As discussed in the Introduction, we consider a model in which the eigenvectors \( \xi _i (X)\) are advected by the flow map \(g_t\), giving the time-dependent vector fields \( \zeta _i (t,x)\). Mathematically, the advection of a vector field by a smooth invertible map \(g_t\) corresponds to the push-forward operation,
$$\begin{aligned} \zeta _i (t)= (g_t) _ {*}\xi _i , \end{aligned}$$
(3.1)
meaning that
$$\begin{aligned} \zeta _i( t,g_t(X))= Dg_t (X) \cdot \xi _i ( X), \quad \text {for all } X\in {\mathcal {D}} . \end{aligned}$$
From a Lie group point of view, the push-forward is given by the adjoint action Ad\(:G\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}\) of the Lie group G on its Lie algebra \( {\mathfrak {g}}\), that is,
$$\begin{aligned} \zeta _i(t) = (g _t)_*\xi _i = {\text {Ad}}_{g_t} \xi _i . \end{aligned}$$
The notation used in this section is general enough to make our developments valid for any Lie group G. In our examples, we will take G to be the either the group of diffeomorphisms or the group of volume preserving diffeomorphisms. The corresponding coadjoint operators are
$$\begin{aligned} {\text {ad}}^*_u m= u\cdot \nabla m+ \nabla u^\mathsf {T} \cdot m+ m {\text {div}}u\qquad \text {and}\qquad {\text {ad}}^*_u m= \mathbb {P} ( u\cdot \nabla m+ \nabla u^\mathsf {T} \cdot m).\qquad \end{aligned}$$
(3.2)
The stochastic model considered here for advection of the eigenvectors is obtained by modifying the expression and the symmetries of the stochastic Hamiltonians \(H_i(g,\pi ; \xi _i )\) in (2.14). Namely, given N vector fields \( \xi _i \), \(i=1,\ldots ,N\), we consider the principle
$$\begin{aligned} \delta \int _0^T \left[ L(g, v)\mathrm{d}t+ \big \langle \pi , \mathrm{d}g- v\mathrm{d}t \big \rangle - \sum _{i=1}^NH_i(g,\pi ; \xi _i )\circ \mathrm{d} W _i(t) \right] =0, \end{aligned}$$
(3.3)
where each of the stochastic Hamiltonians \(H _i (\_,\_;\xi _i ):T^*G \rightarrow {\mathbb {R}}\) is right invariant only under the action of the isotropy subgroup of the eigenvector \( \xi _i \) with respect to the adjoint action (i.e., the push-forward action), namely
$$\begin{aligned} G_{ \xi _i }=\{ g \in G \mid {\text {Ad}}_g \xi _i = \xi _i \}=\{ g \in G \mid g _*\xi _i = \xi _i \} \subset G. \end{aligned}$$
That is, we have \(H _i (gh, \pi h;\xi _i )= H _i( g, \pi ; \xi _i )\), for all \( h \in G_{ \xi _i }\). The SHP principle (3.3) yields equations in the same general form as (2.17), but with stochastic Hamiltonians which are not G-invariant. As we will show, this difference due to symmetry breaking from G to \(G_{\xi _i}\) induces significant changes in the reduced Eulerian representation.
Physically, the symmetry breaking means that the initial conditions for the correlation eigenvectors are “frozen” into the subsequent flow, as a property carried along with individual Lagrangian fluid parcels, and which is not exchanged with other fluid parcels.
