Rossby Wave Propagation in the Tropics and Midlatitudes Kevin Mitchell, David Muraki Department of Mathematics Simon Fraser University Burnaby, British Columbia Kevin Mitchell, David Muraki Rossby Wave Propagation Midlatitudes & Tropics N midlatitudes φ tropics midlatitudes S 0.00 ū 0.25 midlatitudes: strong jetstream westerlies & strong Coriolis tropics: weak tradewind easterlies & weak Coriolis small Rossby # Ro → slow Rossby waves Kevin Mitchell, David Muraki Rossby Wave Propagation Midlatitudes & Tropics φ N S 0.00 ū 0.25 midlatitudes: strong jetstream westerlies & strong Coriolis tropics: weak tradewind easterlies & weak Coriolis small Rossby # Ro → slow Rossby waves Kevin Mitchell, David Muraki Rossby Wave Propagation Three Regimes of Rossby Wave Theory midlatitude (strong shear) critical latitudes no propagation through tropics associated with non-linear absorption (Dickinson 70 / Lindzen 71) tropical (negligible shear) equatorial β-plane (dry) infinite # modes, all wavelength scales (Matsuno 66) global long wavelength waves (planetary scale) finite # of modes observed in climate data (Madden 07) how can these three perspectives be reconciled? Kevin Mitchell, David Muraki Rossby Wave Propagation Rossby Waves in Rotating Shallow Water 2d wind ~u(λ, φ, t) & thin fluid layer height h(λ, φ, t) momentum & mass equations Ro2 D~u − sin φ r̂ × ~u + ∇h = 0 Dt Dh + h∇ · ~u = 0 Dt disturbance wave modes ~u = ū(φ) + ûn (φ)ei(mλ−ωt) linear eigenvalue problem for dispersion relation ω(m, n) small Ro → slowest waves (small ω) are Rossby waves Kevin Mitchell, David Muraki Rossby Wave Propagation Dispersion Relation (no tradewinds) gravity no tradewinds discrete spectrum of wave frequencies ω̄(m, n) Kelvin 0 Rossby slow Rossby waves N ū Rossby-gravity 50 fast gravity waves φ ω̄ 50 gravity 0 5 10 m S 15 20 Kevin Mitchell, David Muraki Rossby Wave Propagation Dispersion Relation gravity no tradewinds 50 50 Kelvin ω̄ Rossby Rossby N ū Rossby-gravity 50 N ū Rossby-gravity 50 gravity gravity 0 continuous spectrum Kelvin 0 φ 0 φ ω̄ tradewinds gravity 5 10 m S 15 20 Kevin Mitchell, David Muraki 0 5 Rossby Wave Propagation 10 m S 15 20 Dispersion Relation (with tradewinds) tradewinds gravity same discrete mode structure ω̄ 50 continuous spectrum Kelvin 0 Rossby added continuous spectrum N ū Rossby-gravity φ 50 gravity 0 Kevin Mitchell, David Muraki 5 Rossby Wave Propagation 10 m S 15 20 Dispersion Relation (with tradewinds) tradewinds same discrete mode structure ω̄ 50 0 added continuous spectrum 50 0 Kevin Mitchell, David Muraki 5 Rossby Wave Propagation 10 m 15 20 Summary of Rossby Wave Modes ω̄ 5 0 5 10 0 n =2 5 n =1 10 n =0 m 15 discrete Rossby modes - (infinite # like β-plane) global longwave modes longest wavelength discrete modes observed (Madden 2007) & computed (Kasahara 1980) tropical shortwave modes very shortwave in tropics, but longwave in midlatitudes negligible north-south group velocity in the tropics continuous Rossby spectrum critical latitudes (no waves in the tropics) analogue of midlatitude waves Kevin Mitchell, David Muraki Rossby Wave Propagation 20 Summary of Rossby Wave Modes ω̄ 5 0 5 10 0 n =2 5 n =1 10 n =0 m 15 discrete Rossby modes - (infinite # like β-plane) global longwave modes longest wavelength discrete modes observed (Madden 2007) & computed (Kasahara 1980) tropical shortwave modes very shortwave in tropics, but longwave in midlatitudes negligible north-south group velocity in the tropics continuous Rossby spectrum critical latitudes (no waves in the tropics) analogue of midlatitude waves Kevin Mitchell, David Muraki Rossby Wave Propagation 20 Discrete Rossby Mode Latitude Structure (m = 