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- Author or Editor: T. C. Chang x

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## Abstract

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## Abstract

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## Abstract

The linear boundary layer solution which includes the effect of temporal acceleration is used to parameterize the CISK (Conditional Instability of the Second Kind) mechanism of tropical waves. By simplifying the first law to a statement of balance between adiabatic cooling and diabatic heating, which is usually valid for weak tropical motions, the model is formulated as a one-level primitive equation expressed at the top of the boundary layer and is solved numerically for its eigenvalues. The growth rates are generally scale-independent and are fairly small. The presence of a quasi-Stokes boundary layer near the equator and a transition zone between this layer and a quasi-Ekman layer poleward seems important only in the wave structures but not the growth rates.

## Abstract

The linear boundary layer solution which includes the effect of temporal acceleration is used to parameterize the CISK (Conditional Instability of the Second Kind) mechanism of tropical waves. By simplifying the first law to a statement of balance between adiabatic cooling and diabatic heating, which is usually valid for weak tropical motions, the model is formulated as a one-level primitive equation expressed at the top of the boundary layer and is solved numerically for its eigenvalues. The growth rates are generally scale-independent and are fairly small. The presence of a quasi-Stokes boundary layer near the equator and a transition zone between this layer and a quasi-Ekman layer poleward seems important only in the wave structures but not the growth rates.

## Abstract

In this study, nonlinear effects of barotropic instability in a downstream varying easterly jet are studied and compared with previous linear model results of Tupaz and others. The barotropic vorticity equation with Rayleigh friction and forcing is solved with finite differences. The initial mean flow is an easterly Bickley jet whose maximum speed and half-width vary downstream; the half-width ranges from 500 to 1200 km and the maximum speed is 30 m s^{−1}. The time-independent forcing makes the initial mean flow, which is unstable in the central jet region, a steady-state solution to the vorticity equation. A disturbance with wavenumber 10, which is predicted to be locally unstable and most dominant based on linear model results, is added to the initial mean flow. The equation is then integrated numerically for 450 days.

The solutions may be separated into two phases: 1) an initial adjustment phase which consists of several ∼50-day cycles wherein an initial wavenumber 10 disturbance grows rapidly in the jet region, and then the disturbance energy shifts to a slightly longer wavelength and decays before the next cycle; and 2) a quasi-equilibrium phase which is achieved after 350 days. Fourier analysis of the disturbance streamfunction at each point during a typical interval in the adjustment phase shows two dominant modes with periods near 3.35 days and 3.58 days, respectively. After entering the quasi-equilibrium phase, a 4-day oscillation develops in the kinetic energy and the main periods of the streamfunction become 4 and 2 days, respectively. The former is the dominant mode and the latter is the result of the nonlinear self-interaction by the former. The frequency of the dominant mode is equal to the frequency of the most unstable mode from a parallel flow calculation based on the outflow region mean flow. However, in most of the unstable region, it is much less than the most unstable local frequency inferred from the parallel flow solution.

The dominant mode in the quasi-equilibrium phase propagates through the modified mean flow essentially as a linear wave, and its behavior can be compared with the linear model results. However, its maximum growth rate is 25% larger than the highest local growth rate for the parallel flow solution. This “enhancement effect” is also larger than was found by Tupaz and others. In addition, there is a hysteresis effect wherein the growth rate curve and the phase structure from the full model are shifted downstream relative to the parallel flow solution, similar to the linear model results. On the other hand, the wavelength is generally short in the jet region and much longer in the outer regions, opposite to the wavelength variation in Tupaz and others. With the help of a generalized Rossby wave formula, it is shown that two effects determine the downstream variation of the disturbance wavelength: 1) the variation of the latitudinal integral of the mean zonal wind and 2) the variation of the latitudinal integral of the mean absolute vorticity gradient. Due to the difference in disturbance scale, the second effect dominates in the quasi-equilibrium phase of this study while the first effect dominates the linear model used by Tupaz and others.

