1
The Primitive Equations
Coordinate System:
The frame of reference in which the primitive equations are presented is of local
Cartesian coordinated placed on a rotating sphere.
The 3 components of the
velocity are defined locally in
Cartesian coordinates.
Atmospheric motions relative to a
spherical rotating planet

u  uiˆ  vjˆ  wkˆ
y
z
x
a
y'
z'


x’
(1)
We should not confuse
the above velocities with the
ones we would have used in
plane spherical
coordinates.
d
u  zonal  a cos 
dt
d
v  meridional  a
dt
dz
w  vertical 
dt
(2)
The depth of the atmosphere is very small compared with the radius of earth,
a  REarth  40,000km / 2
It is useful to write
r = a + z ; z << a
where z is the height above mean sea level.
(3)
(4)
Newton’s Second Law
Written per unit mass, Newton’s second law is given by
Du
  Fi
(5)
Dt
i
Where,
D 




  u   u  v  w
(6)
Dt t
t
x
y
z
Newton’s second law simply states that the acceleration per unit mass of a parcel
of air is equal to the vector sum of the forces acting on it. We should list the forces most
relevant in atmospheric motions:
2
Gravity: the gravitational force (per unit mass) is spherically symmetric and therefore
only vertical. It is given by
GM
GM E
GM
Fg   2 E zˆ  
zˆ  2 E zˆ  g
(7)
2
r
a
a  z
Where G is Newton’s Constant and ME is the mass of earth. It is often written as the
gradient of the Gravitational potential
Fg  
(8)
GM E
 gz
(9)
r
Pressure: the force exerted by pressure acting on a unit area is
Fp  p  A
(10)
For a given unit area, the net force will be,
(r ) 
F
p
p(s0+s)A
p(s0)A
s0
s
 p(s0 )A  p(s0   s)A
(11)
In order for the net force to be in ‘force per
mass’ units we divide by a unit mass. A unit
mass is given by A s so that when we divide
the net force by a unit mass we get
1 p( s0   s)  p( s0 )
(12)

s
Since s is a general coordinate, we generalize
the expression and get
1
(13)
Fp   p
Fp  

Friction: aside for stating that friction always acts to halt the wind at any given location,
there is no general empirical or otherwise accurate expression for this force. In global
models where an analytical expression for the friction force is required we would often
find an expression relating the friction force to some positive power of the velocity. The
simplest approximation is derived by a linear parametrization of the variables,
u u
F 
(14)
t  D
Where D is a typical time scale (for earth, this typical time scale is of the order of five
days). Such a friction force is often used because it yields an exponential decay of the
velocity in the absence of other forces.
 t
u
u
F 
   u(t )  e  D
(15)
t
D
Coriolis Force: as mentioned above, the frame of reference we’re using is of Cartesian
coordinates on a rotating sphere. The fact that our Inertial frame of reference is placed on
a rotating frame of reference introduces a ‘correction’ or ‘relative’ force given by
3
 |   r |2 
 Du I   Du R 
(16)




  2  u R

 

2 
 Dt  I  Dt  R

Where the subscript R is used for the rotating frame of reference and the subscript I for
the inertial frame of reference. The second term on the right of (16) is the centripetal
acceleration, presented as the gradient of a scalar. This term is often negligible with
respect to other forces and may be written as a small correction to the gravitational
potential. The third term on the right is the ‘Coriolis force’. The ‘Coriolis force’ is often
described as a ‘pseudo force’ resulting from the geometry of the reference frame we
chose. For practical uses, it being ‘real’ or ‘fictitious’ makes no difference in the way we
write the equations.
When we sum all the above forces, Newton’s second law reads
u
1
(17)
 (u  )u    2  u  p  F
t

Equation (17) is better understood when broken into components. We begin by treating
the left side of (17):

u


 
 (u  )u    u  v  w  (uiˆ  vjˆ  wkˆ)
t
x
y
z 
 t
Du ˆ Dv ˆ Dw
Diˆ
Djˆ
Dkˆ
 iˆ
 j
k
u
v
w
Dt
Dt
Dt
Dt
Dt
Dt
Du
Diˆ
Djˆ
Dkˆ

u
v
w
Dt
Dt
Dt
Dt
 iˆ
Du
iˆ
iˆ
iˆ   ˆj
ˆj
ˆj
ˆj 

 u  u  v  w   v  u  v  w 
Dt
x
y
z   t
x
y
z 
 t
 kˆ
kˆ
kˆ
kˆ 
 w  u  v  w 
 t
x
y
z 

(18)
Du
is the ‘regular’ differential of the velocity. The rest of the terms result from our
Dt
choice of coordinates. The way each unit vector changes in the above 12 terms needs to
be addressed one by one. First of all, the unit vectors do not change in time
iˆ ˆj kˆ
ĵ
 
0
(19)
t t t
Next, motion in the vertical direction does not
affect the horizontal unit vectors,
iˆ ˆj kˆ


0
(20)
z z z
We are now left with 6 terms.
iˆ
k̂
we see that the latitude lines are always
horizontal, therefore:
4
iˆ
0
y
(21)
iˆ '
iˆ
iˆ
iˆ ' iˆ
On the other hand, when we move along a latitude
line, the direction of iˆ changes. On the left, looking
from above at any of the latitude lines, we observe that
on the equator,
iˆ iˆ ' iˆ
(22)
limx 0

x
x
Infinitesimally, we get
iˆ ' iˆ kˆ
(23)
| iˆ | | x |
It is easy to see that
and we get (on the

a
| iˆ |
iˆ kˆ
 . On latitudes other than the equator
equator)
x a
we get
iˆ
1
kˆ
(24)

