SOME OCEAN MODEL FUNDAMENTALS
3 1
In this equation, A 4 = pdV is the parcel's mass, p is its in situ density, and dV is its infinitesimal volume. The time derivative is taken
following the parcel, and is known as a material or Lagmngian time
derivative. Writing dV = dx dy dz, and defining the
v = dx/dt = (u, w) leads to
d lnp
= -V . v.
dt
Note that the horizontal coordinates xh = (x, y) can
parcel's velocity as
(2)
generally be spherical coordinates (A, q5), or any other generalized horizontal coordinate
appropriate for the sphere, such as those illustrated in Figures 1 and 2
(see chapters 20 and 21 of Griffies, 2004 for a presentation of generalized
horizontal coordinates).
For many purposes in fluid mechanics as well as ocean model design,
it is useful to transform the frame of reference from the moving parcel to
a fixed point in space. This transformation takes us from the material
or Lagrangian frame to the Eulerian frame. It engenders a difference in
how observers measure time changes in a fluid parcel's properties. In
particular, the material time derivative picks up a tmnsport or advective
term associated with motion of the parcel
This relation allows us to write the Lagrangian
conservation in an Eulerian conservation form4
P,t + V . (pv) = 0.
(3)
expression (2) for mass
(4)
Fluids that conserve mass are said to be compressible since the volume of a mass conserving fluid parcel can expand or contract based on
pressure forces acting on the parcel, or properties such as temperature
and salinity. However, in many circumstances, it is useful to consider
the kinematics of a parcel that conserves its volume, in which case
The non-divergence condition V . v = 0 provides a constraint on the
parcel's velocity that must be satisfied at each point of the fluid. Fluid
4Throughout these lectures, a comma is used as a shorthand for partial derivative. Hence,
p,t = ap/&. This notation follows Griffies, 2004, and is commonly used in mathematical
physics. It is a useful means to distinguish a derivative from some of the many other uses of
subscripts, such as a tensor component or as part of the name of a variable such as the fresh
water flux q, introduced in equation (27).
3 1
In this equation, A 4 = pdV is the parcel's mass, p is its in situ density, and dV is its infinitesimal volume. The time derivative is taken
following the parcel, and is known as a material or Lagmngian time
derivative. Writing dV = dx dy dz, and defining the
v = dx/dt = (u, w) leads to
d lnp
= -V . v.
dt
Note that the horizontal coordinates xh = (x, y) can
parcel's velocity as
(2)
generally be spherical coordinates (A, q5), or any other generalized horizontal coordinate
appropriate for the sphere, such as those illustrated in Figures 1 and 2
(see chapters 20 and 21 of Griffies, 2004 for a presentation of generalized
horizontal coordinates).
For many purposes in fluid mechanics as well as ocean model design,
it is useful to transform the frame of reference from the moving parcel to
a fixed point in space. This transformation takes us from the material
or Lagrangian frame to the Eulerian frame. It engenders a difference in
how observers measure time changes in a fluid parcel's properties. In
particular, the material time derivative picks up a tmnsport or advective
term associated with motion of the parcel
This relation allows us to write the Lagrangian
conservation in an Eulerian conservation form4
P,t + V . (pv) = 0.
(3)
expression (2) for mass
(4)
Fluids that conserve mass are said to be compressible since the volume of a mass conserving fluid parcel can expand or contract based on
pressure forces acting on the parcel, or properties such as temperature
and salinity. However, in many circumstances, it is useful to consider
the kinematics of a parcel that conserves its volume, in which case
The non-divergence condition V . v = 0 provides a constraint on the
parcel's velocity that must be satisfied at each point of the fluid. Fluid
4Throughout these lectures, a comma is used as a shorthand for partial derivative. Hence,
p,t = ap/&. This notation follows Griffies, 2004, and is commonly used in mathematical
physics. It is a useful means to distinguish a derivative from some of the many other uses of
subscripts, such as a tensor component or as part of the name of a variable such as the fresh
water flux q, introduced in equation (27).