Wave effects on currents (WEC). In studies of the ocean mixing layer (OML) the problem of wave effects on currents (WEC) arises, because surface gravity waves can have important influences on near-surface currents, particularly in driving Langmuir circulations [40] . Langmuir circulations are horizontal pairs of opposing vortex cells aligned generally along the direction of the wind, as reviewed, e.g., in [36, 17, 55] . Because they represent arrays of organised fluid transport, Langmuir circulation patterns can strongly entrain sediment and detritus into the OML. Thus, for example, they can have a strong effect on the dispersion of oil spills in a shallow sea, [54] .
Most numerical studies of turbulence in the OML have been performed in the context of wave-averaged dynamics. 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 Navier-Stokes equations over a time scale longer than the wave period results in a modified set of equations, known as the CL equations, after Craik and Leibovich, [11] .
The CL equations introduce Stokes-drift effects which represent additional wave-averaged forces and material advection terms that emerge from multi-scale asymptotic theories [11, 49] . See also the Hamiltonian formulation of the ideal CL equations in [24] .
The aim of this paper is to develop an energy-preserving stochastic theory of WEC based on the Craik-Leibovich theory. We will take an approach which adapts to selection of the primary conservation laws desired in the stochastic theory to be derived. Specifically, our approach will use a stochastic version of a standard method from classical mechanics known as the Reduced Lagrange-d'Alembert-Pontryagin (RLDP) approach for the formulation of constrained dynamics reduced by a symmetry of the Lagrangian in Hamilton's principle [6, 19] .
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. For introductory surveys of stochastic fluid dynamics, see [13, 20] .
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 are introduced in the geometrical framework.
The main result of the paper is the development of a constrained variational formulation of ideal fluid dynamics through which noise may be introduced as external forces. By choosing appropriate form of the noise, augmented dynamics will preserve whatever conservation law one may desire among those of the ideal fluid equations in any number of dimensions and they will be referred as Stochastic Forcing by Lie Transport (SFLT). Moreover, as discussed in section 3.3, this formulation will enable stochastic deviations from passive transport may result for inertial fluid parcels whose density has a certain probability of being heavier or lighter than the ambient (or average, or expected) density. Accommodating the effects of uncertainty in other fluid properties other than mass density is of course also possible.
For example, a stochastic version of the Craik-Leibovich (CL) vortex force arises as a result of choosing to conserve the ideal fluid energy which presumably could generate stochastic Langmuir circulation. This result is introduced for the Euler fluid equations for 3D incompressible flow in Proposition 3.2 and Corollary 3.2.1. The corresponding theory for ideal flows in general is also treated in section 3.2. Finally, explicit examples of stochastic motion under gravity with SFLT noise are given: in section 4.1 for the heavy top, which is a finite degree-of-freedom subsystem of fluid motion; in section 4.2 for rotating shallow water dynamics; and in section 4.3 for the stochastic Euler-Boussinesq equations.
In section 2 we explain the mathematical foundations of deterministic semidirect-product coadjoint motion for ideal fluids. After that, we will briefly discuss the modelling approach via stochastic advection by Lie transport (SALT), which has been developed in previous works, [28, 15, 29, 30] .
In section 3 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, including the SALT noise.
In section 4 we present several examples of conservative noise types. These include the finite-dimensional case of the heavy top (which is a gyroscopic analog for collective motion of a stratified fluid), and the infinitedimensional fluid cases of rotating shallow water (RSW) dynamics in 2D and Euler-Boussinesq (EB) dynamics in 3D.
The course of the paper emphasises the utility of the RLDP approach 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. For example, the RLDP approach admits a unified variational formulation which combines both SALT and SFLT noise, as explained in remark 3.10.
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 [47] . 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. 1 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 [42, 43, 44] . 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 [37] . 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 [38] is commonly used in computational simulations to feed energy from the burgeoning unstable development of highwavenumber excitations into large-scale coherent structures at low frequencies, so as to enhance the formation of eddies in ocean flows, [4] .