## Abstract

In this study, nonlinear effects of barotropic instability in a downstream varying easterly jet are studied and compared with previous linear model results of Tupaz and others. The barotropic vorticity equation with Rayleigh friction and forcing is solved with finite differences. The initial mean flow is an easterly Bickley jet whose maximum speed and half-width vary downstream; the half-width ranges from 500 to 1200 km and the maximum speed is 30 m s^{−1}. The time-independent forcing makes the initial mean flow, which is unstable in the central jet region, a steady-state solution to the vorticity equation. A disturbance with wavenumber 10, which is predicted to be locally unstable and most dominant based on linear model results, is added to the initial mean flow. The equation is then integrated numerically for 450 days.

The solutions may be separated into two phases: 1) an initial adjustment phase which consists of several ∼50-day cycles wherein an initial wavenumber 10 disturbance grows rapidly in the jet region, and then the disturbance energy shifts to a slightly longer wavelength and decays before the next cycle; and 2) a quasi-equilibrium phase which is achieved after 350 days. Fourier analysis of the disturbance streamfunction at each point during a typical interval in the adjustment phase shows two dominant modes with periods near 3.35 days and 3.58 days, respectively. After entering the quasi-equilibrium phase, a 4-day oscillation develops in the kinetic energy and the main periods of the streamfunction become 4 and 2 days, respectively. The former is the dominant mode and the latter is the result of the nonlinear self-interaction by the former. The frequency of the dominant mode is equal to the frequency of the most unstable mode from a parallel flow calculation based on the outflow region mean flow. However, in most of the unstable region, it is much less than the most unstable local frequency inferred from the parallel flow solution.

The dominant mode in the quasi-equilibrium phase propagates through the modified mean flow essentially as a linear wave, and its behavior can be compared with the linear model results. However, its maximum growth rate is 25% larger than the highest local growth rate for the parallel flow solution. This “enhancement effect” is also larger than was found by Tupaz and others. In addition, there is a hysteresis effect wherein the growth rate curve and the phase structure from the full model are shifted downstream relative to the parallel flow solution, similar to the linear model results. On the other hand, the wavelength is generally short in the jet region and much longer in the outer regions, opposite to the wavelength variation in Tupaz and others. With the help of a generalized Rossby wave formula, it is shown that two effects determine the downstream variation of the disturbance wavelength: 1) the variation of the latitudinal integral of the mean zonal wind and 2) the variation of the latitudinal integral of the mean absolute vorticity gradient. Due to the difference in disturbance scale, the second effect dominates in the quasi-equilibrium phase of this study while the first effect dominates the linear model used by Tupaz and others.

## Abstract

The structure and behavior of barotropically unstable and stable waves in the vicinity of a zonally varying easterly jet are studied numerically with a linearized barotropic vorticity equation on a β plane. The easterly jet is approximated by a Bickley jet with a slow zonal variation. The numerical results are also compared with a simple mechanistic analytical model using the local phase speed and growth rate concepts. In several aspects the results are grossly similar to that expected from the parallel flow theory of barotropic instability. However, in the unstable region the resultant structure of the waves causes a spatial growth rate greater than predicted by the local growth rates computed with a parallel flow model. In the stable region, the structure leads to a strong dynamic damping. When a uniform advective velocity is added to a variable mean flow, the difference between the magnitude of the growth rate of the computed waves and that implied by the parallel flow theory is somewhat reduced. However, in this case a stronger zonal asymmetry in the spatial growth rate curve with respect to the jet maximum occurs as a result of slower adjustment of the wave structure to the local stability conditions.

## Abstract

The structure and behavior of barotropically unstable and stable waves in the vicinity of a zonally varying easterly jet are studied numerically with a linearized barotropic vorticity equation on a β plane. The easterly jet is approximated by a Bickley jet with a slow zonal variation. The numerical results are also compared with a simple mechanistic analytical model using the local phase speed and growth rate concepts. In several aspects the results are grossly similar to that expected from the parallel flow theory of barotropic instability. However, in the unstable region the resultant structure of the waves causes a spatial growth rate greater than predicted by the local growth rates computed with a parallel flow model. In the stable region, the structure leads to a strong dynamic damping. When a uniform advective velocity is added to a variable mean flow, the difference between the magnitude of the growth rate of the computed waves and that implied by the parallel flow theory is somewhat reduced. However, in this case a stronger zonal asymmetry in the spatial growth rate curve with respect to the jet maximum occurs as a result of slower adjustment of the wave structure to the local stability conditions.

