by the kinetic energy, we
expect Eq. (2.7) to arise, in the Lagrangian description, via a stochastic extension of
Hamilton’s principle (2.11). This is indeed the case if one proceeds formally here and
below by considering the stochastic Hamilton–Pontryagin (SHP) principle1
L(g, v)dt + π, dg − vdt −
ξi g ◦ dWi (t )
= 0,
(2.12)
for variations δg, δv, δπ . The variables v and π are, respectively, the material fluid
velocity and material fluid momentum. In (2.12), ·, · denotes the pairing between
elements in Tg∗G, and Tg G, the cotangent and tangent space to G at g. The notation
ξi g indicates the composition of the vector field ξi on the right by the diffeomorphism
g. Stochastic Hamilton–Pontryagin principles (SHP) have been considered for finite
dimensions in
Bou-Rabee and Owhadi (2009)
. The present paper considers SHP in
infinite dimensions for the first time. The SHP affords a systematic derivation of the
stochastic equations that preserves their deterministic mathematical properties, both
geometrical and analytical.
In the present paper, we shall consider stochastic variational principles in infinite
dimensions only in a formal sense for the purpose of modelling time-dependent spatial
correlations. The corresponding questions in analysis, for example, the questions of
local in time existence and uniqueness of solutions answered in
Crisan et al. (2017)
for the stochastic 3D Euler fluid model, will all be left open for the two new stochastic
geometric fluid models that are introduced in this paper.
Note that (2.12) imposes the stochastic process (1.1) as a constraint on the variations
by using the Lagrange multiplier π . From the G-invariance of both the Lagrangian
and the constraint, this principle can be equivalently written formally in the reduced
Eulerian description as
δ
T ⎡
0 ⎣
δ
0
with respect to variations δu, δg, δm, and where u = vg−1 ∈ g, m = π g−1 ∈ g∗.
This is the reduced stochastic Hamilton–Pontryagin (RSHP) principle.
One then directly checks that the stochastic variational principle (2.13) also yields
the stochastic equation (2.7). Thus, the two variational principles (2.3) and (2.13) both
yield the same stochastic equations. Moreover, in absence of stochasticity, Eq. (2.12)
recovers the Hamilton–Pontryagin principle for Lagrangian mechanics, see
Yoshimura
and Marsden (2006)
.
Remark 2.2 The RSHP principle in (2.13) has several interesting properties: (i) it
allows a formulation of reduction by symmetry in the stochastic context; (ii) it does
not need the introduction of the extra advected quantities q, p; and (iii) it does not
restrict the values of the Eulerian fluid momentum m ∈ g∗ to be of the form, m = p q.
In addition, as we will show later, the unreduced SHP principle (2.12) allows us to
consistently implement the new Model 2 and Model 3, in which the spatial correlation
eigenvectors ξi (x ) which are fixed functions of the spatial coordinates in Model 1
become time dependent through their flow dependence in Model 2 and Model 3.
Observation 2 Stochastic Hamiltonian formulations. We note that the SHP
principle (2.12) can be equivalently written as
N
i=1
L(g, v) + π, dg − vdt −
for the G-invariant functions Hi (_, _; ξi ) : T ∗G → R defined by
Hi (g, π ; ξi ) := π, ξi g = π g−1, ξi , i = 1, . . . , N .
g
(2.15)
The principle in (2.14) yields the following stochastic extension of the Euler–Lagrange
equations with Lagrangian L:
Consequently, we can call the functions Hi the stochastic Hamiltonians. Stochastic
Hamiltonian systems of the form (2.17) have been developed in
Bismut (1982)
. The
intrinsic form of Eqs. (2.16) and (2.17) on the Lie group G would require introducing
a covariant derivative. The formulation above is only valid locally.
These equations can be written in terms of the canonical Poisson bracket { ·, · }can
on T ∗G as
d F = {F, H }candt +
{F, Hi }can ◦ dWi (t ),
for arbitrary functionals F = F (g, π ) : T ∗G → R.
Consistently with this observation, we note that the stochastic equation (2.7) can
also be written in Hamiltonian form as
N
i=1
N
i=1
dm + ad∗δh m dt +
δm
ad∗δhi m ◦ dWi (t ) = 0,
δm
where h : g∗ → R and hi : g∗ → R are the reduced Hamiltonians associated to H
and Hi in (2.15), i.e., H (g, π ) = h(π g−1) and Hi (g, π ; ξi ) = hi (π g−1). We find
h(m) =
1
D 2
|m(x )|2dn x and hi (m) =
m, ξi g =
m(x )·ξi (x ) dn x . (2.20)
D
The expression (2.19) is the reduced (or Eulerian) formulation of the Hamiltonian
formulation (2.17).
(2.18)
(2.19)
In terms of the Lie–Poisson bracket {, }LP on g∗, given by
and with the stochastic Hamiltonians
Equation (2.7) and hence (2.19) can be formulated in the Stratonovich–Lie–Poisson
form
d f = { f, h}LPdt +
{ f, hi }LP ◦ dWi (t ),
(2.21)
for arbitrary functions f : g∗ → R, which is just the reduced form of (2.18).
For example, the stochastic 2D Euler equations (2.10) can be written in the
Stratonovich–Lie–Poisson form (2.21) with the Lie–Poisson bracket written on the
space of vorticities as
Marsden and Weinstein (1983)
{ f, h}LP(m) =
m,
δ f δh
,
δm δm
g
,
δ f δg
,
δω δω
d2x
hi (ω) =
ω(x )ψi (x ) d2x .