An opportunity for further theoretical understanding of the interactions of disparate scales in GFD (and, for example, in astrophysics and planetary physics [52] ) arises when the non-dissipative part of the fluidic system dynamics under consideration in a domain D can be derived from Hamilton's principle, δS = 0, for an action time-integral given by S =´ℓ(u, a)dt, where the fluid Lagrangian ℓ(u, a) depends on Eulerian fluid variables comprising the fluid velocity vector field u ∈ X(D) and some set of advected quantities, a ∈ V * (D), dual in L 2 pairing to a vector space, or tensor space, V , defined over the domain D with appropriate boundary conditions. This approach leads directly to a Kelvin-Noether circulation theorem arising from the symmetry of the Lagrangian ℓ(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 [31] . In this case, the averaging over high frequencies in the solution may be applied by substituting a WKB (slow mean amplitude times fast 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 [1] and developed further in [21, 25, 26] . Many of these ideas are also standard in the stability analysis of fluid equilibria in the Lagrangian picture. See, e.g., the classic works [5, 16, 51, 22] .
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 ℓ(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 [28] . In [28] , the fast motion of the Lagrangian trajectory is represented by a stochastic process, whose correlate statistics are to be calibrated from data as in [9, 8] . The stochastic decomposition proposed in [28] was later derived using multi-time homogenization by Cotter et al. [10] . 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 [12] for initial conditions in appropriate Sobolev spaces.
Energy versus circulation in 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 SDiff(D) acting on the flow domain D. This result of V. I. Arnold [2] introduced a fundamentally new geometric way of understanding energy and circulation in fluid dynamics. The incompressible Euler fluid case in [2] 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 SDiff(D). Later, Arnold [3] noticed the topological nature of another conservation law for the Euler fluid equations known as the helicity. Namely, 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 [32] 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, [31] . In physical fluids, the particle-relabelling symmetry of the fluid Lagrangian in the Eulerian representation is broken from the full diffeomorphism group G = Diff(D) to its subgroup G a = Diff(D)| a0 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 asẋ t =ġ t x 0 = u(g t x 0 , t) = u(x t , t), where the Eulerian velocity vector field given by u :=ġ t g −1 t is right-invariant under the particle-relabelling x 0 → y 0 = h 0 x 0 for any fixed h 0 ∈ G.
In the 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 X div (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 X(D) V (D). As discussed in [31] , the fluid motion in each case represents the coadjoint action of the corresponding Lie algebra on its dual space.
Advective transport. The action of the full diffeomorphism group G = Diff(D) on its order parameter variables a ∈ V (D) represents fluid advection. Advection occurs by push-forward of functions on V (D) (by right action by the inverse of the Lagrange-to-Euler map) so the time-dependence of an advected quantity is given by the following push-forward relation for composition of functions a(t) = g t * a 0 := a 0 g −1 t and a(t) evolves by the Lie chain rule for advection
in which L ut a denotes the Lie derivative of the advected quantity a ∈ V (D) by the time-dependent Eulerian velocity vector field, u t :=ġ t g −1 t ∈ X(D). For more details about performing this type of calculation, see appendix B.
2 Stochastic advection by Lie transport (SALT) 2.1 Deterministic semidirect-product coadjoint motion for ideal fluids Proposition 2.1 (Euler-Poincaré theorem [31] ). The Euler-Poincaré (EP) equations of a reduced Lagrangian ℓ(u, a), ℓ : X×V * → R defined over the space of smooth vector fields with elements u ∈ X acting by Lie derivative on elements a ∈ V * of a vector space V * are written as [31] d dt
In the EP equation in (2.1), the variational derivative is defined as usual by,
The quantities a ∈ V * and δℓ/δa ∈ V in (2.2) are dual to each other under the L 2 pairing · , · V : V × V * → R. Likewise, the velocity vector field u ∈ X and momentum 1-form density m := δℓ/δu ∈ X * are dual to each other under the L 2 pairing · , · X : X × X * → R. In terms of these pairings and the Lie derivative operator L u with respect to the vector field u ∈ X, the coadjoint operator ad * u and the diamond operator (⋄) in (2.1) are defined by
where v ∈ X, and ad :
As we shall see later the coadjoint operator ad * u and the diamond operator (⋄) enter together in (2.1) as a form of coadjoint motion for semidirect product action of the Lie algebra
The EP equation in (2.1) can be derived from the following Hamilton-Pontryagin variational principle (HPVP)
where m ∈ X * , b ∈ V , g ∈ G and the variations are taken to be arbitrary.
