Zdunkowski W., Bott A. Dynamics of the atmosphere: A course in theoretical meteorology
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M3.6 Problems 61
(c) By using the simplifications stated in (b), find the eigenvalues of J.
(d) Show that the eigenvectors corresponding to (c) form an orthogonal system.
(e) In this orthogonal system J can be written in the form J =
3
i=1
λ
i
Ψ
i
Ψ
i
,
where Ψ
i
are the eigenvectors corresponding to the eigenvalues λ
i
. By assuming
additionally that ∂v/∂z = 0, show that the tensor ellipsoid is a hyperbola.
M3.7: By evaluating the expression ∂g
ij
/∂q
k
show the validity of the relation
∂q
i
∂q
j
=
m
ij
q
m
M3.8: Show that the following relations are valid:
(a)
n
in
=
1
√
g
∂
√
g
∂q
i
, (b) g
mn
∂g
mn
∂q
i
=−
2
√
g
∂
√
g
∂q
i
M3.9: Show that, in generalized coordinates, the following expression is valid:
∇·
J =
1
√
g
∂
∂q
n
(
√
gJ
mn
q
m
) with J = J
mn
q
m
q
n
Hint: Start with equation (M3.53). The properties of the vector A have not been
used in the derivation. Can A be replaced by a dyadic?
M3.10: Suppose that the vectors M and N satisfy the expressions
∇
2
M + M = 0, ∇
2
N + N = 0 with ∇·M = 0, ∇·N = 0
Show that a consequence of these relations is that
M =∇×N, N =∇×M
M4
Coordinate transformations
M4.1 Transformation relations of time-independent coordinate systems
M4.1.1 Introduction
As mentioned earlier, all physically relevant quantities such as mass, time, force,
and work are defined without reference to any coordinate system. This implies that
they are invariant with regard to coordinate transformations. On the other hand,
there are other quantities such as the components of vectors and of tensors and par-
ticularly the basis vectors q
i
, q
i
that strongly depend on the choice of a coordinate
system. The question of the way in which these quantities are changed by changing
the coordinate system arises quite naturally. In order to derive the desired trans-
formation relations we will consider two time-independent but otherwise arbitrary
coordinate systems q
i
and a
i
as stated in
q
i
= q
i
(a
1
,a
2
,a
3
),a
i
= a
i
(q
1
,q
2
,q
3
)(M4.1)
In general, these relations are nonlinear. Extensive quantities that are invariant with
regard to coordinate transformations may be expressed in either coordinate system.
For instance we may write
q
i
system a
i
system
(a) ψ = ψ(q
i
,t) ψ = ψ(a
i
,t)
(b) A = A
q
n
(q
i
,t)q
n
= A
q
n
(q
i
,t)q
n
A = A
a
n
(a
i
,t)a
n
= A
a
n
(a
i
,t)a
n
(c) dr = q
n
dq
n
= q
n
dq
n
dr = a
n
da
n
= a
n
da
n
(d)
√
g
q
= [q
1
, q
2
, q
3
]
√
g
a
= [a
1
, a
2
, a
3
]
(e)
1
√
g
q
= [q
1
, q
2
, q
3
]
1
√
g
a
= [a
1
, a
2
, a
3
]
(f) ∇=q
n
∇
q
n
= q
n
∂
∂q
n
∇=a
n
∇
a
n
= a
n
∂
∂a
n
(M4.2)
62
M4.1 Transformation relations of time-independent coordinate systems 63
While the coordinate systems are assumed to be independent of time, the scalar
ψ and the vector components A
q
i
,A
a
i
in general are time-dependent. The time
dependency is evident on considering such scalar quantities as temperature and the
components of the velocity vector.
We will always assume that the relations (M4.1) are uniquely invertible, which
is guaranteed if the functional determinant is nonzero, that is
∂(a
1
,a
2
,a
3
)
∂(q
1
,q
2
,q
3
)
=
∂a
1
∂q
1
∂a
1
∂q
2
∂a
1
∂q
3
∂a
2
∂q
1
∂a
2
∂q
2
∂a
2
∂q
3
∂a
3
∂q
1
∂a
3
∂q
2
∂a
3
∂q
3
=
√
g
q
[q
n
∇
q
n
a
1
, q
m
∇
q
m
a
2
, q
r
∇
q
r
a
3
]
=
√
g
q
J
q
(a
1
,a
2
,a
3
) = 0
(M4.3)
The form stated to the right of the second equality sign is easily established by
substituting A = q
n
∇
q
n
a
1
, B = q
m
∇
q
m
a
2
, C = q
r
∇
q
r
a
3
into equation (M1.96).