D
2.1 Aim of the Paper
By exploiting the two observations discussed above, this paper will develop two new
stochastic models by appropriate modifications of the stochastic Hamiltonians Hi and
of their symmetries.
3 Model 2: Frozen-in Correlations for Non-stationary Statistics
As discussed in the Introduction, we consider a model in which the eigenvectors ξi (X )
are advected by the flow map gt , giving the time-dependent vector fields ζi (t, x ).
Mathematically, the advection of a vector field by a smooth invertible map gt corresponds
to the push-forward operation,
ζi (t ) = (gt )∗ξi ,
(3.1)
meaning that
ζi (t, gt (X )) = Dgt (X ) · ξi (X ), for all X ∈ D.
From a Lie group point of view, the push-forward is given by the adjoint action
Ad: G × g → g of the Lie group G on its Lie algebra g, that is,
ζi (t ) = (gt )∗ξi = Adgt ξi .
The notation used in this section is general enough to make our developments valid for
any Lie group G. In our examples, we will take G to be the either the group of
diffeomorphisms or the group of volume preserving diffeomorphisms. The corresponding
coadjoint operators are
adu∗ m = u · ∇m + ∇uT · m + m div u and adu∗ m = P(u · ∇m + ∇uT · m).
(3.2)
The stochastic model considered here for advection of the eigenvectors is obtained
by modifying the expression and the symmetries of the stochastic Hamiltonians
Hi (g, π ; ξi ) in (2.14). Namely, given N vector fields ξi , i = 1, . . . , N , we consider
the principle
δ
0
T
(u)dt + m, dgg−1 − udt g −
where each of the stochastic Hamiltonians Hi (_, _; ξi ) : T ∗G → R is right invariant
only under the action of the isotropy subgroup of the eigenvector ξi with respect to
the adjoint action (i.e., the push-forward action), namely
Gξi = {g ∈ G | Adg ξi = ξi } = {g ∈ G | g∗ξi = ξi } ⊂ G.
That is, we have Hi (gh, π h; ξi ) = Hi (g, π ; ξi ), for all h ∈ Gξi . The SHP
principle (3.3) yields equations in the same general form as (2.17), but with stochastic
Hamiltonians which are not G-invariant. As we will show, this difference due to
symmetry breaking from G to Gξi induces significant changes in the reduced Eulerian
representation.
Physically, the symmetry breaking means that the initial conditions for the
correlation eigenvectors are “frozen” into the subsequent flow, as a property carried along
with individual Lagrangian fluid parcels, and which is not exchanged with other fluid
parcels.
Being only Gξi -invariant, the stochastic Hamiltonian function Hi induces, in the
Eulerian description, the reduced stochastic Hamiltonians
hi = hi (m, ζi ) : (T ∗G)/Gξi
g∗ × Oξi →
R,
hi (m, ζi ) = Hi (g, π ; ξi ), for m = π g−1, ζi = Adg ξi ,
where Oξi := {Adg ξi | g ∈ G} = {g∗ξi | g ∈ G} ⊂ g is the adjoint orbit of ξi . The
SHP principle (3.3) can thus be written in the reduced Eulerian form as
(3.4)
(3.5)
The stationarity conditions with respect to the variations δu, δm, δg, yield
δ
δu : δu − m = 0,
δm :
dgg−1 − udt −
δ δ
δg : d δu + adu∗ δu dt −
δhi
adζ∗i δζi ◦ dWi (t ) = 0,
N
i=1
N
i=1
δhi
δm ◦ dWi (t ) = 0,
(3.7)
where the advected eigenvector ζi := Adg ξi obeys the auxiliary equation dζi =
[dgg−1, ζi ], obtained from its definition. The expression of the coadjoint operator
ad∗ is given in (3.2). Upon using the second equation in (3.7), this auxiliary equation
becomes
N
j=1
ζi ,
δh j
δm
dζi + [ζi , u]dt +
◦ dW j (t ) = 0.
(3.8)
Remark 3.1 In our applications of this model, we will always assume that the
stochastic Hamiltonians hi in (3.8) do not depend on m. That is, δh j /δm = 0, so that the
eigenvectors ζi are advected only by the drift velocity, u. That is,
dζi + [ζi , u]dt = 0.
(3.9)
This model using correlation eigenvectors frozen into the drift velocity is quite different
from the model in
Holm (2015)
, which chose hi (m) = m, ξi g and all fluid properties
were advected by a velocity vector field comprising the sum of both the drift component
and the stochastic component.
3.1 Hamiltonian Structure
Denoting by h : g∗ → R the Hamiltonian associated to , the above system can be
equivalently written as
⎧
⎪⎪⎪⎪⎪⎨ dm + ad∗δδmh m dt −
⎡
δh
⎪⎪⎪⎪⎪⎩ dζi + ⎣ ζi , δm
dt +
N
i=1
N
j=1
δhi
adζ∗i δζi ◦ dWi (t ) = 0,
δh j ⎤
ζi , δm ⎦ ◦ dW j (t ) = 0.
One may check that this system is Hamiltonian with respect to the Poisson bracket
(3.10)
m,
−
N
i=1
δg
ζi , δm
,
δ f
δζi g
(3.11)
on g∗ × Oξ (m, ζ1, . . . , ζN ), where Oξ ⊂ gN is the orbit of ξ := (ξ1, . . . , ξN )
under the adjoint action of G. This reduced Poisson bracket is inherited from Poisson
reduction of the canonical Poisson bracket { ·, · }can on T ∗G, by the isotropy subgroup
Gξ ⊂ G of ξ := (ξ1, . . . ., ξN ). Namely, the map
(g, π ) ∈ T ∗G → (m, ζ1, . . . , ζN )
= (π g−1, Adg ξ1, . . . , Adg ξN ) ∈ (T ∗G)/Gξ
g∗Oξ
is Poisson with respect to the Poisson brackets { ·, · }can and { ·, · }red, see Appendix.