## Abstract

An important issue in the formation of concentric eyewalls in a tropical cyclone is the development of a symmetric structure from asymmetric convection. It is proposed herein, with the aid of a nondivergent barotropic model, that concentric vorticity structures result from the interaction between a small and strong inner vortex (the tropical cyclone core) and neighboring weak vortices (the vorticity induced by the moist convection outside the central vortex of a tropical cyclone). The results highlight the pivotal role of the vorticity strength of the inner core vortex in maintaining itself, and in stretching, organizing, and stabilizing the outer vorticity field. Specifically, the core vortex induces a differential rotation across the large and weak vortex to strain out the latter into a vorticity band surrounding the former. The straining out of a large, weak vortex into a concentric vorticity band can also result in the contraction of the outer tangential wind maximum. The stability of the outer band is related to the Fjørtoft sufficient condition for stability because the strong inner vortex can cause the wind at the inner edge to be stronger than the outer edge, which allows the vorticity band and therefore the concentric structure to be sustained. Moreover, the inner vortex must possess high vorticity not only to be maintained against any deformation field induced by the outer vortices but also to maintain a smaller enstrophy cascade and to resist the merger process into a monopole. The negative vorticity anomaly in the moat serves as a “shield” or a barrier to the farther inward mixing the outer vorticity field. The binary vortex experiments described in this paper suggest that the formation of a concentric vorticity structure requires 1) a very strong core vortex with a vorticity at least 6 times stronger than the neighboring vortices, 2) a large neighboring vorticity area that is larger than the core vortex, and 3) a separation distance between the neighboring vorticity field and the core vortex that is within 3 to 4 times the core vortex radius.

## Abstract

An important issue in the formation of concentric eyewalls in a tropical cyclone is the development of a symmetric structure from asymmetric convection. It is proposed herein, with the aid of a nondivergent barotropic model, that concentric vorticity structures result from the interaction between a small and strong inner vortex (the tropical cyclone core) and neighboring weak vortices (the vorticity induced by the moist convection outside the central vortex of a tropical cyclone). The results highlight the pivotal role of the vorticity strength of the inner core vortex in maintaining itself, and in stretching, organizing, and stabilizing the outer vorticity field. Specifically, the core vortex induces a differential rotation across the large and weak vortex to strain out the latter into a vorticity band surrounding the former. The straining out of a large, weak vortex into a concentric vorticity band can also result in the contraction of the outer tangential wind maximum. The stability of the outer band is related to the Fjørtoft sufficient condition for stability because the strong inner vortex can cause the wind at the inner edge to be stronger than the outer edge, which allows the vorticity band and therefore the concentric structure to be sustained. Moreover, the inner vortex must possess high vorticity not only to be maintained against any deformation field induced by the outer vortices but also to maintain a smaller enstrophy cascade and to resist the merger process into a monopole. The negative vorticity anomaly in the moat serves as a “shield” or a barrier to the farther inward mixing the outer vorticity field. The binary vortex experiments described in this paper suggest that the formation of a concentric vorticity structure requires 1) a very strong core vortex with a vorticity at least 6 times stronger than the neighboring vortices, 2) a large neighboring vorticity area that is larger than the core vortex, and 3) a separation distance between the neighboring vorticity field and the core vortex that is within 3 to 4 times the core vortex radius.

## Abstract

The interactions between monsoon circulations and tropical disturbances in the Northwest Pacific, where the low-level mean flow is westerly in the west and easterly in the east, are studied with a barotropic model. The authors’ model results suggest that the scale contraction by the confluent background flow, the nonlinear dynamics, the *β* effect, and the large-scale convergence are important for the energy and enstrophy accumulation near the region where the zonal flow reverses. The energy/enstrophy accumulation can be maintained with a continuous Rossby wave emanation upstream. The largest accumulation occurs when the emanating zonal wavelength is around 2000 km. Longer Rossby waves experience less scale contraction and nonlinear effects while shorter Rossby waves cannot hold a coherent structure against dispersive effects.

