24
DYNAMICAL OCEANOGRAPHY
2kmtoadepthof5km.
e. What is the in-situ temperature and the potential temperature of the parcel
at the final depth?
(1.3) Buoyancy frequency
Consider a stratified water column with a density profile ρ(z).
a. Derive the equation of motion for a water parcel that, without any exchange
of heat and salt with its environment, is subjected to a small initial vertical
displacement.
b. At t =0 , the position of the parcel is z = z 0 and the velocity of the parcel
is zero. Show that the buoyancy frequency N can be seen as the characteristic
oscillation frequency of the water parcel in a stably stratified water column.
c. Consider now the same situation but in the presence of friction that is linearly related to the velocity of the water parcel. Derive in this case also the
equation of motion for the water parcel. When is the water column unstably
stratified?
(1.4) Neutral surface and neutral density
The concept of a neutral surface is important when we consider the (small
scale) mixing processes between water masses.
a. Argue that the local mixing in the ocean is much larger along a surface of
constant density than perpendicular to this surface.
Neutral directions are the directions in which a parcel can move in an adiabatic
and isohaline manner without altering its buoyancy, i.e., neutral directions are
parallel to lines of constant buoyancy.
b. For a linear equation of state, show that neutral directions are orthogonal
to the vector ∇ρ where ρ is the local density. How can one identify these
directions in a typical T -S diagram such as Fig. 1.8?
c. With a nonlinear equation of state, it is necessary to perform the gradient
operation that removes pressure effects, just as is done for static stability. Show
DYNAMICAL OCEANOGRAPHY
2kmtoadepthof5km.
e. What is the in-situ temperature and the potential temperature of the parcel
at the final depth?
(1.3) Buoyancy frequency
Consider a stratified water column with a density profile ρ(z).
a. Derive the equation of motion for a water parcel that, without any exchange
of heat and salt with its environment, is subjected to a small initial vertical
displacement.
b. At t =0 , the position of the parcel is z = z 0 and the velocity of the parcel
is zero. Show that the buoyancy frequency N can be seen as the characteristic
oscillation frequency of the water parcel in a stably stratified water column.
c. Consider now the same situation but in the presence of friction that is linearly related to the velocity of the water parcel. Derive in this case also the
equation of motion for the water parcel. When is the water column unstably
stratified?
(1.4) Neutral surface and neutral density
The concept of a neutral surface is important when we consider the (small
scale) mixing processes between water masses.
a. Argue that the local mixing in the ocean is much larger along a surface of
constant density than perpendicular to this surface.
Neutral directions are the directions in which a parcel can move in an adiabatic
and isohaline manner without altering its buoyancy, i.e., neutral directions are
parallel to lines of constant buoyancy.
b. For a linear equation of state, show that neutral directions are orthogonal
to the vector ∇ρ where ρ is the local density. How can one identify these
directions in a typical T -S diagram such as Fig. 1.8?
c. With a nonlinear equation of state, it is necessary to perform the gradient
operation that removes pressure effects, just as is done for static stability. Show