The system (3.10) admits the Stratonovich–Poisson formulation,
for arbitrary functions f : g∗ × O → R. Note that in (3.12) the Hamiltonians h and
hi , i = 1, . . . , N , depend a priori on all the variables (m, ζ1, . . . , ζN ). The system
(3.10) is recovered when h depends only on m, while the hi depend only on m and ζi
(not on ζ j , for j = i ). The Poisson tensor at (m, ζ1, . . . , ζN ) reads
− adζN
· · ·
0
adζ∗N ⎤
⎥⎥ .
⎥
⎦
(3.13)
Remark 3.2 (Itô form) In the special case that δhi /δm = 0, as assumed in Remark
3.1, the Itô form of Eqs. (3.10) does not introduce any additional drift terms. That
is, in this special case, the Itô form of (3.10) is obtained by simply removing the
Stratonovich symbol ( ◦ ). However, in the general case, if δhi /δm = 0, the Itô form
does contain additional drift terms. These additional drift terms in the Itô form for
the general case can be computed in a standard way, but the equations then may take
a more complicated form. By making use of the Poisson formulation in terms of the
bracket { ·, · }red in (3.12), we can write the additional drift terms in the Itô form for
the general case in a concise way as
d f =
#
1 $
{ f, h}red − 2 {hi , {hi , f }red}red dt +
N
i=1
{ f, hi }red ◦ dWi (t ).
(3.14)
Example 3.3 (Incompressible 2D models) In the 2D incompressible case, we can
identify the Lie algebra g with the space of differentiable functions on D, modulo
constants. These are the stream functions, denoted ψ . We use the L2 duality pairing
, ψ g = (x )ψ (x )d2x and identify g∗ with the space of functions on D with
zero integral.DThese are the absolute vorticities, denoted , as explained in
Marsden
and Weinstein (1983)
.
Let ψi0(X ) be the stream function associated to the eigenvector ξi (X ), i = 1, . . . , N .
The stream function ψi (t, x ) of the advected eigenvector ζi (t ) = (gt )∗ξi is found to
be
ψi (t, gt (X )) = ψi0(X ).
The stochastic model (3.10) applied to 2D incompressible fluid dynamics with
Hamiltonian h( ) and stochastic Hamiltonians hi ( , ψi ) = hi (ψi ) is given by
d
+
,
δh
δ
dt +
N
i=1
δhi
ψi , δψi
(3.15)
where we have used the formula ad∗ψ = { , ψ } for the coadjoint operator for 2D
incompressible fluids. For example, with the appropriate choice of the Hamiltonian,
we can write the stochastic model (3.15) for the following cases:
(a) 2D perfect fluid: ψ = δδh = − −1 ;
(b) 2D rotating perfect fluid: ψ = δδh = − −1(
(c) 2D rotating quasigeostrophy (QG): ψ = δδh
− f );
= −(
− F )−1(
− f );
where F and f denote, respectively, the square of the inverse Rossby radius and the
rotation frequency. For the stochastic Hamiltonians hi , i = 1, . . . , N , one may choose
◦ dWi (t ) = 0,
dψi +
δh
ψi , δ
dt = 0,
D
ψi (x )ψi (x ) d2x , (no sum),
in which case δδψhii = − ψi .
Similarly to the discussion in Remark 3.2, the Itô form of equations (3.15) takes the
same expression, since we have chosen stochastic Hamiltonians for which δhi /δ =
0. In this case, one may obtain the Itô forms by simply replacing the Stratonovich
noise ◦ dWi (t ) by the Ito noise dWi (t ) without modifying the drift terms.
Remark 3.4 (Conserved correlation enstrophies) The stochastic model (3.15) does
not preserve the well-known vorticity enstrophies (ω) = (ω)d2x , which are
preserved in the deterministic case. However, the stochastic eqDuation for ψi in (3.15)
does preserve the following correlation enstrophy functionals
i (ψi ) =
D
(ψi )d2x ,
when the stochastic model (3.10) is applied to 2D incompressible fluid dynamics. In
general, for the case h = h(m) and hi = hi (ζi ), by Eq. (3.10), the corresponding
functionals i (ζi ) verify
d i =
{ i , h}redd2x dt = −
D
D
δh
ζi , δm
,
δ i
δζi g
d2x dt
which vanishes for 2D incompressible fluids after integration by parts and imposition
of homogeneous boundary conditions.
Example 3.5 (3D incompressible Euler) We consider the stochastic Hamiltonians
hi (ζi ) =
Fi (ζi (x ))d3x , i = 1, . . . , N ,
D
where Fi are smooth functions. Upon using the formula from (3.2) for incompressible
flows,
adu∗ m = P(u · ∇m + ∇uT · m) = P(curl m × u),
where P is the Hodge projector onto divergence free vector fields, the stochastic model
(3.10) reads
⎧
⎪⎨ du + P(u · ∇u)dt =
i=1
⎪⎩ dζi + curl(ζi × u)dt = 0.