To add noise in the SALT form to the deterministic HPVP in (2.5), one introduces the following stochastic variational principle, cf. [28] ,
For brevity of notation, where the meaning is clear, we will suppress the subscript labels in the pairings, as in (2.6). The symbol d has been introduced to abbreviate stochastic time integrations with respect to the Stratonovich noise, •dW i t . The choice of Stratonovich noise enables the standard calculus chain rule and product rule to be used, which are needed for the operations of differentiation and integration by parts in which variational principles are defined.
The Hamiltonian functions h i (m, a) on X * ×V * in (2.6) are taken to be prescribed here without discussing how they would be obtained in practice. Taking arbitrary variations in the stochastic HPVP in equation (2.6) yields the following determining relations among the variables
Here, η = δg g −1 and natural boundary terms have been assumed. Collecting terms among the variational relations gives the following set of equations
(2.7)
Thus the SALT EP equations are found to be
where the definition of dg g −1 are taken from (2.7). Using the Legendre transform, h(m, a) = m , u − ℓ(u, a) and taking variations yields
and the corresponding SALT Lie-Poisson (LP) Hamiltonian equations obtained after a Legendre transform are
where the Lagrangian path dx t = dg g −1 Legendre-transforms to the Hamiltonian side as
By using the the ad * operator for semidirect product Lie algebras defined in appendix B, the LP equations (2.10) can be written equivalently in the following compact form,
Remark 2.2 (Stochastic reduced Hamiltonian phase space variational principle for SALT). The SALT equations (2.10) can be derived from a stochastic reduced Hamilton phase-space variational principle, namely,
where the variations δm, δg, δa and δ(b) are taken to be arbitrary.
The SALT Hamiltonian equations in (2.10) can be arranged into the Lie-Poisson (LP) operator form
(2.12)
The Lie-Poisson operator in (2.12) preserves the Casimirs of its deterministic counterpart, since the Poisson structures remains the same. However, the Hamiltonian is now a semimartingale, so energy depends explicitly on time and, hence, is no longer preserved.
Kelvin-Noether theorem. The SALT Lie-Poisson (LP) Hamiltonian equations in (2.10) also possess a Kelvin-Noether theorem. To understand this statement, consider the following G-equivariant map K : C ×V * → X * * as explained in [31] ,
for a manifold C on which G acts from the left.
Given solutions m(t), a(t) satisfying the LP equations (2.10) and fixed c 0 ∈ C, the associated Kelvin-Noether quantity K(g(t)c 0 , a(t)) , m(t) satisfies the following stochastic Kelvin-Noether relation.
where we identify dg g −1 =: dx t .
Remark 2.4. The fluid mechanics counterpart of the Kelvin-Noether theorem is expressed as
where the material loop c t = g(t)c 0 moves with stochastic velocity dg g −1 = dx t in equation (2.11) and the quantity D is the mass density of the fluid which is also advected as D(t) = g t * D 0 . That is, the mass density D satisfies the stochastic continuity equation dD + L dxt D = 0.
Remark 2.5 (SALT Hamiltonians). In [28] , the SEP equations are derived from a stochastic Clebsch variational principle where the advection of phase space Lagrangian variables are through a stochastic vector field dx t = u dt
Compared to the stochastic vector field defined via dg g −1 in (2.7), they coincide with those in [28] when the noise Hamiltonians h i are linear in m, i.e. h i (m, a) = ξ i (x) , m . The SEP equations then becomes
For data assimilation purposes, the choice h i (m, a) = i m, ζ i (x) was made in [9, 8] .
