Divergence and vorticity
1
Divergence and vorticity
A two-dimensional world
If we consider large-scale atmospheric flows, then the horizontal length scale (L  10010000 km) is a factor of 10 to 1000 larger than the vertical length scale (H  10 km). On
this scale the atmosphere is very “flat” and we may restrict ourselves to two instead of
three dimensions. This does not imply that the third dimension (the vertical direction) is
not important, but only that vertical motions (w) are smaller than horizontal motions (u,v)
by the same factor of 100-1000. For large-scale systems we may use as a reasonably
good approximation that w << u or v. We call this the “quasi-horizontal” approximation.
For calculations of the flow quantities vorticity and divergence the vertical component of
the wind speed (w):
w
z
t
(1)
in the (x,y,z) coordinate system, or

p
  gw
t
(2)
in the (x,y,p) coordinate system is neglected from here on.
Understanding divergence
The divergence of a vector field is relatively easy to understand intuitively. Imagine that
the vector field in Figure 1a gives the velocity of some fluid flow. It appears that the fluid
is exploding outward from the origin.
Figure 1. Vector field with (a) pure divergence (left) and (b) pure convergence (right).
This expansion of fluid flowing with a velocity field U = (u,v) is captured by the
divergence of U, which we denote div U. The divergence of the above vector field is
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positive since the flow is expanding. In contrast, the vector field in Figure 1b represents
fluid flowing so that it compresses as it moves toward the origin, we have what is called
convergence. Since this compression of fluid is the opposite of expansion, the
divergence of this vector field is negative.
From Figure 1a it is easy to see that both u x  0 and v y  0 as u is increasing in
the positive x-direction and v is increasing in the positive y-direction. Therefore
mathematically the two-dimensional expression for the divergence (D) is given by:
 u v 
 u v 
D  div U  
  or D  div U    
 x y  z
 x y  p
(3a,b)
where the subscript z or p indicates that this quantity is estimated on a surface with
constant height (z) or constant pressure (p). Usually the wind components u and v are
given on pressure surfaces and we use Equation (3b). There is just a very small
difference between the value of D on a pressure surface and on a surface with constant
height, and this difference is neglected in many applications. On a latitude-longitude grid
Equation (3) needs adjustment (see Appendix A). We shall not use this more
complicated expression. One last observation about the divergence: the divergence is a
scalar. At a given point, the divergence of a vector field is just a single number that
represents how much the flow is expanding at that point.
Understanding vorticity
The rotation or vorticity of a vector field is slightly more complicated than the
divergence. It captures the idea of how a fluid may rotate. Imagine that the vector field in
Figure 2 represents fluid flow.
Figure 2. Vector field with pure rotation. In this
case the rotation is counter clockwise or cyclonic.
It appears that fluid is circulating in a counter clockwise fashion. This macroscopic
circulation of fluid around circles (i.e., the rotation you can easily view in the above
graph) isn’t exactly what vorticity measures. But, it turns out that this vector field also has
vorticity, which we might think of as “microscopic circulation”. We denote the rotation or
vorticity by rot U. Unlike divergence vorticity is a vector and its direction in this case can
be found with the “right hand rule”: curl the fingers of your right hand in the direction of
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the rotation; your thumb will point in the direction of rot U, in this case out of the paper
(or screen). If the flow was not two but rather three dimensional, then the vorticity vector
could point in another direction. As large-scale atmospheric flows are approximately
horizontal it follows that we are mainly interested in the vertical component () of the
vorticity vector.
Mathematically the expressions for the vertical component () of the vorticity are:
 v u 
 v u 
     , or      .
 x y  z
 x y  p
(4a,b)
Simple calculations
Often the wind data are given on an almost rectangular grid. The calculation of the
divergence or vorticity is then straightforward (Figure 3).
Figure 3. Wind vectors are given on a rectangular grid.
In Figure 3 a part of a rectangular grid is shown and in each grid point a wind vector is
given. For the divergence in point P(i,j) Equation (3) can be used. For discrete values
we have:
u  u i 1 v j 1  v j 1
 u v 
 u v 
  i 1
Di , j      



2x
2y
 x y  i , j  x y  i , j
u  A  u B  vC   vD 


2x
2y
(5)
Note that only the components of the wind speed which are at right angles to the sides
of the square around P are important because only the flow in or out of the square
contributes to the divergence. A flow parallel to the sides of the square will not flow into
or out of the square and hence does not contribute to the divergence. However, the flow
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parallel to the sides of the square is very important for the possible rotation or vorticity of
the air around point P. For the vorticity in point P(i,j) Equation (4) transforms into:
 v
u 
 v
u 
 
 i , j       
 x y  i , j  x y  i , j
vi 1  vi 1 u j 1  u j 1


2x
2y
v A  vB  u C   u D 


2x
2y
(6)
Application of Equations (5) and (6) is rather simple. However before we perform this
application it is necessary to gain additional understanding in what exactly vorticity and
divergence are.
Divergence in natural coordinates
An alternative coordinate system giving a different view on divergence and vorticity is the
so-called natural coordinate system (Figure 4).
Figure 4. Definition of the natural coordinate system. The two unit vectors n and s are pointed
perpendicular (normal) and parallel (streamwise) to the local wind vector V, respectively.
In the natural coordinate system the direction of the unit vectors is not constant as in the
(x,y)-system in Figure 3, but this direction is determined by the direction of the wind. The
s-component (the streamwise direction) is parallel to the wind direction and the ncomponent (the normal direction) is perpendicular to it and points to the left.
As a consequence we have for the two-dimensional wind vector:
V  V ,0 ,
(7)
with the s-component V  0. If the wind direction reverses the coordinate system
reverses also but we still have: V  0. Hence by definition (!) it is impossible to have
V  0 . For the same reason the n-component of the wind is always exactly equal to 0.
In natural coordinates the expression for the divergence reads:
D
V

