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## Abstract

A method was recently proposed for evaluating the impact of a perturbation, such as air pollution or urbanization, on the precipitation at a location by calculating the ratio between the precipitation at the perturbed location and that at a location believed to be unperturbed. However, this method may be inappropriate because of the high degree of variability of precipitation at each of the stations. To explore the validity of this approach, noisy annual rainfall records are generated numerically in an upwind, unperturbed station and in a downwind, perturbed station, and the time series of ratio between the annual rainfalls in the two stations is analyzed. The noisy rainfall records are 50 yr long, and the imposed trend for the downwind, perturbed station is âˆ’2 mm yr^{âˆ’1} while at the upwind station the variations in annual rainfall are purely noisy. Many pairs of noisy rainfall records are numerically generated (each pair constitutes an experiment), and in every experiment the slope of the linear best fit to the rainfall ratio yields an estimate of the trend of rainfall at the perturbed station. In the absence of noise, the trend of the rainfall ratio is explicitly related to the trend of rainfall at the perturbed station, but the natural rainfall variation at the stations completely masks this explicit relationship. The results show that in some experiments the trend line of the rainfall ratio has the opposite sign to the imposed trend and that in only about one-half of the experiments does the ratioâ€™s trend line lie within Â±75% of the imposed trend. Trend estimates within Â±25% of the imposed trend are obtained in less than one-quarter of the experiments. This result casts doubt on the generality and validity of using trends of rainfall ratio between two stations to estimate trends of precipitation in one of these stations.

## Abstract

A method was recently proposed for evaluating the impact of a perturbation, such as air pollution or urbanization, on the precipitation at a location by calculating the ratio between the precipitation at the perturbed location and that at a location believed to be unperturbed. However, this method may be inappropriate because of the high degree of variability of precipitation at each of the stations. To explore the validity of this approach, noisy annual rainfall records are generated numerically in an upwind, unperturbed station and in a downwind, perturbed station, and the time series of ratio between the annual rainfalls in the two stations is analyzed. The noisy rainfall records are 50 yr long, and the imposed trend for the downwind, perturbed station is âˆ’2 mm yr^{âˆ’1} while at the upwind station the variations in annual rainfall are purely noisy. Many pairs of noisy rainfall records are numerically generated (each pair constitutes an experiment), and in every experiment the slope of the linear best fit to the rainfall ratio yields an estimate of the trend of rainfall at the perturbed station. In the absence of noise, the trend of the rainfall ratio is explicitly related to the trend of rainfall at the perturbed station, but the natural rainfall variation at the stations completely masks this explicit relationship. The results show that in some experiments the trend line of the rainfall ratio has the opposite sign to the imposed trend and that in only about one-half of the experiments does the ratioâ€™s trend line lie within Â±75% of the imposed trend. Trend estimates within Â±25% of the imposed trend are obtained in less than one-quarter of the experiments. This result casts doubt on the generality and validity of using trends of rainfall ratio between two stations to estimate trends of precipitation in one of these stations.

## Abstract

The linear instability of circular quasigeostrophic vortices with horizontal cross sections of angular velocity identical to those that fit the observations of submesoscale eddies in the ocean is investigated analytically and numerically. The theoretical necessary conditions for instability are formulated as a single condition on the mean potential vorticity or mean angular velocity rather than the streamfunction, which is more readily applicable to oceanic lenses, eddies, and meddies. It is shown that the suggested cross sections that best fit the observed angular velocity of several long-lived vortices are all unstable to small, wavelike perturbations, and that the *e*-folding time for perturbation growth at all wavenumbers and cross sections is on the order of 1 day. For all cross sections considered, azimuthal wavenumber 1 is stable while all higher azimuthal wavenumbers, as well as all vertical wavenumbers, are unstable. The main contribution to the instability comes from the jump in potential vorticity at the radius of maximum angular velocity. When the potential vorticity is continuous at this radius, the growth rates become smaller and the details of potential vorticity distribution become important. The fast growth rates obtained by the numerical calculations clearly emphasize the insufficient spatial resolution of existing observations for deciphering the exact velocity cross sections of submesoscale oceanic vortices, especially near the radius of maximum angular velocity.

