The Geometry of Stochastic Fluid Dynamics

Abstract: Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric climate' as the ensemble expectation, while the atmosphericweather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean--Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.

• The paper develops a Lie-group variational framework in which SALT stochastic transport preserves Kelvin–Noether circulation, Casimir invariants, and potential-vorticity structure, although energy is generally not conserved.
• LA-SALT separates deterministic climate evolution from stochastic weather fluctuations, producing closed mean and scalar-moment equations and proving global strong solutions for the 2D Euler–Boussinesq case.
• The paper extends the framework to coupled stochastic ocean–atmosphere models, establishing local or martingale well-posedness results and numerically confirming Hasselmann-predicted red-shifting of ocean energy spectra.

This paper is a pedagogical review of stochastic geometric mechanics (SGM) as a framework for modelling uncertainty in geophysical fluid dynamics (GFD), culminating in stochastic ocean–atmosphere models (2608.15321). The unifying idea is that fluid equations should be derived from Lie group-invariant variational principles, so that approximations and stochastic parametrisations preserve the geometric structure — most importantly the Kelvin–Noether circulation theorem — rather than being imposed ad hoc on the Eulerian equations.

Deterministic foundations: Poincaré, Noether, and the diffeomorphism group

The review begins with the classical observation, due to Arnold, that ideal incompressible flow is geodesic motion on the manifold $\mathfrak{D}^s$ of volume-preserving diffeomorphisms with respect to the $L^2$ kinetic-energy metric. The variational machinery rests on two pillars: Poincaré's 1901 derivation of the Euler–Poincaré equations for Lagrangians invariant under a transitive Lie group action, and Noether's theorem, which associates to each continuous symmetry a momentum map $J = p \diamond q \in \mathfrak{g}^*$ via the diamond operation defined by $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$.

For continuum mechanics the configuration space is not itself a Lie group once advected quantities are present. Each advected quantity (mass density $D\,d^3x$, buoyancy $b$, entropy $s$) reduces the symmetry group $\mathfrak{D}^s$ to its isotropy subgroup $\mathfrak{D}^s_{a_0}$, so reduction proceeds over the semidirect product $\mathfrak{D}^s \circledS V$. The paper emphasises that this symmetry breaking is not merely structural: the diamond terms $L^2$0 appearing in the Euler–Poincaré equations are precisely the forces generated by broken symmetry, e.g., buoyancy forces from gravity acting on horizontal buoyancy gradients.

The 3D Euler–Boussinesq equations as a template

The second part develops the deterministic 3D Euler–Boussinesq (EB) equations for rotating, stratified, incompressible flow, written intrinsically as

$L^2$1

with $L^2$2 the circulation one-form including Coriolis potential. Two consequences are highlighted. First, the Kelvin–Stokes theorem shows circulation is generated exactly when $L^2$3 fails to align with the vertical $L^2$4, i.e., $L^2$5. Second, Ertel's theorem — conservation of potential vorticity $L^2$6 on parcels — follows purely from the product rule for Lie derivatives, independent of the detailed form of the Lagrangian's dependence on $L^2$7. The paper also derives rotating shallow water and adiabatic compressible flow as further Euler–Poincaré examples, and presents the semidirect-product Lie–Poisson bracket for 2D EB, proving that $L^2$8 constitutes a momentum map for the symplectic vector fields and verifying the Casimir family $L^2$9. A key methodological claim is that the entire genealogy of approximate GFD models descending from 3D EB can be obtained by sequential approximation of the Lagrangian alone, with each descendant inheriting the parent's geometric properties.

SALT: stochastic advection by Lie transport

The third part introduces transport noise by replacing the deterministic reconstruction equation $J = p \diamond q \in \mathfrak{g}^*$0 with the Stratonovich semimartingale

$J = p \diamond q \in \mathfrak{g}^*$1

where the prescribed data vector fields $J = p \diamond q \in \mathfrak{g}^*$2 encode unresolved-scale velocity correlations and can be estimated from observations via empirical orthogonal function analysis. The technical backbone is the Kunita–Itô–Wentzell (KIW) formula, which extends the Lie chain rule to semimartingale-valued $J = p \diamond q \in \mathfrak{g}^*$3-forms and underpins both the constrained variations $J = p \diamond q \in \mathfrak{g}^*$4 and the stochastic Kelvin–Noether theorem:

