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CAI AND HUANG
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A New Look at the Physics of Rossby Waves: A Mechanical–Coriolis Oscillation
MING CAI
Department of Earth, Ocean, and Atmospheric Science, The Florida State University, Tallahassee, Florida
BOHUA HUANG
Department of Atmospheric, Oceanic, and Earth Sciences, College of Science, George Mason University,
Fairfax, Virginia, and Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland
(Manuscript received 22 March 2012, in final form 2 August 2012)
ABSTRACT
The presence of the latitudinal variation of the Coriolis parameter serves as a mechanical barrier that causes
a mass convergence for the poleward geostrophic flow and divergence for the equatorward flow, just as a
sloped bottom terrain does to a crossover flow. Part of the mass convergence causes pressure to rise along the
uphill pathway, while the remaining part is detoured to cross isobars out of the pathway. This mechanically
excited cross-isobar flow, being unbalanced geostrophically, is subject to a ‘‘half-cycle’’ Coriolis force that
only turns it to the direction parallel to isobars without continuing to turn it farther back to its opposite
direction because the geostrophic balance is reestablished once the flow becomes parallel to isobars. Such
oscillation, involving a barrier-induced mass convergence, a mechanical deflection, and a half-cycle Coriolis
deflection, is referred to as a mechanical–Coriolis oscillation with a ‘‘barrier-induced half-cycle Coriolis
force’’ as its restoring force. Through a complete cycle of the mechanical–Coriolis oscillation, a new geostrophically balanced flow pattern emerges to the left of the existing flow when facing the uphill (downhill)
direction of the barrier in the Northern (Southern) Hemisphere. The b barrier is always sloped toward the
pole in both hemispheres, responsible for the westward propagation of Rossby waves. The b-induced
mechanical–Coriolis oscillation frequency can be succinctly expressed as v 5 2beffective /k, where
beffective 5 b cos2 l cos2 a, and l is the angle of a sloped surface along which the unbalanced flow crosses isobars,
a is the angle of isobars with the barrier’s slope, and k is the wavenumber along the direction of the barrier’s
contours.
1. Introduction
Rossby waves, named after the pioneering work of
Rossby (1939), play a fundamental role in large-scale atmospheric and oceanic motions that affect weather and
climate. Rossby wave dynamics is the cornerstone of all
modern textbooks on atmospheric and ocean dynamics
(e.g., Gill 1982; Pedlosky 1987; Holton 2004; Mak 2011).
Rossby waves owe their existence to the meridional gradient of planetary vorticity and/or topography on the rotating Earth. It is straightforward to derive the dispersion
relation of Rossby waves from the quasigeostrophic potential vorticity (PV) conservation equation without referencing to ageostrophic flow. This is possible because the
Corresponding author address: Ming Cai, Department of Earth,
Ocean, and Atmospheric Science, The Florida State University,
Tallahassee, FL 32306.
E-mail: [email protected]
DOI: 10.1175/JAS-D-12-094.1
Ó 2013 American Meteorological Society
net effect of ageostrophic flow to Rossby waves has been
already incorporated into quasigeostrophic potential
vorticity (QGPV) dynamics. The oscillation mechanism of
Rossby waves is generally understood through the conservation of QGPV, and the potential vorticity gradient is
identified as its restoring force (e.g., Hoskins et al. 1985),
although it is not clear what is its mechanical restoring
force from the prospective of the Newton’s second law, as
in other types of waves in fluid mechanics. The role of
ageostrophic flow in Rossby wave dynamics is discussed
mainly in the context of integral features, such as the
averaged energy balance within a cycle (e.g., LonguetHiggins 1964; Longuet-Higgins and Gill 1967; Gill 1982).
As an exception, Gill (1982) has qualitatively described
the role of ageostrophic motions in Rossby waves.
In addition to Gill (1982), there are also a few publications in the literature to explain the propagation
mechanism of Rossby waves that is not based on the
principle of QGPV conservation. Bjerknes (1937)
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argued that the convergence/divergence owing to the
latitudinal variation of the Coriolis parameter causes
variations in the mass field, responsible for the westward
propagation of Rossby waves. Bjerkness and Holmboe
(1944) further substantiated this point by examining
the b effect on the balance of forces. Using a kinematic
method, they found that a convergence occurs in front
of a propagating midlatitude trough because of the increasing Coriolis force on the poleward flow, which is
countered by the divergence due to extra anticyclonic
curvature in a gradient wind balance. They also deduced
that this balance makes the Rossby wave phase speed
slower than the prevailing westerly and called the
b-induced westward component of phase speed as the
‘‘critical velocity.’’ This argument was further elaborated by Palmen and Newton (1969) in their famous
treatise of the atmospheric circulation.
It has been argued that the divergence is not the essential factor for the existence of Rossby waves since
Rossby waves can exist in a nondivergent barotropic
model, as illustrated in the original publication of Rossby
(1939). Because of the existence of Rossby waves in a
nondivergent barotropic model, Rossby (1940, p. 69)
further pointed out that ‘‘the factors determining the
stationary or progressive character of the motion are to
be found in the vorticity distribution and that the displacement of the pressure field is a secondary effect.’’
Platzman (1968) examined the momentum equations of
a nondivergent barotropic model. He concluded that the
pressure gradient along the wave propagation direction
is exactly in geostrophic balance, while the transverse
pressure gradient is not.
Durran (1988) substantiated this intriguing finding
of Platzman (1968). By utilizing the normal-mode solution obtained from the vorticity equation, he identified
the pressure pattern associated with Rossby wave motions from the momentum equations of a nondivergent
barotropic model. He demonstrated clearly that ‘‘geostrophically balanced meridional windfield periodically
reverses in response to a small meridional pressure gradient arising from the latitudinal variation of the Coriolis parameter’’ (Durran 1988, p. 4021). In this sense,
the b-induced pressure gradient force is identified as
the restoring force for Rossby waves. Though quite
successful in giving an explanation for the mechanism
of nondivergent barotropic Rossby waves, a potential
shortcoming of this approach is the total elimination
of the ageostrophic flow. As a result, the notion of the
b-induced pressure gradient force as the restoring force
for Rossby waves cannot be easily generalized to the
case with the presence of convergence/divergence
without an apparent ambiguity in partitioning the total
flow into geostrophic and ageostrophic components, as
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reported in Durran (1988). As far as b-induced Rossby
waves are concerned, one could attribute the unbalance
part of the total flow either to pressure as in Durran
(1988) or to motion itself as to be shown below in our
work. However, for topographic Rossby waves on an
f plane, there is no reason for unbalanced pressure to
exist because the Coriolis parameter does not vary with
latitude. As a result, Durran’s explanation could not be
generalized to explain the mechanism of topographic
Rossby waves. It is of importance to point out that
the momentum equations of a nondivergent barotropic
model used in the existing studies (e.g., Longuet-Higgins
1964; Platzman 1968; Durran 1988) should still be regarded as ‘‘primitive equations.’’ Although its corresponding vorticity equation is not distinguishable from
the quasigeostrophic (QG) vorticity equation of a nondivergent barotropic model, the momentum equations
between the two are quite different. This difference
seems to explain why Durran’s non-PV based explanation is more like a standalone explanation that cannot be
easily reconciled with the PV-based explanation.
