GEF 1100 Klimasystemet. Chapter 7: Balanced flow
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1 GEF1100 Autumn GEF 1100 Klimasystemet Chapter 7: Balanced flow Prof. Dr. Kirstin Krüger (MetOs, UiO) 1
2 Lecture Outline Ch. 7 Ch. 7 Balanced flow 1. Motivation 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 6. Take home message *With add ons. 2
3 1. Motivation Today s weather chart Where does the wind come from in Oslo? x Varmfront Kaldfront Okklusjon 3
4 1. Motivation Today s 500 hpa (~5 km) weather map GFS: Global Forecast System-Model How is the wind blowing? Which forces are acting? 4
5 2. Geostrophic motion Scale analysis geostrophic balance -1- First consider magnitudes of the first 2 terms in momentum eq. for a fluid on a rotating sphere (Eq Du + f z u + 1 p + Φ = F) for horizontal Dt ρ components in a free atmosphere (F=0, Φ=0): Du Dt +f z u = u t U T u u + f z u U 2 L 10 4 fu 10 3 Large-scale flow magnitudes in atmosphere: u and v U, U 10 m/s, length scale: L 10 6 m, time scale: T 10 5 s, U / T U 2 /L 10-4 ms -2, f s -1 Rossby number R 0 : - Ratio of acceleration terms (U 2 /L) to Coriolis term (f U), - R 0 =U/f L - R for large-scale flows in atmosphere (R in ocean, see Chapter 9) 5
6 2. Geostrophic motion Scale analysis geostrophic balance -2- Coriolis term is left ( smallness of R 0 ) together with the pressure gradient term (~10-3 ): f z u + 1 p = 0 Eq. 7-2 geostrophic balance ρ It can be rearranged to (z z u = u): u g = 1 z p Eq. 7-3 geostrophic flow fρ z p = x y z p x p y p In vector component form: z (u g, v g )= 1 p, 1 p fρ y fρ x = x p y + y p x Eq. 7-4 Note: For geostrophic flow the pressure gradient is balanced by the Coriolis term; to be approximately satisfied for flows of small R 0. 6
7 2. Geostrophic motion Geostrophic flow (NH: f>0) isobar High: clockwise flow Low: anticlockwise flow Marshall and Plumb (2008) 7
8 2. Geostrophic motion Geostrophic flow u g = 1 p = 1 δp fρ fρ δs s: distance 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
9 2. Geostrophic motion Isobars Geostrophic wind L FP u g F C H Balance between: u g = 1 z p fρ F P = -F C 9
10 2. Geostrophic motion Isobars Geostrophic wind L FP u g Balance between: F C H Which rules can be laid down for the geostrophic wind? 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 motion. The geostrophic wind closely describes the horizontal wind above the boundary layer! F P = -F C 10
11 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): ρ can be neglected, f~const. for planetary scales, then geostrophic wind components (Eq. 7-4) gives: u g + v g = 0 Eq. 7-5 ``Horizontal flow is non divergent. x y Comparison with continuity equation ( u = 0) => w g = 0 => w g z = 0 => geostrophic flow is horizontal. For compressible fluids (like the atmosphere) ρ varies, see next slide. 11
12 2. Geostrophic motion Geostrophic wind pressure coordinates p We derive from Eq. 7.3 in pressure coordinates: u g = g f z p p z Eq. 7-7 (u g, v g )= g f z y, g f z x Eq. 7-8 Marshall and Plumb (2008) p u g = u x + v y = 0 Eq. 7-9 Geostrophic wind is non-divergent in pressure coordinates, if f~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
13 2. Geostrophic motion 500 hpa wind and geopotential height (gpm), 12 GMT June 21, 2003 over USA u g = g f z p p z z (u g, v g )= g, g f y f z x 5 knots 10 knots Note: Geopotential height (z GH 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). Marshall and Plumb (2008) 13
14 2. Geostrophic motion Summary: The geostrophic wind u g 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=u g v=v g u g 1 p f ẑ u g 1 p f y v g 1 f p x ρ: density f = 2 Ω sinφ; at equator f = 0 at Pole f is maximum 14
15 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 GFS: Global Forecast System-Model 15 How is the wind blowing? Which forces are acting?
16 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 GFS: Global Forecast System-Model 16 How is the wind blowing? Which forces are acting?
