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## Abstract

This paper reports on a study of quasi-linear baroclinic adjustment in a quasigeostrophic two-layer model with a constant baroclinic shear and an Ekman damping that is equal at the upper and lower boundaries. An analytic solution is obtained for a system consisting of a linearly unstable wave and a nontruncated zonally symmetric flow. The numerical calculations have confirmed that exclusion of the higher harmonics of an unstable wave in the analytic analysis does not alter the underlying physics of quasi-linear baroclinic adjustment.The “dynamically most efficient wave” (the wave that has the minimum equilibrated mean baroclinic shear) deduced from the analytic analysis is found to be mostly responsible for the baroclinic adjustment in the fully nonlinear model of the same baroclinic system. The dynamically most efficient wave is neither necessarily the most unstable wave nor the wave that has the largest amplitude in the fully nonlinear model. The wavelength of the most efficient wave becomes longer than that of the most unstable wave shortly after the baroclinic forcing parameter exceeds its critical value. Such a shift toward a longer wave continues as the forcing parameter increases. Thus, it is possible to predict from the analytic analysis which wave is mostly responsible for the fully nonlinear baroclinic adjustment in this two-layer model.

## Abstract

This paper reports on a study of quasi-linear baroclinic adjustment in a quasigeostrophic two-layer model with a constant baroclinic shear and an Ekman damping that is equal at the upper and lower boundaries. An analytic solution is obtained for a system consisting of a linearly unstable wave and a nontruncated zonally symmetric flow. The numerical calculations have confirmed that exclusion of the higher harmonics of an unstable wave in the analytic analysis does not alter the underlying physics of quasi-linear baroclinic adjustment.The “dynamically most efficient wave” (the wave that has the minimum equilibrated mean baroclinic shear) deduced from the analytic analysis is found to be mostly responsible for the baroclinic adjustment in the fully nonlinear model of the same baroclinic system. The dynamically most efficient wave is neither necessarily the most unstable wave nor the wave that has the largest amplitude in the fully nonlinear model. The wavelength of the most efficient wave becomes longer than that of the most unstable wave shortly after the baroclinic forcing parameter exceeds its critical value. Such a shift toward a longer wave continues as the forcing parameter increases. Thus, it is possible to predict from the analytic analysis which wave is mostly responsible for the fully nonlinear baroclinic adjustment in this two-layer model.

## Abstract

The relationship between the local shape of an unstable disturbance and the basic deformation field has been put forward by Mak and Cai as a general condition for barotropic instability of a zonally varying nondivergent basic flow. The general condition states that an unstable disturbance has to be elongated locally at an angle of less than 45° along the axis of contraction of the basic deformation field. The conventional condition for barotropic instability of a zonally uniform basic flow (“an unstable disturbance necessarily leans against the basic shear”) is a special case of the general condition.

To physically interpret the general condition, we have analyzed the immediate subsequent evolution of a localized elliptic-shaped disturbance (defined in terms of streamfunction) embedded in a purely deformation flow. The localized disturbance has the minimum kinetic energy and enstrophy when its shape is circular. Under the influence of the basic deformation, the disturbance tends to shrink along the axis of contraction and to expand along the axis of dilatation. Hence, the disturbance with the major axis along the axis of contraction would deform toward a circle shape. The change in eccentricity of such a disturbance alone acts to reduce its total energy and enstrophy. Because of the conservation constraint of the total perturbation enstrophy, the amplitude of the disturbance has to increase as its eccentricity decreases. The energy change due to the change in amplitude overwhelms that resulting from the change in eccentricity. Therefore, the overall kinetic energy of the localized disturbance tends to increase with time during the course of its evolution. The same arguments also explain why the disturbance with major axis along the axis of dilatation is decaying.

## Abstract

The relationship between the local shape of an unstable disturbance and the basic deformation field has been put forward by Mak and Cai as a general condition for barotropic instability of a zonally varying nondivergent basic flow. The general condition states that an unstable disturbance has to be elongated locally at an angle of less than 45° along the axis of contraction of the basic deformation field. The conventional condition for barotropic instability of a zonally uniform basic flow (“an unstable disturbance necessarily leans against the basic shear”) is a special case of the general condition.

