Zdunkowski W., Bott A. Dynamics of the atmosphere: A course in theoretical meteorology
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21.4 Problems 571
It should be understood that ˙η is the vertical velocity of a point mass. Comparison
of (21.21) with (21.22) shows that the contravariant form has a less complicated
structure than the covariant form and, therefore, is easier to apply. It should also
be noted that the gradient of the geopotential does not appear in the horizontal
components of the contravariant system. The covariant system, however, does
contain contributions from the geopotential.
21.3 The continuity equation in the η system
The general form of the continuity equation (18.48) for rigid rotation is easily
adapted to the η system since the functional determinant has already been given by
(21.10). The following result is obtained:
dρ
dt
+
ρ
H − h
∂
∂x
[(H − h) ˙x] +
∂
∂y
[(H − h) ˙y] +
∂
∂η
[(H − h) ˙η]
= 0(21.23)
If covariant velocity components are desired in the continuity equation, we need
to substitute equation (21.11) into (21.23). This gives a very lengthy equation,
which will not be presented here.
In this chapter we have demonstrated that it is very easy to obtain the equation
of motion for a special coordinate system due to our knowledge of the general
equations for the covariant and contravariant forms given by (18.23), (18.25),
(18.33) and (18.34). It appears that Sommerville and Gal-Chen (1974) were the
first to introduce the orography following-coordinates giving the exact form of the
equation of motion. In this context we also recommend the important paper by
Gal-Chen and Sommerville (1975). Many other papers have followed. Very often
various approximations were used, such as forced orthogonalization. This can be
done quite simply by setting the off-diagonal elements in the metric tensors g
ij
and
g
ij
in (21.6) and (21.9) equal to zero.
21.4 Problems
21.1: Rewrite the first equation of (21.21) by using covariant velocity components.
To keep the analysis simple, simplify the metric tensor by a forced orthogonaliza-
tion.
22
The stereographic system with a generalized vertical
coordinate
In the previous chapter we introduced the vertical coordinate η to handle orographic
effects in mesoscale models. In the synoptic-scale models we are going to replace
the height coordinate z which extends to infinity by a generalized vertical coordinate
ξ. The introduction of ξ is motivated by the fact that we cannot integrate the
predictive equations using z as a vertical coordinate to infinitely large heights.
Replacing z by the atmospheric pressure p, for example, results in a finite range
of the vertical coordinate. We will see that another advantage of the (x,y,p)-
coordinate system is that the continuity equation is time-independent. There are
other specific coordinate systems that we are going to discuss. Therefore, it seems
of advantage to first set up the atmospheric equations in terms of the unspecified
generalized vertical coordinate ξ. Later we will specify ξ as desired. We wish to
point out that the introduction of the generalized coordinate is of advantage only if
the hydrostatic equation is a part of the atmospheric system.
We will briefly state the consequences of the transformation from the stereo-
graphic (x,y,z)-coordinate system to the stereographic (x,y,ξ)-coordinate sys-
tem, which henceforth will be called the ξ system.
(i) The hydrostatic approximation is not restricted to the hydrostatic equation itself but
enters implicitly into the horizontal equations of motion and the continuity equation.
(ii) The boundary conditions at the earth’s surface can be formulated rather easily. A
surface ξ = constant may be arranged in such a way that it coincides with th
e
orographi
c surface of the earth.
(iii) The infinite height range in the z-coordinate is usually replaced by a finite height
range in the ξ -coordinate. Large-scale models normally employ covariant and phys-
ical velocity measure numbers to formulate the equation of motion. For this reason
and to prevent the following chapters from becoming too lengthy, we will omit any
discussion of contravariant forms. Nevertheless, we have given a sufficient background
for any interested reader to formulate thes
e by using the proper transformation rules.
Furthermore, we will neglect friction in the large-scale flow.
572
22.1 The ξ transformation and resulting equations 573
22.1 The ξ transformation and resulting equations
As the original system we consider the orthogonal stereographic Cartesian system.