The symbol J
q
(a
1
,a
2
,a
3
) is known as the Jacoby operator of the q
i
system. By
interchanging the symbols q
i
and a
i
we immediately find the inverse relation
∂(q
1
,q
2
,q
3
)
∂(a
1
,a
2
,a
3
)
=
∂q
1
∂a
1
∂q
1
∂a
2
∂q
1
∂a
3
∂q
2
∂a
1
∂q
2
∂a
2
∂q
2
∂a
3
∂q
3
∂a
1
∂q
3
∂a
2
∂q
3
∂a
3
=
√
g
a
[a
n
∇
a
n
q
1
, a
m
∇
a
m
q
2
, a
r
∇
a
r
q
3
]
=
√
g
a
J
a
(q
1
,q
2
,q
3
) = 0
(M4.4)
The relation between the two functional determinants is given by
∂(a
1
,a
2
,a
3
)
∂(q
1
,q
2
,q
3
)
∂(q
1
,q
2
,q
3
)
∂(a
1
,a
2
,a
3
)
= 1(M4.5)
This formula, which is derived in many textbooks on analysis, will be verified later.
M4.1.2 Transformation of basis vectors and coordinate differentials
Let us now consider the following equations:
dr(q
1
,q
2
,q
3
) = dr[a
1
(q
1
,q
2
,q
3
),a
2
(q
1
,q
2
,q
3
),a
3
(q
1
,q
2
,q
3
)]
dr(a
1
,a
2
,a
3
) = dr[q
1
(a
1
,a
2
,a
3
),q
2
(a
1
,a
2
,a
3
),q
3
(a
1
,a
2
,a
3
)]
(M4.6)
64 Coordinate transformations
where we have used the functional relations (M4.1). After taking the total differ-
entials of both sides of these expressions, we compare the coefficients of dq
i
and
da
i
.Thisleadsto
∂r
∂q
i
=
∂r
∂a
n
∂a
n
∂q
i
,
∂r
∂a
i
=
∂r
∂q
n
∂q
n
∂a
i
(M4.7)
By employing the definition (M3.11), we immediately obtain the important trans-
formation rules
q
i
=
∂a
n
∂q
i
a
n
, a
i
=
∂q
n
∂a
i
q
n
(M4.8)
for the covariant basis vectors. Using the definition (M3.27), we find from (M4.2f)
the transformation rules for the contravariant basis vectors
q
i
=∇q
i
=
∂q
i
∂a
n
a
n
, a
i
=∇a
i
=
∂a
i
∂q
n
q
n
(M4.9)
Now we will discuss the relations among the differentials da
i
,da
i
,dq
i
, and dq
i
.
Owing to (M4.2c) and (M4.9) we may first write
(a) dr = q
n
dq
n
= a
n
da
n
= q
m
∂a
n
∂q
m
da
n
(b) dr = a
n
da
n
= q
n
dq
n
= a
m
∂q
n
∂a
m
dq
n
(M4.10)
Scalar multiplication of (M4.10a) by the basis vector q
i
and scalar multiplication
of (M4.10b) by the basis vector a
i
immediately gives the transformations
dq
i
=
∂a
n
∂q
i
da
n
,da
i
=
∂q
n
∂a
i
dq
n
(M4.11)
Comparison of (M4.11) with (M4.8) shows that the differentials da
i
and dq
i
trans-
form in exactly the same way as the covariant basis vectors a
i
and q
i
.
We proceed analogously to find the relations for the differentials of the con-
travariant quantities da
i
and dq
i
. Starting with
(a) dr = q
n
dq
n
= a
n
da
n
= q
m
∂q
m
∂a
n
da
n
(b) dr = a
n
da
n
= q
n
dq
n
= a
m
∂a
m
∂q
n
dq
n
(M4.12)
and multiplying (M4.12a) by q
i
and (M4.12b) by a
i
yields the transformations
dq
i
=
∂q
i
∂a
n
da
n
,da
i
=
∂a
i
∂q
n
dq
n
(M4.13)
Comparison of (M4.13) with (M4.9) shows that the contravariant differentials are
transformed in the same way as the corresponding contravariant basis vectors.
