BRIEF REVIEW OF U/G DYNAMICS
ASSUMED BACKGROUND…CHAPTERS 1-4 OF JRH
(PLUS CAR!)
THE GOAL OF DYNAMICS…
USE THE GOVERNING EQUATIONS (OFTEN
SIMPLIFIED) TO UNDERSTAND DYNAMIC
PHENOMENA, INCLUDING OBSERVED STRUCTURES IN
THE ATMOSPHERE (E.G., TEMPERATURE FIELD FROM
POLE-EQUATOR, GROUND-MESOSPHERE)
MAIN FOCUS…
 MID-LATITUDES
 SYNOPTIC-SCALES
 TROPOSPHERE
THE GOVERNING EQUATIONS1 (IN SPHERICAL,
ISOBARIC COORDINATES) ARE:
DV
 f k  V  
Dt
MOMENTUM EQN.
(6.1)

RT
   
p
p
HYDROSTATIC EQN.
(6.2)
1
aka PRIMITIVE EQUATIONS
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 V 

0
p
CONTINUITY
J


  V    T  S p 
cp
 t

(6.3)
THERMODYNAMIC EQ.
(6.4)
WITH
D 


  V     
Dt   t
p
p
V = (u,v)  VH
k = unit vector directed upwards
f = Coriolis parameter = 2sin,  = latitude
(and  = longitude)
and  
df 2

cos  ,
dy
a
a = earth’s radius (6371 km)
 = geopotential =
 

 i
 x y
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
z
0
gdz

j   H

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 = 1/
R  Rd
 = pressure vertical “velocity” = Dp/Dt
and   gw
where w=Dz/Dt
Sp is a static stability measure

p 
 (ln  )
, where   T  o  ,  R / c p ,
S p  T
p
 p
and po = 1000 mb typically
A statically stable atmosphere has
 / z  0
 / p  0
S p  0.
We will always demand Sp > 0.
J = diabatic heating rate (per unit mass, pum)
Adiabatic  J = 0
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Additional notes:
 We assume frictionless unless otherwise specified (friction
is a second-order effect for us!)
 Spherical coordinates
dx  a cos  .d 
dy  a.d
 Curvature terms have been neglected via scale analysis
e.g.,
uv
tan  term
a
 Isobaric coordinates
o Easy correspondence with observations on e.g.,
500 mb ( 500 hPa) level
o Some equations are simplified (e.g., time derivative
drops out of continuity equation)
o Density “vanishes” from equations
e.g.,
1

 z p   p
o But…Sp varies strongly with height/pressure (and we
usually set it to a constant)
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Where do the governing equations come from?
1. Conservation laws of physics written in mathematical form
e.g., F = ma
a
F
DV F
Fi

 
m
Dt m
i m
2. Need to adapt equations to a non-inertial frame of reference
 Coriolis term & modification of gravitation  gravity
3. Expand into component form in chosen coordinate system
[(x,y,z), (x,y,p), (x,y,), (,, p) etc.]
4. Simplify – e.g., via scale analysis
Scale analysis
“A systematic examination of equation(s) term-by-term for the
purpose of simplifying the equation(s).”
e.g., vertical momentum eq.  hydrostatic equation
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dw
1 p

g
dt
 z
107 ,10,10
so,0  
1 p
p
 g , or    g
 z
z
this is OK provided H  L , where L is a length scale and
H a depth scale.
e.g., horizontal momentum eqs.  geostrophic wind equations
1
1
k  p  k  
f
f
1 
1 
ug  
, vg 
f y
f x
Vg 
Observations indicate VH  Vg , and that the atmosphere
“behaves geostrophically”.
This means that studying the geostrophic equations
(actually the quasi-geostrophic equations) tells us a lot
about how the atmosphere behaves.
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The Rossby number
Ro 
U
fL
where U is a speed scale, and L a length scale.
Ro is the ratio of acceleration (the inertial term) to Coriolis
terms in an equation.
So, Ro << 1 means Du/Dt << fv, for example.
This in turn means fv  -x (geostrophic).
Thus…Ro “small”  VH  Vg.
mid-lats, synoptic-scale, Ro  0.1
tropics, synoptic-scale, Ro  1.0
The Thermal Wind
Connects vertical changes in the horizontal wind with horizontal
temperature gradients.
If V H  V g ,
then
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V g
R
  k   pT
lnp
f
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Thus, T = 0  V/p = 0 … no shear
Barotropic atmosphere…=(p) only
Thus on a pressure surface,  = constant
 T = constant
 T = 0
 no shear!
Baroclinic atmosphere…=(p,T) only
Vorticity
A measure of the rotation of a fluid parcel about an axis
Natural coordinates:
 
V V

R n
where R = radius of curvature (!)
v u

x y
xy coordinates:
 
absolute vorticity:
   f
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Potential vorticity…is always some measure of the ratio
absolutevorticity
depthof vortex
incompressible atmosphere: q 
compressible atmosphere:
f
H
q    f 

p
205A…more to come!
The vorticity equation
Predictive eq. for vorticity…derived from momentum equations
D
(  f )  divergence term + twisting term
Dt
+ solenoidal term
(JRH Eq 4.17)
Often simplified via scale analysis (JRH section 4.4.3)
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The simplest form is the Barotropic Vorticity Equation (BVE)
DH
( g  f )  0 ,
Dt
where
DH  

   VH   
Dt  t

This is only strictly valid for an incompressible fluid and
horizontal motions in a barotropic atmosphere.
Divergence
Defined as
u v
 ,
x y
V


V
s
n
 V 
There is a predictive equation for V = , but it is not used
much.
Streamfunction
Is defined in the usual way…
If V = 0, then we can write
V  k  ,
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where  is the streamfunction
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so
u


,v 
.
y
x
Compare to:
ug  
1 
1 
,v 
.
f y
f x
Note that the streamfunction can be defined in any two
dimensions…not just in the xy-plane.
As an example, consider the time- and zonally-averaged northsouth overturning circulations such as the Hadley cell (more in
Chapter 10)(remind me to show a Figure!)
The continuity equation in this case for the mean circulation
( v, ) is:
 v 

 0,
y p
which is 2-D non-divergence.
We can therefore define a mean streamfunction ( ) via
v


,and   
.
p
y
The distribution of ( ) – via its gradients – therefore determines
the ( v, ) field … see Fig.10.7 (p.324).
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