1:45 PM - 3:15 PM
[O11-P86] Atmospheric dynamics on a Riemannian manifold and considerations of approximations leading to the Primitive equations
Keywords:Atmospheric dynamics, Primitive equations, Riemannian manifold
Purpose
Atmospheric dynamics is usually described as an ideal fluid on a sphere. However, many celestial bodies such as large asteroids (e.g., Sylvia) and rotating planets have irregular shapes or non-uniform curvature. Even Earth is sometimes approximated as a cylinder to analyze perturbations. In this study, we treat such irregular celestial bodies as Riemannian manifolds and mathematically derive the equations governing atmospheric motion on them. To obtain primitive equations, curvature terms must be discarded. We identify the conditions under which this approximation holds. While detailed examples are omitted here due to space, we apply our framework to various bodies including ellipsoids, cylinders, and cones.
Method
A non-spherical planet, Planet, is regarded as a submanifold in three-dimensional Euclidean space. Its boundary ∂Planet is given a Riemannian metric, forming a 2D manifold. At each point, we define a local basis of the tangent bundle T∂Planet and set ∂/∂x^3 as the cross product ∂/∂x^1 × ∂/∂x^2, establishing a 3D atmospheric coordinate system E_Planet.
Rotation is introduced as a linear map preserving the metric and orientation, yielding inertial forces. Using the velocity field X_t, angular velocity Ω, covariant derivative ∇, density ρ, pressure p, and total force G (gravity and centrifugal), we derive Equation 1. Due to its complexity, we assume: G lies in (T∂Planet)^⊥, metric terms involving ∂/∂x^1 and ∂/∂x^2 vanish, and ∂/∂x^1 points longitudinally. With component notation using superscripts, we obtain Equation 2. The first two terms on the left of Equation 2 represent curvature effects, originating from covariant derivatives involving Christoffel symbols.
Computation
Equation 2 enables approximation toward the primitive equations. Let U be horizontal wind speed, L the horizontal scale, and coordinate components dimension [L]. The curvature term coefficient u^j u^k has dimension [L^-1], matching the inverse of radius of curvature A. Thus, acceleration scales as U^2/L, while curvature terms scale as U^2/A.
On Earth, A >> L, justifying neglect of curvature. However, when L ≈ A or larger, curvature becomes significant. For example, if neglected terms must be under 1/10 of acceleration, the condition 10L > A must hold. Hence, primitive equations require sufficiently large A; while locally valid, they fail globally on bodies with strong curvature variation.
Conclusion
By defining bases on a manifold and introducing rotation, we derived Equation 1. With reasonable assumptions, this reduces to Equation 2, generalizing curvature terms in atmospheric dynamics.
We also presented conditions for using primitive equations in terms of horizontal scale L and curvature radius A. Though omitted here, various geometries (e.g., ellipsoids, cylinders) confirm that atmospheric equations on manifolds can be systematically derived via Christoffel symbols.
Acknowledgments
I am deeply grateful to Professor Yoshihiko Mimatsu of Chuo University for answering my many questions on fluid dynamics on closed compact Riemannian manifolds.
Atmospheric dynamics is usually described as an ideal fluid on a sphere. However, many celestial bodies such as large asteroids (e.g., Sylvia) and rotating planets have irregular shapes or non-uniform curvature. Even Earth is sometimes approximated as a cylinder to analyze perturbations. In this study, we treat such irregular celestial bodies as Riemannian manifolds and mathematically derive the equations governing atmospheric motion on them. To obtain primitive equations, curvature terms must be discarded. We identify the conditions under which this approximation holds. While detailed examples are omitted here due to space, we apply our framework to various bodies including ellipsoids, cylinders, and cones.
Method
A non-spherical planet, Planet, is regarded as a submanifold in three-dimensional Euclidean space. Its boundary ∂Planet is given a Riemannian metric, forming a 2D manifold. At each point, we define a local basis of the tangent bundle T∂Planet and set ∂/∂x^3 as the cross product ∂/∂x^1 × ∂/∂x^2, establishing a 3D atmospheric coordinate system E_Planet.
Rotation is introduced as a linear map preserving the metric and orientation, yielding inertial forces. Using the velocity field X_t, angular velocity Ω, covariant derivative ∇, density ρ, pressure p, and total force G (gravity and centrifugal), we derive Equation 1. Due to its complexity, we assume: G lies in (T∂Planet)^⊥, metric terms involving ∂/∂x^1 and ∂/∂x^2 vanish, and ∂/∂x^1 points longitudinally. With component notation using superscripts, we obtain Equation 2. The first two terms on the left of Equation 2 represent curvature effects, originating from covariant derivatives involving Christoffel symbols.
Computation
Equation 2 enables approximation toward the primitive equations. Let U be horizontal wind speed, L the horizontal scale, and coordinate components dimension [L]. The curvature term coefficient u^j u^k has dimension [L^-1], matching the inverse of radius of curvature A. Thus, acceleration scales as U^2/L, while curvature terms scale as U^2/A.
On Earth, A >> L, justifying neglect of curvature. However, when L ≈ A or larger, curvature becomes significant. For example, if neglected terms must be under 1/10 of acceleration, the condition 10L > A must hold. Hence, primitive equations require sufficiently large A; while locally valid, they fail globally on bodies with strong curvature variation.
Conclusion
By defining bases on a manifold and introducing rotation, we derived Equation 1. With reasonable assumptions, this reduces to Equation 2, generalizing curvature terms in atmospheric dynamics.
We also presented conditions for using primitive equations in terms of horizontal scale L and curvature radius A. Though omitted here, various geometries (e.g., ellipsoids, cylinders) confirm that atmospheric equations on manifolds can be systematically derived via Christoffel symbols.
Acknowledgments
I am deeply grateful to Professor Yoshihiko Mimatsu of Chuo University for answering my many questions on fluid dynamics on closed compact Riemannian manifolds.