Being only \(G_{ \xi _i }\)-invariant, the stochastic Hamiltonian function \(H _i \) induces, in the Eulerian description, the reduced stochastic Hamiltonians
$$\begin{aligned} h _i = h _i (m , \zeta _i ):(T^*G)/G_{ \xi _i }\simeq {\mathfrak {g}} ^*\times {\mathcal {O}} _{ \xi _i } \rightarrow {\mathbb {R}}, \end{aligned}$$
(3.4)
defined by
$$\begin{aligned} h _i ( m , \zeta _i )=H _i ( g, \pi ; \xi _i ), \quad \text {for}\quad m = \pi g ^{-1}, \;\;\zeta _i = {\text {Ad}}_g\xi _i ,\end{aligned}$$
(3.5)
where \( {\mathcal {O}} _{\xi _i }:= \{{\text {Ad}}_g \xi _i \mid g \in G\}=\{g _*\xi _i \mid g \in G\} \subset {\mathfrak {g}}\) is the adjoint orbit of \( \xi _i \). The SHP principle (3.3) can thus be written in the reduced Eulerian form as
$$\begin{aligned} \delta \int _0 ^T \left[ \ell (u) \mathrm{d}t+ \big \langle m , \mathrm{d}g g^{-1} - u \mathrm{d}t \big \rangle _ {\mathfrak {g}} - \sum _{i=1}^Nh_i( m , {\text {Ad}}_g \xi _i )\circ \mathrm{d} W _i(t) \right] =0. \end{aligned}$$
(3.6)
The stationarity conditions with respect to the variations \( \delta u\), \( \delta m\), \( \delta g\), yield
$$\begin{aligned} \begin{aligned} \delta u:&\quad \frac{\delta \ell }{\delta u} - m = 0 ,\qquad \delta m:\quad \mathrm{d}g g ^{-1} - u \mathrm{d}t - \sum _{i=1}^N\frac{\delta h _i }{\delta m } \circ \mathrm{d}W _i (t) = 0 , \\ \delta g:&\quad \mathrm{d} \frac{\delta \ell }{\delta u} + {\text {ad}}^*_ u \frac{\delta \ell }{\delta u} \mathrm{d}t - \sum _{i=1}^N {\text {ad}}^*_ { \zeta _i } \frac{\delta h _i }{\delta \zeta _i } \circ \mathrm{d}W _i (t) =0, \end{aligned} \end{aligned}$$
(3.7)
where the advected eigenvector \( \zeta _i := {\text {Ad}}_g \xi _i \) obeys the auxiliary equation \(\mathrm{d}\zeta _i = [\mathrm{d}gg^{-1},\zeta _i ]\), obtained from its definition. The expression of the coadjoint operator \( {\text {ad}} ^*\) is given in (3.2). Upon using the second equation in (3.7), this auxiliary equation becomes
$$\begin{aligned} \mathrm{d} \zeta _i +[ \zeta _i ,u]\mathrm{d}t+\sum _{j=1}^N\left[ \zeta _i , \frac{\delta h _j }{\delta m }\right] \circ \mathrm{d}W _j (t) =0. \end{aligned}$$
(3.8)
Remark 3.1
In our applications of this model, we will always assume that the stochastic Hamiltonians \(h _i \) in (3.8) do not depend on m. That is, \(\delta h _j /\delta m =0\), so that the eigenvectors \(\zeta _i \) are advected only by the drift velocity, u. That is,
$$\begin{aligned} \mathrm{d} \zeta _i +[ \zeta _i ,u]\mathrm{d}t=0. \end{aligned}$$
(3.9)
This model using correlation eigenvectors frozen into the drift velocity is quite different from the model in Holm (2015), which chose \(h_i(m)=\langle m,\,\xi _i\rangle _ {\mathfrak {g}} \) and all fluid properties were advected by a velocity vector field comprising the sum of both the drift component and the stochastic component.
