Transcript math

MET 61
Topic 2
Atmospheric Dynamics
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Goal(s)…
Main goal = investigate motions = winds
– Wind = air in motion
1. What causes air motions?
–
–
–
–
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What causes any object to move?
A force!
“F = ma”
All we need to do is indentify forces in the
atmosphere → ideas about why air moves
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2. What air motions result?
– Range from “simple” to highly complex
– e.g., geostropic wind is a simple wind that
we can derive
– Sometimes a good approximation to real
(observed) winds
3. How can we characterize air motions?
– What properties of air motions are useful to
know?
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Kinematics of large-scale flows
VIP note: we will focus on flows with
“on the order of”
1) horizontal scales O(1000 km +)
•
e.g., not a tornado
2) vertical scales O(10 km)
•
E.g., not a Cu cloud
3) time scales O(one day +)
•
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e.g., not a tornado
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Properties are listed in Table 7.1
Need to understand:
Physics
what the property is
what information this conveys
Math
how to express / compute the property
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Shear
Physically
A change in wind speed or direction in space
Example:
Winds across a front
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Example:
Associated with a thunderstorm
Microburst example 
A microburst is a very localized
column of sinking air, producing
damaging divergent and straight-line
winds at the surface that are similar to
but distinguishable from tornadoes
which generally have convergent
damage.
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http://en.wikipedia.org/wiki/Eastern_Air_Lines_Flight_66
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Math
•
•
•
We invent and use natural coordinates
Flow-following coordinates
Fig. 7.1
•
•
s = direction along the flow
n = direction  to flow (and to the left)
Lower cases!
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Math
Thus:
V
shear  
n
Here, V = wind speed (scalar)
Wind direction is tracked via the coordinate
system!
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Curvature
Physically
A change in wind direction as one travels
downstream
Sign convention:
Curvature is called positive if flow direction
(vector) is turning anticlockwise in the NH.
aka…cyclonic curvature (NH)
anticyclonic curvature (SH) !!!
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Math

curvature  V
s
where  is an angle which defines the flow
direction (relative to something…Fig. 7.1)
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Diffluence / confluence
Physically
Relates to some measure of “spreading out” of
flow direction  the flow (and vice versa)
Example:
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Math

diffluence  V
n
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Stretching
Physically
Relates to wind speed changes downstream
Example:
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Math
V
stretching 
s
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Streamlines
Streamlines are lines drawn such that at any
point, the actual (horizontal) wind is parallel
to the streamline.
Indicate flow direction.
Can be made to indicate flow speed via spacing
(closer spacing  higher wind speed)/
Only valid at an instant in time!
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http://weather.unisys.com/surface/sfc_con_stream.html
http://www.weatheronline.co.uk/cgibin/expertcharts?LANG=en&MENU=0000000000&CONT=samk&MOD
ELL=gfs&MODELLTYP=1&BASE=&VAR=w010&HH=48&ZOOM=0&ARCHIV=0
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Vorticity
Physically
• A measure of spin of an individual fluid parcel
• NOT (necessarily) rotation in the entire fluid!
•
The sum of a shear effect and a curvature
effect
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Math
 V v u
vorticity    V

 
s n x y
Where
• u = east-west wind (u > 0 for eastward flow)
• v = north-south wind (v > 0 for northward
flow)
Typical values:  10-5 s-1.
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Fig. 7.2a – shear  vorticity
imagine a “paddlewheel” in the flow
spin indicates vorticity
Fig. 7.2b – shear + curvature  vorticity
Fig. 7.2c – zero shear, zero curvature  zero
vorticity
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Vorticity distributions and forecasting?
Earliest forecasts were for future vorticity
distributions!!
http://www.aip.org/history/sloan/gcm/prehistory.html
J.G. Charney et al integrated (solved) an
equation for the evolution of vorticity
 a primitive forecast (c. 1950, using ENIAC)
ENIAC… http://en.wikipedia.org/wiki/ENIAC
and… http://en.wikipedia.org/wiki/Numerical_weather_prediction
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Vorticity distributions and “weather”?
http://www.met.sjsu.edu/weather/gfsp.html
High positive vorticity associated with troughs
and regions of active “weather”
High negative vorticity associated with ridges and
regions of calm “weather”
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Divergence
Physically
• A measure of the tendency of flow to spread
out (diverge) from a location
• or converge towards a location
• negative divergence = convergence
•
The sum of diffluence and stretching effects.
Example: Fig. 7.2c
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Math
 V u v
divergence    V