3 Stochastic forcing by Lie transport (SFLT)
As SALT introduces stochasticity into the transport velocity u ∈ X, it is natural to consider the corresponding stochastic modification to the momentum m ∈ X * . By Newton's force law, this modification can be achieved through external forces. One variational principle which permits external forces is the reduced Lagrange d'Alembert Pontryagin (RLDP) principle. The general formulation of RLDP is as follows [19] . 2 Consider an arbitrary Lie group G with an associated Lie algebra g, the RLDP principle for reduced Lagrangian ℓ : g → R and external force F is given by
where g ∈ G, the variations δg, δξ, δµ are arbitrary with δg vanishes at the end points t = a, b. The stationary condition (3.1) yields the forced Euler-Poincaré equation with force F , as a 1-form equation,
Deterministic Kelvin-Noether theorem. The deterministic Euler-Poincaré equation with force F in (3.2) implies the following Kelvin-Noether theorem given in the form of a Newtonian force law as
in which c t is a closed material loop moving with the fluid velocity vector field ξ. For the 3D Euler fluid, the force F = ∇p is the gradient of pressure p. Consequently, the circulation integral I(t) is conserved for the 3D Euler fluid. This is Kelvin's circulation theorem [32] .
Adding stochasticity to the RLDP principle with external forces is a straight forward extension. For reduced Lagrangian ℓ : g → R and the set of external forces F i , stochastic RLDP is given by
where g ∈ G, the variations δg, δξ, δµ are arbitrary, δg vanishes at the end points, t = a, b, and η := δg g −1 .
The stationary condition of the variational principle (3.4) is the following stochastically forced Euler-Poincaré (SFEP) equation, 
in which an arbitrary closed material loop c t is advected with the fluid by the push-forward action c t = g t * c 0 of the time-dependent diffeomorphism g t generated by the smooth vector field ξ representing the fluid transport velocity. Unless i F i is a sum of gradient forces in (3.6), the circulation integral I(t) will not be conserved.
Upon passing to the Hamiltonian side via the Legendre transform ℓ(ξ) = µ , ξ − h(µ), where µ = δℓ δξ and h is the reduced Hamiltonian, one finds the reduced Hamilton-d'Alembert phase space principle given by
As before, g ∈ G, the variations δg, δµ are arbitrary and δg vanishes at the endpoints in time, t = a, b. The resulting stochastically forced Lie-Poisson (SFLP) equation is 
in agreement with [15] . Energy preservation may now be immediately verified, since
The last equality here follows because of the anti-symmetry of the commutator, ad ξ ξ = −[ξ, ξ] = 0. Proof. By direct calculation from equation (3.10) we have
where in vector calculus terms, the right hand side is expressed as
Thus, a stochastic version of the CL vortex force i ξ × curlf i • dW i t emerges in the energy-preserving form of the RLDP equations in (3.10), cf. [11] . In the case of 3D Euler fluid equations when δC δµ = dµ and µ is the circulation 1-form, the example here with forces
preserves (only) the helicity,
Proof. In vector calculus terms, one computes ad * ξ µ ,
which is obtained after an integration by parts in the second summand and taking the boundary term to vanish as usual for the helicity.
Proposition 3.5 (Itô form). The Itô form of (3.10) is given by
where the brackets (· , ·) in the definition of σ i denotes contraction, not L 2 pairing.
The derivation of formula (3.12) is given in appendix B.
We next extend the energy-preserving noise for Euler fluids in Example 3.2 and its vortex force Corollary 3.2.1 to the case of coadjoint motion of fluids by semidirect-product action of the diffeomorphisms on advected quantities in vector spaces. 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. For symmetry-reduced semidirect-product motion, the stochastic RLDP principle in (3.4) becomes
where the variations δg, δu, δm, δb, δa are arbitrary with δg vanishing at the endpoints, t = a, b. Taking the indicated variations in (3.13) yields the following result,
where we denote η = δg g −1 , as before. Vanishing of the coefficients of like variations implies the following relations,
(3.14)
Assembling these relations yields the SFEP equations with advected quantities 
The proof of this statement is a direct calculation following the same pattern as for the EP derivation.
