# Using the Equation of Motion

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```							http://www.physics.curtin.edu.au/teaching/units/2003/Av
p201/?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.
du    1p
-      fv
dt    x
dv    1p
-     - fu
dt    x
1p
0 -      -g
 z
Hydrostatic Equation
• The final term of the equation can be further
simplified to give us the following result;

p
 - g
z
And if we re - arrange we can get the following
1 p
g-
 z
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

1 p                1 p
0  -   fvg  vg         
 x                f x
1 p                 1 p
0  -  - fu g  u g  -      
 y                 f y
Where vg and u g are componentgeostrophic winds
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;

1 p
Vg    
f n
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)

PGF  2hPa/100km  200pa/100km  2x10-4 pa/m
  1.2 kgm-3
f  2  sin 
1        200
 for 20 Vg 

 4
1.2 x 2 sin 20 10
 3.35 ms-1
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
Co
Vg
PGF
1012hPa
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;
g h
Vg 
f n
Where h is height distance between contours
and n is distance between contours
• Wind which results when the Centrifugal
Force resulting from curved flow is exactly
balanced by the Coriolis and Pressure
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)
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;

Cyclonic flow
r 2 f 2  4 r f vg
Vgr  - rf 
2
Where r is the radius of curvatureand vg is geostrophic wind
Anti - cyclonic flow
r 2 f 2 - 4 r f vg
Vgr  rf 
2
There are some limiting factors to gradient flow
around high pressure systems when we look at the
equation closely.
r f - r 2 f 2 - 4 r f vg
Vgr 
2
There is a maximum value to Vgr when    0
i.e. when r 2 f 2  4 r f vg
r 2f 2 r f
 vg        
4rf     4
Substituting back into the original equation we have
Vgr max  2Vg
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.
Next weconsider the square root termitself.
In order for theequation to make sense, the number
inside the bracket must be 0 or greater.
 r 2 f 2  4 r f vg
 1 p 
4rf    f  n 

          
4 r p
    
 n
 rf2       p
           
4         n
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.
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
Cef
Wind
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.
curvature is very small (such as found around
tornadoes), then the centrifugal force and pressure
This is Cyclostrophic flow.
Cef
Wind
direction

Pgf
Cof

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
Wind veers as
Frictional effects reduce Wind
Coriolis is
speed
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 Effects
1012
Vgr (3000ft)

1010                               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
Low

Ce
PGF
Vgr

Cof

High
Effects of Friction
Low

PGF
Vf Ce
V           F

Cof

High

Friction [F] reduces gradient Speed [V]
Cof now reduced
PGF becomes dominant force and so wind VEERS
Effect of Friction
Cross - Isobar Flow
H

              Co

PGF
L
Wind speed reduced by friction and so Co decreases. PGF>Co
Over water   10 0 V  2 Vg
3

Over land   30 0 V  1 Vg
3
Friction Effects
Cross-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) Minimum (~ 25kt)
Veered (~20)    Veered (~5)

3am Minimum (~ 5kt)    Maximum (~ 35kt)
Veered (~ 30)   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                                  270/60

Wind Shear

5000 FT                                   270/25

Wind Shear = 270/60 - 270/25
= 270/35
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
A                                      B

W                                      C
A       P = 1013 hPa
O
R                                      L
M                                      D
Equator                                               Pole

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

W                                          C
A                                          O
R            Constant Height AGL
L
M                                          D
Equator                                                     Pole

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)
Pgf
A                                     B

W                                      C
A                                      O
R            Constant Height AGL
L
M                                      D
Equator                                                   Pole

This sets up a Pressure Gradient Force between A and B
Co

Pgf
A                                        B

W                                         C
A                                         O
R            Constant Height AGL
L
M                                         D
Equator                                                   Pole

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

A                                      B

W                                      C
A                                      O
R           Constant Height AGL
L
M                                      D
Equator                                                Pole

Giving us the upper level westerly which we observe as being
the pre-dominant wind in the upper atmosphere
Web References
• www.met.tamu.edu/teach.html

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

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