n s x y
Typical values:  10-6 s-1.
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Divergence values – positive or negative –
difficult to discern from looking at wind obs
Examples:
http://weather.uwyo.edu/models/fcst/ukmet.html
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Imagine putting an elastic sheet into a divergent
flow.
•
•
The flow will stretch & deform the sheet.
The area of the sheet will increase.
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It can be shown that divergence ( ) is related to
area change by:
u v 1 dA
  
x y A dt
Divergence ( > 0)  dA/dt > 0
 area increases
Convergence ( < 0)  dA/dt < 0
 area decreases
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Flow deformation
Consider again this “stretchy area” idea…
Fig. 7.3 shows how the area could be deformed
by a complex flow involving all the elements
in Table 7.1
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Flow deformation can lead to sharpened
gradients…
Fig. 7.4a … north-south temperature gradient is
sharpened by the flow
 Frontal zone
…zone in which frontal disturbances could
develop (“frontogenesis”)
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Streamlines and trajectories
Streamlines are only valid at an instant in time!
As time evolves, a parcel’s motion is defined by a
trajectory.
See example on the board…
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Horizontal Equation of Motion
As discussed earlier, we can:
• develop an equation of motion using
• Newton’s 2nd Law of Motion…
And use it to understand and forecast
motions
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Newton’s 2nd Law of Motion
F  ma
dV F
a
 
dt m
F
m
i
(*)
where “Fi” refers to the various forces at
work.
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Step 1 = identify the forces
•
physically
–
–
•
e.g., Newton’s 2nd Law of Motion
e.g., Law of Gravitation
mathematically
–
–
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F = m.a
g = see below!
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Step 2 = slot expressions into eqn. (*) above
Step 3 = solve the equation
•
Solution is wind vector at any location
and future time
•
Vector wind → u, v components
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Real Forces
These are very basic forces:
a) gravity (same as gravitation??? No!)
b) friction
c) (air) pressure gradient force
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Pressure gradient force
We met the hydrostatic equation in §3.2
p
1 p
  g  
g
z
 z
Equates the
• (downward directed) force of gravity with the
• (upward directed) pressure gradient force
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The horizontal pressure gradient forces (PGF)
can be expressed similarly:
1 p
1 p
( PGF ) x  
, ( PGF ) y  
 x
 y
In vector form:
( PGF )  P  
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1

p
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In MET 121A,B we will show that the PGF can be
written as:

1

p
 gz
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when “z” is the vertical
coordinate (p = pressure)
when “p” is the vertical
coordinate (z = altitude)
 g oZ
when “p” is the vertical
coordinate (Z = geopotential
height, Z  z)
 
when “p” is the vertical
coordinate ( = geopotential)
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Gravitational force
Given by Newton’s Law of Gravitation:
Gravitational force between two objects


product of two masses (e.g., earth and moon)
1 / (distance separating the masses)2
What we experience is pure gravitation modified
by earth’s rotation … see below!
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Frictional force
Maximizes @ surface where wind speeds MUST
be zero!
Insignificant above the boundary layer (typically
about 1 km)
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The value of the force is (see MET 121A,B etc.):
1 
F 
 z
Here,  is the shear stress = the stress
associated with vertical wind shear
Thus, stress  vertical wind shear
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Apparent Forces
These arise due to Earth’s rotation:
a) Coriolis
b) A centrifugal force which modifies pure
gravitation → gravity (which we
experience)
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Coriolis force
Causes a deflection of moving air
Acts in a direction  direction of motion (i.e.,
left/right, up/down)
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Value:
CF   f k V
With
f  2 sin( latitude)  Coriolis parameter
  earth' s rotation rate
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f = 0 @ equator
no Coliolis deflection @ equator
hurricanes do not form @ equator
f has max value @ poles
 =7.292 x 10-5 s-1 = 2 / “day”
f > 0 in northern hemisphere (NH)
f < 0 in SH
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Centrifugal force
Due to planet’s rotation, a parcel is subject to
gravity and a centrifugal force
Fig. 7.6
The net effect is what we call GRAVITY.
g  ga   R
2
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Equation of Motion
Substituting all these expressions into (*) above
gives:
dV
1
  p  f k  V  F
dt

