Calculations
With the above qualitative discussion in mind, some simple numerical values will be considered.
The
equation of motion for an ideal fluid of density p, horizontal velocity V, in a frame of reference fixed to earth
ole
is
V=-=Vp+28 sin (Vxk),
where Vp is horizontal pressure ascendent,
vertical vector.
(1)
@ is earth's angular speed, ¢ is latitude, and k is local unit
At the equator the last term vanishes and the problem becomes one-dimensional along the
equator (x direction).
Thus equation (1) becomes
Dire
du
dt
ap
(2)
ax
By finding expressions for p and p as functions of t, and holding x fixed, the goal of computing u will be
realized,
An expression for the variation of pressure as a function of time and distance along the equator can be
written as
p-p,= (Ap) oy cos (
x - xX)
a
t-t,
Ut am
)
;
(3)
where Py is mean pressure, (AP) ax is peak pressure difference, a is geocentric radius to level of interest,
x- Xx) is in 7a units east along the equator, b is time for one period (24 hours = 8.64 x10
4
sec).
Since
pressure and density change in phase with each other, a similar expression results for density:
(- ~ Xo
P
=
+
Po
(AP) vax cos
t - 2)
+
a
27
.
b
4
(4)
Integration of equation (2) after substitution from (3) and (4) yields
t
x - x
(Ap) max sin
us :
a
t - to
+ 20
b
x - XO
hy
Po
+
(Ap) vax cos(
a
dt.
(5)
t - tS
+ 27
b
}
Evaluating the right-hand side of (5) with boundary conditions that u = 0 at x = Xj t=t,, where
P-P
u
max
=P 7 PS = Zero:
=
b(Ap)
max
27a(Ap ) na
In
p
(is)
p. + (Ap) a
x
Now (6) can be evaluated with use of some typical numbers
b= 8.64 x 10° sec
+
a= 7,.0x 10° em
-
Ap = 3.85 x 10
2
wb (dynes/cm ),
Ap = 3,90 x io 38 em/em*
>
1
p, = 4.70 x io '® gm/om>
a
x
.
(6)