M4.1 Transformation relations of time-independent coordinate systems 65
M4.1.3 Transformation of vectors and dyadics
The transformation relations for the covariant and the contravariant measure num-
bers of the vector A are found in the same way as for the various differentials of
the previous section. First we write A in the four different forms
(a) A = q
n
A
q
n
= a
n
A
a
n
= q
m
∂a
n
∂q
m
A
a
n
(b) A = a
n
A
a
n
= q
n
A
q
n
= a
m
∂q
n
∂a
m
A
q
n
(c) A = q
n
A
q
n
= a
n
A
a
n
= q
m
∂q
m
∂a
n
A
a
n
(d) A = a
n
A
a
n
= q
n
A
q
n
= a
m
∂a
m
∂q
n
A
q
n
(M4.14)
Scalar multiplication of (M4.14a) by q
i
, (M4.14b) by a
i
, (M4.14c) by q
i
,and
(M4.14d) by a
i
yields the desired results
A
q
i
=
∂a
n
∂q
i
A
a
n
,A
a
i
=
∂q
n
∂a
i
A
q
n
A
q
i
=
∂q
i
∂a
n
A
a
n
,A
a
i
=
∂a
i
∂q
n
A
q
n
(M4.15)
Again, as expected, the covariant and the contravariant measure numbers of A
transform in the same way as the corresponding basis vectors in (M4.8) and (M4.9).
The transformation of the measure numbers of
B is a little more complex. We
will derive in detail the transformation rules for the contravariant measure numbers.
In the two coordinate systems B may be written as
B = B
a
mn
a
m
a
n
= B
q
rs
q
r
q
s
= B
q
rs
∂a
u
∂q
r
∂a
v
∂q
s
a
u
a
v
(M4.16)
Scalar multiplication of this expression from the right first by a
j
andthenbya
i
gives
B
a
mn
δ
i
m
δ
j
n
= B
q
rs
∂a
u
∂q
r
∂a
v
∂q
s
δ
i
u
δ
j
v
(M4.17)
which may be further evaluated, yielding
B
a
ij
= B
q
rs
∂a
i
∂q
r
∂a
j
∂q
s
(M4.18)
66 Coordinate transformations
The transformation of the covariant and the mixed measure numbers proceeds
analogously. All results are summarized in
B
q
ij
= B
a
mn
∂q
i
∂a
m
∂q
j
∂a
n
,B
a
ij
= B
q
mn
∂a
i
∂q
m
∂a
j
∂q
n
B
q
ij
= B
a
mn
∂a
m
∂q
i
∂a
n
∂q
j
,B
a
ij
= B
q
mn
∂q
m
∂a
i
∂q
n
∂a
j
B
q
i
j
= B
a
m
n
∂q
i
∂a
m
∂a
n
∂q
j
,B
a
i
j
= B
q
m
n
∂a
i
∂q
m
∂q
n
∂a
j
B
q
i
j
= B
a
m
n
∂a
m
∂q
i
∂q
j
∂a
n
,B
a
i
j
= B
q
m
n
∂q
m
∂a
i
∂a
j
∂q
n
(M4.19)
Utilizing the transformation rules for the covariant basis vectors (M4.8), it is
easy to find the transformation relations for the functional determinants of the q
i
system and the a
i
system. According to (M1.37) and (M4.8) we may write
√
g
q
= [q
1
, q
2
, q
3
] =
∂a
n
∂q
1
a
n
,
∂a
m
∂q
2
a
m
,
∂a
r
∂q
3
a
r
= [a
1
, a
2
, a
3
]
∂(a
1
,a
2
,a
3
)
∂(q
1
,q
2
,q
3
)
=
√
g
a
∂a
i
∂q
j
(M4.20)
The inverse relation can be found in the same way or by simply interchanging the
symbols q and a so that
√
g
q
=
√
g
a
∂a
i
∂q
j
,
√
g
a
=
√
g
q
∂q
i
∂a
j
(M4.21)
Occasionally the chain rule for the functional determinant is of great usefulness.
With the help of (M4.21) this rule can be derived very easily by recalling that, for
the Cartesian system, the functional determinant
√
g
x
= 1. Setting in the first and
the second equation of (M4.21) a
i
= x
i
and q
i
= x
i
, respectively, we obtain
√
g
q
=
∂x
i
∂q
j
,
√
g
a
=
∂x
i
∂a
j
(M4.22)
Substitution of these expressions into (M4.21) gives
√
g
q
=
∂x
i
∂a
j
∂a
k
∂q
l
,
√
g
a
=
∂x
i
∂q
j
∂q
k
∂a
l
(M4.23)
M4.2 Transformation relations of time-dependent coordinate systems 67
The interpretation of the chain rule is quite simple. The functional determinant
√
g
q
corresponds to an initial transformation from the Cartesian x
i
system into the a
i
system followed by the transformation into the q
i
system. For
√
g
a
the interpretation
is analogous.