V
s
n
where β is the wind direction relative to the coordinate system.
(8)
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The quantity V s is a measure for the stretching (shrinking) of an air parcel in the
direction of the wind when his quantity is positive (negative) (Figure 5).
Figure 5. (a) Stretching and (b) shrinking in a natural coordinate system. In (a) the wind speed
increases in the direction of the flow (V/s > 0); in (b) it decreases (V/s < 0).
Positive values of the quantity  n indicate diffluence: streamlines are getting further
apart, while negative values of  n indicate confluence streamlines are getting closer
together (Figure 6).
Figure 6. (a) Diffluence and (b) confluence in a natural coordinate system. In (a) β increases
in the normal direction (β/n > 0); in (b) β decreases in the normal direction (β/n < 0).
Diffluence indicates that streamlines are getting further apart; confluence indicates that
streamlines are getting closer together.
In large-scale flow in the middle latitudes divergence due to diffluence is often
counteracted by a downstream decrease of the wind speed (Figure 7a). In the same way
convergence due to confluence is often compensated by a downstream increase of the
wind speed (Figure 7b). The total divergence, Equation (8), is the sum of two large but
opposing terms that nearly cancel. In active synoptic systems the absolute values of the
divergence D lies between 1x10-5 and 4x10-5 s-1.
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Figure 7. (a) Diffluent and (b) confluent flow with total divergence D = 0 in both cases. In (a)
the diffluence is compensated by the decrease in wind speed in the downstream direction; in
(b) the confluence is compensated by an increase of the wind speed in the downstream
direction.
For the calculation of both terms of the divergence when the wind speed is given on a
rectangular x-y grid (this is usually the case, see Figure 3), the simplest way is to
calculate the acceleration term V s . The expression is (see Appendix B):
V u V v V u P  V  A  V B  vP  V C   V D 




s V x V y V P 
2x
V P 
2y
(9)
The diffluence/confluence-term ( V  n ) should NOT be calculated directly in a similar
manner, but can be calculated with the aid of Equation (8) and the calculation of the
divergence (D) from Equation (5).
Vorticity in natural coordinates
In the natural coordinate system the expression for the vorticity reads:
 V
 V

s n
(10)
According to this equation vorticity is determined by curvature and shear, respectively.
The quantity  V n , also called shear vorticity, is a measure for the change of the
wind speed perpendicular to the wind vector. Positive (negative) shear vorticity means
positive (negative) values of  V n (Figure 8). The quantity V  s , the curvature
vorticity, is positive (negative) in a clockwise (counterclockwise) flow (Figure 9).
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Figure 8. (a) Negative and (b) positive (lateral) windshear in a natural coordinate system. In
(a) the wind speed decreases in the normal direction (V/n < 0); in (b) the wind speed
increases in the normal direction (V/n > 0). In (a) we have positive shear vorticity, in (b) we
have negative shear vorticity.
Figure 9. (a) Counterclockwise (cyclonic) flow and (b) clockwise (anticyclonic) flow in a natural
coordinate system. In (a) β increases downstream (β/s > 0) and we have positive curvature
vorticity; in (b) β decreases downstream (β/s < 0) and we have negative curvature vorticity.
For calculating both terms of the vorticity if the wind speed is given on a regular x-y grid
(this is usually the case), it is, just as with divergence, easiest to calculate the
downstream changes hence the shear vorticity  V n . This reads (see Appendix B):

V v V u V vP  V  A  V B  u P  V C   V D 




n V x V y V P 
2x
V P 
2y
(11)
The curvature term ( V  s ) follows directly from Equation (10) and the calculation of
the vorticity () from Equation (6).
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Appendix A. Expressions on a sphere
Because the earth is a sphere to a good approximation and the wind components are
usually given on a latitude () longitude () grid, Equations (3) and (4) should be
corrected by using the following expressions:
D
1
r cos 
 u v cos   




 
p
(A1)
 
1
r cos
 v u cos  




 
p
(A2)
where r is the average radius of the earth (6.37106 m). These equations only given for
completeness and will not be used in this unit.
Appendix B. Derivations
Divergence
For the divergence the easiest way is to calculate the acceleration term V s (Figure
10). We will do the calculations in the simple x-y coordinate system.
Figure 10. Wind components and the unit vector (s) in x-y coordinates.
For the shear of the wind speed (V) in the x-y coordinate system we have:
 V V 
 .
V  
,
 x y 
(A3)
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The unit vector s in natural coordinates is by definition parallel to the wind vector (V) and
has the x-y system the following coordinates:
 u v u v
s  s x , s y   s cos  , s sin     s , s    ,  .
 V V  V V 
(A4)
The component of the wind shear (A3) in the direction of the unit vector s can be
calculated directly by:
s  V 
u V v V

V x V y
(A5)
Vorticity
For the vorticity the calculation is analogous, but now we need the wind shear along the
n-vector and not the s-vector (Figure 11).
The unit vector n in natural coordinates is perpendicular to the wind vector (V) pointing
to the left. Its components in the x-y system are:
v
u  v u 

n  n x , n y    n sin  , n cos      n , s  n   
, .
V
V V V

(A6)
Figure 11. Wind components and the unit vector (n) in x-y coordinates.
The component of the windshear (A3) in the direction of the unit vector n now follows
directly from
n  V 
 v V u V

V x V y
(A7)