GEF 2500 Problem set 4
1) Using approximate, but realistic values for the observed stratification, what is the
buoyancy period (i.e. ⁄ , where N is the Brunt-Väisälä frequency) for:
a) The mid-latitude troposphere
b) The equatorial troposphere
c) The lower part of the stratosphere
d) The oceanic thermocline in the mid-latitudes
e) The oceanic thermocline in the tropics
f) The deep ocean (1000m-4000m)
Use the figures below and on the next page to determine realistic values
The potential density profile of the deep ocean. Labels denote potential density
values.
(figure from http://paoc.mit.edu/labweb/)
The potential density profile of the upper ocean. Labels denote potential density
values.
(figure from http://paoc.mit.edu/labweb/)
2) Stability and the Brunt-Väisälä frequency
In this exercise we consider the topic discussed in chapter 2.1 in the compendium.
a) Consider a density profile as given in the figure below (the only feature you should
note from this profile is the sign of the gradient; do not attempt to extract any values).
A fluid particle with density
is initially situated at
, where
. The
particle is suddenly lifted a small distance upwards. Take into account that the fluid
may be compressible, and use a Tailor approximation of the first order to derive an
expression for the density of the particle after it has been displaced to
.
Then do the same for a particle displacement downwards, i.e. to
.
b) What would happen if the fluid was incompressible?
c) The density profile in the figure above shows the density of the environment fluid,
. If the density at
is , derive an expression for the environmental
density a small distance δz above and below
.
d) If the density of a fluid particle is different than the density of the surrounding fluid, it
will be accelerated. Explain this principle, and use the previously derived results to
derive the stability conditions (2.1.3)-(2.1.5) in the compendium.
e) In the case of an incompressible fluid, the density profile in the figure shows a stable
situation. However, if the fluid is compressible, the profile might be unstable. Explain!
f) Assume that N is constant, and solve equation (2.1.8) in the compendium with the
following initial conditions:
i)
ii)
(
(
)
)
g) How will the solutions develop in the case of
3) (Vallis ex. 1.18)
and
? Explain!
A parcel of water is added to the ocean surface that is denser than any water in the
ocean. Suppose the parcel sinks adiabatically to the ocean bottom. Estimate the change
in temperature that the parcel undergoes, being explicit about the assumptions you
make.
Hint: use eqn. 2.3.1 from the compendium and the table below
T0
Reference temperature
283K
ρ0
Reference density
1027kg m-3
γT
First thermal expansion
1.67 10-4 K-1
coefficient
cp
Specific heat at constant
3986 J K-1 kg-1
pressure
OPTIONAL:
4) Another take at the continuity equation
a) Derive the conservation of mass equation (1.4.8) in the compendium.
b) If the density is constant everywhere, it is trivial to show that the continuity equation may
be written as
⃗
. In the atmosphere, this approximation is not necessarily fulfilled,
as air is compressible. However we will now look at an example where the continuity
equation still applies.
Consider large scale atmospheric motion. Typical horizontal length scale L=1000 km,
vertical length scale H=10 km and horizontal velocities U=10 m/s. Typical time scale is 1
( )
day. The density is separated as
̂(
), where ( )
̂ . We
assume that
|
|
|
|
Use conservation of mass to show that under these circumstances, the continuity
equation ⃗
is a valid approximation, even though the density is not constant.
c) Use scale analysis to show that in the case described above, the hydrostatic
approximation is a valid one. Insert realistic values (scale values) into the continuity
equation to obtain a characteristic value for the vertical velocity.
d) Consider the vertical component of Navier Stokes equation, i.e. eqn. 1.6.5 and 1.6.8c
in the compendium. Insert realistic values (scale values) and show that the dominate
balance indeed is the hydrostatic balance.
e) Consider a convective cumulus cloud. The vertical velocities are in the range of 1025m/s. Use typical values of U~W~20m/s, L~1km, H~500m and T~100sek. Would
you assume hydrostatic conditions in such a system?