Zdunkowski W., Bott A. Dynamics of the atmosphere: A course in theoretical meteorology
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16
Rossby waves
The daily weather maps for any extended part of the globe in middle and high
latitudes show well-defined dynamic systems, which normally move from west to
east. The speed of displacement of these systems differs from the mean wind speed
of the air current. Moreover, the structure of these systems varies considerably with
height. Near the earth’s surface the motion systems exhibit much complexity, their
predominant features being closed cyclonic and anticyclonic patterns of irregular
shape. In contrast to this, in the middle and upper troposphere as well as in the lower
stratosphere (200–700 hPa), the dynamic systems usually consist of relatively
simple smooth wave-shaped patterns. Typically one finds about four or five waves
around the hemisphere. Their thermal structure is characterized by cold troughs
and warm ridges. The dynamics of these relatively long waves was first studied by
Rossby (1939) and in some generalization by Haurwitz (1940). Therefore, these
waves are called Rossby–Haurwitz waves or more simply Rossby waves.
16.1 One- and two-dimensional Rossby waves
Rossby employed the filtered barotropic model which we have studied in some de-
tail in the previous chapter. This model assumes that there is horizontal frictionless
flow between plane-parallel plates. The medium is thought to be an incompress-
ible homogeneous fluid of constant density so that, due to the upper boundary
condition, sound waves and gravity waves cannot form. Nevertheless, horizontal
pressure gradients are possible.
The starting point of the analysis is the divergence-free barotropic vorticity
equation (15.70), which is restated here for convenience:
∇
2
h
∂ψ
∂t
+
∂ψ
∂x
∂
∂y
∇
2
h
ψ + f
−
∂ψ
∂y
∂
∂x
∇
2
h
ψ + f
= 0
with f = f
0
+ βy, β =
2 cos ϕ
a
= constant
(16.1)
where a is the mean radius of the earth.
471
472 Rossby waves
With Rossby we shall assume that the Coriolis parameter varies linearly in the
y-direction; see also equation (15.104). In order to obtain an analytic solution for
the speed of displacement of the long waves we will linearize the vorticity equation.
We assume the validity of a linearization of the form
ψ =
ψ(y) + ψ
(x,y, t), ψ = constant − uy, u =−
∂
ψ
∂y
= constant
(16.2)
where
u is the mean wind speed of the air current. The corresponding linear variant
of the vorticity equation in the so-called β-plane is
∂
∂t
+
u
∂
∂x
∇
2
h
ψ
+ β
∂ψ
∂x
= 0(16.3)
Substituting the wave-type disturbance
ψ
(x,y, t) = A exp[i(k
x
x + k
y
y − ωt)] (16.4)
into (16.3) gives the characteristic or frequency equation
ω = k
x
u −
βk
x
k
2
x
+ k
2
y
(16.5)
Division of the circular frequency by the wavenumber k
x
or by k
y
yields the
components of the phase velocity c in the x-andy-directions:
c
x
=
ω
k
x
= u −
β
k
2
x
+ k
2
y
c
y
=
ω
k
y
=
k
x
k
y
u −
βk
x
/k
y
k
2
x
+ k
2
y
=
k
x
k
y
c
x
(16.6)
From the mathematical point of view it is interesting to remark that the Rossby
waves satisfy not only the linearized equation (16.3) but also the complete nonlin-
earized equation (16.1). By combining terms (16.1) may be written as
∇
2
h
∂ψ
∂t
+ J
ψ, ∇
2
h
ψ
+ β
∂ψ
∂x
= 0(16.7)
The Jacobian is given by
J
ψ, ∇
2
h
ψ
=
∂ψ
∂x
∂
∂y
∇
2
h
ψ
−
∂ψ
∂y
∂
∂x
∇
2
h
ψ
−
∂ψ
∂y
∂
∂x
∇
2
h
ψ
=−ik
x
u
k
2
x
+ k
2
y
ψ
(16.8)
16.1 One- and two-dimensional Rossby waves 473
Substitution of (16.8) into (16.7) together with (16.2) and (16.4) again yields the
frequency equation (16.5).