(cos  kˆ  sin  ˆj )    tan  ˆj
x
a cos 
a
iˆ
ĵ
Calculating
on the equator is equivalent to calculating
anywhere, therefore,
y
x
ĵ kˆ

(25)
y a
In a manner similar to the one used in order to obtain (24) we find that
kˆ
| kˆ | x
kˆ iˆ
: kˆ iˆ ;



x
a
x a
| kˆ |
kˆ
| kˆ | y
kˆ ˆj
: kˆ ˆj ;



y
a
y a
kˆ

b
ĵ
ˆj '
ˆj ' ˆj
ĵ
is a little bit trickier:
x
we see that ˆj iˆ and using some
geometry we find that
| ˆj | x
a

; b
(28)
ˆ
b
tan 
| j|
And finally we get
ˆj
iˆ
  tan 
(29)
x
a
It is left now to group the terms.
(18) is now written as:
(26)
(27)
5
 kˆ
 iˆ   ˆj
u
Du
ˆj 
kˆ 
 (u  )u 
 u  u   v  u  v   w u  v 
 x
t
Dt
y 
y 
 x   x

2
 ˆ  u 2  v2 
Du ˆ  uw uv
 ˆ  vw u

i
 tan    j 
 tan    k  

Dt
a
a
a 
 a

 a


Du
is the regular differential in spherical coordinates:
Dt
Du u  u
 v 
 



 w u
Dt t  a cos   a 
z 
Where we used

1  ˆ
1
 ˆ
(32)
  rˆ 


r
r 
r cos  
In spherical coordinates.
(30)
(31)
We now need to break the Coriolis force into components:
We see that the  changes locally as we move along the longitude curve.  is given by
  cos  ˆj  sin  kˆ
(33)
Locally. Therefore,
ˆj
iˆ
kˆ
2  u  2u    2 u
v
w  iˆ2  v sin   w cos  
(34)
0 cos  sin 
 ˆj 2  u sin    kˆ 2  u cos  
It is left now to group the terms according to their components. (18) broken into
its components reads
u
u u v u
u uv
uw


 w  tan  
t a cos   a 
z a
a
(35a) 
1
p
 2v sin   2w cos  
 F1
 a cos  
v
u v v v
v u 2
vw


 w  tan  
t a cos   a 
z a
a
(35b)
1 p
 2u sin 
 F2
 a 
6
w
u w v w
w  u  v 


w

t a cos   a 
z
a
(35c)
1 p
 2u cos   g 
 F3
 z
Finally, (35a), (35b) and (35c) are simplified when we take into account that the vertical
component of velocity is negligible with respect to the horizontal components.
Furthermore, the vertical component of the equation of motion, (35c), is dominated by
the pressure gradient and g. We also define the ‘coriolis parameter’, f,
f  2 sin 
(36)
Therefore we may write
u
u u v u uv
1
p
(37a) 


 tan   fv 
 F1
t a cos   a  a
 a cos  
v
u v v v u 2
1 p
(38b)


 tan    fu 
 F2
t a cos   a  a
 a 
1 p

(39c)
 g 
 z
z
(39c) is also called the Hydrostatical Equilibrium equation.
2
2
Continuity Equation
The continuity equation for conservation of matter is

D
 (  u )  0 
    u
(40)
t
Dt
We may write the density as
(41)
  R   A
Where R is the mean density at any height and A is the departure from this mean
density. In planetary atmospheres the vertical changes in density are much larger than the
horizontal fluctuations,      z  . Scale analysis also shows that

 (  u )
(42)
t
Or in other words, the speed in which the air responds to local perturbations, (the speed
of sound) is much larger than the wind speed such that locally, the density is close to
constant over time. Under the above conditions the continuity equation may be reduced to
w  R w
v 
0
(43)
 R z
Where v(u,w) is the horizontal component of the velocity.
Thermodynamic equations:
The first law of thermodynamics is stated as
dQ  dU  dW  cv dT  pdv  c p dT  vdp
(44)
where v is the volume per mass unit and cp=cv+R (R being the ideal gas constant). Many
atmospheric processes occur at constant pressure and the use of cp is often useful.
7
For Adiabatic processes (dQ = 0) using the ideal gas equation,
RT
v
(45)
p
we get
k
P
R
c p dT  vdp  T     ; k 
cp
 Pr 
We may use (46) and rewrite the first law
(46)
k
P
dQ  c pTd (ln  )  c p   d
(47)
 Pr 
is a constant of integration. It is referred to as the potential temperature (or the
temperature at P=Pr) and Pr is usually taken to be 105Pa.
When heat is added or taken from the system, differentiating (47) produces
k
d 1  P  dQ
  
l
dt c p  Pr  dt
(48)
Where l denotes the rate of change of due to heating. The above is the Lagrangian
derivative of a block of air. If we want to work in an Eulerian frame of reference we need
to add the rate of chance due to advection, u   ,

 u    l
(49)
t
The Primitive Equations
Equations of (horizontal) motion
u
u u v u
u uv
1
p


 w  tan   fv 
 F1
t a cos   a 
z a
 a cos 
v
u v v v
v u 2
1 p


 w  tan    fu 
 F2
t a cos   a 
z a
 a 
Hydrostatic Equilibrium equation
1 p

 g 
 z
z
Continuity Equation
1 u
1 (v cos  )
w    ( z )w


0
a cos   a cos 

 ( z)
z
Thermodynamic equation

u  v 



w
l
t a cos   a 
z
(a) 
(b)
(c)
(d)
(f)