Transcript Chapter 3. Applications of the Basic Equations

Chapter 3. Applications of the Basic Equations
3.1 Isobaric Coordinates
3.1.1 Horizontal Momentum Equation
Revisit  2.19  20  :
Du vu tan  uw
1 p


 2v sin   2w cos  
 Frx
Dt
r
r
 x
Dv u 2 tan  vw
1 p


 2u sin  
 Fry
Dt
r
r
 y
Neglecting the small terms, the horizontal components become:
Du
1 p
 2v sin  
Dt
 x
Dv
1 p
 2u sin  
or in vectorial form:
Dt
 y
DV
1
p where
V  iu  jv.  3.1
3  fk  V  
Reinisch_85.515
1
Dt

Horizontal Momentum Equation
To express the momentum equation in isobaric coordinates,
we use 1.22  for s=p (constant pressure surface):
 p 
 p  z   p 

 
     
 x  p  z  x  p  x  z
 p 
 p 
 p  z   p 
 p   z 
But    0         ,        
 x  p
 x  z
 z  x  p  y  z
 z   y  p
 z  
 p  z 
 p   z 
 p   z 
p      i      j       i    j
 z  x  p
 z   y  p
 z   x  p
 y  p 
 z 
 z  
  g   i    j   p  ( p = gradient on surface p= const).
 y  p 
 x  p
DV
 fk  V   p 
 3.2 
Dt 3
Reinisch_85.515
2
Total Derivative
Total derivative in isobaric coordinates:
D  Dx  Dy  Dp 
 


Dt t Dt x Dt y Dt p






  u  v 
  V 
t
x
y
p t
p
Dp
where  
is the "omega" velocity.
Dt
3
Reinisch_85.515
 3.3
3
Geostrophic Relations
 DV

The geostrophic 
 0  relation in isobaric
 Dt

coordinates becomes from  3.2  :
fVg  k   p , Apply  
 3.4 
 p  Vg  0
The divergence of Vg on a constant pressure surface is zero.
3
Reinisch_85.515
4
3.1.2 Continuity Equation
In Isobaric Coordinate System
Lagrangian control volume
 M    x y z     x y
p
1
   x y p.
g
g
1

D   x y p 
D  M 
g

 0

Dt
Dt
D  Dx  Dy  Dp 
But
 


, therefore with g=const:
Dt t Dt x Dt y Dt p
  x y p 

Dx

Dy

Dp
  y p  x
  x p  y
  x y  p
0
t
x
Dt
y
Dt
p
Dt
Dx
Dy
Dp



 u  v 
0  Dt  Dt  Dt  0 


 0.
x
y
p
x y p
u v 

For


0
:



0


V

0
3
Reinisch_85.515
p
x y p
p
5
3.2 Balanced Flow
To simplify the math, we usually introduce approximations, and
assume initially that the vertical velocity is negligible. We generally
start with steady state conditions, assuming  t  0, and study the
horizontal balance of forces. We first express the isobaric form of
the horizontal momentum equation in "natural" coordinates.
3
Reinisch_85.515
6
Several Types of Balanced Flow
Name
Approximation
Balance

0
s
FCo  FPG
Geostrophic Flow
R  ;
Inertial Flow
 p  0
CyclostrophicFlow
V
Ro 
fR
Gradient Wind
3
finite R;
FCo  FCentr
1
FPG  FCentr

0
s
FCo  FCentr  FPG
Reinisch_85.515
7
3.2.1 Natural Coordinate System
n
s(x,y,t)
ds
t
streamlines
Starting with the horizontal momentum equation  3.2 
DV
 fk  V   p  , we use the natural coordinates:
Dt
Ds
V  Vt, where V 
. then
Dt
DV DV
Dt

t V
Dt
Dt
Dt
3
Reinisch_85.515
8
Acceleration in Natural Coordinates
DV DV
Dt

t V
Dt
Dt
Dt
Dt V
The book has a good discussion showing that
 .
Dt R
Dt
Here R is the radius of curvature. The direction of
Dt
is toward the center of curvature, i.e., in the direction of n.
DV DV
V2

t
n
Dt
Dt
R
change of speed
along s
3
 3.8 
centripetal
acceleration
Reinisch_85.515
9
Coriolis Force in Natural Coordinates
The Coriolis force  fk  V becomes:
 fk  V   fkVn
The pressure gradient force   p  becomes:
 
 
 p    
t
n  . Substitute this in  3.2  :
n 
 s
DV
 
 
  fk  V   p    fVn  
t
n
Dt
n 
 s
DV DV
V2
Compare with

t
n:
Dt
Dt
R
DV
 V 2

V2



,
  fV 
or
 fV  
Dt
s R
n
R
n
3
Reinisch_85.515
10
3.2.2 Geostrophic Flow
DV
 V 2


,
 fV  
Dt
s
R
n
Straight line motion: R    fVg    n

Motion along height contour 
 0  Vg  const
s
Such motion is called geostrophic.
FPG
Vg
FCo
3
Reinisch_85.515
11
3.2.3 “Inertial” Flow const)
DV
 V 2

We had:

,
 fV  
 3.9  10 
Dt
s
R
n
We now consider the case where  is constant on the
isobaric surface:

