Download Equations of Motion - School of Engineering

http://www.physics.curtin.edu.au/teaching/units/2003/Av
p201/?plain
Lec02.ppt
Equations of Motion
Or: How the atmosphere moves
Objectives
• To derive an equation which describes the
horizontal and vertical motion of the
atmosphere
• Explain the forces involved
• Show how these forces produce the
equation of motion
• Show how the simplified equation is
produced
Newtons Laws
• The fundamental law used to try and
determine motion in the atmosphere is
Newtons 2nd Law
Force = Mass x Acceleration
• Meteorology is a science that likes the
KISS principle, and to further simplify
matters we shall consider only a unit
mass.
i.e. Force = Acceleration
Frame of Reference
If we were to consider absolute
acceleration relative to “fixed” stars (i.e. a
non-rotating Earth )
Then our equation would read something
like this;
Dv
Dt
F
“The rate of change of velocity with time is
equal to the sum of the forces acting on
the parcel”
Frame of Reference
• For a non-rotating Earth, these forces are:
Pressure gradient force (Pgf)
Gravitational force (ga) and
Friction force (F)
Frame of Reference
• However, we don’t live on a non-rotating
Earth, and we have to consider the
additional forces which arise due to this
rotation, and these are:
Centrifugal Force (Ce) and
Coriolis Force (Cof)
Equation of Motion
• We now have a new equation which states
that:
Dv
 Pgf  g a  F  Ce  Cof
Dt
• i.e. The relative acceleration relative to the
Earth is equal to the real forces (Pressure
gradient, Gravity and Friction) plus the
“apparent forces” Centrifugal force and
Coriolis force
A Useable form
• If we now consider the centrifugal force we
can combine it with the gravitational force
(ga) to produce a single gravitational force
(g), since the centrifugal force depends
only on position relative to the Earth.
• Hence, g = ga + Ce
A Useable form
• We can now write our equation as:
Dv
 Pgf  g  F  C of
Dt
• We now look at the equation in its
component forms, since we are
considering the atmosphere as a 3Dimensional entity
Conventions
• In Meteorology, the conventions for the
components in the horizontal and vertical are;
x = E-W flow
y = N-S flow
Z = Vertical motion
• Also, the conventions for velocity are
u = velocity E-W
v = velocity N-S
w = Vertical velocity
Pressure Gradient Force
• “Force acting on air by virtue of spatial
variations of pressure”
• These changes in pressure (or Pressure
Gradient) are given by;
Change of pressure p 
Change in distance n 
Pressure Gradient Force
If we now consider this pressure gradient
acting on a unit cubic mass of air with
volume given by x y z we can say that
the Pressure gradient (Pg) on this cube is
given by:
(Pg) = Force/Volume.
Pressure Gradient Force
We can also say that the volume of this unit
mass is the specific volume which is given
by 1/, where  is density.
This has the dimensions of Volume/Mass.
The dimensions of the Pressure gradient are
Force/Volume
Pressure Gradient Force
1 p Volume Force
Force



