GEF 1100 – Klimasystemet Chapter 7: Balanced Flow

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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 Ch. 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:

휕푢 휕푣 g + g = 0 Eq. 7-5 ``Horizontal flow is non divergent.’’ 휕x 휕y 휕푤 Comparison with continuity equation (훻 ∙ 풖 = 0) => 푤 = 0 => g = 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

x x Oslo

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

u g

 FC

Balance H 1 between: 풖 = 풛 × 훻푝 g 푓휌 FP = -FC 13 2. Geostrophic motion Geostrophic wind

L Isobars

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

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

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 - - - - Which the rules from thermal wind if deduceapproximation, can I

The geostrophic right rotation with increasing height for geostrophic left rotation with increasing height for whereby the y geostr u . wind . wind change with height Thermal Thermal wind approximation g

( colder air remains the to left on the NH, z cold cold air advection +

∆ ∆ z

z u

) g

cold

u g

( z)

is parallel the to mean isotherms x

u g

( z) warmair advection

cold warm ∆ u z cold air advection

u warm air advection

( g z

+ ∆ z

)

u g <10 m/s ,

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 풖 = 풛 × 훻푝 Eq. 7-3 f y g 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

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 Ch. 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. 32 5. Subgeostrophic 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:

R0 small, ℱ exists, then horizontal component of momentum (geostrophic) balance (Eq. 7-2) can be written as:

1 푓풛 × 풖 + 훻푝 = ℱ Eq. 7-25 휌

33 5. Subgeostrophic flow Surface (friction) wind

1 푓풛 × 풖 + 훻푝 = ℱ Eq. 7-25 휌

ℱ − 푓풛 × 풖 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. 34 5. Sub-geostrophic flow

The ageostrophic flow uag

The ageostrophic flow is the difference between the geostrophic

flow (ug) and the horizontal flow (uh): uag

uh uh = ug+ uag Eq. 7-26 u g

푓풛 × 풖ag = ℱ Eq. 7-27

The ageostrophic component is always directed to the right of ℱ in the NH.

35 4. Sub-geostrophic flow Surface pressure (hPa) and wind (m/s or kn ?), 12 GMT June 21, 2003

5 knots

Marshall and Plumb (2008) 10 knots 36 4. Sub-geostrophic flow Today’s surface pressure (hPa) map

www.yr.no www.met.fu-berlin.de

37 4. Sub-geostrophic flow

Surface (friction) wind

Low pressure (NH): L H High pressure (NH): - anticlockwise flow - clockwise flow - often called - often called “cyclone” “anticyclone”

Marshall and Plumb (2008)

38 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: 휕푤 훻 ∙ 풖 + = 0 푝 푎푔 휕푝

• Low: convergent flow • High: divergent flow • Continuity equation requires vertical motions: - “Ekman pumping“ > ascent - “Ekman suction” > descent

Marshall and Plumb (2008) 39 4. Sub-geostrophic flow Characteristics of horizontal wind

Combination of horizontal and vertical vergences

40 4. Sub-geostrophic flow The Ekman spiral – theory

Ekman spiral: simplified theoretical calculation

1 푓푧 × 풖 + 훻푝 = ℱ Eq. 7-25 휌

u0 ℱ = 휇 휕2풖/휕푧2

휇: constant eddy viscosity

Roedel, 1987

The Ekman spiral was calculated for the oceanic friction layer by the Swedish oceanographer in 1905. In 1906, a possible application in Meteorology was developed. 41

4. Sub-geostrophic flow Wind measurement in the boundary layer

uh m

The “Leipzig wind profile” Kraus, 2004

42 4. Sub-geostrophic flow

Marshall and Plumb (2008) 43 4. Sub-geostrophic flow

44 Marshall and Plumb (2008) Chapters 6 and 7 Summary

휕풖 푓 = 푎푔 풛 × 훻푇 휕푧

45/36 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.