## Abstract

The linear instability of circular quasigeostrophic vortices with horizontal cross sections of angular velocity identical to those that fit the observations of submesoscale eddies in the ocean is investigated analytically and numerically. The theoretical necessary conditions for instability are formulated as a single condition on the mean potential vorticity or mean angular velocity rather than the streamfunction, which is more readily applicable to oceanic lenses, eddies, and meddies. It is shown that the suggested cross sections that best fit the observed angular velocity of several long-lived vortices are all unstable to small, wavelike perturbations, and that the *e*-folding time for perturbation growth at all wavenumbers and cross sections is on the order of 1 day. For all cross sections considered, azimuthal wavenumber 1 is stable while all higher azimuthal wavenumbers, as well as all vertical wavenumbers, are unstable. The main contribution to the instability comes from the jump in potential vorticity at the radius of maximum angular velocity. When the potential vorticity is continuous at this radius, the growth rates become smaller and the details of potential vorticity distribution become important. The fast growth rates obtained by the numerical calculations clearly emphasize the insufficient spatial resolution of existing observations for deciphering the exact velocity cross sections of submesoscale oceanic vortices, especially near the radius of maximum angular velocity.

## Abstract

Nonlinear, dispersive perturbations are applied to a single layer, geostrophic, density current of zero potential vorticity, which flows along a vertical wall (coast) and is bounded by a free streamline in the seaward direction. The temporal evolution of these perturbations is shown to be governed by the, well known, Korteweg-deVries equation. The amplitude of the solitary wave (standing wave) solutions of this equation depends on their width (wavelength) and on the undisturbed velocity of the free streamline.

The correlation between the free streamline displacement and the velocity perturbation shows that the observable displacements will be in the seaward direction and will propagate *upstream* relative to the mean flow.

The theoretical calculations of the amplitude-wavelength relations are shown to be consistent with winter observations of the Davidson Current (a northern extension of the California Current System) and with observations of the African current. Comparison with the finite amplitude, longwave, theory shows that dispersion eliminates the blocking waves but retains the breaking waves, bores and wedges. These solutions are encountered only for specific values of the undisturbed free streamline velocity. In the finite amplitude, long-wave theory these waves are permissible solutions for any value of that velocity.

## Abstract

Nonlinear, dispersive perturbations are applied to a single layer, geostrophic, density current of zero potential vorticity, which flows along a vertical wall (coast) and is bounded by a free streamline in the seaward direction. The temporal evolution of these perturbations is shown to be governed by the, well known, Korteweg-deVries equation. The amplitude of the solitary wave (standing wave) solutions of this equation depends on their width (wavelength) and on the undisturbed velocity of the free streamline.

The correlation between the free streamline displacement and the velocity perturbation shows that the observable displacements will be in the seaward direction and will propagate *upstream* relative to the mean flow.

The theoretical calculations of the amplitude-wavelength relations are shown to be consistent with winter observations of the Davidson Current (a northern extension of the California Current System) and with observations of the African current. Comparison with the finite amplitude, longwave, theory shows that dispersion eliminates the blocking waves but retains the breaking waves, bores and wedges. These solutions are encountered only for specific values of the undisturbed free streamline velocity. In the finite amplitude, long-wave theory these waves are permissible solutions for any value of that velocity.

## Abstract

It is shown that tidal perturbations of a geopotential height in an inviscid, barotropic atmosphere can turn a purely inertial, predictable trajectory of a Lagrangian particle chaotic. Hamiltonian formulation of both the free, inertial, and the tidally forced problems permitted the application of the twist and KAM theorems, which predicts the existence of chaotic trajectories in the latter case The chaotic behavior manifests itself in extreme sensitivity of both the trajectory and the energy spectra to initial conditions and to the precise value of the perturbation's amplitude. In some cases dispersion of initially close particles can be very fast, with an *e*-folding time of the rms particle separation as high as one day. A vigorous mixing is induced by the chaotic advection associated with the tidal forcing through the stretching and folding of material surfaces.

