# Stochastic effects of waves on currents in the ocean mixed layer

Darryl D. Holm and Ruiao Hu
Department of Mathematics
Imperial College London SW7 2AZ, UK

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###### Abstract

This paper introduces an energy-preserving stochastic model for studying wave
effects on currents in the ocean mixing layer. The model is called stochastic
forcing by Lie transport (SFLT). The SFLT model is derived here from a
stochastic constrained variational principle, so it has a Kelvin circulation
theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating
shallow water dynamics and the Euler-Boussinesq equations. In each example,
one sees the effect of stochastic Stokes drift and material entrainment in the
generation of fluid circulation. We also present an Eulerian-averaged SFLT
model (EA SFLT), based on decomposing the Eulerian solutions of the energy-
conserving SFLT model into sums of their expectations and fluctuations.

##  1 Introduction

#### Wave effects on currents (WEC).

In studies of the ocean mixing layer (OML) the problem of wave effects on
currents (WEC) arises. For example, surface gravity waves can drive Langmuir
circulations which have important influences on near-surface currents [50].
Langmuir circulations are horizontally oriented pairs of oppositely
circulating vortex tubes aligned generally along the direction of the wind, as
reviewed, e.g., in [46, 26, 70, 24]. Because they represent arrays of
organised fluid transport, Langmuir circulation patterns can produce vertical
transport which strongly entrains sediment and detritus into the OML from both
above and below. This means they can have a strong effects, for example, on
the dispersion of oil spills in a shallow sea, [69].

Most previous studies of turbulence in the OML have been performed in the
context of wave-averaged dynamics. For a recent review, see [1]. The
underlying assumption is that the surface gravity waves represent the fastest
component in the system and are only weakly modulated by the other components
(turbulence and currents). Averaging the stratified Euler equations in three
spatial dimensions over a time scale longer than the wave period produces a
modified set of equations, known as the CL equations, after Craik and
Leibovich, [15].

The temporal-averaging basis of the CL equations introduces Stokes-drift
effects which represent additional wave-averaged forces and material advection
terms that emerge from multi-scale asymptotic theories [15, 61]. Remarkably,
the ideal CL equations preserve many of the properties of the original
stratified Euler equations. For example, the ideal CL equations conserve
energy and have a Hamiltonian formulation [34].

The aim of the present paper is to develop a stochastic theory of WEC which
encompasses the Craik-Leibovich theory and still preserves the original
energy. Indeed, in deriving the stochastic theory, we will take an approach
which can be adapted to select whichever primary conservation laws of the
deterministic theory are desired. Specifically, our approach will use a
stochastic version of a method from classical mechanics known as the Reduced
Lagrange-d’Alembert-Pontryagin (RLDP) formulation of constrained dynamics
which is reduced by a symmetry of the Lagrangian in Hamilton’s principle.
Stochastic applications of the RLDP formulation are discussed, e.g., in [10,
27].

#### RLDP result for the stochastic Craik-Leibovich theory for the Euler-
Boussinesq equations.

Specific results of the theoretical developments in this paper can be assessed
by simply examining the example of applying the RLDP approach to the ideal CL
equations themselves in section 4.3.

One finds in this example that the energy-preserving stochastic EB equations
in (4.31) produce a stochastic contribution to the vortex force in the Craik-
Leibovich equations [15] whose deterministic formulation with Hamilton’s
principle is given in [34]. Namely, they reduce as follows, where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}$ denotes the Stratonovich stochastic time differential of Eulerian
fluid velocity ${\bf u}$, the Stokes drift velocity is ${\bf u}^{S}({\bf x})$,
the Coriolis parameter is $2\bm{\Omega}={\rm curl}{\bf R}({\bf x})$, the
pressure is $p$, the volume element is $D$, the buoyancy is $b$ with
gravitational constant $g$, and the stochastic spatial modes obtained via data
calibration are denoted ${\bf f}^{k}({\bf x})$

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}-{\bf u}\times{\rm curl}\big{(}{\bf u}-{\bf u}^{S}({\bf x})+{\bf
R}({\bf x})\big{)}dt&=-\,\nabla\Big{(}p+\frac{1}{2}|{\bf u}|^{2}+{\bf
u}\cdot{\bf u}^{S}\Big{)}dt-gb\,\mathbf{\hat{z}}\,dt\\\
&\qquad+\sum_{k>0}\Big{(}{\bf u}\times{\rm curl}\,{\bf
f}^{k}-\,\nabla\big{(}{\bf u}\cdot{\bf f}^{k}\big{)}\Big{)}\circ
dW^{k}_{t}\,,\\\ \partial_{t}D+{\rm div}(D{\bf u})&=0\,,\quad\hbox{with}\quad
D=1\,,\\\ \partial_{t}b+{\bf u}\cdot\nabla b&=0\,,\end{split}$ (1.1)

where one interprets the semimartingale
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}^{S}={\bf u}^{S}({\bf x})\,dt+\sum_{k}{\bf f}^{k}({\bf x})\circ
dW^{k}_{t}$ as a stochastic augmentation of the usual steady prescribed Stokes
drift velocity and one recovers the deterministic Craik-Leibovich equations
when the stochastic terms proportional to $\circ dW^{k}_{t}$ are absent.

The energy-preserving stochastic EB equations in (4.31) imply the following
equation for potential vorticity density, defined by $q:=(\hbox{curl}{\bf
m})\cdot{\bm{\nabla}}b=(\bm{\omega}+2\bm{\Omega})\cdot{\bm{\nabla}}b$, where
${\bf m}={\bf u}+{\bf R}$ for $D=1$. Namely,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}q+{\bf u}\cdot{\bm{\nabla}}q\,dt=:-\,\sum_{k}{\rm div}\mathbf{J}^{k}\circ
dW^{k}_{t}\,,$ (1.2)

in which the “$J$-fluxes of PV” on the right-hand side are discussed, e.g., in
[32, 56]. See also [8] for LES turbulence interpretations of these fluxes.

In summary, while the energy-preserving stochastically-augmented CL vortex
force and entrainment effects in this example can locally create stochastic
Langmuir circulations, the total volume-integrated potential vorticity
$Q=\int_{\mathcal{D}}q\,dV$ will be preserved for appropriate boundary
conditions.

###  1.1 Motivating question and main results of the paper

The present paper addresses the geometric interplay between energy and
circulation, when stochasticity is introduced into fluid dynamics, in both the
incompressible flow of an ideal Euler fluid and in the flows of ideal fluids
with advected quantities. This is a burgeoning area of research in fluid
dynamics. For recent introductory surveys of stochastic fluid dynamics with
applications, see, e.g., [17, 23, 28].

The question underlying the present work is, “What types of noise
perturbations can be added to fluid dynamics which will preserve its
fundamental properties of energy conservation, Kelvin-Noether circulation
dynamics and conserved properties resulting from invariance under Lagrangian
particle relabelling?” Since these properties all arise from the geometric
structure of fluid dynamics, the noise perturbations we consider will be
introduced in a geometrical framework.

The RLDP formulation of fluid dynamics leads to a constrained variational
principle for stochastic ideal fluid dynamics through which noise may be
introduced as a prescribed stochastic force. By _choosing_ an appropriate form
of the stochastic force, the RLDP formulation can be designed to preserve
whatever conservation law one may desire among those of the deterministic
ideal fluid equations in any number of dimensions.

For applications in fluid dynamics, the RLDP formulation results in the
procedure mentioned above, called Stochastic Forcing by Lie Transport (SFLT).
This is constructed by choosing the stochastic forces to Lie derivative of the
transport velocity vector field acting on some prescribed one-form density
which conserves the ideal fluid energy. Potential applications include a
stochastic version of the Craik-Leibovich (CL) vortex force which presumably
could generate stochastic Langmuir circulation. This result is introduced for
the Euler fluid equations for 3D incompressible flow in Example 2.1 and
Proposition 2.2. The corresponding theory for ideal flows in general is also
treated in section 2.2 where stochastic advected quantities are introduced.
Using RLDP, the existence of Kelvin-Noether theorem is automatic due to the
variational nature of the formulation. When the advection of the fluid density
is assumed, the Kelvin circulation theorem becomes a simple corollary of the
Kelvin-Noether theorem as proven in Theorem 2.7.

#### SFLT has dual design capabilities.

Besides adding stochastic _forces_ which drive the fluid motion equation, the
SFLT framework can also be designed to distribute stochastic _sources_ in the
advective transport equations, as discussed in section 2.2. The stochastic
sources distributed in the advective transport equations by SFLT can be
designed to model, for example, stochastic changes in the material properties
of inertial fluid parcels which may be embedded in the flow. In particular,
one can use SFLT to model the stochastic dynamics of a mixture of heavier and
lighter parcels whose fluid paths deviate from passive tracers, which are
carried by the drift flow velocity. In this case, the mass density would be
changing stochastically in the material frame of the flow drift velocity,
because of entrainment or detrainment of sediments. One can observe the
entrainment or detrainment of sediments in Langmuir circulations, see for
example, [50]. One can also imagine using SFLT to model the transport of a
stochastically evolving, spatially distributed, algae bloom which has a
distribution of shapes, so it is only partially embedded in a flow around an
obstacle, such as an island in the ocean [30, 63].

Thus, SFLT has dual design capabilities. It can transport stochastic _sources_
in the material frame, and it can also impose stochastic _non-inertial forces_
arising from stochastic changes of the frame of motion, such as the Craik-
Leibovich _vortex force_ [15, 34]. The dual design capabilities of SFLT could
potentially lead to a variety of important applications. For example, the use
of SFLT for modelling entrainment and detrainment of various materials into
flows in the ocean mixed layer (OML) can in principle be instrumental in
modelling some important components of natural processes such as gas and
nutrient exchanges. Besides its potential importance in modelling natural
processes in the transport of materials such as sediment or algae blooms in
the OML, the SFLT stochastic modelling framework may also find a role in data
assimilation for predicting the transport of pollution such as oil droplets,
microplastics, etc.

The energy preserving SFLT framework is extended to the Eulerian Averaged SFLT
(EA SFLT) framework by applying an _Eulerian Average_ on the Eulerian
quantities in the equations. These systems are _non-local_ in probability
space in the sense that the expected momentum density and expected advected
quantities are assumed to replace the drift momentum density and advected
quantities respectively. These equations retain the fundamental properties of
fluid dynamics and in addition, they have potential uses in climate change
science since the quantities of interests have a clear sense of expectation
and fluctuation, and the expected fluid motion is deterministic. For quadratic
Hamiltonians, we find closed-form equations for the dynamics of the
expectations and fluctuations of the Eulerian fluid variables. Total energy
conservation enables us to show that, over time, the energy of the expected
quantities is converted into the energy of the fluctuations, while the sum
remains the same.

To illustrate the dual design capability of the proposed SLFT framework and
the EA SFLT framework, explicit applications for the motion of fluids under
gravity with SFLT noise will be given in the last part of the paper. Section
4.1 deals with the heavy top, which is a finite degree-of-freedom subsystem of
fluid motion [33]; in section 4.2 for rotating shallow water dynamics; and in
section 4.3 for the stochastic Euler-Boussinesq equations.

#### List of abbreviations

  * •

Geophysical Fluid Dynamics (GFD)

  * •

Wave Effects on Currents (WEC)

  * •

Reduced Lagrange d’Alembert Principle (RLDP)

  * •

Craik-Leibovich (CL)

  * •

Generalised Lagrangian Mean (GLM)

  * •

Stochastic Forcing by Lie Transport (SFLT)

  * •

Eulerian Averaged Stochastic Forcing by Lie Transport (EA SFLT)

  * •

Stochastic Advection by Lie Transport (SALT)

  * •

Lagrangian Averaged Stochastic Advection by Lie Transport (LA SALT)

  * •

Stochastic Forced Euler-Poincaré (SFEP)

  * •

Stochastic Forced Lie-Poisson (SFLP)

  * •

Rotating Shallow Water (RSW)

  * •

Euler-Boussinesq (EB)

###  1.2 Plan of the paper

In section 2 we introduce SFLT noise in the RLDP form and show that this
formulation is flexible enough to accommodate a variety of different types of
noise perturbations with potential applications to stochastic fluid dynamics.
In section 3 we introduce the EA SFLT framework as an extension of the SFLT
framework. We show that this modification to SFLT have potential applications
to climate change science. In section 4 we present several examples of
conservative noise types for semidirect-product coadjoint motion. These
include the finite-dimensional case of the heavy top (which is a gyroscopic
analog for collective motion of a stratified fluid [33]), and the infinite-
dimensional fluid cases of rotating shallow water (RSW) dynamics in 2D and
Euler-Boussinesq (EB) dynamics in 3D. The paper emphasises the utility of the
RLDP formulation in introducing a variety of stochastic perturbations which
may be chosen to preserve the properties of the deterministic solutions of the
fluid dynamics equations that are of most concern and value to the modeller.
In particular, RLDP admits a unified variational formulation which combines
the variational principle used in [38] and SFLT, as explained in remark 2.16.

Three appendices have been provided. These appendices are meant to supply
supporting details without interfering with the main flow of the paper. They
contain further discussions of the following topics related to the main part
of the paper: A Coadjoint operator of semidirect-product Lie-Poisson brackets;
B Stochastic advection by Lie transport (SALT); C Itô form of the SFLP
equation;

#### Slow + Fast decompositions of fluid flows.

In modelling geophysical fluid dynamics (GFD) in ocean, atmosphere or climate
science, one tends to focus on balanced solution states which are near certain
observed equilibrium [59, 24]. These equilibria include hydrostatic and
geostrophic balanced states, for example, in both the ocean and the
atmosphere. Upsetting these balances can introduce both fast and slow temporal
behaviour. The response depends on the range of the frequencies in the
spectrum of excitations of the system away from equilibrium under the
perturbations. When a separation in time scales exists in the response of the
system to perturbations of one of its equilibria, then one may propose to
average over the high frequency response and retain the remaining slow
dynamics which remains near the equilibrium. This happens, for example, in the
quasigeostrophic response to disturbances of geostrophic equilibria in the 2D
rotating shallow water equations. Averaging over the high frequencies also
often produces a slow _ponderomotive force_ , due for example to a slowly
varying envelope which modulates the high frequency response.111In taking
these averages over the fast behaviour of the dynamics one must also deal
properly with any resonances which would occur. However, the effects of
resonances will be neglected in our discussion here. In most situations in
GFD, though, the solution only stays near the low-frequency _slow manifold_
for a rather finite time before developing a high-frequency response, See,
e.g., Lorenz [52, 53, 54]. In practice, for example in numerical weather
prediction, the emergence of the high-frequency disturbances of the devoutly
wished slow manifold introduces undeniable uncertainty which historically has
often been handled by some sort of intervention, such as nonlinear re-
initialisation [47]. Sometimes the effects of the emergence of high
frequencies can be treated to advantage in computational simulations. For
example, the stochastic back-scatter approach of Leith [48] is commonly used
in computational simulations to feed energy from the burgeoning unstable
development of high-wavenumber excitations into large-scale coherent
structures at low frequencies, so as to enhance the formation of eddies in
ocean flows, [6].

#### Hamilton’s principle and Kelvin’s circulation theorem.

An opportunity for further theoretical understanding of the interactions of
disparate scales in GFD (and, for example, in astrophysics and planetary
physics [65]) arises when the non-dissipative part of the fluidic system
dynamics under consideration in a domain $\mathcal{D}$ can be derived from
Hamilton’s principle, $\delta S=0$, for an action time-integral given by
$S=\int\ell(u,a)dt$, where the fluid Lagrangian $\ell(u,a)$ depends on
Eulerian fluid variables comprising the fluid velocity vector field
$u\in\mathfrak{X}(\mathcal{D})$ and some set of advected quantities, $a\in
V^{*}(\mathcal{D})$, dual in $L^{2}$ pairing to a vector space, or tensor
space, $V$, defined over the domain $\mathcal{D}$ with appropriate boundary
conditions. This approach leads directly to a Kelvin-Noether circulation
theorem arising from the symmetry of the Lagrangian $\ell(u,a)$, written in
terms of Eulerian fluid variables, under transformations of the initial
material labels which preserve the initial conditions of the advected
quantities along the particle trajectories in the flow [41]. In this case, the
averaging over high frequencies in the solution may be applied by substituting
a WKB (slowly varying complex amplitude times a fast but slowly varying phase)
decomposition of the fluid parcel trajectory into the Lagrangian, then
averaging over the fast phase before taking variations. A prominent example of
this approach for applications in GFD is the General Lagrangian Mean (GLM)
phase-averaged description of the interaction of fluctuations with a mean flow
introduced in [2] and developed further in [29, 35, 36, 71, 1]. Many of the
ideas underlying GLM are also standard in the stability analysis of fluid
equilibria in the Lagrangian picture. See, e.g., the classic stability
analysis papers of [7, 25, 64, 31].