17 Quiz How is the wind blowing? Where are the low and high pressure systems? Which forces are acting? L Tues UTC, GFS model L L L H H 17
18 2. Geostrophic motion The geostrophic wind u g 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
19 2. Geostrophic motion Pressure [hpa] W E Zonal mean zonal wind u (m/s) January average SPARC climatology E E W 1000 Randel et al
20 GEF1100 Autumn GEF 1100 Klimasystemet Chapter 7: Balanced flow Prof. Dr. Kirstin Krüger (MetOs, UiO) 20
21 Lecture Outline Ch. 7 Ch. 7 Balanced flow 1. Motivation 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 6. Take home message *With add ons. 21
22 2. Geostrophic motion Rossby number R 12 GMT June 21, 2003 R 0 > R 0 : Ratio of acceleration term (U/T U 2 /L) to Coriolis term (f U) - R 0 =U/f L - R for large-scale flows in atmosphere Marshall and Plumb (2008) 22
23 2. Geostrophic motion (Other) Balanced flows Geostrophic balance holds, when small positive R 0 (~ 0.1). Pressure gradient force and Coriolis force are in balance. Marshall and Plumb (2008) Gradient wind balance (R 0 1) Pressure gradient force, Coriolis force and Centrifugal force play a role. Cyclostrophic wind balance (R 0 >1) Pressure gradient force and Centrifugal force play a role. 23
24 2. Geostrophic motion Gradient wind u G 2 cases with same F p Case 1: sub-geostrophic Case 2: super-geostrophic streamline L u G streamline u G Balance between the forces: F pressure = -(F Coriolis + F Z(centrifugal) ) Kraus (2004) Balance between the forces: F pressure + F Z(centrifugal) = -F Coriolis 25
25 2. Geostrophic motion Cyclostrophic wind L F Z Balance between: F P = -F Z - 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
26 Altitude z 3. Thermal wind equation The thermal wind Geostrophic wind change with altitude z? Thermal wind ( z u g ) 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) ẑ u g (z+δz) T Δz u g (z) u g : geostrophic wind vector, T: temperature ρ: density p: pressure; R: gas constant for dry air, g: gravity acceleration, f: Coriolis parameter, z: geometric height, z: unit vector in z direction 27
27 3. Thermal wind equation Altitude z Thermal wind approximation ẑ In many cases: 1) u g <10 m/s; or 2) strong horizontal temperature gradient; then the first term can be neglected. z u g u g ( z z) u g ( z) g ft z zˆ h T z u g zˆ h T u g (z+δz) z+δz T Δz u g (z) z Note: In this approximation, the wind change is always parallel to the average isotherms (lines of const. temperature). 28
28 3. Thermal wind equation y Thermal wind approximation warm z u g cold z u g u g (z) u g (z) u g (z+ z ) u g (z+ z ) cold cold air advection warm warm air advection x Which rules can I deduce from the thermal wind approximation? - 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. 29
29 y 3. Thermal wind equation Thermal wind approximation z u g cold z u g isotherm u g (z) u g (z) u g (z+ z ) u g (z+ z ) cold cold air advection x warm warm air advection Which rules can I deduce from the thermal wind approximation, if u g <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. 30
30 3. Thermal wind equation - Summary Thermal wind: u g z = ag f z T Eq a: Thermal expansion coefficient (Ch. 4) g: Gravity acceleration u z v g g z u v g g ( z z) u ( z z) v g g ( z) ( z) ag f ag f T x T y Geostrophic wind: u g = 1 z p Eq. 7-3 fρ u g = g z f p p z Eq. 7-7 u g v g 1 p f y 1 p f x Compare: - Thermal wind is determined by the horizontal temperature gradient, - geostrophic wind through the horizontal pressure gradient. 31
31 3. Thermal wind equation Thermal wind examples C W C W W 90S 90N W W Marshall and Plumb (2008) 32
32 3. Thermal wind equation Temperature ( 12 GMT June 21, 2003 cold air warm air x x Oslo 33 Marshall and Plumb (2008)
33 3. Thermal wind equation W E Westerly wind Easterly wind Warm Cold C W 70N 20N 34 Marshall and Plumb (2008)
34 3. Thermal wind equation Stratospheric polar vortex Marshall and Plumb (2008) 35
