ROSSBY WAVE PROPAGATION (PHH lecture 4) The presence of a gradient of PV (or q.-g. p.v.) allows slow wave motions generally called Rossby waves These waves arise through the Rossby restoration mechanism, which limits displacements in the direction perpendicular to PV gradients 2 simple examples: (from 2-D vortex dynamics) 1. Piecewise constant vorticity vorticity gradient concentrated along certain curves. For demonstration of Rossby restoration mechanism in this case, see Kida computer demonstration [Sufficiently strong vortex patch can resist external strain field] PHH 4 / 1 2. Uniform flow U on a β plane: disturbances of the form exp{ ikx + ily iωt} have dispersion relation ω = ku βk / (k 2 + l 2 ) phase propagation w ward relative to flow advection by mean flow due to p.v. gradient c g = U + β(k2 l 2 ) (k 2 + l 2 ) 2, 2βkl (k 2 + l 2 ) 2 group velocity c g c g y > 0 c p phase lines c g y < 0 c p c g phase crests slope westward in direction of group propagation PHH 4 / 2
Radiation of Rossby-waves away from a localised region of eddies Stream function Vorticity (from Rhines, 1977, in The Sea) PHH 4 / 3 Effects of stratification: single-layer models For a single-layer model, linearised about a state of rest on a β-plane, the q.-g. p.v. equation reduces to ψ t 2 ψ 2 L D + β ψ x = 0 dispersion relation (for disturbances e ikx+ ily iωt ) ω = βk k 2 + l 2 + L D 2 ω(k) βl D /2 (l = 0) ω k = 1 8 βl 2 D L D 1 k 2 cm s 1 for first ω k = βl 2 baroclinic mode D 10 years to cross Atlantic c p < 0 c g < 0 long waves > 0 short waves asymmetry has important implications for structure of ocean circulation (e.g. response to forcing, reflection from lateral boundaries) PHH 4 / 4
Rossby wave propagation mechanism (scales >> L D ) Relative vorticity may be neglected, conservation of q.-g. p.v. βy f 0 h constant following each particle H [more generally f 0 + βy H + h conserved] H L H Contours of q.-g. p.v. H L = surface elevation = surface depression Velocity field follows from geostrophic balance and tends to move the pattern of displacement towards the west PHH 4 / 5 Single-layer models can be used to make useful predictions about the behaviour of disturbances in the real (3-D) ocean. E.g. dominant part of many lowfrequency motions is the 1st baroclinic mode shape is such as to raise or lower all density surfaces in the thermocline, weak (but non-negligible) displacement of the surface itself. Gravity-wave speed for 1st baroclinic mode c 1 3 ms 1 L D 3 ms 1 2Ωsin φ long Rossby-wave speed 10 2 cos φ sin 2 φ ms 1 [different expression for Rossby-wave speed near equator fastest would be 1 3 c 1 1ms 1 ] Propagation speed decreases with latitude Possible mechanism for multi-year persistence of El Nino effects? (Jacobs et al., 1994, Nature) PHH 4 / 6
Time-longitude sections of time-filtered sea-level in Pacific Ocean (Chelton & Schlax, 1996) 39 o N 32 o N 21 o N Phase speed decreases with latitude, but is larger than predicted by simplest, resting ocean, theory. See Killworth et al (1997) for improved theory PHH 4 / 7 Ocean Rossby waves arising from El Nino (Jacobs et al, 1994) Model simulations August 82 January 83 May 83 Eastward propagating equatorial Kelvin wave hits coast of S. America (a) and propagates n'ward along coast as Kelvin wave (b). Disturbance then peels away from coast as w'ward propagating Rossby wave (c). PHH 4 / 8
Multi-year evolution (Jacobs et al, 1994) March 84- Feb 85 May 87- April 88 April 92- March 93 Rossby wave propagates w'ward, faster at low latitudes, giving perturbation to Kuroshio extension current (E of Japan) 10 yrs after original El Nino. Stationary Rossby wave patterns in upper troposphere Averages of displayed quantities for December, January, February 1979 89 PHH 4 / 9 height = geopotential height on a pressure surface What is needed to explain this picture? PHH 4 / 10
Charney and Eliassen s solution: (1-D model see computer demo) Topography calculated as mean of values at 40, 42, 44, 46, 48 and 50 Wind at surface = 0.4 wind at 500mb L D = 8 km σ = 0.25 corresponds to Ekman layer with vertical eddy diffusivity 10m 2 s 1 Charney and Eliassen found that l the meridional wavenumber could not be taken as zero they chose l = π / (25 of latitude) Good agreement considering simplicity of model but response sensitive to choice of friction, and to assumed value of L D. 2-D Rossby wave dispersion on a sphere PHH 4 / 11 Localised source propagation in latitude and longitude Analyse using direct solution of shallow-water equations or using ray tracing techniques based on slowly-varying wave theory needs scale separation between waves and medium (see e.g. Lighthill, Waves in Fluids, 4.5) Hoskins and Karoly (1981): ray-tracing on sphere, basic flow in solid body rotation PHH 4 / 12
