Particle relabelling symmetries and Noether's theorem for vertical slice models
aa r X i v : . [ m a t h - ph ] A ug Particle relabelling symmetries and Noether’s theorem for vertical slicemodels
C. J. Cotter and M.J.P. Cullen August 17, 2018
Abstract
We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc RoySoc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closedloops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can bederived directly from this circulation theorem. In this paper, we show that this property is due to these modelshaving a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, notjust those diffeomorphisms that preserve the potential temperature. This is developed using the methodologyof Cotter and Holm (Foundations of Computational Mathematics, 2012).
Contents
Vertical slice models are models of 3D fluids that assume that all fields are independent of y , except for the(potential) temperature, which is assumed to take the form θ ( x, y, z, t ) = θ S ( x, z, t ) + sy , where s is a time-independent coefficient, this allows for a model of North-South temperature gradient s . These models providea simplification of the equations that allows to study geophysical phenomena such as frontogenesis in idealisedgeometries. They also provide a useful testbed for numerical algorithms for atmospheric dynamical cores, sincethey only require computation with 2D data, and so can be run quickly on a desktop computer.A hierarchy of vertical slice models have been used to study the formation and evolution of fronts. The verticalslice non-hydrostatic incompressible Boussinesq equations, the hydrostatic Boussinesq equations and the corre-sponding semi-geostrophic equations all exhibit frontogenesis, whilst representing solutions of the full 3D equations.As summarised in [Cul06], the semi-geostrophic equations have an optimal transport interpretation. The optimaltransport formulation proves that geostrophic and hydrostatic balance can be achieved while respecting Lagrangianconservation properties. The optimal transport formulation has also been used to develop numerical algorithms forthe equations in Lagrangian form, that form fronts even at low resolution. These solutions provide a useful com-parison for standard Eulerian numerical algorithms for the non-hydrostatic incompressible Boussinesq equations,by considering their solutions in the semi-geostrophic limit, as described in [VCC14] who found that the limitingEulerian solutions deviated from the semi-geostrophic solution, suggesting that fronts need some extra parameteri-sation in Eulerian models. The limiting Eulerian numerical solutions satisfied geostrophic and hydrostatic balanceto the expected extent, but the Lagrangian conservation properties were systematically violated. This shows thatthe computed solution was fundamentally diffusive, which is not physically realistic on this scale. Thus, there is a Department of Mathematics, Imperial College London. London SW7 2AZ, UK. [email protected] Met Office, Fitzroy Road, Exeter EX1 3PB. α -regularised vertical slice model, and a compressiblevertical slice model, which fixed the lack of conservation in the compressible model of [Cul08].In [CH13b], it was noted that the slice Euler-Poincar´e equations with advected density D and potential tem-perature θ S have a Kelvin circulation theorem ddt I c ( u S ) (cid:18) s (cid:18) D δlδu S (cid:19) − (cid:18) D δlδu T (cid:19) ∇ θ S (cid:19) · d x = 0 , (1.1)where c ( u S ) is a loop advected by the in-slice velocity u