First let us discuss the original Rossby formula by assuming that k
y
= 0in
(16.4). In this case and with u = 0 we speak of pure Rossby waves. We then find
for the more general case with
u = 0
c
x
= u −
β
k
2
x
= u − β
L
2
x
4π
2
(16.9)
In the midlatitudes, for a wide range of values of L
x
, this formula is in reasonable
agreement with observations.
The simplest situation occurs if we assume that the wind speed u vanishes. In this
case the Rossby waves propagate from east to west as implied by the minus sign
in (16.9). Usually the zonal flow is large enough with
u>0 that the Rossby waves
propagate to the east. However, relative to the basic zonal current, the waves still
move to the west. Furthermore, we recognize that the phase velocity of the Rossby
wave equals the mean wind speed
u if the Coriolis parameter is not permitted to
vary with latitude. For high geographical latitudes and very short waves this is
in rough agreement with reality. For lower geographical latitudes and increasing
wavelength L
x
the wave lags behind the zonal air current.
There exists a certain wavelength L
x,stat
L
x,stat
= 2π
u
β
= 2π
au
2 cos ϕ
(16.10)
at which the wave becomes stationary. For a latitude ϕ = 45
◦
and u = 10 m s
−1
the stationary wavelength is about 5000 km and for u = 20 m s
−1
it amounts to
7000 km.
Introducing (16.10) into (16.9), the wave speed may be expressed as
c
x
=
β
4π
2
L
2
x,stat
− L
2
x
(16.11)
From this equation it follows that waves of length L
x
>L
x,stat
are retrogressive,
i.e. they are moving from east to west. Let N represent the number of waves around
a latitude circle so that
NL
x
= 2πa cos ϕ (16.12)
The so-called velocity deficit of the retrogressive wave c
x
− u may be found, for
example, from the equation
c
x
− u =−
2a cos
3
ϕ
N
2
(16.13)
474 Rossby waves
If, for a given latitude, the wave speed is zero, we may calculate the so-called
stationary wavenumber N
stat
so that the velocity deficit may also be determined
with the help of
c
x
= 2a cos
3
ϕ
1
N
2
stat
−
1
N
2
(16.14)
The velocity deficit may assume considerable values. For example, for a latitude
ϕ = 45
◦
and N = 3 the deficit amounts to −36.5ms
−1
;forN = 6 and for the
same latitude the deficit is only −9.1ms
−1
. The largest deficits are found at low
latitudes and for very long waves. For ϕ = 30
◦
and N = 3 the deficit is −67 m s
−1
.
Compared with observations, particularly for the very long waves, the velocity
deficit is clearly too high. This very unrealistic behavior of the Rossby displacement
formula may be traced back to the simplicity of the model which assumed the
existence of a fixed upper plate. This resulted in a vanishing horizontal divergence
of the velocity field. If the rigid lid is replaced by a free surface, then the Rossby
formula (16.9) assumes a modified form, as will be shown next. For convenience
we restate the two equations needed for the analysis. These are the barotropic
vorticity equation
dη
dt
+ η ∇
h
· v
h
= 0(16.15)
repeated from (10.146), and the time-change equation for a free surface H
dH
dt
+ H ∇
h
· v
h
= 0(16.16)
repeated from (15.23a). Since we wish to find a simple analytic solution to the
problem we assume that the ground is flat. By eliminating the divergence of the
horizontal wind field between these two equations we find
∂η
∂t
+ u
∂η
∂x
+ v
∂η
∂y
−
η
H
∂H
∂t
+ u
∂H
∂x
+ v
∂H
∂y
= 0(16.17a)
which can also be written as the conservation statement
d
dt
η
H
= 0
(16.17b)
This is the potential vorticity equation which was derived earlier; see Section 10.5.8.
It simply states that the potential vorticity of each individual air parcel is conserved
in a divergent barotropic flow. Hence, the number of minima and maxima of the
absolute vorticity cannot change, so a true cyclogenesis cannot be predicted by the
theory of barotropic flow. It should be emphasized, however, that in many regions
for long time periods the atmosphere acts nearly as a barotropic medium. Many
16.1 One- and two-dimensional Rossby waves 475
developments that appear to be new simply result from a redistribution of already
existing extremals of the absolute vorticity.
