http://www.physics.curtin.edu.au/teaching/units/2003/Avp201/?plain

Lec03.ppt

Using the Equation of Motion

Objectives

• Revision of last weeks lecture

• Applying Equation of motion to derive wind models

• “Anomalous” wind flows

• Impact of friction on air flow

• Thermal wind

Revision

• Last week we obtained a complete equation of motion which incorporated both “real” and “apparent” forces.

• Stated that the total rate of change of velocity with time (acceleration) was due to a combination of Pressure Gradient Force, Gravity, Centrifugal Force, Coriolis Force and Friction.

Revision

• We simplified matters further by combining Gravity and Centrifugal force into a single Gravity force.

• We also eliminated friction by assuming flow in a friction free environment, e.g. 3000ft above the Earth’s surface.

Revision

• By resolving into the different components of a 3-dimensional system and using scale analysis we obtained the general equation of motion as below.

g - z

p

1 - 0

fu - x

p

1 -

dt

dv

fv x

p

1 -

dt

du

Hydrostatic Equation

• The final term of the equation can be further simplified to give us the following result;

z

p

1 - g

following get thecan wearrange-re weif And

g - z

p

Hydrostatic Equation

• This equation tells us that as gravity is a constant, then the rate of change of pressure with height is greater for cold dense air than for warm less dense air.

• We can say therefore that the rate of change of pressure with height is dependent on temperature.

Use of Hydrostatic Equation

• Main use is in measurement of height above ground

• If a ‘standard’ atmosphere is assumed whereby mean sea level temperature is 15°C and the lapse rate is 6.5°C/km, then a ‘standard’ distribution of pressure with height results

• This ‘standard’ is used in pressure altimeters, which sense pressure but read out height.

Wind Equations

• The horizontal components of Newton’s 2nd law are sometimes called the wind equations.

• For both N-S and E-W flow the only forces we need consider are the Pressure gradient force, the Coriolis force and Friction.

• We can neglect friction if we assume flow in a friction free environment, i.e. above 3000ft.

Wind Equations

• We have also seen from our scale analysis that we have an acceleration which is an order of magnitude less than the forces which cause it.

• Therefore we can disregard these accelerations, and if we have no local curvature effects such as those found around lows and highs we can state the following.

Geostrophic Wind

• Geostrophic motion occurs when there is an exact balance between the HPGF and the Cof, and the air is moving under the the action of these two forces only.

• It implies– No acceleration

• eg Straight, parallel isobars

– No other forces• eg friction

– No vertical motion• eg no pressure changes

Geostrophic Wind

As geostrophic conditions imply no acceleration or friction, we can set these terms to zero in the simplified equations of motion to get the Geostrophic wind equations

windscgeostrophicomponent are u and vWhere

y

p

f

1 - u fu -

y

p

1 - 0

x

p

f

1 vfv

x

p

1 - 0

gg

gg

g g

Geostrophic Wind

We can combine these results to give an equation for the geostrophic wind on a surface chart if we know the perpendicular distance n between isobars.

The equation is as follows;

n

p

f

1 Vg

An Example

What is the Geostrophic wind speed for a pressure gradient of 2hPa/Km and density of 1.2kgm-3 at a latitude of 20° ? ( = 7.272 x 10-5 ,2 = 1.45x10-4)

1-

4g

3-

-4

ms 3.35

10

200

20sin 2 x 1.2

1 V 20for

sin 2 f

kgm 1.2

pa/m 2x10 m200pa/100k 2hPa/100km PGF

The Nature of Vg

• Geostrophic wind acts Parallel to the Isobars

• If you have your back to the wind then Low Pressure is on your RIGHT

1016hPa

1012hPa

Vg

Co

PGF

Upper level charts

It can be shown that the geostrophic argument works for upper level charts as well as for surface charts.

The equation for geostrophic wind at upper levels loses the density term and becomes;

contoursbetween distance isn and

contoursbetween distanceheight ish Where

n

h

f

g Vg

Gradient Wind Vgr

• Wind which results when the Centrifugal Force resulting from curved flow is exactly balanced by the Coriolis and Pressure Gradient Forces

Ce= Co- PGF

• 3 Cases of Vgr exist– Anti-clockwise flow (a High)– Clockwise flow (A Low)

– Straight Flow (Vgr=Vg which is a special case)

Gradient Wind Equations

The equations for the gradient wind depend on whether the flow is cyclonic or anti-cyclonic, but it can be shown they are as follows;

2

vfr 4 - f r rf V

flow cyclonic-Anti

windcgeostrophi is vand curvature of radius theisr Where2

vfr 4 f r rf- V

flow Cyclonic

g22

gr

g

g22

gr

Gradient Wind

There are some limiting factors to gradient flow around high pressure systems when we look at the equation closely.

gmaxgr

22

g

g22

gr

g22

gr

2V V

have eequation w original theintoback ngSubstituti4

fr

fr 4

fr v

vfr 4 f r when i.e.

0 when V to valuemaximum a is There

2

vfr 4 - f r - fr V

Gradient Wind

This tells us that there is a limit to how fast the wind can move around an anti-cyclone, and that limit is twice the speed of the Geostrophic wind.

There is no limit to the speed a cyclonic circulation can achieve.

