Regions of freshwater influence (ROFIs) - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 268

Given that the pressure in such a steady flow will be hydrostatic ( Equation 3.21 ), the
pressure gradient can be written as:

0
@

1
A ¼ @
@

0
@

1
A

ð
z

ð ð

0

0
z

@ p

x ¼ @

þ

gdz gdz gdz

@ @ x x

ð 9 : 2 Þ

ð

0

@
@

þ s g @
@

gz @
@ x þ s g @

¼

x gdz x ¼ @ x

z

where, in the final step, the depth-uniform x has been taken outside the integral

and r s is the density at the surface. Assuming ( Equation 3.40 ) that the stress is related
to the velocity shear by t x ¼

@ r/ @

N z 0 @

u

z where N z is the eddy viscosity, the force @

balance can now be written as:

:

@ p

gz @
@ x þ s g @ x ¼ @

t x
@ z ¼ 0 @

N z @

u

x ¼

ð 9 : 3 Þ

@ @ @ z @ z

If N z is independent of depth and making the reasonable approximation that r s r 0

the reference density, this can be arranged to give:
@

2 u

g
N z

g
N z z 2 ¼

z ð 9 : 4 Þ

@

where w ¼ @ / @ x and x ¼ (1/r 0 ) @ r/ @ x. After integrating twice with respect to z,

we have:

2N z z 2

g

6N z z 3

g

u ð z Þ¼ þ Az þ B ð 9 : 5 Þ

where A and B are constants of integration to be determined from the boundary
conditions: (i) at the surface (z ¼ 0) the stress t s ¼ 0 and (ii) at the bottom (z ¼ h)

the velocity u ¼ 0. Applying these conditions gives the constants as:

h 2
2N z

h 3
6N z

g g

A ¼ 0 ; B ¼ ð 9 : 6 Þ

so that the solution for u is:

2N z ð

g

6N z ð

g

z 2 h 2 z 3 h 3

u ð z Þ¼ Þ þ Þ: ð 9 : 7 Þ

Note that an alternative bottom boundary condition, setting the bed shear stress to
match the bottom stress given by a quadratic drag law t b ¼

u b , yields similar
results which you can see in a numerical simulation at the website. In a steady state,
the gradients w and x are not independent (Officer, 1976 ) .
In a steady state, which you can see in a numerical simulation at the website, the
gradients w and x are not independent. They are related through the condition that the
net transport must match the freshwater input R w per unit width from the river, i.e.

k b r j u b j