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(7.13a)
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In component form:
du
1 p

 fv  Fx
dt
 x
dv
1 p

 fu  Fy
dt
 y





(7.13b)
NEXT…seek solutions!
Too complicated 
Instead – make simplifying assumptions 
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One more thing…
du
1 p

 fv  Fx
dt
 x
Looks like an ODE
Is actually a PDE – much harder to solve
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We can show (METR 121) that
d




 u v  w
dt t
x
y
z
  
d 
  
 
i.e.,
 
 u    v    w 
dt  t  x , y , z
 x  y , z ,t
 z  x , y ,t
 y  x , z ,t
Time derivative
moving with the
flow =
Lagrangian
derivative
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Time derivative at a
fixed location (e.g.,
as measured by
instruments) =
Eulerian derivative
Spatial derivative at a fixed
time = (negative of)
advection
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Example with u wind (u) in our Equation of Motion on the
LHS…
 u 
du  u 
 u 
 u 
 
 u    v    w 
dt  t  x , y , z
 x  y , z ,t
 z  x , y ,t
 y  x , z ,t
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2nd example with temperature (T) …
 T 
dT  T 
 T 
 T 


u

v

w







dt  t  x , y , z
 x  y , z ,t
 z  x , y ,t
 y  x , z ,t
 T 
dT
 T 
 T 
 T 
or 


u

v

w







 t  x , y , z dt
 x  y , z ,t
 z  x , y ,t
 y  x , z ,t
advection
Rate of change of T
at a fixed location
Rate of change of T
following the motion
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Large-scale winds
Examine:
http://www.met.sjsu.edu/weather/models/gfsone-00/all500relh.html
We see:
• Flow roughly west → east with low heights on
left (NH)
• With meanders
• Winds  contours “everywhere”
• Winds stronger when contours closelyspaced
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What do the contours indicate?
The height of the 500 hPa surface above MSL
•
•
Compare to earth topo maps…
http://www.digital-topo-maps.com/
•
Compare to 500 hPa maps…
•
http://www.met.sjsu.edu/weather/models/gfsp/avort500f00.gif
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Large-scale winds
Examine:
http://www.met.sjsu.edu/weather/models/gfsone-00/all500relh.html
We see:
• Flow roughly west → east with low heights on
left (NH) (why?)
• With meanders (why?)
• Winds  contours “everywhere” (why?)
• Winds stronger when contours closelyspaced (why?)
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The geostrophic wind
1. Suppose we look at typical values of winds
etc. for larger-scale motions in mid-latitudes.
2. And compute typical values of terms in (7.13)
3. And neglect smaller terms in (7.13)
4. And analyze the solution to modified-(7.13)
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•
•
Wind speed  10 m/s
And wind variations take place over a day 
105 seconds
Thus,  dV/dt   10-4 m/s2
•
But, f  10-4, so  f V   10-3 m/s2
Now look back at (7.13) and neglect friction
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du
1 p

 fv
dt
 x
10-4 m/s2
10-3 m/s2
We conclude that the pressure gradient term
MUST also be of size  10-3 ms-2
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So an approximate equation is:
1 p

 fv  0
 x
similarly,

1 p
v  vg 
f  x
1 p
u  ug  
f  v
In this case, we can actually solve for the wind
components – ug and vg!
This wind is called the geostrophic wind.
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1 p
1 p
ug  
and vg 
f  v
f  x
1
1
Vg 
k p  k 
f
f
The geostrophic wind.
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Properties…
The geostrophic wind blows parallel to isobars
(height lines) with lower pressures (heights)
on the left in the NH (right/SH).
Closely spaced isobars (height lines)
 stronger pressure gradient
 stronger winds
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Comparing with observations away from the
surface (friction)

