1
Despite the apparent complexity of atmospheric motion systems as depicted on synoptic weather charts, the pressure and velocity distributions of weather disturbances are actually related by rather simple approximate force balances. in order to gain a qualitative understanding of the horizontal balance of forces in atmospheric motions it is useful to idealize the equations by considering flows which are steady state (i.e., time independent\ דימע בצמ) and have no vertical component of velocity. Furthermore it is useful to describe the flow field by expanding the horizontal momentum equation,
(where
D V
V
 f k V
1
 p (1)
Dt
 ui
ˆ  vj
ˆ
is the horizontal velocity vector) into its components in a so-called natural coordinate system.
The natural coordinate system is defined by the orthogonal set of unit vectors t , n and k . Unit vector t is oriented parallel to the horizontal velocity at each point; unit vector n is normal to the horizontal velocity and is oriented so that it is positive to the left of the flow direction; unit vector k is directed vertically upward.
For those of you who recall the right hand rule, in this coordinate system the index finger points at the direction of the velocity, t , the middle finger at the direction of positive
Fig 1 n , and your thumb at the direction of k .
In this system the horizontal velocity may be written as V

V t where V , the horizontal speed, is a nonnegative scalar defined by V

/ where s ( x,y,t ) is the curve followed by a parcel moving in the horizontal plane. The acceleration following the motion is thus
D V
 t
DV

V
D t
(2)
Dt Dt Dt
With the aid of Fig 2, we derive the rate of change of t from geometrical considerations.
 

R s

|
 t |

|
 t | (3) here R is the radius of curvature following the parcel motion, and we have used the fact that being a unit vector | | 1.
By convention R is taken to be positive when the center of curvature is in the positive n direction. Thus, for R>0 the air parcel turns toward the left following the motion, and for R<0, the air parcel turns toward the right following the motion.
Noting that in the limit
 s

0,
 t is directed parallel to n , the above relationship yields
D t
 n
Ds R
(4)

2 thus,
D t

D Ds
 n
Dt Ds Dt R
V and comparing (4) with (2) we have
D V
Dt
 t
DV
Dt
 n
V
R
2
 t t
  t

(5)
(6)

 s t
R n
Fig 2
Therefore, the acceleration following the motion is the sum of the rate of change of speed of the air parcel and its centripetal acceleration due to the curvature of the trajectory. Since the Coriolis force always acts normal to the direction of motion, its natural coordinate form is simply
 f k V fV n (7) while the pressure gradient force can be expressed as
    p  t

 s
 n

 n

The horizontal momentum equation may thus be expanded into the following component equations in the natural coordinate system:
DV
Dt
 
1

 p
 s
 

 s
(9)
V
R
2
 fV
 
1

 p
 n
 

 n
(8)
(10)
Equations (9) and (10) express the force balances parallel to and normal to the direction of flow, respectively. For motion parallel to the geopotential height contours,
/ s 0 and the speed is constant following the motion. If, in addition, the geopotential gradient normal to the direction of motion is constant along a trajectory, (10) implies that the radius of curvature of the trajectory is also constant. In that case, the flow can be classified into several simple categories depending on the relative contributions of the three terms in (10) to the net force balance.

3
Flow in a straight line ( R
 
) parallel to the height contours is referred to as geostrophic motion . In geostrophic motion the horizontal components of the Coriolis force and pressure gradient force are in exact balance so that V = V g
where the geostrophic wind, V g
, is defined by assuming R
  in equation (10) and obtaining fV g
 
1

 p
 n
 

 n
(11)
This balance is indicated schematically in Fig 3. The actual wind can be in exact geostrophic motion only if the height contours are parallel latitude circles. As discussed previously (see Holton, section 2.4.1), the geostrophic wind is generally a good approximation to the actual wind in extra-tropical synoptic-scale disturbances. However, in some of the special cases to be treated below this is not true. p
0
  p p
V g
Fig 3 : The balance of forces for geostrophic equilibrium. The pressure gradient force is designated by p and the Coriolis force by
Co. Geostrophic motion is the case when the pressure and
Coriolis force balance each other. p
0
Co
If the geopotential field is uniform on an isobaric surface so that the horizontal pressure gradient vanishes, (10) reduces to a balance between the Coriolis and the centrifugal forces
V
2
R
  fV R
V f
(12) if we neglect the longitudinal dependence of f , since the speed is constant in this case, so is the radius of curvature. Thus, the air parcels follow circular paths in the anticyclonic sense (clockwise rotation in the northern hemisphere and counterclockwise in the southern hemisphere). The period of this oscillation is
P

