A simple model of topographic Rossby waves in the ocean

Transcription

1 A simple model of topographic Rossby waves in the ocean V.Teeluck, S.D.Griffiths, C.A.Jones University of Leeds April 18, 2013 V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

2 1.Outline of talk 1 Introduction and motivation 2 Analytical model for topographic Rossby waves 3 Comparison of analytical and numerical work V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

3 Introduction and motivation 2. Surface signature of topographic Rossby waves es over the Newfoundland Shelf-break / 103 tive ours km Elev: Modes 1 & 2 in O 1 Assimilative t=12 hours km Fig 2d from Wright and Xu (2004, A-O) 0 ours V.Teeluck (University of Leeds) 800 Topographic Rossby waves (TRW) in t=16 the ocean hours April 18, /

4 Introduction and motivation 3. Motivation Environmental conditions Rapid depth change from continental shelf through continental slope to deep ocean Temperature variations (warm sea surface), salinity variations, sea ice... Tidal forcing, wind forcing, discharge from rivers With the aid of applied mathematics Start from equations of motion describing underlying physics (conservation of mass, conservation of momentum, etc.) Derive a reduced model incorporating what is thought to be the essential physics Derive wave-like solutions in this model, which can be compared to observations and used to predict behaviour elsewhere Mathematical approach gives understanding of the structure of the solutions V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

5 Introduction and motivation 4. Navier-Stokes Equations Figure : local plane called the f-plane at a fixed latitude θ 0 where u = (u, v, w). u t + (u )u + 2Ω u = 1 ρ p + g + ν 2 u u = 0 V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

6 Introduction and motivation 4. Navier-Stokes Equations Figure : local plane called the f-plane at a fixed latitude θ 0 u t + (u )u + 2Ω u = 1 ρ p + g + ν u u = 0 where u = (u, v, w). V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

7 Introduction and motivation 5. Laplace s tidal equations (1776) Figure : thin shell Hydrostatic in z, so p = p 0 ρg(z h(x, y, t)) giving u t v t fv = g h x + fu = g h y h t + (Hu) = 0 where f = 2Ω sin θ 0, θ 0 (constant) is latitude, H is undisturbed depth. V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

8 Introduction and motivation 6. The topographic Rossby wave (TRW) mechanism Motion of columns of water Step profile DQ Dt = 0 where Q = f + v x u y H+h Longuet-Higgins, JFM (1969) ω = f H 1 H 2 H 1 + H 2 V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

9 7. Scaling analysis Introduction and motivation Scale lengths by L, velocities by U, and depth by H 0. To target TRWs, we scale time by f 1 and h by ful/g: ( L L R ) 2 h t + u t v = h x v t + u = h y ( (Hu) x + (Hv) y ) = 0 where L R = gh 0 / f is the Rossby radius of deformation and a typical value is 2000km. Since L L R, we drop the h/ t term. V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

10 Analytical model for topographic Rossby waves 8. Wave-like solutions u t fv = g h x, v t So we introduce a streamfunction + fu = g h y, (Hu) x + (Hv) y = 0. Hu = ψ y Hv = ψ x. When H = H(y), we seek wave-like solutions of the form ) ψ(x, y, t) = Re (H 1/2 φ(y)e i(kx ωt). Eliminating h gives [ 1 φ + 2 H H 3 4 ( H H ) 2 + fk ω ( H H ) k 2 ] φ = 0. (1) This is an eigenvalue problem for ω given k and suitable boundary conditions. V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

11 Analytical model for topographic Rossby waves 9. A model slope: H = H 0 + H tanh(y/l s ) Writing ŷ = y/l s, (1) becomes d 2 [ ( φ H dŷ 2 + k 2 L 2 s H tanh ŷ Figure : hyperbolic tangent profile ( H H ) 2 sech 2 ŷ fk HL ) ] s sech 2 ŷ φ = 0. (2) ωh V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

12 Analytical model for topographic Rossby waves 10. Small topography assumption: H H 0 d 2 ( φ dŷ 2 + k 2 L 2 s + fk HL ) s sech 2 ŷ φ = 0, to leading order. ωh 0 In an unbounded domain, this has solutions of the form φ = sech kls ŷ p C n tanh n ŷ n=0 When this series terminates, we obtain an eigenvalue relation for ω: 4fkL s H ω = 2 ] where n N 0. H 0 [(2n kL s ) 1 So there are infinitely many eigenvalues as n ω 0 when n = 0 and L s 0, we obtain Longuet-Higgins (1969) solution ω f H sgn(k) H 0. V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

13 Analytical model for topographic Rossby waves 11. Cross-shore structure Modes with even/odd n are symmetric/anti-symmetric in ŷ n = 0, φ = sech kls ŷ n = 1, φ = sech kls ŷ tanh ŷ [ n = 2, φ = 1 3 ] 4 (4 + 3 kl s ) tanh 2 ŷ sech kls ŷ Figure : Fixed H 1 with H = 0.5km and H = 3km V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

14 Comparison of analytical and numerical work 12. Comparison of analytical/numerical work Figure : kl s = 1 H 1 = 4km L s = 50km V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

15 Comparison of analytical and numerical work 13. Why do analytics/numerics match so well? n = 0 mode Rearranging (2) so that we have the leading order solution on the left gives, [ ] d 2 φ dŷ + k 2 L 2 2 s + fk HLs sech 2 ŷ φ ωh 0 ( H = H 0 Expanding ω and φ in powers of H/H 0 we have we find tanh ŷ + 3( H)2 4H 2 0 [ ( ) ( ) 2 ] H H ω = ω a 1 + a 2 + H 0 H 0 a 1 = 0 and a 2 = ) sech 2 ŷ sech 2 ŷ φ + (3) kl s (2 + kl s ) 2 2 (3 + 2 kl s ) (1 + kl s ) 2 (4) V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

16 Comparison of analytical and numerical work 14. Comparing analytical/numerical results for a 2 a 2analytical = kl s (2 + kl s ) 2 ( ) ( ω H 2 (3 + 2 kl s ) (1 + kl s ) 2, a 2 numerical = 1 / ω 0 H 0 ) 2. Note that a 2 < Figure : a 2 against kl s for step values 50m to 400m V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

17 Figure : Comparison of TRW dispersion relations V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19 Comparison of analytical and numerical work 15. TRWs with a shelf and coast Coastal step formula (Larsen 1969, Journal of Marine Research): ω = f (H 2 H 1 )(1 e 2 k L ) H 2 (1 e 2 k L ) + H 1 (1 + e 2 k L ).

18 Comparison of analytical and numerical work 16. Cross-shore structure with a coastline TRW n=0, H 0 = 2150 m, H = 1850 m, k = m 1 V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19

19 Comparison of analytical and numerical work 17. Summary Topographic Rossby waves are likely to be excited in the ocean above steep topographic slopes. We studied the form of these waves above a smooth slope with a tanh profile. There exist infinitely many topographic Rossby waves. Analytical predictions (based on a small topographic amplitude expansion) for frequency and cross-slope structure are excellent. These same predictions also work well in the presence of a coastline, providing the wave is sufficiently trapped above the continental slope. Future work will be directed towards understanding how these trapped barotropic waves may generate radiating internal waves, at tidal frequencies. V.Teeluck (University of Leeds) Topographic Rossby waves (TRW) in the ocean April 18, / 19