Hamiltonian Structure
Denoting by \(h: {\mathfrak {g}} ^*\rightarrow {\mathbb {R}} \) the Hamiltonian associated to \(\ell \), the above system can be equivalently written as
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{d} m + {\text {ad}}^*_ {\frac{\delta h}{\delta m }} m \, \mathrm{d}t- \sum _{i=1}^N {\text {ad}}^*_ { \zeta _i } \frac{\delta h _i }{\delta \zeta _i } \circ \mathrm{d}W _i (t) =0 ,\\ \displaystyle \mathrm{d} \zeta _i +\left[ \zeta _i , \frac{\delta h}{\delta m} \Big ] \mathrm{d}t+\sum _{j=1}^N\Big [ \zeta _i , \frac{\delta h _j }{\delta m }\right] \circ \mathrm{d}W _j (t) =0 . \end{array} \right. \end{aligned}$$
(3.10)
One may check that this system is Hamiltonian with respect to the Poisson bracket
$$\begin{aligned}&\{f,g\}_\mathrm{red}( m , \zeta _1 , \ldots , \zeta _N )= \left\langle m , \left[ \frac{\delta f}{\delta m }, \frac{\delta g}{\delta m }\right] \right\rangle _ {\mathfrak {g}} + \sum _{i=1}^N \left\langle \left[ \zeta _i , \frac{\delta f}{\delta m } \right] , \frac{\delta g}{\delta \zeta _i } \right\rangle _{\mathfrak {g}}\nonumber \\&\quad - \sum _{i=1}^N \left\langle \left[ \zeta _i , \frac{\delta g}{\delta m } \right] , \frac{\delta f}{\delta \zeta _i } \right\rangle _{\mathfrak {g}} \end{aligned}$$
(3.11)
on \( {\mathfrak {g}} ^*\times {\mathcal {O}} _{ \xi }\ni ( m , \zeta _1 , \ldots , \zeta _N )\), where \({\mathcal {O}} _{ \xi } \subset {\mathfrak {g}} ^N\) is the orbit of \( \xi := ( \xi _1 , \ldots , \xi _N )\) under the adjoint action of G. This reduced Poisson bracket is inherited from Poisson reduction of the canonical Poisson bracket \(\{\, \cdot ,\,\cdot \, \}_\mathrm{can}\) on \(T^*G\), by the isotropy subgroup \(G_ \xi \subset G\) of \(\xi :=( \xi _1 , \ldots ., \xi _N)\). Namely, the map
$$\begin{aligned}&(g, \pi ) \in T^*G \mapsto ( m, \zeta _1,\ldots , \zeta _N )\\&\quad = ( \pi g ^{-1} ,{\text {Ad}}_g \xi _1 ,\ldots ,{\text {Ad}}_ g \xi _N )\in (T^* G)/G_ \xi \simeq {\mathfrak {g}} ^*{\mathcal {O}}_{ \xi } \end{aligned}$$
is Poisson with respect to the Poisson brackets \(\{\, \cdot ,\,\cdot \, \}_\mathrm{can}\) and \(\{\, \cdot ,\,\cdot \, \}_\mathrm{red}\), see Appendix. The system (3.10) admits the Stratonovich–Poisson formulation,
$$\begin{aligned} \mathrm{d}f = \{f,h\} _\mathrm{red}\mathrm{d}t+\sum _{i=1}^N \{f, h_i \}_\mathrm{red}\circ \mathrm{d} W _i (t) , \end{aligned}$$
(3.12)
for arbitrary functions \(f: {\mathfrak {g}} ^*\times {\mathcal {O}} \rightarrow {\mathbb {R}} \). Note that in (3.12) the Hamiltonians h and \(h _i \), \(i=1,\ldots ,N\), depend a priori on all the variables \((m ,\zeta _1 , \ldots ,\zeta _N)\). The system (3.10) is recovered when h depends only on m, while the \( h _i \) depend only on m and \(\zeta _i \) (not on \( \zeta _j \), for \(j\ne i\)). The Poisson tensor at \((m, \zeta _1 ,\ldots , \zeta _N )\) reads
$$\begin{aligned} \left[ \begin{array}{cccc} \displaystyle - {\text {ad}}^*_{\square } m&{}\quad {\text {ad}}^*_{ \zeta _1 } &{}\quad \cdots &{}\quad {\text {ad}}^*_{ \zeta _N } \\ \displaystyle - {\text {ad}}_{ \zeta _1 } &{}\quad &{}\quad &{}\quad \\ \displaystyle \vdots &{}\quad &{}\quad 0&{}\quad \\ \displaystyle - {\text {ad}}_{ \zeta _N } &{}\quad &{}\quad &{}\quad \end{array} \right] . \end{aligned}$$
(3.13)
Remark 3.2