The nonlinear energy/enstrophy accumulation mechanism is significantly different from previous linear energy accumulation theories. In the linear theories this is primarily accomplished by the slowdown of the Doppler-shifted group velocity through the convergence of mean zonal advection, while in nonlinear dynamics the contraction of the zonal wave scale plays the crucial role. More importantly, after the initial energy increase by the wave accumulation, linear dynamics will lead to an eventual loss of wave energy to the mean flow due to the increase of zonal wavenumber near the critical longitude. Thus, without the presence of other forcing processes such as diabatic heating, the disturbances will decay. In nonlinear dynamics, the sharpening of the vorticity gradient as the waves approach the confluence zone leads to the development of disturbance asymmetries with respect to the central latitude. This effect is through the nonlinear interaction of Rossby waves with the planetary vorticity gradient. This development leads to a pair of vorticity centers that straddles the central latitude with the cyclone (anticyclone) in the north (south), and an elongated, weak westerly flow along the central latitude. This elongated westerly flow, which possesses a zonal wavenumber smaller than that in the linear cases, reverses the sign of the Reynold’s stress and allows the energy to grow near the critical longitude, leading to intensified disturbances.

With a more realistic monsoonlike background flow, a northwestward propagation pattern with an approximately 8-day period and 3000-km wavelength is produced, in general agreement with observed disturbances in the Northwest Pacific. The intensified disturbance may disperse energy upstream, leading to a series of trailing anticyclonic and cyclonic cells along the northwestward propagation path. When an opposing current is present, the energy dispersion leads to the formation of new disturbances in the confluence zone by vortex axisymmetrization dynamics. Thus, our results indicate that the scale contraction and nonlinear effects may cause a succession of tropical disturbances to develop without disturbance-scale diabatic effects.

## Abstract

The interactions between monsoon circulations and tropical disturbances in the Northwest Pacific, where the low-level mean flow is westerly in the west and easterly in the east, are studied with a barotropic model. The authors’ model results suggest that the scale contraction by the confluent background flow, the nonlinear dynamics, the *β* effect, and the large-scale convergence are important for the energy and enstrophy accumulation near the region where the zonal flow reverses. The energy/enstrophy accumulation can be maintained with a continuous Rossby wave emanation upstream. The largest accumulation occurs when the emanating zonal wavelength is around 2000 km. Longer Rossby waves experience less scale contraction and nonlinear effects while shorter Rossby waves cannot hold a coherent structure against dispersive effects.

The nonlinear energy/enstrophy accumulation mechanism is significantly different from previous linear energy accumulation theories. In the linear theories this is primarily accomplished by the slowdown of the Doppler-shifted group velocity through the convergence of mean zonal advection, while in nonlinear dynamics the contraction of the zonal wave scale plays the crucial role. More importantly, after the initial energy increase by the wave accumulation, linear dynamics will lead to an eventual loss of wave energy to the mean flow due to the increase of zonal wavenumber near the critical longitude. Thus, without the presence of other forcing processes such as diabatic heating, the disturbances will decay. In nonlinear dynamics, the sharpening of the vorticity gradient as the waves approach the confluence zone leads to the development of disturbance asymmetries with respect to the central latitude. This effect is through the nonlinear interaction of Rossby waves with the planetary vorticity gradient. This development leads to a pair of vorticity centers that straddles the central latitude with the cyclone (anticyclone) in the north (south), and an elongated, weak westerly flow along the central latitude. This elongated westerly flow, which possesses a zonal wavenumber smaller than that in the linear cases, reverses the sign of the Reynold’s stress and allows the energy to grow near the critical longitude, leading to intensified disturbances.

With a more realistic monsoonlike background flow, a northwestward propagation pattern with an approximately 8-day period and 3000-km wavelength is produced, in general agreement with observed disturbances in the Northwest Pacific. The intensified disturbance may disperse energy upstream, leading to a series of trailing anticyclonic and cyclonic cells along the northwestward propagation path. When an opposing current is present, the energy dispersion leads to the formation of new disturbances in the confluence zone by vortex axisymmetrization dynamics. Thus, our results indicate that the scale contraction and nonlinear effects may cause a succession of tropical disturbances to develop without disturbance-scale diabatic effects.