To physically interpret the general condition, we have analyzed the immediate subsequent evolution of a localized elliptic-shaped disturbance (defined in terms of streamfunction) embedded in a purely deformation flow. The localized disturbance has the minimum kinetic energy and enstrophy when its shape is circular. Under the influence of the basic deformation, the disturbance tends to shrink along the axis of contraction and to expand along the axis of dilatation. Hence, the disturbance with the major axis along the axis of contraction would deform toward a circle shape. The change in eccentricity of such a disturbance alone acts to reduce its total energy and enstrophy. Because of the conservation constraint of the total perturbation enstrophy, the amplitude of the disturbance has to increase as its eccentricity decreases. The energy change due to the change in amplitude overwhelms that resulting from the change in eccentricity. Therefore, the overall kinetic energy of the localized disturbance tends to increase with time during the course of its evolution. The same arguments also explain why the disturbance with major axis along the axis of dilatation is decaying.

## Abstract

It is shown that when the basic state has zonal variation, the linearized operator of an atmospheric spectral model acts on the perturbation spectral coefficients of both the nonnegative zonal indexes and their conjugates. Mathematically, this implies that the complex linear system is no longer analytic (i.e., the Cauchy–Riemann condition is not satisfied). This paper presents a method that solves the steady response and eigenvalue problems in the complex domain. It is also suggested that for a given computer memory capacity, the linear forced problem of a zonally varying basic state could be solved by the new method at a resolution twice as high as the methodologically more straightforward real matrix method.

## Abstract

It is shown that when the basic state has zonal variation, the linearized operator of an atmospheric spectral model acts on the perturbation spectral coefficients of both the nonnegative zonal indexes and their conjugates. Mathematically, this implies that the complex linear system is no longer analytic (i.e., the Cauchy–Riemann condition is not satisfied). This paper presents a method that solves the steady response and eigenvalue problems in the complex domain. It is also suggested that for a given computer memory capacity, the linear forced problem of a zonally varying basic state could be solved by the new method at a resolution twice as high as the methodologically more straightforward real matrix method.

## Abstract

This paper proposes a mechanism that explains how coupled dynamics alone can spontaneously give rise to a realistic west–east asymmetric mean state and an ENSO-like interannual variability without requiring the existence of an external preexisting west–east asymmetry in circulation. The essence of the newly proposed mechanism is that the basinwide ocean–atmosphere coupling acts to reduce the effective restoring force. As a result, the coupled oceanic waves travel more and more slowly within the equatorial ocean basin as the coupling strength increases. When the coupling strength reaches a critical value, the zonally leveled thermocline becomes unstable as a result of the weakening of the effective restoring force, at which the theoretical limit of the traveling timescale would be infinite without nonlinearity. Due to nonlinearity in the coupled system, this primary air–sea interaction instability leads to a west–east asymmetric mean state in which the atmosphere has a prevailing easterly and the ocean basin has a deep-in-west–shallow-in-east thermocline with a warm-west–cold-east sea surface temperature. The direction of the west–east asymmetry in the mean state is dictated by a planetary factor of the earth, namely, that the Coriolis parameter changes sign at the equator. As the coupling strength further increases, the asymmetry in the mean state amplifies and the phase speeds of the coupled equatorial oceanic waves begin to decrease gradually toward an asymptotic limit equal to the full speed in the uncoupled situation.

Using the coupling coefficient that is consistent with the observation, the fully coupled model can produce a realistic mean state in which the basinwide SST (thermocline depth) difference is 4.2°C (116 m) and the westward wind stress at the central Pacific basin is 0.54 dyn cm^{–2}. The self-sustained oscillation has a primary period of 3.7 yr. The SST in the west (east) oscillates between 27.5° and 28.5°C (between 25.2° and 22.5°C).

