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    Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) Rossby waves (f0 > 0, β > 0, and δy > 0). The y axis points to the uphill direction of the β barrier (which is north for f0 > 0 and south for f0 < 0), and the x axis is parallel to contours of the β 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 β-induced divergent/convergent flow. Dashed black arrows indicate the geostrophic flow tendency caused by the β-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 β 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.”

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    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 , the required compensating flow of the β-induced convergence/divergence of the uphill/downhill flow. Dashed blue arrows represents , which is along a sloped surface whose angle with the horizontal is λ, and the solid blue arrows correspond to the projection of on the horizontal surface (i.e., ). Dashed red arrows represent , which is along a sloped surface whose angle with the horizontal is , and the solid red arrows correspond to the projection of on the horizontal surface (i.e., ). 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 , which is the projection of 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.”

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    Illustration of various projections for generalizing the 1D solution to the 2D solution (f0 > 0, β > 0). The angle that the isobars make with the uphill direction of the β barrier (solid black arrow) is α, and n represents a unit vector pointing to the gradient direction of pressure X. The actual β-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 β 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.”

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    Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) topographic Rossby waves (f0 > 0, β = 0, and δy > 0). The shading gradient represents the latitudinal shallowing of water depth, (ɛ > 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 topographic-induced 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.”

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A New Look at the Physics of Rossby Waves: A Mechanical–Coriolis Oscillation

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  • 1 Department of Earth, Ocean, and Atmospheric Science, The Florida State University, Tallahassee, Florida
  • 2 Department of Atmospheric, Oceanic, and Earth Sciences, College of Science, George Mason University, Fairfax, Virginia, and Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland
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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 β barrier is always sloped toward the pole in both hemispheres, responsible for the westward propagation of Rossby waves. The β-induced mechanical–Coriolis oscillation frequency can be succinctly expressed as , where , and λ is the angle of a sloped surface along which the unbalanced flow crosses isobars, α is the angle of isobars with the barrier’s slope, and k is the wavenumber along the direction of the barrier’s contours.

Corresponding author address: Ming Cai, Department of Earth, Ocean, and Atmospheric Science, The Florida State University, Tallahassee, FL 32306. E-mail: mcai@fsu.edu

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 β barrier is always sloped toward the pole in both hemispheres, responsible for the westward propagation of Rossby waves. The β-induced mechanical–Coriolis oscillation frequency can be succinctly expressed as , where , and λ is the angle of a sloped surface along which the unbalanced flow crosses isobars, α is the angle of isobars with the barrier’s slope, and k is the wavenumber along the direction of the barrier’s contours.

Corresponding author address: Ming Cai, Department of Earth, Ocean, and Atmospheric Science, The Florida State University, Tallahassee, FL 32306. E-mail: mcai@fsu.edu

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

Let us first look at the simplest possible case for Rossby waves, namely, zonally propagating waves in a linearized 1D 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 are
e1
In (1) f0 is the midlatitude (at y0) Coriolis parameter; β is the meridional gradient of planetary vorticity at y0; is geostrophic streamfunction; and the subscript a denotes ageostrophic flow, which is related to ageostrophic streamfunction in the nondivergent model as and . Applying geostrophic and ageostrophic streamfunction to the first equation of (1), we obtain
e2
In deriving (2), we have utilized the fact that ψ does not vary with y in the 1D Rossby wave model. The total (geostrophic plus ageostrophic) flow along isobars is
e3
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 β-plane Coriolis parameter within the accuracy of the β-plane approximation, that is, , 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):
e4
Considering a normal-mode solution, we obtain the following dispersion relation:
e5

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 f0. 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 is positive and faster where 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 β barrier, along which the Coriolis parameter or f changes) that slows down the balanced flow when it moves uphill crossing the f contours (), causing the convergence of mass along the uphill pathway (we will formally prove the equivalency of the β effect to a physical barrier in section 4, where we discuss topographic Rossby waves). The reverse can be said along the downhill pathway () of the balanced flow. The 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.

Fig. 1.
Fig. 1.

Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) Rossby waves (f0 > 0, β > 0, and δy > 0). The y axis points to the uphill direction of the β barrier (which is north for f0 > 0 and south for f0 < 0), and the x axis is parallel to contours of the β 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 β-induced divergent/convergent flow. Dashed black arrows indicate the geostrophic flow tendency caused by the β-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 β 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.”

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-094.1

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.

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 to the strength of its restoring force and inversely proportional to the speed of the movement, namely, .2 Therefore, for the mechanical–Coriolis oscillation, we have
e6
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 β barrier due to the latitudinal variation of the Coriolis parameter has a slope equal to β/f0. 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/f0), where the negative sign is because corresponds to an uphill pathway and 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
e7
Substituting (7) into (6) yields
e8
Obviously, for a normal-mode solution, we have , 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 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.