The uniqueness of the transformation between the (x,y, z)-system and the (x, y,ξ)-
system is guaranteed by the requirement that ξ is a monotonically increasing or
decreasing function of z. Now the function z = z(x,y, ξ,t) represents the height
of the time-dependent ξ -surface. Although the original system is orthogonal, the
effect of the coordinate transformation is that the new system is not orthogonal. In
order to retain the convenience of the orthogonal systems we are forced to make
a simplifying assumption regarding the metric tensor. We assume that the height
of the ξ -surface is independent of the horizontal coordinates (x,y). We call this
assumption the second metric simplification since it is used only to find a simplified
form of the metric tensor. Hence, we have
z = z(ξ,t) =⇒ ˙z =
∂z
∂t
ξ
+
∂z
∂ξ
t
˙
ξ (22.1)
We obtain the metric tensor by replacing ˙z
2
in the original system (20.34) with the
help of (22.1). The absolute kinetic energy is given by K
A
= K + K
P
with
K =
1
2
˙x
2
+ ˙y
2
m
2
0
+
∂z
∂ξ
2
t
˙
ξ
2
K
P
=
(x ˙y − ˙xy)
m
2
0
+
(x
2
+ y
2
)
2
2m
2
0
+
1
2
∂z
∂t
2
ξ
+
∂z
∂t
ξ
∂z
∂ξ
t
˙
ξ
(22.2)
from which we obtain the covariant form of the metric tensor according to
g
ij
=
∂
2
K
A
∂ ˙q
i
∂ ˙q
j
=⇒ g
11
=
1
m
2
0
,g
22
=
1
m
2
0
,g
33
=
∂z
∂ξ
2
t
,g
ij
= 0,i= j
(22.3)
Since all off-diagonal elements of g
ij
are zero, we are dealing with an orthogonal
system as desired. What we have done, in fact, is implemented a forced orthog-
onalization of the metric tensor. Had we permitted the height z of the ξ -surface
to vary with the horizontal coordinates (x, y), every element g
ij
would have been
nonzero.
As the next step we are going to obtain the components of the velocity vector v
P
which can be split into the two velocity vectors v
and v
D
describing the rotation
of the earth and the deformation of the material surfaces on which the point P is
located. The components W
i
of v
P
can be found from
W
i
=
∂K
P
∂ ˙q
i
=⇒ W
1
=−
y
m
2
0
,W
2
=
x
m
2
0
,W
3
=
∂z
∂t
ξ
∂z
∂ξ
t
(22.4)
574 The stereographic system with a generalized vertical coordinate
Utilizing this information, the components of v
P
are given by
(W
1
,W
2
,W
3
) =
−
y
m
2
0
,
x
m
2
0
, 0
, (W
1
,W
2
,W
3
) = (−y, x, 0)
(W
1
D
,W
2
D
,W
3
D
) =
0, 0,
∂z
∂t
ξ
∂z
∂ξ
t
, (W
1
D
,W
2
D
,W
3
D
) =
0, 0, −
∂ξ
∂t
z
(22.5)
From the contravariant measure numbers of the relative velocity (v
1
,v
2
,v
3
) =
( ˙x, ˙y,
˙
ξ) we obtain the covariant and physical measure numbers with the help of
(22.3) as
(v
1
,v
2
,v
3
) =
˙x
m
2
0
,
˙y
m
2
0
,
∂z
∂ξ
2
t
˙
ξ
, (u, v, w) =
˙x
m
0
,
˙y
m
0
,
∂z
∂ξ
t
˙
ξ
(22.6)
Since the ξ system is deformational, in contrast to the Cartesian stereographic
system, we have to evaluate the general forms (18.23) (covariant) and (18.36)
(physical) of the equations of motion. From (22.5) it can be seen that only the third
component of the deformational velocity differs from zero. Furthermore, due to
the second metric simplification W
3
D
is independent of (x, y). As a consequence of
this we have, see (M5.48),
ω
ij
D
= 0,i,j= 1, 2, 3,
∂
∂x
v
2
D
2
= 0,
∂
∂y
v
2
D
2
= 0,
∂
∂t
W
1
D
= 0,
∂
∂t
W
2
D
= 0
(22.7)
The rotational tensor ω
ij
is identical with the corresponding tensor of the stereo-
graphic system and is given by (20.42). Comparison of (22.3) with (20.26) shows
that g
11
and g
22
are also identical in these two systems since we are using m = m
0
.
With this information the evaluation of (18.50) shows that the horizontal com-
ponents of the equations of motion in the ξ system are the same as those in the
Cartesian stereographic system. However, it should be observed that, in the ξ sys-
tem, all partial derivatives with respect to x, y, or t are taken at constant values
of ξ , in contrast to the stereographic coordinate system, in which the independent
variable z is held constant in these operations.