## Abstract

It is shown that tidal perturbations of a geopotential height in an inviscid, barotropic atmosphere can turn a purely inertial, predictable trajectory of a Lagrangian particle chaotic. Hamiltonian formulation of both the free, inertial, and the tidally forced problems permitted the application of the twist and KAM theorems, which predicts the existence of chaotic trajectories in the latter case The chaotic behavior manifests itself in extreme sensitivity of both the trajectory and the energy spectra to initial conditions and to the precise value of the perturbation's amplitude. In some cases dispersion of initially close particles can be very fast, with an *e*-folding time of the rms particle separation as high as one day. A vigorous mixing is induced by the chaotic advection associated with the tidal forcing through the stretching and folding of material surfaces.

## Abstract

The Lagrangian description of cross-equatorial flow under any steady, strictly meridional, pressure gradient forcing is shown to comprise an integrable two-degree-of-freedom Hamiltonian system. The system undergoes a pitchfork bifurcation when the angular momentum passes through a critical value. It is shown that, even when the full variation of the Coriolis parameter is taken into account, the dynamical system is fully integrable, which implies that its evolution from any initial state can be calculated with sufficient accuracy (depending on the accuracy of the initial state) to any desired time. The role of the driving pressure gradient is merely to shift the latitude of the fixed points from their location in the inertial case.

When zonal variation or time dependence of the pressure field is allowed, the system becomes nonintegrable and chaotic bands appear where nearby trajectories diverge exponentially.

## Abstract

The Lagrangian description of cross-equatorial flow under any steady, strictly meridional, pressure gradient forcing is shown to comprise an integrable two-degree-of-freedom Hamiltonian system. The system undergoes a pitchfork bifurcation when the angular momentum passes through a critical value. It is shown that, even when the full variation of the Coriolis parameter is taken into account, the dynamical system is fully integrable, which implies that its evolution from any initial state can be calculated with sufficient accuracy (depending on the accuracy of the initial state) to any desired time. The role of the driving pressure gradient is merely to shift the latitude of the fixed points from their location in the inertial case.

When zonal variation or time dependence of the pressure field is allowed, the system becomes nonintegrable and chaotic bands appear where nearby trajectories diverge exponentially.

## Abstract

The interaction between a simple meandering jet such as the Gulf Stream, and an eddy is shown to greatly enhance the mixing and dispersal of fluid parcels in the jet. This enhanced mixing is quantified by calculating the rate of increase of the root-mean-square pair separation of Lagrangian particles (e.g., floats) launched in the jet's immediate vicinity. In the presence of an eddy, particles can escape from the regions in which they were initially launched. Comparisons with observations show a markedly improved qualitative agreement when the eddy is allowed to interact with the meandering jet.

## Abstract

The interaction between a simple meandering jet such as the Gulf Stream, and an eddy is shown to greatly enhance the mixing and dispersal of fluid parcels in the jet. This enhanced mixing is quantified by calculating the rate of increase of the root-mean-square pair separation of Lagrangian particles (e.g., floats) launched in the jet's immediate vicinity. In the presence of an eddy, particles can escape from the regions in which they were initially launched. Comparisons with observations show a markedly improved qualitative agreement when the eddy is allowed to interact with the meandering jet.

## Abstract

The linear instability of a zonal geostrophic jet with a cosh^{âˆ’2} meridional profile on an *f* plane is investigated in a reduced-gravity, shallow-water model. The stability theory developed here extends classic quasigeostrophic theory to cases where the change of active-layer depth across the jet is not necessarily small. A shooting method is used to integrate the equations describing the cross-stream structure of the alongstream wave perturbations. The phase speeds of these waves are determined by the boundary conditions of regularity at infinity. Regions exist in parameter space where the waves that propagate along the jet will grow exponentially with time. The wavelength of the most unstable waves is 2*Ï€*
*R,* where *R* is the internal deformation radius on the deep side, and their *e*-folding time is about 25 days.