We conclude this section by presenting the transformation rule for the measure
numbers of the nabla operator which, according to (M4.2f), may be stated in various
ways. Application of (M4.15) immediately yields the transformation rules
∂
∂q
i
=∇
q
i
=
∂a
n
∂q
i
∇
a
n
=
∂a
n
∂q
i
∂
∂a
n
,
∂
∂a
i
=∇
a
i
=
∂q
n
∂a
i
∇
q
n
=
∂q
n
∂a
i
∂
∂q
n
(M4.24)
which are quite useful for handling various problems.
M4.2 Transformation relations of time-dependent coordinate systems
M4.2.1 The addition theorem of the velocities
In order to derive the equations of air motion relative to the rotating earth, we
must consider two coordinate systems. The first system is a time-independent
absolute coordinate system or inertial system, which is assumed to be at rest with
respect to the fixed stars.
1
This coordinate system will be described in terms of
the Cartesian coordinates x
i
. The second coordinate system, also known as the
relative coordinate system, is time-dependent and is moving with respect to the
absolute system. Motion in the relative system will be described with the help
of the q
i
-coordinates, which may be curvilinear and oblique. The transformation
relation for the two coordinate systems is given by
x
i
= x
i
(q
1
,q
2
,q
3
,t),i= 1, 2, 3(M4.25)
In contrast to the transformation relation (M4.1), we have now admitted an explicit
time dependency.
Physical quantities (scalars, vectors, etc.) are invariant with respect to coordinate
transformations and may be expressed either in the x
i
system or in the q
i
system.
Two examples are
ψ = ψ(x
1
,x
2
,x
3
,t) = ψ(q
1
,q
2
,q
3
,t)
A = A
x
n
(x
1
,x
2
,x
3
,t)i
n
= A
q
n
(q
1
,q
2
,q
3
,t)q
n
(q
1
,q
2
,q
3
,t)
(M4.26)
The reader should note that, in the q
i
system, the basis vectors q
i
depend not only
on the coordinates q
i
but also explicitly on time t.
1
The system may also move with a constant velocity with respect to the fixed stars.
68 Coordinate transformations
Let us consider the scalar quantity ψ which may be expressed by the functional
relation
ψ = ψ(q
1
,q
2
,q
3
,t) = ψ[x
1
(q
1
,q
2
,q
3
,t),x
2
(q
1
,q
2
,q
3
,t),x
3
(q
1
,q
2
,q
3
,t),t]
(M4.27)
Forming the partial derivative of this expression with respect to time, we obtain
the local change with time of ψ which differs by one term in the two coordinate
systems,
∂ψ
∂t
q
i
=
∂ψ
∂t
x
i
+
∂ψ
∂x
n
∂x
n
∂t
q
i
(M4.28)
The total derivative of ψ with respect to time yields the two equations
dψ
dt
=
∂ψ
∂t
x
i
+
∂ψ
∂x
n
˙x
n
with ˙x
n
=
dx
n
dt
dψ
dt
=
∂ψ
∂t
q
i
+
∂ψ
∂q
n
˙q
n
with ˙q
n
=
dq
n
dt
(M4.29)
Analogous expressions are also obtained for vectors and higher-degree extensive
functions. The total time derivative d/dt is an invariant operator, i.e. it is inde-
pendent of any particular coordinate system. Thus, d/dt expresses the individual
changes with time of physical quantities such as temperature, density and velocity.
We now form the individual change with time of the position vector r expressed
in the absolute system
dr
dt
= i
n
dx
n
dt
= i
n
∂x
n
∂t
x
i
+ i
n
∂x
n
∂x
m
˙x
m
= 0 + i
n
˙x
n
= v
A
(M4.30)
First we observe that the partial derivative with respect to time vanishes since the x
i
-
coordinates are held constant. The second part of (M4.30) leads to the introduction
of the velocity in the absolute coordinate system v
A
which is also called the absolute
velocity.
Intheq
i
-coordinate system the individual change with time of the position vector
r is given by
dr
dt
=
∂r
∂t
q
i
+ ˙q
n
∂r
∂q
n
=
∂r
∂t
q
i
+ ˙q
n
q
n
= v
P
+ v (M4.31)
where use of (M3.11) has been made. Now the local change with time of r does
not vanish since the q
i
system is time-dependent. Instead, the first term of (M4.31)
expresses the motion of a fixed point in the q
i
system moving with velocity v
P
relative to the Cartesian system. The second part of (M4.31) represents the velocity
M4.2 Transformation relations of time-dependent coordinate systems 69
v of a fluid particle relative to the q
i
system, which is known as the relative velocity.