S , u T is the transverse velocity (the component orthogonalto the slice), the potential temperature is written as θ ( x, y, z, t ) = θ S ( x, z, t ) + sy, (1.2)where s is the constant temperature gradient in the y -direction, and l [ u S , u T , θ S , D ] is the reduced Lagrangian. Forthe case of the incompressible Euler-Boussinesq slice model, this becomesdd t I C ( t ) ( su S − ( u T + f x ) ∇ θ S ) · d x = 0 . (1.3)This is curious because the circulation theorem for EP equations on the diffeomorphism group (instead of thesemidirect product used in the slice models) has a baroclinic term that only vanishes which the curve is on a θ -isosurface. In the slice model case, this term is replaced by an additional term in the circulation, leading to aconserved circulation on any loop , θ -isosurface or not. Since the contour used to defined the circulation theorem is amaterial contour, it cannot cross fronts in the semi-geostrophic limits. In the semi-geostrophic solutions, fronts aresingularities in the optimally transporting map where the boundary of the domain has been mapped into the interior.Thus solutions which are discontinuous in physical space are still compatible with the circulation theorem, as long asthis restriction is made. These formulations will preserve Lagrangian conservation properties. However, computingthe solutions remains challenging. Care with use potential vorticity conservation near the semi-geostrophic limitfor standard models is therefore also required.In this paper, we show that this Kelvin circulation theorem occurs because of relabelling symmetries that occurin the variational description of [CH13b], by using the variational tools for applying Noether’s Theorem to theEuler-Poincar´e theory discussed in [CH13a]. The rest of this paper is structured as follows. In Section 2, we reviewthe variational formulation of vertical slice models. In Section 3, we show how the conserved potential vorticityarises from relabelling symmetries that are specific to the slice geometry. Finally in Section 4 we provide a summaryand outlook. In this section, we briefly review the variational formulation of slice models. In a vertical slice model, we assume thatthe velocity field is independent of y , but still has a component in the y -direction. This means that the Lagrangianflow map can be written Φ( X, Y, Z, t ) = ( x ( X, Z, t ) , y ( X, Z, t ) +
Y, z ( X, Z, t )) , (2.1)where X, Y, Z are Lagrangian labels, ( x, y, z ) are particle locations and t is time. Equivalently, the Lagrangian flowmap can be represented by a diffeomorphism of the vertical slice x − z plane combined with an ( x, z )-dependent The assumption that the invertible map Φ is smooth is necessary to compute the potential vorticity equation. However, the optimaltransport formulation relaxes the smoothness requirements, enabling weaker solutions such as the semigeostrophic frontal solutionswhich are weak Lagrangian solutions. The Lagrangian formulation is likely to have some sort of global existence, based on conservationproperties, but the Eulerian formulation may only work in smooth cases, which may not be generic for weather phenomena such asfronts. displacement in the y -direction, which we write as an element ( φ, f ) of the semidirect product Diff(Ω) s F (Ω), where s denotes the semidirect product, and F (Ω) denotes an appropriate space of smooth functions on Ω that specifythe displacement of Lagrangian particles in the y -direction at each point in Ω. Then, if we write X = ( x, z ), thetransformation