N
P #curl δ Fi
δζi
$
× ζi ◦ dWi (t )
The stochastic terms can be written equivalently with the help of the stress tensors
σi = ζi ⊗ ∂∂Fζii + Fi (ζi )I, i.e., (σi )ab = ζia ⊗ ∂∂ζFibi − Fi (ζ )δba ,
and pressures pi , i = 1, . . . , N . With these definitions, Eq. (3.18) becomes
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
du + (u · ∇u + ∇ p)dt =
(div σi + ∇ pi ) ◦ dWi (t ),
N
i=1
where the divergence is defined as (div σi )b = ∂a (σi )ab for all i , and the individual
pi are each found by solving a Poisson equation, with boundary conditions given by
nˆa (div σi )a = 0. Recall that the vector fields ζi (t, x ) are obtained from the given
eigenvectors field ξi (X ) by the push-forward operation (3.1). As mentioned earlier in
Remark 3.2, the Itô forms of the equations have the same expression.
Remark 3.6 The three-dimensional system (3.18) is reminiscent of incompressible
magnetohydrodynamics (MHD), except it has a stochastic “ J × B ” force depending
on the sum over all of the ζi . In following this analogy with MHD, we may introduce
vector potentials αi by writing ζi =: curl αi . Having done so, one notices that evolution
under the stochastic system (3.18) preserves the integrals
i =
αi · ζi d3x (No sum).
Proof By the second equation in the equation set (3.18), we have
d i = d
αi · ζi d3x = −2
αi · curl(ζi × u) d3x dt = 0.
D
D
The i integrals are topological quantities known as correlation helicities which
measure the number of linkages of the lines of each vector field ζi with itself.
Conservation of the correlation helicity i means that evolution by the stochastic system
(3.18) cannot unlink the linkages of each divergence free vector field ζi with itself.
This conclusion is the analogue of conservation of magnetic helicity in MHD.
3.2 Inclusion of Additional Advected Tensor Fields
More generally, suppose that the fluid model involves a tensor field q(t, x ) advected
by the fluid flow as in (2.2). The evolution of this advected field is this given by
q(t ) = (gt )∗q0, where q0(X ) is the initial value and gt ∈ G is the fluid flow. In this
case, the variational principle is written in reduced Eulerian form as
δ
0
T
(u, g∗q0)dt + m, dgg−1 − udt g −
(3.21)
for variations δu, δm, and δg. The stationarity conditions yield the same first two
equations of (3.7), whereas the third one becomes
N
i=1
N
i=1
δ δ
d δu + adu∗ δu dt −
δhi δ
adζ∗i δζi ◦ dWi (t ) = δq
q dt.
(3.22)
From its definition, the quantity q(t ) = (gt )∗q0 verifies dq + £dgg−1 q = 0. For the
case that δhi /δm = 0, this quantity is governed by the ordinary advection equation,
dq + £u q dt = 0.
Remark 3.7 (Itô form) For the case that δhi /δm = 0, passing to the Ito formulation
does not introduce any change in the drift terms.
Example 3.8 (Rotating shallow water) Equations (3.22) and (3.23) for the inclusion
of such advected quantities into Model 2 may be illustrated with the example of the
(3.23)
rotating shallow water equation. In this case D is a two-dimensional domain and G is
the group of diffeomorphisms of D. Let us denote by η(t, x ) the water depth, by B(x )
the bottom topography, by R(x ) the Coriolis vector field, and x = (x1, x2) ∈ D. The
Lagrangian of the rotating shallow water system is
(u, η) =
D
1 1
2 η|u|2 + η R · u − 2 g(η − B)2 d2x ,
(3.24)
where g is the gravity acceleration. Taking the stochastic Hamiltonians hi (ζi ) in (3.16),
the stochastic variational principle in (3.21) produces the equations
⎧ 1 N
⎪⎨ du + (u · ∇u + curl R × u + g∇(η − B))dt = η
⎪⎩ dη + div(ηu)dt = 0,
dζi + [ζi , u]dt = 0,
i=1
div σi ◦ dWi (t )
(3.25)
where the stochastic stress σi is defined as above in (3.19). The effect of σi can be seen
by writing the Kelvin circulation theorem obtained by integrating the first equation in
(3.25) around a loop c(u) moving with the drift velocity u(t, x ), to find
d %
c(u)
&u + R' · dx =
N
div σi ◦ dWi (t )) · dx .
Thus, the total stochastic stress generates circulation of the total velocity (u + R)
around any loop moving with the relative fluid velocity u in the rotating frame.
Example 3.9 (Rotating compressible barotropic fluid) The above developments easily
extend to other fluid models such as compressible barotropic fluid flow in a rotating
frame, whose Lagrangian is
(u, ρ) =
D
ρ( 21 |u|2 + u · R − e(ρ) − gz)d3x ,
(3.26)
where z = x3 is the vertical coordinate, ρ is the mass density and e is the specific
internal energy. With the choice (3.16), one gets from (3.22), the stochastic balance
of momentum
1 1 N
du + (u · ∇u + curl R × u + gz + ρ ∇ p)dt = ρ
i=1
div σi ◦ dWi (t ),
with advection equations
4 Model 3: Eigenvectors Depending on Advected Quantities
The stochastic model detailed in this section applies to fluids with advected quantities.