## Abstract

This paper proposes a mechanism that explains how coupled dynamics alone can spontaneously give rise to a realistic west–east asymmetric mean state and an ENSO-like interannual variability without requiring the existence of an external preexisting west–east asymmetry in circulation. The essence of the newly proposed mechanism is that the basinwide ocean–atmosphere coupling acts to reduce the effective restoring force. As a result, the coupled oceanic waves travel more and more slowly within the equatorial ocean basin as the coupling strength increases. When the coupling strength reaches a critical value, the zonally leveled thermocline becomes unstable as a result of the weakening of the effective restoring force, at which the theoretical limit of the traveling timescale would be infinite without nonlinearity. Due to nonlinearity in the coupled system, this primary air–sea interaction instability leads to a west–east asymmetric mean state in which the atmosphere has a prevailing easterly and the ocean basin has a deep-in-west–shallow-in-east thermocline with a warm-west–cold-east sea surface temperature. The direction of the west–east asymmetry in the mean state is dictated by a planetary factor of the earth, namely, that the Coriolis parameter changes sign at the equator. As the coupling strength further increases, the asymmetry in the mean state amplifies and the phase speeds of the coupled equatorial oceanic waves begin to decrease gradually toward an asymptotic limit equal to the full speed in the uncoupled situation.

Using the coupling coefficient that is consistent with the observation, the fully coupled model can produce a realistic mean state in which the basinwide SST (thermocline depth) difference is 4.2°C (116 m) and the westward wind stress at the central Pacific basin is 0.54 dyn cm^{–2}. The self-sustained oscillation has a primary period of 3.7 yr. The SST in the west (east) oscillates between 27.5° and 28.5°C (between 25.2° and 22.5°C).

## Abstract

Both the global precipitation and evaporation in global warming simulations increase at 1%–3% K^{−1}, much smaller than the rate suggested from the Clausius–Clapeyron (C–C) relation (6%–6.5% K^{−1}). However, the reduction of surface sensible heat flux over the global ocean (5.2% K^{−1}) matches the difference between the fractional increase of evaporation and the C–C relation, implying that the fractional decrease of the Bowen ratio over the global ocean follows the C–C relation closely. The analysis suggests that the stabilization of the atmospheric boundary layer (ABL) in response to global warming is the main factor responsible for the simultaneous reduction of the surface sensible flux and the muted increase in the surface latent heat. Because the stabilization of the ABL causes the same amount of fractional change in both the sensible and latent heat fluxes, the fractional decrease of the Bowen ratio closely follows the C–C relation. The ABL stabilization mechanism for the muted increase in the global hydrological cycle in response to global warming is physically consistent with two other proposed mechanisms, namely, the atmospheric energy constraint and the reduction of convective mass flux.

## Abstract

Both the global precipitation and evaporation in global warming simulations increase at 1%–3% K^{−1}, much smaller than the rate suggested from the Clausius–Clapeyron (C–C) relation (6%–6.5% K^{−1}). However, the reduction of surface sensible heat flux over the global ocean (5.2% K^{−1}) matches the difference between the fractional increase of evaporation and the C–C relation, implying that the fractional decrease of the Bowen ratio over the global ocean follows the C–C relation closely. The analysis suggests that the stabilization of the atmospheric boundary layer (ABL) in response to global warming is the main factor responsible for the simultaneous reduction of the surface sensible flux and the muted increase in the surface latent heat. Because the stabilization of the ABL causes the same amount of fractional change in both the sensible and latent heat fluxes, the fractional decrease of the Bowen ratio closely follows the C–C relation. The ABL stabilization mechanism for the muted increase in the global hydrological cycle in response to global warming is physically consistent with two other proposed mechanisms, namely, the atmospheric energy constraint and the reduction of convective mass flux.