$J = p \diamond q \in \mathfrak{g}^*$5

The resulting stochastic Euler–Poincaré equations preserve the full semidirect-product structure: the stochastic Lie–Poisson system uses the same $J = p \diamond q \in \mathfrak{g}^*$6 operator in drift and diffusion, so the Casimir conservation laws survive stochasticity verbatim. Circulation is conserved when only mass density is advected; additional advected quantities generate circulation through their diamond terms, exactly as in the deterministic case. The cost is explicit: energy is no longer conserved because the Legendre-transformed Hamiltonian becomes a semimartingale with explicit time dependence. A notable analytical point, drawn from Crisan, Flandoli and Holm, is that because the noise consists of Lie derivatives whose commutators admit suitable estimates, the local well-posedness theory of the 3D stochastic Euler equations with SALT closely parallels the deterministic case. Extensions to geometric rough paths recover results such as a Beale–Kato–Majda-type blowup criterion.

LA-SALT: climate as expectation, weather as fluctuation

The fourth part develops Lagrangian-averaged SALT, in which the drift velocity of the stochastic transport is replaced by its expectation:

$J = p \diamond q \in \mathfrak{g}^*$7

Three structural results follow. First, the expectation dynamics form a closed deterministic system — answering Lorenz's question about determinism of climate affirmatively within this model class. Second, the fluctuation variables satisfy linear stochastic transport equations slaved to the deterministic mean, which are well-posed; global strong solutions are established for the 2D EB case. Third, closed evolution equations exist for covariance tensors of advected quantities,

$J = p \diamond q \in \mathfrak{g}^*$8

and, for scalar fields, an iterated closed hierarchy for all $J = p \diamond q \in \mathfrak{g}^*$9-th central moments. For the 2D LA-SALT Boussinesq system, closure of the velocity–buoyancy cross-covariance is achieved by adjoining the equation for $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$0, driven by the buoyancy-gradient covariance $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$1. Under specific choices of $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$2 the space-averaged buoyancy variance grows at most exponentially, bounded by the mean-field gradient energy. The paper is candid that closure fails in general: for tensorial advected quantities the non-commutativity of the tensor product obstructs closed $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$3-th moment systems, and the momentum covariance generally does not close outside special cases such as 2D EB.

Stochastic ocean–atmosphere models and Hasselmann's paradigm

The final part couples a stochastic atmospheric component to a deterministic oceanic component in a Gill–Matsuno-type intermediate coupled model, relevant to ENSO dynamics. Both SALT and LA-SALT versions retain Kelvin circulation theorems for atmosphere and ocean separately, coupled through the Leray–Helmholtz decomposition of atmospheric velocity into solenoidal and gradient parts acting on the ocean. Local well-posedness is proved for the deterministic model, martingale solutions exist for the SALT version, and unique local solutions hold for LA-SALT. Recent discrete exterior calculus simulations of the LA-SALT coupled model verify the red-shift of the oceanic energy spectrum predicted by Hasselmann's 1976 stochastic climate theory — the mechanism by which stochastic atmospheric forcing renders long-time ocean statistics computable at coarse resolution.

Limitations and open questions

Several caveats bear directly on the results. Energy conservation is lost under SALT noise, a deliberate trade-off for circulation preservation; whether this loss is physically acceptable across regimes remains a modelling judgement. Global well-posedness of the deterministic 2D inviscid Boussinesq equations is still open, and the SALT version inherits this gap, with only local well-posedness and blow-up criteria available. Moment closure in LA-SALT holds for scalar advected quantities but not for general tensors, and the momentum-variable covariance closes only in special cases. The data vector fields $$\langle b \diamond a, \xi\rangle := -\langle b, \mathcal{L}_\xi a\rangle$$4 must be supplied externally from observations or high-resolution simulations; the framework does not determine them internally. Finally, the coupled ocean–atmosphere analysis is restricted to intermediate-complexity Gill–Matsuno-class models, leaving the extension to primitive-equation couplings untreated.

Conclusion

The paper consolidates a coherent programme in which stochasticity enters fluid dynamics exclusively through the transport velocity in a Lie group-invariant variational principle, thereby preserving Kelvin–Noether circulation, Casimirs, and the semidirect-product Hamiltonian structure while quantifying uncertainty from unresolved scales. The progression from deterministic Euler–Poincaré theory through SALT and LA-SALT to coupled stochastic ocean–atmosphere models demonstrates that Hasselmann's deterministic-plus-stochastic decomposition can be implemented without sacrificing the geometric identities on which GFD diagnostics such as potential vorticity depend.