#### Stochastic variational principles.

The present paper discusses yet another opportunity for introducing a slow-
fast decomposition for the sake of further understanding of GFD. This
opportunity arises when the separation in time scales in the symmetry-reduced
Lagrangian $\ell(u,a)$ for Eulerian fluid variables can be posed as the
decomposition of the fluid velocity $u$ into the sum of a deterministic drift
velocity modelling the computationally resolvable scales and a stochastic
velocity vector field modelling the correlates at the resolvable scales of the
computationally unresolvable _sub-grid_ scales of motion. A stochastic
variational principle using the slow-fast decomposition of the Lagrangian flow
map led to the derivation of stochastic Euler–Poincaré (SEP) equations in
[38]. In [38], the fast motion of the Lagrangian trajectory is represented by
a stochastic process, whose correlate statistics are to be calibrated from
data as in [13, 12]. The stochastic decomposition proposed in [38] was later
derived using multi-time homogenization by Cotter et al. [14]. Transport of
fluid properties along an ensemble of these stochastic Lagrangian trajectories
is called Stochastic Advection by Lie Transport (SALT). The well-posedness of
the Euler fluid version of the Euler–Poincaré SALT equations in three
dimensions was established in [16] for initial conditions in appropriate
Sobolev spaces. The mathematical framework of semimartingale-driven stochastic
variational principles was established in [66].

#### The geometric interplay between energy and circulation in ideal fluid
dynamics.

Fluid dynamics transforms energy into circulation. As it turns out, this
transformation is quite geometric. In particular, the solutions of Euler’s
fluid equations for ideal incompressible flow describe geodesic curves
parametrised by time on the manifold of volume-preserving diffeomorphisms
(smooth invertible maps). These geodesic curves are defined with respect to
the metric provided by the fluid’s kinetic energy, defined on the smooth,
divergence-free, velocity vector fields which comprise the tangent space of
the volume-preserving diffeomorphisms ${\rm SDiff}(\mathcal{D})$ acting on the
flow domain $\mathcal{D}$. This 1966 result of V. I. Arnold [4] introduced a
fundamentally new geometric way of understanding energy and circulation in
fluid dynamics. The incompressible Euler fluid case in [4] possesses the well-
known conservation laws of energy and circulation, both of which arise via
Noether’s theorem from symmetries of the Lagrangian in Hamilton’s variational
principle under the right action of ${\rm SDiff}(\mathcal{D})$. Later, Arnold
[5] noticed the topological nature of another conservation law for the Euler
fluid equations which is known as helicity. In particular, the conserved
helicity measures the topological linkage number of the vorticity field lines
in Euler fluid dynamics.

Some of this geometric interplay between energy and circulation in ideal fluid
dynamics already shows up in Kelvin’s circulation theorem [43] for ideal Euler
fluids, which emerges as the Kelvin-Noether theorem from right-invariance
(relabelling symmetry) of the Lagrangian in Hamilton’s principle when written
in terms of the Eulerian representation, [41]. In physical fluids, the
particle-relabelling symmetry of the fluid Lagrangian in the Eulerian
representation is broken from the full diffeomorphism group $G={\rm
Diff}(\mathcal{D})$ to its subgroup $G_{a}={\rm Diff}(\mathcal{D})|_{a_{0}}$
which leaves invariant the initial conditions $a_{0}$ for advected fluid
variables, denoted $a$, such as the mass density and thermodynamic properties.
The Lagrangian histories $x_{t}=g_{t}x_{0}$ evolve as
$\dot{x}_{t}=\dot{g}_{t}x_{0}=u(g_{t}x_{0},t)=u(x_{t},t)$, where the Eulerian
velocity vector field given by $u:=\dot{g}_{t}g_{t}^{-1}$ is right-invariant
under the particle-relabelling $x_{0}\to y_{0}=h_{0}x_{0}$ for any fixed
$h_{0}\in G$.

In this geometrical setting for fluids in the Eulerian representation, the
Legendre transform maps the Lagrangian variational formulation to the
Hamiltonian formulation in which a Lie-Poisson bracket governs the motion
generated by the Hamiltonian. This Lie-Poisson bracket is defined on the dual
space of the Lie algebra of divergence-free vector fields $\mathfrak{X}_{\rm
div}(\mathcal{D})$ for Euler fluids. For ideal fluids with advected
quantities, the Lie-Poisson bracket is defined on the dual space of the SDP
Lie algebra $\mathfrak{X}(\mathcal{D})\circledS V(\mathcal{D})$. As discussed
in [41], the fluid motion in each case represents the coadjoint action of the
corresponding Lie algebra on its dual space. For more details in the present
context, refer to appendices A and B, particularly B.3.

#### Geometric formulation of advective transport.

The action of the full diffeomorphism group $G={\rm Diff}(\mathcal{D})$ on its
_order parameter_ variables $a\in V(\mathcal{D})$ represents fluid advection.
Advection occurs by _push-forward_ of functions on $V(\mathcal{D})$ (by right
action by the inverse of the Lagrange-to-Euler map). This means the time-
dependence of an advected quantity is given by the push-forward relation for
composition of functions; namely,

$a(t)=g_{t\,*}a_{0}:=a_{0}g_{t}^{-1}\,.$

Thus, an advected quantity $a(t)$ evolves by the _Lie chain rule_

${\partial}_{t}a=-\mathcal{L}_{\dot{g}_{t}g_{t}^{-1}}a_{t}=-\mathcal{L}_{u_{t}}a\,,$

in which $\mathcal{L}_{u_{t}}a$ denotes the _Lie derivative_ of the advected
quantity $a\in V(\mathcal{D})$ by the time-dependent Eulerian velocity vector
field, $u_{t}:=\dot{g}_{t}g_{t}^{-1}\in\mathfrak{X}(\mathcal{D})$. For more
details about performing this type of calculation for _Lie transport_ , see
appendix A. The applications of these ideas in developing the SALT approach
are discussed in appendix B.

##  2 Stochastic forcing by Lie transport (SFLT)

After briefly surveying in section 2.1 a few of the available capabilities
arising from the RLDP principle for the sake of other potential research
directions, we will continue in section 2.2 toward our primary objective to
develop the energy preserving SFLT theory for applications to semidirect-
product fluid motion and material entrainment in the remainder of the paper.

###  2.1 Reduced Lagrange d’Alembert Pontryagin (RLDP) Principle

#### Stochastic non-inertial frames.

Newton’s law of motion in a non-inertial frame redefines momentum so as to
introduce an additional force. The canonical example is the Coriolis force,
which arises from redefining the momentum in a rotating frame of motion in
terms of the velocity as viewed from an inertial frame. The CL vortex force
arises in the fluid momentum equation by the addition of Stokes drift velocity
to the frame of motion of the fluid momentum variable $m\in\mathfrak{X}^{*}$.
In the present setting, it is natural to consider a stochastic addition to the
momentum which corresponds to a stochastic change of referemce frame.

One variational principle which permits such additional forces is the _reduced
Lagrange d’Alembert Pontryagin_ (RLDP) principle. The general construction of
fluid dynamics using the RLDP formulation is as follows [27].222See [10] for
the corresponding result for the finite-dimensional Euler-Lagrange equation in
the absence of symmetry. Consider the Lie group $G={\rm Diff}(\mathcal{D})$
with an associated Lie algebra $\mathfrak{X}$, the RLDP principle for reduced
Lagrangian $\ell:\mathfrak{X}\rightarrow\mathbb{R}$ and external force
$F\in\mathfrak{X}^{*}$ is given by

$\displaystyle\delta\int_{b}^{a}\ell(u)+{\left\langle{m}\,,\,{\dot{g}\,g^{-1}-u}\right\rangle}\,dt-\int_{b}^{a}{\left\langle{F}\,,\,{\delta
g\,g^{-1}}\right\rangle}\,dt=0\,,$ (2.1)

where $g\in G$, the variations $\delta g,\delta u,\delta m$ are arbitrary with
$\delta g$ vanishes at the end points $t=a,b$. The stationary condition (2.1)
yields the forced Euler-Poincaré equation with force $F$, as a 1-form density
equation,

$\displaystyle\frac{\partial}{\partial t}\frac{\delta\ell}{\delta u}+{{\rm
ad}}^{*}_{u}\frac{\delta\ell}{\delta u}+F=0\,.$ (2.2)

Adding stochasticity to the RLDP principle with external forces can be done,
as follows. For reduced Lagrangian $\ell:\mathfrak{X}\rightarrow\mathbb{R}$
and the set of external forces $F_{i}\in\mathfrak{X}^{*}$, stochastic RLDP is
given by

$\displaystyle\delta\int_{b}^{a}\ell(u)+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u}\right\rangle}\,dt-\sum_{i}\int_{b}^{a}{\left\langle{F_{i}}\,,\,{\eta}\right\rangle}\circ
dW_{t}^{i}=0\,,$ (2.3)

where $g\in G$, the variations $\delta g,\delta u,\delta m$ are arbitrary,
$\delta g$ vanishes at the end points, $t=a,b$, and $\eta:=\delta g\,g^{-1}$.
The stationary condition of the variational principle (2.3) yields the
following _stochastically forced Euler-Poincaré_ (SFEP) equation,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta\ell}{\delta u}+{{\rm ad}}^{*}_{u}\frac{\delta\ell}{\delta
u}\,dt+\sum_{i}F_{i}\circ dW^{i}_{t}=0\,.$ (2.4)

#### Lie-Poisson Hamiltonian formulation.

Upon passing to the Hamiltonian side via the Legendre transform
$\ell(u)={\left\langle{m}\,,\,{u}\right\rangle}-h(m)$, where
$m=\frac{\delta\ell}{\delta u}$ and $h$ is the reduced Hamiltonian, one finds
the _reduced Hamilton-d’Alembert Pontryagin phase space principle_ given by

$\displaystyle\delta\int_{b}^{a}\left[{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\right\rangle}-h(m)\right]\,dt-\sum_{i}\int_{b}^{a}{\left\langle{F_{i}}\,,\,{\eta}\right\rangle}\circ
dW_{t}^{i}=0\,.$ (2.5)

As before, $g\in G$, the variations $\delta g,\delta m$ are arbitrary and
$\delta g$ vanishes at the endpoints in time, $t=a,b$. The resulting
_stochastically forced Lie-Poisson_ (SFLP) equation is

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{u}m\,dt+\sum_{i}F_{i}\circ
dW^{i}_{t}=0\,,\quad\text{where }u=\frac{\delta h}{\delta m}\,.$ (2.6)

Within the construction of class of SFLP equations, choices of the external
forces $F_{i}$ exists. With specific choices of the external forces $F_{i}$,
the resulting equations can be either energy-preserving, or Casimir preserving
as shown by subsequent examples.

######  Example 2.1 (Energy-preserving SFLP equations).

Let $F_{i}={{\rm ad}}^{*}_{u}f_{i}$ where $f_{i}\in\mathfrak{X}^{*}$, then the
SFLP equation (2.6) becomes

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{u}m\,dt+\sum_{i}{{\rm ad}}^{*}_{u}f_{i}\circ
dW^{i}_{t}=0\,,\quad\text{where }u=\frac{\delta h}{\delta m}\,,$ (2.7)

in agreement with [21]. Energy preservation may now be immediately verified,
since

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m}\,,\,{\frac{\delta h}{\delta m}}\right\rangle}={\left\langle{-{{\rm
ad}}^{*}_{u}m\,dt-\sum_{i}{{\rm ad}}^{*}_{u}f_{i}\circ
dW^{i}_{t}}\,,\,{u}\right\rangle}=0\,,$ (2.8)

in which the last equality follows because of the anti-symmetry of the
commutator, ${\rm ad}_{u}u=-[u,u]=0$. However, the Casimirs are no longer
conserved, because the Lie-Poisson operator in equation (2.7) has been changed
by the addition of noise. For completeness, the Itó form of the equation (2.7)
is presented in Appendix C.

######  Proposition 2.2 (Vortex force).

The energy preserving SFLP equation (2.7) contains stochastic vortex forces.

###### Proof.

The stochastic term in (2.7) can be expressed as

$\displaystyle\sum_{i}{{\rm ad}}^{*}_{u}f_{i}\circ
dW^{i}_{t}=\sum_{i}\Big{(}\bm{u}\times{\rm
curl}\bm{f}_{i}-\nabla(\bm{u}\cdot\bm{f}_{i})-\bm{f}_{i}\hbox{div}{\,{\bf
u}}\Big{)}\cdot d\bm{x}\circ dW^{i}_{t}\,,$ (2.9)

where $\bm{u}$ and $\bm{f}_{i}$ are the coefficients of the vector field $u$
and 1-form densities $f_{i}$. i.e.
$u=\bm{u}\cdot\frac{\partial}{\partial\bm{x}}$ and $f_{i}=\bm{f}_{i}\cdot
d\bm{x}\,d^{3}x$. Thus, a stochastic version of the CL vortex force
$\sum_{i}\bm{u}\times{\rm curl}\bm{f}_{i}\circ dW^{i}_{t}$ with a stochastic
contribution to the pressure
$\sum_{i}{\bm{\nabla}}(\bm{u}\cdot\bm{f}_{i})\circ dW^{i}_{t}$ emerges in the
energy-preserving form of the RLDP equations in (2.7), cf. [15]. Note that the
term $\bm{f}_{i}\hbox{div}\bm{u}$ vanishes for incompressible flow. ∎

For a specific choice of Hamiltonian, the forces presented in example 2.1 may
not be the only forces that conserve energy. However, the ${{\rm
ad}}_{u}f_{i}^{*}$ form of the noise is geometric, so the energy preserving
property does apply to all fluid systems that have a variational formulation.
Thus, it is possible to construct a class of energy preserving stochastic
systems using the geometric formulation. As discussed further in section 2.2,
the noise terms in this class of energy preserving stochastic systems appear
the form of a ‘frozen’ (constant coefficient) Lie-Poisson bracket, whose
properties are discussed, e.g., in Appendix B of [42].

######  Example 2.3 (Casimir-preserving SFLP equations).

Given a Casimir function $C(m)$, the Casimir preserving forces satisfy
$F_{i}={{\rm ad}}^{*}_{\frac{\delta C}{\delta m}}f_{i}$ where $f_{i}$ are
arbitrary. Then

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}C={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m}\,,\,{\frac{\delta C}{\delta m}}\right\rangle}={\left\langle{-{{\rm
ad}}^{*}_{u}m\,dt-\sum_{i}{{\rm ad}}^{*}_{\frac{\delta C}{\delta m}}f_{i}\circ
dW^{i}_{t}}\,,\,{{\frac{\delta C}{\delta m}}}\right\rangle}=0,$

_provided_ both ${\left\langle{{{\rm ad}}^{*}_{u}m}\,,\,{{\frac{\delta
C}{\delta m}}}\right\rangle}=0$ and ${\left\langle{{{\rm
ad}}^{*}_{\frac{\delta C}{\delta m}}f_{i}}\,,\,{{\frac{\delta C}{\delta
m}}}\right\rangle}=0$, because of the degeneracy of the LP bracket and the
anti-symmetry of the commutator, respectively. One concludes that the choice
of external forces $F_{i}={{\rm ad}}^{*}_{\frac{\delta C}{\delta m}}f_{i}$
cannot in general preserve all of the Casimirs seen in the unperturbed LP
equation, unlike the case with SALT, where _all of the Casimirs_ are
preserved. Note that the energy are no longer conserved, since

$\displaystyle{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m}\,,\,{\frac{\delta h}{\delta m}}\right\rangle}={\left\langle{-{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt-\sum_{i}{{\rm ad}}^{*}_{\frac{\delta
C}{\delta m}}f_{i}\circ dW^{i}_{t}}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}={\left\langle{-\sum_{i}{{\rm ad}}^{*}_{\frac{\delta
C}{\delta m}}f_{i}\circ dW^{i}_{t}}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}\,,$

does not vanish trivially.