Gradient Wind

n

p

4

fr

n

p

r 4

n

p

f

1 fr 4

vfr 4 f r

greater.or 0 bemust bracket theinside

number thesense, make oequation t for theorder In

itself. root term square heconsider t Next we

2

g22

Gradient Wind

This tells us that when the radius of curvature is small, then so must the rate of change of pressure with distance.

In other words the isobars must get further apart the closer you get towards the centre of the anticyclone.

There is no limit to the spacing of the isobars around the centre of cyclonic flow.

Gradient Wind

The previous slide shows us the balance of forces required to make Gradient flow occur.

From our knowledge we can now say that gradient flow around a cyclone is sub-geostrophic, and that gradient flow around an anti-cyclone is super-geostrophic.

Cof Pgf

CefWind direction

In this situation, it is impossible to achieve balanced flow, as all the forces are acting in the same direction. Therefore it is impossible for clockwise flow to exist around a high in the Southern Hemisphere.

Cyclostrophic Flow

As mentioned previously there are no restrictions to the strength of the pressure gradient around low pressure systems.

This can lead to situations whereby if the radius of curvature is very small (such as found around tornadoes), then the centrifugal force and pressure gradient forces balance each other.

This is Cyclostrophic flow.

Cef

PgfCof

Wind direction

In order for balanced flow to occur, the Cef must balance the Cof and PGF. This can only happen with large amounts of Cef, eg small radius of curvature and large speeds. Therefore can only occur with small scale systems such as dust devils and tornadoes.

Friction

So far we have chosen to ignore the effects that friction has on air in motion, by looking at motion above 3000ft.

However, we have to take it into account when looking at motion closer to the surface.

Frictional Effects

Frictional effects reduce Wind speed

Wind veers as Coriolis is reduced

Cross Isobar flow towards Low Pressure Region

Flow outwards from High Pressure

Flow inwards to Low Pressure As friction reduces with height, wind flow will BACK with height

Frictional Effects1012

1010

Vgr (3000ft)

Sfc wind

Friction at Surface is greatest and we get a reduction in speed, which in turn leads to a reduction in Cof. PGF becomes dominant and so wind blows towards LP.

Vgr = W’ly SFC = NW’ly

From Vgr to SFC, winds have veered.

From SFC to Vgr, winds have backed.

Effects of Friction Balanced Gradient flow

High

Low

Vgr

Ce PGF

Cof

Effects of Friction

Low

High

CePGF

Cof

V

Vf

F

Friction [F] reduces gradient Speed [V]

Cof now reduced

PGF becomes dominant force and so wind VEERS

Effect of FrictionCross - Isobar Flow

H

L

Co

PGF

Wind speed reduced by friction and so Co decreases. PGF>Co

g0

g0

V 31 V 30 landOver

V 32V 10 Over water

Friction EffectsCross-Isobar Flow

Diurnal Variation of Wind

• Wind shows a marked diurnal variation– Peak during daytime and lull overnight

• Variation more marked with existence of low level temperature inversion.– During night inversion acts as “lid” and prevents

energy transfer downwards and thus have (relatively) stronger winds above inversion.

– During day convective currents break inversion down allowing sfc winds to increase and transfer turbulence aloft to decrease upper winds.

Diurnal Variation of wind

Surface 1000-3000 ft AGL

3pm Maximum (~ 15kt)Veered (~20 )

Minimum (~ 25kt)Veered (~5 )

3am Minimum (~ 5kt)Veered (~ 30 )

Maximum (~ 35kt)Backed (~ 5 )

Terrain Induced Turbulence

• Varying terrain will cause changes in the depth of the turbulence

• Be aware of changes in turbulence depth with changes in surface features– i.e. Sea surfaces will have a lesser

depth of turbulence than land surfaces

Wind Shear

• The variation of wind between 2 points

• Vertical wind shear is the variation in the wind between 2 layers

• In particular it is the – Wind at top of layer minus wind at bottom of

layer

Wind Shear

10000 FT

5000 FT

270/60

270/25

Wind Shear = 270/60 - 270/25

= 270/35

Wind Shear

Thermal Wind

• This shear is given the name Thermal wind.

• Use of the Hydrostatic assumption and gradient wind equation can show us how the vertical shear will vary due to horizontal temperature gradients.

Equator Pole

P = 1013 hPa

A B

WARM

COLD

Two columns of air, A and B exert the same pressure (1013hPa) at the surface

Equator Pole

Constant Height AGL

A B

WARM

COLD

However, because A is warmer and therefore less dense than B, the pressure at the constant height surface at A is greater than at B.(By using the hydrostatic assumption and remembering that pressure drops more rapidly in cold air than warm air)

Equator Pole

Constant Height AGL

A B

WARM

COLD

This sets up a Pressure Gradient Force between A and B

Pgf

Equator Pole

Constant Height AGL

A B

WARM

COLD

Co

Coriolis force now acts on the moving air to deflect it to the left in the Southern Hemisphere to achieve balanced Geostrophic flow

Pgf

Equator Pole

Constant Height AGL

A B

WARM

COLD

Giving us the upper level westerly which we observe as being the pre-dominant wind in the upper atmosphere

Resultant Wind

Web References

• www.met.tamu.edu/teach.html

• www.page.ucar.edu/pub/education_res/presearch/meteortoc.htm