takes the form ( x , y ) = Φ( X , Y ) = ( φ ( X ) , Y + f ( X )) . (2.2)In this representation, composition of two slice flow maps ( φ , f ) and ( φ , f ) is obtained from the semi-directproduct formula [HMR98], ( φ , f ) · ( φ , f ) = ( φ ◦ φ , f ◦ φ + f ) . (2.3)We see that the vertical slice diffeomorphisms φ and φ compose in the normal way, whilst the combined verticaldeflection is obtained by moving f with φ before adding f .We represent Eulerian velocity fields by splitting into two components ( u S , u T ) where u S is the “slice” componentin the x - z plane, and u T is the “transverse” component in the y direction. ( u S , u T ) is considered as an element inthe semidirect product Lie algebra X (Ω) s F (Ω) where X (Ω) denotes the vector fields on Ω, representing the twocomponents of the velocity u S ∈ X (Ω) and u T ∈ F (Ω). This Lie algebra has a Lie bracket, which we shall makeuse of below, given by [( u S , u T ) , ( w S , w T )] = ([ u S , w S ] , u S · ∇ w T − w S · ∇ u T ) , (2.4)where [ u S , w S ] = u S · ∇ w S − w S · ∇ u S is the Lie bracket for the time-dependent vector fields ( u S , w S ) ∈ X (Ω), and ∇ denotes the gradient in the x - z plane.A time-dependent Lagrangian flow map must satisfy the equation ∂∂t ( φ, f ) = ( u S , u T )( φ, f ) = ( u S ◦ φ, u T ◦ φ ) , (2.5)for some slice vector field ( u S , u T ) ∈ X (Ω) s F (Ω). Similarly, if within Hamilton’s principle we consider a one-parameter family of perturbations ( φ ǫ , f ǫ ) (parameterised by ǫ ) to ( φ, f ), then δ ( φ, f ) = lim ǫ → ( φ ǫ , f ǫ ) − ( φ, f ) ǫ = ( w S , w T )( φ, f ) = ( w S ◦ φ, w T ◦ φ ) , (2.6)for some time-dependent slice vector field ( u S , u T ) that generates the infinitesimal perturbations at ǫ = 0. Taking ǫ and time-derivatives of these two expressions and comparing leads to δ ( u S , u T ) = ∂∂t ( w S , w T ) + [( u S , u T ) , ( w S , w T )] , (2.7)which gives us a formula telling us how perturbations in ( φ, f ) lead to perturbations in ( u S , u T ) that we can use inHamilton’s principle.In order to build geophysical models, we also need to consider advected tracers and densities. In this modellingframework, we assume that densities are independent of y , so that the continuity equation becomes ∂∂t D + ∇ · ( u S D ) = 0 , (2.8)which we write in geometric notation as ∂∂t D d S + L u S D d S = 0 , (2.9)where d S is the area form (so that D d S ∈ Λ (Ω), the space of 2-forms on Ω) and L u S is the Lie derivative withrespect to u S , given via Cartan’s Magic Formula as L u S D d S = d( u S D d S ) , (2.10)recalling that d( D d S ) = 0. Under an infinitesimal perturbation to ( φ, f ) the density changes according to δD d S + L w S D d S = 0 . (2.11)We assume that tracers can be written as sum of a y -independent component θ S plus a time-independentcomponent sy with linear dependence on y (geophysically this allows for North-South temperature gradients thatare necessary for the baroclinic instability that leads to frontogenesis). This means that the flow map ( φ, f )transports initial conditions ( θ S, , s ) according to θ S = φ ∗ ( θ S − s f ) , s = s . (2.12)Time-differentiation then leads to the transport equation ∂∂t θ S + u S · ∇ θ S + sw T = 0 , ∂s∂t = 0 , (2.13)which we write in geometric notation as ∂∂t ( θ S , s ) + L ( u S ,u T ) ( θ S , s ) = 0 , (2.14)where L ( u S ,u T ) ( θ S , s ) = ( u S d θ S + su T , , (2.15)and where we have considered ( θ S , s ) ∈ F (Ω) × R ). Similarly, infinitesimal perturbations to ( φ, f ) lead to infinites-imal perturbations