Positivity for the curvature of the diﬀeomorphism group corresponding to the incompressible Euler equation with Coriolis force

We investigate the geometry of the central extension of the group of volume-preserving diﬀeomorphisms of the 2-sphere equipped with an L 2 -metric, for which geodesics correspond to solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misio lek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature. . . .. .. ..


Introduction
An example of a stable multiple zonal jet flow can be observed on Jupiter's surface. Despite attracting many researchers over the years, its mechanism has not yet been well understood. The incompressible 2D-Navier-Stokes equations on a rotating sphere form one of the simplest models of this flow. Williams [24] was the first to find that turbulent flow evolves into multiple jet flows for such models. However, he assumed a high degree of symmetry for the flow field. Later, Yoden and Yamada [26] and Nozawa and Yoden [14] made further progress. In particular, Obuse, Takehiro, and Yamada [15] calculated non-forced 2D-Navier-Stokes flow (without symmetry assumptions) on a rotating sphere, observing multiple zonal jet flows merging with each other, and finally, obtaining only two or three broad zonal jets remaining in the overall flow (see [16][17][18] for further progress). Therefore, it seems that we need to develop totally different ideas to clarify the existence of stable multiple zonal jet flow on the sphere. For recent developments in this field, see the work of Sasaki, Takehiro, and Yamada [19,20], who apply a spectral method to the linearized fluid equations. For a mathematical analysis on the resonant interaction of Rossby waves on a beta plane (approximating the curved surface as a 2D plane), see for example [7,25].
However, as far as the authors are aware, few mathematical studies have attempted to investigate the effect of the Coriolis force on a sphere (see [3] for one of the few), and hence our motivation for the present mathematical study becomes clarifying the mechanism underlying the Coriolis force on various manifolds. Let us explain more precisely. The Euler equations for incompressible flow on a two-dimensional sphere S 2 are expressed as follows: We drop the viscosity term because we only focus on the stability of the largest-scale zonal flow with large-scale perturbation. This is considered the simplest model for Jupiter (if we also add the Coriolis force). However, rigorously, Jupiter is not a sphere; a perceptible bulge appears around its equatorial middle and the poles are flattened (see [4] and [21, Table 4]). As a first step, we studied equation (1) (without the Coriolis force) not only on S 2 but also on a two-dimensional manifold M = M s , which is defined by for s ≥ 1 ( [22]). Note that M s is an ellipsoid if s > 1 and S 2 if s = 1. As for the sequence of study, in this paper, we investigate the effect of the Coriolis force using the same method in [22], namely, the Arnol'd method, which we outline below. Let G be a (infinite-dimensional) Lie group and g its Lie algebra with an inner product , . By right translation, we extend , to a right-invariant metric on G. Then, the geodesic equation on G is equivalent to the differential equation on g called the Euler-Arnol'd equation [23] (see Appendix A): where u denotes a curve on g, [, ] the Lie bracket of g, and [, ] * is defined by if it exists. For certain G, it is known that this differential equation becomes a physically and mathematically important differential equation. For example, if G is the group of volume preserving diffeomorphisms D µ (M ) of a compact Riemannian manifold M , (3) is the incompressible Euler equation (1), which was discovered by Arnol'd [1]. Moreover, (3) is the KdV equation if G is the Bott-Virasoro group D(S 1 ), which is the central extension of the diffeomorphism group D(S 1 ) of the unit circle (e.g., see [13,23]). Therefore, the geometry of a group with a right-invariant metric has attracted much attention. In particular, its sectional curvature is important because the structure of geodesics is closely related to it. For example, Misio lek showed that the non-positivity of the sectional curvature of G implies the Lagrangian instability of solutions in [11,Lem. 4.2]. The explicit calculation of the curvature appears in [1,9,13,27] where G is D µ (M ) for some M .
Given the direction of this study, the existence of a conjugate point on G also has attracted some attention because in a Lagrangian sense it can be thought of as the stability of the corresponding solution of the Euler-Arnol'd equation. Indeed, the existence of a conjugate 2/18 point p ∈ G on a geodesic γ(t) means that a family of geodesics that despite initially diverging have almost the same initial value of γ converge to p at some later moment. Definition 1.1 (Conjugate points). Let D be a Riemannian manifold and η(t) := exp p (tV ) a geodesic for some V ∈ T p D, for which exp p : T p D → D is the exponential map at p ∈ D. Then, we say that η(1) is a conjugate point or conjugate to p along η if the differential T V exp p : T V (T p D) → T η(1) D of the exponential map at V is not bijective. (For dim D = ∞, there are two reasons for a point to be conjugate to another; see [6].) We note that, in principle, greater positivity of the sectional curvature on G creates a conjugate point, and greater negativity implies the absence of a conjugate point. For a study of the existence of conjugate points, see for example [5,[11][12][13].
In [12], Misio lek calculated the second variation of a geodesic corresponding to a certain stationary solution X of (3) and showed the existence of a conjugate point along it if G is the diffeomorphism group of the flat torus T 2 . Moreover, he also revealed the importance of the value where Y ∈ g. Specifically, he essentially proved Fact 2.1, which states that the M C g X,Y > 0 ensures the existence of a conjugate point on G. We call this important value M C g X,Y the Misio lek curvature and study when it is positive or otherwise.
In this article, we calculate the Misio lek curvature for the incompressible Euler equation with the Coriolis force az (a > 0) on M s : where ⋆ denotes the Hodge operator. In this case, the solutions correspond to geodesics on the central extension D µ (M s ) of the group of volume-preserving diffeomorphisms of M s (see Section 3), for which the Lie algebra is identified with g ⊕ R. Our main result is the positivity of the Misio lek curvature M C g⊕R with respect to a west-facing zonal flow: for some function F . Moreover, if F ≤ 0, we call Z a west-facing zonal flow.
Note that the definition of the west-facing zonal flow is just for simplicity. See Corollary 4.3. Actually, we just need the positivity of the second term in the right-hand side of (47). This means that we can generalize Theorem 1.3 to cover oscillatory situations. Here, we say that a zonal flow Z = F (r)∂ θ is oscillating if the sets {r | F (r) > 0} and {r | F (r) < 0} are nonempty. For example, Theorem 1.3 is still true for any oscillating zonal flow Z = F (r)∂ θ if we only consider perturbations (Y, b) that are sufficiently large on the set {r | F (r) < 0} and |Y (r)| ≪ 1 on {r | F (r) > 0}. We also note that any zonal flow is a stationary solution of (1) and (6). Then, our main results are the following: 3/18 Theorem 1.3. Let Z be a nonzero west-facing zonal flow and a ∈ R >0 . Then we have The theorem states that for any west-facing zonal flow Z, the Misio lek curvature of Z regarded as a solution of (6) is greater than the Misio lek curvature regarded as a solution of (1), which can be understood as the stability of Z under Coriolis force. We note that the positivity of the Misio lek curvature directly implies the positivity of the sectional curvature on the corresponding group (see Definition B.4 and Lemma B.6 in Appendix). Therefore, the corollary implies the positivity of the sectional curvature on D µ (M s ) under some support condition.