The upper-layer thickness of the basic state in the system has a spatial structure resembling that of the isopycnals across the Gulf Stream. The unstable waves obtained in the present analysis have a wavelength that is in agreement with some recent observationsâ€”based on infrared imaging of the sea surface temperature fieldâ€”of the fastest- growing meandersâ€™ wavelength. Calculated growth rates fall toward the low end of the range of values obtained from these infrared observations on the temporal evolution of Gulf Stream meanders.

## Abstract

The linear instability of a zonal geostrophic jet with a cosh^{âˆ’2} meridional profile on an *f* plane is investigated in a reduced-gravity, shallow-water model. The stability theory developed here extends classic quasigeostrophic theory to cases where the change of active-layer depth across the jet is not necessarily small. A shooting method is used to integrate the equations describing the cross-stream structure of the alongstream wave perturbations. The phase speeds of these waves are determined by the boundary conditions of regularity at infinity. Regions exist in parameter space where the waves that propagate along the jet will grow exponentially with time. The wavelength of the most unstable waves is 2*Ï€*
*R,* where *R* is the internal deformation radius on the deep side, and their *e*-folding time is about 25 days.

The upper-layer thickness of the basic state in the system has a spatial structure resembling that of the isopycnals across the Gulf Stream. The unstable waves obtained in the present analysis have a wavelength that is in agreement with some recent observationsâ€”based on infrared imaging of the sea surface temperature fieldâ€”of the fastest- growing meandersâ€™ wavelength. Calculated growth rates fall toward the low end of the range of values obtained from these infrared observations on the temporal evolution of Gulf Stream meanders.

## Abstract

Finite-wavelength instabilities of a coupled density front with zero potential vorticity are found for the single-layer and the two-layer problems. These instabilities result from the resonance between two distinct waves whose real phase speeds coalesce. In the single-layer problem, the range of wavenumbers over which the coalescence takes place decreases with increasing wavenumber; consequently, the instability exponents and the growth rates also decrease. For shallow lower layers, the coalescence range increases with increasing wavenumber; at large wavenumbers, the coalescence range becomes continuous, while the instability exponent is approaching a constant value. The growth rate in the two-layer problem increases, therefore, linearly with wavenumber and the short waves fastest. These short-wave instabilities are qualitatively reminiscent of small-scale features along coastal fronts and in laboratory experiments.

## Abstract

Finite-wavelength instabilities of a coupled density front with zero potential vorticity are found for the single-layer and the two-layer problems. These instabilities result from the resonance between two distinct waves whose real phase speeds coalesce. In the single-layer problem, the range of wavenumbers over which the coalescence takes place decreases with increasing wavenumber; consequently, the instability exponents and the growth rates also decrease. For shallow lower layers, the coalescence range increases with increasing wavenumber; at large wavenumbers, the coalescence range becomes continuous, while the instability exponent is approaching a constant value. The growth rate in the two-layer problem increases, therefore, linearly with wavenumber and the short waves fastest. These short-wave instabilities are qualitatively reminiscent of small-scale features along coastal fronts and in laboratory experiments.

## Abstract

A Lagrangian model is employed to study the characteristics of a horizontal cross-equatorial flow. The Coriolis force and the mean meridional pressure field assumed here render the dynamics of particle flow across the equator a nonlinear Hamiltonian system of a bistable potential that has a local maximum at the equator. In the absence of any additional forces this local maximum at the equator prohibits particles from flowing from one hemisphere to the other. When all other (i.e., in addition to the mean meridional pressure gradient) forces are introduced into the system as stochastic forcing, modeled by Gaussian white noise, anomalous diffusion up the mean pressure gradient occurs and particles launched in one hemisphere can reach the other. Spectral estimations of equator crossing events show that at low noise intensity the spectral peak is low, narrow, and situated at low frequencies, and that as the amplitude of the noise increases, the peak becomes higher, wider, and shifts toward higher frequencies. At very large noise intensities the spectral peak flattens out, which implies that the process of equator crossing becomes noise dominated. The results demonstrate the existence of an optimal noise intensity where the signal-to-noise ratio of equator crossings exhibits a sharp maximum, and this optimal noise intensity is insensitive to the precise value of the mean meridional pressure gradient. These findings are applicable to the terrestrial atmosphere where the mean meridional geopotential height gradient and the (zonal and temporal) deviations from it are of the same order.