######  Proposition 2.4 (Helicity preservation for a stochastic 3D Euler
fluid).

The deterministic 3D Euler fluid equations preserve the helicity,

$C(m)=\langle m,{\bf d}m\rangle=\frac{1}{2}\int\bm{m}\cdot{\rm
curl}\bm{m}\,d^{3}x\,,$

where $m=\bm{m}\cdot d\bm{x}$ is the circulation 1-form. The SFLT model will
preserve the helicity $C(m)$ when the constraint force is taken to be

$F_{i}={{\rm ad}}^{*}_{\frac{\delta C}{\delta m}}f_{i}\in\mathfrak{X}^{*}$

where $\frac{\delta C}{\delta m}={\bf d}m$ is the vorticity 2-form, ${\bf
d}m=({\rm curl}\bm{m})\cdot d{\bf S}$ and .

###### Proof.

In vector calculus terms, one computes

${\left\langle{{{\rm ad}}^{*}_{u}m}\,,\,{{\frac{\delta C}{\delta
m}}}\right\rangle}={\Big{\langle}{-\bm{u}\times{\rm
curl}\bm{m}+\nabla(\bm{u\cdot m})}\,,\,{{\rm
curl}\bm{m}}\Big{\rangle}}=\oint_{{\partial}\mathcal{D}}(\bm{u\cdot m})\,{\rm
curl}\bm{m}\cdot\bm{\widehat{n}}dS=0\,,$

which is obtained after an integration by parts in the second summand and
taking the boundary term to vanish as usual for the helicity. ∎

######  Example 2.5 (Proto-SALT SFLP equation).

Let us choose $F_{i}={{\rm ad}}^{*}_{f_{i}(x)}m$ in equation (2.6), where
$f_{i}\in\mathfrak{X}$ are arbitrary functions taking values in the Lie
algebra of smooth vector fields $\mathfrak{X}$. Then, the SFLP equation (2.6)
recovers an expression similar to the momentum terms in the SALT fluid motion
equation (B.10); namely,

$\displaystyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}m+{{{\rm
ad}}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}m=0\,,\quad\text{where }u=\frac{\delta h}{\delta
m}\quad\hbox{and}\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}=u\,dt+\sum_{i}f_{i}\circ dW^{i}_{t}\,.$ (2.10)

This is the SALT fluid motion equation (B.10) in the absence of advected
quantities.

###  2.2 Energy preserving SFLT for semidirect-product motion and material
entrainment

We next extend the SFLT approach to case of coadjoint motion of fluids
carrying advected quantities such as mass and heat which are associated with
potential energy. 333Advected quantities are also known as order parameters in
condensed matter physics. The motion of advected quantities arises from the
semidirect-product action of the diffeomorphisms on the vector spaces
containing the advected quantities. The vector spaces containing the advected
quantities comprise coset spaces obtained from symmetry-breaking of the full
diffeomorphism group to the remaining isotropy subgroup of the initial
conditions of the advected quantities.

Keeping in sight the objective of the paper to derive a class of energy-
preserving stochastic models of fluid dynamics, the same energy-preserving
noise as the Euler fluids in Example 2.1 and its vortex force 2.2 will be
chosen in the construction that follows. This extension enables the derivation
of stochastic vortex forces which model the uncertainty of unresolved slow-
fast interaction effects as energy-preserving stochastic perturbations of
fluid models which possess a potential energy. In addition to vortex forces,
the introduction of stochasticity in the passive advection relations allows
the modelling of quantities that do not passively follow the drift velocity
field. Physical applications of this type of stochastic modification are
discussed in Remark 2.12 For symmetry-reduced semidirect-product motion,
considering the energy preserving noise of the form $F_{i}={{\rm
ad}}^{*}_{u}f_{i}$, the stochastic RLDP principle in (2.3) becomes

$\displaystyle 0=\delta
S=\delta\int^{b}_{a}\ell(u,a)\,dt+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}-\int_{a}^{b}\sum_{i}{\left\langle{{{\rm
ad}}^{*}_{u}f_{i}}\,,\,{\delta g\,g^{-1}}\right\rangle}\circ dW^{i}_{t},$
(2.11)

where the variations $\delta g,\delta u,\delta
m,\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b,\delta a$ are arbitrary with $\delta g$ vanishing at the endpoints,
$t=a,b$. Taking the indicated variations in (2.11) yields the following
result,

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}{\left\langle{\frac{\delta\ell}{\delta
u}}\,,\,{\delta u}\right\rangle}\,dt+{\left\langle{\frac{\delta\ell}{\delta
a}}\,,\,{\delta a}\right\rangle}\,dt+{\left\langle{\delta
m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}\delta g^{-1}-\delta
a}\right\rangle}+{\left\langle{\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}$
$\displaystyle\quad+{\left\langle{m}\,,\,{\delta({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1})-u\,dt}\right\rangle}-\sum_{i}{\left\langle{{{\rm
ad}}^{*}_{u}f_{i}}\,,\,{\delta g\,g^{-1}}\right\rangle}\circ dW_{t}^{i}$
$\displaystyle=\int_{a}^{b}{\left\langle{\frac{\delta\ell}{\delta
u}-m}\,,\,{\delta u}\right\rangle}\,dt+{\left\langle{\frac{\delta\ell}{\delta
a}}\,,\,{\delta a}\right\rangle}\,dt+{\left\langle{\delta
m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{-a\eta-\delta
a}\right\rangle}+{\left\langle{\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}$
$\displaystyle\quad+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta-{{\rm
ad}}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\eta}\right\rangle}-\sum_{i}{\left\langle{{{\rm
ad}}^{*}_{u}f_{i}}\,,\,{\eta}\right\rangle}\circ dW_{t}^{i},$

where we denote $\eta=\delta g\,g^{-1}$, as before. Vanishing of the
coefficients of the variations implies the following relations,

$\displaystyle\begin{split}&\frac{\delta\ell}{\delta
u}=m,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}=u\,dt,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b=\frac{\delta\ell}{\delta a}\,dt\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m=-{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}m+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\diamond a-\sum_{i}{{\rm ad}}^{*}_{u}f_{i}\circ
dW^{i}_{t},\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}a\,.\end{split}$ (2.12)

Assembling these relations yields the SFEP equations with advected quantities

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta\ell}{\delta u}=-{{\rm ad}}^{*}_{u}\frac{\delta\ell}{\delta
u}\,dt-\sum_{i}{{\rm ad}}^{*}_{u}f_{i}\circ
dW^{i}_{t}+\frac{\delta\ell}{\delta a}\diamond
a\,dt,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{u}a\,dt\,.$ (2.13)

Now, an application of the Legendre transform
$h(m,a)={\big{\langle}{m}\,,\,{u}\big{\rangle}}-\ell(u,a)$, followed by
calculations similar to those made in the deterministic case arrives at the
SFLP equations with advected quantities

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\sum_{i}{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f_{i}\circ dW^{i}_{t}+\frac{\delta
h}{\delta a}\diamond
a\,dt=0,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{u}a\,dt\,.$ (2.14)

These equations can also be written in the form of a Poisson operator, as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}m\\\ a\end{bmatrix}=-\begin{bmatrix}{{\rm
ad}}^{*}_{\framebox{}}\left(m\,dt+\sum_{i}f_{i}\circ
dW^{i}_{t}\right)&{\framebox{}}\diamond a\,dt\\\
\mathsterling_{\framebox{}}\,a\,dt&0\end{bmatrix}\begin{bmatrix}{\delta
h}/{\delta m}\\\ {\delta h}/{\delta a}\end{bmatrix}\,.$ (2.15)

######  Remark 2.6 (Alternative formulation for a reduced Hamilton-d’Alembert
principle with advected quantities).

The SFLP equation with advected quantities (2.14) can also be derived from the
following reduced Hamilton-d’Alembert phase space principle with advected
quantities,

$\displaystyle 0=\delta
S=\delta\int^{b}_{a}{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}-h(m,a)\,dt-\int^{b}_{a}\sum_{i}{\left\langle{{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f_{i}}\,,\,{\delta
g\,g^{-1}}\right\rangle}\circ dW^{i}_{t}.$

The proof of this statement is a direct calculation following the same pattern
as for the EP derivation.

######  Theorem 2.7 (SFLT Kelvin-Noether theorem).

The Kelvin-Noether quantity
${\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}$
associated with equation (2.14) satisfies the following stochastic Kelvin-
Noether relation.

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}={\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{-\frac{\delta
h}{\delta a}\diamond a\,dt-\sum_{i}{{\rm ad}}^{*}_{u}f^{m}_{i}\circ
dW^{i}_{t}}\right\rangle},$ (2.16)

where the identification
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}=\frac{\delta h}{\delta m}\,dt$ is obtained from the stochastic
RLDP constrained variational principle in (2.11).

######  Remark 2.8.

Upon assuming that the fluid density $D$ is also advected by the flow, so that
$\partial_{t}D+\mathcal{L}_{u}D=0$, the Kelvin circulation theorem may be
expressed as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\frac{m}{D}=\oint_{c(u)}\frac{1}{D}\left[-\frac{\delta
h}{\delta a}\diamond a\,dt-\sum_{i}{{\rm ad}}^{*}_{u}f^{m}_{i}\circ
dW^{i}_{t}\right],$ (2.17)

where the notation of Lagrangian loop $c(u)$ denotes the Lagrangian loop
moving with the deterministic fluid velocity $u$, as in the deterministic
case.

#### Natural generalisation of SFLT

Note that the SFLP equations with advected quantities in (2.14) modifies the
momentum equation alone, while keeping the advection equation of $a\in V^{*}$
unchanged. In terms of co-adjoint motion, the natural generalisation of (2.7)
to semidirect-product Lie group action would be

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(m,a)=-{{\rm ad}}^{*}_{\left(\frac{\delta h}{\delta m},\frac{\delta
h}{\delta a}\right)}(m,a)\,dt-\sum_{i}{{\rm ad}}^{*}_{\left(\frac{\delta
h}{\delta m},\frac{\delta h}{\delta a}\right)}(f^{m}_{i},f^{a}_{i})\circ
dW_{t}^{i}\,,$ (2.18)

where $(m,a),(f^{m}_{i},f^{a}_{i})\in\mathfrak{s}^{*}$, $\left(\frac{\delta
h}{\delta m},\frac{\delta h}{\delta a}\right)\in\mathfrak{s}$ and ${{\rm
ad}}^{*}$ is the coadjoint operator on $\mathfrak{s}^{*}$. The maps
$f^{m}_{i}:\mathfrak{X}\times V^{*}\rightarrow\mathfrak{X}^{*}$ and
$f^{a}_{i}:\mathfrak{X}\times V^{*}\rightarrow V^{*}$ are arbitrary for all
$i$. Using definition of semidirect product coadjoint action discussed in
appendix A, the individual equations are then

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{\delta h}{\delta
a}\diamond a\,dt+\sum_{i}{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f^{m}_{i}\circ dW^{i}_{t}+\frac{\delta h}{\delta a}\diamond f^{a}_{i}\circ
dW^{i}_{t}=0\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{\frac{\delta h}{\delta
m}}a\,dt+\sum_{i}\mathcal{L}_{\frac{\delta h}{\delta m}}f^{a}_{i}\circ
dW_{t}^{i}=0\,.\end{split}$ (2.19)

These individual equations can be written in the form of a Poisson operator,
as follows,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}m\\\ a\end{bmatrix}=-\begin{bmatrix}{{\rm
ad}}^{*}_{\framebox{}}\left(m\,dt+\sum_{i}f^{m}_{i}\circ
dW^{i}_{t}\right)&\framebox{}\diamond\left(a\,dt+\sum_{i}f^{a}_{i}\circ
dW_{t}^{i}\right)\\\
\mathsterling_{\framebox{}}\left(a\,dt+\sum_{i}f^{a}_{i}\circ
dW_{t}^{i}\right)&0\end{bmatrix}\begin{bmatrix}{\delta h}/{\delta m}\\\
{\delta h}/{\delta a}\end{bmatrix}\,.$ (2.20)

Energy is conserved since the Poisson operator is skew-symmetric. Thus,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m,a)$
$\displaystyle={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(m,a)}\,,\,{\left(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a}\right)}\right\rangle}$ $\displaystyle={\left\langle{{{\rm
ad}}^{*}_{\left(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a}\right)}(m,a)\,dt-\sum_{i}{{\rm ad}}^{*}_{\left(\frac{\delta h}{\delta
m},\frac{\delta h}{\delta a}\right)}(f^{m}_{i},f^{a}_{i})\circ
dW_{t}^{i}}\,,\,{\left(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a}\right)}\right\rangle}=0,$

where the last equality uses the anti-symmetry of the ${{\rm ad}}$ operator.
The class of SFLP equations in (2.18) can be obtained via a phase-space
variational principle, as we discuss next.

#### Variational principle for semidirect product SFLP equation

The SFLP equation (2.18) can be derived from a reduced Hamilton-d’Alembert
phase space variational principle in terms of the full semidirect product
group $S=G\circledS V$ with the associated semidirect-product Lie algebra
$\mathfrak{s}=\mathfrak{X}\circledS V$. This phase-space variational principle
reads

$\displaystyle\delta\int_{t_{1}}^{t_{2}}{\left\langle{(m,a)}\,,\,{({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)}\right\rangle}_{\mathfrak{s}}-h(m,a)\,dt-\sum_{i}\int_{t_{1}}^{t_{2}}{\left\langle{{{\rm
ad}}^{*}_{(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a})}(f^{m}_{i},f^{a}_{i})}\,,\,{(\eta,w)}\right\rangle}_{\mathfrak{s}}\circ
dW^{i}_{t}=0\,,$ (2.21)

for arbitrary variations of $\delta(m,a)$ of $(m,a)\in\mathfrak{s}^{*}$ and
constrained variations
$\delta({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)$ of
$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)\in\mathfrak{s}$. The constrained variation of
$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)$ takes the form

$\displaystyle\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\,,\,{\eta}\right]}\quad\hbox{and}\quad\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}w-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\,\eta+w{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u\,,$

in which $\eta\in\mathfrak{X}$ and $w\in V$ are arbitrary. Here the notation
${\left\langle{\cdot}\,,\,{\cdot}\right\rangle}_{\mathfrak{s}}$ is the
semidirect product pairing where

$\displaystyle{\left\langle{(m,a)}\,,\,{(u,b)}\right\rangle}_{\mathfrak{s}}:={\left\langle{m}\,,\,{u}\right\rangle}_{\mathfrak{X}}+{\left\langle{a}\,,\,{b}\right\rangle}_{V}.$

In the following we will continue to suppress the pairing subscript when the
context is clear. Note that the constrained variations of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm d}}u$
and ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b$ are related to the adjoint action of $\mathfrak{s}$ by

$\displaystyle\delta({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)=\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\,,\,{\eta}\right]},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}w-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\,\eta+w{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u\right)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(\eta,w)-{{\rm
ad}}_{({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)}(\eta,w).$

Computing the variations and applying the constrained variations yields

$\displaystyle 0=$
$\displaystyle\int_{t_{2}}^{t_{1}}{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u-\frac{\delta h}{\delta m}\,dt}\,,\,{\delta
m}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b-\frac{\delta h}{\delta a}\,dt}\,,\,{\delta
a}\right\rangle}+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\,,\,{\eta}\right]}}\right\rangle}+{\left\langle{a}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}w-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\eta+w\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\right\rangle}$ $\displaystyle\quad-{\left\langle{{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f^{m}_{i}+\frac{\delta h}{\delta
a}\diamond f^{a}_{i}}\,,\,{\eta}\right\rangle}\circ
dW^{i}_{t}-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}f^{a}_{i}}\,,\,{w}\right\rangle}\circ dW^{i}_{t}$ $\displaystyle=$
$\displaystyle\int_{t_{2}}^{t_{1}}{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u-\frac{\delta h}{\delta m}\,dt}\,,\,{\delta
m}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b-\frac{\delta h}{\delta a}\,dt}\,,\,{\delta
a}\right\rangle}+{\left\langle{-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m-{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}m}\,,\,{\eta}\right\rangle}+{\left\langle{-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a-a\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\,,\,{w}\right\rangle}+{\left\langle{a\diamond{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{\eta}\right\rangle}$ $\displaystyle\quad-{\left\langle{{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f^{m}_{i}+\frac{\delta h}{\delta
a}\diamond f^{a}_{i}}\,,\,{\eta}\right\rangle}\circ
dW^{i}_{t}-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}f^{a}_{i}}\,,\,{w}\right\rangle}\circ dW^{i}_{t}\,.$

Consequently, one may collect terms to find the following system of motion and
advection equations

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}m+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\,\diamond a+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f^{m}_{i}\circ
dW^{i}_{t}+\frac{\delta h}{\delta a}\diamond f^{a}_{i}\circ dW^{i}_{t}=0\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u=\frac{\delta h}{\delta
m}\,dt,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b=\frac{\delta h}{\delta
a}\,dt,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}a+\mathcal{L}_{\frac{\delta h}{\delta m}}f^{a}_{i}\circ
dW_{t}^{i}=0\,.\end{split}$ (2.22)

######  Remark 2.9 (Reduced Hamilton-d’Alembert Pontryagin phase space
variation principle).

The choice of variation principle (2.21) is not the Hamiltonian version of the
RLDP variation principle used previous sections. It is, however, equivalent to
the reduced Hamilton-d’Alembert Pontryagin phase space variation principle

$\displaystyle\delta\int_{t_{1}}^{t_{2}}{\left\langle{(m,a)}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(g,v)\,(g,v)^{-1}}\right\rangle}_{\mathfrak{s}}-h(m,a)\,dt-\sum_{i}\int_{t_{1}}^{t_{2}}{\left\langle{{{\rm
ad}}^{*}_{(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a})}(f^{m}_{i},f^{a}_{i})}\,,\,{\delta(g,v)\,(g,v)^{-1}}\right\rangle}_{\mathfrak{s}}\circ
dW^{i}_{t}=0\,,$ (2.23)

where the variations $\delta(g,v)\in S$ and $\delta(m,a)\in\mathfrak{s}$ are
arbitrary.