(Itô form) In the special case that \({\delta h_i}/{\delta m }=0\), as assumed in Remark 3.1, the Itô form of Eqs. (3.10) does not introduce any additional drift terms. That is, in this special case, the Itô form of (3.10) is obtained by simply removing the Stratonovich symbol \((\,\circ \,)\). However, in the general case, if \({\delta h_i}/{\delta m }\ne 0\), the Itô form does contain additional drift terms. These additional drift terms in the Itô form for the general case can be computed in a standard way, but the equations then may take a more complicated form. By making use of the Poisson formulation in terms of the bracket \(\{\, \cdot ,\,\cdot \, \}_\mathrm{red}\) in (3.12), we can write the additional drift terms in the Itô form for the general case in a concise way as
$$\begin{aligned} \mathrm{d}f =\left( \{f,h\} _\mathrm{red}- \frac{1}{2} \{h_i,\{ h_i,f\}_\mathrm{red}\} _\mathrm{red} \right) \mathrm{d}t+\sum _{i=1}^N \{f, h_i \}_\mathrm{red}\circ \mathrm{d} W _i (t) .\end{aligned}$$
(3.14)
Example 3.3
(Incompressible 2D models) In the 2D incompressible case, we can identify the Lie algebra \( {\mathfrak {g}} \) with the space of differentiable functions on \( {\mathcal {D}} \), modulo constants. These are the stream functions, denoted \( \psi \). We use the \(L ^2 \) duality pairing \( \left\langle \varpi , \psi \right\rangle _ {\mathfrak {g}} =\int _ {\mathcal {D}} \varpi (x) \psi (x) \mathrm{d}^2x\) and identify \({\mathfrak {g}} ^*\) with the space of functions on \( {\mathcal {D}} \) with zero integral. These are the absolute vorticities, denoted \( \varpi \), as explained in Marsden and Weinstein (1983).
Let \( \psi _i^0(X)\) be the stream function associated to the eigenvector \( \xi _i (X)\), \(i=1,\ldots ,N\). The stream function \( \psi _i (t,x)\) of the advected eigenvector \( \zeta _i (t)= (g_t) _*\xi _i \) is found to be
$$\begin{aligned} \psi _i (t,g_t(X))= \psi _i ^0(X). \end{aligned}$$
The stochastic model (3.10) applied to 2D incompressible fluid dynamics with Hamiltonian \(h( \varpi )\) and stochastic Hamiltonians \( h _i ( \varpi , \psi _i )= h _i ( \psi _i )\) is given by
$$\begin{aligned} \mathrm{d}\varpi + \left\{ \varpi , \frac{\delta h}{\delta \varpi } \right\} \mathrm{d}t+\sum _{i=1}^N\left\{ \psi _i ,\frac{\delta h _i }{\delta \psi _i } \right\} \circ \mathrm{d}W _i (t) =0 , \qquad d \psi _i + \left\{ \psi _i , \frac{\delta h}{\delta \varpi } \right\} \mathrm{d}t=0, \end{aligned}$$
(3.15)
where we have used the formula \( {\text {ad}}^*_ \psi \varpi =\{\varpi , \psi \}\) for the coadjoint operator for 2D incompressible fluids. For example, with the appropriate choice of the Hamiltonian, we can write the stochastic model (3.15) for the following cases:
-
(a)
2D perfect fluid: \( \psi =\frac{\delta h}{\delta \varpi }= - \Delta ^{-1} \varpi \);
-
(b)
2D rotating perfect fluid: \( \psi =\frac{\delta h}{\delta \varpi }= - \Delta ^{-1} (\varpi -f)\);
-
(c)
2D rotating quasigeostrophy (QG): \( \psi =\frac{\delta h}{\delta \varpi }= -( \Delta - \mathcal {F} ) ^{-1} (\varpi -f)\);
where \( {\mathcal {F}} \) and f denote, respectively, the square of the inverse Rossby radius and the rotation frequency. For the stochastic Hamiltonians \(h _i \), \(i=1,\ldots ,N\), one may choose
$$\begin{aligned} h_i ( \psi _i )= -\, \frac{1}{2} \int _ {\mathcal {D}} \Delta \psi _i (x) \psi _i (x) \,\mathrm{d} ^2 x,\;\; \text {(no sum)}, \end{aligned}$$
in which case \( \frac{\delta h _i }{\delta \psi _i } = - \,\Delta \psi _i \).