to ( θ S , s ) given by δ ( θ S , s ) + L ( w S ,w T ) ( θ S , s ) = 0 . (2.16)Given a Lagrangian l [( u S , u T ) , D, ( θ S , s )], we define δl [( u S , u T ) , D, ( θ S , s ); ( δu S , δu T ) , δD, ( δθ S , δs )]= lim ǫ → ǫ ( l [( u S , u T ) + ǫ ( δu S , δu T ) , D + ǫδD, ( θ S , s ) + ǫ ( δθ S , δs )] − l [( u S , u T ) , D, ( θ S , s )]) , = R Ω δlδu S · δu S + δlδu T δu T + δlδD δD + δlδθ S δθ S + δlδs δs d S, (2.17)for all ( δu S , δu T ) ∈ X (Ω) s F (Ω) , δD ∈ Λ (Ω) , ( δθ S , δs ) ∈ F (Ω) × R . Geometrically, we interpret δlδu S · d x ⊗ d S asbeing a 1-form-density in Λ (Ω) ⊗ Λ (Ω), the dual space of vector fields X (Ω). δlδv T d S and δlδθ S d S are interpretedas being densities in Λ (Ω), whilst δlδD is interpreted as being a function in F (Ω), and δlδs ∈ R .Proceeding with Hamilton’s principle δS = 0, where δS = Z t t l [( u S , u T ) , D, ( θ S , s )] d t, (2.18)considering variations δ ( u S , u T ) = ∂∂t ( w S , w T ) + [( u S , u T ) , ( w S , w T )] , (2.19) δD d S = −L w S D d S, (2.20) δ ( θ S , s ) = −L ( w S ,w T ) ( θ S , s ) , (2.21)for perturbation-generating velocity fields ( w S , w T ) that vanish at t = t and t = t , we obtain0 = δS = Z t t Z Ω δlδu S · δu S + δlδu T δu T + δlδD δD + δlδθ S δθ S + δlδs δs d S d t, = Z t t Z Ω δlδu S · (cid:18) ∂∂t w S + [ u S , w S ] (cid:19) d S + δlδu T (cid:18) ∂∂t w T + L u S w T − L w S u T (cid:19) d S − δlδD L w S ( D d S ) − L ( w S ,w T ) θ S δlδθ S d S d t, = − Z t t Z Ω w S · (cid:18)(cid:18) ∂∂t + L u S (cid:19) δlδu S · d x ⊗ d S + δlδu T d u T ⊗ d S + δlδθ S d θ S ⊗ d S − D d δlδD ⊗ d S (cid:19) + Z Ω w T (cid:18)(cid:18) ∂∂t + L u S (cid:19) δlδu T d S + δlδθ S s d S (cid:19) d t, (2.22)where we have integrated by parts in time and space, and have used the identity Z Ω m · [ u, v ] ⊗ d S = − Z Ω v · L u S m · x ⊗ S, (2.23)for all u, v ∈ X (Ω), m · d x ⊗ d S ∈ Λ (Ω) ⊗ Λ (Ω), and where the Lie derivative of a one-form density is defined as L u S ( m · d x ⊗ d S ) = ( L u S m · d x ) ⊗ d S + m · d x ⊗ d S. (2.24)Since ( w S , w T ) are arbitrary, save for the endpoint conditions, we obtain (for sufficiently smooth solutions), (cid:18) ∂∂t + L u S (cid:19) δlδu S · d x ⊗ d S + δlδu T d u T ⊗ d S + δlδθ S d θ S ⊗ d S − D d δlδD ⊗ d S = 0 , (2.25) (cid:18) ∂∂t + L u S (cid:19) δlδu T d S + δlδθ S s d S = 0 . (2.26)It becomes useful to write δlδu S · x ⊗ d S = 1 D δlδu S · x ⊗ D d S, (2.27)in which case (cid:18) ∂∂t + L u S (cid:19) D δlδu S · x ⊗ D d S = (cid:18)(cid:18) ∂∂t + L u S (cid:19) D δlδu S · x (cid:19) ⊗ D d S, (2.28)after making use of the continuity equation for D d S . Similarly, we have (cid:18) ∂∂t + L u S (cid:19) D δlδu T D d S = (cid:18)(cid:18) ∂∂t + L u S (cid:19) D δlδu T (cid:19) D d S. (2.29)Hence we obtain (cid:18) ∂∂t + L u S (cid:19) D δlδu S · d x + 1 D δlδu T d u T + 1 D δlδθ S d θ S − d δlδD = 0 , (2.30) (cid:18) ∂∂t + L u S (cid:19) D δlδu T + 1 D δlδθ S s = 0 . (2.31)Translating back into vector calculus notation, we get (cid:18) ∂∂t + u S · ∇ + ( ∇ u S ) T · (cid:19) D δlδu S + 1 D δlδu T ∇ u T + 1 D δlδθ S ∇ θ S − ∇ δlδD = 0 , (2.32) (cid:18) ∂∂t + u S · ∇ (cid:19) D δlδu T + 1 D δlδθ S s = 0 . (2.33)The Lagrangian for the incompressible Euler-Boussinesq slice equations is l = Z Ω D (cid:0) | u S | + u T (cid:1) + Df u T x + gθ D (cid:18) z − H (cid:19) θ S + p (1 − D ) d S, (2.34)where f is the Coriolis parameter, g is the acceleration due to gravity, θ is a reference potential temperature, H isthe height of the vertical slice, and p is a Lagrange multiplier introduced