35 4. Sub-geostrophic flow Sub-geostrophic flow: The Ekman layer Large scale flow in free atmosphere and ocean is close to geostrophic and thermal wind balance. Boundary layers have large departures from geostrophy due to friction forces. Ekman layer: - Friction acceleration F becomes important, - roughness of surface generates turbulence in the first ~1 km of the atmosphere, - wind generates turbulence at the ocean surface down to first 100 meters. Sub-geostrophic flow: R 0 small, F exists, then horizontal component of momentum (geostrophic) balance (Eq. 7-2) can be written as: fz u + 1 p = F Eq ρ 36
36 4. Subgeostrophic flow Surface (friction) wind fz u + 1 p = F Eq ρ F fz u and Marshall and Plumb (2008) 1. Start with u 2. Coriolis force per unit mass must be to the right of u 3. Frictional force per unit mass acts as a drag and must be opposite to u. 4. Sum of the two forces (dashed arrow) must be balanced by the pressure gradient force per unit mass 5. Pressure gradient is not normal to the wind vector or the wind is no longer directed along the isobars, however the low pressure system is still on the left side but with a deflection. => The flow is sub-geostrophic (less than geostrophic); ageostrophic component directed from high to low. 37
37 4. Sub-geostrophic flow The Ekman spiral theory Ekman spiral: simplified theoretical calculation u h fz u + 1 p = F Eq ρ u 0 F = μ 2 u/ z 2 μ: constant eddy viscosity u g Roedel, 1987 The Ekman spiral was first calculated for the oceanic friction layer by the Swedish oceanographer in In 1906, a possible application in Meteorology was developed. 38
38 4. Sub-geostrophic flow Wind measurement in the boundary layer u h m u g The Leipzig wind profile Kraus,
39 4. Sub-geostrophic flow The ageostrophic flow u ag The ageostrophic flow is the difference between the geostrophic flow (u g ) and the horizontal flow (u h ): u ag u h = u g + u ag Eq u h u g fz u ag = F Eq The ageostrophic component is always directed to the right of F in the NH. 40
40 4. Sub-geostrophic flow Surface (friction) wind Low pressure (NH): - anticlockwise flow - often called cyclone L H High pressure (NH): - clockwise flow - often called anticyclone Marshall and Plumb (2008) Note: At the surface the wind is blowing from the high to the low pressure system. 41
41 Quiz Today s weather chart x Oslo Varmfront Kaldfront Okklusjon x What is the wind direction and approximately wind strength in Oslo? a) Weak northeasterly. b) Fresh southwesterly. c) Storm from the west. d) Storm from the east. 42
42 Beaufort wind scale «Sir Francis Beaufort var en irsk hydrograf og offiser i den britiske marinen. I 1806 satte han navn på vindens styrke. Denne skalaen brukes i dag i all vanlig værvarsling.» 44
43 4. Sub-geostrophic flow Vertical motion induced by Ekman layers Ageostrophic flow is horizontally divergent. Convergence/divergence drives vertical motions. In pressure coordinates, if f is const., the continuity equation (Eq. 6-12) becomes: p u ag + w p = 0 Low: convergent flow High: divergent flow Continuity equation requires vertical motions: - Ekman pumping > ascent - Ekman suction > descent Marshall and Plumb (2008) 45
44 4. Sub-geostrophic flow Characteristics of horizontal wind Combination of horizontal and vertical vergences 46
45 4. Sub-geostrophic flow Surface pressure (hpa) and wind (kn), 12 GMT June 21, 2003, USA 5 knots 10 knots Marshall and Plumb (2008) 47
46 4. Sub-geostrophic flow Tomorrow s surface pressure (hpa) map 30. Sept 12 UTC, DWD ICON model x x Oslo 48
47 4. Sub-geostrophic flow See also chapter 8. Marshall and Plumb (2008) 49
48 5. Summary Summary of chapters 6 and 7 f u z = ag z T 50
49 Take home message Balanced flows for horizontal fluids (atmosphere and ocean). Balanced horizontal winds: Geostrophic wind balance good approximation for the observed wind in the free troposphere. Gradient wind balance and cyclostrophic wind balance occur with higher Rossby number. Surface wind (subgeostrophic flow) balance occurs within the Ekman layer. Thermal wind is good approximation for the geostrophic wind change with height z. 51
50 Pressure [hpa] Quiz Which winds play a role in the below shown climatology? Zonal mean temperature ( C) Zonal mean zonal wind (m/s) 52
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