Stochastic Parametrization of the Richardson Triple

A Richardson triple is an ideal fluid flow map $g_{t/\ep,t,\ep t} = h_{t/\ep}k_t l_{\ep t}$ composed of three smooth maps with separated time scales: slow, intermediate and fast; corresponding to the big, little, and lesser whorls in Richardson's well-known metaphor for turbulence. Under homogenisation, as $\lim \ep\to0$, the composition $h_{t/\ep}k_t $ of the fast flow and the intermediate flow is known to be describable as a single stochastic flow $\dd g$. The interaction of the homogenised stochastic flow $\dd g$ with the slow flow of the big whorl is obtained by going into its non-inertial moving reference frame, via the composition of maps $(\dd g)l_{\ep t}$. This procedure parameterises the interactions of the three flow components of the Richardson triple as a single stochastic fluid flow in a moving reference frame. The Kelvin circulation theorem for the stochastic dynamics of the Richardson triple reveals the interactions among its three components. Namely, (i) the velocity in the circulation integrand acquires is kinematically swept by the large scales; and (ii) the velocity of the material circulation loop acquires additional stochastic Lie transport by the small scales. The stochastic dynamics of the composite homogenised flow is derived from a stochastic Hamilton's principle, and then recast into Lie-Poisson bracket form with a stochastic Hamiltonian. Several examples are given, including fluid flow with stochastically advected quantities, and rigid body motion under gravity, i.e., the stochastic heavy top in a rotating frame.

1 How do Big and Little Whorls affect Intermediate Scale Whorls? 1
Here, we are not interested in regarding the Richardson cascade as a turbulence closure metaphor. Instead, we are interested in rephrasing it in terms of Kelvin's circulation theorem and then modelling the effects of the larger and smaller whorls on the circulation of an intermediate size whorl as spatially dependent stochastic processes. The effects of the smaller whorls are modelled using Stratonovich stochastic Lie transport as in [18,7], while the effects of the larger whorls are modelled using an Ornstein-Uhlenbeck (OU) process to boost the fluid motion into a non-inertial frame, although the reference frame velocity could be represented by any other semimartingale.
The present approach results in stochastic partial differential equations (SPDEs) for the motion of an intermediate scale flow, stochastically transported by the sequence of nested smaller scales, in the non-inertial frame of the sequence of stochastically eddying larger scales. We expect that this approach may be useful in multi-scale geophysical fluid dynamics. For example, it may be useful in quantifying predictability and variability of sub-mesoscale ocean flow dynamics in the random frame of one or several unspecified mesoscale eddies interacting with each other. This approach may also be useful in investigating industrial flows, e.g., to quantify uncertainty and variability in multi-scale flow processes employed in industry.
Plan. In section 1.2, we recast the Richardson scale cascade in a new way by reformulating it in terms of Kelvin's circulation theorem. We then use the reformulation to propose a stochastic closure model, based on the pullback of the stochastic Lagrange-to-Euler map. The stochastic Euler vorticity equation in 3D is given as an example.
A related Kelvin circulation theorem for stochastic incompressible fluid models in an inertial frame was obtained in [6] based on the back-to-labels map (the inverse of the map used here) and applied there in a derivation of the Navier-Stokes equations.
In section 2, we present a variational formulation of the stochastic Richardson cascade closure proposed in section 1.2. In the corresponding equations of fluid motion, two types of stochasticity appear. Namely, (i) Non-inertial stochasticity due to random sweeping by the larger whorls, which is added to the momentum density as an OU process for the reference frame velocity, and (ii) Stochastic Lie transport due to the smaller whorls, which is added to the transport velocity of the intermediate scales as a cylindrical Stratonovich process.
The stochastic Euler vorticity equation in 3D is again given as an example in section 2.
A variational formulation of the stochastic Kelvin Theorem based on the back-to-labels map which led to the derivation of the Navier-Stokes equations in [6] was given in [11], thereby showing that this Kelvin theorem, regarded as a stochastic conservation law, arises from particle-relabelling symmetry of the corresponding action principle.
