The shelf edge system - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 305

10.2 Bathymetric steering and slope currents

...................................................................................
An important and fundamental aspect of the flow at the shelf-deep ocean boundary is
that the steep topography of the slope, in combination with the Earth's rotation,
imposes a particular discipline on the currents. This constraint on the flow, which is
known as the Taylor-Proudman theorem after its discoverers, will now be derived
from the equations of motion following the approach of Brink (Brink, 1998 ).

10.2.1 The Taylor-Proudman theorem

We start by assuming that the flow is steady and in geostrophic balance so that from
Equation (3.16) we have:

1 p 1 p

] ]

fu ¼

y ;

fv ¼

x :

ð 10 : 1 Þ

] ]

We are aiming at understanding how geostrophic flows might be influenced by a sloping
seabed, which could lead to flows converging or diverging. So we want to know how
geostrophic flows change horizontally, while at the same time being constrained by flow
continuity. Differentiating Equations ( 10.1 ) with respect to x and y respectively and
combining with the continuity Equation (3.1) ,wehave

u v w

]

]

x þ ]

¼ ]

z ¼

0 ð 10 : 2 Þ

y

] ]

i.e. the vertical velocity is everywhere independent of depth. Since w ¼ 0 at the

surface (z ¼ 0), it follows that w ¼ 0 throughout the water column. This condition

that w 0 everywhere means that the flow has to be parallel to the isobaths; cross-

isobath flow would require a component of flow upward or downward. We can
express this formally as

¼

u

:r

h ¼ w ¼ 0 ð 10 : 3 Þ

i.e. the flow vector u at the bed must be perpendicular to ∇ h and hence parallel to the

isobaths.
In addition, for a steady state and neglecting diffusion, Equation ( 4.35 ) for the
transport of density becomes

u ]
]

v ]
]

w ]
] x þ y þ z ¼

0 : ð 10 : 4 Þ

Since the flow is geostrophic, we can use the thermal wind balance ( Equation 3.30 )to
write the density gradients in terms of the velocity shear to give:

]
]

x ¼ 0 f

]
]

y ¼ 0 f

] v ] u

z ; z :

ð 10 : 5 Þ

g g

] ]