The primary objective of this paper is to take a new
look from the mass conservation and QG momentum
prospective at the old question, what is the exact form of
the physical restoring force responsible for the oscillation that causes Rossby waves to propagate only in one
direction, namely, along the direction with large background potential vorticity on the right? Such a physical
explanation would have to be applicable to cases with
and without the inclusion of divergent flow, and be
equally applicable to topographical Rossby waves, and
be easily reconciled with the well-established PV-based
explanation. Furthermore, such an explanation, if it is
physical, should lead to the solution of Rossby waves
without obtaining the solution in prior from a QGPV
equation. The physical explanation to be developed in
our study, if successful, would provide a bridge between
geophysical fluid QGPV dynamics of large-scale motions and classic fluid mechanics.
In the next section, we explicitly examine the role of
ageostrophic motions induced by the latitudinal variation of the Coriolis parameter in giving rise to the restoring force for an oscillation responsible for Rossby
wave motions in the simplest possible model, namely,
a 1D nondivergent barotropic model. The delineation
of the exact form of the restoring force for Rossby waves
enables us to put forward a mechanics-based derivation
of the dispersion relation of Rossby waves as well as the
complete QG solution of the 1D nondivergent model
without assuming a normal-mode solution and without
explicitly solving for the partial differential equations.
The physics principles applied in the mechanics-base
derivation are (i) the conservation principles of mass
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and energy, (ii) Newton’s second law, and (iii) the geostrophy. The mechanics-based derivation is equivalent
to a parcel method and therefore its solution is a local
solution that does not have to be in the form of a normalmode solution. In section 3, we apply the mechanics-based
derivation to obtain the dispersion of Rossby waves and
a complete solution of a linearized QG shallow water
(QGSW) model without explicitly solving for the corresponding partial differential equations. The ability of
obtaining Rossby wave solution from the simplest possible model using the mechanics-based derivation and
generalizing it to a more realistic model enables us to
identify succinctly the most intrinsic physical factors
responsible for Rossby wave motions and their time
scales. In section 4, we consider the case of topographic
Rossby waves, which allows us to illustrate that the
same form of restoring force is also applicable to topographic Rossby waves in an f-plane QG model.
Summary and discussions are provided in section 5.
Particularly, we will reconcile our non-PV-based explanation on the origin of Rossby waves with the PV-based
explanation.
2. 1D nondivergent Rossby waves
Let us first look at the simplest possible case for Rossby
waves, namely, zonally propagating waves in a linearized
1D b-plane nondivergent QG barotropic model with a
motionless mean state. In the 1D model, the geostrophic
flow y is aligned with the orientation of planetary vorticity
gradient (y coordinate) and does not vary with y.1 The
corresponding governing equations are
b(y 2 y0 )y 1 f0 ya 5 0
where y 5
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›c
.
›x
and
›y
5 2f0 ua ,
›t
(1)
In (1) f0 is the midlatitude (at y0) Coriolis parameter;
b is the meridional gradient of planetary vorticity at
y0; c is geostrophic streamfunction; and the subscript
a denotes ageostrophic flow, which is related to ageostrophic streamfunction ca in the nondivergent model
as ua 5 2›ca /›y and ya 5 ›ca /›x. Applying geostrophic
1
Alternatively, we could set up the coordinate for the 1D model
in such a way that the geostrophic flow is aligned with contours
of planetary vorticity. However, with this alternative setting, the
b-plane model is reduced to an f-plane model and the only possible
solution is a time-independent geostrophic mode because the
flow does not cross planetary vorticity contours. The negligence
of tendencies of ua and y a in (1) is in accordance with the QG
approximation.
and ageostrophic streamfunction to the first equation
of (1), we obtain
ca 5 2b(y 2 y0 )c/f0 , ua 5 bc/f0 ,
and
y a 5 2b(y 2 y0 )y/f0 .
(2)
In deriving (2), we have utilized the fact that c does
not vary with y in the 1D Rossby wave model. The total
(geostrophic plus ageostrophic) flow along isobars is
b(y 2 y0 ) ›c
f0
›c
’
.
y 1 ya 5 1 2
›x f0 1 b(y 2 y0 ) ›x
f0
(3)
According to (3), the total flow parallel to isobars is in
the geostrophic balance with a latitudinally varying
Coriolis parameter that approximately equals the local
b-plane Coriolis parameter within the accuracy of the
b-plane approximation, that is, jb(y 2 y0 )/f0 j 1, as
indicated by the wavy equal sign in (3). For this reason,
the total flow parallel to isobars is referred to as the total
‘‘balanced flow’’ and the departure of the total balanced
flow from the geostrophic flow is called the ‘‘balanced
ageostrophic flow.’’ Obviously, the existence of the balanced ageostrophic flow is entirely due to our defining the
geostrophic flow with the domain-mean Coriolis parameter in a QG model even when the actual Coriolis parameter
varies with latitude. Because the balanced ageostrophic
flow is always proportional to the geostrophic flow, it
suffices to predict just the geostrophic flow alone in a
QG model and the total balanced flow can then be obtained by summing up the geostrophic flow and the
balanced ageostrophic flow. The other component of the
ageostrophic flow is ua. Because it crosses isobars, it is
not geostrophically balanced (referred to as the unbalanced ageostrophic flow). This decomposition of
ageostrophic flow follows that of Gill (1982), who called
the unbalanced component as the isallobaric part following Brunt and Douglas (1928), because it is directly
associated with the pressure tendency, as shown in (4):
›y ›2 c
5
5 2f0 ua 5 2bc:
›t ›x›t
(4)
Considering a normal-mode solution, we obtain the
following dispersion relation:
v 5 2b/k
(5)
Equation (4) clearly suggests that the restoring force
for the oscillation associated with Rossby waves is related to the Coriolis force acting on the unbalanced
ageostrophic flow. To understand the nature of the underlying restoring force responsible for the oscillation
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causing Rossby wave motions, let us take a close look at
the origin of the unbalanced ageostrophic flow in this 1D
nondivergent QG barotropic model. Under the QG
approximation, the geostrophic flow is in balance with
the pressure gradient force using the domain-mean f0.
As indicated by the negative sign in front of the b
term in (3) and also discussed in Gill (1982), the total
balanced flow (sum of the solid black and red arrows
in Fig. 1) is slower than the geostrophic flow where
b(y 2 y0 )/f0 is positive and faster where b(y 2 y0 )/f0 is
negative. In other words, the speed of the balanced flow
decreases along the gradient direction of the (absolute)
local Coriolis parameter even through the pressure
gradient remains invariant. In this sense, the latitudinal
variation of the Coriolis parameter acts as a physical
barrier (referred to as the b barrier, along which the
Coriolis parameter or f changes) that slows down the
balanced flow when it moves uphill crossing the f contours (y/f0 . 0), causing the convergence of mass along
the uphill pathway (we will formally prove the equivalency of the b effect to a physical barrier in section 4,
where we discuss topographic Rossby waves). The reverse can be said along the downhill pathway (y/f0 . 0)
of the balanced flow. The b-induced convergence along
the uphill pathway causes a rise in pressure there. In
a nondivergent model, the rise in pressure along the
uphill pathway takes place instantly (without requiring
mass relocation because the pressure is exerted to the
fluid through the lid on top of the fluid). The mass
convergence along the uphill pathway needs to be detoured out of the pathway along the direction perpendicular to the pathway as required by the mass
conservation law, causing mass fluxes in the direction
perpendicular to isobars. Therefore, the unbalanced
flow that crosses isobars is just the detour flow of the
total balanced flow when it is convergent or divergent.