We stress that the goal of the present work is to use the metaphor of the Richardson cascade to inspire the derivation of SPDEs for stochastic fluid dynamics, as done in [18]. In contrast to [6,11], it is not our intention to derive the Navier-Stokes equations in the present context. For a discussion of the history of variational derivations of stochastic fluid equations and their relation to the Navier-Stokes equations, one should consult the original sources, some of which are cited in [18].
In section 3, we present the Lie-Poisson Hamiltonian formulation of the stochastic Richardson cascade closure derived in the earlier sections. We find that while the Hamiltonian obtained from the Legendre transform of the Lagrangian in the stochastic Hamilton's principle in section 2 also becomes stochastic, the semidirect-product Lie-Poisson Hamiltonian structure of the deterministic ideal fluid equations [20] persists.
This persistence of Hamiltonian structure implies the preservation of the standard potential vorticity (PV) invariants even for the stochastic Lie transport dynamics of fluids in a randomly moving reference frame. To pursue the geometric mechanics framework further, we show in section 3 that the Lie-Poisson Hamiltonian structure of ideal fluid mechanics is preserved by the introduction of either, or both types of stochasticity treated here. As a final example, we consider the application of these ideas to the finite dimensional example of the stochastic heavy top. This is an apt example, because of the well-known gyroscopic analogue with stratified fluids, as discussed, e.g., in [16,9] and references therein. Moreover, the effects of transport stochasticity alone on the dynamical behaviour of the heavy top in the absence of the OU noise, was investigated in [2]. Future work will investigate the further effects of the non-inertial OU noise and its interaction with the transport noise in the examples of the rigid body and heavy top.

Transport Noise in a Stochastic Noninertial Frame
Let u be the velocity of an intermediate-scale (IS) whorl relative to the reference frame moving with the sum of the velocities of the larger nested whorls. From the Kolmogorov viewpoint, one may regard u as a velocity somewhere in the inertial range. Then the Kelvin theorem for the circulation of u that follows literally from Taylor's hypothesis and Richardson's cascade would be given by That is, the circulation of the sum of velocities of whorls of intermediate scale and larger (u + u > ), when integrated around a closed material loop c moving with the total velocity of the whorls of intermediate scale and smaller (u + u < ) changes with time in proportion to the circulation around the material loop of the total external and thermodynamic forces per unit mass, D −1 F , where F is the sum of all external and thermodynamic forces and D is the mass density of the fluid. The sum of velocities u + u > is measured relative to an inertial frame; while u and u + u < are measured in the frame moving with u > . From this viewpoint of the Richardson cascade, one could generalise to the case in which the Eulerian coordinates of the fluid flow at each scale are taken as the Lagrangian coordinates of the next successively smaller scale, as formulated mathematically in [22]. However, the latter formulation introduces coordinate systems at multiple spatial scales, while the present approach requires only one spatial coordinate system Taking the time derivative inside the circulation integral in equation (1.1) leads to the following equation of motion Of course, this equation is far from being closed; since nothing is known about the evolution of the sums of velocities either larger, u > , or smaller, u < , than the IS scale velocity, u. Here, we explore the implications of the rather unconventional step of replacing both of the unknown sums of velocities larger and smaller than the IS velocity, u, by stochastic processes, albeit of two different types.
Larger whorls. One might reasonably assume that the noise representing the influence of the sequence of larger whorls (eddies) would have a longer correlation time than that for the smaller scales. To introduce the correlation time in modelling the non-inertial effects of the larger whorls, we will use an Ornstein-Uhlenbeck (OU) process, although any other semimartingale process could also have been used. Namely, we will set where η(x) is a smooth spatially dependent co-vector (coordinate index down) and N (t) is the solution path of the stationary Gaussian-Markov OU process whose evolution is given by the following stochastic differential equation, with long-term mean N , and real-valued constants θ and σ. The solution of (1.4) is known to be in which we assume an initially normal distribution, N (0) ≈ N (N , σ 2 /(2θ)).
