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- Author or Editor: Matthias Münnich x

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## Abstract

We study the behavior of an iterative map as a model for El Niño and the Southern Oscillation (ENSO). This map is derived from a model that combines linear equatorial beta-plane ocean dynamics with a version of the Bjerknes hypothesis for ENSO. It differs from the linear model of Cane et al. only in that the coupling from ocean to atmosphere is idealized as a *nonlinear* relation *τ*(*h _{e}*) between a wind stress

*τ*of fixed spatial form and

*h*, the thermocline displacement at the eastern end of the equator. The model sustains finite amplitude periodic and aperiodic oscillations. A period doubling bifurcation leads from a period of less than 2 years to the 3–4 year one observed in nature. Other principal results are: the resulting period depends on the curvature of the function away from the unstable equilibrium at

_{e}*h*= 0, and not solely on its linear instability; at least two Rossby modes must be included in the model for aperiodic oscillations to appear; no stochastic term is needed for this aperiodicity, but it appears more readily if the model background state includes an annual cycle.

_{e}## Abstract

We study the behavior of an iterative map as a model for El Niño and the Southern Oscillation (ENSO). This map is derived from a model that combines linear equatorial beta-plane ocean dynamics with a version of the Bjerknes hypothesis for ENSO. It differs from the linear model of Cane et al. only in that the coupling from ocean to atmosphere is idealized as a *nonlinear* relation *τ*(*h _{e}*) between a wind stress

*τ*of fixed spatial form and

*h*, the thermocline displacement at the eastern end of the equator. The model sustains finite amplitude periodic and aperiodic oscillations. A period doubling bifurcation leads from a period of less than 2 years to the 3–4 year one observed in nature. Other principal results are: the resulting period depends on the curvature of the function away from the unstable equilibrium at

_{e}*h*= 0, and not solely on its linear instability; at least two Rossby modes must be included in the model for aperiodic oscillations to appear; no stochastic term is needed for this aperiodicity, but it appears more readily if the model background state includes an annual cycle.

_{e}## Abstract

We analyze the linearized version of an analytical model, which combines linear ocean dynamics with a simple version of the Bjerknes hypothesis for El Niño. The ocean is represented by linear shallow water equations on an equatorial beta-plane. It is driven by zonal wind stress, which is assumed to have a fixed spatial form. Stress amplitude is set to be proportional to the thermocline displacement at the eastern boundary.

It is shown that, for physically plausible parameter values, the model system can sustain growing Oscillations. Both growth rate and period scale directly with the time that an oceanic Kelvin wave needs to crow the basin. They are quite sensitive to the coupling parameter between thermocline displacement and wind stress, and the zonal location and meridional width of the wind.

The most important parameter determining this behavior of the system is the coupling constant. For strong coupling the system exhibits exponential growth without oscillation. As the coupling is decreased the growth rate decreases until a transition value is reached. For smaller values of the coupling the growing modes of the system oscillate, with a period which is infinite at the transition value and decreases for decreasing coupling. The inviscid system has growing modes for any positive feedback, no mater how weak, though the growth rate rapidly becomes very small. For very weak coupling the period approaches the first resonance period of the free ocean. The model can also be expressed as a nondifferential delay equation, The components of dfis equation are easy to interpret physically and allow some insights into the nature of the oscillations. The relation of our results to other recent work and its implications for El Niño and the Southern Oscillation are discussed.

## Abstract

We analyze the linearized version of an analytical model, which combines linear ocean dynamics with a simple version of the Bjerknes hypothesis for El Niño. The ocean is represented by linear shallow water equations on an equatorial beta-plane. It is driven by zonal wind stress, which is assumed to have a fixed spatial form. Stress amplitude is set to be proportional to the thermocline displacement at the eastern boundary.

It is shown that, for physically plausible parameter values, the model system can sustain growing Oscillations. Both growth rate and period scale directly with the time that an oceanic Kelvin wave needs to crow the basin. They are quite sensitive to the coupling parameter between thermocline displacement and wind stress, and the zonal location and meridional width of the wind.

The most important parameter determining this behavior of the system is the coupling constant. For strong coupling the system exhibits exponential growth without oscillation. As the coupling is decreased the growth rate decreases until a transition value is reached. For smaller values of the coupling the growing modes of the system oscillate, with a period which is infinite at the transition value and decreases for decreasing coupling. The inviscid system has growing modes for any positive feedback, no mater how weak, though the growth rate rapidly becomes very small. For very weak coupling the period approaches the first resonance period of the free ocean. The model can also be expressed as a nondifferential delay equation, The components of dfis equation are easy to interpret physically and allow some insights into the nature of the oscillations. The relation of our results to other recent work and its implications for El Niño and the Southern Oscillation are discussed.