Similarly to the discussion in Remark 3.2, the Itô form of equations (3.15) takes the same expression, since we have chosen stochastic Hamiltonians for which \(\delta h _i / \delta \varpi = 0\). In this case, one may obtain the Itô forms by simply replacing the Stratonovich noise \( \circ \, \mathrm{d}W _i (t)\) by the Ito noise \( \mathrm{d}W _i (t) \) without modifying the drift terms.
Remark 3.4
(Conserved correlation enstrophies) The stochastic model (3.15) does not preserve the well-known vorticity enstrophies \(\Lambda ( \omega )=\int _ {\mathcal {D}} \Phi ( \omega ) \mathrm{d}^2 x,\) which are preserved in the deterministic case. However, the stochastic equation for \(\psi _i\) in (3.15) does preserve the following correlation enstrophy functionals
$$\begin{aligned} \Lambda _i ( \psi _i )=\int _ {\mathcal {D}} \Phi ( \psi _i ) \mathrm{d}^2 x, \end{aligned}$$
when the stochastic model (3.10) is applied to 2D incompressible fluid dynamics. In general, for the case \(h=h(m)\) and \( h _i = h _i ( \zeta _i )\), by Eq. (3.10), the corresponding functionals \( \Lambda _i ( \zeta _i )\) verify
$$\begin{aligned} \mathrm{d} \Lambda _i = \int _ {\mathcal {D}} \{ \Lambda _i ,h\}_\mathrm{red}\mathrm{d}^2x\,\mathrm{d}t = - \int _ {\mathcal {D}} \left\langle \left[ \zeta _i , \frac{\delta h}{\delta m} \right] ,\frac{\delta \Lambda _i }{\delta \zeta _i } \right\rangle _ {\mathfrak {g}}\mathrm{d}^2x\, \mathrm{d}t \end{aligned}$$
which vanishes for 2D incompressible fluids after integration by parts and imposition of homogeneous boundary conditions.
Example 3.5
(3D incompressible Euler) We consider the stochastic Hamiltonians
$$\begin{aligned} h _i (\zeta _i )= \int _ {\mathcal {D}} F_i ( \zeta _i (x)) \mathrm{d} ^3 x, \quad i=1,\ldots ,N, \end{aligned}$$
(3.16)
where \(F_i\) are smooth functions. Upon using the formula from (3.2) for incompressible flows,
$$\begin{aligned} {\text {ad}}^*_u m= \mathbb {P} ( u \cdot \nabla m+ \nabla u^\mathsf {T} \cdot m)= \mathbb {P} ( {\text {curl}}m\times u) , \end{aligned}$$
(3.17)
where \( \mathbb {P} \) is the Hodge projector onto divergence free vector fields, the stochastic model (3.10) reads
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{d}u+ \mathbb {P} ( u\cdot \nabla u ) \mathrm{d}t=\sum _{i=1}^N\mathbb {P}\left( {\text {curl}} \frac{\delta F _i}{\delta \zeta _i }\times \zeta _i \right) \circ \mathrm{d}W _i (t) \\ \mathrm{d}\zeta _i + {\text {curl}}( \zeta _i \times u)\mathrm{d}t=0. \end{array} \right. \end{aligned}$$
(3.18)
The stochastic terms can be written equivalently with the help of the stress tensors
$$\begin{aligned} \sigma _i= \zeta _i \otimes \frac{\partial F _i }{\partial \zeta _i }+ F _i ( \zeta _i )\mathbf {I} , \quad \text {i.e.,} \quad (\sigma _i)_b^a = \zeta _i ^a \otimes \frac{\partial F _i }{\partial \zeta ^b _i }- F _i ( \zeta ) \delta ^a _b , \end{aligned}$$
(3.19)
and pressures \( p _i \), \(i=1,\ldots ,N\). With these definitions, Eq. (3.18) becomes
$$\begin{aligned} \mathrm{d}u+ (u\cdot \nabla u+ \nabla p)\mathrm{d}t= \sum _{i=1}^N({\text {div}} \sigma _i +\nabla p _i ) \circ \mathrm{d}W _i (t), \end{aligned}$$
(3.20)
where the divergence is defined as \(({\text {div}} \sigma _i ) _b = \partial _a ({\sigma _i}) ^a _b \) for all i, and the individual \(p_i\) are each found by solving a Poisson equation, with boundary conditions given by \(\hat{n}_a ({\text {div}}{\sigma _i}) ^a = 0\). Recall that the vector fields \(\zeta _i (t, x)\) are obtained from the given eigenvectors field \( \xi _i (X)\) by the push-forward operation (3.1). As mentioned earlier in Remark 3.2, the Itô forms of the equations have the same expression.