As a mnemonic, we may write curl R(x, t) =: 2Ω(x)N (t) , to remind ourselves that R(x, t) represents the velocity of a stochastically moving reference frame, and that curl η(x) =: 2Ω(x) suggests the Coriolis parameter for a spatially dependent angular rotation rate. That is, u(x, t) is the fluid velocity of interest relative to a moving reference frame with stochastic velocity R(x, t).
Smaller whorls. We select a Brownian motion process to represent the influence of smaller whorls as a deltacorrelated stochastic process. Namely, we take the following cylindrical stochastic Stratonovich process (denoted with •) for a contravariant vector (coordinate index up); that is, for the components of a vector field, The spatially smooth vector functions ξ i , i = 1, . . . , n, may be taken, for example, as n spatially dependent eigenvectors of a proper orthogonal decomposition of certain observed spatial velocity-velocity correlation data.
Lifting to the diffeos. One may lift the stochastic process in Eulerian space defined in (1.6) to a stochastic process on the Lie group G of diffeomorphisms acting on the reference configuration of the fluid parcels. This lift may be accomplished by defining the Eulerian trajectories of the Lagrangian paths as x = y t = g t y 0 with x an Eulerian position in the domain of flow through which the Lagrangian trajectory passes, with g(t) ∈ G and g(0) = Id. Thus, we may write the stochastic vector field in (1.6) equivalently as With the two assumptions (1.3) and (1.6) for replacing, respectively, the sums of velocities over larger and smaller whorls by two different types of stochastic velocities, the previously unclosed Kelvin circulation theorem for an IS whorl in (1.2) now transforms into the closed, but stochastic, expression, 1 for given forces F and an evolution equation for stochastic advection of density D in the moving fame of the larger whorls, given by, Hence, Kelvin's circulation theorem for the closure in (1.8) now has noise in both its integrand and in its material loop velocity. According to equation (1.8), the noise that models effects of the larger scales, in the integrand is given by the OU process (1.4), and the material loop velocity is perturbed by the cylindrical noise, The smooth spatial functions η and ξ i , with i = 1, 2, . . . , N , and the constant parameters θ and σ in the OU process (1.4) should be obtained from either data assimilation for a certain observed flow, or from other information about the flow under consideration.
Taking the evolution operator d inside the circulation integral in equation (1.8) now leads to the following stochastic co-vector equation of motion Example 1 (Stochastic Euler fluid equation in 3D). For an ideal incompressible Euler fluid flow, D is the unit volume element and D −1 F = −∇p, where p(x, t) is the pressure. Upon taking the curl of (1.12) in this case and defining total vorticity as ̟ := curl(u + R(x, t)) we find the stochastic total vorticity equation, which has the familiar form for the evolution of total vorticity ̟ in fluid dynamics.
Remark 1 (Analytical properties). The analytical properties of the stochastic 3D Euler fluid equation in (1.13) (its local in time existence and uniqueness properties, as well as its Beale-Kato-Majda blow up conditions) in the absence of R(x, t) are treated in [7]. Future work will determine whether these analytical properties persist for fluid motion in a stochastic non-inertial frame, modelled as the OU process in equation (1.13).

Variational formulation of stochastic Richardson cascade closure
We have been considering two types of noise for the closure of the Richardson cascade, one representing large scale stochastic sweeping by the larger whorls, and another representing stochastic transport by the smaller whorls. These two types of noise can both be implemented and generalized at the same time by invoking the reduced Hamilton-Pontryagin variational principle for continuum motion δS = 0 [20,17,21], upon choosing the stochastic action integral as follows [18,12], Here, g(t) ∈ G is a stochastic time-dependent curve in the manifold of diffeomorphisms G; the quantities u(x, t), ξ(x) and dgg −1 are Eulerian vector fields defined appropriately over the domain of flow, and a 0 ∈ V for a vector space V . The Lagrangian ℓ(u, a 0 g −1 , D) is a general functional of its arguments; the term involving R(x, t) transforms the motion into the moving frame with co-vector velocity R(x, t) given in (1.3); the co-vector Lagrange multiplier µ enforces the vector field constraint (1.7) on the variations and the brackets · , · denote L 2 pairing of spatial functions. The space and time dependence of the Lagrangian in the action integral (2.1) prevents conservation of energy and momentum, and the presence of the initial condition a 0 for the Eulerian advected quantity a = a 0 g −1 restricts particle-relabelling symmetry to the isotropy subgroup of the diffeomorphisms that leaves the initial condition a 0 invariant. Actually, the density D = D 0 g −1 (a volume form) is also an advected quantity, although we have separated it out because it multiplies the co-vector R(x, t), which plays a special role as the reference frame velocity. In the deterministic case, this breaking of a symmetry to an isotropy subgroup of a physical order parameter leads to semidirect-product Lie-Poisson Hamiltonian structure, As we shall see, this semidirect-product structure on the Hamiltonian side holds true for the type of stochastic deformations we use in the present case, as well.