Grose and Hoskins (1979) SWE on sphere response to circular mountain at 30 N, with basic flow solid body rotation Disturbance vorticity (Note split wave train) Total streamfunction Disturbance geopotential (Note small amplitude near equator) PHH 4 / 13 Hoskins and Karoly (1981) Ray-tracing on a sphere with basic state corresponding to NH 300 mb wintertime flow Source at 15 N + = every 2 days Ray paths roughly follow great circles (details depend on basic state) + = every phase change of π Waves are refracted by variations in basic state PHH 4 / 14
Teleconnection patterns from Wallace and Gutzler (1981) One-point correlations with +, in 500 mb height field, DJF for 28 years, with climatological monthly averages removed. PHH 4 / 15 Lagged 1-point correlations (5 day means in 500 mb height) from Wallace and Hsu, 1983 Five days earlier Five days later Wave pattern is quasi-stationary, but clear sense of group propagation, particularly for pattern in SW hemisphere See Ambrizzi & Hoskins (1997) and references for more recent theoretical work, including 3-D simulations PHH 4 / 16
A simple model of forced threedimensional Rossby waves u 0 ( z) β-plane channel rigid walls at y = 0, L y = L y = 0 Linearise about flow in longitudinal direction ψ = u 0 (z) y + ψ Linearised q.-g. p.v. equation t + u 0( z) 2 f ψ xx + ψ yy + 0 x N 2 ψ z 2 f + β 0 N u ψ 2 0 z z x = 0 1 4 442 4 4 43 Background gradient of q.-g. potential vorticity z Boundary conditions: ψ x = 0 y = 0, L Side t + u 0 ( z) ψ x z = N 2 2 f u h 0 (0) 0 x Lower PHH 4 / 17 Steady state: assume h = Re( ˆ h 0 sin πy L eikx ) hence ψ = Re( ˆ ψ (z)sin πy L eikx ) Ordinary differential equation for vertical structure: f 0 2 N 2 ˆ ψ z z β f 2 0 u N 2 0 z z + k 2 π 2 u 0 L 2 ψ ˆ = 0 Lower boundary condition: ψ ˆ z = N2 h ˆ 0 ( z = 0) f 0 Are solutions oscillatory in the vertical? (upward propagating waves) Trapped in the vertical? PHH 4 / 18
[Neglecting vertical curvature of wind profile] Consider β u 0 (z) k2 + π2 L 2 > 0 vertical propagation < 0 vertical trapping Vertical propagation only if 0 < u 0 (z) < β k 2 + π 2 / L 2 i.e. only if winds eastward and weak. For given u 0 (z) > 0 (and < βl 2 /π 2 ), only sufficiently long waves propagate in vertical. Summertime stratospheric circulation (westward flow around pole negligible large-scale wave disturbances) PHH 4 / 19 (from FUB observations) 4 July 1989: 30 mb ~ 24 km PHH 4 / 20
Increase in scale of waves with height 30mb (FUB) ( 23 km) 50 mb ( 20 km) (ECMWF) (ECMWF) ( 9 km) 300mb (All 21 Jan 1983) Vertical propagation in another simple flow u 0 piecewise linear PHH 4 / 21 N 2 piecewise constant (Held, 1983) Solution for δ-function mountain Height at which trapping occurs PHH 4 / 22
Vertical Rossby wave propagation in the equatorial ocean (Kessler and McCreary, 1993) Annual harmonic observed isotherm displ. 4 N PHASE AMPLITUDE LONGITUDE Theory: needs equatorial β-plane, see e.g. Gill or Andrews, Holton & Leovy Dispersion relation l=latitudinal mode number ( =1 in this case ) βk ω = k 2 + (2l + 1)β m / N Nk (2l + 1) m (long-wave) PHH 4 / 23 Annual harmonic observed isotherm displ. Latitudinal cross-section (170 E) PHASE AMPLITUDE PHH 4 / 24
Breaking of Rossby waves Just as gravity waves propagate on an internal density gradient, and then, when they break, they overturn density surfaces, so Rossby waves propagate on a (quasi-horizontal) gradient of potential vorticity and then when they break they overturn (in a sideways sense) potential vorticity contours. Breaking gravity waves stir up vertical density gradients and lead to 3-D turbulence. Breaking Rossby waves stir up quasi-horizontal potential vorticity gradients and may lead to a sort of turbulence (quasi-2-d). Effect on mean flow: From an averaged viewpoint, a breaking Rossby wave redistributes potential vorticity on an isentropic surface (or σ θ surface). There is therefore a corresponding change in the wind and density or temperature fields (by invertibility). Waves have exerted a force on the mean state. PHH 4 / 25