1. Transport and mixing
1.1 The material derivative
Let
be the velocity of a fluid at the point
and time . Consider also
some scalar field
such as the temperature or density.
We are interested not only in the
partial derivative of
with respect to time,
, but also in the time derivative following the
motion of the fluid,
.
The latter is the so-called
material derivative
:
.
(1.1)
In Cartesian coordinates,
,
and
.
(1.2)
We will also be using spherical coordinates. The notation
is conventional for the
(eastward, northward, radially outward) components of the velocity field. In addition to the radial
distance from the origin, , we use the symbol
for latitude (
at the south pole,
at the
north pole) and
for longitude (ranging for zero to
).
The gradient operator in these coordi-
nates is
,
(1.3)
so that
.
(1.4)
The material derivative of a vector, such as the velocity
itself, is defined just as in 1.1.
But care is required when considering the material derivative of a component of a vector if the
unit vectors of one’s coordinate system are position dependent, as they are in spherical coordi-
nates. For example, the radial component of the material derivative of the velocity is not equal to
the material derivative of the radial component of the velocity; rather,
.
(1.5)
The second term on the rhs is sometimes referred to as the
metric term
.
After obtaining
expressions for
, etc., one arrives at the spherical-coordinate expression for the acceleration
of a fluid parcel (see Problem 1.1):
V x
t
,
(
29
x
x y z
, ,
(
29
=
t
χ
x
t
,
(
29
χ
χ
t
∂
∕
∂
d
χ
dt
∕
d
χ
dt
∕
χ
x
V x
t
,
(
29
t
δ
+
t
t
δ
+
,
(
29
χ
x
t
,
(
29
–
[
]
t
0
→
δ
lim
t
δ
∕
=
∂ ∂
t
∕
V
∇
⋅
+
(
29χ
=
V
u v w
, ,
(
29
=
∇
∂
x
∂
y
∂
z
,
,
(
29
=
d
χ
dt
∕
χ
t
∂
∕
u
χ
x
∂
∕
v
∂χ
y
∂
∕
w
∂χ
z
∂
∕
+
+
∂
+
∂
=
u v w
, ,
(
29
r
θ
π
2
∕
–
π
2
∕
λ
2
π
∇
λ
ˆ
r
θ
cos
(
29
1
–
∂ ∂λ
∕
(
29
θ
ˆ
r
1
–
∂ ∂θ
∕
(
29
r
ˆ
∂ ∂
r
∕
(
29
+
+
=
d dt
∕
∂ ∂
t
∕
r
θ
cos
(
29
1
–
u
∂ ∂λ
∕
(
29
r
1
–
v
∂ ∂θ
∕
(
29
w
∂ ∂
r
∕
(
29
+
+
+
=
V
r
ˆ
d
V
dt
-------
⋅
d
dt
----
r
ˆ
V
⋅
(
29
V
d
r
ˆ
dt
-----
⋅
–
=
d
r
ˆ
dt
∕