Taking variations of the action integral in equation (2.1) yields Hamilton's principle, with advected quantities a = a 0 g −1 , whose temporal and variational derivatives satisfy similar equations; namely, where £ (·) denotes Lie derivative with respect to a vector field. In particular, δD = − div(Dw). A quick calculation of the same type yields two more similar equations, Taking the difference of the two equations in (2.4) yields the required expression for δ(dgg −1 ) in terms of the variational vector field, w := δgg −1 , where ad v w := vw − wv denotes the adjoint action (commutator) of vector fields. Finally, one defines the diamond (⋄) operation as [20] δℓ δa ⋄ a , w := δℓ δa , −£ w a . (2.6) Inserting these definitions into the variation of the action integral in (2.2) yields (2.7) Integrating by parts in space and time (2.7) yields Putting all this together yields the following results for the variations in Hamilton's principle (2.2), upon setting the variational vector field, w := δgg −1 equal to zero at the endpoints in time, where the advected quantities a and D satisfy as in (2.3) and (1.9), respectively. In the last line of (2.9) one defines the coadjoint action of a vector field dgg −1 on an element µ in its L 2 dual space of 1-form densities as Equations (2.9) now have stochasticity in both the momentum density µ and in the Lie transport velocity dgg −1 , while the advection equation (2.10), of course, still has only stochastic transport. Now, the coadjoint action of a vector field on a 1-form density turns out to be the same as the Lie derivative of the 1-form density with respect for that vector field, so that The advection of the mass density D is also a Lie derivative, cf. (2.10). Consequently, the last line of (2.9) implies the following stochastic equation of motion in 3D coordinates, where we have defined the 1-form and dgg −1 is given explicitly in equation (2.9). Equation (2.13) allows us to write Kelvin's circulation theorem as since the contribution to the loop integral from the last term in (2.13) vanishes. Therefore, as discussed above, Kelvin's circulation theorem now has noise in both its integrand and in its material loop velocity. The noise in the integrand is an OU process, and the material loop velocity contains cylindrical noise.
Remark 2 (More general forces). More general forces than those in equation (2.15) may be included into the Kelvin circulation, by deriving it from Newton's Law, as done in [7].
Example 2 (Euler's fluid equation in 3D). For an ideal incompressible Euler fluid flow, we have D = 1 and ℓ(u) = 1 2 u 2 L 2 ; so δℓ δu = u and D −1 δℓ δa ⋄ a = −dp, where p is the pressure. Upon taking the curl of equation (2.13) in this case and defining total vorticity as ̟ := curl(u + R(x, t)) we recover the stochastic total vorticity equation, Thus, the total vorticity in this doubly stochastic version of Euler's fluid equation in 3D evolves by the same stochastic Lie transport velocity as introduced in [18], but now the total vorticity also has an OU stochastic part, which arises from the curl of the OU stochastic velocity of the large-scale reference frame, relative to which the motion takes place.
Remark 3 (Next steps, other examples, Hamiltonian structure). Having established their Hamilton-Pontryagin formulation, the equations of stochastic fluids (2.13) and advection equations (2.10) now fit into the Euler-Poincaré mathematical framework laid out for deterministic continuum dynamics in [20]. Further steps in that framework will follow the patterns laid out in [20] with minor adjustments to incorporate these two types of stochasticity into any fluid theory of interest. In particular, as we shall see, the Lie-Poisson Hamiltonian structure of ideal fluid mechanics is preserved by the introduction of either, or both types of stochasticity treated here.
