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GEF 1100 – Klimasystemet Chapter 7: Balanced Flow

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GEF 1100 – Klimasystemet Chapter 7: Balanced Flow GEF1100–Autumn 2014 30.09.2014 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. Taylor-Proudman theorem 4. Thermal wind equation* 5. Subgeostrophic flow: The Ekman layer 5.1 The Ekman layer* 5.2 Surface (friction) wind 5.3 Ageostrophic flow 6. Summary Lecture Outline Lecture 7. Take home message *With add ons. 2 1. Motivation Motivation x Oslo wind forecast for today 12–18: 2 m/s light breeze, east-northeast x www.yr.no 3 2. Geostrophic motion Scale analysis – geostrophic balance Consider magnitudes of first 2 terms in momentum eq. for a fluid on a rotating 퐷퐮 1 sphere (Eq. 6-43 + f 풛 × 풖 + 훻푝 + 훻훷 = ℱ) for horizontal components 퐷푡 휌 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, 5 2 -4 -2 푇 time scale: 풯 ∼10 s, U/ 풯≈ 풰 /L ∼10 ms , 퐿 f ∼ 10-4 s-1 10−4 10−4 10−3 45° • Rossby number: - Ratio of acceleration terms (U2/L) to Coriolis term (f U), - R0=U/f L - R0≃0.1 for large-scale flows in atmosphere -3 (R0≃10 in ocean, Chapter 9) 4 2. Geostrophic motion Scale analysis – geostrophic balance • 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 • With 풛 × 풛 × 풖 = −풖 it can be rearranged to: 1 풖 = 풛 × 훻푝 Eq. 7-3 geostrophic flow g 푓휌 푥 푦 푧 휕푝 휕푝 풛 × 훻푝 = 0 0 1 = 푥 − + 푦 휕푦 휕푥 푢푔 푣푔 0 • In component form: 1 휕푝 1 휕푝 (푢 푣 )= − , Eq. 7-4 g, g 푓휌 휕푦 푓휌 휕푥 Note: For geostrophic balance the pressure gradient is balanced by the Coriolis term; to be approximately satisfied for flows of small R0. 5 2. Geostrophic motion Geostrophic flow (NH: f>0) High: Low: clockwise flow anticlockwise flow Marshall and Plumb (2008) 6 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). 7 2. Geostrophic motion Geostrophic flow – vertical component Defined as horizontal flow (z–component of Eq. 7-3 is zero), which can not be deduced from geostrophic balance (Eq. 7-2). • For 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. 8 2. Geostrophic motion Geostrophic wind – pressure coordinates “p” 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 훻 ∙ 풖 = + = 0 Eq. 7-9 푝 g 휕x 휕y Marshall and Plumb (2008) Geostrophic wind is non-divergent in pressure coordinates, if 푓~const. (≤ 1000 km). Note: 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). 9 2. Geostrophic motion 500 hPa wind and geopotential height (gpm), 12 GMT June 21, 2003 5 knots 10 knots 10 Marshall and Plumb (2008) 2. Geostrophic motion Today’s 500 hPa GH (dam) map L x x Oslo H www.ecmwf.int 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 12 2. Geostrophic motion Geostrophic wind L Isobars FP u g FC Balance H between: 1 풖 = 풛 × 훻푝 g 푓휌 FP = -FC 13 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! 14 2. Geostrophic motion The geostrophic wind ug . Geostrophic wind balance (GB): Simultaneous wind and pressure measurements in the open atmosphere show that mostly the GB 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. (→ weather forecast) 15 2. Geostrophic motion W E Zonal mean zonal wind u (m/s) E W January average SPARC climatology E Pressure [hPa] Pressure 1000 16 2. Geostrophic motion Rossby number @500hPa, 12 GMT June 21, 2003 R0 > 0.1 Marshall and Plumb (2008) 17 2. Geostrophic motion (Other) Balanced flows When geostrophic balance does not hold, then R0 ≥ 1: • 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. Marshall and Plumb (2008) 18 2. Geostrophic motion Gradient wind uG – 2 cases 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 19 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). 20 3. Taylor-Proudman theorem Taylor-Proudman theorem - see Ch. 10 & 11 For geostrophic motions and incompressible flows (휌=const.): Taylor-Proudman theorem: (Ω ∙ 훻) 풖 = 0 Eq.7-14 (Ω ∙ 훻 gradient operator in direction of Ω, i.e., for 푧 : 휕풖 = 0 Eq. 7-15 휕푧 Note: For geostrophic motion, if the fluid is incompressible (휌 const.), the flow is two-dimensional and does not vary in the direction of the rotation vector. Marshall and Plumb (2008) 21 4. Thermal wind equation The thermal wind Geostrophic wind change with altitude z? →”Thermal wind” (∆zug) Applying geostrophic wind equation together with ideal gas law; insert finite vertical layer ∆z* with average temperature T, leads to: (*∆: finite difference) zˆ ug(z+Δz) T Δz Altitudez ug(z) 22 4. 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 isotherm (lines of const. temperature). 23 4. Thermal wind equation Thermal wind approximation cold ∆z ug ∆z ug u (z) g ug (z) ug (z+∆z ) ug (z+∆z ) y cold warm cold air advection x warm air advection Which rules can I deduce from the thermal wind approximation, if ug<10 m/s ? - The geostr. wind change with height is parallel to the mean isotherms, - whereby the colder air remains to the left on the NH, - geostrophic left rotation with increasing height for cold air advection, - geostrophic right rotation with increasing height for warm air advection. 25 4. Thermal wind equation - Summary Thermal wind: u ag T g u (z z) u (z) 휕풖 푎g g g g = 풛 × 훻푇 Eq. 7-18 z f y 휕푧 푓 vg ag T vg (z z) vg (z) z f x 푎: Thermal expansion coefficient (Ch. 4) g: Gravity acceleration Geostrophic wind: 1 p u 1 g f y 풖g= 풛 × 훻푝 Eq. 7-3 푓휌 1 p 푔 vg 풖 = 풛 × 훻 푧 Eq. 7-7 f x g 푓 푝 푝 Compare: - Thermal wind is determined by the horizontal temperature gradient, - geostrophic wind through the horizontal pressure gradient. 26 4. Thermal wind equation Thermal wind - examples W W C W C W W Marshall and Plumb (2008) 27 4. Thermal wind equation Temperature (°C) @500hPa, 12 GMT June 21, 2003 x x Oslo 28 Marshall and Plumb (2008) 4. Thermal wind equation W C W 29 Marshall and Plumb (2008) 4. Thermal wind equation Stratospheric polar vortex Marshall and Plumb (2008) 30 GEF1100–Autumn 2014 02.10.2014 GEF 1100 – Klimasystemet Chapter 7: Balanced flow Prof. Dr. Kirstin Krüger (MetOs, UiO) 31 Ch. 7 – Balanced flow 1. Motivation Ch. 7 2. Geostrophic motion 2.1 Geostrophic wind – 2.2 Synoptic charts 2.3 Balanced flows 3. Taylor-Proudman theorem 4. Thermal wind equation 5.
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