######  Proposition 2.10 (Itô form of semidirect product SFLP equation).

The Itô form of (2.19) are

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}\left(m\,dt+\sum_{i}f^{m}_{i}\,dW_{t}^{i}\right)+\frac{\delta h}{\delta
a}\diamond\left(a\,dt+\sum_{i}f^{a}_{i}\,dW_{t}^{i}\right)+\frac{1}{2}\sum_{i}\left({{\rm
ad}}^{*}_{\sigma_{i}}f^{m}_{i}+\theta_{i}\diamond f^{a}_{i}\right)\,dt=0,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathsterling_{\frac{\delta h}{\delta
m}}\left(a\,dt+\sum_{i}f^{a}_{i}\,dW_{t}^{i}\right)+\frac{1}{2}\sum_{i}\mathcal{L}_{\sigma_{i}}f^{a}_{i}\,dt=0,\end{split}$
(2.24)

where one defines

$\displaystyle\sigma_{i}:={\left({\frac{\delta^{2}h}{\delta
m^{2}}}\,,\,{-{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f^{m}_{i}-\frac{\delta
h}{\delta a}\diamond
f^{a}_{i}}\right)}\quad\hbox{and}\quad\theta_{i}:={\left({\frac{\delta^{2}h}{\delta
a^{2}}}\,,\,{-\mathcal{L}_{\frac{\delta h}{\delta m}}f^{a}_{i}}\right)}.$

Again the notation ${\left({\cdot}\,,\,{\cdot}\right)}$ denotes contraction,
not $L^{2}$ pairing.

The proof is similar to the case without advected quantities are it is given
in appendix C.

######  Remark 2.11.

The external forces ${{\rm ad}}_{(\frac{\delta h}{\delta m},\frac{\delta
h}{\delta a})}(f^{m}_{i},f^{a}_{i})$ introduced in (2.21) are energy
preserving only. For a general sets of forces $F^{m}_{i}$ and $F^{a}_{i}$, the
semidirect product SFLP equation will be derived from the variational
principle

$\displaystyle\delta\int_{t_{1}}^{t_{2}}{\left\langle{(m,a)}\,,\,{({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)}\right\rangle}-h(m,a)\,dt-\sum_{i}\int_{t_{1}}^{t_{2}}{\left\langle{(F^{m}_{i},F^{a}_{i})}\,,\,{(\eta,w}\right\rangle}\circ
dW^{i}_{t}=0\,,$

where the variations have the same condition as principle (2.21). The
resulting equations are the following

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{\delta h}{\delta
a}\diamond a\,dt+\sum_{i}F^{m}_{i}\circ
dW^{i}_{t}=0\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{\frac{\delta h}{\delta m}}a\,dt+\sum_{i}F^{a}_{i}\circ
dW_{t}^{i}=0\,.$

######  Remark 2.12 (Stochastic material entrainment modelled in equation
(2.19)).

When stochastic processes $f^{a}_{i}$ are included in the Lie-derivative
action on the fluid variables, $a(t)$, then one can no longer say that $a(t)$
is passively advected by the flow $g\in G$, i.e., $a(t)\neq a_{0}g^{-1}(t)$.
Consider the case that the variable $a(t)$ represents the mass density of the
fluid. Then, the class of stochastic equations in (2.19) or (2.20) could model
a fluid containing parcels whose density does not quite passively follow the
drift velocity flow. Examples of such deviations from passive transport might
include inertial fluid parcels whose density has a certain probability of
being heavier or lighter than the ambient (or average, or expected) density.
The motion of these inertial parcels would then be uncertain, as modelled by a
stochastic variation in their density, relative to parcels undergoing passive
advection by the flow. We include this feature of equation (2.19) because it
may provide a useful single-fluid approach to dealing with stochastic
entertainment of material particles into fluid flows such as Langmuir
circulations. This stochastic model of material entrainment into fluid flows
introduces probabilistic aspects into the theory, rather than dealing with the
intricacies of multiphase flow models. The applications of this feature may
include, for example, ice slurries in the Arctic Ocean, or dust clouds, or
debris in tornadoes, or fluid flows with gas bubbles, or well-mixed oil
spills, or transport of algae, or plastic detritus in the ocean. In the case
where $f^{a}_{i}=0$ for all $i$, then $a(t)$ would be passively advected and
would satisfy the standard relation $a(t)=a_{0}g^{-1}(t)$. For a recent review
of deterministic LES turbulent models of this type of mixed-buoyancy fluid
transport, see [11].

######  Remark 2.13 (SFLT Kelvin-Noether theorem).

Note that the inclusion of forcing terms $f^{a}_{i}$ on advected quantities
$a$ implies $a(t)\neq a_{0}g^{-1}$. This means one cannot take the pull back
the Kelvin-Noether quantity
${\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}$ by
$g(t)$ to the Kelvin-Noether quantity defined by the initial conditions
$c_{0}$ and $a_{0}$, i.e.
${\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}\neq{\left\langle{\mathcal{K}(c_{0},a_{0})}\,,\,{{{\rm
Ad}}^{*}_{g}m(t)}\right\rangle}$. Hence there is no simple modification of the
Kelvin-Noether theorem associated with (2.19). For fluids, the map
$\mathcal{K}$ is the circulation integral around a material loop $c$ which is
independent of the advected quantities $a$. Thus a Kelvin circulation theorem
exists as formulated below.

######  Theorem 2.14 (SFLT Kelvin circulation theorem with entrainment).

Upon assuming that the fluid density $D$ is also advected by the flow, so that
$\partial_{t}D+\mathcal{L}_{u}D=0$, the Kelvin circulation theorem associated
with (2.19) may be expressed as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\frac{m}{D}=\oint_{c(u)}\frac{1}{D}\left[-\sum_{i}{{\rm
ad}}^{*}_{u}f^{m}_{i}\circ dW^{i}_{t}-\frac{\delta h}{\delta
a}\diamond\left(a\,dt+\sum_{i}f^{a}_{i}\circ dW^{i}_{t}\right)\right],$ (2.25)

where the notation of Lagrangian loop $c(u)$ denotes the Lagrangian loop
moving with the deterministic fluid velocity $u$, as in the deterministic
case.

######  Remark 2.15.

Note that the presence of the stochastic material entrainment terms
$f^{i}_{a}$ in equation (2.25) may have a significant effect as a source of
circulation of the fluid flow.

######  Remark 2.16 (An unified variational approach).

By introducing stochastic Hamiltonians into the variational principle (2.21),
one can formulate a stochastic RLDP principle which encompasses both SALT and
SFLT. The augmented stochastic RLDP principle becomes

$\displaystyle\begin{split}&\delta\int_{t_{1}}^{t_{2}}{\left\langle{(m,a)}\,,\,{({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)}\right\rangle}-h(m,a)\,dt-\sum_{i}h_{i}(m,a)\circ dW^{i}_{t}\\\
&\qquad\qquad-\sum_{i}\int_{t_{1}}^{t_{2}}{\left\langle{(F^{m}_{i},F^{a}_{i})}\,,\,{(\eta,w)}\right\rangle}\circ
dW^{i}_{t}=0\,.\end{split}$ (2.26)

Here variations of $\delta(m,a)$ of $(m,a)\in\mathfrak{s}^{*}$ are arbitrary
and variations $\delta(g,v)$ of $(g,v)\in S$ are given by

$\displaystyle\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u}\,,\,{\eta}\right]}\quad\hbox{and}\quad\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}w-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\,\eta+w{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}u\,,$

where $(\eta,w)\in\mathfrak{s}$ are arbitrary are vanishes at the boundaries.
The set of Hamiltonians $h_{i}$ generates SALT type noise and the set of
forces $(F^{m}_{i},F^{a}_{i})$ generates SFLT type of noise.

##  3 Eulerian Averaged SFLT

Ed Lorenz captured the essence of the climate science problem in his
celebrated unpublished paper [55] in which he motivated his discussion by
invoking the old adage that

> “Climate is what you expect. Weather is what you get.”

Lorenz’s lesson was that climate science is fundamentally probabilistic. Much
later, this lesson inspired the derivation of the LA SALT fluid model, which
also exploited an idea introduced in [21] to apply Lagrangian-averaging (LA)
in _probability space_ to the fluid equations governed by stochastic advection
by Lie transport (SALT) which were introduced in [38]. The general theory of
LA SALT and applications to 2D Euler-Boussinesq equations can be found in [22]
and [3] respectively.

Here, we apply the probabilistic approach to derive the corresponding
Eulerian-averaged SFLT model (EA SFLT) by decomposing the Eulerian solutions
of the energy-preserving SFLT model into the sums of their expectations and
their fluctuations. As for the energy-preserving SFLT models, the EA SFLT
equations admit a Kelvin circulation theorem and preserve the deterministic
energy. For systems resulting from quadratic Hamiltonians, this modification
of the SFLT model allows the dynamics of the statistical properties of the
solutions of EA-SFLT such as the evolution of the expectation of energy to be
considered explicitly. As for the LA-SALT models, the EA-SFLT models can be
viewed as the interaction of the expected quantities and the fluctuations in a
conservative system. In this system, the energy of the expected quantities is
dynamically converted into the energy of fluctuations whilst keeping the total
energy invariant as shown in Theorem 3.2. The Kelvin circulation theorem for
EA-SFLT differs from that of SFLT by the presence of an additional forcing.
This feature makes the EA-SFLT approach particularly apt for the examples of
quadratic fluid Hamiltonians which are discussed in section 4.

###  3.1 EA SFLT

In Eulerian Averaged SFLT, the expectation of the _Eulerian_ quantity $m$ from
equation (2.7) is the taken over the underlying probability space, and the
Eulerian Averaged Stochastic Forced Lie Poisson (EA SFLP) equation is
proposed, as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+\sum_{i}{{\rm
ad}}^{*}_{u}f^{i}\circ dW^{i}_{t}=0\,,\quad u=\frac{\delta h}{\delta m}.$
(3.1)

This modification still preserves the deterministic energy of the original
SFLP equation since

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m)={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m}\,,\,{u}\right\rangle}=-{\left\langle{{{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+\sum_{i}{{\rm
ad}}^{*}_{u}f^{i}\circ dW^{i}_{t}}\,,\,{u}\right\rangle}=0,$ (3.2)

by anti-symmetry of of the ${{\rm ad}}$ operation. The evolution of the
expectation $\mathbb{E}[m]$ can be determined by considering the Itô form of
the equation (3.1)

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+\sum_{i}{{\rm
ad}}^{*}_{u}f^{i}\,dW^{i}_{t}+\frac{1}{2}\sum_{i}{{\rm
ad}}^{*}_{\sigma_{i}}f^{i}\,dt=0,$ (3.3)

where $\sigma_{i}={\left({\frac{\delta^{2}h}{\delta m^{2}}}\,,\,{-{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f_{i}}\right)}$. The proof of the Itô form
is similar to the case of the SFLP equation in appendix C. Taking the
expectation of (3.3) yields the following partial differential equation (PDE)
for the expectation of the momentum density, $\mathbb{E}\left[{m}\right]$,

$\displaystyle\partial_{t}\mathbb{E}\left[{m}\right]+{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}\mathbb{E}\left[{m}\right]+\frac{1}{2}\sum_{i}{{\rm
ad}}^{*}_{\mathbb{E}\left[{\sigma_{i}}\right]}f^{i}=0\,.$ (3.4)

The expectation of (3.3) has produced a deterministic PDE because ${{\rm
ad}}^{*}$ is a linear operation and the expectation of $dW^{i}_{t}$ vanishes
by Itô’s Lemma. The PDE (3.4) closes, whenever the Hamiltonian is quadratic in
$m$, i.e. $h=\frac{1}{2}{\left\langle{m}\,,\,{\mathbb{I}^{-1}m}\right\rangle}$
where $I:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ is a constant invertible
symmetric operator which commutes with taking the expectation. Furthermore,
$\mathbb{I}$ is assumed to be positive definite. In this case, one finds the
vector field

$\displaystyle\mathbb{E}\left[{\sigma_{i}}\right]={\left({-{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}}\,,\,{\frac{\delta^{2}h}{\delta
m^{2}}}\right)}=-\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}\,.$

Assuming quadratic Hamiltonian $h$, one can computes also the Hamiltonian
function of the expectation of $m$, $\mathbb{E}\left[{m}\right]$ to have

$\displaystyle\partial_{t}h(\mathbb{E}\left[{m}\right])={\left\langle{\mathbb{E}\left[{u}\right]}\,,\,{\partial_{t}\mathbb{E}\left[{m}\right]}\right\rangle}=-\frac{1}{2}{\left\langle{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}}\right\rangle}<0\,,$ (3.5)

Define fluctuations $u^{\prime}=u-\mathbb{E}\left[{u}\right]$,
$m^{\prime}=m-\mathbb{E}\left[{m}\right]$ and
$\sigma_{i}^{\prime}=\sigma_{i}-\mathbb{E}\left[{\sigma_{i}}\right]$ and
obtain the fluctuation dynamics of $m$ by subtracting (3.4) from (3.3) to
obtain

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m^{\prime}+{{\rm
ad}}^{*}_{u^{\prime}}\mathbb{E}\left[{m}\right]\,dt+\sum_{i}{{\rm
ad}}^{*}_{u}f^{i}\,dW^{i}_{t}+\frac{1}{2}\sum_{i}{{\rm
ad}}^{*}_{\sigma_{i}^{\prime}}f^{i}\,dt=0\,.$ (3.6)

The time derivative of the Hamiltonian of the fluctuation of $m$,
$h(m^{\prime})$ is found as

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m^{\prime})&={\left\langle{\frac{\delta h}{\delta
m^{\prime}}}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m^{\prime}}\right\rangle}+\frac{1}{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{m^{\prime}}\,,\,{\frac{\delta h}{\delta
m^{\prime}}}\right\rangle}_{t}\\\ &={\left\langle{\frac{\delta h}{\delta
m^{\prime}}}\,,\,{-{{\rm
ad}}^{*}_{u^{\prime}}\mathbb{E}\left[{m}\right]\,dt-\sum_{k}{{\rm
ad}}^{*}_{u}f^{k}\,dW^{k}_{t}-\frac{1}{2}{{\rm
ad}}^{*}_{\sigma_{k}^{\prime}}f^{k}\,dt}\right\rangle}+\frac{1}{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{m^{\prime}}\,,\,{\frac{\delta h}{\delta
m^{\prime}}}\right\rangle}_{t}\,dt\end{split}$ (3.7)

This expression simplifies dramatically when the Hamiltonian is quadratic,
where

$\displaystyle\frac{\delta h}{\delta
m^{\prime}}=\mathbb{I}^{-1}m^{\prime}=u^{\prime},\quad\sigma_{i}^{\prime}=-\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u^{\prime}}f^{i}.$