As for the model described in §3, the model in this section also introduces a
modification of the symmetries of the stochastic Hamiltonians Hi in the SHP principle
(2.14). Namely, in the presence of an advected tensor field q(t, x ), evolving as by the
push-forward q(t ) = (gt )∗q0, we assume that Hi depends on q0. This dependence
breaks the G-symmetry of Hi . In particular, we shall take Hi (_, _; q0) : T ∗G → R
to be a Gq0 -invariant function, i.e.,
Hi (gh, π h; q0) = Hi (g, π ; q0), for all h ∈ Gq0 ,
where Gq0 = {g ∈ G | g∗q0 = q0} ⊂ G is the isotropy subgroup of the tensor
field q0 under the pull-back action. This is similar to the symmetry assumed on the
Lagrangian L(_, _; q0) : T G → R for such fluids, studied in
Holm et al. (1998)
. With
this assumption, the SHP principle (2.14) becomes
L(g, v; q0) + π, dg − vdt −
Hi (g, π ; q0) ◦ dWi (t )
= 0.
(4.1)
If we also assume that Hi is linear in the material fluid momentum π , then it necessarily
takes the form
Hi (g, π ; q0) = π g−1, ξi (g∗q0)
g
(4.2)
for a function ξi : Oq0 → g defined on the G-orbit of q0, Oq0 = {g∗q0 | g ∈ G}.
In the simplest case, the vector fields ξi only depend on the pointwise values of the
tensor fields, denoted as ξi (q(x ), x ), but more general dependencies are possible, in
which, for example, the vector fields ξi would depend on the spatial gradients of the
tensor fields.
The reduced Eulerian version of this SHP principle reads
δ
0
T
δ
0
T
N
i=1
N
i=1
(u, g∗q0) + m, dgg−1 − udt −
ξi (g∗q0) ◦ dWi (t ) g
= 0.
(4.3)
The stochastic Hamilton–Pontryagin principle (4.1) and its reduced form (4.3) extend
to the stochastic case the Hamilton–Pontryagin principles with advection developed
in
Gay-Balmaz and Yoshimura (2015)
.
By comparing (4.3) with (2.13), we observe that after the introduction of symmetry
breaking in this SHP principle (4.1) in allowing the stochastic Hamiltonians in (4.2)
to depend functionally on the initial advected quantities q0 the reduction process has
allowed the vector fields ξi , i = 1, . . . , N in the RSHP (4.3) to depend on the advected
tensor field q(t ) = (gt )∗q0, as we have sought.
(4.4)
(4.5)
(4.6)
ρ , v =
⎧
⎪⎪⎪⎪⎨ dm + ad∗δδmh m dt +
⎪⎪⎪ dq + £ δh q dt +
⎪⎩ δm
N
N i=1
£ δhi q ◦ dWi (t ) = 0.
i=1 δm
ad∗δhi m ◦ dWi (t ) = −
δm
* δh
dt +
δq
N
i=1
δhi
δq ◦ dWi (t )
# δ
δq
dt −
( δ ∂ξi )
δu · ∂q
$
q,
N
i=1
where dxt := u dt +
verifies
N
i=1 ξi (q) ◦ dWi (t ) and the advected tensor field q(t ) = (gt )∗q0
dq + £dxt q = 0.
The expression of the coadjoint operators ad∗ in the compressible and incompressible
cases are recalled in Eq. (3.2).
δ ∂ξi ) is the composition of two linear maps. The
In (4.4), the dot product ( δu · ∂q
contraction in the dot product is taken on the vector indices of the variational derivative
δu with the derivative of the vector field ∂∂ξqi of ξi (q) with respect to q, so that, using
δ
(2.6),
=
,
for v ∈ g. For a density, q = ρ, we have φ ρ , v = φ, div(ρv) , so φ
− ρ∇φ, v for a scalar function φ. Hence, in this example, we have
# δ ∂ξi $
δu · ∂ρ
ρ , v
= − ρ∇
g
# δ ∂ξi $
δu · ∂ρ
, v
g
.
4.1 Hamiltonian Structure
From right-invariance of the Hamiltonian (4.2) under the isotropy subgroup Gq0 , we
obtain the reduced stochastic Hamiltonians
hi : g∗ × Oq0 →
R, hi (m, q) :=
m, ξi (q) g , i = 1, . . . , N ,
(4.7)
see Appendix for more discussion. Upon denoting by h : g∗ × Oq0 → R, the
Hamiltonian associated to , the above system can be written equivalently as
Applying Hamilton’s principle to the reduced action integral in (4.3) now results
in the stochastic motion equation for the model that we will discuss in this section,
(4.8)
One may check that system (4.8) is Hamiltonian with respect to the Poisson bracket
{ f, g}red(m, q) =
m,
g
+ £ δf q,
δm
δg
δq V
− £ δg q,
δm
δ f
δq V
,
(4.9)
on g∗ × Oq0 (m, q). This Poisson bracket is inherited by Poisson reduction of the
canonical Poisson bracket on T ∗G, by the isotropy subgroup Gq0 ⊂ G of q0, namely
the map
(g, π ) ∈ T ∗G → (m, q) = (π g−1, g∗q0) ∈ (T ∗G)/Gq0
g∗ × Oq0
is Poisson with respect to these brackets, as discussed in Appendix. Thus, the system
(4.8) admits the Stratonovich–Poisson formulation
N
d f = { f, h}reddt +
{ f, hi }red ◦ dWi (t ) = 0,
(4.10)
where both h and hi depend on m and q.