These results demonstrate, for the first time, the occurrence of stochastic resonance in a Hamiltonian system.

## Abstract

A Lagrangian model is employed to study the characteristics of a horizontal cross-equatorial flow. The Coriolis force and the mean meridional pressure field assumed here render the dynamics of particle flow across the equator a nonlinear Hamiltonian system of a bistable potential that has a local maximum at the equator. In the absence of any additional forces this local maximum at the equator prohibits particles from flowing from one hemisphere to the other. When all other (i.e., in addition to the mean meridional pressure gradient) forces are introduced into the system as stochastic forcing, modeled by Gaussian white noise, anomalous diffusion up the mean pressure gradient occurs and particles launched in one hemisphere can reach the other. Spectral estimations of equator crossing events show that at low noise intensity the spectral peak is low, narrow, and situated at low frequencies, and that as the amplitude of the noise increases, the peak becomes higher, wider, and shifts toward higher frequencies. At very large noise intensities the spectral peak flattens out, which implies that the process of equator crossing becomes noise dominated. The results demonstrate the existence of an optimal noise intensity where the signal-to-noise ratio of equator crossings exhibits a sharp maximum, and this optimal noise intensity is insensitive to the precise value of the mean meridional pressure gradient. These findings are applicable to the terrestrial atmosphere where the mean meridional geopotential height gradient and the (zonal and temporal) deviations from it are of the same order.

These results demonstrate, for the first time, the occurrence of stochastic resonance in a Hamiltonian system.

## Abstract

An axially symmetric inviscid shallow-water model (SWM) on the rotating Earth forced by off-equatorial steady differential heating is employed to characterize the main features of the upper branch of an ideal Hadley circulation. The steady-state solutions are derived and analyzed and their relevance to asymptotic temporal evolution of the circulation is established by comparing them to numerically derived time-dependent solutions at long times. The main novel feature of the steady-state solutions of the present theory is the existence of a tropical region, associated with the rising branch of the Hadley circulation, which extends to about half the combined width of the Hadley cells in the two hemispheres and is dominated by strong vertical advection of momentum. The solutions in this tropical region are characterized by three conditions: (i) the meridional temperature gradient is very weak but drastically increases outside of the region, (ii) moderate easterlies exist only inside this region and they peak off the equator, and (iii) angular momentum is not conserved there. The momentum fluxes of the new solutions at the tropics differ qualitatively from those of existing nearly inviscid theories and the new flux estimates are in better agreement with both observations and axially symmetric simulations. As in previous nearly inviscid theories, the steady solutions of the new theory are determined by a thermal Rossby number and by the latitude of maximal heating.

## Abstract

An axially symmetric inviscid shallow-water model (SWM) on the rotating Earth forced by off-equatorial steady differential heating is employed to characterize the main features of the upper branch of an ideal Hadley circulation. The steady-state solutions are derived and analyzed and their relevance to asymptotic temporal evolution of the circulation is established by comparing them to numerically derived time-dependent solutions at long times. The main novel feature of the steady-state solutions of the present theory is the existence of a tropical region, associated with the rising branch of the Hadley circulation, which extends to about half the combined width of the Hadley cells in the two hemispheres and is dominated by strong vertical advection of momentum. The solutions in this tropical region are characterized by three conditions: (i) the meridional temperature gradient is very weak but drastically increases outside of the region, (ii) moderate easterlies exist only inside this region and they peak off the equator, and (iii) angular momentum is not conserved there. The momentum fluxes of the new solutions at the tropics differ qualitatively from those of existing nearly inviscid theories and the new flux estimates are in better agreement with both observations and axially symmetric simulations. As in previous nearly inviscid theories, the steady solutions of the new theory are determined by a thermal Rossby number and by the latitude of maximal heating.