Note that the SFLP equations with advected quantities in (3.16) modifies the momentum equation alone, while keeping the advection equation of a ∈ V * unchanged. In terms of of co-adjoint motion, the natural generalisation of (3.10) to semidirect-product Lie group action would be
where the maps f m i : X × V * → X * and f a i : X × V * → V * are arbitrary for all i. The individual equations are then
These individual equations can be written in the form of a Poisson operator, as follows, [ ad *
Here δm and δa are arbitrary. The constrained variations of u and b take the form δu = ∂ t η − [u , η] and δb = ∂ t w − bη + wu , respectively, in which η ∈ X and w ∈ V are arbitrary. Applying these variational formulas yields
Consequently, one may collect terms to find the following system of motion and advection equations 
where one defines
Again the notation (· , ·) denotes contraction, not L 2 pairing. The proof is similar to the case without advected quantities. The reduced Hamilton-d'Alembert phase space principle presented here has been written in terms of the individual components of the semidirect-product Lie algebra s = g V . However, it can be written a bit more succinctly as (m, a) ,
for arbitrary variations of δ(m, a) of (m, a) ∈ s * and variations δ(g, v) of (g, v) ∈ S. This one yields stationary conditions
Remark 3.8 (Stochastic material entrainment modelled in equation (3.18) ). 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 ∈ G, i.e., a(t) = 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 (3.18) or (3.19) could model a fluid containing parcels whose density does not quite passively follow the 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 (3.18) 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 [7] .
Kelvin-Noether theorem for equation (3.18) . The Kelvin-Noether theorem for equation (3.18) can be written as
where the identification dg g −1 = δh δm dt is obtained from the stochastic RLDP constrained variational principle in (3.27) . Upon assuming that the fluid density D is also advected by the flow, so that ∂ t D + L u D = 0, then the Kelvin circulation theorem may be expressed as
(3.26)
where the notation of Lagrangian loop c(u) denotes the Lagrangian loop moving with the deterministic fluid velocity u, as in the deterministic case. 
(3.27)
Here δm and δa are arbitrary. The constrained variations of u and b take the form
respectively, in which η ∈ X and w ∈ V are arbitrary. The set of Hamiltonians h i generates SALT type noise and the set of forces f i generates the new energy-preserving type of noise.
Following, e.g., [27] , let S = SO(3) R 3 . The associated Lie algebra and its dual are s = so(3) R 3 ∼ = R 3 R 3 and s * = so(3) * R 3 ∼ = R 3 R 3 respectively. The natural pairing between R 3 and its dual is the dot product pairing, i.e. there is no density in g * . The heavy top Hamiltonian H(Π, Γ) is taken to be the rotational kinetic energy plus gravitational potential energy written in terms of Π and Γ as
where I is the moment of inertia in the body frame. The variational derivatives are dH/dΠ = I −1 Π = Ω and dH/dΓ = mgχ. Identifying the Lie derivative and ad * operation as the cross product in the R 3 representation as the deterministic LP equation is written as
which can also be written in Poisson operator form d dt
Defining the force vectors f m i , f a i ∈ R 3 , the SFLP equations corresponding to (3.18) are
When written in Poisson bracket form, these equations become
One observes that these equations preserve the Hamiltonian in (4.1), which follows because of skew symmetry of the Poisson operator. . We denote u · ∂ ∂x = u ∈ X(D) and m = m · dx ⊗ dV ∈ Λ 1 (D) ⊗ Den(D). In this notation, the LP equations can be written in Cartesian coordinates as the following partial differential equations, 4) and the LP operator can be written as ∂ ∂t
For rotating shallow water equations (RSW) in R 2 , the Hamiltonian H is given by 