Remark 4.1 (Itô forms) The Itô forms of the equations in (4.8) are rather involved. In
particular, the Itô correction in the drift term of the momentum equation is
1
MItodt = − 2
∂ξi )
adξ∗i (q) adξ∗i (q) m + adξ∗i (q) #(m · ∂q
q + #((m · ∂∂ξqi ) q) · ∂∂ξqi $
∂2ξi ∂ξi
+ ad∗∂∂ξqi ·£ξi (q)q m + m · ∂q2 ( ·, £ξi (q)q) q + m · ∂q
q
£ξi (q)q dt
Likewise, the Itô correction in the drift term of the advection equation is
1
AItodt = − 2
£ξi (q)£ξi (q)q + £ ∂∂ξqi ·£ξi (q)q q dt.
The drift terms MItodt and AItodt above follow from the double bracket terms in
Eq. (3.14).
Example 4.2 (Rotating shallow water) Upon choosing the Lagrangian (3.24) for the
rotating shallow water equation, we obtain from (4.4)
du + £dxt (u + R) = ∇( 21 |u|2 + R · u − g(η − B))dt
N
−
i=1
∇(η(u + R) · ∂∂ξηi ) ◦ dWi (t ),
(4.11)
(4.12)
where dxt := udt +
as
N
i=1 ξi (η)◦dWi (t ). This equation may be written more explicitly
du + (u · ∇u + curl R × u) dt +
£ξi (u + R) ◦ dWi (t )
N
= − g∇(η − B)dt −
N
i=1
∂ξi )
∇(η(u + R) · ∂η
◦ dWi (t ).
The advection equation for the surface elevation in this model is
dη + div(ηdxt ) = 0.
Taking the curl of the momentum equation yields the corresponding RSW vorticity
equation
dω + div(ωdxt ) = 0 for ω = curl
· zˆ = curl(u + R) · zˆ.
# 1 δ $
η δu
This, together with the advection equation (4.14) yields the potential vorticity (PV)
equation,
d Q + dxt · ∇ Q = 0 for Q := ω/η,
which expresses conservation of potential vorticity along the stochastic fluid particle
path dxt .
Remark 4.3 We shall now consider a specific expression for ξi (h) that yields a
simplified expression for the stochastic PDE (4.13). First, we note that by assembling the
stochastic terms in (4.13), we get, in slightly abbreviated notation,
(4.13)
(4.14)
∂ξi )
£ξi (u + R) + ∇(η(u + R) · ∂η
= ξi · ∇(u + R) + ∇ξiT · (u + R)
∂ξi )
= ξi · ∇(u + R) + ∇(ξi · (u + R)) − ∇(u + R)T · ξi + ∇(η(u + R) · ∂η
= curl(u + R) × ξi + ∇(ξi · (u + R) + η(u + R) · ∂∂ξηi ).
1, . . . , N , we have ∂∂ξηi = η−2 Xi and ∂∂ξηi ∗ · m = η−2(m · Xi ), so that
Upon making the choice ξi (η) := −η−1 Xi , for a set of fixed vector fields Xi , i =
∇(ξi · (u + R) + η(u + R) · ∂∂ξηi )
= 0,
N
i=1
N
i=1
in the expression above. In this case, the stochastic RSW equations (4.13) take the
simpler form
N
du + (u · ∇u + curl R × u) dt = −g∇(η − B)dt −
curl(u + R) × ξi (η) ◦ dWi (t ).
curl(u + R) × ξi (η) ◦ dWi (t ) = − Q Xi⊥ ◦ dWi (t ).
Example 4.4 (Rotating compressible barotropic fluid) Upon choosing the tensor field
q(t, x ) to be the mass density ρ(t, x ), the stochastic model (4.4) yields
δ δ
d δu + £dxt δu = ρ∇
( δ
δρ
dt −
( ∂∂ξρi · δδu ) ◦ dWi (t )),
where dxt := udt +
N
i=1 ξi (ρ) ◦ dWi (t ). Upon using the advection equation,
dρ + div(ρ dxt ) = 0,
the previous equation can be written equivalently as
d
# 1 δ $
ρ δu
Upon taking the Lagrangian (3.26) for the compressible barotropic fluid, we obtain
the stochastic motion equation,
N
i=1
N
i=1
−
1
du + (u · ∇u + curl R × u + gz + ρ ∇ p)dt
= −
( curl(u + R) × ξi + ∇(ξi · (u + R))) ◦ dWi (t )
# ∂ξi $
∂ρ · ρ(u + R) ◦ dWi (t ),
in which the thermodynamic pressure is defined by p = ρ2 ∂∂ρe .
Taking the curl of the momentum equation and using the advection equation dρ +
div(ρ dxt ) = 0 now implies PV conservation as
1
d Q + dxt · ∇ Q = 0, where Q := ρ curl
# 1 δ $
ρ δu
1
· ∇ρ = ρ curl(u + R) · ∇ρ .
Similarly to the case of the RSW equations, with the choice ξi (ρ) = − Xi /ρ for a set
of fixed vector fields Xi , i = 1, . . . , N , the previous motion equation simplifies to
Remark 4.5 The last two examples preserve the integrals
N
i=1
curl(u + R) × ξi (ρ) ◦ dWi (t ).
C =
D
ρ
(Q) d3x ,
for smooth functions , as may be seen either from the Hamiltonian formulation (4.10)
with Poisson bracket (4.9) with q = ρ, or by direct verification. These conserved
quantities are Casimirs for the Poisson bracket. They are the same as the conserved
potential vorticity integrals in the deterministic case, because the Poisson bracket
persists in passing to the stochastic case.