For the vertical component of the equation of motion we obtain from (18.23)
dv
3
dt
−
˙
ξ
2
2
∂
∂ξ
∂z
∂ξ
2
t
+
∂
∂t
W
3
D
−
1
2
∂
∂ξ
∂z
∂t
2
ξ
∂z
∂ξ
2
t
=−
1
ρ
∂p
∂ξ
−
∂φ
∂ξ
(22.8)
22.1 The ξ transformation and resulting equations 575
Had we permitted the height z of the ξ -surface to vary with (x,y) then this equation
as well as the horizontal components would have assumed a very complicated form.
Even in the simplified form the left-hand side of (22.8) is difficult to evaluate.
In Section 2.3 we discussed the hydrostatic approximation and showed by means
of scale analysis that, for large-scale frictionless flow, the vertical equation of
motion may be realistically approximated by the hydrostatic equation. However,
for mesoscale motion the use of the hydrostatic equation is not permitted. We may
find the hydrostatic equation in terms of the generalized coordinate ξ from
∂p
∂z
=
∂p
∂ξ
∂ξ
∂z
=−gρ ,
∂φ
∂ξ
=
∂φ
∂z
∂z
∂ξ
= g
∂z
∂ξ
=⇒
∂p
∂ξ
=−ρ
∂φ
∂ξ
(22.9)
The same equation is obtained by setting the left-hand side of (22.8) equal to zero.
Assuming that we have hydrostatic conditions and frictionless flow, the covariant
form of the equations of motion in the ξ system may be written as
dv
1
dt
+
v
2
1
+ v
2
2
2
∂m
2
0
∂x
− fv
2
=−
1
ρ
∂p
∂x
−
∂φ
∂x
dv
2
dt
+
v
2
1
+ v
2
2
2
∂m
2
0
∂y
+ fv
1
=−
1
ρ
∂p
∂y
−
∂φ
∂y
0 =−
1
ρ
∂p
∂ξ
−
∂φ
∂ξ
with
d
dt
=
∂
∂t
+ m
2
0
v
1
∂
∂x
+ v
2
∂
∂y
+
˙
ξ
∂
∂ξ
(22.10)
After substituting the identities u = m
0
v
1
and v = m
0
v
2
into (22.10), by obvious
steps we obtain the equations of motion in physical velocity components:
du
dt
+ v
v
∂m
0
∂x
− u
∂m
0
∂y
− fv =−
m
0
ρ
∂p
∂x
− m
0
∂φ
∂x
dv
dt
− u
v
∂m
0
∂x
− u
∂m
0
∂y
+ fu=−
m
0
ρ
∂p
∂y
− m
0
∂φ
∂y
0 =−
1
ρ
∂p
∂ξ
−
∂φ
∂ξ
with
d
dt
=
∂
∂t
+ m
0
u
∂
∂x
+ v
∂
∂y
+
˙
ξ
∂
∂ξ
(22.11)
As in the covariant case, the horizontal equations of motion agree with the corre-
sponding equations (20.48) of the stereographic (x,y, z) system. Recall, however,
576 The stereographic system with a generalized vertical coordinate
that all partial derivatives with respect to x, y, or t in (22.11) are calculated at
constant values of ξ .
Any realistic flow problem requires the involvement of the continuity equa-
tion, which will be derived next. The necessary prerequisite is knowledge of the
functional determinant whose elements were given in (22.3) so that
√
g =
|g
ij
|=
1
m
2
0
∂z
∂ξ
t
=
1
m
2
0
g
∂φ
∂ξ
t
=−
1
m
2
0
gρ
∂p
∂ξ
t
(22.12)
follows immediately. Since the density ρ is time-dependent, the functional deter-
minant
√
g is time-dependent also, so the last term in (18.47) does not vanish, that
is the ξ system is deformative. Rewriting (18.47) as
1
ρ
√
g
d
dt
(ρ
√
g) +
∂ ˙q
n
∂q
n
= 0(22.13)
yields the contravariant form of the continuity equation:
m
2
0
∂p
∂ξ
t
d
dt
1
m
2
0
∂p
∂ξ
t
+
∂ ˙x
∂x
+
∂ ˙y
∂y
+
∂
˙
ξ
∂ξ
= 0(22.14)
With the help of (22.6) we obtain the covariant form
m
2
0
∂p
∂ξ
t
d
dt
1
m
2
0
∂p
∂ξ
t
+
∂
∂x
m
2
0
v
1
+
∂
∂y
m
2
0
v
2
+
∂
˙
ξ
∂ξ
= 0
(22.15)
and the form with physical velocity components
m
2
0
∂p
∂ξ
t
d
dt
1
m
2
0
∂p
∂ξ
t
+
∂
∂x
(m
0
u) +
∂
∂y
(m
0
v) +
∂
˙
ξ
∂ξ
= 0
(22.16)
For convenience the contravariant velocity component
˙
ξ has been retained in the
latter two equations.
