Physics of the EUC: Preliminaries
333
au
ay = f3(y- Yo).
(6.2.13)
The shear of the flow becomes negative as the column moves south of its initial
position, and relative vorticity is produced as the planetary vorticity is diminished. Integrating (6.2.13) and insisting that u is also zero (i.e., negligible) at
the starting latitude where y = yo, yields:
u = ~ (y- y0 ) 2
(6.2.14)
2
which yields an eastward velocity at the equator where y = 0:
f3y2
u(y= 0) = - 0
2 .
( 6.2.15)
With f3 = 2 X w- 13 cm-'s- 1 , and choosing Yo to be the width of the observed undercurrent, i.e. y0 = 200 km, ( 6.2.15) yields as an estimate for
u(y = 0) of 40 cm/s, which is rather low. If y0 were chosen to be 300 km, the
equatorial speed would be 90 cm/s and closer to the values observed.
Although the solution of Fofonoff and Montgomery is valuable in illustrating how potential vorticity conservation produces an eastward current at
the equator, the sensitivity to the starting value, y 0 , is unacceptable. The
starting latitude for the calculation is clearly arbitrary. What ( 6.2.15) really
yields is a relation between the lateral scale and the equatorial speed. A complete theory must also yield the value for y0 unconnected with the arbitrary
choice of the starting latitude for the integration.
The form of the solution is also inadequate. If we consider the EUC as an
inertial boundary current produced at the equator by the emergence of higher
order physics due to the breakdown of the geostrophic balance, the profiles of
fields such as the relative vorticity should smoothly join to the extra-equatorial
fields. Instead, in the Fofonoff-Montgomery solution, shown in Fig. 6.2.2, the
relative vorticity abruptly starts its linear increase at the arbitrary latitude 80 .
It is important to recognize the distinction between potential vorticity
conservation and angular momentum conservation in the argument leading to
( 6.2.15). Angular momentum is not conserved since a zonal pressure gradient is
implicitly required for the consistency of (6.2.14). This can be seen by using
(6.2.13) in the zonal momentum equation. Thus the momentum equation:
uau+v(au_[Jy) =-~ap
(6.2.16)
ax
ay
pax
becomes:
vf3yo = .!!__ [E + ~u 2 ]
ax p 2
( 6.2.17)
The southward flow feeding the undercurrent and producing it by carrying
fluid elements which exchange planetary for relative vorticity is possible only if
the total Bernoulli function, pj p + u 2 /2 is a function of longitude. A ring of
333
au
ay = f3(y- Yo).
(6.2.13)
The shear of the flow becomes negative as the column moves south of its initial
position, and relative vorticity is produced as the planetary vorticity is diminished. Integrating (6.2.13) and insisting that u is also zero (i.e., negligible) at
the starting latitude where y = yo, yields:
u = ~ (y- y0 ) 2
(6.2.14)
2
which yields an eastward velocity at the equator where y = 0:
f3y2
u(y= 0) = - 0
2 .
( 6.2.15)
With f3 = 2 X w- 13 cm-'s- 1 , and choosing Yo to be the width of the observed undercurrent, i.e. y0 = 200 km, ( 6.2.15) yields as an estimate for
u(y = 0) of 40 cm/s, which is rather low. If y0 were chosen to be 300 km, the
equatorial speed would be 90 cm/s and closer to the values observed.
Although the solution of Fofonoff and Montgomery is valuable in illustrating how potential vorticity conservation produces an eastward current at
the equator, the sensitivity to the starting value, y 0 , is unacceptable. The
starting latitude for the calculation is clearly arbitrary. What ( 6.2.15) really
yields is a relation between the lateral scale and the equatorial speed. A complete theory must also yield the value for y0 unconnected with the arbitrary
choice of the starting latitude for the integration.
The form of the solution is also inadequate. If we consider the EUC as an
inertial boundary current produced at the equator by the emergence of higher
order physics due to the breakdown of the geostrophic balance, the profiles of
fields such as the relative vorticity should smoothly join to the extra-equatorial
fields. Instead, in the Fofonoff-Montgomery solution, shown in Fig. 6.2.2, the
relative vorticity abruptly starts its linear increase at the arbitrary latitude 80 .
It is important to recognize the distinction between potential vorticity
conservation and angular momentum conservation in the argument leading to
( 6.2.15). Angular momentum is not conserved since a zonal pressure gradient is
implicitly required for the consistency of (6.2.14). This can be seen by using
(6.2.13) in the zonal momentum equation. Thus the momentum equation:
uau+v(au_[Jy) =-~ap
(6.2.16)
ax
ay
pax
becomes:
vf3yo = .!!__ [E + ~u 2 ]
ax p 2
( 6.2.17)
The southward flow feeding the undercurrent and producing it by carrying
fluid elements which exchange planetary for relative vorticity is possible only if
the total Bernoulli function, pj p + u 2 /2 is a function of longitude. A ring of