GEF1100–Autumn 2016 27.09.2016 GEF 1100 – Klimasystemet Chapter 7: Balanced flow Prof. Dr. Kirstin Krüger (MetOs, UiO) 1 Ch. 7 – Balanced flow 1. Motivation Ch. 7 2. Geostrophic motion – 2.1 Geostrophic wind 2.2 Synoptic charts 2.3 Balanced flows 3. Thermal wind equation* 4. Subgeostrophic flow: The Ekman layer 5.1 The Ekman layer* 5.2 Surface (friction) wind 5.3 Ageostrophic flow 5. Summary Lecture Outline Lecture 6. Take home message *With add ons. 2 1. Motivation Today‘s weather chart Where does the wind come from in Oslo? x Varmfront Kaldfront Okklusjon 3 www.yr.no 1. Motivation Today‘s 500 hPa (~5 km) weather map www.wetteronline.de GFS: Global Forecast System-Model 4 How is the wind blowing? Which forces are acting? 2. Geostrophic motion Scale analysis – geostrophic balance -1- First consider magnitudes of the first 2 terms in momentum eq. for a fluid on a 퐷퐮 1 rotating sphere (Eq. 6-43 + f 풛 × 풖 + 훻푝 + 훻훷 = ℱ) for horizontal 퐷푡 휌 components in a free atmosphere (ℱ=0,훻훷=0): 퐷퐮 휕푢 • +f 풛 × 풖 = + 풖 ∙ 훻풖 + f 풛 × 풖 퐷푡 휕푡 Large-scale flow magnitudes in atmosphere: 푈 푈2 fU 푢 and 푣 ∼ 풰, 풰∼10 m/s, length scale: L ∼106 m, 푇 퐿 time scale: 풯 ∼105 s, 풰 / 풯≈ 풰2/L ∼10-4 ms-2, −4 −4 −3 -4 -1 10 10 10 f45° ∼ 10 s 2 • Rossby number R0: - Ratio of acceleration terms (U /L) to Coriolis term (f U), - R0=U/f L - R0≃0.1 for large-scale flows in atmosphere -3 (R0≃10 in ocean, see Chapter 9) 5 2. Geostrophic motion Scale analysis – geostrophic balance -2- • Coriolis term is left (≙ smallness of R0) together with the pressure gradient term (~10-3): 1 f 풛 × 풖 + 훻푝 = 0 Eq. 7-2 geostrophic balance 휌 • It can be rearranged to (풛 × 풛 × 풖 = −풖): 1 풖 = 풛 × 훻푝 Eq. 7-3 geostrophic flow g 푓휌 푥 푦 푧 0 0 1 휕푝 휕푝 풛 × 훻푝 = = 푥 − + 푦 휕푝 휕푝 휕푝 휕푦 휕푥 휕푥 휕푦 휕푧 • In vector component form: 1 휕푝 1 휕푝 (푢 푣 )= − , Eq. 7-4 g, g 푓휌 휕푦 푓휌 휕푥 Note: For geostrophic flow the pressure gradient is balanced by the Coriolis term; to be approximately satisfied for flows of small R0. 6 2. Geostrophic motion Geostrophic flow (NH: f>0) isobar High: Low: clockwise flow anticlockwise flow Marshall and Plumb (2008) 7 2. Geostrophic motion Geostrophic flow 1 1 훿푝 풖 = 훻푝 = s: distance g 푓휌 푓휌 훿푠 Marshall and Plumb (2008) Note: The geostrophic wind flows parallel to the isobars and is strongest where the isobars are closest (pressure maps > winds). 8 2. Geostrophic motion Geostrophic wind L Isobars FP u g FC Balance H between: 1 풖 = 풛 × 훻푝 g 푓휌 FP = -FC 9 2. Geostrophic motion Geostrophic wind L Isobars FP u g Which rules can be laid down for the F geostrophic wind? C 1. The wind blows isobars parallel. 2. Seen in direction of the wind, the low pressure lies to the left. 3. The pressure gradient force and the Corolis force balance each other Balance H balance motion. between: The geostrophic wind closely describes the horizontal wind above the boundary FP = -FC layer! 10 2. Geostrophic motion Geostrophic flow – vertical component Vertical component of geostrophic flow, as defined by Eq. 7-3, is zero. This can not be deduced directly from geostrophic balance (Eq. 7-2). • Consider incompressible fluids (like the ocean): 휌 can be neglected, 푓~const. for planetary scales, then geostrophic wind components (Eq. 7-4) gives: 휕푢g 휕푣g + = 0 Eq. 7-5 ``Horizontal flow is non divergent.’’ 휕x 휕y 휕푤g Comparison with continuity equation (훻 ∙ 풖 = 0) => 푤 = 0 => = 0 g 휕z => geostrophic flow is horizontal. • For compressible fluids (like the atmosphere) 휌 varies, see next slide. 11 2. Geostrophic motion Geostrophic wind – pressure coordinates “p” We derive from Eq. 7.3 in pressure coordinates: g 풖 = 풛 × 훻 z Eq. 7-7 g 푓 푝 푝 g 휕푧 g 휕푧 (푢 푣 )= − , Eq. 7-8 g, g 푓 휕푦 푓 휕푥 휕푢 휕푣 훻푝 ∙ 풖g = + = 0 Eq. 7-9 Marshall and Plumb (2008) 휕x 휕y Geostrophic wind is non-divergent in pressure coordinates, if 푓~const. (≤ 1000 km). Note: Geopotential height (z) contours are streamlines of the geostrophic flow on pressure surfaces; geostrophic flow streams along z contours (as p contours on height surfaces) (Fig. 7-4). 12 2. Geostrophic motion 500 hPa wind and geopotential height (gpm), 12 GMT June 21, 2003 over USA g 풖 = 풛 × 훻 z Eq. 7-7 g 푓 푝 푝 g 휕푧 g 휕푧 (푢 푣 )= − , Eq. 7-8 g, g 푓 휕푦 푓 휕푥 5 knots 10 knots Note: Geopotential height (zGH≈z; see Chapter 5 Eq. 5.5) contours are streamlines of the geostrophic flow on pressure surfaces; geostrophic flow streams along z contours (as p contours on height surfaces) (Fig. 7-4). 