## Abstract

This paper investigates the modal and nonmodal instability of a barotropic jet streak. The normal mode analysis reveals that all the unstable modes are either stationary or propagating local modes. The more localized the jet is, the more dominant the stationary unstable mode would be. An exact analysis of the local energetics shows that the energy generation rate depends upon the local structure of the disturbance and the basic deformation field. The energy redistribution processes are the mechanical work done by the ageostrophic pressure and the energy advection by the basic flow. They affect not only the phase speed but also the growth rate of a normal mode disturbance by virtue of the zonal inhomogeneity of the basic flow. The local energy generation rate is maximum in the near exit region of the jet streak. The pressure work process contributes to an additional downstream shift of the maximum energy center and the advection process causes a further downstream displacement of the center. These three processes have comparable magnitude and tend to oppose one another locally. The compensating and yet accumulative effects of those three processes result in the downstream localization of an unstable disturbance.

Our nonmodal analysis confirms that an isolated disturbance not only has to have a favorable orientation but also has to be in the downstream position with respect to the jet core before it can develop rapidly. Furthermore, a disturbance with a localized structure in the downstream region of the jet core can emerge from a zonally unbiased disturbance within a few days. The same mechanisms of local energetics account for the downstream localization of the disturbances during this transient adjustment as in the normal modes. The maximum instantaneous growth rate of such a nonmodal disturbance can be several times larger than that of the most unstable normal mode. The transitional stage can be understood in terms of simultaneous growth of and interference among the multiple unstable modes.

## Abstract

This paper investigates the modal and nonmodal instability of a barotropic jet streak. The normal mode analysis reveals that all the unstable modes are either stationary or propagating local modes. The more localized the jet is, the more dominant the stationary unstable mode would be. An exact analysis of the local energetics shows that the energy generation rate depends upon the local structure of the disturbance and the basic deformation field. The energy redistribution processes are the mechanical work done by the ageostrophic pressure and the energy advection by the basic flow. They affect not only the phase speed but also the growth rate of a normal mode disturbance by virtue of the zonal inhomogeneity of the basic flow. The local energy generation rate is maximum in the near exit region of the jet streak. The pressure work process contributes to an additional downstream shift of the maximum energy center and the advection process causes a further downstream displacement of the center. These three processes have comparable magnitude and tend to oppose one another locally. The compensating and yet accumulative effects of those three processes result in the downstream localization of an unstable disturbance.

Our nonmodal analysis confirms that an isolated disturbance not only has to have a favorable orientation but also has to be in the downstream position with respect to the jet core before it can develop rapidly. Furthermore, a disturbance with a localized structure in the downstream region of the jet core can emerge from a zonally unbiased disturbance within a few days. The same mechanisms of local energetics account for the downstream localization of the disturbances during this transient adjustment as in the normal modes. The maximum instantaneous growth rate of such a nonmodal disturbance can be several times larger than that of the most unstable normal mode. The transitional stage can be understood in terms of simultaneous growth of and interference among the multiple unstable modes.

## Abstract

This paper investigates the dynamics of regional cyclogenesis from the perspective of local instability of a zonally inhomogeneous baroclinic jet streak in a two-layer quasi-geostrophic beta-plane channel model. When such a representative jet streak is embedded in a background uniform vertical shear *U _{T}
*, there are both local and global unstable normal modes. In the absence of such a background shear (

*U*= 0), only the local modes are unstable. The shorter the jet is, the fewer local modes would there be. A local mode consists of a group of dominant waves that jointly give rise to a maximum local energy downstream of the jet core. Its existence is independent of the cyclical boundary condition. The growth rate of a local mode diminishes rapidly when the constant part of the basic zonal wind

_{T}*U*

_{0}is increased. A global mode, on the other hand, largely consists of a single wave and its growth rate is much less sensitive to

*U*

_{0}. These properties are qualitatively similar to those in the WKB solution. The structural characteristics of these modes are identifiable with those of three classes of unstable modes of an observed atmospheric flow reported in Frederiksen and Bell.

Our nonmodal analysis shows that a localized disturbance naturally emerges from a zonally unbiased initial state in a relatively short time. The excitation of a local mode within a few days from an initially isolated disturbance also depends strongly upon its initial position relative to the jet core.