Substituting into (4.4) and using the relation (∇×u)×v + ∇(u · v) = (u · ∇)v + u i ∇v i yields the standard set of RSW equations governing motion and continuity,
Consider the Poisson operator of the form (3.19) applied to s * . It reads
where f k i are the components of f k such that f k = f k · dx ⊗ dV ∈ Λ 1 (D) ⊗ Den(D) and g k ∈ F (D) for all k. Then the SFLP equations become
(4.9)
Remark 4.1. When noise is introduced into the continuity equation for density η, one can still write the momentum equation in terms of m/η. Namely, it reads
where the inhomogenous term −η −2 m j ∂ i (g k u j ) in the second line of (4.10) is due to the modified advection relation for η dV . The SFLP equations for RSW dynamics in (4.9) written more succinctly using Lie derivatives and denoting the momentum 1-form as α = m · dx, then
(4.11)
In vector calculus notation, the stochastic RSW equation obtained by subsituting the variational derivatives of H into equation (4.8) yields
where π RSW = η−B ǫF − ǫ 2 |u| 2 − u · R . The corresponding Kelvin circulation theorem follows easily from (4.11) in geometric form, as
The stochastic RSW equations in (4.12) enable the Kelvin circulation equation (4.13) to be written in vector calculus form as d˛c (u) (ǫu + R) · dx (4.14)
= −˛c
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 (4.13) to
The vector form of the motion equation also simplifies for g k = 0 to
whose right hand side may be regarded as a compressible version of the CL vortex force. 
The stochastic extension of equation (3.19) to include buoyancy reads
(4.18)
Here, f k i are the components of f k such that f k = f k · dx ⊗ dV ∈ Λ 1 (D) ⊗ Den(D), g k ∈ F (D) and a k ∈ F (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, ∇·u = 0.
The Euler-Boussinesq Hamiltonian is
where the momentum density m = D u + R(x) and the pressure p is a Lagrannge multiplier which enforces incompressibility. The variational derivatives of H are given by
The deterministic Euler-Boussinesq equations then follow as
(4.21)
One can obtain the energy-preserving stochastic equations by substituting the variational derivatives into stochastic Poisson operator in (4.18) to find du − u×curl(u + R) dt + ∇ dp + 1 2 |u| 2 dt + gbẑ dt Remark 4.2. The change of pressure p → dp = p dt + i p i • dW i t has been made, so that the incompressibility condition, ∇·u = 0 will be enforced in both the drift and stochastic parts of u.
Similar to the case of S = Diff(D) Den(D) for energy-preserving stochastic rotating shallow water, 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 α = m · dx. These equations then read
where the 1-form β k := f k · dx. The Kelvin circulation theorem for these equations is immediate, as
(4.24)
In vector calculus notation, this Kelvin circulation theorem is written as
In the case where f 0 = −u S (x), dW 0 t = dt, and a k = 0 for k = 0, . . ., one finds that the energy-preserving stochastic EB equations in (4.22) produce a stochastic contribution to the vortex force in the Craik-Leibovich equations [11] whose formulation with Hamilton's principle is discussed in [24] . Namely, they reduce as follows,
where one interprets the semimartingale du S = u S (x) dt + k f k (x) • dW k t as a stochastic augmentation of the usual steady prescribed Stokes drift velocity.
The energy-preserving stochastic EB equations in (4.22) imply the following equation for potential vorticity, q, defined by q = dα ∧ db = (∇×m) · ∇b d 3 x.
(4.27)
In vector calculus notation, after using the incompressibility condition ∇·u = 0, the potential vorticity equation can be written in terms of q = (∇×m) · ∇b as dq + u · ∇q dt = k curl(u×curlf k ) · ∇b + curlm · (curl∇a k )×u + ∇(u · ∇a k )) + g∇z · (∇a k ×∇b) • dW k t (4.28)
Thus the stochastic EB equations with energy-preserving noise no longer conserve the standard potential vorticity (PV) along Lagrangian fluid parcels, i.e., dq + u · ∇q dt = 0. That is, Ertel's theorem is no longer satisfied when this type of energy-preserving noise is added.