Let us first consider a 1D linearized 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 , where h is the surface height of a shallow-water 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 β-induced convergent/divergent flow is diverted to cross isobars. We denote the maximum possible deflection flow as , whose divergence is exactly equal to (υβ/f0), the β-induced convergence. According to (7), we have
e9
As discussed above, is just a continuation of the geostrophic flow that crosses the β barrier, which is the meridional component of the geostrophic flow, and is always along the direction parallel to the contours of the β 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 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 in the 1D divergent barotropic model as . The net unbalanced flow that crosses isobars can be written as . The opposite of , that is, , is used to change surface height at the rate equal to . 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,
e10
Next, we determine how is partitioned into and by a geometry consideration. In a nondivergent barotropic model, the unbalanced flow resulting from the mechanical deflection—that is, —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 also not along the horizontal surface. For this reason, we use a vector form, namely, and , to denote them, where the horizontal components of and are and , respectively. The opposite of 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 corresponds to the part that crosses isobars along a sloped surface in a divergent model. Note that both and are on the same vertical–horizontal cross-section plane whose horizontal axis is parallel to the pressure gradient direction. Their vector sum is equal to , and so is the sum of their projections on the horizontal surface. As a result, we can assume that is along a sloped surface with an angle of λ from the horizontal surface, whereas is along a surface that is perpendicular to . As illustrated in Fig. 2, the maximum possible deflection flow is diverted into two parts: one is at a speed of and the other part is at a speed of . The projections of and 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,
e11
The angle λ can be determined by substituting (11) into (10), which yields
e12
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 , the negative sign in front of ensures that the ratio of to is always positive definite. Based on (12), we have
e13
Fig. 2.
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 , the required compensating flow of the β-induced convergence/divergence of the uphill/downhill flow. Dashed blue arrows represents , which is along a sloped surface whose angle with the horizontal is λ, and the solid blue arrows correspond to the projection of on the horizontal surface (i.e., ). Dashed red arrows represent , which is along a sloped surface whose angle with the horizontal is , and the solid red arrows correspond to the projection of on the horizontal surface (i.e., ). 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 , which is the projection of 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.”

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-094.1

Applying the second equation in (11) to (6), we obtain the mechanical–Coriolis oscillation frequency, or the dispersion of the 1D QGSW model as
e14
Now we are ready to discuss the physical meaning 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 from the direction perpendicular to the pathway along a sloped surface of 90° − λ. The convergence/divergence of 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 λ—that is, —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 and 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 is gentle but the slope of 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
e15
Substituting (15) into (14), we recover the dispersion relation of the 1D QGSW Rossby waves obtained from the potential vorticity conservation equation, namely,
e16
Next, let us consider 2D plane Rossby waves in a 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 . The strength of the divergence of the balanced flow due to the β barrier is equal to , where 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), is
e17
where 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 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 are given in (11) by replacing with . We can use (13) to determine λ for the 2D case by either replacing x with X, or replacing with , namely,
e18
The value of 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 in the second equation of (11) with in (17) yields the net unbalanced ageostrophic flow along the direction perpendicular to isobars for the 2D case as shown:
e19
Replacing in (6) with in (19) and υ with V yields the mechanical–Coriolis oscillation frequency for the 2D case as shown:
e20
The term on the right-hand side of (20) is obtained from the relation , as indicated in Fig. 3. By projecting 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
e21
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:
e22
where . 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.
Fig. 3.
Fig. 3.

Illustration of various projections for generalizing the 1D solution to the 2D solution (f0 > 0, β > 0). The angle that the isobars make with the uphill direction of the β barrier (solid black arrow) is α, and n represents a unit vector pointing to the gradient direction of pressure X. The actual β-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 β 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.”

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-094.1

For a normal-mode solution, the two angles α and λ are
e23
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,
e24
With the normal-model solution, the net unbalanced ageostrophic flow given in (19) becomes
e25

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, , where both H0 and ɛ are constant and .

Again, we first consider the 1D nondivergent situation, in which the geostrophic flow, , 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 (υ > 0 for ɛ > 0) and divergence for a downhill flow (υ < 0 for ɛ > 0). The amount of mass divergence is equal to . 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 , the mass convergence along the direction perpendicular to isobars can be approximated as . By applying the mass conservation law [i.e., ] and , we obtain
e26
Substituting (26) into (6) yields
e27
Applying a normal-mode solution to (27) yields the dispersion of the 1D nondivergent topographic Rossby waves on an f plane, namely, . Therefore, the 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, , is equivalent to , the 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 for the topographic barrier and for the β barrier.
Fig. 4.
Fig. 4.

Illustration of the mechanical–Coriolis oscillation and (1D nondivergent) topographic Rossby waves (f0 > 0, β = 0, and δy > 0). The shading gradient represents the latitudinal shallowing of water depth, (ɛ > 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 topographic-induced 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.”

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-094.1

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 () with the physical slope of the topography ().

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.”

The latitudinal variation of the Coriolis parameter acts as a physical barrier, referred to as the β barrier. The latitudinal slope of the β barrier is . Large large-scale motions are in geostrophic balance with the local Coriolis parameter. In a β-plane QGSW model, the locally balanced geostrophic flow is approximated as
e28
where , which is an equally valid approximation as the β-plane approximation under the condition . Therefore, the β-plane QG approximation effectively approximates the full Coriolis parameter on a β 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 (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 is , where 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.

We consider both the β barrier and a latitudinally sloping topography, , as an example. The combined slope of these two barriers is . The mechanical–Coriolis oscillation frequency due to the two barriers can be succinctly expressed as
e29
For 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 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 barrier-induced half-cycle Coriolis force. This explains a slower mechanical–Coriolis oscillation as the spatial wavelength increases or as λ approaches 90°.
Using the normal-mode solution, (29) becomes
e30
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 , represents a reduction to the base frequency due to a nondirect passing through 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 , 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 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, . 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 topographic-induced 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 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.

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 (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.

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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 ua and υa in (1) is in accordance with the QG approximation.

2

Note that , where ω is a real number and is the frequency as conventionally defined in a normal model solution in the form of . One needs to multiply ω* in the mechanics-based solution with to convert it to .

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 (no vertical motion) and crosses isobars horizontally at a speed of .

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