A dynamical expression can be calculated as

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m^{\prime})&={\left\langle{u^{\prime}}\,,\,{-{{\rm
ad}}^{*}_{u^{\prime}}\mathbb{E}\left[{m}\right]\,dt-\sum_{k}{{\rm
ad}}^{*}_{u}f^{k}\,dW^{k}_{t}-\frac{1}{2}{{\rm
ad}}^{*}_{\sigma_{k}^{\prime}}f^{k}\,dt}\right\rangle}+\frac{1}{2}\sum_{k}{\left\langle{{{\rm
ad}}^{*}_{u}f^{k}}\,,\,{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dt\\\ &=\sum_{k}{\left\langle{-{{\rm
ad}}^{*}_{u}f^{k}\,dW^{k}_{t}-\frac{1}{2}{{\rm
ad}}^{*}_{\sigma_{k}^{\prime}}f^{k}\,dt}\,,\,{u^{\prime}}\right\rangle}+\frac{1}{2}{\left\langle{{{\rm
ad}}^{*}_{u}f^{k}}\,,\,{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dt\\\
&=\sum_{k}{\left\langle{u^{\prime}}\,,\,{-{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dW^{k}_{t}+{\left\langle{\sigma_{k}^{\prime}}\,,\,{\frac{1}{2}{{\rm
ad}}^{*}_{u^{\prime}}f^{k}}\right\rangle}\,dt+\frac{1}{2}{\left\langle{{{\rm
ad}}^{*}_{u}f^{k}}\,,\,{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dt\\\
&=\sum_{k}{\left\langle{u^{\prime}}\,,\,{-{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dW^{k}_{t}+{\left\langle{-\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}+\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\,,\,{\frac{1}{2}{{\rm
ad}}^{*}_{u^{\prime}}f^{k}}\right\rangle}\,dt+\frac{1}{2}{\left\langle{{{\rm
ad}}^{*}_{u}f^{k}}\,,\,{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dt\\\
&=\sum_{k}{\left\langle{u^{\prime}}\,,\,{-{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dW^{k}_{t}+\frac{1}{2}{\left\langle{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{u}f^{k}}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\right\rangle}\,dt-\frac{1}{2}{\left\langle{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\,,\,{{{\rm
ad}}^{*}_{u}f^{k}}\right\rangle}\,dt\\\
&\qquad\qquad\qquad+\frac{1}{2}{\left\langle{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\right\rangle}\,dt\,.\end{split}$
(3.8)

Taking the expectation yields

$\displaystyle\partial_{t}\mathbb{E}\left[h(m^{\prime})\right]=\frac{1}{2}{\left\langle{\mathbb{I}^{-1}{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{k}}\right\rangle}\,.$ (3.9)

The above calculation with (3.5) have proved the following.

######  Theorem 3.1 (Energy balance for EA SFLT).

The sum of energies
$h(\mathbb{E}\left[{m}\right])+\mathbb{E}\left[{h(m^{\prime})}\right]$ is
preserved by the dynamics of EA SFLT in (3.1),

$\displaystyle\partial_{t}h(\mathbb{E}\left[{m}\right])+\partial_{t}\mathbb{E}\left[{h(m^{\prime})}\right]=0\,.$
(3.10)

###### Proof.

The previous calculation demonstrates this theorem by direct calculation.
Having done so, an alternative proof suggests itself, for quadratic
Hamiltonians, one has
$h(m)=h(\mathbb{E}\left[{m}\right]+m^{\prime})=const\Longrightarrow\mathbb{E}\left[{h(m)}\right]=h(\mathbb{E}\left[{m}\right])+\mathbb{E}\left[{h(m^{\prime})}\right]$.
∎

###  3.2 EA SFLT with advected quantities

It is straight forward to extend the EA SFLT framework to SFLT systems with
advected quantities. Starting with the energy conserving SFLP equation with
advected quantities (2.19), one can take the average of $m$ and $a$ in the
underlying probability space to have the EA SFLP equation with advected
quantities

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+b\diamond\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\left({{\rm
ad}}^{*}_{u}f^{i}\circ dW^{i}_{t}+b\diamond f^{a}_{i}\circ
dW^{i}_{t}\right)=0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{u}\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\mathcal{L}_{u}f^{a}_{i}\circ
dW^{i}_{t}=0\,,\quad(u,b)=\left(\frac{\delta h}{\delta m},\frac{\delta
h}{\delta a}\right)\,,\end{split}$ (3.11)

which can be written more succintly by using the ${{\rm ad}}^{*}$ action on
the semidirect product Lie algebra, more specifically,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(m,a)+{{\rm ad}}^{*}_{(u,b)}\mathbb{E}[(m,a)]\,dt+\sum_{i}{{\rm
ad}}^{*}_{(u,b)}(f^{i},f^{a}_{i})\circ dW^{i}_{t}=0\,.$ (3.12)

Energy preservation is inherited from the SFLP equation with advected
quantities,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m,a)={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(m,a)}\,,\,{\left(\frac{\delta h}{\delta m},\frac{\delta h}{\delta
a}\right)}\right\rangle}=-{\left\langle{{{\rm
ad}}^{*}_{(u,b)}\mathbb{E}[(m,a)]\,dt+\sum_{i}{{\rm
ad}}^{*}_{(u,b)}(f^{i},f^{a}_{i})\circ
dW^{i}_{t}}\,,\,{(u,b)}\right\rangle}=0\,,$ (3.13)

where the last equality uses the anti-symmetry of ${{\rm ad}}^{*}$ of the
semidirect product Lie algebra. The Itô form of (3.11) is

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+b\diamond\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\left({{\rm
ad}}^{*}_{u}f^{i}\,dW^{i}_{t}+b\diamond
f^{a}_{i}\,dW^{i}_{t}+\frac{1}{2}\left({{\rm
ad}}^{*}_{\sigma_{i}}f^{i}+\theta_{i}\diamond
f^{a}_{i}\right)\,dt\right)=0,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{u}\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\left(\mathcal{L}_{u}f^{a}_{i}\,dW^{i}_{t}+\frac{1}{2}\mathcal{L}_{\sigma_{i}}f^{a}_{i}\,dt\right)=0,\end{split}$
(3.14)

where one defines

$\displaystyle\sigma_{i}:={\left({\frac{\delta^{2}h}{\delta
m^{2}}}\,,\,{-{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f^{i}-\frac{\delta
h}{\delta a}\diamond
f^{a}_{i}}\right)}\quad\hbox{and}\quad\theta_{i}:={\left({\frac{\delta^{2}h}{\delta
a}}\,,\,{-\mathcal{L}_{\frac{\delta h}{\delta m}}f^{a}_{i}}\right)}.$

The proof is similar to the case of SFLP equation with advected quantities
which is included in appendix C. Taking expectation of (3.14) to have temporal
evolution of the expectation of $m$ and $a$. These equations are deterministic
because of the linearity of ${{\rm ad}}^{*}$ and $\diamond$ operators, as well
as the $dW^{i}_{t}$ terms vanishing by Itô’s Lemma.

$\displaystyle\begin{split}&\partial_{t}\mathbb{E}\left[{m}\right]+{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}\mathbb{E}\left[{m}\right]+\mathbb{E}\left[{b}\right]\diamond\mathbb{E}\left[{a}\right]+\frac{1}{2}\sum_{i}\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{\sigma_{i}}\right]}f^{m}_{i}+\mathbb{E}\left[{\theta_{i}}\right]\diamond
f^{a}_{i}\right)=0\\\
&\partial_{t}\mathbb{E}\left[{a}\right]+\mathcal{L}_{\mathbb{E}\left[{u}\right]}\mathbb{E}\left[{a}\right]+\frac{1}{2}\sum_{i}\mathcal{L}_{\mathbb{E}\left[{\sigma_{i}}\right]}f^{a}_{i}=0.\end{split}$
(3.15)

Equations (3.15) closes when the Hamiltonian is assumed to be quadratic in $m$
and $a$, i.e., $\frac{\delta^{2}h}{\delta m^{2}}=\mathbb{I}^{-1}$ and
$\frac{\delta^{2}h}{\delta a^{2}}=\mathbb{J}^{-1}$, where the
$\mathbb{I}:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ and
$\mathbb{J}:V\rightarrow V^{*}$ are the inertia tensors for the vector fields
and advected quantities respectively, which are also assumed to be positive
definite.

$\displaystyle\frac{\delta h}{\delta
m^{\prime}}=\mathbb{I}^{-1}m^{\prime}=u^{\prime}\,,\quad\frac{\delta h}{\delta
a^{\prime}}=\mathbb{J}^{-1}a^{\prime}=b^{\prime}$

In this case one have the vector fields

$\displaystyle\begin{split}&\mathbb{E}\left[{\sigma_{i}}\right]={\left({-\frac{\delta^{2}h}{\delta
m^{2}}}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}}\right)}=-\mathbb{I}^{-1}\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}\right)\\\
&\mathbb{E}\left[{\theta_{i}}\right]={\left({-\frac{\delta^{2}h}{\delta
a^{2}}}\,,\,{\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}}\right)}=-\mathbb{J}^{-1}\left(\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}\right)\,,\end{split}$
(3.16)

and the equations (3.15) closes. Using the evolution of the expectations
$\mathbb{E}\left[{m}\right]$ and $\mathbb{E}\left[{a}\right]$, we have the
following time derivative of a quadratic Hamiltonian of
$\mathbb{E}\left[{m}\right]$ and $\mathbb{E}\left[{a}\right]$

$\displaystyle\begin{split}\partial_{t}h(\mathbb{E}\left[{m}\right],\mathbb{E}\left[{a}\right])&=-\sum_{i}\frac{1}{2}\left({\left\langle{\mathbb{E}\left[{u}\right]}\,,\,{{{\rm
ad}}^{*}_{\mathbb{E}\left[{\sigma_{i}}\right]}f^{m}_{i}+\mathbb{E}\left[{\theta_{i}}\right]\diamond
f^{a}_{i}}\right\rangle}+{\left\langle{\mathbb{E}\left[{b}\right]}\,,\,{\mathcal{L}_{\mathbb{E}\left[{\sigma_{i}}\right]}f^{a}_{i}}\right\rangle}\right)\\\
&=-\sum_{i}\frac{1}{2}\left({\left\langle{\mathbb{I}^{-1}\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}\right)}\,,\,{\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}\right)}\right\rangle}+{\left\langle{\mathbb{J}^{-1}\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}}\,,\,{\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}}\right\rangle}\right)<0\,.\end{split}$
(3.17)

The fluctuation $m^{\prime}=m-\mathbb{E}\left[{m}\right]$ and
$a^{\prime}=a-\mathbb{E}\left[{a}\right]$ are computed in Itô form using
(3.14) and (3.15) as

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m^{\prime}+{{\rm
ad}}^{*}_{u^{\prime}}\mathbb{E}\left[{m}\right]\,dt+b^{\prime}\diamond\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\left({{\rm
ad}}^{*}_{u}f^{i}\,dW^{i}_{t}+b\diamond
f^{a}_{i}\,dW^{i}_{t}+\frac{1}{2}\left({{\rm
ad}}^{*}_{\sigma_{i}^{\prime}}f^{i}+\theta_{i}^{\prime}\diamond
f^{a}_{i}\right)\,dt\right)=0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a^{\prime}+\mathcal{L}_{u^{\prime}}\mathbb{E}\left[{a}\right]\,dt+\sum_{i}\left(\mathcal{L}_{u}f^{a}_{i}\,dW^{i}_{t}+\frac{1}{2}\mathcal{L}_{\sigma_{i}^{\prime}}f^{a}_{i}\,dt\right)=0\,,\end{split}$
(3.18)

where the notations $b^{\prime}$, $\sigma_{i}^{\prime}$ and
$\theta_{i}^{\prime}$ are defined as
$b^{\prime}=b-\mathbb{E}\left[{b}\right]$,
$\sigma_{i}^{\prime}=\sigma_{i}-\mathbb{E}\left[{\sigma_{i}}\right]$ and
$\theta_{i}^{\prime}=\theta_{i}-\mathbb{E}\left[{\theta_{i}}\right]$. Then the
stochastic time derivative of quadratic Hamiltonians of fluctuations can be
computed in similar fashion as in the case without advected quantities.

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(m^{\prime},a^{\prime})&={\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m^{\prime}}\,,\,{\frac{\delta h}{\delta
m^{\prime}}}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a^{\prime}}\,,\,{\frac{\delta h}{\delta
a^{\prime}}}\right\rangle}+\frac{1}{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{m^{\prime}}\,,\,{\frac{\delta h}{\delta
m^{\prime}}}\right\rangle}_{t}+\frac{1}{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{a^{\prime}}\,,\,{\frac{\delta h}{\delta
a^{\prime}}}\right\rangle}_{t}\\\ &=-\sum_{i}{\left\langle{{{\rm
ad}}^{*}_{u}f^{i}+b\diamond
f^{a}_{i}}\,,\,{u^{\prime}}\right\rangle}\,dW^{i}_{t}+{\left\langle{\mathcal{L}_{u}f^{a}_{i}}\,,\,{b^{\prime}}\right\rangle}dW^{i}_{t}\\\
&\qquad-\frac{1}{2}{\left\langle{\mathbb{I}^{-1}\left({{\rm
ad}}^{*}_{u^{\prime}}f^{i}+b^{\prime}\diamond
f^{a}_{i}\right)}\,,\,{\left({{\rm
ad}}^{*}_{u^{\prime}}f^{i}+b^{\prime}\diamond
f^{a}_{i}\right)}\right\rangle}\,dt-\frac{1}{2}{\left\langle{\mathbb{J}^{-1}\mathcal{L}_{u^{\prime}}f^{a}_{i}}\,,\,{\mathcal{L}_{u^{\prime}}f^{a}_{i}}\right\rangle}\,dt\\\
&\qquad+\frac{1}{2}{\left\langle{\mathbb{I}^{-1}\left({{\rm
ad}}^{*}_{u}f^{i}+b\diamond f^{a}_{i}\right)}\,,\,{\left({{\rm
ad}}^{*}_{u}f^{i}+b\diamond
f^{a}_{i}\right)}\right\rangle}\,dt+\frac{1}{2}{\left\langle{\mathbb{J}^{-1}\mathcal{L}_{u}f^{a}_{i}}\,,\,{\mathcal{L}_{u}f^{a}_{i}}\right\rangle}\,dt\end{split}$
(3.19)

Taking expectation gives

$\displaystyle\partial_{t}\mathbb{E}\left[{h(m^{\prime},a^{\prime})}\right]=\sum_{i}\frac{1}{2}\left({\left\langle{\mathbb{I}^{-1}\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}\right)}\,,\,{\left({{\rm
ad}}^{*}_{\mathbb{E}\left[{u}\right]}f^{i}+\mathbb{E}\left[{b}\right]\diamond
f^{a}_{i}\right)}\right\rangle}+{\left\langle{\mathbb{J}^{-1}\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}}\,,\,{\mathcal{L}_{\mathbb{E}\left[{u}\right]}f^{a}_{i}}\right\rangle}\right)\,.$
(3.20)

The above calculations with (3.17) proves the following theorem

######  Theorem 3.2 (Energy balance for EA SFLT with advected quantities).

Equations (3.11) lead to the energy balance,

$\displaystyle\partial_{t}h(\mathbb{E}\left[{m}\right],\mathbb{E}\left[{a}\right])+\partial_{t}\mathbb{E}\left[{h(m^{\prime},a^{\prime})}\right]=0\,.$
(3.21)

###### Proof.

The previous calculation demonstrates this theorem by direct calculation.
Having done so, an alternative proof suggests itself for quadratic
Hamiltonians, since in this case one has
$h(m,a)=h(\mathbb{E}\left[{m}\right]+m^{\prime},\mathbb{E}\left[{a}\right]+a^{\prime})=const\Longrightarrow\mathbb{E}\left[{h(m,a)}\right]=h(\mathbb{E}\left[{m}\right],\mathbb{E}\left[{a}\right])+\mathbb{E}\left[{h(m^{\prime},a^{\prime})}\right]$.
∎

######  Theorem 3.3 (EA SFLT Kelvin circulation theorem).

Upon assuming that the fluid density $D$ is also advected by the flow, so that
$\partial_{t}D+\mathcal{L}_{u}D=0$, the Kelvin circulation theorem may be
expressed as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\frac{m}{D}=\oint_{c(u)}\mathcal{L}_{u}\left(\frac{m}{D}\right)\,dt+\frac{1}{D}\left[\frac{m}{D}\mathcal{L}_{u}D\,dt-{{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt-b\diamond\mathbb{E}\left[{a}\right]\,dt-\sum_{i}\left({{\rm
ad}}^{*}_{u}f^{m}_{i}\circ dW^{i}_{t}+b\diamond f^{i}_{a}\circ
dW^{i}_{t}\right)\right],$ (3.22)

where the notation of Lagrangian loop $c(u)$ denotes the Lagrangian loop
moving with the deterministic fluid velocity $u$, as in the deterministic
case.