actual winds  geostrophic winds
in speed & direction
(to within about 10%)
Thus we often “replace” actual winds with
geostrophic winds
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Adding friction…
Observations
http://www.met.sjsu.edu/weather/models/gfsone-12/allsfcdwpf.html
Results same as with geostrophic wind BUT:
•
•
Cross-isobar flow towards low pressure
Cross-isobar flow away from high pressure
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•
•
•
“F” acts to oppose flow
3-way force balance (“F”, “CF”, “PGF”)
Fig. 7.10
Net results of adding friction:
•
Speeds?
–
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Reduced (as expected!)
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•
Direction?
–
Friction induces flow towards lower pressure
–
This gives cross-isobar flow
–
–
Towards low pressure
Away from high pressure
Cross-isobar flow angle can be as much as 45
Depends on surface roughness (parameter)
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The gradient wind
Similar to the geostrophic wind BUT with
curvature effects included
Flow around a LOW implies an extra force of the
form:
2
V
R
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“cyclostrophic” component
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Since flows are often curved (isobars and height
lines are typically NOT straight), the gradient
wind is a better approximation to the real
wind.
However – the math in this case is more
complicated.
{See 121A for derivation.
See Eq. (7.17) for solution}
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Some relationships we can derive:
For cyclonic flow (NH),
Gradient wind speed < Geostrophic wind speed
Calculated from isobar spacing
Calculated from isobar
spacing and V2/R term
We say the wind speed is subgeostrophic.
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For anti cyclonic flow (NH),
Gradient wind speed > Geostrophic wind speed
Calculated from isobar spacing
Calculated from isobar
spacing and V2/R term
Here, the wind speed is supergeostrophic.
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The thermal wind
Observations show that when there is a
north-south temperature gradient
There is also a
vertical wind shear.
Ditto with east-west temp. gradient
WHY???
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1st - observations
Best example = zonal average winds and
temperatures (Fig. 1.11)
= east-west average
Strongest temperature gradient is in mid-latitudes
Strongest vertical wind shear @ same location!
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2nd – “math”
a) We take the geostrophic wind expressions
(7.15b)
b) We take the “vertical” derivatives
c) We get a relationship between:
ug/p and T/x
and between vg/p and T/y
(derive in 121A)
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These allow us to write (7.19b):
g 
(u g ) 2  (u g )1  
( Z 2  Z1 )
f o y
g 
(v g ) 2  (v g )1  
( Z 2  Z1 )
f o x
where Z is geopotential height (Z  z) and “2” and
“1” refer to two different pressure levels.
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Again…
g 
(u g ) 2  (u g )1  
( Z 2  Z1 )
f o y
Can use to find u @ one
level, given u @ second level
AND if the north-south height
gradient is known.
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Can use to find the northsouth height gradient if
winds are known at two
levels.
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3rd – “physics”
pressure  as z  - and more rapidly in a colder
column of air
Example: Consider the 1000-500 hPa layer
Imagine this layer from the (cold) pole to the
(warm) equator
500 hPa
Warm, “deep”
layer
Conclusion: slope = Z/y  T/y
Cold, shallow
layer
1000 hPa
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Example
Ex. 7.3 on p. 284
Given: zonally-averaged temperature gradient
Given: geostrophic wind @ surface  zero
Compute: geostrophic wind @ 200 hPa
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First, expand Eq. 7.20 → components
Rd  1000  T
(u g ) 250  (u g )1000   ln

f  250  y
Rd  1000  T
(vg ) 250  (vg )1000  
ln
0

f  250  x
Only in this case
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Vorticity prediction
As we saw earlier, vorticity patterns have some
association with “weather”.
Regions of high positive vorticty = regions of
“active” or “stormy” weather
Stormy weather @ surface typically east of
upper-level* vort max
* 500 hPa, 250 hPa etc.
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high negative vorticty = region of calm
weather
Thus – it should be useful to predict future
vorticity patterns!
To do this, we need an equation to predict
vorticity evolution.
solve this → prediction
as done in about 1950.
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Process to get (see 121B)…
take


equation to predict u   equation to predict v
x
y

equation to predict  

x
Result is Eq. 7.21a. = vorticity equation
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Solution procedure…
Write vorticity equation as :

 ( RHS )
t
now approximate as :
 (t2 )   (t1 )
t2  t1
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 ( RHS )t1
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approximate this as :
 (t2 )   (t1 )
t2  t1
 ( RHS )t1
where (t2 – t1) = timestep - might be 20 mins, say
heavily constrained – see 121B
(RHS) is value at current time, which can be measured
(e.g., radiosonde network etc.)
t1 is the value of vorticity at current time, and thus…
t2 is the value of vorticity in 20 mins = forecast !!
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This process is the basis of computer-based
weather forecasting.
Called Numerical Weather Prediction
NWP
Top-line forecast and climate models solve
several equations – not one.
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The full set of primitive equations
Let’s think about the atmosphere
Specifically – a single parcel.
What quantities do we need to
• know about
• measure
• Forecast
to fully understand and forecast?
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1. Wind
•
•
3-components
(u,v,w)
4. Temperature (T)
5. Density ()
6. Pressure (p)
These are the bare minimum.
We need equations to connect and forecast
these.
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1. Wind
•
•
3-components
(u,v,w)
4. Temperature (T)
5. Density ()
6. Pressure (p)
Ideal Gas Law:
p = RdT – connects three of the variables.
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1. Wind
•
•
3-components
(u,v,w)
4. Temperature (T)
5. Density ()
6. Pressure (p)
Hydrostatic Equation:
p
z
  g
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connects p and .
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1. Wind
•
•
3-components
(u,v,w)
4. Temperature (T)
5. Density ()
6. Pressure (p)
Equations of Motion:
du
1 p