6) global tropical û(φ) n =25 û(φ) n =6 S φ N S φ global planetary-scale waves tropical shortwave latitude dependent wavelength Kevin Mitchell, David Muraki Rossby Wave Propagation N Discrete Rossby Mode Latitude Structure (m = 6) tropical û(φ) û(φ) n =25 S φ S N φ N n =25 û(φ) n =6 û(φ) no tradewinds tradewinds global n =6 S φ N S φ global planetary-scale waves tropical shortwave latitude dependent wavelength Kevin Mitchell, David Muraki Rossby Wave Propagation N Discrete Rossby Mode Latitude Structure (m = 6) global tropical û(φ) n =25 û(φ) n =6 102 n →∞ 101 100 10-1 S n =6 φ S N φ N local wavenumber local wavenumber S N φ 102 n →∞ 101 100 10-1 S n =25 φ midlatitude maximum wavenumber unbounded wavenumber at equator Kevin Mitchell, David Muraki Rossby Wave Propagation N Continuous Rossby Mode Latitude Structure (m = 6) continuous spectrum tropical û(φ) û(φ) n =25 φ 102 101 100 10-1 S φ S N N local wavenumber local wavenumber S N φ 102 n →∞ 101 100 10-1 S n =25 φ N exceed discrete spectrum maximum wavenumber critical latitudes: wave speed matches background flow Kevin Mitchell, David Muraki Rossby Wave Propagation Wave Packet Dynamics û(φ) S 0 180 N t =0 S û(φ) N 360 λ ū φ φ tropical N S 0 180 N t =0 360 λ critical latitude ū φ φ continuous ū φ S t =0 N φ global N S û(φ) S 0 180 λ Kevin Mitchell, David Muraki 360 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 1.6 1.2 0.8 0.4 0.0 0.4 0.8 1.2 1.6 circular mask northern hemisphere southward group velocity 3.2 2.4 1.6 0.8 0.0 0.8 1.6 2.4 3.2 Rossby Wave Propagation Conclusion: Three Rossby Wave Behaviours û(φ) S 0 180 N t =9.74145 S û(φ) N 360 λ ū φ φ tropical N S 0 180 N t =19.9112 360 λ critical latitude ū φ φ continuous ū φ S t =3.14183 N φ global N S û(φ) S 0 180 λ Kevin Mitchell, David Muraki 360 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 1.6 1.2 0.8 0.4 0.0 0.4 0.8 1.2 1.6 3.2 2.4 1.6 0.8 0.0 0.8 1.6 2.4 3.2 planetary scale pass through tropics shortwave in tropics longwave in midlat. very slow in tropics critical latitudes small scale midlat. cannot enter tropics Rossby Wave Propagation Linear eigenvalue problem Ro2 ∂ m (cos φ u) v̂ + Ro η̂, Ro ω̄û = sin φ − cos φ ∂φ cos φ ∂ η̂ Ro2 ω̄v̂ = sin φ + Ro2 2 tan φ u û + Ro , ∂φ m Ro ∂ Ro2 ω̄ η̂ = Ro hû − (cos φ hv̂), cos φ cos φ ∂φ 2 reduced frequency ω̄(φ) = ω − m Kevin Mitchell, David Muraki u cos φ Rossby Wave Propagation Methods Numerical spectral eigenvalue computation on sphere careful convergence analysis de-singularised equations for continuous spectrum WKB/Geometric Optics as Ro → 0 zonal flow with fast phase, slow amplitude perturbation ~ (φ)ei(mλ+ρ(φ)−ωt)/Ro ~u = u(φ) + U latitude phase ρ(φ) satisfies Ro-independent eikonal ODE m m sin2 φ m2 sin φ 2 ω− + u + ρ + h =0 φ cos φ cos2 φ cos φ h h φ . . . midlatitude replacements . . . m →m cos φ ρφ → ` sin2 φ → f2 h h sin φ h →β φ . . . gives Rossby dispersion relation (Hoskins, Karoly 1981) ω = mu − mβ m2 + `2 + f 2 ~ (φ) amplitude functions are also determined the U Kevin Mitchell, David Muraki Rossby Wave Propagation zerocrossings of û(φ) no tradewinds tradewinds 20 20 15 15 n 25 n 25 10 10 5 5 0 −π/2 0 0 φ π/2 Kevin Mitchell, David Muraki −π/2 0 φ Rossby Wave Propagation π/2 Eigenvalues for fixed m −ω̄ 101 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 0 no tradewinds Nφ =16 Nφ =32 no tradwinds Nφ =64 tradewinds Nφ =512 Nφ =1024 Nφ =2048 10 20 n 30 algebraic decay with n doubling resolution doubles number of converged solutions tradewinds 40 50 Kevin Mitchell, David Muraki exponential decay with n doubling resolution increases converged solution by 4 Rossby Wave Propagation
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