5 Conclusions
This paper has used standard methods from symmetry breaking in geometric
continuum mechanics, as discussed, for example, in
Holm et al. (1998)
and
Gay-Balmaz
and Tronci (2010)
, to set out three different approaches for incorporating stochastic
transport into ideal fluid dynamics. As mentioned in the Introduction, our approach
preserves the relabelling symmetry of fluid dynamics, up to isotropy of the advected
quantities. Therefore, we may summarise the results of these approaches, simply by
comparing their advection laws and the symmetry breaking terms of their respective
Kelvin–Noether circulation theorems.
5.1 Model 1—Time-Independent Spatial Correlation Eigenfunctions
In Model 1, reviewed in Sect. 2, the time-independent spatial statistical correlation
eigenvectors ξi (x ) are obtained as eigenvectors of an appropriate correlation
function, which is assumed to be time independent. From Eqs. (2.19) and (2.20), possibly
extended to include advected quantities q, the corresponding Kelvin circulation
theorem is given by
d %
1 δ
c(dxt ) ρ δu · dx =
where ρ is the mass density obeying the continuity equation, dρ + div(ρdxt ) = 0,
and the advected tensor field q = (gt )∗q0 satisfies
(5.1)
(5.2)
where £vq denotes Lie derivative of advected quantity a by the stochastic vector field
dxt = u(x , t )dt +
ξi (x ) ◦ dWi (t ).
N
This is the model introduced in
Holm (2015)
.
5.2 Model 2—Time-Dependent Advected Statistical Correlation Eigenvectors
From Eq. (3.22) for Model 2, discussed in Sect. 3 we have the circulation dynamics
d %c(u) ρ1 δδu · dx =
( 1 N
c(u) ρ i=1
δhi
adζ∗i δζi ◦ dWi (t )) · dx ,
and with δhi /δm = 0 the advected tensor fields q(t ) = (gt )∗q0 and ζi (t ) = Adgt ξi
verify
dq + £u q dt = 0, and dζi + [ζi , u]dt = 0.
In particular, the mass density verifies the ordinary continuity equation dρ +
div(ρu)dt = 0.
This means that in Model 2, treated in Sect. 3, the choice of stochastic Hamiltonians
satisfying δhi /δm = 0 implies that the transport velocity of the circulation loop in
(5.3) reduces to the drift velocity, u(x , t )dt , alone, rather than the entire stochastic
velocity dxt , as in Model 1. Moreover, for δhi /δm = 0, the Stratonovich and Itô
representations of Model 2 take the same form.
Remark 5.1 In deriving the Kelvin theorem in Eq. (5.3) from (3.22), we have used the
relation
N
i=1
δhi
adζ∗i δζi =
N
i=1
δhi
δζi
ζi ,
which arises from equivalence of the definitions of ad∗ and the diamond operator ( )
when the advected quantity is a vector field. This notation makes it clear that the drift
term and the stochastic term on the right-hand side of Eq. (5.3) both arise from the
same source, namely symmetry breaking of the Lie group of diffeomorphisms to the
isotropy subgroup of the initial condition for an advected quantity.
(5.3)
(5.4)
(5.5)
5.3 Model 3—Time-Dependent Spatial Correlation Eigenvectors Depending on
Advected Quantities
From Eq. (4.4) for Model 3, discussed in Sect. 4, we have the circulation dynamics
d %
1 δ
c(dxt ) ρ δu · dx =
%
c(dxt ) ρ
i=1
N
i=1 ξi (q) ◦ dWi (t ) and the advected tensor fields q = (gt )∗q0
where £dxt q denotes Lie derivative of advected quantity q by the stochastic vector
field
N
dxt := udt +
ξi (q) ◦ dWi (t ).
In particular the mass density satisfies dρ + div(ρdxt ) = 0.
In Model 3, treated in Sect. 4, the velocity of the circulation loop is dxt . As in Model
1 this is the full stochastic velocity, except now its spatial correlation eigenvectors
ξi (q) respond to the motion of the advected quantities which also contribute to the
thermodynamic and potential energy properties of the fluid.
Remark 5.2 The Itô forms of these three models have been computed in the body of
the paper, although the additional Itô drift terms may take a more complicated form
than in the Stratonovich case. An example is the additional Itô drift term MItodt in
(4.11) for Model 3. Nonetheless, the equivalent Itô forms of their circulation theorems
may be computed by the standard methods from their Stratonovich forms, namely by
transforming both the integrand and the velocity of the loop into Itô form, using the
corresponding motion equations.
Remark 5.3 Besides the initial and boundary conditions, the key to applying the
methods of this paper efficiently begins with the choice of the spatial correlation
eigenvectors. Once a data assimilation step for determining the eigenvectors is
completed, such as the one discussed in Remark 2.1 based on
Cotter et al. (2017
a), then the
stochastic Hamilton’s principle provides the dynamics of the drift velocity. Thus, the
present approach may be regarded as a data-driven approach to dynamically and
selfconsistently separating the Lagrangian paths into their drift dynamics and stochastic
parts. This question needs to be confronted in any application to any data set. Model 1
chooses to model the stochastic parts by using optimal eigenvectors of an equilibrium
spatially-constant time-independent correlation function. Models 2 and 3 provide two
different approaches for making these correlation eigenvectors time dependent. For a
detailed discussion of methods for modelling the separation between drift and
stochasticity in comparing numerical simulations of fluid flows at fine and coarse resolutions,
see
Cotter et al. (2017
a).
In applications, these three models of the stochastic fluid velocity vector field could
be used either separately, or in combinations, by taking combinations of the correlation
eigenvectors to evolve in any way that might be needed for tuning the stochastic
transport, e.g., in applying them for data assimilation.