to enforce that the density stays constant.In this case we have 1 D δlδu S = u S , D δlδu T = u T + f x ,δlδD = 12 (cid:0) | u S | + u T (cid:1) + f u T x − p + gθ θ S (cid:18) z − H (cid:19) , D δlδθ S = gθ (cid:18) z − H (cid:19) . (2.35)Substituting these formula and rearranging leads us to the Euler-Boussinesq vertical slice equations ∂ t u S + u S · ∇ u S − f u T ˆ x = −∇ p + gθ θ S ˆ z,∂ t u T + u S · ∇ u T + f u S · ˆ x = − gθ (cid:18) z − H (cid:19) s, ∇ · u S = 0 ,∂ t θ S + u S · ∇ θ S + u T s = 0 , (2.36)where ˆ x and ˆ z are the unit normals in the x - and z - directions, respectively. In addition to providing a variationalderivation of the incompressible Euler-Boussinesq slice model, [CH13b] also provided Lagrangians that lead to analpha-regularised Euler-Boussinesq slice model, and a compressible Euler slice model. Since these models have avariational derivation, they all have a conserved energy; they also have a conserved potential vorticity as we shallnow discuss.Returning to (2.30-2.31), [CH13b] made the following direct calculation to derive Kelvin’s circulation theoremand thus conservation of potential vorticity. Using the fact that the exterior derivative d commutes with ∂/∂t and L u S , we deduce that (cid:18) ∂∂t + L u S (cid:19) d θ S = − s d u T . (2.37)Combining with (2.31), we obtain that (cid:18) ∂∂t + L u S (cid:19) D δlδu T d θ S = 1 D δlδu T (cid:18) ∂∂t + L u S (cid:19) d θ S + d θ S (cid:18) ∂∂t + L u S (cid:19) D δlδu T , = − s (cid:18) D δlδu T d u T + 1 D δlδθ S d θ S (cid:19) , = (cid:18) ∂∂t + L u S (cid:19) D δlδu S · d x − d δlδD , (2.38)where we used (2.30) in the last equality. Hence, we obtain (cid:18) ∂∂t + L u S (cid:19) (cid:18) s D δlδu S · d x − D δlδθ S d θ S (cid:19) = d δlδD . (2.39)Integrating this around a closed curve C ( t ) that is moving with velocity u S , we obtain a Kelvin circulation theoremdd t I C ( t ) (cid:18) s D δlδu S · d x − D δlδθ S d θ S (cid:19) = 0 . (2.40)In the case of the incompressible Euler Boussinesq slice model, the circulation theorem readsdd t I C ( t ) ( su S − ( u T + f x ) ∇ θ S ) · d x = 0 . (2.41)Applying d to Equation (2.39), we obtain (cid:18) ∂∂t + L u S (cid:19) d (cid:18) s D δlδu T − D δlδθ S d θ S (cid:19) = 0 . (2.42)Finally combining with the continuity equation we obtain conservation of potential vorticity,( ∂ t + L u S ) q = 0 , (2.43) q = 1 D d (cid:18) s D δlδu S − D δlδu T d θ S (cid:19) . (2.44)In the case of the incompressible Euler-Boussinesq slice model this becomes q = s ∇ ⊥ · ( u S − ( u T + f x ) ∇ θ S ) . (2.45)The circulation theorem is curious, because usually in the presence of advected temperatures, we obtain a baroclinicsource term, so that circulation is only preserved on isosurfaces of θ S . In the slice model case, this baroclinic termcan be replaced by the Lie derivative of an additional quantity, so that we get a conservation law for any circulationloop. The new contribution of this paper is to show that these extra conservation laws arise from new relabellingsymmetries that exist in the slice model framework. In this section we describe the relabelling symmetry for slice models and compute the corresponding conservedquantities via Noether’s theorem. Relabelling symmetry in fluid dynamics is the statement that the referenceconfiguration for the Lagrangian flow map is arbitrary. In the slice framework, this means that we can arbitrarilyselect an alternative reference configuration, which can be transformed back to the original reference configurationby