We refer to the transition of the b-induced convergent/
divergent flow from the direction along isobars to the
direction perpendicular to isobars as a ‘‘mechanical deflection.’’ The mechanical deflection essentially is just
a continuation of the geostrophic flow that crosses the b
barrier and is always along the direction parallel to the
contours of the b barrier, which is the x axis. In the 1D
nondivergent barotropic model in which isobars are
perpendicular to contours of the b barrier, all of the
mass convergence/divergence along the geostrophic
flow pathway is subject to such a mechanical deflection
and the deflection is along the direction perpendicular to
isobars, giving rise to an unbalanced ageostrophic flow
that crosses isobars at its maximum strength.
The unbalanced ageostrophic flow resulting from the
mechanical deflection has a perfect positive correlation
with the existing pressure field [blue arrows in Fig. 1 and
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cf. Fig. 12.2 in Gill (1982)], as indicated by (2). The
Coriolis force only turns the unbalanced flow to the direction parallel to isobars without continuing to turn it
to its opposite direction because of the balance nature of
the flow parallel to isobars. We refer to the transition of
the unbalanced ageostrophic flow from the direction
perpendicular to isobars back to the direction parallel to
isobars as the ‘‘Coriolis deflection.’’ In this sense, the
Coriolis force acts as a restoring force that turns the
unbalanced flow only to one direction, corresponding to
a half-cycle Coriolis force. The strength of the half-cycle
Coriolis force is determined by the b-induced mass
convergence rate. Because the mass convergence along
the geostrophic flow pathway is due to the presence of
a mechanical barrier (such as the b barrier due to the
latitudinal variation of the Coriolis parameter) and only
a half cycle of a complete oscillation involves the Coriolis force, we name the restoring force ‘‘barrier-induced
half-cycle Coriolis force.’’
As illustrated in Fig. 1, the Coriolis deflection acts to
relocate the existing geostrophic flow (black arrows)
toward its left. The instant relocation of the pressure
field in a nondivergent model automatically ensures that
the new flow pattern along isobars is in geostrophic
balance. Again, the new balanced flow is in geostrophic
balance with the local Coriolis parameter, resulting in the
same pattern of the b-induced convergence/divergence
along the constant phase of the new balanced flow, which
is the beginning of the next round of the oscillation. The
mechanical deflection of the b-induced convergence/
divergence of the balanced flow and subsequent Coriolis
deflection form a complete oscillation cycle. For this
reason, this oscillation is referred to as a mechanical–
Coriolis (deflection) oscillation, responsible for Rossby
wave motions propagating to the left of the Coriolis
parameter gradient.
The discussions above effectively point to an alternative but probably to a more physical way to derive the
dispersion relation of Rossby waves as well as the
complete solution for the unbalanced ageostrophic flow
in a QG model without going through the normal-mode
solution of the partial different equations. The alternative derivation is based on the following physics principles: (i) the conservation principles of mass and energy,
(ii) Newton’s second law, and (iii) the geostrophic balance for large-scale motions. We refer to the alternative
derivation as the mechanics-based derivation to differentiate it from the conventional dynamics-based derivation that utilizes the QGPV conservation principle.
To demonstrate this, we again first consider the 1D
nondivergent QG barotropic case, and in the next section we will extend it to the general case. The frequency
of an oscillation (any oscillation) is linearly proportional
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FIG. 1. Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) Rossby waves
(f0 . 0, b . 0, and dy . 0). The y axis points to the uphill direction of the b barrier (which is north
for f0 . 0 and south for f0 , 0), and the x axis is parallel to contours of the b barrier. Thin solid
black lines labeled with ‘‘low pressure’’ and ‘‘high pressure’’ are the lines of constant phases for
the minimum and maximum pressure, respectively. Solid black arrows are the geostrophic flow,
and solid red arrows are the balanced ageostrophic flow. The sum of solid black and solid red
arrows is the total balanced flow. Blue arrows correspond to the unbalanced ageostrophic flow
resulting from the mechanical deflection of the b-induced divergent/convergent flow. Dashed
black arrows indicate the geostrophic flow tendency caused by the b-barrier-induced half-cycle
Coriolis force (i.e., the Coriolis deflection of the unbalanced flow), whereas dashed red arrows are
the tendency of the balanced ageostrophic flow. The geostrophic flow tendency indicates
a propagation of the geostrophic flow pattern toward the left when facing the uphill direction of
the b barrier in the Northern Hemisphere. For the Southern Hemisphere situation, one needs to
swap between ‘‘high pressure’’ and ‘‘low pressure’’ and replace ‘‘uphill’’ with ‘‘downhill.’’
to the strength of its restoring force and inversely proportional to the speed of the movement, namely,
v* 5 (dy/dt)/y [or v* 5 (du/dt)/u].2 Therefore, for the
mechanical–Coriolis oscillation, we have
v* 5
dy/dt 2f0 uua
5
.
y
y
(6)
Note that in (6), we have explicitly used the subscript
‘‘ua’’ to denote ‘‘unbalanced ageostrophic flow’’ instead
of a generic ‘‘a’’ for ageostrophic flow. Similarly, below
we will use the subscript ‘‘ba’’ to denote ‘‘balanced
ageostrophic flow.’’ The sum of the two is the total
ageostrophic flow.
The convergence/divergence rate along the geostrophic flow pathway is proportional to the slope of the
barrier and the speed and angle of the flow at which the
flow crosses the barrier. The b barrier due to the latitudinal variation of the Coriolis parameter has a slope
equal to b/f0. In the 1D case, the geostrophic flow passes
through the b barrier along its gradient direction at
speed y. The divergence due to the passing through the b
barrier by the geostrophic flow is equal to (2yb/f0),
where the negative sign is because y/f0 . 0 corresponds
to an uphill pathway and y/f0 , 0 is downhill. By the
mass conservation law, the divergence along isobars in
a nondivergent model is exactly compensated by the
convergence along the direction perpendicular to isobars. Therefore, we have
›u
›c
2 ua 5 2y(b/f0 ) 5 2(b/f0 ) ,
›x
›x
or uua 5 (b/f0 )c.
(7)
pffiffiffiffiffiffiffi
Note that v* 5 2 21v, where v is a real number and is the
frequency as conventionally defined in a normal model solution in
pffiffiffiffiffiffiffi
the form of exp[ 21(kx 1 ly 2 vt)]. One needs to multiply v* in
pffiffiffiffiffiffiffi
the mechanics-based solution with 21 to convert it to v.
2
Substituting (7) into (6) yields
b
.
v* 5 2
(›c/›x)/c
(8)
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Obviously, for a normal-mode solution, we have
(›c/›x)/c 5 k, where k is the zonal wavenumber, which
reduces (8) to (5).