When one sets the material entrainment terms a k = 0, the PV equation (4.28) simplifies to
This equation shows that the energy-preserving stochastically augmented CL vortex force has the potential to stimulate the formation of stochastic Langmuir circulations, even in the absence of the material entrainment terms proportional to a k in equation (4.28) . See [53] for more information about Langmuir circulations and their importance in the mixing processes in the upper ocean boundary layer. The resulting sub-mesoscale excitations are a subject of intense present research [47, 45, 46, 48, 41] .
The motivation of this paper has been to answer what type of stochastic perturbations can be added to fluid dynamics which 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 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. These stochastic external force can be seen as the slow + fast decomposition of external forces corresponding to the slow + fast decomposition of fluid flow. For Euler type 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 RLDP principle. In comparison with the location uncertainty approach (LU) by Mémin [50] , the present paper gives an alternative set of fluid equations which are energy preserving and their relation to LU is a part of future work. Numerical simulations of the SFLP equations (3.18) will be needed to classify solution behaviours of these new stochastic extension of classical fluid equations. As in the applications of the SALT and LU approaches, computational simulations of Langmuir fluid circulations and their material entrainment using equations (4.22) and (4.26) will require the calibration of the fields (f m i , f a i ) via data assimilation similar to the approach detailed in [8, 9] . The computational study of these equations has been left for future work.
In the integral notation, the Kunita-Itô-Wenzell (KIW) formula is written aŝ
So, in the differential notation the KIW formula 'transfers' to the equivalent differential form
Remark A.2. In applications, one sometimes expresses equation (A.2) using the differential notation
where dx t is the stochastic vector field dx t (x) = b(t, x)dt + N i=1 ξ i (t, x) • dB i t . Importantly for fluid dynamics, this formula is also valid when K is a vector field rather than a k-form.
Stochastic Kelvin circulation theorem for the SALT theory. Having understood the differential notation for stochastic integrals, now we may assemble the stochastic fluid equations via the stochastic Kelvin circulation theorem for the SALT theory [28] . For this purpose, we shall make the argument that the stochastic Kelvin circulation theorem is fundamentally a stochastic form of Newton's law of motion,
This formula corresponds to the motion equation derived from Hamilton's principle
along with the law of advection of mass expressed in KIW form
where the flow velocity is given by the stochastic vector field dφ t (x) := u(φ t (x), t)dt + i ξ i (φ t (x)) • dW i t .
Thus, the stochastic Kelvin's circulation theorem for SALT simply describes the rate of change of momentum of a stochastically moving material loop.
Consider a Lie Group G 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 V has group multiplication (g 1 , v 1 )(g 2 , v 2 ) = (g 1 g 2 , v 2 + v 1 g 2 ), (B.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
The Lie algebra bracket on the semidirect product Lie algebra s = g V is given by
where the induced action of g on V is denoted by concatenation vξ. The operation AD : S × S → S is defined by AD (g1,v1) (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 ) .
Taking the time derivatives of g 2 and v 2 , then evaluating them at the identity t = 0 yields the Adjoint operation Ad : S × s → s which is defined by Ad (g,v) (ξ, a) = d dt t=0 AD (g,v) (g(t),ṽ(t)) = (Ad g ξ, (a + vξ)g −1 ), (B.4) where d dt t=0g (t) = ξ and d dt t=0ṽ (t) = a. The coAdjoint operation Ad * is the formal adjoint of Ad with respect to the pairings ·, · g and ·, · V which can be computed as Ad * (g,v) (µ, a) = (gµ + (vg −1 ) ⋄ (ag −1 ), ag −1 ), (B.5)
where the diamond operator ⋄ is defined as
The notation ag −1 denotes the inverse of the dual isomorphism defined by g ∈ G (so that g → ag −1 is a right action). Note that the adjoint and coadjoint actions are left actions. In this case, the g-actions on g * and V * are defined as before to be minus the dual map given by the g-actions on g and V and are denoted, respectively, by ξµ (left action) and aξ (right action). Taking time derivative of (g, v) in the definition of Ad (g,v) (ξ, u) and evaluating at the identity gives the adjoint operator ad which coincides with the Lie algebra bracket. where the brackets (· , ·) in the definition of σ i denotes contraction, not L 2 pairing.