3.11 The Pressure-Gradient Force
41
3.10.11 Additional Exercises for the Reader
Repeat this exercise with the inclusion of friction and vary the friction coefficien in
a range between 0.001 s
−1
and 0.5 s
−1
. Advanced readers can also try to implement
a quadratic friction term of the form −R | W | W , where R has now units of 1/m.
Use a semi-implicit approach for this term; that is −R | W
n
| W
n+1
, and choose
different values of R. Use the FORTRAN function “ABS()” to calculate the absolute
value | W
n
|.
3.11 The Pressure-Gradient Force
3.11.1 The Hydrostatic Balance
For a flui at rest, the downward acting gravity force is balanced by an upward acting
pressure-gradient force. This balance, called the hydrostatic balance or hydrostatic
approximation, can be written as:
0 = −
1
ρ
∂ P
∂z
− g
(3.28)
where P is pressure, z is vertical coordinate, ρ is local density, and g is acceleration
due to gravity. The minus sign in the pressure-gradient terms arises to make this
term positive if pressure decreases with height.
3.11.2 Which Processes are Hydrostatic?
It can be shown that processes of a horizontal scale large compared with their vertical scale are hydrostatic. Otherwise the dynamics are said to be nonhydrostatic,
which implies that the pressure fiel is modifie by the f ow. This book exclusively
deals with hydrostatic processes.
3.11.3 The Hydrostatic Pressure Field in the Ocean
Hydrostatic pressure in the ocean has three contributions: atmospheric pressure,
pressure excess or defici owing to elevated or lowered sea level, and pressure owing
to the density stratificatio in the ocean itself. Atmospheric pressure has no impact,
for the sea surface tends to adjust instantaneously to atmospheric pressure variations, such that the pressure below the sea surface remains virtually the same. This
is known as the inverted barometer effect. The pressure fiel associated with the
mean density and a plane sea level has no dynamical consequences for it is void of
horizontal gradients.
41
3.10.11 Additional Exercises for the Reader
Repeat this exercise with the inclusion of friction and vary the friction coefficien in
a range between 0.001 s
−1
and 0.5 s
−1
. Advanced readers can also try to implement
a quadratic friction term of the form −R | W | W , where R has now units of 1/m.
Use a semi-implicit approach for this term; that is −R | W
n
| W
n+1
, and choose
different values of R. Use the FORTRAN function “ABS()” to calculate the absolute
value | W
n
|.
3.11 The Pressure-Gradient Force
3.11.1 The Hydrostatic Balance
For a flui at rest, the downward acting gravity force is balanced by an upward acting
pressure-gradient force. This balance, called the hydrostatic balance or hydrostatic
approximation, can be written as:
0 = −
1
ρ
∂ P
∂z
− g
(3.28)
where P is pressure, z is vertical coordinate, ρ is local density, and g is acceleration
due to gravity. The minus sign in the pressure-gradient terms arises to make this
term positive if pressure decreases with height.
3.11.2 Which Processes are Hydrostatic?
It can be shown that processes of a horizontal scale large compared with their vertical scale are hydrostatic. Otherwise the dynamics are said to be nonhydrostatic,
which implies that the pressure fiel is modifie by the f ow. This book exclusively
deals with hydrostatic processes.
3.11.3 The Hydrostatic Pressure Field in the Ocean
Hydrostatic pressure in the ocean has three contributions: atmospheric pressure,
pressure excess or defici owing to elevated or lowered sea level, and pressure owing
to the density stratificatio in the ocean itself. Atmospheric pressure has no impact,
for the sea surface tends to adjust instantaneously to atmospheric pressure variations, such that the pressure below the sea surface remains virtually the same. This
is known as the inverted barometer effect. The pressure fiel associated with the
mean density and a plane sea level has no dynamical consequences for it is void of
horizontal gradients.