Misio lek curvature
We next define the Misio lek curvature and explain its importance. We refer to [12,22].
Let G be an (infinite-dimensional) Lie group with right-invariant metric , , and g the Lie algebra of G. Then, we define the Misio lek curvature M C X, The initial importance of this value is that the positivity of M C directly implies that of the curvature (see Definition B.4 and Lemma B.6 in Appendix). Note that this formula for M C seems to be simpler than the general formula of the curvature on the group with right-invariant metric (see Lemma B.2). The main importance of M C is Fact 2.1 given below. In [12], this fact is proved for G being the group D s µ (T 2 ) of volume-preserving H s -diffeomorphisms of the 2-dimensional flat torus T 2 . (For D s µ (M ), where M is a compact n-dimensional Riemannian manifold, see also [22].) The essential point of the proof in [12] is that the inverse function theorem holds for the Riemannian exponential map exp : T e D s µ (M ) → D s µ (M ). Here, we say that the inverse function theorem holds for exp if exp is an isometry near X ∈ T e D s µ (M ) whenever the differential of exp is an isomorphism at X. Therefore, we obtain the following.
Fact 2.1. Suppose that there exists the (Riemannian) exponential map exp : g → G and the inverse function theorem holds for exp. Let X ∈ g be a stationary solution of the Euler-Arnol'd equation. Suppose that there exists Y ∈ g satisfying M C X,Y > 0. Then, there exists a point conjugate to the identity element e ∈ G along the geodesic corresponding to X on G.

Central extension of volume-preserving diffeomorphism group
We now briefly recall basics regarding the central extension of the volume-preserving diffeomorphism group by a Lichnerowicz cocycle. Our main references are [11,23]. 4/18 Let (M, g) be a compact n-dimensional Riemannian manifold and D µ (M ) the group of volume-preserving C ∞ -diffeomorphisms with L 2 -metric where µ is the volume form. We write g for the space of divergence-free vector fields on M , which is identified with the tangent space of D µ (M ) at the identity element. For a closed 2-form η, we define a Lichnerowicz 2-cocycle Ω on g by Consider a (n − 2)-form B satisfying η = ι B (µ) or, equivalently, where B × X := ⋆(B ∧ X). Note that this can be rewritten as where P : X(M ) → g is the projection to the divergence-free part. Then, the Euler-Arnol'd equation of D µ (M ) is Remark 3.1. This formula is slightly different from [23] because the sign convention differs.
To clarify, we present our conventions in Appendix.
We state the content of this section setting dim M = 2.
Proof. Note that u × B = u ⋆ B if dim M = 2. Moreover, the assertion of the Misio lek curvature follows from definition, (12) and (13).