In conclusion, we mention that the three models of stochastic fluid dynamics treated
here may also be transferred into data assimilation methods in biomedical image
analysis for computational anatomy. In particular, the approach of
Arnaudon (2017
b)
for stochastic image analysis based on Model 1 could be extended to create new
approaches to computational anatomy based on the ideas underlying Model 2 and
Model 3.
Acknowledgements We are very grateful to C. J. Cotter, D. Crisan, E. Mémin, S. Olhede, W. Pan, V.
Resseguier, I. Shevshenko, and A. Sykulski for valuable discussions and presentations during the course
of this work. During this work, FGB was partially supported by the ANR project GEOMFLUID
14-CE230002-01 and DDH was partially supported by the European Research Council Advanced Grant 267382
FCCA and EPSRC Standard Grant EP/N023781/1.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix
Appendix provides additional background for some of the standard notations from
geometric mechanics used in the paper and gives some details on the structure of
the reduced Poisson brackets { ·, · }red arising in Model 2 and Model 3. For further
geometric mechanics background, see
Marsden and Ratiu (1994)
and
Holm (2011)
.
Lie Group Notation
Given a Lie group G, and two elements g, h ∈ G, we denote by gh the Lie group
multiplication. The spaces Tg G and Tg∗G refer to the tangent and cotangent spaces of
G at g. For v ∈ Tg G and π ∈ Tg∗G, we denote by vh ∈ Tgh G and π h ∈ Tg∗h G, the
right translation of v and π by h ∈ G. These right translations are defined as follows:
d
dε ε=0
gε h
and
π h, v :=
π, vh−1 , for all v ∈ Tgh G,
where gε ∈ G is a path with gε=0 = g and ddε ε=0 gε = v, and ·, · is the duality
pairing between the tangent and cotangent spaces of G. We have g = Te G and g∗ =
Te∗G, where e is the neutral element in G. The adjoint action of g ∈ G on u ∈ g is
where hε ∈ G is a path with hε=0 = e and ddε ε=0 hε = u.
When G is the group of diffeomorphism of the fluid domain D, the multiplication
gh corresponds to the composition of diffeomorphisms. Elements v ∈ Tg G are vector
field on D covering the diffeomorphism g and vh is the composition on the right by
the diffeomorphism h. In particular, elements in the Lie algebra g are vector fields on
D. In the paper, we have extensively used the fact that on groups of diffeomorphisms,
Adg u = g∗u, the push-forward of vector fields.
Reduced Poisson Brackets
The Poisson brackets (3.11) and (4.9) of models 2 and 3 both have the same structure,
which we now explain. Both brackets are inherited from the canonical Poisson bracket
{ ·, · }can on T ∗G, by a reduction by symmetry associated to a subgroup of G.
Consider a right action of G on a manifold P, denoted as follows (n, g) ∈ P ×G →
φg(n) ∈ P. Given n0 ∈ P, we denote by Gn0 = {g ∈ G | φg(n0) = n0} ⊂ G the
isotropy group of n0 and by On0 = {φg(n) | g ∈ G} ⊂ P the orbit of n0. The Lie
algebra action of u ∈ g on n ∈ N is written as
d
n ·u := dε ε=0 φgε (n) ∈ Tn N ,
weTrehgeεqu∈oGtieinst asppaactehowfiTth∗gGε=b0y =Gne0 arenldatidvdεe εto=0theεa=ctiuo.n given by (g, π ) ∈ T ∗G →
g
(gh, π h) ∈ T ∗G, h ∈ Gn0 , is denoted as (T ∗G)/Gn0 . One observes that we have the
identification (T ∗G)/Gn0 g∗ × On0 given by
[g, π ]Gn0 ∈ (T ∗G)/Gn0 → (m, n) = (π g−1, φg−1 (n0)) ∈ g∗ × Oq0 .
h : g∗ × On0 →
A Gn0 -invariant Hamiltonian H : T ∗G → R thus induces the reduced Hamiltonian
R defined by
H (g, π ) = h (π g−1, φg−1 (n0)) .
This formula has been used several times in the paper, for instance in (3.5) and (4.7)
for the stochastic Hamiltonians. The reduced Poisson bracket on g∗ × On0 induced by
the canonical Poisson bracket { ·, · }can on T ∗G is given by
{ f, g}red(m, n) =
m,
g
δh δ f
+ δn , n · δm
δ f δg
− δn , n · δm ,
(5.8)
(5.9)
for functions f, g : g∗ ×On0 → R, see, e.g.,
Gay-Balmaz and Tronci (2010)
(Theorem
2.7) for a proof of this fact. The last two terms involve the pairing ·, · between T ∗On0
and T On0 . We now explain how this Poisson reduction T ∗G → g∗ × On0 applies to
Models 2 and 3.
For Model 2, we have P = g × · · · × g (ζ1, . . . , ζN ), n0 = (ξ1, . . . , ξN ) = ξ
and the action φg given by the push-forward of vector fields Adg ξ = g∗ξ on each
term. With these choices, the reduced Poisson bracket (5.9) recovers the expression
(3.11). For the system (3.22), we take P = V × g × · · · × g (q, ζ1, . . . , ζN ),
n0 = (q0, ξ1, . . . , ξN ) and the action φg given by the push-forward action on each
terms.
For Model 3, we have P = V q, n0 = q0 and the action φg given by the
push-forward of tensor fields, φg (q) = g∗q. With these choices, the reduced Poisson
bracket (5.9) recovers the expression (4.9).
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