2

V
R

|
2 f

|

1
2 day

(13)
P is equivalent to the time required for a Foucault pendulum to turn through an angle of
180°. Hence, it is often referred to as one-half pendulum day . Since both the Coriolis and the centrifugal forces owing to the motion are caused by the inertia of the fluid, this type of motion is referred to as inertial oscillation , and the circle of radius | R | is called the inertia circle. It is important to realize, however, that the "inertial flow" governed by (12)

4 is not the same as inertial motion in an absolute reference frame. In the former, the force of gravity, acting orthogonal to the plane of motion, keeps the oscillation on a horizontal surface. In the latter, all forces vanish and the velocity maintains a uniform absolute velocity.
If the horizontal scale of a disturbance is small enough, the Coriolis force may be neglected in (10) compared to the pressure gradient force and the centrifugal force. The force balance normal to the direction of flow is then
V
R
2
 
1


 p n
 

 n
(14)
Therefore we obtain the speed of cyclostrophic wind
V


  n


1
2

R

 n

1
2
(15) which may be either cyclonic or anticyclonic. In both cases, the pressure gradient force is directed toward the center of curvature and the centrifugal force away from the center of curvature. An example of such a cyclostrophic scale motion is a typical tornado. The condition for such a motion is that
V
2
R
 fV

V fR

1 (16)
We recall that the Rossby number is defined as
Ro

V fR
(17) so that condition (16) refers to a large Rossby number.
The above three special cases,
Geostrophic flow fV g
 
1

 p
 n
 

 n
(11)
Inertial flow
V
2
  fV
 
1

 

 n
(12)
Cyclostrophic flow
V
R
R
2  p
 n
(14) as in most problems of a physical nature, are greatly simplified when we work with dimensionless variables. If we divide (10) by f
2
R we obtain
V
2

V
  f R
2 fR
1


(18)
We use the definition of V g
from (11) to write
Ro 2 
Ro

Rog where
Rog

V g fR
(19)
(20)

5
Fig 4 . classification of winds as a function of the Rossby number Ro .
Using this dimensionless equation it is easy to see that geostrophic flow corresponds to
Ro=Rog , meaning Ro <<1 such that Ro
2 
Ro . As we see in Fig 4, for Ro> -0.5 the wind flow may be approximated as being geostrophic, hence the name regular . For Ro <-0.5 the wind flow cannot be approximated as geostrophic, hence the name anomalous .
The minima of Rog is 0.25, below which there is no solution to (19) the division into High and Low refers the wind circling around a high or a low pressure zone,
High: Rog
Low: Rog
R
R

 n

 n
  for Ro
  for Ro


0
0
R
R
 n

 n



0

0

0
; for Ro
0
; for Ro


0
0
R
 n


R
 n


0

0

0
0
(21)
Low
Anomalous Regular
High Anticyclonic Anticyclonic
Antibaric cyclonic
*Baric flow : coriolis and pressure gradient forces are oppositely directed.
* Cyclonic flow : fR > 0
It is also easy to see that inertial flow refers to Rog << Ro and that cyclostrophic flow refers to Ro >>1.

6
Homework Assignment:
1 . A tornado rotates with constant angular velocity

. a) Show that the surface pressure at the center of the tornado is given by p
 p
0 exp




2 r
2
0
2 RT

 where p
0
is the surface pressure a distance r
0
from the center, R is the ideal gas constant and T is the temperature (assumed constant).
 b) If the temperature is 288K and pressure and wind speed at 100 m from the center are 10
2 kPa and 100ms
-1
respectively, what is the central pressure. c) Show that the pressure gradient at r = 0 is zero. Explain qualitatively , why the pressure gradient must be zero at r=0.
* hint1: a) which of the 3 flow cases does tornado corresponds to b) you will need to use the identity :
 dp
 ln p
 const .
p
2 . Use the synoptic map below to estimate the radius of curvature and wind speeds at points A, B and C. (map handed out at class)