The two processes that locally generate the perturbation energy depend upon the structural properties of the disturbance relative to the basic thermal and deformation fields. The two processes that redistribute the perturbation energy are the advection of energy by the basic flow and the convergence of energy flux associated with the ageostrophic component of the perturbation. These four processes are comparably important and greatly counteract one another resulting in a net intensification of a disturbance centered downstream of the jet core. The feedback effects of the most unstable mode on the basic flow resemble the observed geopotential tendencies induced by the transient eddies. The feedback results of this analysis differ noticeably from the WKB counterparts.

## Abstract

This paper investigates the dynamics of regional cyclogenesis from the perspective of local instability of a zonally inhomogeneous baroclinic jet streak in a two-layer quasi-geostrophic beta-plane channel model. When such a representative jet streak is embedded in a background uniform vertical shear *U _{T}
*, there are both local and global unstable normal modes. In the absence of such a background shear (

*U*= 0), only the local modes are unstable. The shorter the jet is, the fewer local modes would there be. A local mode consists of a group of dominant waves that jointly give rise to a maximum local energy downstream of the jet core. Its existence is independent of the cyclical boundary condition. The growth rate of a local mode diminishes rapidly when the constant part of the basic zonal wind

_{T}*U*

_{0}is increased. A global mode, on the other hand, largely consists of a single wave and its growth rate is much less sensitive to

*U*

_{0}. These properties are qualitatively similar to those in the WKB solution. The structural characteristics of these modes are identifiable with those of three classes of unstable modes of an observed atmospheric flow reported in Frederiksen and Bell.

Our nonmodal analysis shows that a localized disturbance naturally emerges from a zonally unbiased initial state in a relatively short time. The excitation of a local mode within a few days from an initially isolated disturbance also depends strongly upon its initial position relative to the jet core.

The two processes that locally generate the perturbation energy depend upon the structural properties of the disturbance relative to the basic thermal and deformation fields. The two processes that redistribute the perturbation energy are the advection of energy by the basic flow and the convergence of energy flux associated with the ageostrophic component of the perturbation. These four processes are comparably important and greatly counteract one another resulting in a net intensification of a disturbance centered downstream of the jet core. The feedback effects of the most unstable mode on the basic flow resemble the observed geopotential tendencies induced by the transient eddies. The feedback results of this analysis differ noticeably from the WKB counterparts.

## Abstract

It is hypothesized that the low-frequency planetary scale waves and the high-frequency cyclone-scale waves in an equilibrated state of the atmosphere are symbiotically dependent upon one another. This is demonstrated with an analysis of a dissipative atmospheric model driven by a zonally symmetric forcing. Under a geophysically relevant parameter condition, the synoptic scale waves in the equilibrated state of this system intermittently extract a sufficient amount of energy from the modified instantaneous zonal flow to compensate not only for their own dissipative loss of energy but also for a net supply of energy to the planetary scale waves through the upscale energy cascade process. The planetary scale waves gain this energy in the barotropic form. The planetary scale waves, in turn, create localized strong baroclinic regions whereby the synoptic scale waves may preferentially intensify downstream of the model planetary jet streams. Such cyclone scale eddies collectively give rise to two model stormtracks that have a coherent statistical relation with the zonally traveling planetary scale waves. The zonal propagation of the planetary waves is slowed down due to interaction with the synoptic scale waves.

These findings are deduced from a multifacet diagnosis of a long record of the model evolution. First, some salient statistical characteristics and the energetics of the equilibrated flow are analysed. Then, the dominant planetary scale wave in the equilibrated state is used as a reference to construct a phase-shifted composite flow and a corresponding record of the synoptic scale anomaly flow. Using a complete local energetics analysis of the synoptic scale anomalies, we delineate how the planetary scale jet streams statistically organize the synoptic scale eddies downstream of the jet cores. The phase-shifted composite flow is finally analysed for its linear instability properties. Evidence is shown to relate the nonlinear synoptic scale anomaly flow to an unstable local normal mode with similar structural and energetic properties.