## Abstract

The second WetNet Precipitation Intercomparison Project (PIP-2) evaluates the performance of 20 satellite precipitation retrieval algorithms, implemented for application with Special Sensor Microwave/Imager (SSM/I) passive microwave (PMW) measurements and run for a set of rainfall case studies at full resolution–instantaneous space–timescales. The cases are drawn from over the globe during all seasons, for a period of 7 yr, over a 60°N–17°S latitude range. Ground-based data were used for the intercomparisons, principally based on radar measurements but also including rain gauge measurements. The goals of PIP-2 are 1) to improve performance and accuracy of different SSM/I algorithms at full resolution–instantaneous scales by seeking a better understanding of the relationship between microphysical signatures in the PMW measurements and physical laws employed in the algorithms; 2) to evaluate the pros and cons of individual algorithms and their subsystems in order to seek optimal “front-end” combined algorithms; and 3) to demonstrate that PMW algorithms generate acceptable instantaneous rain estimates.

It is found that the bias uncertainty of many current PMW algorithms is on the order of ±30%. This level is below that of the radar and rain gauge data specially collected for the study, so that it is not possible to objectively select a best algorithm based on the ground data validation approach. By decomposing the intercomparisons into effects due to rain detection (screening) and effects due to brightness temperature–rain rate conversion, differences among the algorithms are partitioned by rain area and rain intensity. For ocean, the screening differences mainly affect the light rain rates, which do not contribute significantly to area-averaged rain rates. The major sources of differences in mean rain rates between individual algorithms stem from differences in how intense rain rates are calculated and the maximum rain rate allowed by a given algorithm. The general method of solution is not necessarily the determining factor in creating systematic rain-rate differences among groups of algorithms, as we find that the severity of the screen is the dominant factor in producing systematic group differences among land algorithms, while the input channel selection is the dominant factor in producing systematic group differences among ocean algorithms. The significance of these issues are examined through what is called “fan map” analysis.

The paper concludes with a discussion on the role of intercomparison projects in seeking improvements to algorithms, and a suggestion on why moving beyond the “ground truth” validation approach by use of a calibration-quality forward model would be a step forward in seeking objective evaluation of individual algorithm performance and optimal algorithm design.

## Abstract

The second WetNet Precipitation Intercomparison Project (PIP-2) evaluates the performance of 20 satellite precipitation retrieval algorithms, implemented for application with Special Sensor Microwave/Imager (SSM/I) passive microwave (PMW) measurements and run for a set of rainfall case studies at full resolution–instantaneous space–timescales. The cases are drawn from over the globe during all seasons, for a period of 7 yr, over a 60°N–17°S latitude range. Ground-based data were used for the intercomparisons, principally based on radar measurements but also including rain gauge measurements. The goals of PIP-2 are 1) to improve performance and accuracy of different SSM/I algorithms at full resolution–instantaneous scales by seeking a better understanding of the relationship between microphysical signatures in the PMW measurements and physical laws employed in the algorithms; 2) to evaluate the pros and cons of individual algorithms and their subsystems in order to seek optimal “front-end” combined algorithms; and 3) to demonstrate that PMW algorithms generate acceptable instantaneous rain estimates.

It is found that the bias uncertainty of many current PMW algorithms is on the order of ±30%. This level is below that of the radar and rain gauge data specially collected for the study, so that it is not possible to objectively select a best algorithm based on the ground data validation approach. By decomposing the intercomparisons into effects due to rain detection (screening) and effects due to brightness temperature–rain rate conversion, differences among the algorithms are partitioned by rain area and rain intensity. For ocean, the screening differences mainly affect the light rain rates, which do not contribute significantly to area-averaged rain rates. The major sources of differences in mean rain rates between individual algorithms stem from differences in how intense rain rates are calculated and the maximum rain rate allowed by a given algorithm. The general method of solution is not necessarily the determining factor in creating systematic rain-rate differences among groups of algorithms, as we find that the severity of the screen is the dominant factor in producing systematic group differences among land algorithms, while the input channel selection is the dominant factor in producing systematic group differences among ocean algorithms. The significance of these issues are examined through what is called “fan map” analysis.