M = M s case
We apply the results of Section 3 with M = M s . Recall Proposition 4.1. Let B := z and η := zµ. Then, there exists a group D µ (M r ) for which the Euler-Arnol'd equation is (17).
Thus, the proposition follows from Proposition 3.2.
Consider a "spherical coordinate" of M s : in such a way that c 2 (0) = 0, c 1 (r) > 0,ċ 2 (r) > 0, and thatċ 2 1 +ċ 2 2 = 1. Note that (c 1 , c 2 , d) = (cos(r), sin(r), π/2) for s = 1 (M 1 = S 2 ). Then, we obtain This implies and for X = X 1 ∂ r + X 2 ∂ θ and Y = Y 1 ∂ r + Y 2 ∂ θ , which are elements of g. Moreover, we have for a function f on M and u = u 1 ∂ r + u 2 ∂ θ . Recall that we call a vector field Z on M s a zonal flow if Z has the form for some function depending only on variable r. Moreover, if F ≤ 0, we call Z a west-facing zonal flow.

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Proof. Recall that B = z = c 2 (r) and that The last equality follows with dim M = 2. In contrast, This expression implies the existence of function f satisfying grad f = B ⋆ Z. Therefore, we have P (B ⋆ Z) = 0, which implies the first equality. For the second equality, we have Moreover, This is equal to Thus, a divergence-free Y implies By Stokes' theorem, this is equal to This completes the proof.

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Corollary 4.3. Let Z = F (r)∂ θ be a zonal flow. Then, we have Proof. This is a consequence of (18)

Final remark
Note that we do not know whether D µ (M s )-the existence of which is guaranteed by Proposition 4.

A.1. Sign conventions
To clarify our sign conventions, we briefly derive formula (16). All content in this section is well known. We refer to [8,Section 46] or [23, Section 2].
A.1.1. Right-invariant Maurer-Cartan form. Let G be a (possibly infinite-dimensional) Lie group and g its Lie algebra.
Definition A.1. The right-invariant Maurer-Cartan form ω is the g-valued 1-form on G defined by where r g −1 denotes the differential of the right translation map R g −1 (h) := hg −1 .
For X ∈ g, we write X R for the right-invariant vector fields on G with X R (e) = X. Note that Proof. By (A2), we have which completes the proof. Proof. Recall that This equation describes g-valued 2-forms and, therefore, holds for any U and V .
Corollary A.4. For smooth vector fields U, V on G, we have Proof. This is obvious from the preceding lemma. Proof. Consider the energy function of a curve η on G: For a proper variation η s of η, define c s (t) := r η −1 (η) = ω(η) ∈ g, X s (t) := ∂ s γ s (t), and x s (t) := ω(X s ) where ω is the right-invariant Maurer-Cartan form. Then, the first variation is Thus, This completes the proof.

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Definition A.6. We define the Euler-Arnol'd equation of G by for u : [0, t 0 ] → g. In other words, where ad u v = −[u, v] and ad u v, w = v, ad * u w .
A.1.3. Euler-Arnol'd equation of D µ (M ). Let D µ (M ) be the group of volume-preserving C ∞ -diffeomorphisms of a n-dimensional compact Riemannian manifold (M, g) with rightinvariant Riemannian metric where X, Y ∈ g := T e D µ (M ). Let P : X(M ) → g be the projection onto the divergence-free part, where X(M ) is the space of vector fields.
where ∇ M is the Levi-Civita connection on M .
Proof. The Koszul formula and the right-invariance imply This completes the proof using the Koszul formula on M and the right-invariance of , .
Lemma A.8. For X ∈ g, we have Proof. By the Koszul formula, we have This completes the proof.
or, equivalently, A.1.5. Formulae on Riemannian manifold. For the convenience of readers, we briefly review the formulae concerning Riemannian manifolds. Let (M, g) be a n-dimensional Riemannian manifold and µ the volume form. Write X p (M ) for the space of p-vector fields on M , and E q (M ) for the space of q-forms on M .
Definition A.14. Define , X : where ι is the interior derivative. Similarly, define , E : Proof. By Definition A.13 and A.14, we have This completes the proof.
Definition A. 16. We define the Hodge star operator ⋆ : X p (M ) → X n−p (M ) and ⋆ : Proof. The Koszul formula and right-invariance imply This completes the proof.
For simplicity, put Proof. By the properties of the Levi-Civita connection and right-invariance, we have Similarly, Moreover, Then, we have Note that [X, X] * = 0 implies that X is a stationary solution of the Euler-Arnol'd equation of G.
Lemma B.3. Let X ∈ g satisfying [X, X] * = 0 and η a geodesic corresponding to X ∈ T e G.
For Y ∈ g and f ∈ C ∞ (G) with f (e) = f (t 0 ) = 0 for some t 0 > 0, define a vector field Y along η by Y (η(t)) = f (t) · Y R (η(t)). Then, the second variation E ′′ of the energy function of η is Proof. Note that, by assumption,η = X R . The general formula for the second variation of the energy function implies For the first term, we have Thus, In addition, This completes the proof.
Definition B.4. Let X ∈ g satisfying [X, X] * = 0. Define the Misio lek curvature M C X,Y := M C G X,Y by for Y ∈ g. 16/18