Equations (16.17a) and (16.17b) were obtained from the equations for the
barotropic vorticity and the free surface without any additional assumptions, so
gravity waves are not eliminated. A reasonable filter for this type of noise is the
geostrophic approximation of the wind field. By introducing the geostrophic wind
components
u
g
=−
g
f
0
∂H
∂y
,v
g
=
g
f
0
∂H
∂x
(16.18)
we find the following expressions for the absolute geostrophic vorticity η
g
:
η
g
= ζ
g
+ f with ζ
g
=
∂v
g
∂x
−
∂u
g
∂y
=
g
f
0
∇
2
h
H (16.19)
It can be seen that the Coriolis parameter f has been replaced by the average value
f
0
in the expression for ζ
g
but not in the expression for η
g
. This treatment ensures
that the β-effect will still be accounted for in the model and that at the same time
analytic solutions may be obtained.
By substituting (16.18) and (16.19) into (16.17a) we obtain the barotropic model
equation
∇
2
h
∂H
∂t
−
∂H
∂y
∂
∂x
g
f
0
∇
2
h
H + f
+
∂H
∂x
∂
∂y
g
f
0
∇
2
h
H + f
−
f
0
gH
g
f
0
∇
2
h
H + f
∂H
∂t
= 0
(16.20)
This partial differential equation for the tendency ∂H/∂t is of the Helmholtz type
and can be solved, for example, by a numerical procedure known as the relaxation
method.
In order to obtain an analytic solution for the displacement of a wave-type dis-
turbance, we must linearize equation (16.20). We assume that a simple disturbance
(perturbation) is embedded in a zonal current. For this purpose we decompose the
H -fieldintoapart
H (y) plus a perturbation H
(x,t):
H (x,y,t) =
H (y) + H
(x,t)(16.21)
In analogy to the geostrophic wind component u
g
we may write for the zonal
current
−
g
f
0
∂H
∂y
=
u = constant (16.22a)
Integration yields
H (y) = H
0
− uf
0
y/g (16.22b)
where H
0
is a constant.
476 Rossby waves
Substituting (16.21) into (16.20) gives the linearized equation
∂
2
∂x
2
−
f
2
0
gH
0
∂H
∂t
+ β
∂H
∂x
+ u
∂
3
H
∂x
3
= 0(16.23)
Wherever H appears in undifferentiated form we have replaced it by the mean
height H
0
of the free surface. Moreover, we have approximated ff
0
≈ f
2
0
.
Assuming that the disturbance is a wave propagating in the x-direction,
H
= A exp[ik
x
(x − ct)] (16.24)
we find that the phase velocity of a Rossby wave, modified by gravitational effects,
is given by
c
x
=
u − β
L
2
x
4π
2
1 +
f
2
0
L
2
x
gH
0
4π
2
(16.25)
Sometimes this wave is also called a mixed Rossby wave.
We now compare (16.25) with (16.9), which applies to a fixed upper boundary.
We recognize that the stationary wavelength has not changed. For all nonstationary
waves the magnitude of the phase velocity c
x
is smaller than that predicted by
(16.9). The velocity deficit with respect to the basic current increases somewhat
for waves propagating from west to east. However, the speed of the extremely
fast retrogressive waves is considerably less, resulting in better agreement with
observations. For the above example, assuming a latitude ϕ = 30
◦
, N = 3, and
H
0
= 8000 m, the deficit is now −56.5ms
−1
instead of −67 m s
−1
as computed
from (16.13). Had we chosen H
0
= 2000 m, the velocity deficit would have been
reduced further, to a value of only −40 m s
−1
.
16.2 Three-dimensional Rossby waves
The starting point of the analysis is the baroclinic vorticity equation (10.77) in the
pressure coordinate system, which is rewritten as
∂ζ
∂t
+ v
h
·∇
h
η + ω
∂ζ
∂p
=
e
r
×
∂v
h
∂p
·∇
h
ω + η
∂ω
∂p
(16.26)
For simplicity we have omitted the suffix p since confusion is unlikely to occur.