Beside the fact that only the geostrophic flow is considered to cross over the b barrier, two more approximations within the QG approximation are invoked
implicitly in deriving (6) and (7). In reality, the crossisobar flow, that is, the mechanical deflection, is driven
by the pressure gradient tendency built by the b-induced
convergence along the uphill pathway and divergence
along the downhill pathway. A complete deflection of all
b-induced convergent/divergent flow requires a finite
amount of time, although it is much faster than the time
to build the new geostrophically balanced flow pattern.
Under the QG approximation, it takes place instantly
to establish the unbalanced flow at its full strength so
that mass is immediately transported by the unbalanced
flow from the uphill geostrophic pathway, where the
balanced flow is convergent, to the downhill pathway,
where the balanced flow is divergent. At the same time,
the Coriolis force acts to turn the unbalanced flow from
the direction of crossing isobars back to the direction
parallel to isobars. As a result, all of the unbalanced flow
is turned back to the direction parallel to isobars, giving
rise to a new geostrophically balanced flow that crosses
the same b barrier at a longitude on the west of the
current geostrophic flow. It should be noted that the
seemingly passive unbalanced ageostrophic flow, due to
its total dependency on the geostrophic flow, plays the
essential role of generating the tendency of the geostrophic flow, which is accomplished by transporting
mass across isobars out of the current geostrophic flow
pathway to build the future pathway at a different longitude zone via the Coriolis deflection.
3. From the 1D nondivergent solution to the
general solution in a QGSW model
The mechanics-based derivation of the dispersion
of the 1D nondivergent Rossby waves can be extended
to a general case of 2D/3D Rossby waves by just using
(6)–(8) without going through their corresponding partial differential equations. The key is to figure out the
strength of the unbalanced ageostrophic flow that
crosses isobars on the horizontal surface.
Let us first consider a 1D linearized b-plane QGSW
model with a motionless mean state that extends the 1D
nondivergent barotropic model with the inclusion of divergence. Note that in a QGSW, we need to replace
c 5 (g/f0 )h, where h is the surface height of a shallowwater model with a constant mean depth of H0. In the 1D
nondivergent model, the unbalanced ageostrophic flow
has its maximum possible strength, because all of the
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b-induced convergent/divergent flow is diverted to cross
isobars. We denote the maximum possible deflection flow
as umax deflection , whose divergence is exactly equal to (yb/f0),
the b-induced convergence. According to (7), we have
umax
2
deflection 5 bgh/f0 .
(9)
As discussed above, umax deflection is just a continuation of
the geostrophic flow that crosses the b barrier, which is
the meridional component of the geostrophic flow, and
umax deflection is always along the direction parallel to the
contours of the b barrier, which is the x axis.
When the surface height is allowed to change (by removing the lid in a nondivergent case), part of the
b-induced convergence causes the surface height to rise
along the uphill geostrophic flow and vice versa. The remaining part of the b-induced convergence/divergence
is subject to mechanical deflection as the compensating
ageostrophic flow that crosses isobars. This causes a
reduction in the strength of the cross-isobar flow. Let
us denote the reduction part from umax deflection in the
1D divergent barotropic model as uua reduction . The
net unbalanced flow that crosses isobars can be written as uua 5 umax deflection 2 uua reduction . The opposite of
uua reduction , that is, 2uua reduction , is used to change surface
height at the rate equal to 2H0 [›(2uua reduction )/›x]. Such
restoration of the pressure field requires a finite amount
of time, unlike the nondivergent case in which the restoration takes place instantly. Under the QG approximation, the time scale for restoring the pressure field has
to be exactly synchronized with the Coriolis deflection of
the net unbalanced flow in such a way that the restored
flow along isobars by the Coriolis deflection is in balance
with the newly restored pressure gradient force under
the global-mean Coriolis parameter, namely,
2
gH › (2uua reduction )
g › ›h
52 0
5 2f0 uua
f0 ›x ›t
f0
›x2
5 2f0 (umax
deflection 2 uua reduction ) 5
›y
.
›t
(10)
Next, we determine how umax deflection is partitioned
into uua and uua reduction by a geometry consideration. In
a nondivergent barotropic model, the unbalanced flow
resulting from the mechanical deflection—that is,
umax deflection —is forced to be along the horizontal surface because of the presence of a lid on top of the rotating fluid. In a divergent barotropic (or baroclinic)
model, the unbalanced flow that crosses isobars is not
along the horizontal surface because it also involves
a vertical component. Similarly, the reduction flow is
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maximum possible deflection flow is diverted into two
parts: one is uua reduction at a speed of umax deflection sinl
and the other part is uua at a speed of umax deflection cosl.
The projections of umax deflection sinl and umax deflection cosl
to the horizontal are the net reduction from the maximum
possible deflection flow (solid red arrow) and the actual
unbalanced flow that crosses isobars horizontally (solid
blue arrow), respectively—namely,
uua
FIG. 2. Illustration of the vertical orientation of the unbalanced
horizontal flow in the direction perpendicular to isobars (x axis) in
a QGSW model for the North Hemisphere situation (f0 . 0). All
thin dotted lines are auxiliary lines for the purpose of illustrating
the angles of various surfaces with the vertical and horizontal.
Maximum and minimum surface height perturbations are located
at the longitude marked with ‘‘High’’ and ‘‘Low.’’ The longitude
location between ‘‘High’’ and ‘‘Low’’ on the left-hand side corresponds to the downhill pathway of the geostrophic flow with the
maximum downhill flow located at ‘‘Max. Downhill Flow’’ whereas
that on the right-hand side corresponds to the uphill pathway of the
geostrophic flow with the maximum uphill flow located at ‘‘Max.
Uphill Flow.’’ The thick black arrows (underneath solid red and
blue arrows) correspond to umax deflection , the required compensating flow of the b-induced convergence/divergence of the uphill/
downhill flow. Dashed blue arrows represents uua , which is along
a sloped surface whose angle with the horizontal is l, and the solid
blue arrows correspond to the projection of uua on the horizontal
surface (i.e., uua ). Dashed red arrows represent uua reduction , which is
along a sloped surface whose angle with the horizontal is
l* 5 908 2 l, and the solid red arrows correspond to the projection
of uua reduction on the horizontal surface (i.e., uua reduction ). The
vector sum of dashed red and blue arrows exactly equals the solid
black arrows, and so does the scalar sum of solid blue and red arrows. The convergence of 2uua , which is the projection of
2uua reduction on the horizontal surface, is responsible for the new
surface height pattern. For the Southern Hemisphere situation, one
needs to swap between ‘‘High’’ and ‘‘Low.’’
also not along the horizontal surface. For this reason, we
use a vector form, namely, uua and uua reduction , to denote
them, where the horizontal components of uua and
uua reduction are uua and uua reduction , respectively. The
opposite of uua reduction represents the actual flow that is
convergent into the uphill geostrophic pathway and divergent away from the downhill pathway to account for
the needed change in the mass field, whereas uua corresponds to the part that crosses isobars along a sloped
surface in a divergent model. Note that both uua and
uua reduction are on the same vertical–horizontal crosssection plane whose horizontal axis is parallel to the
pressure gradient direction. Their vector sum is equal to
umax deflection , and so is the sum of their projections on the
horizontal surface. As a result, we can assume that uua is
along a sloped surface with an angle of l from the horizontal surface, whereas uua reduction is along a surface
that is perpendicular to uua . As illustrated in Fig. 2, the
2
reduction 5 umax deflection sin l
uua 5 umax
5 umax
and
2
deflection cos l
deflection 2 uua reduction .
