Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere

Transcription

1 Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere Gordon E. Swaters University of Alberta, Canada gswaters@ualberta.ca, c.math.ualberta.ca/gordon CMOS Congress, Montreal, Canada 2012

2 Outline of talk. Introduction Derivation of nonlinear planetary-geostrohic model General steady-state solution Some general properties Example solution with upslope and downslope groundings Properties of the example solution Conclusions Swaters G. E., Geophys. Astrophys. Fluid Dynamics, submitted, 2012

3 Outline Intro Model derivation Steady-state solution General properties Example solution Example properties Conclusions Physical geometry Sverdrup vorticity balance predicts equatorward ow in source region. Geostrophically-balanced grounded equatorward ow over sloping topography. Basin length scales suggest planetary e ects important.

4 Spherical reduced-gravity shallow water equations u t + uu λ R cos θ + vu θ R v t + uv λ R cos θ + vv θ R u v tan θ R v 2 tan θ R 2Ω v sin θ = + 2Ω u sin θ = g 0 h t + 1 R cos θ [(h u) λ + (h v cos θ) θ ] = 0, p = ρ 1 g 0 (h + h B ). g 0 R cos θ (h + h B ) λ, R h θ,

5 Introduce the scalings λ = L R e λ, t = R V e t, (u, v) = V (h, h B, p) = 2ΩVL g 0 Assuming typical scales L eu, ev, R e h, eh B, g 0 ρ 1 ep, V ' m/s, L ' 10 5 m and g 0 ' m/s 2, suggests that the time, zonal velocity and abyssal height scalings are given by, respectively, R V ' 20 years, LV R ' m/s and 2ΩVL g 0 ' 145 m.

6 One gets (after dropping the tildes) ε δ 2 u t + uu λ cos θ + v u θ ε v t + uv λ cos θ + v v θ u v tan θ v sin θ = 1 cos θ (h + h B ) λ, v 2 tan θ + u sin θ = h θ, h t + 1 cos θ [(h u) λ + (h v cos θ) θ ] = 0, where ε and δ are, respectively, the Rossby number and aspect ratio ε = V 2ΩL ' 10 4 and δ = L R ' 10 2, and where the dynamic pressure is given by p = h + h B.

7 Thus, to leading order in ε, the model reduces to v = u = 1 sin θ h θ, 1 sin θ cos θ (h + h B ) λ, sin 2 θ h t + tan θ h Bλ h θ h h λ = h Bλ h. This is just the PV equation t + (u, v) 1 cos θ λ, θ sin θ = 0, h Automatically ensures that the kinematic condition on a grounding λ = eλ (θ, t) is satis ed, i.e., 1 e t + (u, v) cos θ λ, θ λ (θ, t) λ = 0 on λ = eλ (θ, t).

8 Steady-state solution h (λ, θ) will be determined by tan θ h θ subject to the boundary condition The solution is given by h h Bλ h λ = h, h (λ, θ 0 ) = h 0 (λ). h (λ, θ) = sin θ sin θ 0 h 0 (τ), h B (τ) + sin θ 0 sin θ h 0 (τ) = h B (λ). sin θ 0 The streamlines are co-parallel with the characteristics p h B (λ) + h (λ, θ) = h B (τ) + h 0 (τ).

9 General Properties Westward intensi cation. Along the characteristics dθ dλ = h0 B (λ) sin θ 0 τ=constant cos θ h 0 (τ) > 0. Groundings are constant w.r.t. θ h λ e (θ), θ = 0 () eλ (θ) = λ where h 0 (λ ) = 0. Abyssal height decreases lim h (λ, θ) = 0. θ!0 But the meridional transport is constant w.r.t. θ T m Z λ2 λ 1 h (λ, θ) v (λ, θ) cos θ dλ = 1 sin θ 0 No shock forms provided v (λ, θ 0 ) < 0. Z λ2 λ 1 h 0 (τ) h 0 B (τ) dτ.

10 We take Example solution h B (λ) = s(λ 5a/4), H 1 λ h 0 (λ) = 2 /a 2 for jλj < a, 0 for jλj > a, where s = 1.92 ( in m/m), H = 1.38 (200 m), a = 2 (170 km) and θ 0 = π/3. The solution is given by h (λ, θ) = sin θ sin θ 0 H 1 τ 2 /a 2 for jτj < a, 0 for jτj > a, with τ (λ, θ) given by 8 >< p s+ s 2 +4H (sin θ 0 sin θ)[sλ+h (sin θ 0 sin θ)/ sin θ 0 ]/(a 2 sin θ 0 ) 2H (sin θ 0 sin θ)/(a τ = 2 sin θ 0, ) >: for jλj < a, and λ for jλj > a..

11 Example solution properties h 0 v h B (λ) + h 0 (λ) 6 8 v (λ, θ 0 ) h τ (λ, θ) h (λ, θ) h (θ)

12 v v (λ, θ) v (θ) u u (λ, θ) u (θ)

13 T m v (λ, θ) h (λ, θ) T m (θ) Tz u (λ, θ) h (λ, θ) T z (λ)

14 Conclusions A nonlinear planetary-geostrophic model has been derived for hemispheric-scale grounded abyssal ow along zonally-sloping topography. The solution exhibits westward intensi cation as the ow moves equatorward. The abyssal current height decreases to zero and the velocities become unbounded (but the volume uxes remain nite) as the current ows toward the equator. The meridional volume transport is constant and equatorward with respect to latitude. The zonal volume transport is not constant but westward (but is zero along the groundings). No shock forms in the solution if the ow is everywhere equatorward along the northern boundary condition. Further work is required to determine the transport streamlines as the ow encounters the equator.