With sufficient accuracy we will now replace the velocity vector by the geostrophic
wind vector and the vorticity by the geostrophic vorticity
v
h
−→ v
g
,ζ−→ ζ
g
=
1
f
0
∇
2
h
φ, η −→ η
g
= ζ
g
+ f (16.27)
16.2 Three-dimensional Rossby waves 477
In view of this approximation and a consequential treatment of the various terms
appearing in the vorticity equation, we obtain the quasi-geostrophic approximation
of the vorticity equation in the form
∂
∂t
+ v
g
·∇
h
(ζ
g
+ f ) = f
0
∂ω
∂p
(16.28)
As before, on the left-hand side the Coriolis parameter f is permitted to vary with
latitude to include the β-effect. The steps leading to (16.28) will be given later
when we discuss the quasi-geostrophic theory in some detail.
In order to include in a rough approximation the vertical structure of the
atmosphere, we must involve the first law of thermodynamics. The approximate
adiabatic form for dry air is given by
ρc
p,0
dT
dt
=
dp
dt
= ω (16.29)
It is useful to introduce the definition of the static stability σ
0
into the previous
equation:
σ
0
=−
1
ρ
∂ ln θ
∂p
=
R
0
p
R
0
T
c
p,0
p
−
∂T
∂p
(16.30)
After a few easy steps we find
∂
∂t
+ v
g
·∇
h
T −
p
R
0
σ
0
ω = 0(16.31)
By means of the hydrostatic equation we replace the temperature T in terms of
the geopotential φ:
T =−
p
R
0
∂φ
∂p
(16.32)
so that the first law of thermodynamics can be written as
∂
∂t
+ v
g
·∇
h
∂φ
∂p
+ σ
0
ω = 0 (16.33)
Recall that we are using the p system, so the operators ∂/∂t and ∇
h
are applied at
constant pressure.
In order to obtain an analytic solution to our problem we must linearize
equations (16.28) and (16.33). With v
g
= u
g
i + v
g
j, u
g
= u we have
v
g
·∇
h
ζ
g
=⇒ u
∂
∂x
1
f
0
∇
2
h
φ
, v
g
·∇
h
f =⇒
β
f
0
∂φ
∂x
(16.34)
478 Rossby waves
With this linearization the vorticity equation obtains the form
∂
∂t
+
u
∂
∂x
∇
2
h
φ
+ β
∂φ
∂x
= f
2
0
∂ω
∂p
(16.35)
while the first law of thermodynamics may be written in linearized form as
∂
∂t
+
u
∂
∂x
∂φ
∂p
+ σ
0
ω
= 0(16.36)
By eliminating the generalized vertical velocity, equations (16.35) and (16.36) may
be combined to give the single equation
∂
∂t
+
u
∂
∂x
∇
2
h
φ
+
f
2
0
σ
0
∂
2
φ
∂p
2
+ β
∂φ
∂x
= 0(16.37)
In order to obtain this equation we have replaced the stability parameter σ
0
by an
average value
σ
0
for the vertical layer under consideration. Assuming as a solution
of (16.37) the three-dimensional wave in the form
φ
(x,y, p, t) = A exp[i(k
x
x + k
y
y + k
p
p − ωt)] (16.38)
we obtain the desired dispersion relation for the intrinsic frequency χ:
χ = ω −
uk
x
=−
βk
x
k
2
x
+ k
2
y
+
f
2
0
σ
0
k
2
p
(16.39)
The component of the phase velocity c of the three-dimensional wave along the
x-axisisgivenby
c
x
=
ω
k
x
= u −
β
k
2
x
+ k
2
y
+
f
2
0
σ
0
k
2
p
(16.40a)
Compared with the two-dimensional phase velocity of (16.5), the value of c
x
is
modified due to the presence of the vertical wavenumber k
p
which has the dimension
hPa
−1
. Along the y-axis, in analogy to the two-dimensional case (16.6), we find
c
y
=
ω
k
y
=
u −
β
k
2
x
+ k
2
y
+
f
2
0
σ
0
k
2
p
k
x
k
y
= c
x
k
x
k
y
(16.40b)
The phase speed c
p
along the vertical pressure axis,
c
p
=
ω
k
p
=
u −
β
k
2
x
+ k
2
y
+
f
2
0
σ
0
k
2
p
k
x
k
p
= c
x
k
x
k
p
(16.40c)
has the units hPa s
−1
.