(11)
The angle l can be determined by substituting (11) into
(10), which yields
(f02 /gH0 )
2f h/H0
sin2 l
.
5 0
5
2
2
2
cos l [2(› h/›x )/h] (›y/›x)
(12)
Therefore, the slope of the surface along which the unbalanced flow crosses isobars is proportional to the
square root of the ratio of the thickness portion of PV of
Rossby waves to the relative vorticity portion, or the
ratio of the scale of geostrophic motions to the Robby
radius of deformation.3 Because relative vorticity always
has the opposite polarity of f0 h, the negative sign in front
of f0 h ensures that the ratio of sin2 l to cos2 l is always
positive definite. Based on (12), we have
cos2 l 5
2(›2 h/›x2 )/h
[2(›2 h/›x2 )/h] 1 (f02 /gH0 )
sin2 l 5
f02 /gH0
.
[2(›2 h/›x2 )/h] 1 (f02 /gH0 )
and
(13)
Applying the second equation in (11) to (6), we obtain
the mechanical–Coriolis oscillation frequency, or the
dispersion of the 1D QGSW model as
v* 5
2f0 uua
2
maximum cos l
y
b cos2 l
.
52
(›h/›x)/h
(14)
Now we are ready to discuss the physical meaning of l.
Without the QG approximation, the convergence along
the uphill geostrophic flow should immediately result in
3
In a nondivergent barotropic model, the notion of Rossby radius of deformation is no longer relevant, corresponding to the case
l 5 0, in which 2uua reduction 5 0 (no vertical motion) and uua
crosses isobars horizontally at a speed of umax deflection .
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pressure rising there, which in turns acts to accelerate the
unbalanced flow that crosses isobars downward. The
opposite can be said along the downhill pathway. Under
the QG approximation, the geostrophic flow defined with
the domain-mean Coriolis parameter is nondivergent. As
a result, the rising of pressure along the uphill geostrophic
flow and falling along the downhill pathway are done by
the mass convergence carried out by 2uua reduction from
the direction perpendicular to the pathway along a sloped
surface of 908 2 l. The convergence/divergence
of 2uua reduction gives rise to vertical motion, corresponding to surface height changes. Similarly, the mass
carried out by the unbalanced flow that crosses isobars
downward/upward at a slope of l—that is, uua —does
not directly come from the uphill/downhill geostrophic
pathway under the QG approximation. Instead, it comes
from the direction perpendicular to the geostrophic
pathway (Fig. 2). The slopes of both 2uua reduction and uua
depend on the ratio of the spatial scale of motions to the
Rossby radius of deformation. When the spatial scale of
motions is longer, the slope of 2uua reduction is gentle but
the slope of uua is steeper, responsible for a slower rate at
which the mass field along the existing geostrophic pathway and the geostrophic flow along the new pathway are
restored or a slower mechanical–Coriolis oscillation frequency. The reverse can be said about short-scale geostrophic disturbance. For a normal-mode solution of the
1D QGSW model, we have
cos2 l 5 k2 /(k2 1 f02 /gH0 ) and
sin2 l 5 (f02 /gH0 )/(k2 1 f02 /gH0 ) .
(15)
FIG. 3. Illustration of various projections for generalizing the 1D
solution to the 2D solution (f0 . 0, b . 0). The angle that the
isobars make with the uphill direction of the b barrier (solid black
arrow) is a, and n represents a unit vector pointing to the gradient
direction of pressure X. The actual b-barrier slope that the geostrophic flow is crossing is represented by the dashed black arrow.
The dashed red arrows represent the unbalanced flow, which are
parallel to the normal direction of isobars. The projections of
dashed red arrows onto the directions that are parallel to and
perpendicular to contours of the b barrier are indicated by the solid
red and blue arrows, respectively. ‘‘High’’ and ‘‘Low’’ represent
lines of constant phase for the maximum ‘‘peak’’ and minimum
‘‘valley’’ of the free surface of the shallow-water-equation model
for the Northern Hemisphere situation. For the Southern Hemisphere situation, one needs to swap between ‘‘High’’ and ‘‘Low.’’
b barrier along the geostrophic pathway is (b/f0 ) cosa.
The strength of the divergence of the balanced flow due
to the b barrier is equal to 2V(b/f0 ) cosa, where V is the
speed of the geostrophic flow. The maximum possible
mechanical deflection of the b-induced divergence in
the 2D case, according to (7), is
Substituting (15) into (14), we recover the dispersion
relation of the 1D QGSW Rossby waves obtained from
the potential vorticity conservation equation, namely,
bk
v52 2
.
k 1 f02 /gH0
(16)
Next, let us consider 2D plane Rossby waves in a
b-plane QGSW model with a free surface. Again, we will
first seek a generic solution based on (6), (7), (11), and
(13) before applying it with a normal-mode solution.
The 1D case is a special case of the 2D in which the
geostrophic flow crosses the b barrier directly. In the 2D
case, the geostrophic flow crosses the b barrier at an
angle. This implies that for the same spatial variability of
pressure, the geostrophic flow crosses the b barrier at
a slower speed. As a result, the resultant unbalanced
flow is weaker and responsible for a slower oscillation
frequency according to (6). Let a be the angle of isobars
with the b-barrier slope (Fig. 3). The actual slope of the
VOLUME 70
Umax
2
deflection 5 (gb cosa/f0 )h ,
(17)
where Umax deflection is the maximum possible deflection
flow along the direction X on the horizontal surface,
where X denotes the direction parallel to the pressure
gradient. The reduction Uua reduction from the maximum
possible deflection flow along the direction X, which is
needed to rebuild the pressure field, and the actual unbalanced horizontal flow along the direction parallel to
the pressure gradient Uua are given in (11) by replacing
umax deflection with Umax deflection . We can use (13) to determine l for the 2D case by either replacing x with X,
or replacing ›2 h/›2 x with ›2 h/›2 x 1 ›2 h/›2 y, namely,
cos2 l 5
2(›2 h/›x2 1 ›2 h/›y2 )/h
,
[2(›2 h/›x2 1 ›2 h/›y2 )/h] 1 (f02 /gH0 )
sin2 l 5
f02 /gH0
.
[2(›2 h/›x2 1 ›2 h/›y2 )/h] 1 (f02 /gH0 )
(18)
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CAI AND HUANG
The value of l determined from (18) again ensures that
changes in the pressure field and in the geostrophic flow
would still satisfy the geostrophic balance with the domain
Coriolis parameter. Replacing umax deflection in the second
equation of (11) with Uua maximum in (17) yields the net
unbalanced ageostrophic flow along the direction perpendicular to isobars for the 2D case as shown:
Uua 5 b cosa cos2 l(g/f02 )h .