V2
0
 fV  0
n
R

and
 0  V  const. Therefore
s
 3.12 
R  V f  const
(negleting the small variation of f with lat.)
This means circular motion of the air parcel. Since R < 0, the center
of curvature is on the right side of the stream line: anticyclonic motion
3
Reinisch_85.515
12
Period of Inertial Oscillation
The period of one oscillation, i.e., the time to move
through one full circle is
1
day
2 R
2 R 2
2
P



2
V
fR
f
2 sin 
sin 
 3.13
This result derived from the absence of pressure gradient forces
when  is uniform (constant) on the isobaric surface. Then Coriolis
and centrifugal forces balance. Since these 2 forces are based
on the inertia of the fluid, the oscillation are called "inertial" oscillations.
They rarely occur in the atmosphere.
3
Reinisch_85.515
13
3.2.4 Cyclostrophic Flow (Fco 0)
We start again with
DV
 V 2


,
 fV  
 3.9  10 
Dt
s
R
n
and neglect the Coriolis force (OK for small scale disturbances):
V2



 V  R
cyclostrophic wind speed  3.14 
R
n
n
The cyclostrophic balance is valid for Fcentr. FCo
Fcentr .
1.
V2
V
FCo 

, which is the Rossby number RRo .
RfV Rf

must be < 0.
n
This is cyclonic flow in northern hemisphere (Fig.3.4)
For R  0, i.e., flow curving toward left ,
3
Reinisch_85.515
14
Tornado, An Example Of Cyclostrophic Flow
Scale numbers for a tornado
V
V  30 ms , R ~ 300 m, f  10 s  RRo 
~ 103
Rf
1
4
1
1, OK
The tornado is a typical cyclostrophic flow. Although R can be
positive or negative, tornados are always cyclonic (R< 0).
Apparently the general environmental conditions favor cyclonic flow.
3
Reinisch_85.515
15
3.2.5 Gradient Wind ( s  0 )
If the flow is parallel to the height contours, then  s  0.
DV 


0
Dt
s
Such flow is called gradient flow. The real wind is generally not at
constant height. But we can take the horizontal component of the
real wind, called the gradient wind, as a good approximation of the
real wind. The gradient wind speed must satisfy  3.10  :
V2

 fV  
. Solve for V:
R
n
3
Reinisch_85.515
16
Speed of the Gradient Wind
fR

 fR 
V 
   R
2
n
 2 
2
 3.15
Discussion: V is real and positive. The values for R, f and can be
positive or negative, but only values that assure a real and positive
V are physical solutions. The resulting gradient wind speeds are a
bit more accurate than the geostrophic approximation, which neglects
the centrifugal force. At mid latitude the differences are 1-20%.
V2

Combining  3.10  and  3.11 ,
 fV  
, fVg   n :
R
n
Vg
V2
V
 fV  fVg  0 
1
 1  RRo  3.17 
R
V
fR
3
Reinisch_85.515
17
3.3 Trajectories and Streamlines
The path of an air parcel is called the trajectory, i.e., s(x,y,t) is the
trajectory, and
Ds
 V  x, y , t  .
 3.18
Dt
The radius of curvature R of the path s(x,y,t) is the curvature of the
trajectory. As an estimate, we generally use the curvature of the
height contours for R.
The streamlines are a snapshot of the slope (or direction) of the
velocity field over the synoptic map at a given time t 0 :
dy v  x, y, t0 

dx u  x, y, t0 
3
 3.19 
Reinisch_85.515
18

For steady state motion, i.e.,
 0, streamlines=trajectory,
t
and the streamlines of the height contours, along which the
gradient winds flows, gives the right curvature radius.

In general, pressure systems are moving, and
 0.
t
Assume   x,y,t  is the direction of the wind on any point
D
of the isobaric surface. Then
is the change of  along
Ds
the trajectory.
3
Reinisch_85.515
19
Radius of Curvature
From Figure 3.6
D
1

1
 s  R 

and

Ds Rt  x, y , t 
s Rs  x, y , t 
D  D  Ds D 
V


V
 3.21
Dt
Ds Dt Ds
Rt
The rate of change of the wind direction following parcel trajectory is
And
D  
  V

V


Dt
t
s
t Rs

 3.22 
V  V
 V V



 
Rt t Rs
t Rt Rs
1 1 

V   
t
 Rt Rs 
 3.23
As expected, for steady state solutions, Rs  Rt .
3
Reinisch_85.515
20
3.4 The Thermal Wind
Assume a geostrophic wind directed along the positive
y axis that increases in magnitude with altitude. Use
Figure 3.8 to show that such geostrophic winds are
associated with a horizontal temperature gradiant that
increases in the positive x direction
3
Reinisch_85.515
21
Vertical Wind Shear


In general fVg  k   p, or fug  
, fvg 
y
x
Differentiate re p:
ug
  vg
 
1
RT
f

,f

;    p  
p
p
y p
p x p

p
ug
vg
T
T
fp
R
, fp
 R
(remember p=const), or
p
y
p
x
ug
vg
R  T 
R  T 
 
,




 ln p f  y  p  ln p
f  x  p
Vg
R
 k   pT thermal wind equation
 ln p f
3
Reinisch_85.515
 3.30
22
Vertical Wind Shear
Vg
R
Integrate
  k   pT from p0 to p1 :
 ln p
f
p1
R
VT  Vg  p1   Vg  p0     k   pTd ln p
f p0
R
VT   k   p T
f
p1
 d ln p
p0
R
p0
VT  k   p T ln
f
p1
3
Reinisch_85.515
 3.31
23