 fv  Fx
dt
 x
dv
1 p

 fu  Fy
dt
 y
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





connect u,v, p
and .
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1. Wind
•
•
3-components
(u,v,w)
4. Temperature (T)
5. Density ()
6. Pressure (p)
We still need an equation to predict
temperature…of the form:
dT
 ...
dt
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We get this from the 1st Law of Thermodynamics (MET 60):
dq  c p dT   dp
1 dq dT
dp



c p dt dt
dt
dT 1 dq
dp



dt c p dt
dt
Recall: dq = heat added
dq → work done (pd) + internal energy change (cvdT)
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As it happens…
dp

dt
where “” is the vertical “velocity” when we use
pressure coordinates
 has units mb/s or hPa/s
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As it happens…
    gw
where “w” is the “regular” vertical velocity when
we use z-coordinates
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As a result, our equation becomes:
dT
J
  
dt
cp
Where we use the symbol “J” to represent heat
added.
This is called the Thermodynamic Equation
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There is a sixth equation – the equation of
continuity:
Physically – see Fig. 7.17
Math:
 u v 

      divergence !!
p
 x y 
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Convergence and vertical motions!
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The full set of predictive equations are called the
Primitive Equations
They are the basis for forecasting.
When there is moisture, we need an equation
for “total moisture”:
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d
total vapor + liquid + ice 
dt
effects of evaporation, condensation etc.
+ rain, snow etc.
+ advection
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Forecasting and chaos
Figure 7.26
•
Each yellow dot represents a “state” of the
atmosphere
–
–
–
•
Cold, cool, warm, hot
Humid vs. dry
Stormy vs. settled
Actually a very simplified atmosphere
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•
•
•
•
•
When we forecast, we make multiple
forecasts with a model
Each starts from a slightly different initial
condition
Resulting set of forecasts is called an
ensemble (“ensemble forecasting”)
On some days, all forecasts are about the
same
→ high confidence in forecast
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•
•
•
•
On some days, forecasts diverge somewhat
from each other as time progresses
→ moderate confidence in forecast
On some days, forecasts diverge dramatically
from each other as time progresses
→ zero confidence in forecast
Figure 7.26 shows these 3 situations
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Sadly, we cannot tell ahead of time which of
these situations we will be in.
This is why we look at multiple forecasts from
multiple models (GFS, ECMWF etc.)
e.g., http://weather.unisys.com/index.html
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•
•
•
When we forecast, we make multiple
forecasts with a model
Each starts from a slightly different initial
condition
Resulting set of forecasts is called an
ensemble (“ensemble forecasting”)
Case (a)
• Each yellow dot represents a “state” of the
atmosphere
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The General Circulation
What is it?
The time- and zonally-averaged large-scale
motions, structure, and behavior of the
atmosphere.
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What drives it?
•
Constraints such as: motions must prevent
huge equator-pole temperature gradients due
to heating.
–
–
•
Net radiative heating @ equator
Net radiative cooling @ poles
All atmospheric and oceanic motions work to
prevent the N-S temperature gradient!
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How do we measure it?
•
Take zillions of observations
•
Form averages over a month or longer
–
This “averages out” synoptic-scale systems
•
Form zonal averages (west-east)
•
Plot, study, understand, try to reproduce and
explain with models etc.
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We saw the average u-wind and temperature
fields in Fig. 1.11
Average v- and w-fields?
•
Indicate three overturning cells in each
hemisphere
•
•
http://en.wikipedia.org/wiki/File:Earth_Global_Circulation.jpg
http://en.wikipedia.org/wiki/File:Omega-500-july-era40-1979.png
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