(19)
Replacing uua in (6) with Uua in (19) and y with V yields
the mechanical–Coriolis oscillation frequency for the
2D case as shown:
f U
b cosa cos2 l(g/f0 )h
b cos2 a cos2 l
v* 5 2 0 ua 5 2
.
52
(›h/›x)/h
V
V
(20)
The term on the right-hand side of (20) is obtained from
the relation V 5 (g/f0 )(›h/›X) 5 [(g/f0 )(›h/›x)]/cosa, as
indicated in Fig. 3. By projecting Uua onto the x and the y
directions, we obtain the x and y components of the net
unbalanced ageostrophic flow in the 2D QGSW model,
written as
uua 5 Uua cosa 5 (gb cos2 a cos2 l/f02 )h,
y ua 5 Uua sina 5 (gb cosa sina cos2 l/f02 )h .
(21)
In addition to the unbalanced ageostrophic flow given in
(21), there is a balanced ageostrophic flow that is related
to the geostrophic flow as shown:
uba 5 2(b/f0 )(y 2 y0 )u and
y ba 5 2(b/f0 )(y 2 y0 )y ,
(22)
where u 5 2(g/f0 )(›h/›y) and y 5 (g/f0 )(›h/›x). We will
further discuss the exact nature of the balanced ageostrophic flow in section 5. The sum of (21) and (22) is
the total ageostrophic flow in the 2D QGSW model. The
total flow of the 2D QGSW model is the sum of the
geostrophic flow and the total ageostrophic flow.
For a normal-mode solution, the two angles a and l
are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
l2
and
sina
5
,
cosa 5
k2 1 l2
k2 1 l2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2 1 l2
and
cosl 5
k2 1 l2 1 f02 /gH0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f02 /gH0
sinl 5
,
(23)
k2 1 l2 1 f02 /gH0
where k and l are zonal and meridional wavenumbers,
respectively. Substituting (23) into (20), we recover the
dispersion relation of the 2D QGSW Rossby waves obtained from the potential vorticity conservation equation,
namely,
bk
.
v52 2
k 1 l2 1 f02 /gH
(24)
With the normal-model solution, the net unbalanced
ageostrophic flow given in (19) becomes
uua 5
bk2 (gH0 /f02 )(h/H0 )
k2 1 l2 1 f02 /gH0
y ua 5
bkl(gH0 /f02 )(h/H0 )
.
k2 1 l2 1 f02 /gH0
and
(25)
4. Topographic Rossby waves
Since the pioneering work of Robinson and Stommel
(1959), Phillips (1965), and Rhines (1969, 1970), who
were among the first to analytically derive the solution
of topographic Rossby waves from the QGPV equation,
the planetary vorticity gradient has been generalized
to the PV gradient as the restoring force for Rossby
waves. Many factors, including the planetary vorticity gradient, topography, and spatial variations of
temperature/density of the mean flow, constitute the
PV gradient. Therefore, the PV view of Rossby waves
has broad applicability to a wide range of mean states
for weather and climate variability. In this section, we
wish to demonstrate the generality of the mechanical–
Coriolis oscillation for Rossby waves by applying it
to topographic Rossby waves. Again, we derive the
solution of topographic Rossby waves purely based
on physics without going through the partial differential governing equations. Without losing the generality, let us consider a shallow-water model with
a mean depth H that varies linearly along the y-axis
direction on an f plane (we can always rotate the coordinate on an f plane so that its y coordinate is perpendicular to contours of the mean depth), namely,
H 5 H0 2 «y, where both H0 and « are constant and
0 , j«yj H0 .
Again, we first consider the 1D nondivergent situation, in which the geostrophic flow, y 5 ›c/›x, which
does not vary in the y direction, crosses the topography
along the y direction. As sketched in Fig. 4, the immediate consequence of the shallowing of the water
depth with latitude is a mass convergence for an uphill
flow (y . 0 for « . 0) and divergence for a downhill
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JOURNAL OF THE ATMOSPHERIC SCIENCES
VOLUME 70
FIG. 4. Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) topographic
Rossby waves (f0 . 0, b 5 0, and dy . 0). The shading gradient represents the latitudinal
shallowing of water depth, H 5 H0 2 «y (« . 0), due to a bottom topography that is sloped
toward the north (y axis). Thin solid black lines labeled with ‘‘low pressure’’ and ‘‘high pressure’’ are the lines of constant phases for the minimum and maximum pressure, respectively.
Solid black arrows are the mass fluxes carried out by the geostrophic flow that crosses the
topography. Blue arrows correspond to the mass transport along contours of the topography
carried out by the unbalanced flow, resulting from the mechanical deflection of the topographicinduced mass divergence/convergence. Dashed black arrows indicate the mass flux tendency
along the sloping direction of the topography, resulting from the Coriolis force acting on the
unbalanced flow (i.e., the Coriolis deflection of the unbalanced flow). The mass flux tendency
along the sloping direction of the topography indicates a propagation of the pattern of the
geostrophic mass fluxes toward the left when facing the uphill direction of the topography in the
Northern Hemisphere. For the Southern Hemisphere situation, one needs to swap between
‘‘high pressure’’ and ‘‘low pressure’’ and replace ‘‘uphill’’ with ‘‘downhill.’’
flow (y , 0 for « . 0). The amount of mass divergence
is equal to 2«y. In the nondivergent case, all of the
mass convergence/divergence in the slopping direction has to be diverted to the direction along topographic contours by the mass conservation principle,
resulting in an unbalanced flow that crosses isobars at
its maximum possible speed. The strength of the unbalanced flow can be determined by requiring that the
mass divergence along the pathway of the geostrophic flow
is identical to the mass convergence along the direction
that is perpendicular to the pathway. Considering the
condition j«yj H0 , the mass convergence along the
direction perpendicular to isobars can be approximated
as 2›uua (H0 2 «y)/›x ’ 2H0 (›uua /›x). By applying the
mass conservation law [i.e., 2H0 (›uua /›x) 5 2«y] and
y 5 ›c/›x, we obtain
uua 5
«
c.
H0
Substituting (26) into (6) yields
(26)
~
b
,
v* 5 2
(›c/›x)/c
~5
where b
f0 «
.
H0
(27)
Applying a normal-mode solution to (27) yields the
dispersion of the 1D nondivergent topographic Rossby
~ Therefore, the
waves on an f plane, namely, v 5 2b/k.
mechanical–Coriolis oscillation mechanism for Rossby
waves associated with the latitudinal variation of the Coriolis parameter is equally applicable to topographic Rossby
waves. As far as the excitation of Rossby waves is concerned, the slope of topography, that is, «/H0 , is equivalent
to b/f0 , the b-barrier slope. Both are capable of exciting the
unbalanced flow that crosses isobars when the geostrophic
flow crosses over the barrier. The Coriolis deflection of the
unbalanced flow restores the balanced flow, completing
a whole cycle of the mechanical–Coriolis oscillation. The
mechanical–Coriolis oscillation frequency is proportional
to the product of the Coriolis parameter and the physical/
dynamical slope of the barrier, which is f0 («/H0 ) for the
topographic barrier and f0 (b/f0 ) for the b barrier.