In order to solve atmospheric flow problems, it is sometimes of advantage
to combine the equation of motion and the continuity equation to give the so-
called stream-momentum form of the equation of motion. This form is obtained
by multiplying the covariant horizontal equations of motion by ∂p/∂ξ and the
22.2 The pressure system 577
continuity equation by v
k
,k= 1, 2. Adding the resulting equations yields
∂
∂t
∂p
∂ξ
v
1
+ m
2
0
∂
∂x
∂p
∂ξ
v
2
1
+
∂
∂y
∂p
∂ξ
v
1
v
2
+
∂
∂ξ
∂p
∂ξ
v
1
˙
ξ
+
∂p
∂ξ
v
2
1
+ v
2
2
2
∂m
2
0
∂x
− f
∂p
∂ξ
v
2
=−
1
ρ
∂p
∂ξ
∂p
∂x
−
∂p
∂ξ
∂φ
∂x
∂
∂t
∂p
∂ξ
v
2
+ m
2
0
∂
∂x
∂p
∂ξ
v
2
v
1
+
∂
∂y
∂p
∂ξ
v
2
2
+
∂
∂ξ
∂p
∂ξ
v
2
˙
ξ
+
∂p
∂ξ
v
2
1
+ v
2
2
2
∂m
2
0
∂y
+ f
∂p
∂ξ
v
1
=−
1
ρ
∂p
∂ξ
∂p
∂y
−
∂p
∂ξ
∂φ
∂y
(22.17)
22.2 The pressure system
For any practical applications we need to specify ξ in order to obtain various
prediction systems. In this section we choose p as the generalized vertical coor-
dinate and thus obtain the so-called pressure system or simply the p system.In
the following we will see that the atmospheric equations assume a simpler form
in the p system than they do in any other ξ system. Therefore, the p system is
often used for analytic and numerical studies. The (x,y,z)-coordinate system is a
right-handed system. For the (x, y,ξ)-coordinate system to be right-handed also,
the generalized coordinate ξ should also increase with height so that ξ =−p.Itis
customary, however, to set ξ = p so that we may view this system as left-handed
since z and p increase in opposite directions. It can be shown that the resulting
equations of the p system are identical irrespective of whether we choose ξ =−p
or ξ = p. Therefore, we stick to the conventional notation and choose ξ = p.
It should be observed that the p system as used here was subjected to a forced
orthogonalization.
Since the pressure cannot vary along any constant-pressure surface, the covariant
form of the equations of motion (22.10) simplifies to
dv
1
dt
+
v
2
1
+ v
2
2
2
∂m
2
0
∂x
− fv
2
=−
∂φ
∂x
dv
2
dt
+
v
2
1
+ v
2
2
2
∂m
2
0
∂y
+ fv
1
=−
∂φ
∂y
∂φ
∂p
=−
1
ρ
with
d
dt
=
∂
∂t
+ m
2
0
v
1
∂
∂x
+ v
2
∂
∂y
+ ω
∂
∂p
(22.18)
578 The stereographic system with a generalized vertical coordinate
where ω = dp/dt. The system (22.11) with physical velocity components assumes
the form
du
dt
+ v
v
∂m
0
∂x
− u
∂m
0
∂y
− fv =−m
0
∂φ
∂x
dv
dt
− u
v
∂m
0
∂x
− u
∂m
0
∂y
+ fu=−m
0
∂φ
∂y
∂φ
∂p
=−
1
ρ
with
d
dt
=
∂
∂t
+ m
0
u
∂
∂x
+ v
∂
∂y
+ ω
∂
∂p
(22.19)
No further comments are necessary. Likewise the continuity equations (22.15) and
(22.16) reduce to
m
2
0
∂v
1
∂x
+
∂v
2
∂y
+
∂ω
∂p
= 0(22.20)
and
m
0
∂u
∂x
+
∂v
∂y
+
∂ω
∂p
−
u
∂m
0
∂x
+ v
∂m
0
∂y
= 0(22.21)
From (22.6) we see that, for the special case m
0
= 1, the contravariant velocities
( ˙x, ˙y), the covariant velocities (v
1
,v
2
), and the physical velocity components (u, v)
are identical. In this case the equations of motion and the continuity equation reduce
to
du
dt
− fv =−
∂φ
∂x
dv
dt
+ fu =−
∂φ
∂y
∂φ
∂p
=−
1
ρ
(22.22)
and
∂u
∂x
+
∂v
∂y
+
∂ω
∂p
= 0
(22.23)
describing the flow on the tangential plane.