16.3 Normal-mode considerations 479
Let us now return to equation (16.40a) to obtain the condition for the standing
wave. Setting c
x
equal to zero, we find
Standing wave: k
2
p,s
=
β
u
−
k
2
x
+ k
2
y
σ
0
f
2
0
(16.41)
showing how the vertical and the horizontal wavenumbers must be related for
standing waves to form. Compare this result with (16.10). Inspection of (16.38)
shows that vertical propagation of Rossby waves is possible only if k
p
is a real
quantity, i.e. k
2
p
> 0. In order to state this condition for the more general case we
solve (16.40a) and obtain
k
2
p
=
β
u − c
x
−
k
2
x
+ k
2
y
σ
0
f
2
0
(16.42)
For the large-scale motion which is being considered here, the vertical stability
is larger than zero. From (16.42) we recognize that the condition k
2
p
> 0 holds
whenever the inequality
β
u − c
x
>k
2
x
+ k
2
y
(16.43)
is satisfied. This requires that, for a large positive value of the velocity deficit
u − c
x
, vertically propagating Rossby waves cannot occur as might be the case
for strong westerly flow. For easterly flows with a negative value of the velocity
deficit, the vertical propagation of Rossby waves is not permitted by (16.42). These
conclusions are supported by observations.
Finally we observe that (16.40a) reduces to (16.6) if k
p
= 0.
16.3 Normal-mode considerations
Additional information about vertical Rossby waves can be obtained by simplifying
the basic equations (16.35) and (16.36). Following Wiin-Nielsen (1975) we set the
basic flow velocity
u = 0 and ignore any y-dependency of the pertinent variables.
However, we permit the amplitudes of the geopotential and the generalized vertical
velocity to depend on pressure. The system of equations to be solved is then given
by
(a)
∂
∂t
∂
2
φ
∂x
2
+ β
∂φ
∂x
= f
2
0
∂ω
∂p
(b)
∂
∂t
∂φ
∂p
+ σ
0
ω
= 0
(16.44)
Now we consider perturbations of the form
φ
ω
=
A(p)
B(p)
exp[ik
x
(x − ct)] (16.45)
480 Rossby waves
which are referred to as normal-mode solutions. By substituting (16.45) into (16.44)
we obtain the first-order differential-equation system
k
3
x
c + k
x
β
iA = f
2
0
∂B
∂p
,ik
x
c
∂A
∂p
= σ
0
B
(16.46)
Since A and B depend only on the single variable p we may replace the partial
by the total derivative. By obvious steps we combine this system to give a single
second-order differential equation:
d
2
B
dp
2
−
c + β/k
2
x
c(f
0
/k
x
)
2
σ
0
B = 0(16.47)
We now introduce the abbreviations
c
R
=
β
k
2
x
,c
I
=
f
0
k
x
,a
2
= σ
0
p
2
0
,p
r
=
p
p
0
(16.48)
into (16.47) and obtain
d
2
B
dp
2
r
+ l
2
(p
r
)B = 0withl
2
(p
r
) =−
c + c
R
cc
2
I
a
2
(p
r
)(16.49)
Note that, according to (16.30) and (16.48), a
2
= a
2
(p
r
)sothatwealsohave
l
2
= l
2
(p
r
). We recognize that −c
R
and c
I
represent the phase velocities of the pure
Rossby waves and of the inertial waves. The amplitude B of the vertical velocity
is now defined with respect to the relative pressure p
r
. Obviously the solution of
(16.49) depends on the behavior of l
2
(p
r
).
A simple wave-type solution can be obtained by requiring that l(p
r
) is a constant
and that l
2
> 0. In this case the characteristic values of (16.49) are imaginary, thus
resulting in the solution
B(p
r
) = C
1
cos(lp
r
) + C
2
sin(lp
r
)(16.50)
Imposing the boundary conditions
B(p
r
= 0) = 0,B(p
r
= 1) = 0 =⇒ C
1
= 0,l= l
n
= nπ, n = 0, 1,...
(16.51)
which require that the generalized vertical velocities vanish at the top and at the
base of the atmosphere, means that we must select l = l
n
= nπ to make the sine
function vanish.
With l
n
= nπ, n = 0, 1,... from (16.49) we find a whole spectrum of phase
velocities:
c
n
=−
a
2
c
R
a
2
+ n
2
π
2
c
2
I
(16.52)