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CAI AND HUANG
It is straightforward to generalize the solution of the
1D nondivergent topographic Rossby waves for the 2D
case with a free surface by repeating the same procedures, (9)–(25), in section 3. All we need to do is replace
the dynamical slope of the b barrier (b/f0 ) with the
physical slope of the topography («/H0 ).
5. Summary and discussion
In this paper, we have deduced the oscillation mechanism and physical restoring force responsible for Rossby
waves. A minimal model for illustrating the oscillation
mechanism of Rossby wave motions is a nondivergent
barotropic model on an f plane with bottom topography.
The presence of bottom topography is a physical barrier
when the geostrophic flow crosses over it. There is a mass
convergence when the geostrophic flow climbs the topography and a mass divergence when it descends along
the topography. The mass convergence along the geostrophic flow pathway is compensated exactly in a nondivergent model by mass divergence in the direction
perpendicular to isobars, giving rise to an unbalanced
ageostrophic flow that crosses isobars at its maximum
possible speed. We refer to the deflection of the barrierinduced convergent flow from the direction along isobars
to the direction perpendicular to isobars as a ‘‘mechanical
deflection.’’ The Coriolis force then acts to turn the
ulocally
balanced 5 2f
y locally
balanced 5 f
313
unbalanced flow back to the direction parallel to isobars,
but it cannot continue to turn it to its opposite direction
because of the balance nature of the flow parallel to isobars. The restoration of the flow along isobars by the
Coriolis force acting on the unbalanced flow corresponds
to a ‘‘Coriolis deflection.’’ The mechanical deflection of
the barrier-induced convergence/divergence of the balanced flow and subsequent Coriolis deflection of the unbalanced flow form a complete oscillation cycle, referred
to as a mechanical–Coriolis oscillation. The new balanced
flow restored by the Coriolis deflection has the same
spatially alternating pattern of uphill flow and downhill
flow as before, but it is shifted to its left when facing the
uphill (downhill) direction, giving rise to a wave motion
propagating to the left of the uphill (downhill) direction in
the Northern (Southern) Hemisphere. Because the mass
convergence along the geostrophic flow pathway is due
to the presence of a physical barrier and only a half cycle
of a complete mechanical–Coriolis oscillation involves
the Coriolis force, we name the restoring force ‘‘barrierinduced half-cycle Coriolis force.’’
The latitudinal variation of the Coriolis parameter
acts as a physical barrier, referred to as the b barrier.
The latitudinal slope of the b barrier is b/f0 . Large largescale motions are in geostrophic balance with the local
Coriolis parameter. In a b-plane QGSW model, the locally balanced geostrophic flow is approximated as
g
›h
g
›h
g ›h
’ 2 [1 2 b(y 2 y0 )/f0 ] 5 2
,
f0
›y
fQG ›y
0 1 b(y 2 y0 ) ›y
g
›h g
›h
g ›h
’ [1 2 b(y 2 y0 )/f0 ] 5
,
f0
›x fQG ›x
0 1 b(y 2 y0 ) ›x
where fQG 5 f0 /[1 2 b(y 2 y0 )/f0 ] ’ f0 1 b(y 2 y0 ), which
is an equally valid approximation as the b-plane
approximation under the condition jb(y 2 y0 )/f0 j 1.
Therefore, the b-plane QG approximation effectively
approximates the full Coriolis parameter on a b plane as
fQG, a local QG Coriolis parameter. The locally balanced geostrophic flow, which is parallel to the geostrophic flow defined with f0, is divergent. Therefore, the
divergence/convergence of the true geostrophic flow
defined with the local Coriolis parameter has been approximated by the divergence/convergence of the local
geostrophic flow defined with the local QG Coriolis
parameter. The departure of the local geostrophic flow
from the geostrophic flow is the balanced ageostrophic
flow. Because of this, we need to explicitly include the
balanced ageostrophic flow in the continuity equation,
which gives rise to a (b-induced) mass source/sink term
(28)
for local change in the mass as well as for the unbalanced
ageostrophic flow, as we did it implicitly in (2) and explicitly in (7) for the 1D nondivergent QG barotropic
model. As the geostrophic flow, the balanced ageostrophic
flow is not subject to the Coriolis deflection because it
is part of the total balanced flow as defined in (28).
Therefore, only the Coriolis deflection of the unbalanced
ageostrophic flow contributes to the change in the geostrophic flow as shown in (1) for the 1D nondivergent
model. The mass source/sink term in the continuity
equation due to both the b barrier and a latitudinally
sloping topography H 5 H0 2 «y is (b/f0 1 «/H0 )H0 y,
where y is the meridional component of the geostrophic
flow that crosses the barriers. This vividly shows that the
role of the latitudinal-varying Coriolis parameter is
identical to topography. They all act as a mechanical
barrier that slows down the (total) balanced flow when it
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JOURNAL OF THE ATMOSPHERIC SCIENCES
runs into the barrier, causing a convergence of mass
and vice versa. When passing through the b barrier, the
balanced flow causes mass convergence/divergence and
excites the unbalanced ageostrophic flow via mechanical
deflection, which in turns is restored to the balanced
flow by the half-cycle Coriolis force, forming a complete
cycle of a mechanical–Coriolis oscillation with the
b-induced half-cycle Coriolis force as its restoring force.
Again, the pattern of the new flow parallel to isobars
generated by the b-induced mechanical–Coriolis oscillation is identical to the balanced flow on its right when
facing the uphill (downhill) direction of the b barrier in
the Northern (Southern) Hemisphere, giving rise to
a wave pattern that propagates to the left. Because the
b barrier is always sloped toward the pole in both
hemispheres, the wave pattern excited by the b-induced
mechanical–Coriolis oscillation always propagates westward in both hemispheres.
Here, we have proved the exclusive role of mass
convergence/divergence resulting from the geostrophic
flow passing through the b barrier or a physical barrier in
exciting Rossby wave motions, as originally envisioned
in Bjerknes (1937). The b-induced and/or topographicinduced mass convergence effectively is mass transport
of the total balanced flow along the isobars, which in
turn is redistributed by unbalanced flow crossing the
isobars. The identification of the restoring force for the
mechanical–Coriolis oscillation associated with Rossby
wave motions helps us to put forward a mechanics-based
derivation of the dispersion relation of Rossby waves as
well as the complete solution for all components of
the total flow associated with Rossby waves without
explicitly solving for a normal-model solution of the
corresponding partial differential equations. The
physics principles applied in the mechanics-based
derivation are (i) the conservation principles of mass
and energy, (ii) Newton’s second law, and (iii) the
geostrophy. The mechanics-based derivation is equivalent to a parcel method and therefore its solution is a
local solution that does not have to be in the form of
the normal-mode solution. The normal-mode solution
is only a special case of the mechanics-based derivation. The mechanics-based derivation should be easily
applicable to individual cyclones and anticyclones
since it does not explicitly require a specific form of
solution, such as a plane wave solution. In this paper,
we demonstrate the mechanics-based derivation in a
shallow-water model. This can be naturally extended to
a baroclinic model, which will be the subject of a separate paper.
