2. ®
®
®
®
®
®
+
+
Ñ
-
´
W
-
= r
F
g
P
U
dt
U
d
r
1
2
0
.
. =
Ñ
+
Ñ
+
¶
¶ ®
®
u
u
t
r
r
r
.
Q
dt
dp
dt
dT
Cp =
-a
Complete set of model equations when moisture (q = specific humdity) is included
There are seven unknowns: U (u, v, w); T, p, (ρ or α) and q; and seven equations
3 Momentum equations for
the 3 components (u, v, w)
Continuity equation
Thermodynamic energy equation
RT
p =
a Equation of state
( ) )
(
. C
E
Uq
t
q
-
+
-Ñ
=
¶
¶
r
r
r
Moisture equation
E = Evaporation (moisture source)
C = Condensation (moisture sink)
α = Inverse of density
= Specific volume
3. Recall Newton’s second law;
Momentum Equations
Equation of Motion
®
®
®
®
®
®
+
+
Ñ
-
´
W
-
= r
F
g
P
U
dt
U
d
r
1
2
Rate of change of velocity following the motion = Sum of forces acting per unit
mass.
®
®
´
W
- U
2 Coriolis force
®
Ñ
- P
r
1
Pressure gradient force
®
®
*
®
W
+
= R
g
g 2
Gravity force per unit mass = Sum of the
gravitational and centrifugal force terms
®
r
F
Frictional force in the fluid
4. Pressure Gradient Force
x-component of the PGF per unit mass
!!
"
= ⎯
#
$
%&
%'
y- and z components of the PGF per unit mass:
!"
"
= ⎯
#
$
%&
%(
!#
"
= ⎯
#
$
%&
%)
!
"
= ⎯
#
$
∇𝑃
It is important to note that:
• The pressure gradient points from low to high pressure, but the pressure gradient
force points from high to low pressure.
• The pressure gradient force is proportional to the gradient of the pressure field, not
to the pressure itself.
5. Geostrophic motions
ug = -
1
rf
¶p
¶y
vg =
1
rf
¶p
¶x
with the Coriolis parameter f = 2W sinf.
!!
"!
vg =
1
rf
"!
k ´ Ñp
Component form:
Vector form:
Ø The geostrophic wind describes the dominant balance
between the pressure gradient force and the Coriois force.
• Just above the top of the boundary layer the atmosphere is close to geostrophic balance…
8. Tropical M. D. Eastin
Land / Ocean Forcing:
Land
• Major elevation features deflect air
over (clouds and precipitation) and
around (cyclonic / anticyclonic flow)
• Differences in elevation create thermal
gradients due to surface heating
(e.g. Indian Monsoon)
Ocean
• Oceans are a heat and moisture
reservoir that the atmosphere “taps”
(ocean has a large heat capacity)
• Differential solar heating leads to
thermal gradients and ocean currents
Land-Ocean
• Large heat / moisture gradients often
help force atmospheric circulations
(nor-easterlies and land-sea breezes)
12. ☞ Incoming solar radiation
• Stronger at low latitudes
• Weaker at high latitudes
☞ Tropics receive more solar radiation per unit area than
Poles.
☞ Global Atmospheric Circulations are driven by the uneven horizontal
distribution of the net incoming radiation.
13. Differential heating
Specific heat is the
amount of heat required to
change the heat content of
exactly 1 gram of a
material by exactly 1°C.
Water – 4.18 J/ °C/g
Land ~ 1 J/°C/g
14. Local wind Systems-Land and Sea Breezes
Ø Thermal Circulations: Circulations brought on by changes in air temperature, in
which warmer air rises and colder air sinks.
Ø It’s a diurnal wind activity
Ø Specific heat plays an important role in it.
How they form:
• During the day, the land heats more quickly than the adjacent water, and the
intensive heating of the air above, produces a shallow thermal low.
• The air over the water remains cooler than the air over the land;
• Hence, a shallow thermal high exists above the water. The overall effect of this
pressure distribution is a sea breeze that blows from the sea toward the land.
• At night, the land cools more quickly than the water. The air above the land
becomes cooler than the air over the water.
• With higher surface pressure now over the land, the wind reverses itself and
becomes a land breeze—a breeze that flows from the land toward the sea
15. Ø Sea breeze: from sea to
land
Ø Land breeze: land to sea
Ø Land and Sea Breezes also
occur near the shores of
large lakes
Local wind Systems-Land and Sea Breezes
16. General Definition of the Tropics
“…the latitudes at which the Coriolis force, horizontal
temperature gradients, and horizontal pressure
gradients are all relatively weak…”
Defining the Tropics
