## 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 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., Longuet-Higgins 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) 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 *β* 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 *β*-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 *β*-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 *β*-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 reported in Durran (1988). As far as *β*-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 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 normal-mode 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

*β*-plane nondivergent QG barotropic model with a motionless mean state. In the 1D model, the geostrophic flow

*υ*is aligned with the orientation of planetary vorticity gradient (

*y*coordinate) and does not vary with

*y*.

^{1}The corresponding governing equations areIn (1)

*f*

_{0}is the midlatitude (at

*y*

_{0}) Coriolis parameter;

*β*is the meridional gradient of planetary vorticity at

*y*

_{0};

*a*denotes ageostrophic flow, which is related to ageostrophic streamfunction

*ψ*does not vary with

*y*in the 1D Rossby wave model. The total (geostrophic plus ageostrophic) flow along isobars isAccording to (3), the total flow parallel to isobars is in the geostrophic balance with a latitudinally varying Coriolis parameter that approximately equals the local

*β*-plane Coriolis parameter within the accuracy of the

*β*-plane approximation, that is,

*u*. 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):Considering a normal-mode solution, we obtain the following dispersion relation:

_{a}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 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 *f*_{0}. As indicated by the negative sign in front of the *β* 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 *β* barrier, along which the Coriolis parameter or *f* changes) that slows down the balanced flow when it moves uphill crossing the *f* contours (*β* effect to a physical barrier in section 4, where we discuss topographic Rossby waves). The reverse can be said along the downhill pathway (*β*-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 *β*-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 *β* barrier and is always along the direction parallel to the contours of the *β* barrier, which is the *x* axis. In the 1D nondivergent barotropic model in which isobars are perpendicular to contours of the *β* 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 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 *β*-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 *β* 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 *β*-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 *β*-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.

^{2}Therefore, for the mechanical–Coriolis oscillation, we haveNote 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.

*β*barrier due to the latitudinal variation of the Coriolis parameter has a slope equal to

*β/f*

_{0}. In the 1D case, the geostrophic flow passes through the

*β*barrier along its gradient direction at speed

*υ*. The divergence due to the passing through the

*β*barrier by the geostrophic flow is equal to (−

*υβ/f*

_{0}), where the negative sign is because

*k*is the zonal wavenumber, which reduces (8) to (5).

Beside the fact that only the geostrophic flow is considered to cross over the *β* barrier, two more approximations within the QG approximation are invoked implicitly in deriving (6) and (7). In reality, the cross-isobar flow, that is, the mechanical deflection, is driven by the pressure gradient tendency built by the *β*-induced convergence along the uphill pathway and divergence along the downhill pathway. A complete deflection of all *β*-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 *β* 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.

*β*-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

*h*is the surface height of a shallow-water model with a constant mean depth of

*H*

_{0}. In the 1D nondivergent model, the unbalanced ageostrophic flow has its maximum possible strength, because all of the

*β*-induced convergent/divergent flow is diverted to cross isobars. We denote the maximum possible deflection flow as

*υβ/f*

_{0}), the

*β*-induced convergence. According to (7), we haveAs discussed above,

*β*barrier, which is the meridional component of the geostrophic flow, and

*β*barrier, which is the

*x*axis.

*β*-induced convergence causes the surface height to rise along the uphill geostrophic flow and vice versa. The remaining part of the

*β*-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

*λ*from the horizontal surface, whereas

*λ*can be determined by substituting (11) into (10), which yieldsTherefore, 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

*λ*. Without the QG approximation, the convergence along the uphill geostrophic flow should immediately result in 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

*λ*. The convergence/divergence of

*λ*—that is,

*β*-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

*β*barrier directly. In the 2D case, the geostrophic flow crosses the

*β*barrier at an angle. This implies that for the same spatial variability of pressure, the geostrophic flow crosses the

*β*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

*α*be the angle of isobars with the

*β*-barrier slope (Fig. 3). The actual slope of the

*β*barrier along the geostrophic pathway is

*β*barrier is equal to

*V*is the speed of the geostrophic flow. The maximum possible mechanical deflection of the

*β*-induced divergence in the 2D case, according to (7), iswhere

*X*on the horizontal surface, where

*X*denotes the direction parallel to the pressure gradient. The reduction

*X*, which is needed to rebuild the pressure field, and the actual unbalanced horizontal flow along the direction parallel to the pressure gradient

*λ*for the 2D case by either replacing

*x*with

*X*, or replacing

*λ*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

*υ*with

*V*yields the mechanical–Coriolis oscillation frequency for the 2D case as shown:The term on the right-hand side of (20) is obtained from the relation

*x*and the

*y*directions, we obtain the

*x*and

*y*components of the net unbalanced ageostrophic flow in the 2D QGSW model, written asIn addition to the unbalanced ageostrophic flow given in (21), there is a balanced ageostrophic flow that is related to the geostrophic flow as shown:where

*α*and

*λ*arewhere

*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,With the normal-model solution, the net unbalanced ageostrophic flow given in (19) becomes

## 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*_{0} and *ɛ* are constant and

*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 (

*υ*> 0 for

*ɛ*> 0) and divergence for a downhill flow (

*υ*< 0 for

*ɛ*> 0). The amount of mass divergence is equal to

*f*plane, namely,

*β*-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

*β*barrier.