Remark 3.6
The three-dimensional system (3.18) is reminiscent of incompressible magnetohydrodynamics (MHD), except it has a stochastic “\(J\times B\,\)” force depending on the sum over all of the \(\zeta _i\). In following this analogy with MHD, we may introduce vector potentials \( \alpha _i\) by writing \(\zeta _i=:\mathrm{curl\,} \alpha _i\). Having done so, one notices that evolution under the stochastic system (3.18) preserves the integrals
$$\begin{aligned} \Lambda _i = \int _{{\mathcal {D}}} \alpha _i\cdot \zeta _i \,\mathrm{d}^3x \quad \hbox {(No sum).} \end{aligned}$$
Proof
By the second equation in the equation set (3.18), we have
$$\begin{aligned} \mathrm{d}\Lambda _i = \mathrm{d} \int _{{\mathcal {D}}} \alpha _i\cdot \zeta _i \,\mathrm{d}^3x = -2\int _{{\mathcal {D}}} \alpha _i \cdot {\text {curl}}( \zeta _i \times u)\,\mathrm{d}^3x \,\mathrm{d}t= 0 . \end{aligned}$$
\(\square \)
The \(\Lambda _i\) integrals are topological quantities known as correlation helicities which measure the number of linkages of the lines of each vector field \(\zeta _i\) with itself. Conservation of the correlation helicity \(\Lambda _i\) means that evolution by the stochastic system (3.18) cannot unlink the linkages of each divergence free vector field \(\zeta _i\) with itself. This conclusion is the analogue of conservation of magnetic helicity in MHD.
Inclusion of Additional Advected Tensor Fields
More generally, suppose that the fluid model involves a tensor field q(t, x) advected by the fluid flow as in (2.2). The evolution of this advected field is this given by \(q(t)= (g_t)_*q_0 \), where \(q_0(X)\) is the initial value and \(g_t \in G\) is the fluid flow. In this case, the variational principle is written in reduced Eulerian form as
$$\begin{aligned} \delta \int _0^T\left[ \ell (u, g_*q_0) \mathrm{d}t+ \big \langle m , \mathrm{d}g g^{-1} - u \mathrm{d}t \big \rangle _ {\mathfrak {g}} - \sum _{i=1}^Nh_i(m , {\text {Ad}}_g \xi _i )\circ \mathrm{d} W _i(t) \right] =0, \end{aligned}$$
(3.21)
for variations \( \delta u\), \( \delta m\), and \( \delta g\). The stationarity conditions yield the same first two equations of (3.7), whereas the third one becomes
$$\begin{aligned} \mathrm{d} \frac{\delta \ell }{\delta u}+ {\text {ad}}^*_ u \frac{\delta \ell }{\delta u} \mathrm{d}t- \sum _{i=1}^N {\text {ad}}^*_ { \zeta _i } \frac{\delta h _i }{\delta \zeta _i } \circ \mathrm{d}W _i (t) = \frac{\delta \ell }{\delta q} \diamond q \,\mathrm{d}t. \end{aligned}$$
(3.22)
From its definition, the quantity \(q (t)= (g_t)_*q_0\) verifies \( \mathrm{d}q+ \pounds _{\mathrm{d}g g ^{-1} } q=0\). For the case that \(\delta h _i /\delta m=0\), this quantity is governed by the ordinary advection equation,
$$\begin{aligned} \mathrm{d}q+ \pounds _u q \, \mathrm{d}t=0. \end{aligned}$$
(3.23)
Remark 3.7
(Itô form) For the case that \(\delta h _i /\delta m=0\), passing to the Ito formulation does not introduce any change in the drift terms.