###### Proof.

When $m\in\mathfrak{X}^{*}$, one can identify the ${{\rm ad}}^{*}$ action as
the Lie derivative, i.e. ${{\rm ad}}^{*}_{u}m=\mathcal{L}_{u}m$ for all
$u\in\mathfrak{u}$. Let $m=D\alpha$ where $\alpha\in\Lambda^{1}$, the
stochastic time derivative of $m$ can be written as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m$
$\displaystyle=\alpha\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}D+D\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha=-\alpha\mathcal{L}_{u}D\,dt+D\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha,$ (3.23)

thus

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha=\frac{\alpha}{D}\mathcal{L}_{u}D\,dt-\frac{1}{D}\left({{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+b\diamond\mathbb{E}\left[{a}\right]\,dt+\sum_{i}({{\rm
ad}}^{*}_{u}f^{i}\circ dW^{i}_{t}+b\diamond f^{a}_{i}\circ dW^{i}_{t})\right)$
(3.24)

Inserting the expression of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha$ into the circulation integral to have the result.

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\alpha&=\oint_{c(u)}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}+\mathcal{L}_{u\,dt}\right)\alpha\\\
&=\oint_{c(u)}\mathcal{L}_{u}\alpha\,dt+\frac{\alpha}{D}\mathcal{L}_{u}D\,dt-\frac{1}{D}\left({{\rm
ad}}^{*}_{u}\mathbb{E}\left[{m}\right]\,dt+b\diamond\mathbb{E}\left[{a}\right]\,dt+\sum_{i}({{\rm
ad}}^{*}_{u}f^{i}\circ dW^{i}_{t}+b\diamond f^{a}_{i}\circ
dW^{i}_{t})\right)\end{split}$ (3.25)

∎

######  Remark 3.4.

This section has responded to Lorentz’s lesson in [55] that climate is
essentially probabilistic. Accordingly, one might imagine that climate change
science would be predicated on the dynamics of variances and higher moments of
the fluctuations, which might even apply to the considerations introduced
here. Investigations of the potential for applications of the EA SFLT
formulation to climate science have been left for future work.

##  4 Examples of SFLT applications

###  4.1 Heavy Top

The motion of a heavy top under gravity is a good first example application
for SFLT, because it is a finite degree-of-freedom subsystem of Euler-
Boussinesq fluid motion [33]. Following, e.g., [37], the configuration space
of the heavy top can be taken as the semidirect-product Lie group of Euclidean
motions by rotation and translation, $S={\rm SO}(3)\circledS\mathbb{R}^{3}$.
The associated Lie algebra and its dual are, respectively,

$\mathfrak{s}=\mathfrak{so}(3)\circledS\mathbb{R}^{3}\cong\mathbb{R}^{3}\circledS\mathbb{R}^{3}\quad\hbox{and}\quad\mathfrak{s}^{*}=\mathfrak{so}(3)^{*}\circledS\mathbb{R}^{3}\cong\mathbb{R}^{3}\circledS\mathbb{R}^{3}\,.$

The natural pairing between $\mathbb{R}^{3}$ and its dual is the dot-product,
for the pairing of Euclidean vectors in $\mathbb{R}^{3}$. The heavy top
Hamiltonian $H(\mbox{\boldmath{$\Pi$}},\mbox{\boldmath{$\Gamma$}})$ comprises
the sum of its rotational kinetic energy and its gravitational potential
energy,

$\displaystyle
H(\mbox{\boldmath{$\Pi$}})=\frac{1}{2}\mbox{\boldmath{$\Pi$}}\cdot\mathbb{I}^{-1}\mbox{\boldmath{$\Pi$}}+mg\mbox{\boldmath{$\Gamma$}}\cdot\mbox{\boldmath{$\chi$}}\,.$
(4.1)

The Hamiltonian for the heavy top is written in the body frame, in terms of
the body angular momentum, $\Pi$, and the vertical unit vector as seen from
the body,
$\mbox{\boldmath{$\Gamma$}}=\mbox{\boldmath{$\widehat{z}$}}O(t)^{-1}$, where
$O(t)\in SO(3)$ is the time-dependent rotation from the reference
configuration of the top to its current configuration. Here, $\mathbb{I}$ is
the moment of inertia in the body frame and $\chi$ is the vector from the
point of support to the centre of mass in the body frame. Finally, $m$ is the
mass of the body and $g$ is the constant acceleration of gravity.
The variational derivatives of the heavy top Hamiltonian are

${dH}/{d\mbox{\boldmath{$\Pi$}}}=\mathbb{I}^{-1}\mbox{\boldmath{$\Pi$}}=\mbox{\boldmath{$\Omega$}}\quad\hbox{and}\quad{dH}/{d\mbox{\boldmath{$\Gamma$}}}=mg\mbox{\boldmath{$\chi$}}\,.$

Upon using the ‘hat’ map isomorphism [37] to identify the Lie derivative and
${{\rm ad}}^{*}$ operation as the cross product in the $\mathbb{R}^{3}$
representation, the deterministic LP equation is written as

$\displaystyle\frac{d\mbox{\boldmath{$\Pi$}}}{dt}=\mbox{\boldmath{$\Pi$}}\times\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$\Gamma$}}\times
mg\mbox{\boldmath{$\chi$}},\quad\frac{d\mbox{\boldmath{$\Pi$}}}{dt}=\mbox{\boldmath{$\Gamma$}}\times\mbox{\boldmath{$\Omega$}}\,,$

which can also be written in Poisson operator form

$\displaystyle\frac{d}{dt}\begin{bmatrix}\mbox{\boldmath{$\Pi$}}\\\
\mbox{\boldmath{$\Gamma$}}\end{bmatrix}=\begin{bmatrix}\mbox{\boldmath{$\Pi$}}\times&\mbox{\boldmath{$\Gamma$}}\times\\\
\mbox{\boldmath{$\Gamma$}}\times&0\end{bmatrix}\begin{bmatrix}\delta
H/\delta\mbox{\boldmath{$\Pi$}}\\\ \delta
H/\delta\mbox{\boldmath{$\Gamma$}}\end{bmatrix}.$

Defining the force vectors
$\mbox{\boldmath{$f^{m}_{i}$}},\mbox{\boldmath{$f^{a}_{i}$}}\in\mathbb{R}^{3}$,
the SFLP equations corresponding to (2.19) may be written as

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Pi$}}&=\left(\mbox{\boldmath{$\Pi$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\circ
dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}+\left(\mbox{\boldmath{$\Gamma$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t}\right)\times mg\mbox{\boldmath{$\chi$}}\,,\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Gamma$}}&=\left(\mbox{\boldmath{$\Gamma$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}\,.\end{split}$ (4.2)

When written in Poisson bracket form, these equations become

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}\mbox{\boldmath{$\Pi$}}\\\
\mbox{\boldmath{$\Gamma$}}\end{bmatrix}=\begin{bmatrix}(\mbox{\boldmath{$\Pi$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\circ
dW^{i}_{t})\times&(\mbox{\boldmath{$\Gamma$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t})\times\\\
(\mbox{\boldmath{$\Gamma$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t})\times&0\end{bmatrix}\begin{bmatrix}\delta
H/\delta\mbox{\boldmath{$\Pi$}}\\\ \delta
H/\delta\mbox{\boldmath{$\Gamma$}}\end{bmatrix}.$ (4.3)

One observes that these equations preserve the Hamiltonian in (4.1), which
follows because of skew symmetry of the matrix Poisson operator.

#### Eulerian averaged heavy top

Consider the EA SFLP equations associated to (4.2), which read

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Pi$}}&=\left(\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\circ
dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}+\left(\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t}\right)\times mg\mbox{\boldmath{$\chi$}}\,,\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Gamma$}}&=\left(\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\circ
dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}\,.\end{split}$ (4.4)

Energy conservation is immediate since the Poisson struture is preserved. In
Itô form, these equation reads

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Pi$}}&=\left(\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\,dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}+\left(\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\,dW^{i}_{t}\right)\times
mg\mbox{\boldmath{$\chi$}}\\\
&\qquad\qquad+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi$}}\right)\,,\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Gamma$}}&=\left(\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\,dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega$}}+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi$}}\right)\,.\end{split}$ (4.5)

Taking expectations to have the time evolution of expectations
$\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]$ and
$\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]$ as

$\displaystyle\begin{split}\partial_{t}\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]&=\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\times
mg\mbox{\boldmath{$\chi$}}+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\,,\\\
\partial_{t}\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]&=\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\,.\end{split}$
(4.6)

As the Hamiltonian is quadratic, the above equations close and we have

$\displaystyle\partial_{t}H(\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right],\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right])=-\frac{1}{2}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\,.$ (4.7)

If the inertial tensor $\mathbb{I}$ is positive definite, the energy of the
expectations $\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]$ and
$\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]$ will decay to zero.
Defining the fluctuation of $\Pi$ and $\Gamma$ as
$\mbox{\boldmath{$\Pi^{\prime}$}}:=\mbox{\boldmath{$\Pi$}}-\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right]$
and
$\mbox{\boldmath{$\Gamma^{\prime}$}}:=\mbox{\boldmath{$\Gamma$}}-\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]$
respectively, the evolution of the fluctuation can be written in Itô form as

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Pi^{\prime}$}}&=\left(\mbox{\boldmath{$\Pi^{\prime}$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\,dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega^{\prime}$}}+\left(\mbox{\boldmath{$\Gamma^{\prime}$}}\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\,dW^{i}_{t}\right)\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\\\
&\qquad\qquad+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\right)\,,\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\mbox{\boldmath{$\Gamma^{\prime}$}}&=\left(\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right]\,dt+\sum_{i}\mbox{\boldmath{$f^{a}_{i}$}}\,dW^{i}_{t}\right)\times\mbox{\boldmath{$\Omega^{\prime}$}}+\frac{1}{2}\sum_{i}\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\right)\,.\end{split}$ (4.8)

The energy of the fluctuations
$h(\mbox{\boldmath{$\Pi^{\prime}$}},\mbox{\boldmath{$\Gamma^{\prime}$}})$ has
evoluation equation after using Itô’s Lemma

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}h(\mbox{\boldmath{$\Pi^{\prime}$}},\mbox{\boldmath{$\Gamma^{\prime}$}})&=\sum_{i}\mbox{\boldmath{$\Omega^{\prime}$}}\cdot\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\right)\,dW^{i}_{t}+mg\mbox{\boldmath{$\chi^{\prime}$}}\cdot\mbox{\boldmath{$f^{a}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}\,dW^{i}_{t}\\\
&\qquad\qquad-\frac{1}{2}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\right)\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega^{\prime}$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi^{\prime}$}}\right)\\\
&\qquad\qquad+\frac{1}{2}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi$}}\right)\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mbox{\boldmath{$\chi$}}\right)\end{split}$ (4.9)

Taking the expectation yields

$\displaystyle\partial_{t}\mathbb{E}\left[{h(\mbox{\boldmath{$\Pi^{\prime}$}},\mbox{\boldmath{$\Gamma^{\prime}$}})}\right]=\frac{1}{2}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\mathbb{I}^{-1}\left(\mbox{\boldmath{$f^{m}_{i}$}}\times\mathbb{E}\left[{\mbox{\boldmath{$\Omega$}}}\right]+\mbox{\boldmath{$f^{a}_{i}$}}\times
mg\mathbb{E}\left[{\mbox{\boldmath{$\chi$}}}\right]\right)\,.$ (4.10)

Together with the expression of
$\partial_{t}H(\mathbb{E}\left[{\mbox{\boldmath{$\Pi$}}}\right],\mathbb{E}\left[{\mbox{\boldmath{$\Gamma$}}}\right])$,
we have exemplified Theorem 3.2 for the EA HT dynamics.

###  4.2 Energy-preserving stochastic rotating shallow water equations (RSW)

Let $S=\text{Diff}(\mathcal{D})\circledS\text{Den}(\mathcal{D})$, where
$\text{Diff}(\mathcal{D})$ denotes the group of diffeomorphisms acting on the
planar domain $\mathcal{D}$. Let $\eta\in\mathcal{F}(\mathcal{D})$, and let
$\eta\,dV\in\text{Den}(\mathcal{D})$ denote the density on $\mathcal{D}$. The
associated Lie algebra and its dual are
$\mathfrak{s}=\mathfrak{X}(\mathcal{D})\circledS\text{Den}(\mathcal{D})$ and
$\mathfrak{s}^{*}=(\Lambda^{1}(\mathcal{D})\otimes\text{Den}(\mathcal{D}))\circledS\text{Den}(\mathcal{D})$.
We denote ${\bf u}\cdot\frac{\partial}{\partial{\bf
x}}=u\in\mathfrak{X}(\mathcal{D})$ and $m={\bf m}\cdot d{\bf x}\otimes
dV\in\Lambda^{1}(\mathcal{D})\otimes\text{Den}(\mathcal{D})$. In this
notation, the LP equations can be written in Cartesian coordinates as the
following partial differential equations,

$\displaystyle\frac{\partial}{\partial t}\left(\frac{{\bf
m}}{\eta}\right)+({\bf u}\cdot{\bm{\nabla}})\frac{{\bf
m}}{\eta}+\frac{m_{j}}{\eta}{\bm{\nabla}}u^{j}+{\bm{\nabla}}\frac{\delta
H}{\delta\eta}=0\,,\qquad\frac{\partial}{\partial
t}\eta+{\bm{\nabla}\cdot}(\eta{\bf u})=0\,,$ (4.11)

and the LP operator can be written as

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}m_{i}\\\
\eta\end{bmatrix}=-\begin{bmatrix}\partial_{j}m_{i}+m_{j}\partial_{i}&\eta\partial_{i}\\\
\partial_{j}\eta&0\end{bmatrix}\begin{bmatrix}\delta H/\delta m_{j}\\\ \delta
H/\delta\eta\\\ \end{bmatrix}\,.$ (4.12)

For rotating shallow water equations (RSW) in $\mathbb{R}^{2}$, the
Hamiltonian $H$ is given by

$\displaystyle H=\int_{\mathcal{D}}\frac{1}{2\epsilon\eta}\left|{\bf
m}-\eta{\bf R}\right|^{2}+\frac{(\eta-B)^{2}}{\epsilon\mathcal{F}}\,d^{2}x\,,$
(4.13)

in which $\epsilon\ll 1$ denotes Rossby number and $\mathcal{F}=O(1)$ denotes
the Froude number. The mean depth is $B$ and the surface elevation is
$(\eta-B)$. The reduced Legendre transform yields ${\bf u}={\delta
H}/{\delta{\bf m}}=({\bf m}-\eta{\bf R})/{\epsilon\eta}$. The variational
derivatives of the RSW Hamiltonian are obtained as

$\displaystyle\delta H=\int_{\mathcal{D}}{\bf u}\cdot\delta{\bf
m}+\left(\frac{\eta-B}{\epsilon\mathcal{F}}-\frac{\epsilon}{2}|{\bf
u}|^{2}-{\bf u}\cdot{\bf R}\right)\cdot\delta\eta\,d^{2}x\,.$

Substituting into (4.11) and using the relation $(\hbox{curl}{\bf
u}){\bm{\times}}{\bf v}+{\bm{\nabla}}({\bf u}\cdot{\bf v})=({\bf
u}\cdot{\bm{\nabla}}){\bf v}+u_{i}{\bm{\nabla}}v^{i}$ yields the standard set
of RSW equations governing motion and continuity,

$\displaystyle\epsilon\frac{\partial{\bf u}}{\partial t}+\left(\hbox{curl}{\bf
R}+\epsilon\,\hbox{curl}{\bf u}\right){\bm{\times}}{\bf
u}+{\bm{\nabla}}\left(\frac{\eta-B}{\epsilon\mathcal{F}}+\frac{\epsilon}{2}|{\bf
u}|^{2}\right)=0\,,\quad\frac{\partial}{\partial
t}\eta+{\bm{\nabla}\cdot}(\eta{\bf u})=0\,.$ (4.14)

Consider the Poisson operator of the form (2.20) applied to
$\mathfrak{s}^{*}$. It reads