However, we will forego this opportunity here, in order to continue following the geometric mechanics framework for deterministic continuum dynamics laid out in [20], by using the Stratonovich representation to derive the Hamiltonian formulation of stochastic Richardson cascade closure.
3 Hamiltonian formulation of stochastic Richardson cascade closure

Legendre transformation to the stochastic Hamiltonian
The stochastic Hamiltonian is defined by the Legendre transformation of the stochastic reduced Lagrangian in the action integral (2.1), as (3.1) whose variations are given by Consequently, the motion equation in the last line of (2.9) and the advection equations in (2.10) may be rewritten equivalently in matrix operator form as  3) The motion and advection equations in (3.3) may also be obtained from the Lie-Poisson bracket for functionals f and h of the flow variables (µ, a) which is the standard semidirect-product Lie-Poisson bracket for ideal fluids, [20].
Thus, perhaps as expected, the presence of stochasticity does not violate the Hamiltonian structure of ideal fluid dynamics. Instead, the Hamiltonian itself becomes stochastic, as in equation (3.1) obtained from the Legendre transform of the stochastic Lagrangian in the action integral (2.1). This preservation of the Lie-Poisson structure under the introduction of stochastic Lie transport means, for example, that the Casimirs c(µ, a) of the Lie-Poisson bracket (3.4), which satisfy {c, f } = 0 for every functional f (µ, a) [20], will still be conserved in the presence of stochasticity. In turn, this conservation of the Casimirs implies the preservation of the standard potential vorticity (PV) invariants, even for stochastic Lie transport dynamics of fluids in a randomly moving reference frame, as treated here.

Gyroscopic analogy
There is a well-known gyroscopic analogy between the spatial moment equations for stratified Boussinesq fluid dynamics and the equations of motion of the classical heavy top, as discussed e.g., in [16,9] and references therein. This analogy exists because the dynamics of both systems are based on the same semidirect-product Lie-Poisson Hamiltonian structure. One difference between the spatial moment equations for Boussinesq fluids and the equations for the motion of the classical heavy top is that for fluids the spatial angular velocity is right-invariant, while the body angular velocity for the heavy top is left-invariant under SO(3) rotations. Because the conventions for the heavy top are more familiar to most readers than those for the Boussinesq fluid gyroscopic analogy, and to illustrate, in passing, the differences between left-invariance and right-invariance, we shall discuss the introduction of OU reference frame stochasticity for the classical heavy top in this example. Moreover, the introduction of stochastic Lie transport for the classical heavy top has already been investigated in [2].
Example 3 (Heavy top). The action integral for a heavy top corresponding to the fluid case in (2.1) is given by, with (3.7) The motion equation for Π and the advection equation for Γ may be rewritten equivalently in matrix operator form as  The motion and advection equations in (3.8) may also be obtained from the Lie-Poisson bracket for functionals f and h of the variables (Π, a) which is the standard semidirect-product Lie-Poisson bracket for the heavy top, with (Π, Γ) ∈ se(3) * ≃ (so(3) R 3 ) * , where denotes semidirect product.
The deterministic Lagrangian for the heavy top is

10)
where I is the moment of inertia, m is mass, g is gravity and χ is a vector fixed in the body. The corresponding stochastic Hamiltonian, obtained by specialising the Hamiltonian in (3.6) is [17], , Ω dt , (3.11) whose variations are given by δΓ : δh δΓ = − mgχ . (3.12) Thus, according to equations Thus, in the dynamics of angular momentum, Π, the effects of two types of noise add together in the motion equations for the Stratonovich stochastic heavy top in an OU rotating frame.
In terms of angular velocity, Ω, equation (3.13) may be compared more easily with its fluid counterpart in (2.13), 

14)
upon substituting dN = θ(N − N (t)) dt + σdW t from (1.4). In (3.14), one sees that the deterministic angular momentum IΩ is enhanced by adding the integrated OU process R = ηN (t) with N (t) given in (1.5), while the deterministic angular transport velocity Ω is enhanced by adding the Stratonovich noise. However, one need not distinguish between Itô and Stratonovich noise in (3.14), since η and ξ are constant vectors in R 3 for the heavy top. The stochastic equation (3.14) in the absence of the OU noise, R(t), was studied in [2] to investigate the effects of transport stochasticity in the heavy top. Future work will investigate the further effects of the OU noise and its interaction with the transport noise in the examples of the rigid body and heavy top.