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 *β* barrier (

## 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 barrier-induced 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 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 “barrier-induced half-cycle Coriolis force.”

*β*barrier. The latitudinal slope of the

*β*barrier is

*β*-plane QGSW model, the locally balanced geostrophic flow is approximated aswhere

*β*-plane approximation under the condition

*β*-plane QG approximation effectively approximates the full Coriolis parameter on a

*β*plane as

*f*

_{QG}, a local QG Coriolis parameter. The locally balanced geostrophic flow, which is parallel to the geostrophic flow defined with

*f*

_{0}, 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 (

*β*-induced) mass source/sink term 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

*β*barrier and a latitudinally sloping topography

*υ*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 runs into the barrier, causing a convergence of mass and vice versa. When passing through the

*β*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

*β*-induced half-cycle Coriolis force as its restoring force. Again, the pattern of the new flow parallel to isobars generated by the

*β*-induced mechanical–Coriolis oscillation is identical to the balanced flow on its right when facing the uphill (downhill) direction of the

*β*barrier in the Northern (Southern) Hemisphere, giving rise to a wave pattern that propagates to the left. Because the

*β*barrier is always sloped toward the pole in both hemispheres, the wave pattern excited by the

*β*-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 *β* barrier or a physical barrier in exciting Rossby wave motions, as originally envisioned in Bjerknes (1937). The *β*-induced and/or topographic-induced 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.

*β*barrier and a latitudinally sloping topography,

*α*= 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

*λ*). As

*α*approaches 90°, the geostrophic flow crosses the barrier at a gentler angle. As a result, the mechanical–Coriolis oscillation approaches zero. The case of

*λ*= 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

*α*). The case of

*λ*≠ 0 corresponds to a divergent barotropic (a baroclinic) model. The ratio

*λ*approaches 90°.

*k*. The term inside the square brackets, which is equal to

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 *β*-induced/topographic-induced 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 *β*-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 *β*-induced/topographic mass convergence per unit depth,

The mechanical–Coriolis oscillation mechanism provides a unified explanation for the origin of *β*-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 one-directional 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 one-directional nature of the restoring force.

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 (ATM-0830068), 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.

## REFERENCES

Bjerknes, J., 1937: Die Theorie der aussertropischen Zyklonenbildung.

,*Meteor. Z.***54**, 462–466.Bjerknes, J., , and J. Holmboe, 1944: On the theory of cyclones.

,*J. Meteor.***1**, 1–22.Brunt, D., , and C. K. M. Douglas, 1928: The modification of the strophic balance for changing pressure distribution, and its effect on rainfall.

,*Mem. Roy. Meteor. Soc.***3**, 29–51.Durran, D. R., 1988: On a physical mechanism for Rossby wave propagation.

,*J. Atmos. Sci.***45**, 4020–4022.Gill, A. E., 1982:

*Atmosphere–Ocean Dynamics.*Academic Press, 662 pp.Holton, J. R., 2004:

*An Introduction to Dynamic Meteorology.*4th ed. Academic Press, 535 pp.Hoskins, B. J., , M. E. McIntyre, , and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps.

,*Quart. J. Roy. Meteor. Soc.***111**, 877–946.Longuet-Higgins, M. S., 1964: On group velocity and energy flux in planetary wave motions.

,*Deep-Sea Res.***11**, 35–42.Longuet-Higgins, M. S., , and A. E. Gill, 1967: Resonant interactions between planetary waves.

,*Proc. Roy. Soc. London***A****229**, 120–140.Mak, M., 2011:

*Atmospheric Dynamics.*Cambridge Press, 486 pp.Palmen, E., , and C. W. Newton, 1969:

*Atmospheric Circulation Systems, Their Structure and Physical Interpretation.*Academic Press, 603 pp.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics.*2nd ed. Springer-Verlag, 710 pp.Phillips, N., 1965: Elementary Rossby waves.

,*Tellus***17**, 295–301.Platzman, G. W., 1968: The Rossby wave.

,*Quart. J. Roy. Meteor. Soc.***94**, 225–248.Rhines, P. B., 1969: Slow oscillations in an ocean of varying depth, I. Abrupt topography.

,*J. Fluid Mech.***37**, 161–189.Rhines, P. B., 1970: Edge-, bottom-, and Rossby waves in a rotating stratified fluid.

,*Geophys. Fluid Dyn.***1**, 273–302.Robinson, A. R., , and H. Stommel, 1959: Amplification of transient response of ocean to storms by the effect of bottom topography.

,*Deep-Sea Res.***5**, 312–314.Rossby, C.-G., 1939: Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action.

,*J. Mar. Res.***2**, 38–55.Rossby, C.-G., 1940: Planetary flow patterns in the atmosphere.

,*Quart. J. Roy. Meteor. Soc.***66**(Suppl.), 68–87.

^{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 *β*-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 *u _{a}* and

*υ*in (1) is in accordance with the QG approximation.

_{a}^{2}

Note that *ω* is a real number and is the frequency as conventionally defined in a normal model solution in the form of *ω** in the mechanics-based solution with

^{3}

In a nondivergent barotropic model, the notion of Rossby radius of deformation is no longer relevant, corresponding to the case *λ* = 0, in which