13 Marshall and Plumb (2008) 2. Geostrophic motion Summary: The geostrophic wind ug • Geostrophic wind: “wind on the turning earth“: Geo (Greek) “Earth“, strophe (Greek) “turning”. • The friction force and centrifugal forces are negligible. • Balance between pressure gradient and Coriolis force → geostrophic balance • The geostrophic wind is a horizontal wind. • It displays a good approximation of the horizontal wind in the free atmosphere; not defined at the equator: - Assumption: u=ug v=vg ρ: density 1 1 p 1 p f = 2 Ω sinφ; ug zˆ p ug vg at equator f = 0 f f y f x at Pole f is maximum 14 Quiz How is the wind blowing? Where are the low and high pressure systems? Which forces are acting? Today‘s 500 hPa (~5 km) weather map www.wetteronline.de GFS: Global Forecast System-Model 15 How is the wind blowing? Which forces are acting? Quiz How is the wind blowing? Where are the low and high pressure systems? Which forces are acting? Today‘s 500 hPa (~5 km) weather map L L H www.wetteronline.de GFS: Global Forecast System-Model 16 How is the wind blowing? Which forces are acting? Quiz How is the wind blowing? Where are the low and high pressure systems? Which forces are acting? Tues 27.09.2016 12 UTC, GFS model L L L L H H www.wetteronline.de 17 2. Geostrophic motion The geostrophic wind ug . Geostrophic wind balance (GWB): Simultaneous wind and pressure measurements in the open atmosphere show that mostly the GWB is fully achieved. Advantage: The wind field can be determined directly from the pressure or height fields. Disadvantage: The horizontal equation of motion becomes purely diagnostic; stationary state. 18 2. Geostrophic motion W E Zonal mean zonal wind u (m/s) E W January average SPARC climatology E Pressure [hPa] Pressure 1000 Randel et al 2004 19 GEF1100–Autumn 2016 29.09.2016 GEF 1100 – Klimasystemet Chapter 7: Balanced flow Prof. Dr. Kirstin Krüger (MetOs, UiO) 20 Ch. 7 – Balanced flow 1. Motivation Ch. 7 2. Geostrophic motion – 2.1 Geostrophic wind 2.2 Synoptic charts 2.3 Balanced flows 3. Thermal wind equation* 4. Subgeostrophic flow: The Ekman layer 5.1 The Ekman layer* 5.2 Surface (friction) wind 5.3 Ageostrophic flow 5. Summary Lecture Outline Lecture 6. Take home message *With add ons. 21 2. Geostrophic motion Rossby number R0 @500hPa, 12 GMT June 21, 2003 R0 > 0.1 - R0 : Ratio of acceleration term (U/T ≈ U2/L) to Coriolis term (f U) - R0=U/f L - R0≃0.1 for large-scale flows in atmosphere Marshall and Plumb (2008) 22 2. Geostrophic motion (Other) Balanced flows • Geostrophic balance holds, when small positive R0 (~ 0.1). Pressure gradient force and Coriolis force are in balance. Marshall and Plumb (2008) • Gradient wind balance (R0∼1) Pressure gradient force, Coriolis force and Centrifugal force play a role. • Cyclostrophic wind balance (R0>1) Pressure gradient force and Centrifugal force play a role. 23 2. Geostrophic motion Gradient wind uG – 2 cases with same Fp Case 1: sub-geostrophic Case 2: super-geostrophic streamline L uG streamline uG Balance between the forces: Balance between the forces: Fpressure = -(FCoriolis+ FZ(centrifugal)) Fpressure+ FZ(centrifugal) = -FCoriolis 25 Kraus (2004) 2. Geostrophic motion Cyclostrophic wind Balance between: L FZ FP = -FZ - Physically appropriate solutions are orbits around a low pressure centre, that can run both cyclonically (counter clockwise) as well as anticyclonically (clockwise). - The cyclostrophic wind appears in small-scale whirlwinds (dust devils, tornados). 26 3. Thermal wind equation The thermal wind Geostrophic wind change with altitude z? →”Thermal wind” (∆zug) Applying geostrophic wind equation together with ideal gas law (Eq. 1-1 p=ρRT); insert finite vertical layer ∆z* with average temperature T, leads to: (*∆: finite difference) zˆ ug(z+Δz) T Δz Altitudez ug(z) 27 풖g: geostrophic wind vector, T: temperature 휌: density p: pressure; R: gas constant for dry air, g: gravity acceleration, 푓: Coriolis parameter, z: geometric height, 풛 : unit vector in z−direction 3. Thermal wind equation Thermal wind approximation zˆ In many cases: 1) ug <10 m/s; g or 2) strong horizontal u u (z z) u (z) z zˆ T temperature gradient; then the z g g g fT h first term can be neglected. zu g zˆ hT ug(z+Δz) z+Δz z T Δz Altitude z ug(z) Note: In this approximation, the wind change is always parallel to the average isotherms (lines of const. temperature). 28 3. Thermal wind equation Thermal wind approximation warm cold ∆z ug ∆z ug u (z) g ug (z) ug (z+∆z ) ug (z+∆z ) y cold warm cold air advection warm air advection x Which rules can I deduce from the thermal wind approximation? - The geostr.