Conclusion/Summary
In this paper, section 1.2 has reinterpreted the Richardson cascade metaphor in terms of the Kelvin circulation theorem and has introduced two different stochastic closures for it. The first stochastic closure accounts for the otherwise unknown effects of the non-inertial forces on the intermediate scale and smaller whorl dynamics arising in the moving reference frame of the larger whorls. Introducing a stochastic model of the effects of the unknown large scales on the resolved intermediate scales may be a new idea, so far untested. However, the physics of deterministically moving reference frames is well known in fluid dynamics and of course goes back to Coriolis for particle dynamics in a rotating frame. The second stochastic closure is a model for the "Brownian transport" of the resolved intermediate scales by the unresolved smaller scales. This stochastic model was introduced in [18], its properties were analysed mathematically in [7] and it was developed further for applications in geophysical fluid dynamics in [12]. Following results of [15] for the introduction of Coriolis effects in Navier-Stokes turbulence, we expect that the introduction of non-inertial stochasticity in a moving reference frame will still be analytically tractable. However, following the results of [5] and [13] for rotating turbulence, we expect that the introduction of noninertial stochasticity could have profound physical effects.
Section 2 established the variational formulation of the stochastic Richardson cascade closure, thereby placing it into the modern mathematical context of geometric mechanics [3,20]. Having established their Hamilton-Pontryagin formulation, the equations of stochastic fluids (2.13) and advection equations (2.10) now fit into the Euler-Poincaré mathematical framework laid out for deterministic continuum dynamics in [20]. In this framework, one may augment the geometrical ideas of [3], in which ideal Euler incompressible fluid dynamics is recognised as geodesic motion on the Lie group of volume preserving diffeomorphisms; namely, to include advected fluid quantities, such as heat and mass. The presence of advected fluid quantities breaks the symmetry of the Lagrangian in Hamilton's principle under the volume preserving diffeos in two ways, one that enlarges the symmetry, and another that reduces the symmetry. First, compressibility enlarges the symmetry group to the full group of diffeos, not just those that preserve fluid volume elements. Second, the initial conditions for the flows with advected quantities reduce the symmetry under the full group of diffeos to the subgroups of diffeos which preserve those initial conditions. These are the isotropy subgroups of the diffeos. Boundary conditions also reduce the symmetry to a subgroup, but that is already known in Arnold's case for the incompressible flows. In the deterministic case, the breaking of a symmetry to an isotropy subgroup of a physical order parameter leads in general to semidirect-product Lie-Poisson Hamiltonian structure [17,20]. This kind of Hamiltonian structure is the signature of symmetry breaking.
Section 3 revealed that this semidirect-product structure on the Hamiltonian side holds true for the two types of stochastic deformations we have introduced in the present case, as well. In addition, we see that either of these two types of stochastic deformation may be introduced independently. Section 3 also provided an example of the application of these two types of stochasticity in the case of the heavy top, which is the classical example in finite dimensions of how symmetry breaking in geometric mechanics leads to semidirect-product Lie-Poisson Hamiltonian structure. For geophysical fluid dynamics (GFD) the preservation of the semidirect-product Lie-Poisson structure under the introduction of the two types of stochasticity treated here implies the preservation of the standard potential vorticity (PV) invariants for stochastic Lie transport dynamics of fluids in a randomly moving reference frame.
As mentioned in the Introduction, we expect that the approach described here will be useful in multi-scale GFD, perhaps in quantifying predictability and variability of sub-mesoscale ocean flow dynamics in the random frame of one or several unspecified mesoscale eddies interacting with each other. We also expect this approach may be useful in investigating industrial flows, e.g., to quantify uncertainty and variability in multi-scale flow processes employed in industry.