## Abstract

It is hypothesized that the low-frequency planetary scale waves and the high-frequency cyclone-scale waves in an equilibrated state of the atmosphere are symbiotically dependent upon one another. This is demonstrated with an analysis of a dissipative atmospheric model driven by a zonally symmetric forcing. Under a geophysically relevant parameter condition, the synoptic scale waves in the equilibrated state of this system intermittently extract a sufficient amount of energy from the modified instantaneous zonal flow to compensate not only for their own dissipative loss of energy but also for a net supply of energy to the planetary scale waves through the upscale energy cascade process. The planetary scale waves gain this energy in the barotropic form. The planetary scale waves, in turn, create localized strong baroclinic regions whereby the synoptic scale waves may preferentially intensify downstream of the model planetary jet streams. Such cyclone scale eddies collectively give rise to two model stormtracks that have a coherent statistical relation with the zonally traveling planetary scale waves. The zonal propagation of the planetary waves is slowed down due to interaction with the synoptic scale waves.

These findings are deduced from a multifacet diagnosis of a long record of the model evolution. First, some salient statistical characteristics and the energetics of the equilibrated flow are analysed. Then, the dominant planetary scale wave in the equilibrated state is used as a reference to construct a phase-shifted composite flow and a corresponding record of the synoptic scale anomaly flow. Using a complete local energetics analysis of the synoptic scale anomalies, we delineate how the planetary scale jet streams statistically organize the synoptic scale eddies downstream of the jet cores. The phase-shifted composite flow is finally analysed for its linear instability properties. Evidence is shown to relate the nonlinear synoptic scale anomaly flow to an unstable local normal mode with similar structural and energetic properties.

## 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 *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.

## 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 *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.

## Abstract

It is shown in this paper that there is no ambiguity in the final form of the governing equations of a quasigeostrophic (QG) model after partitioning the total flow into the geostrophic, balanced ageostrophic, and unbalanced ageostrophic components. The uniqueness of the QG model formulation ensures that the energetics of a QG model is the same as that derived from the QG potential vorticity equation. Particularly, the well-known but somewhat mysterious “missing term” in the energetics of Rossby waves, identified in the literature as the difference between the pressure work and the energy flux transported at the group velocity, can be easily recovered. The missing term is the pressure work on the convergence of the balanced ageostrophic flow, representing a “hidden” conversion between kinetic and potential energy of the geostrophic flow that excites the unbalanced flow. This energy conversion equals the convergence of a one-directional energy flux that always transports energy westward at the zonal phase speed of Rossby waves. The pressure work on the divergence of the unbalanced flow does the actual conversion between kinetic and potential energy of the geostrophic flow and the pressure work on the unbalanced flow causes energy propagation in other directions. Therefore, it is the pressure work on the unbalanced flow that causes Rossby waves to be dispersive, leading to the downstream development. The sum of the energy transported at the zonal phase speed of Rossby waves and the pressure work on the unbalanced flow exactly equals the energy transported at the group velocity of Rossby waves.

## Abstract

It is shown in this paper that there is no ambiguity in the final form of the governing equations of a quasigeostrophic (QG) model after partitioning the total flow into the geostrophic, balanced ageostrophic, and unbalanced ageostrophic components. The uniqueness of the QG model formulation ensures that the energetics of a QG model is the same as that derived from the QG potential vorticity equation. Particularly, the well-known but somewhat mysterious “missing term” in the energetics of Rossby waves, identified in the literature as the difference between the pressure work and the energy flux transported at the group velocity, can be easily recovered. The missing term is the pressure work on the convergence of the balanced ageostrophic flow, representing a “hidden” conversion between kinetic and potential energy of the geostrophic flow that excites the unbalanced flow. This energy conversion equals the convergence of a one-directional energy flux that always transports energy westward at the zonal phase speed of Rossby waves. The pressure work on the divergence of the unbalanced flow does the actual conversion between kinetic and potential energy of the geostrophic flow and the pressure work on the unbalanced flow causes energy propagation in other directions. Therefore, it is the pressure work on the unbalanced flow that causes Rossby waves to be dispersive, leading to the downstream development. The sum of the energy transported at the zonal phase speed of Rossby waves and the pressure work on the unbalanced flow exactly equals the energy transported at the group velocity of Rossby waves.