The paper concludes with a discussion on the role of intercomparison projects in seeking improvements to algorithms, and a suggestion on why moving beyond the “ground truth” validation approach by use of a calibration-quality forward model would be a step forward in seeking objective evaluation of individual algorithm performance and optimal algorithm design.

## Abstract

In the PILPS Phase 2a experiment, 23 land-surface schemes were compared in an off-line control experiment using observed meteorological data from Cabauw, the Netherlands. Two simple sensitivity experiments were also undertaken in which the observed surface air temperature was artificially increased or decreased by 2 K while all other factors remained as observed. On the annual timescale, all schemes show similar responses to these perturbations in latent, sensible heat flux, and other key variables. For the 2-K increase in temperature, surface temperatures and latent heat fluxes all increase while net radiation, sensible heat fluxes, and soil moistures all decrease. The results are reversed for a 2-K temperature decrease. The changes in sensible heat fluxes and, especially, the changes in the latent heat fluxes are not linearly related to the change of temperature. Theoretically, the nonlinear relationship between air temperature and the latent heat flux is evident and due to the convex relationship between air temperature and saturation vapor pressure. A simple test shows that, the effect of the change of air temperature on the atmospheric stratification aside, this nonlinear relationship is shown in the form that the increase of the latent heat flux for a 2-K temperature increase is larger than its decrease for a 2-K temperature decrease. However, the results from the Cabauw sensitivity experiments show that the increase of the latent heat flux in the +2-K experiment is smaller than the decrease of the latent heat flux in the −2-K experiment (we refer to this as the asymmetry). The analysis in this paper shows that this inconsistency between the theoretical relationship and the Cabauw sensitivity experiments results (or the asymmetry) is due to (i) the involvement of the *β*
_{g} formulation, which is a function of a series stress factors that limited the evaporation and whose values change in the ±2-K experiments, leading to strong modifications of the latent heat flux; (ii) the change of the drag coefficient induced by the changes in stratification due to the imposed air temperature changes (±2 K) in parameterizations of latent heat flux common in current land-surface schemes. Among all stress factors involved in the *β*
_{g} formulation, the soil moisture stress in the +2-K experiment induced by the increased evaporation is the main factor that contributes to the asymmetry.

## Abstract

In the PILPS Phase 2a experiment, 23 land-surface schemes were compared in an off-line control experiment using observed meteorological data from Cabauw, the Netherlands. Two simple sensitivity experiments were also undertaken in which the observed surface air temperature was artificially increased or decreased by 2 K while all other factors remained as observed. On the annual timescale, all schemes show similar responses to these perturbations in latent, sensible heat flux, and other key variables. For the 2-K increase in temperature, surface temperatures and latent heat fluxes all increase while net radiation, sensible heat fluxes, and soil moistures all decrease. The results are reversed for a 2-K temperature decrease. The changes in sensible heat fluxes and, especially, the changes in the latent heat fluxes are not linearly related to the change of temperature. Theoretically, the nonlinear relationship between air temperature and the latent heat flux is evident and due to the convex relationship between air temperature and saturation vapor pressure. A simple test shows that, the effect of the change of air temperature on the atmospheric stratification aside, this nonlinear relationship is shown in the form that the increase of the latent heat flux for a 2-K temperature increase is larger than its decrease for a 2-K temperature decrease. However, the results from the Cabauw sensitivity experiments show that the increase of the latent heat flux in the +2-K experiment is smaller than the decrease of the latent heat flux in the −2-K experiment (we refer to this as the asymmetry). The analysis in this paper shows that this inconsistency between the theoretical relationship and the Cabauw sensitivity experiments results (or the asymmetry) is due to (i) the involvement of the *β*
_{g} formulation, which is a function of a series stress factors that limited the evaporation and whose values change in the ±2-K experiments, leading to strong modifications of the latent heat flux; (ii) the change of the drag coefficient induced by the changes in stratification due to the imposed air temperature changes (±2 K) in parameterizations of latent heat flux common in current land-surface schemes. Among all stress factors involved in the *β*
_{g} formulation, the soil moisture stress in the +2-K experiment induced by the increased evaporation is the main factor that contributes to the asymmetry.