Combining (M4.30) and (M4.31) yields
v
A
= v
P
+ v
(M4.32)
which is known as the addition theorem of the velocities.
The relative coordinate system in our case is attached to the solid earth sur-
rounded by the atmosphere. A point fixed in this system will experience the velocity
v
P
which consists of three parts
v
P
= v
T
+ v
+ v
D
(M4.33)
The first part is the translatory velocity v
T
representing the motion of the earth
around the sun. In general, for atmospheric systems the acceleration of this part
of the total motion is ignored. Therefore, v
T
will be omitted from now on. The
second part of v
P
is the rotational velocity v
due to the rotation of the earth
about its axis. Finally, in arbitrary time-dependent coordinate systems a point fixed
on a coordinate surface q
i
= constant may perform motions with respect to the
corresponding x
i
-coordinate of the inertial system. This may be easily recognized
if the vertical coordinate q
3
is given by the pressure p. A point fixed on a pressure
surface q
3
= p = constant will perform vertical motions since usually pressure
surfaces are not stationary. This velocity is known as the deformation velocity v
D
due to the deformation of the coordinate surfaces q
i
= constant. A more detailed
discussion of the velocity v
P
will be given later.
We will conclude this section by considering the divergence of v
P
.Moreover,
we will also show that v
P
can be found with the help of the absolute kinetic energy
of an atmospheric mass particle. According to equation (M3.11) we may first write
∂q
i
∂t
q
i
=
∂
∂t
∂r
∂q
i
q
i
=
∂
∂q
i
∂r
∂t
q
i
=
∂v
P
∂q
i
(M4.34)
With the help of (M4.2f) and (M4.34) we find for the divergence of v
P
∇·v
P
= q
n
·
∂v
P
∂q
n
= q
n
·
∂q
n
∂t
q
i
=
q
2
× q
3
√
g
q
·
∂q
1
∂t
q
i
+
q
3
× q
1
√
g
q
·
∂q
2
∂t
q
i
+
q
1
× q
2
√
g
q
·
∂q
3
∂t
q
i
=
1
√
g
q
∂
∂t
[q
1
·(q
2
× q
3
)]
q
i
=
1
√
g
q
∂
√
g
q
∂t
q
i
(M4.35)
70 Coordinate transformations
where use of the basic definition of the reciprocal systems (M1.55) has been made.
As will be shown later, the divergence of v
is zero so that the divergence of v
P
is
given by
∇·v
P
=
1
√
g
q
∂
√
g
q
∂t
q
i
=∇·v
D
(M4.36)
From physical reasoning this result is also obvious since the angular velocity of
the earth is a constant vector.
We will now show how v
P
can be found from the absolute kinetic energy per
unit mass
K
A
=
v
2
A
2
=
(v + v
P
)·(v + v
P
)
2
=
v
2
2
+ v·v
P
+
v
2
P
2
=
˙q
m
˙q
n
2
q
m
·q
n
+ ˙q
m
q
m
·W
q
n
q
n
+
v
2
P
2
(M4.37)
where W
i
q
are the covariant measure numbers of v
P
,i.e.v
P
= W
m
q
q
m
.Byraising
the index, W
i
q
can also be found very easily, as will be demonstrated later. More
briefly, we may write for K
A
K
A
= K + K
P
with K =
˙q
m
˙q
n
2
g
mn
q
,K
P
= ˙q
n
W
q
n
+
v
2
P
2
(M4.38)
which is composed of the two parts K and K
P
. Thus, K is the relative kinetic energy,
that is the part of the absolute kinetic energy which is quadratically homogeneous
in the relative velocity ˙q
i
and does not contain any part of v
P
. The expression for
K
P
is linear in ˙q
i
.
Inspection of (M4.38) shows that the metric fundamental quantities and, hence,
the functional determinant
√
g
q
can be easily found from the kinetic energy K of
the relative motion
g
ij
q
=
∂
2
K
∂ ˙q
i
∂ ˙q
j
=⇒
√
g
q
=
g
ij
q
=
∂
2
K
∂ ˙q
i
∂ ˙q
j
(M4.39)
where the notation
|
···
|
denotes the determinant. Similarly, it is possible to find the
covariant measure numbers of v
P
from K
P
:
W
q
i
=
∂K
P
∂ ˙q
i
(M4.40)
From (M4.39) and (M4.40) we conclude that the metric fundamental quantities
and the velocity v
P
of an arbitrary q
i
-coordinate system are known if knowledge