Yue-Kin Tsang School of Mathematics and Statistics University of St - PowerPoint PPT Presentation

Ellipsoidal vortices beyond the quasi-geostrophic approximation Yue-Kin Tsang School of Mathematics and Statistics University of St Andrews David G .Dritschel and Jean N. Reinaud

Rotating, Continuously Stratified Flows D � u u = − 1 ρ ( z ) + ρ ′ ( � D t + f 0 ˆ ∇ Φ + b ˆ ρ ( � x ) = ρ 0 + ¯ x ) k × � k ρ 0 b = − ρ − 1 0 g ρ ′ D b D t + N 2 w = 0 ω = ∇ × � � u ∇ · � u = 0 rotating: f -plane approximation, � 2 f 0 ˆ Ω = 1 k ρ stratified: constant buoyancy frequency, N 2 = − ρ − 1 0 g d ¯ d z σ = f 0 N = 0 . 1 potential vorticity anomaly: ω + f 0 ˆ · ∇ b + N 2 ˆ Q ≡ � k k ω + 2 � = 1 + q ∼ ( � Ω ) · ∇ ρ N 2 f 0

Computational Domain and Initial Conditions periodic three-dimensional domain: L × L × H small aspect ratio: H = σ L initial conditions: an ellipsoidal volume of constant PV anomaly q 0 in a near balanced state aspect ratios λ and µ : λ = a ( a < b ) b c µ = √ σ ab

Numerical methods u h } → { q , � change of variables: { b , � A h } � ω h / f 0 + ∇ h b / f 2 A h = � 0 equations of motion: D q D t = 0 D � A h k × � A h = N ( � D t + f 0 ˆ A h , q ) q is materially conserved ⇒ equations can be solved efficiently by the Contour-Advective Semi-Lagrangian (CASL) algorithm (Dritschel & Viúdez 2003 , JFM 488 , 123-150)

Beyond the QG approximation In the limit of asymptotically strong rotation and stratification, geostrophic-hydrostatic balance leads to the quasi-geostrophic (QG) approximation D q QG = 0 D t � A h : a measure of the leading order imbalance in contrast to the QG system, the ellipsoid is not an exact solution to the full non-hydrostatic system extend previous works on ellipsoidal vortices in the QG approximation (much less computational intensive) problem parameters: q 0 , λ and µ

Effects of horizontal aspect ratio λ : q 0 = 0.5

Effects of vertical aspect ratio µ : q 0 = 0.5

Various phases for q 0 = 0.5 unstable 2.4 2.2 2.0 shape oscillation 1.8 1.6 µ 0 1.4 tumbling 1.2 1.0 0.8 quasi-stable 0.6 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ 0

Stability and vortex geometry: q 0 = 0.5 2.4 2.2 2.0 1.8 1.6 µ 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ 0 oblate (small µ ) vortices are more stable vortices with close to circular cross-section (large λ ) are more stable

Stability and the Rossby number q 0 q 0 = 0.5 2.4 2.2 2.0 1.8 1.6 µ 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.3 0.4 0.6 0.7 0.8 0.5 λ 0

Stability and the Rossby number q 0 q 0 = 0.25 2.4 2.2 2.0 1.8 1.6 µ 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.3 0.4 0.6 0.7 0.8 0.5 λ 0

Stability and the Rossby number q 0 q 0 = - -0.25 2.4 2.2 2.0 1.8 1.6 µ 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.3 0.4 0.6 0.7 0.8 0.5 λ 0

Stability and the Rossby number q 0 q 0 = - -0.5 2.4 2.2 2.0 1.8 1.6 µ 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.3 0.4 0.6 0.7 0.8 0.5 λ 0

Stability and the Rossby number q 0 q 0 = 0.5 q 0 = 0.25 2.4 2.4 2.2 2.2 2.0 2.0 1.8 1.8 1.6 1.6 µ 0 µ 0 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ 0 λ 0 q 0 = - -0.25 q 0 = - -0.5 2.4 2.4 2.2 2.2 2.0 2.0 1.8 1.8 1.6 1.6 µ 0 µ 0 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ 0 λ 0 For a given ( λ 0 , µ 0 ), stability generally decreases with q 0 cyclones are more stable than anti-cyclones

Where do the unstable vortices go? q 0 = 0 . 5 q 0 = 0 . 25 2.4 2.4 2.2 2.2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 µ q 0 = − 0 . 25 q 0 = − 0 . 5 2.4 2.4 2.2 2.2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 λ

Vortex rotation rate Ω q 0 = 0 . 5 q 0 = 0 . 25 0.07 0.035 0.06 0.03 0.05 0.025 0.04 0.02 0.03 0.015 0.02 0.01 0.01 0.005 0 0 0.5 1 1.5 2 0.5 1 1.5 2 � Ω � q 0 = − 0 . 25 q 0 = − 0 . 5 0 0 − 0.02 − 0.01 − 0.04 − 0.02 − 0.06 − 0.03 − 0.08 − 0.04 − 0.1 0.5 1 1.5 2 0.5 1 1.5 2 2.5 � µ �

Summary study the stability and evolution of an ellipsoidal vortex in a rotating stratified fluid using the full Boussinesq’s equations oblate vortices with almost circular cross-section are the most stable increase in Rossby number q 0 enhances stability cyclones are more stable than anti-cyclones the vortex rotation rate scales with the vertical aspect ratio