We consider both the b barrier and a latitudinally
sloping topography, H 5 H0 2 «y, as an example. The
combined slope of these two barriers is b/f0 1 «/H0 . The
VOLUME 70
mechanical–Coriolis oscillation frequency due to the
two barriers can be succinctly expressed as
v5
2beffective
,
(›h/›x)/h
where
beffective 5 (b 1 f0 «/H0 ) cos2 l cos2 a .
(29)
For a 5 0, the geostrophic flow is along the barrier’s
gradient direction. As a result, the barrier-induced unbalanced flow is strongest and responsible for the fastest
mechanical–Coriolis oscillation (for a given l). As a
approaches 908, the geostrophic flow crosses the barrier
at a gentler angle. As a result, the mechanical–Coriolis
oscillation approaches zero. The case of l 5 0 corresponds to a nondivergent barotropic flow in which all
barrier-induced mass convergence is required to be
compensated by unbalanced flow, giving rise to the
maximum possible strength of the unbalanced flow, responsible for the fastest mechanical–Coriolis oscillation
(for a given a). The case of l 6¼ 0 corresponds to a divergent barotropic (a baroclinic) model. The ratio
sin2 l/ cos2 l corresponds to the ratio of thickness portion to relative vorticity portion of the potential vorticity
of Rossby waves and is proportional to the square of the
ratio of the spatial wavelength of Rossby waves to the
Rossby radius of deformation. For a spatially large-size
perturbation, it requires more mass to build a new
pressure pattern that is in balance with the new geostrophic flow pattern. As a result, only a smaller amount
of the barrier-induced mass convergence is deflected
to cross isobars, reducing the strength of the barrierinduced half-cycle Coriolis force. This explains a slower
mechanical–Coriolis oscillation as the spatial wavelength increases or as l approaches 908.
Using the normal-mode solution, (29) becomes
)
2 (
b 1 «f0 /H0
k
k2 1 l2
v5 2
k
k2 1 l2 k2 1 l2 1 f02 /gH
(b 1 «f0 /H0 )k
52 2
.
k 1 l2 1 f02 /gH
(30)
With the mechanics-based derivation in mind, we can
explain the physical meaning of the dispersion relation
of a 2D QGSW plane Rossby wave in (30) term by term.
The term inside the parentheses is the mechanical–
Coriolis oscillation frequency in a nondivergent barotropic model with isobars that are parallel to the barrier
slope, representing the base frequency that is fastest for
a given barrier slope and k. The term inside the square
brackets, which is equal to cos2 a, represents a reduction
to the base frequency due to a nondirect passing through
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CAI AND HUANG
of the barrier by the geostrophic flow, which weakens
the excitation of the barrier-induced convergence/
divergence and thereby the subsequent mechanical deflection. When the geostrophic flow is parallel to the
barrier’s contours, there is no mechanical oscillation due
to no barrier-crossing flow. The term inside the curly
brackets, which is equal to cos2 l, represents a reduction
to the base frequency due to a direct reduction in the
unbalanced flow strength for the need of building the
new mass field along the current geostrophic pathway
in a model with free surface. The wider the spatially
alternating pathways are, the more mass is needed to
build a new mass field, the weaker the unbalanced
ageostrophic flow that crosses isobars is, and thereby the
slower the oscillation is.
The barrier-induced mechanical–Coriolis oscillation
mechanism for Rossby waves is consistent with the
QGPV conservation view. In terms of the geostrophic
vorticity, what matters is the total b-induced/topographicinduced mass divergence, regardless of whether only part
or all of the unbalanced ageostrophic flow crosses the
isobars. The mass transport by geostrophic flow when it
crosses the b-barrier/topographic barrier is equivalent to
a vorticity source term. The vorticity source per unit depth
due to convergence of the total balanced flow equals
the product of domain-mean Coriolis parameter and the
b-induced/topographic mass convergence per unit depth,
f0 (b/f0 1 «/H0 )H0 y/H0 5 (b 1 f0 «/H0 )y. Therefore, vorticity source per unit depth due to barrier-induced mass
convergence, which serves as a vorticity torque, is identical
to the advection of planetary vorticity and topographicinduced potential vorticity by the geostrophic flow. In a
nondivergent barotropic model, geostrophic potential
vorticity is the same as vorticity. In a QG model with
divergence, the same vorticity source is used to change
potential vorticity, instead of just vorticity alone, explaining that for the same advection of planetary vorticity,
Rossby wave speed is slower when divergence is included.
Therefore, the two seemingly opposite views of vorticity
and divergence can be reconciled easily by recognizing
that the former is from the vorticity conservation prospective and the latter is from the mass conservation
prospective. By the virtue of the QG approximation, mass
redistribution is the same as vorticity distribution.
The mechanical–Coriolis oscillation mechanism provides a unified explanation for the origin of b-induced and
topographic-induced Rossby waves. In general, the existence of Rossby waves is due to two factors: one is rotation
and the other is some types of geometric constraint along
the fluid pathway, such as rotation rate/direction variation,
topography, edge, narrowing/widening, and shallowing/
deepening. It is the presence of some kind of geometric
constraint that mechanically forces the balanced flow to
change its original course, exciting an unbalanced flow
that crosses isobars. Then the Coriolis deflection of the
unbalanced flow restores the balanced flow in such a way
that it begins to cross the same geometry constraint at a
different location along the contour line of the constraint.
Rossby waves are fundamentally different from most
other waves in nature. Most wave motions are associated
with a bidirectional restoring force and the corresponding
oscillation always oscillates between extreme force (unbalanced) and extreme velocity (no force) by itself. For
Rossby waves, the Coriolis force restores the unbalanced
flow to its balanced point by turning it to another direction
where the unbalanced flow becomes balanced. As a result,
once it is balanced, it has no momentum inertial along
the unbalanced direction, which is unlike other types of
oscillations. Then the way of getting out of the balanced
point is another unique feature of the oscillation responsible for Rossby wave motions. As we have discussed
throughout the entire paper, the balanced flow becomes
unbalanced through a mechanical deflection when meeting a geometry constraint, instead of by itself due to its
inertia as is the case for other oscillations in nature.
Therefore, it is the existence of a geometric constraint
that keeps exciting unbalanced flow and it is the onedirectional Coriolis force that restores the balanced
flow. With this general picture on the origin of Rossby
waves in mind, we refer to such oscillation as ‘‘mechanical–
Coriolis oscillation’’ and name the corresponding restoring
force barrier-induced half-cycle Coriolis force to characterize the two unique features in exciting and restoring
Rossby wave motions: a mechanical excitation and onedirectional nature of the restoring force.
Acknowledgments. Ming Cai is supported in part by
research grants from the National Science Foundation
(Grant ATM-0833001), the NOAA CPO/CPPA program (Grant NA10OAR4310168), and the DOE Office
of Science’s Regional and Global Climate Modeling
(RGCM) program (Grant DE-SC0004974). B. Huang is
supported by the COLA omnibus grant from NSF (ATM0830068), NOAA (NA09OAR4310058), and NASA
(NNX09AN50G). The authors are benefited from discussions with Drs. Mankin Mak, Yi Deng, Zhaohua Wu,
and Huug van den Dool, and Mr. Sergio Sejas on the
earlier version of the paper. The authors greatly appreciate the constructive and informative comments and
suggestions from Dr. Qin Xu and the anonymous reviewer during the peer review process.
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