the slice map represented by the relabelling transformation ( ψ, g ) ∈ Diff(Ω) s F (Ω). After this change of basecoordinates, the Lagrangian flow map is transformed according to( φ, f ) ( φ, f )( ψ, g ) = ( φ ◦ ψ, f ◦ ψ + g ) . (3.1)Relabelling symmetries are such transformations that leave the initial data θ S, , D , s all invariant. In the groupvariable notation of previous sections, relabelling symmetries form a group G D ,θ S, ,s = { ( ψ, g ) ∈ Diff(Ω) s F (Ω) | ψ ∗ ( D d S ) = D d S and g = ( θ S, − ψ ∗ θ S, /s } . (3.2)To compute infinitesimal relabelling transformations, we consider a 1-parameter family of relabellings ( ψ ǫ , g ǫ )for ǫ >
0, with ( ψ , g ) = (Id , ψ ǫ , g ǫ ) = (Id ,
0) + ǫ ( v S , v T ) + O ( ǫ ) . (3.3)Then, δ ( φ, f ) = lim ǫ → ǫ (( φ ◦ ψ ǫ , f ◦ ψ ǫ + g ǫ ) − ( φ, f ))= ( ∇ φ · v S , ∇ f · v S + v T ) . (3.4)Following (2.5), we define ( w S , w T ) to be the unique element of X (Ω) s F (Ω) such that w S ◦ φ = ∇ φ · v S := δφ, w T ◦ φ = ∇ f · v S + v T := δf. (3.5)Time differentiation gives ∂∂t ( w S ◦ φ ) = ∂w S ∂t ◦ φ + u S ◦ φ · ( ∇ w S ) ◦ φ, ( ∂∂t w T ◦ φ ) = ∂w T ∂t ◦ φ + u S ◦ φ · ( ∇ w T ) ◦ φ. (3.6)On the other hand, ∂∂t ( ∇ φ · v S ) = ∇ ( u S ◦ φ ) · v S , = ( ∇ u S ) ◦ φ · ( ∇ φ · v S ) , (3.7)= ( ∇ u S ) ◦ φ · w S ◦ φ,∂∂t ( w T ◦ φ ) = ∇ ( u T ◦ φ ) · v S , = ( ∇ u T ) ◦ φ · ( ∇ φ · v S ) , = ( ∇ u T ) ◦ φ · w S ◦ φ. (3.8)Equating and composing with φ − , we obtain ∂∂t ( w S , w T ) + [( u S , u T ) , ( w S , w T )] = 0 . (3.9)This means that δu S = ˙ w S + [ u S , w S ] = 0 , (3.10) δu T = ˙ w T + u S · ∇ w T − w S · ∇ u T = 0 , (3.11) i.e. u S and u T are left invariant under relabelling transformations.To leave θ S invariant, the infinitesimal symmetries ( w S , w T ) also need to satisfy δθ S = − L ( w S ,w T ) ( θ S , s ) = − w S · ∇ θ S − sw T = 0 . (3.12)In the s = 0 case, this restricts w S to being tangential to the contours of θ , hence the baroclinic torque term.However, if s = 0, we can take any w S and then pick w T = − s w S · ∇ θ S , (3.13)i.e. any change in θ S caused by advection with w S can be corrected by a source term from w T . For this to work,we need equation (3.13) to be compatible with (3.11). This is verified by the following proposition. Proposition 1.
Let w S be a vector field with arbitrary initial condition, and let w T be a function satisfying (3.13) initially. Let ( w S , w T ) satisfy the particle relabelling symmetry conditions (3.10-3.11), and let θ evolve according to ∂θ S ∂t = − L ( u S ,u T ) ( θ S , s ) = − ( u S · ∇ θ S + su T ) . (3.14) Then θ S , w T satisfy (3.13) for all time.Proof. First note that L ( u S ,u T ) ( θ S , s ) is a Lie algebra action of X (Ω) s F (Ω) on F (Ω) × R , i.e. L ( u S ,u T ) L ( w S ,w T ) ( θ S , s ) − L ( w S ,w T ) L ( u S ,u T ) ( θ S , s ) = L [( u S ,u T ) , ( w S ,w T )] , (3.15)where [( u S , u T ) , ( w S , w T )] is the bracket on X (Ω) s F (Ω) defined in [CH13b], given by[( u S , u T ) , ( w S , w T )] = ([ u S , w S ] , u S · ∇ w T − w S · ∇ u T ) . Then we have ∂∂t L ( w S ,w T ) ( θ S , s ) = L ∂∂t ( w S ,w T ) ( θ S , s ) + L ( w S ,w T ) ( ˙ θ S , , = L ∂∂t ( w S ,w T ) ( θ S , s ) + L ( w S ,w T ) L ( u S ,u T ) ( θ S , s ) , = L ∂∂t ( w S ,w T ) ( θ S , s ) − L ( w S ,w T ) L ( u S ,u T ) ( θ S , s ) , = L ∂∂t ( w S ,w T ) ( θ S , s ) − L [( u S ,u T ) , ( w S ,w T )] ( θ S , s ) − L ( u S ,u T ) L ( w S ,w T ) ( θ S , s ) | {z } =0 , = L ∂∂t ( w S , w T ) + [( u S , u T ) , ( w S , w T )] | {z } =0 ( θ S , s ) = 0 , (3.16)as required. Corollary 2.