 n
Mass Volume Mass
Force  Mass x Accelerati on
Force
 Accelerati on 
Mass
Mass  Unity  Accelerati on  Force
Pressure Gradient Force
Therefore we have the Pressure Gradient
Force (Pgf) given by;
1 p
Pgf 
 n
This Pgf acts from High pressure to Low
pressure, and so we have a final equation
which reads;
1 dp
Pgf   dn
Pressure Gradient Force
Because we are dealing in 3-D, there are
components to this Pgf and these are given
as follows;
Pgf x
Pgf y
Pgfz
1 p
 x
1 p
 y
1 p
 z
Pressure Gradient Force
Combining the components we get a total
Pgf of
1  p
p
p 
- i
 j k 
  x
y
z 
The components i and j are the Pgf for
horizontal motion and the k component is
the Pgf for vertical motion.
Horizontal Pgf
We can simplify matters still further if we
take the x axis or y axis normal to the
isobars, i.e. in the direction of the gradient.
We then only have to consider one of the
components as the other one will be zero.
y
1020
1018
x
1 p
Pgf = 
 y
1016
Vertical Pgf
In the synoptic scale (large scale motions
such as highs and lows), the Vertical Pgf is
almost exactly balanced by gravity.
So we can say that p
z
 - g
This is known as the Hydrostatic equation,
and basically states that for synoptic scale
motion there is no vertical acceleration
Coriolis Force
• This is an “apparent” force caused by the
rotation of the Earth.
• It causes a change of direction of air parcels in
motion
• In the Southern Hemisphere this deflection is to
the LEFT.
• Is proportional to | V | sin  where  is the local
latitude
• Its magnitude is proportional to the wind strength
Coriolis Force
• It can be shown that the Coriolis force is given
by
2 sin  V
• The term 2 sin  is known as the Coriolis
parameter and is often written in texts as f.
• Because of the relationship with the sine of the
latitude Cof has a maximum at the Poles and is
zero at the equator (Sin 0° = 0).
Coriolis Force
Frames of referenceRoundabouts (1)
• Our earth is spinning rather
slowly (i.e. once per day)
and so any effects are hard
to observe over short time
periods
• A rapidly spinning
roundabout is better
• From off the roundabout, a
thrown ball travels in a
straight line.
Frames of referenceRoundabouts (2)
• But if you’re on the
roundabout, the ball
appears to take a
curved path.
• And if the roundabout
is spinning clockwise,
the ball is deflected to
the left
Components of Cof
It can be shown that the horizontal and
vertical components of the Cof are as
follows;
x  2  v sin  - 2  w cos 
y  - 2  u cos 
z  2  u cos 
The complete equation
We can now write the equation of motion
which describes the motion of particles on a
rotating Earth.
Remembering that the equation states that;
Acceleration = Pgf + Cof + g + F,
We can write the equation as follows;
The complete equation
(Ignoring frictional
effects)
1 p
du
 2  v sin  - 2  w cos 
 x
dt
dv 1 p
- 2  u sin 

dt  y
1 p
dw
 2  u cos  - g
 z
dt
Scale analysis
Even though the equation has been simplified by
excluding Frictional effects and combining the
Centrifugal force with the Gravitational force, it is
still a complicated equation.
To further simplify, a process known as Scale
analysis is employed.
We simply assign typical scale values to each
element and then eliminate those values which are
SIGNIFICANTLY smaller than the rest
Scale Analysis
Element
Typical Value Magnitude
u,v Horizontal velocity
10-20 ms
w Vertical velocity
1 cms
10 m
L Length (distance)
1000km
10 m
H Depth (Height)
10km
10 m
-1
-1
1
10 m
-2
6
4
3
Horizontal Pressure Change 10 - 20 hPa
10 Pa
Vertical Pressure Change
1000 hPa
10 Pa
L/U (Time)
 (Density)
27 Hours
10 secs
g (Gravity)
 (Angular velocity)
9.8 ms
1 kgm
5
5
-3
10 kgm
0
-2
10 ms
1
5
7.29 x 10
-4
-3
-2
10 Radians s
-1
Scale Analysis
du
1 p
 2  v sin  - 2  w cos 
dt
 x
 
 
101
103
-4
-3
-4
1
-3
-4
-2
6
10

1
10

[10
10
(10
)]
[10
10
(
10
)]
5
6
10
10
dv
1 p
- 2  sin 
dt
 y
101
103
-4
-3
-4
1
-3
(10
)

1
(10
)
10
10
(10
)
5
6
10
10
dw
1 p
  2  cos  - g
dt
 z
10-2
105
-7
1
-4
1
-3
1
(10
)

1
(10
)

10
10
(10
)
10
10-5
10 4
Scale Analysis (Horizontal
Motion)
From the previous slide we can see that for
the horizontal equations of motion du/dt and
dv/dt, the largest terms are the Pgf and the
Coriolis term involving u and v.
The acceleration is an order of magnitude
smaller but it cannot be ignored.
Scale Analysis (Vertical Motion)
For the vertical equation we can see that there are
two terms which are far greater than the other two.
The acceleration is of an order of magnitude so
much smaller than the Pgf and Gravity that it CAN
be ignored
We can say therefore that for SYNOPTIC scale
motion, vertical acceleration can be ignored and
that a state of balance called the Hydrostatic
Equation exists
Simplified Equation of Motion
Using the assumptions of no friction and
negligible vertical motion and using the
Coriolis parameter f = 2sin, we can state
the Simplified equations of motion as
Simplified Equation of Motion
1 p
du
 fv
 x
dt
1 p
dv
- fu
 y
dt
1 p
- g Which becomes the hydrostati c equation
0 z
p
 - g
z
References
• Wallace and Hobbs Atmospheric Science
pp 365 - 375
• Thom Meteorology and Navigation
pp 6.3 - 6.4
• Crowder Wonders of the Weather
pp 52 - 53
• http://www.shodor.org/metweb/session4/se
ssion4.html