Example 3.8
(Rotating shallow water) Equations (3.22) and (3.23) for the inclusion of such advected quantities into Model 2 may be illustrated with the example of the rotating shallow water equation. In this case \( {\mathcal {D}} \) is a two-dimensional domain and G is the group of diffeomorphisms of \( {\mathcal {D}} \). Let us denote by \(\eta (t,x)\) the water depth, by B(x) the bottom topography, by R(x) the Coriolis vector field, and \(x=(x _1 , x _2 ) \in {\mathcal {D}} \). The Lagrangian of the rotating shallow water system is
$$\begin{aligned} \ell (u,\eta )= \int _ {\mathcal {D}} \left[ \frac{1}{2} \eta |u| ^2 + \eta R \cdot u - \frac{1}{2} g(\eta -B) ^2 \right] \mathrm{d} ^2 x, \end{aligned}$$
(3.24)
where g is the gravity acceleration. Taking the stochastic Hamiltonians \(h _i ( \zeta _i )\) in (3.16), the stochastic variational principle in (3.21) produces the equations
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{d}u+ ( u\cdot \nabla u+ {\text {curl}}R \times u + g \nabla (\eta -B)) \mathrm{d}t= \frac{1}{\eta }\sum _{i=1}^N{\text {div}} \sigma _i \circ \mathrm{d}W _i (t)\\ \displaystyle \mathrm{d}\eta + {\text {div}}(\eta u)\mathrm{d}t=0, \qquad \mathrm{d}\zeta _i +[ \zeta _i ,u]\mathrm{d}t=0, \end{array} \right. \end{aligned}$$
(3.25)
where the stochastic stress \( \sigma _i \) is defined as above in (3.19). The effect of \( \sigma _i \) can be seen by writing the Kelvin circulation theorem obtained by integrating the first equation in (3.25) around a loop c(u) moving with the drift velocity u(t, x), to find
$$\begin{aligned} \mathrm{d}\oint _{c(u)} \big (u+R\big )\cdot \mathrm{d}x = \sum _{i=1}^N\oint _{c(u)} \Big ( \frac{1}{\eta }{\text {div}} \sigma _i \circ \mathrm{d}W _i (t) \Big ) \cdot \mathrm{d}x. \end{aligned}$$
Thus, the total stochastic stress generates circulation of the total velocity \((u+R)\) around any loop moving with the relative fluid velocity u in the rotating frame.
Example 3.9
(Rotating compressible barotropic fluid) The above developments easily extend to other fluid models such as compressible barotropic fluid flow in a rotating frame, whose Lagrangian is
$$\begin{aligned} \ell (u, \rho ) = \int _ {\mathcal {D}}\rho \Big ( \frac{1}{2} | u| ^2 + u \cdot R - e(\rho ) - gz \Big ) \mathrm{d} ^3 x, \end{aligned}$$
(3.26)
where \(z=x_3\) is the vertical coordinate, \( \rho \) is the mass density and e is the specific internal energy. With the choice (3.16), one gets from (3.22), the stochastic balance of momentum
$$\begin{aligned} \mathrm{d}u+ \Big ( u \cdot \nabla u+ {\text {curl}} R \times u + g \mathbf {z} + \frac{1}{ \rho }\nabla p \Big ) \mathrm{d}t = \frac{1}{ \rho }\sum _{i=1}^N{\text {div}} \sigma _i \circ \mathrm{d}W _i (t), \end{aligned}$$
with advection equations
$$\begin{aligned} \mathrm{d} \rho + {\text {div}}( \rho u) \mathrm{d}t=0 \quad \hbox {and}\quad \mathrm{d}\zeta _i +[\zeta _i ,u]\mathrm{d}t=0 . \end{aligned}$$