The introduction of ξ = p causes the continuity equation to assume the form of
a diagnostic relation in which the time derivative does not appear explicitly. This
form of the continuity equation implies that the air behaves like an incompressible
fluid in the pressure system, as will be shown next. For simplicity let us assume
22.3 The solution scheme using the pressure system 579
constant
constant
1
2
horizontal
convergence
horizontal
divergence
Fig. 22.1 An example
of mass
flow and c
orresponding vertical displacement of isobars.
The initial pressure distribution is given by the dashed lines.
that m
0
= 1 so that the motion takes place on the tangential plane. Let us consider
the mass M contained between the two constant-pressure surfaces p
2
>p
1
.The
mass contained in a volume defined by an area S and p
1
and p
2
can easily be found
with the help of the hydrostatic equation and is given by
M =
z
2
z
1
S
ρdxdydz=−
1
g
z
2
z
1
S
∂p
∂z
dx dy dz
=
1
g
p
2
p
1
S
dx dy dp =
S(p
2
− p
1
)
g
(22.24)
Since the pressure surfaces p
1
and p
2
are constant, M is also constant, so dM/dt =
0. Therefore, in a hydrostatic system using pressure as the vertical coordinate,
horizontal convergence must be compensated by vertical divergence or vice versa.
Figure 22.1 shows the vertical displacement of the isobars p
1
= constant and p
2
=
constant resulting from horizontal convergence and divergence.
22.3 The solution scheme using the pressure system
In this section we are going to solve in principle the predictive system consisting
of the equation of motion, the continuity equation, and the heat equation. In order
to reduce the problem to the simplest form possible, we assume that we have dry
adiabatic flow so that the heat equation reduces to a conservation equation for the
potential temperature θ:
dθ
dt
= 0(22.25)
580 The stereographic system with a generalized vertical coordinate
The equation of state is always a part of the total atmospheric forecast system. For
dry air we have p = ρR
0
T .
Before we explain the numerical solution procedure we must set up the proper
boundary conditions. The lower boundary is the earth’s surface where the surface
pressure p
s
is observed. Since p
s
varies with the horizontal coordinates (x,y)and
with time t, we may think of the lower boundary as an oscillating pressure surface.
The upper boundary at p = 0 may be thought of as a rigid plane surface.
According to Section 9.4.3, the boundary conditions may be written in the
following form:
Lower boundary: p = p
s
(x,y, t),
dp
s
dt
= ω
s
=
∂p
s
∂t
+ m
2
0
v
1,s
∂p
s
∂x
+ v
2,s
∂p
s
∂y
Upper boundary: p = 0,ω(p = 0) = 0
(22.26)
It should be observed that, in the p system, the pressure surface p = p
s
(x,y, t)
coincides with the time-independent geopotential φ = φ
s
(x,y) of the earth’s sur-
face. Hence, a variation in time of p
s
is equivalent to an apparent vertical displace-
ment of φ
s
; see Figure 22.2. Owing to the vibrating lower boundary surface p
s
external gravity waves are produced. This type of meteorological noise is particu-
larly undesirable from the numerical point of view.
We wish to demonstrate the principle of the integration over time of the atmo-
spheric system consisting of the covariant form of the equation of motion (22.18),
the continuity equation (22.20), the heat equation (22.25) for dry adiabatic pro-
cesses, and the equation of state. If we prefer to work with the physical velocity
components we may proceed analogously.
0, p = 0
0
0
SS
SS
≠
≠
ω
x,y
p
s
(x,y,t
0
), φ
s
(x,y), ω ≠ 0
p
s
(x,y,t
1
), φ
s
(x,y), ω ≠ 0
Fig. 22.2 The time dependency of the lower boundary p = p
s
(x,y,t)ofthep system.