Let u S , u T , θ S , D solve the equations (2.25-2.26). Then the potential vorticity q (weakly) satisfies theLagrangian conservation law ∂q∂t + u S · ∇ q = 0 , (3.17) as a consequence of Noether’s theorem.Proof. Given initial condition D for density, we pick arbitrary ψ ∈ Λ (Ω), compactly supported in the interior ofΩ. Then we choose ψ as the solution of (cid:18) ∂∂t + L u S (cid:19) ψ = 0 , (3.18)with initial condition ψ . We then define w s via w s y D d S = d ψ. (3.19)Then (cid:18) ∂∂t + L u S (cid:19) w s y D d S = (cid:18) ∂∂t + L u S (cid:19) d ψ, (3.20)= d (cid:18) ∂∂t + L u S (cid:19) ψ = 0 , (3.21)and we deduce from the chain rule and Equation (2.11) that w s satisfies Equation (3.10). Further, Equation (3.19)implies that δ ( D d S ) = L w S ( D d S ) = d ( w S y ( D d S )) = 0 , (3.22)for all times, i.e. w s is a relabelling symmetry for D . We then choose w T = − s w S · ∇ θ S , (3.23)which defines a relabelling symmetry for θ S by Lemma 1.Next, we follow the steps of Noether’s Theorem, considering the variations in the action S under the rela-belling transformations generated by ( w S , w T ) defined above. Since these transformations leave ( u S , u T ), D and θ S invariant, the action does not change, and we get0 = δS = Z t t Z Ω δlδu S · δu S + δlδu T δu T + δlδD δD + δlδθ S δθ S + δlδs δs d S d t, = Z t t Z Ω δlδu S · (cid:18) ∂∂t w S + [ u S , w S ] (cid:19) d S + δlδu T (cid:18) ∂∂t w T + L u S w T − L w S u T (cid:19) d S − δlδD L w S ( D d S ) − L ( w S ,w T ) θ S δlδθ S d S d t, = (cid:20)Z Ω δlδu S · w S + δlδu T w T d S (cid:21) t = t t = t − Z t t Z Ω w S · (cid:18)(cid:18) ∂∂t + L u S (cid:19) δlδu S · d x ⊗ d S + δlδu T d u T ⊗ d S + δlδθ S d θ S ⊗ d S − D d δlδD ⊗ d S (cid:19) + Z Ω w T (cid:18)(cid:18) ∂∂t + L u S (cid:19) δlδu T d S + δlδθ S s d S (cid:19) d t, = (cid:20)Z Ω δlδu S · w S + δlδu T w T (cid:21) t = t t = t d S, (3.24)since ( u S , u T ), D and θ S solve (2.25-2.26). For sufficiently smooth solutions in time, we may consider the limit t → t , and we get0 = dd t Z Ω δlδu S · w S + δlδu T · w T d V, = dd t Z Ω (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ w S y d S, = dd t Z Ω D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ w S y D d S, = dd t Z Ω D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ d ψ, = Z Ω ∂∂t D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ d ψ + 1 D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ ∂∂t d ψ, = Z Ω ∂∂t D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ d ψ − D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) ∧ L u S d ψ, = − Z Ω ψ (cid:18) ∂∂t + L u S (cid:19) d 1 D (cid:18) δlδu S · d x − s δlδu T d θ (cid:19) , (3.25)which holds for arbitrary ψ , hence we deduce that0 = (cid:18) ∂∂t + L u S (cid:19) ( qD d S ) , = (cid:18) ∂q∂t + L u S q (cid:19) ( D d S ) , (3.26)hence the result (since D is positive).0 In this paper, we provided a new proof that the conserved potential vorticity in vertical slice models arises throughNoether’s Theorem upon consideration of the relabelling symmetries consisting of rearrangements of the verticalslice combined with transverse motion that restores the original structure of the potential temperature θ S . Theproof applies to a horizontally-periodic geometry with rigid top and bottom boundary but can be easily extendedto other slice geometries.A future direction will be to use this variational structure to build potential vorticity conserving vertical slicemodels, and to make further comparisons with the optimal transport formulation. It would also be interestingto use the relabelling transformations in this paper to derive balanced models using the variational asymptoticsapproach of [Oli06]. References [CH13a] C. J. Cotter and D. D. Holm. On Noethers theorem for the Euler–Poincar´e equation on the diffeomorphismgroup with advected quantities.
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