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}m_{i}\\\
\eta\end{bmatrix}=-\begin{bmatrix}\partial_{j}m_{i}+m_{j}\partial_{i}&\eta\partial_{i}\\\
\partial_{j}\eta&0\end{bmatrix}\begin{bmatrix}\delta H/\delta m_{j}\\\ \delta
H/\delta\eta\\\
\end{bmatrix}\,dt-\sum_{k}\begin{bmatrix}\partial_{j}f^{k}_{i}+f^{k}_{j}\partial_{i}&g^{k}\partial_{i}\\\
\partial_{j}g^{k}&0\end{bmatrix}\begin{bmatrix}\delta H/\delta m_{j}\\\ \delta
H/\delta\eta\\\ \end{bmatrix}\circ dW_{t}^{k}\,,$ (4.15)

where $f^{k}_{i}$ are the components of ${\bf f}^{k}$ such that $f^{k}={\bf
f}^{k}\cdot d{\bf x}\otimes
dV\in\Lambda^{1}(\mathcal{D})\otimes\text{Den}(\mathcal{D})$ and
$g^{k}\in\mathcal{F}(\mathcal{D})$ for all $k$. Then the SFLP equations become

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m_{i}+(\partial_{j}(m_{i}u^{j})+m_{j}\partial_{i}u^{j})\,dt+(\partial_{j}(f^{k}_{i}u^{j})+\sum_{k}f^{k}_{j}\partial_{i}u^{j})\circ
dW_{t}^{k}+\eta\partial_{i}\frac{\delta
H}{\delta\eta}\,dt+g^{k}\partial_{i}\frac{\delta H}{\delta\eta}\circ
dW_{t}^{k}=0,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\bm{\nabla}\cdot}(\eta{\bf
u})\,dt+\sum_{k}{\bm{\nabla}\cdot}(g^{k}{\bf u})\circ
dW_{t}^{k}=0.\end{split}$ (4.16)

######  Remark 4.1.

When noise is introduced into the continuity equation for density $\eta$, one
can still write the momentum equation in terms of ${{\bf m}}/{\eta}$. Namely,
it reads

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{m_{i}}{\eta}&+\left(u^{j}\partial_{j}\frac{m_{i}}{\eta}+\frac{m_{j}}{\eta}\partial_{i}u^{j}\right)\,dt+\partial_{i}\frac{\delta
H}{\delta\eta}\,dt\\\
&+\sum_{k}\frac{1}{\eta}\left(-\frac{m_{i}}{\eta}\partial_{j}(g^{k}u^{j})+\partial_{j}(f^{k}_{i}u^{j})+f^{k}_{j}\partial_{i}u^{j}+g^{k}\partial_{i}\frac{\delta
H}{\delta\eta}\right)\circ dW_{t}^{k}=0,\end{split}$ (4.17)

where the inhomogenous term $-\eta^{-2}{m_{j}}\partial_{i}(g^{k}u^{j})$ in the
second line of (4.17) is due to the modified advection relation for
$\eta\,dV$. The SFLP equations for RSW dynamics in (4.16) written more
succinctly using Lie derivatives and denoting the momentum 1-form as
$\alpha={\bf m}\cdot d{\bf x}$, then

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\alpha}{\eta}+\left(\mathcal{L}_{u}\frac{\alpha}{\eta}+{\bf
d}\frac{\delta
H}{\delta\eta}\right)\,dt+\sum_{k}\frac{1}{\eta\,dV}\left(-\frac{\alpha}{\eta}\mathcal{L}_{u}g^{k}+\mathcal{L}_{u}f^{k}+\frac{\delta
H}{\delta\eta}\diamond g^{k}\right)\circ dW^{k}_{t}=0,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(\eta\,dV)+\mathcal{L}_{u}(\eta\,dV)\,dt+\sum_{k}\mathcal{L}_{u}(g^{k}_{i}\,dV)\circ
dW_{t}^{i}=0\,.\end{split}$ (4.18)

In vector calculus notation, the stochastic RSW equation obtained by
substituting the variational derivatives of $H$ into equation (4.15) yields

$\displaystyle\begin{split}&\epsilon{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}+\left(\hbox{curl}{\bf R}+\epsilon\,\hbox{curl}{\bf
u}\right){\bm{\times}}{\bf
u}\,dt+{\bm{\nabla}}\left(\frac{\eta-B}{\epsilon\mathcal{F}}+\frac{\epsilon}{2}|{\bf
u}|^{2}\right)\,dt\\\ &\quad+\sum_{k}\frac{1}{\eta}\left(-\left(\epsilon{\bf
u}+{\bf R}\right){\bm{\nabla}\cdot}(g^{k}{\bf u})+(\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bf u}+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf u})+{\bf
f}^{k}{\bm{\nabla}\cdot}{\bf u}+g^{k}{\bm{\nabla}}\pi_{RSW}\right)\circ
dW_{t}^{k}=0\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta+{\bm{\nabla}\cdot}(\eta{\bf
u})\,dt+\sum_{k}{\bm{\nabla}\cdot}(g^{k}{\bf u})\circ
dW_{t}^{k}=0,\end{split}$ (4.19)

where
$\pi_{RSW}=\left(\frac{\eta-B}{\epsilon\mathcal{F}}-\frac{\epsilon}{2}|{\bf
u}|^{2}-{\bf u}\cdot{\bf R}\right)$.

The corresponding Kelvin circulation theorem follows easily from (4.18) in
geometric form, as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\frac{\alpha}{\eta}=-\oint_{c(u)}\sum_{k}\frac{1}{\eta\,dV}\left(-\frac{\alpha}{\eta}\mathcal{L}_{u}g^{k}+\mathcal{L}_{u}f^{k}+g^{k}d\frac{\delta
H}{\delta\eta}\right)\circ dW^{k}_{t}\,.$ (4.20)

The stochastic RSW equations in (4.19) enable the Kelvin circulation equation
(4.20) to be written in vector calculus form as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}(\epsilon{\bf u}+{\bf R})\cdot d{\bf x}$ (4.21)
$\displaystyle\quad=-\oint_{c(u)}\sum_{k}\frac{1}{\eta}\left(-\left(\epsilon{\bf
u}+{\bf R}\right){\bm{\nabla}\cdot}(g^{k}{\bf u})+(\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bf u}+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf u})+{\bf
f}^{k}({\bm{\nabla}\cdot}{\bf u})+g^{k}{\bm{\nabla}}\pi_{RSW}\right)\cdot
d{\bf x}\circ dW_{t}^{k}.$

Here, ones sees the effects of the material entrainment terms proportional to
$g^{k}$ appearing in the generation of Kelvin circulation. Setting $g^{k}=0$
simplifies equation (4.20) to

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\frac{\alpha}{\eta}=-\oint_{c(u)}\sum_{k}\frac{1}{\eta\,dV}\left(\mathcal{L}_{u}f^{k}\right)\circ
dW^{k}_{t}\,.$ (4.22)

The vector form of the motion equation also simplifies for $g^{k}=0$ to

$\displaystyle\begin{split}&\epsilon{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}+\left(\hbox{curl}{\bf R}+\epsilon\,\hbox{curl}{\bf
u}\right){\bm{\times}}{\bf
u}\,dt+{\bm{\nabla}}\left(\frac{\eta-B}{\epsilon\mathcal{F}}+\frac{\epsilon}{2}|{\bf
u}|^{2}\right)\,dt\\\ &\qquad=-\sum_{k}\frac{1}{\eta}\left((\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bf u}+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf u})+{\bf
f}^{k}{\bm{\nabla}\cdot}{\bf u}\right)\circ dW_{t}^{k}\,,\end{split}$ (4.23)

whose right hand side may be regarded as a compressible version of the CL
vortex force. The Ito form is

$\displaystyle\begin{split}&\epsilon{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}+\left(\hbox{curl}{\bf R}+\epsilon\,\hbox{curl}{\bf
u}\right){\bm{\times}}{\bf
u}\,dt+{\bm{\nabla}}\left(\frac{\eta-B}{\epsilon\mathcal{F}}+\frac{\epsilon}{2}|{\bf
u}|^{2}\right)\,dt\\\ &\qquad+\sum_{k}\frac{1}{\eta}\left((\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bf u}+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf u})+{\bf
f}^{k}{\bm{\nabla}\cdot}{\bf
u}\right)\,dW_{t}^{k}=\frac{1}{2}\sum_{k}\frac{1}{\eta}\left((\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bm{\sigma}}^{k}+{\bm{\nabla}}({\bf
f}^{k}\cdot{\bm{\sigma}}^{k})+{\bf
f}^{k}{\bm{\nabla}\cdot}{\bm{\sigma}}^{k}\right)\,dt\,,\end{split}$ (4.24)

where ${\bm{\sigma}}^{k}=\frac{1}{\eta}\left((\hbox{curl}{\bf
f}^{k}){\bm{\times}}{\bf u}+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf u})+{\bf
f}^{k}{\bm{\nabla}\cdot}{\bf u}\right)$. By taking the exterior derivative of
equation (4.18) and noting ${\bf d}\frac{\alpha}{\eta}=\eta q\,dV$ where $q$
is the potential vorticity and $dV$ is the area element, one finds that the
vorticity density $\eta q\,dV$ satisfies

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(\eta q\,dV)+\mathcal{L}_{u}(\eta q\,dV)\,dt+\sum_{k}{\bf
d}\frac{1}{\eta\,dV}\left[-\frac{\alpha}{\eta}\mathcal{L}_{u}g^{k}+\mathcal{L}_{u}f^{k}+g^{k}{\bf
d}\frac{\delta H}{\delta\eta}\right]=0\,.$ (4.25)

######  Remark 4.2.

In coordinates, the last two summands in equation (4.25) can be written as
divergences, so one finds the pathwise conservation law
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\int_{\mathcal{D}}(\eta q\,dV)=0$ for appropriate (homogeneous, or
periodic) boundary conditions.

###  4.3 Euler-Boussinesq (EB) equations

Consider the case where
$S=\text{Diff}(\mathcal{D})\circledS\mathcal{F}(\mathcal{D})\circledS\text{Den}(\mathcal{D})$,
where $D\,dV\in\text{Den}(\mathcal{D})$ and $b\in\mathcal{F}(\mathcal{D})$.
The Poisson operator form of the LP equation in this case is

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}m_{i}\\\ D\\\
b\end{bmatrix}=-\begin{bmatrix}\partial_{j}m_{i}+m_{j}\partial_{i}&D\partial_{i}&-b_{,i}\\\
\partial_{j}D&0&0\\\ b_{,j}&0&0\end{bmatrix}\begin{bmatrix}\delta H/\delta
m_{j}\\\ \delta H/\delta D\\\ \delta H/\delta b\end{bmatrix}\,.$ (4.26)

The stochastic extension of equation (2.20) to include buoyancy reads

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}m_{i}\\\ D\\\
b\end{bmatrix}=-\begin{bmatrix}\partial_{j}m_{i}+m_{j}\partial_{i}&D\partial_{i}&-b_{,i}\\\
\partial_{j}D&0&0\\\ b_{,j}&0&0\end{bmatrix}\begin{bmatrix}\delta H/\delta
m_{j}\\\ \delta H/\delta D\\\ \delta H/\delta
b\end{bmatrix}\,dt-\sum_{k}\begin{bmatrix}\partial_{j}f^{k}_{i}+f^{k}_{j}\partial_{i}&g^{k}\partial_{i}&-a^{k}_{,i}\\\
\partial_{j}g^{k}&0&0\\\ a^{k}_{,j}&0&0\end{bmatrix}\begin{bmatrix}\delta
H/\delta m_{j}\\\ \delta H/\delta D\\\ \delta H/\delta b\end{bmatrix}\circ
dW_{t}^{k}\,.$ (4.27)

Here, $f^{k}_{i}$ are the components of ${\bf f}^{k}$ such that $f^{k}={\bf
f}^{k}\cdot d{\bf x}\otimes
dV\in\Lambda^{1}(\mathcal{D})\otimes\text{Den}(\mathcal{D})$,
$g^{k}\in\mathcal{F}(\mathcal{D})$ and $a^{k}\in\mathcal{F}(\mathcal{D})$ for
all $k$. For simplicity, let us consider incompressible flows with $D=1$ and
neglect the stochastic part of the advection of $D$. That is, we set
$g^{k}=0$. This case will yield the standard incompressibility condition,
${\bm{\nabla}\cdot}{\bf u}=0$.

The EB Hamiltonian is

$\displaystyle H=\int_{\mathcal{D}}\bigg{[}\frac{1}{2D}\big{|}{\bf m}-D{\bf
R}\big{|}^{2}+gDbz+p(D-1)\bigg{]}d^{3}x\,,$ (4.28)

where the momentum density ${\bf m}=D\Big{(}{\bf u}+{\bf R}({\bf x})\Big{)}$
and the pressure $p$ is a Lagrange multiplier which enforces
incompressibility. The variational derivatives of $H$ are given by

$\displaystyle\delta H=\int_{\mathcal{D}}{\bf u}\cdot\delta{\bf m}+\delta
D\Big{(}gbz+p-\frac{1}{2}|{\bf u}|^{2}-{\bf u}\cdot{\bf R}({\bf
x})\Big{)}+(gDz)\delta b\,d^{3}x\,.$ (4.29)

The deterministic EB equations then follow as

$\displaystyle\begin{split}\partial_{t}{\bf u}-{\bf u}\times{\rm
curl}\big{(}{\bf u}+{\bf R}({\bf
x})\big{)}&=-\,\nabla\Big{(}p+\frac{1}{2}|{\bf
u}|^{2}\Big{)}-gb\,\mathbf{\hat{z}}\,,\\\ \partial_{t}D+{\rm div}(D{\bf
u})&=0\,,\quad\hbox{with}\quad D=1\,,\\\ \partial_{t}b+{\bf u}\cdot\nabla
b&=0\,.\end{split}$ (4.30)

One can obtain the energy preserving stochastic EB equations by substituting
the variational derivatives into stochastic Poisson operator in (4.27) to find

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}&-{\bf u}{\bm{\times}}\hbox{curl}({\bf u}+{\bf
R})\,dt+{\bm{\nabla}}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}p+\frac{1}{2}|{\bf
u}|^{2}\,dt\right)+gb\mathbf{\hat{z}}\,dt+\sum_{k}\left[-{\bf
u}{\bm{\times}}\hbox{curl}({\bf f}^{k})+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf
u})-gz\cdot{\bm{\nabla}}a^{k}\right]\circ dW_{t}^{k}=0\,,\\\
&\partial_{t}D+{\rm div}(D{\bf u})=0\,,\quad\hbox{with}\quad D=1\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b+{\bf u}\cdot{\bm{\nabla}}b\,dt+\sum_{k}{\bf
u}\cdot{\bm{\nabla}}a^{k}\circ dW_{t}^{k}=0\,.\end{split}$ (4.31)

######  Remark 4.3.

Following [66], the change of pressure
$p\rightarrow{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}p=p\,dt+\sum_{i}p_{i}\circ dW_{t}^{i}$ has been made, so that the
incompressibility condition, ${\bm{\nabla}\cdot}{\bf u}=0$ will be enforced in
both the drift and stochastic parts of ${\bf u}$.

Similar to the case of
$S=\text{Diff}(\mathcal{D})\circledS\text{Den}(\mathcal{D})$ for energy-
preserving stochastic RSW equation, a convenient way of considering the Kelvin
circulation theorem for the Euler-Boussinesq equations is to write the
associated SFLP equations in terms of the momentum 1-form $\alpha={\bf m}\cdot
d{\bf x}$. These equations then read

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha+\left(\mathcal{L}_{u}\alpha-\frac{\delta H}{\delta b}{\bf
d}b\right)\,dt+{\bf
d}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}p-\frac{1}{2}|u|^{2}\,dt-{\bf u}\cdot{\bf
R}\,dt\right)+\sum_{k}\frac{1}{D}\left(\mathcal{L}_{u}\beta^{k}-\frac{\delta
H}{\delta b}{\bf d}a^{k}\right)\circ dW^{k}_{t}=0\,,\\\
&\partial_{t}\left(D\,dV\right)+\mathcal{L}_{u}(D\,dV)=0\,,\quad\hbox{with}\quad
D=1\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b+{\bf u}\cdot{\bm{\nabla}}b\,dt+\sum_{k}{\bf
u}\cdot{\bm{\nabla}}a^{k}\circ dW_{t}^{k}=0\,,\end{split}$ (4.32)

where the 1-form $\beta^{k}:={\bf f}^{k}\cdot d{\bf x}$. The Kelvin
circulation theorem for these equations is immediate, as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}\alpha=\oint_{c(u)}\frac{\delta H}{\delta b}{\bf
d}b\,dt-\sum_{k}\oint_{c(u)}\left(\mathcal{L}_{u}\beta^{k}-\frac{\delta
H}{\delta b}{\bf d}a^{k}\right)\circ dW_{t}^{k}\,.$ (4.33)

In vector calculus notation, this Kelvin circulation theorem is written as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c(u)}({\bf u}+{\bf R})\cdot d{\bf
x}=\oint_{c(u)}gz{\bm{\nabla}}b\cdot d{\bf
x}\,dt-\sum_{k}\oint_{c(u)}\left((\hbox{curl}{\bf f}^{k}){\bm{\times}}{\bf
u}-gz{\bm{\nabla}}a^{k}\right)\cdot d{\bf x}\circ dW^{k}_{t}$ (4.34)

######  Remark 4.4.

In the case where ${\bf f}^{0}=-{\bf u}^{S}({\bf x})$, $dW_{t}^{0}=dt$, and
$a^{k}=0$ for $k=0,\ldots$, one finds that the energy-preserving stochastic EB
equations in (4.31) produce a stochastic contribution to the vortex force in
the Craik-Leibovich equations [15] whose formulation with Hamilton’s principle
is discussed in [34]. Namely, they reduce as follows,

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}-{\bf u}\times{\rm curl}\big{(}{\bf u}-{\bf u}^{S}({\bf x})+{\bf
R}({\bf x})\big{)}dt&=-\,\nabla\Big{(}p+\frac{1}{2}|{\bf u}|^{2}+{\bf
u}\cdot{\bf u}^{S}\Big{)}dt-gb\,\mathbf{\hat{z}}\,dt\\\
&\qquad+\sum_{k>0}\Big{(}{\bf u}\times{\rm curl}\,{\bf
f}^{k}-\,\nabla\big{(}{\bf u}\cdot{\bf f}^{k}\big{)}\Big{)}\circ
dW^{k}_{t}\,,\\\ \partial_{t}D+{\rm div}(D{\bf u})&=0\,,\quad\hbox{with}\quad
D=1\,,\\\ \partial_{t}b+{\bf u}\cdot\nabla b&=0\,,\end{split}$ (4.35)

where one interprets the semimartingale
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}^{S}={\bf u}^{S}({\bf x})\,dt+\sum_{k}{\bf f}^{k}({\bf x})\circ
dW^{k}_{t}$ as a stochastic augmentation of the usual steady prescribed Stokes
drift velocity.

#### Physical interpretation of the functions ${\bf f}^{k}$ and $a^{k}$ in
terms of stochastic PV fluxes

The energy-preserving stochastic EB equations in (4.31) imply the following
equation for potential vorticity density, $q\,dV$, defined by $q\,dV:={\bf
d}\alpha\wedge{\bf d}b=(\hbox{curl}{\bf
m})\cdot{\bm{\nabla}}b\,dV=(\bm{\omega}+2\bm{\Omega})\cdot{\bm{\nabla}}b\,dV$,
where ${\bf m}={\bf u}+{\bf R}$ for $D=1$. Namely,

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(q\,dV)+\mathcal{L}_{u}(q\,dV)\,dt&={\bf
d}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\alpha+\mathcal{L}_{u}\alpha\,dt\right)\wedge{\bf d}b+{\bf
d}\alpha\wedge{\bf
d}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b+\mathcal{L}_{u}b\,dt\right)\\\ &=-\sum_{k}{\bf
d}\bigg{[}\big{(}\mathcal{L}_{u}\beta^{k})-gz{\bf d}a^{k}\big{)}\wedge{\bf
d}b-\big{(}\mathcal{L}_{u}a^{k}\big{)}{\bf d}\alpha\bigg{]}\circ
dW_{t}^{k}\,,\end{split}$ (4.36)

where we recall that $\beta^{k}:={\bf f}^{k}\cdot d{\bf x}$ is a 1-form. In
vector calculus notation, after using the incompressibility condition
${\bm{\nabla}\cdot}{\bf u}=0$, the potential vorticity equation (4.36) can be
written in terms of $q=(\hbox{curl}\,{\bf m})\cdot{\bm{\nabla}}b$ as, cf. [8]

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}q+{\bf u}\cdot{\bm{\nabla}}q\,dt&=\sum_{k}{\rm div}\Big{[}\big{(}{\bf
u}{\bm{\times}}\hbox{curl}\,{\bf f}^{k}+gz\nabla
a^{k}\big{)}\times{\bm{\nabla}}b+(\bm{\omega}+2\bm{\Omega})\cdot{\bm{\nabla}}({\bf
u}\cdot{\bm{\nabla}}a^{k})\Big{]}\circ dW^{k}_{t}\\\ &=\sum_{k}{\rm
div}\Big{[}\mathbf{F}^{k}\times\nabla
b+(\bm{\omega}+2\bm{\Omega})\cdot\big{(}\nabla\mathfrak{D}^{k}\big{)}\Big{]}\circ
dW^{k}_{t}=:-\,\sum_{k}{\rm div}\mathbf{J}^{k}\circ dW^{k}_{t}\,,\end{split}$
(4.37)

where $\hbox{curl}\,{\bf m}=\bm{\omega}+2\bm{\Omega}$ is the total vorticity,
and the quantities $\mathbf{F}^{k}$ and $\mathfrak{D}^{k}$ are defined as

$\displaystyle\mathbf{F}^{k}:={\bf u}{\bm{\times}}\hbox{curl}\,{\bf
f}^{k}+gz\nabla a^{k}\quad\hbox{and}\quad\mathfrak{D}^{k}:={\bf
u}\cdot{\bm{\nabla}}a^{k}\,.$ (4.38)

The summands in $\mathbf{J}^{k}=-\,\mathbf{F}^{k}\times\nabla
b-(\bm{\omega}+2\bm{\Omega})\cdot\big{(}\nabla\mathfrak{D}^{k})$ are called
the “$J$-fluxes of PV” and are identified with “frictional” and “diabatic”
effects, respectively, in [32, 56]. See also [8] for LES turbulence
interpretations of these fluxes.

In summary, while the energy-preserving stochastically-augmented CL vortex
force and entrainment effects in equations (4.31) or (4.35) can locally create
stochastic Langmuir circulations, the total volume-integrated potential
vorticity $Q=\int_{\mathcal{D}}q\,dV$ will be preserved for appropriate
boundary conditions. See [67] for more information about Langmuir circulations
and their importance in the mixing processes in the upper ocean boundary
layer. The sub-mesoscale excitations created by the $J$-fluxes of PV are a
subject of intense present research aimed at understanding the effects of
turbulence on oceanic frontogenesis, as well as wave forcing which transports
materials such as sediment, gases, algae (carbon), oil spills and plastic
detritus, [59, 57, 58, 60, 51, 9, 20, 68].

######  Remark 4.5 (Eulerian averaged Euler Bousinessq equation).

For completeness, we mention that the corresponding Eulerian averaged equation
to the stochastic EB equation (4.31). Namely, the EA SLFT EB equation is given
by

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\bf u}&-{\bf u}{\bm{\times}}\hbox{curl}\,\mathbb{E}\left[{{\bf u}+{\bf
R}}\right]\,dt+{\bm{\nabla}}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}p-\frac{1}{2}|{\bf u}|^{2}\,dt-{\bf u}\cdot{\bf
R}\,dt+\mathbb{E}\left[{{\bf u}+{\bf R}}\right]\cdot{\bf
u}\,dt\right)-gz{\bm{\nabla}}\mathbb{E}\left[{b}\right]\,dt\\\
&\qquad+\sum_{k}\left[-{\bf u}{\bm{\times}}\hbox{curl}({\bf
f}^{k})+{\bm{\nabla}}({\bf f}^{k}\cdot{\bf
u})-gz\cdot{\bm{\nabla}}a^{k}\right]\circ dW_{t}^{k}=0\,,\\\
&\partial_{t}D+{\rm div}(D{\bf u})=0\,,\quad\hbox{with}\quad D=1\,,\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b+{\bf u}\cdot{\bm{\nabla}}\mathbb{E}\left[{b}\right]\,dt+\sum_{k}{\bf
u}\cdot{\bm{\nabla}}a^{k}\circ dW_{t}^{k}=0\,.\end{split}$ (4.39)

Time evolution of expectation of ${\bf u}$ and $b$ can be found by passing to
the Hamiltonian side and we have

$\displaystyle\partial_{t}H(\mathbb{E}\left[{{\bf
m}}\right],\mathbb{E}\left[{b}\right])=-\frac{1}{2}\sum_{i}\int_{\mathcal{D}}\left|-\mathbb{E}\left[{{\bf
u}}\right]{\bm{\times}}\hbox{curl}{\bf
f}^{i}+{\bm{\nabla}}(\mathbb{E}\left[{{\bf u}}\right]\cdot{\bf
f}^{i})-\mathbb{E}\left[{gz}\right]\cdot{\bm{\nabla}}a^{i}\right|^{2}\,d^{3}x$
(4.40)

and

$\displaystyle\partial_{t}\mathbb{E}\left[{H({\bf
m}^{\prime},b^{\prime})}\right]=\frac{1}{2}\sum_{i}\int_{\mathcal{D}}\left|-\mathbb{E}\left[{{\bf
u}}\right]{\bm{\times}}\hbox{curl}{\bf
f}^{i}+{\bm{\nabla}}(\mathbb{E}\left[{{\bf u}}\right]\cdot{\bf
f}^{i})-\mathbb{E}\left[{gz}\right]\cdot{\bm{\nabla}}a^{i}\right|^{2}\,d^{3}x$
(4.41)

which agrees with Theorem 3.2.

##  5 Conclusion and outlook

The motivation of this paper has been to determine what type of stochastic
perturbations can be added to fluid dynamics that will preserve the
fundamental properties of energy conservation, Kelvin circulation theorem and
conserved quantities arising from the Lagrangian particle relabelling
symmetry. The geometric framework employed in this paper introduces stochastic
forcing by Lie transport (SFLT) noise as a series of perturbations which
automatically produce a Kelvin circulation theorem and can be chosen to
satisfy either energy, or Casimir preservation. In this paper, we have mainly
focused on preserving energy conservation. These stochastic external forces
can be seen as the slow $+$ fast decomposition of external forces
corresponding to the slow $+$ fast decomposition of fluid flow. For Euler
fluid equations, the stochastic CL vortex force will always be energy
conserving and its physical interpretation as a wave-averaged forces fits well
into the external forces considered in the reduced Lagrange-d’Alembert-
Pontryagin (RLDP) principle. In comparison with the location uncertainty (LU)
approach by Mémin [62], the present paper gives an _alternative_ set of fluid
equations which are energy preserving. The relation to LU has been left for
future work. Numerical simulations of the stochastically forced Lie-Poisson
(SFLP) equations (2.19) will be needed to classify solution behaviours of
these new stochastic extension of classical fluid equations. As in the
applications of the stochastic advection by Lie transport (SALT) and LU
approaches, computational simulations of Langmuir fluid circulations and their
material entrainment using equations (4.31) and (4.35) will require the
calibration of the functions $(f^{m}_{i},f^{a}_{i})$, perhaps via data
analysis methods similar to those used in the approach detailed in [12, 13].
Computational simulations of the equations resulting from the SFLT and SFLP
modelling approaches introduced here, as well as simulations of the EA SFLT
equations in section 3 have all been left for future work.

## Acknowledgements

We are grateful to our friends and colleagues who have generously offered
their time, thoughts and encouragement in the course of this work during the
time of COVID-19. Thanks to E. Luesink, S. Takao, W. Pan, D. Crisan, O.
Street, F. Gay-Balmaz, E. Mémin, B. Chapron, C. Franzke, J. C. McWilliams, B.
Fox-Kemper, A. J. Roberts and W. Bauer for their thoughtful comments and
discussions. We are also grateful to the anonymous referee for their
constructive comments. DH is also grateful for partial support from ERC
Synergy Grant 856408 - STUOD (Stochastic Transport in Upper Ocean Dynamics).
RH is supported by an EPSRC scholarship [grant number EP/R513052/1].

## Data availability

No data was created or used in writing this paper.

##  Appendix A Coadjoint operator of semidirect-product Lie-Poisson brackets

Following [41] and [18], consider a Lie Group $G$ which acts from the left by
linear maps on a vector space $V$ which induces a left action of $G$ on
$V^{*}$. In the right representation of $G$ on the vector space $V$, the
semidirect product group $S=G\circledS V$ has group multiplication

$\displaystyle(g_{1},v_{1})(g_{2},v_{2})=(g_{1}g_{2},v_{2}+v_{1}g_{2}),$ (A.1)

where the action of $G$ on $V$ is denoted by concatenation $vg$. The identity
element in $S$ is $(e,0)$ where $e$ is the identity in $G$. The inverse of an
element in $S$ is given by

$\displaystyle(g,v)^{-1}=(g^{-1},-vg^{-1})$

The Lie algebra bracket on the semidirect product Lie algebra
$\mathfrak{s}=\mathfrak{g}\circledS V$ is given by

$\displaystyle[(\xi_{1},v_{1}),(\xi_{2},v_{2})]=([\xi_{1},\xi_{2}],v_{2}\xi_{1}-v_{1}\xi_{2})$
(A.2)

where the induced action of $\mathfrak{g}$ on $V$ is denoted by concatenation
$v\xi$. The operation ${\rm AD}:S\times S\to S$ is defined by

$\displaystyle{\rm
AD}_{(g_{1},v_{1})}(g_{2},v_{2})=(g_{1},v_{1})(g_{2},v_{2})(g_{1},v_{1})^{-1}=(g_{1}g_{2}g_{1}^{-1},-v_{1}g_{1}^{-1}+v_{2}g_{1}^{-1}+v_{1}g_{2}g_{1}^{-1})\,.$
(A.3)

Taking the time derivatives of $g_{2}$ and $v_{2}$, then evaluating them at
the identity $t=0$ yields the Adjoint operation ${\rm
Ad}:S\times\mathfrak{s}\to\mathfrak{s}$ which is defined by

$\displaystyle{{\rm Ad}}_{(g,v)}(\xi,a)=\frac{d}{dt}\biggr{|}_{t=0}{\rm
AD}_{(g,v)}(\tilde{g}(t),\tilde{v}(t))=(g\xi g^{-1},(a+v\xi)g^{-1}),$ (A.4)

where $\frac{d}{dt}\big{|}_{t=0}\tilde{g}(t)=\xi$ and
$\frac{d}{dt}\big{|}_{t=0}\tilde{v}(t)=a$. The coAdjoint operation ${{\rm
Ad}}^{*}$ is the formal adjoint of ${{\rm Ad}}$ with respect to the pairings
$\big{<}\cdot,\cdot\big{>}_{\mathfrak{g}}$ and $\big{<}\cdot,\cdot\big{>}_{V}$
which can be computed as

$\displaystyle{{\rm
Ad}}^{*}_{(g,v)}(\mu,a)=(g\mu+(vg^{-1})\diamond(ag^{-1}),ag^{-1}),$ (A.5)

where the diamond operator $\diamond$ is defined as

$\displaystyle{\Big{\langle}{\beta\diamond\alpha}\,,\,{\xi}\Big{\rangle}}_{\mathfrak{g}}:={\Big{\langle}{b}\,,\,{-a\xi}\Big{\rangle}}_{V}.$

The notation $ag^{-1}$ denotes the inverse of the dual isomorphism defined by
$g\in G$ (so that $g\rightarrow ag^{-1}$ is a right action). Note that the
adjoint and coadjoint actions are left actions. In this case, the
$\mathfrak{g}$-actions on $\mathfrak{g}^{*}$ and $V^{*}$ are defined as before
to be minus the dual map given by the $\mathfrak{g}$-actions on $\mathfrak{g}$
and $V$ and are denoted, respectively, by $\xi\mu$ (left action) and $a\xi$
(right action). Taking time derivative of $(g,v)$ in the definition of ${{\rm
Ad}}_{(g,v)}(\xi,u)$ and evaluating at the identity gives the adjoint operator
${{\rm ad}}$ which coincides with the Lie algebra bracket. One computes the
formal adjoint of ${{\rm ad}}$ with respect to the pairing

$\displaystyle{\left\langle{{{\rm