## Abstract

Many aspects of the coupling between the ocean and atmosphere at the mesoscale (on the order of 20–100 km) remain unknown. While recent observations from the Southern Ocean revealed that circular fronts associated with oceanic mesoscale eddies leave a distinct imprint on the overlying wind, cloud coverage, and rain, the mechanisms responsible for explaining these atmospheric changes are not well established. Here the atmospheric response above mesoscale ocean eddies is investigated utilizing a newly developed coupled atmosphere–ocean regional model [Consortium for Small-Scale Modeling–Regional Ocean Modelling System (COSMO-ROMS)] configured at a horizontal resolution of ~10 km for the South Atlantic and run for a 3-month period during austral winter of 2004. The model-simulated changes in surface wind, cloud fraction, and rain above the oceanic eddies are very consistent with the relationships inferred from satellite observations for the same region and time. From diagnosing the model’s momentum balance, it is shown that the atmospheric imprint of the oceanic eddies are driven by the modification of vertical mixing in the atmospheric boundary layer, rather than secondary flows driven by horizontal pressure gradients. This is largely due to the very limited ability of the atmosphere to adjust its temperature over the time scale it takes for an air parcel to pass over these mesoscale oceanic features. This results in locally enhanced vertical gradients between the ocean surface and the overlying air and thus a rapid change in turbulent mixing in the atmospheric boundary layer and an associated change in the vertical momentum flux.

## Abstract

Many aspects of the coupling between the ocean and atmosphere at the mesoscale (on the order of 20–100 km) remain unknown. While recent observations from the Southern Ocean revealed that circular fronts associated with oceanic mesoscale eddies leave a distinct imprint on the overlying wind, cloud coverage, and rain, the mechanisms responsible for explaining these atmospheric changes are not well established. Here the atmospheric response above mesoscale ocean eddies is investigated utilizing a newly developed coupled atmosphere–ocean regional model [Consortium for Small-Scale Modeling–Regional Ocean Modelling System (COSMO-ROMS)] configured at a horizontal resolution of ~10 km for the South Atlantic and run for a 3-month period during austral winter of 2004. The model-simulated changes in surface wind, cloud fraction, and rain above the oceanic eddies are very consistent with the relationships inferred from satellite observations for the same region and time. From diagnosing the model’s momentum balance, it is shown that the atmospheric imprint of the oceanic eddies are driven by the modification of vertical mixing in the atmospheric boundary layer, rather than secondary flows driven by horizontal pressure gradients. This is largely due to the very limited ability of the atmosphere to adjust its temperature over the time scale it takes for an air parcel to pass over these mesoscale oceanic features. This results in locally enhanced vertical gradients between the ocean surface and the overlying air and thus a rapid change in turbulent mixing in the atmospheric boundary layer and an associated change in the vertical momentum flux.

## Abstract

To study the dynamics that may lead to decadal oscillations in the North Pacific a simple coupled model is developed. The ocean is based on the linear, potential vorticity equation for baroclinic planetary waves. The atmosphere is reduced to a nonlocal wind response to thermocline depth anomalies. The wind stress has a spatially fixed structure and its amplitude depends on the thermocline perturbation at one location or in a predefined index region.

Such a simple coupled model produces decadal oscillations for suitable parameter choices. For realistic wind stress patterns, the patterns of oceanic variability are similar to those observed. It is determined by the speed of long Rossby waves at the coupling latitude. The period of the oscillation is rather insensitive to the coupling strength and amounts to approximately twice the time the Rossby wave needs to travel from the center of the wind stress curl anomaly to the coupling location.

A stochastic component to the atmospheric forcing is incorporated by white noise added to the feedback. With such a forcing, typical oceanic spectra become red with a broad peak at decadal timescales superimposed.

## Abstract

To study the dynamics that may lead to decadal oscillations in the North Pacific a simple coupled model is developed. The ocean is based on the linear, potential vorticity equation for baroclinic planetary waves. The atmosphere is reduced to a nonlocal wind response to thermocline depth anomalies. The wind stress has a spatially fixed structure and its amplitude depends on the thermocline perturbation at one location or in a predefined index region.

Such a simple coupled model produces decadal oscillations for suitable parameter choices. For realistic wind stress patterns, the patterns of oceanic variability are similar to those observed. It is determined by the speed of long Rossby waves at the coupling latitude. The period of the oscillation is rather insensitive to the coupling strength and amounts to approximately twice the time the Rossby wave needs to travel from the center of the wind stress curl anomaly to the coupling location.

A stochastic component to the atmospheric forcing is incorporated by white noise added to the feedback. With such a forcing, typical oceanic spectra become red with a broad peak at decadal timescales superimposed.

## Abstract

Winds over the tropical Pacific are interpreted using mixed-layer theory. The theory—which posits that the surface winds can be derived in terms of a force balance among surface drag, pressure gradients, Coriolis forces, and the vertical mixing of momentum into the boundary layer (entrainment)—is very successful in predicting the seasonal climatology of the surface winds. The model is also used as a basis for interpreting previous results. In particular the model illustrates why studies that model the momentum flux divergence as a Rayleigh damping find optimal damping coefficients that are anisotropic. A linear variant of the model, which also incorporates entrainment but neglects the quadratic relation between the wind speed and the surface stress, is also found to predict the surface winds skillfully. In addition to improving the representation of the winds, it leads to realistic representations of the divergence of the vector wind. If the key parameters of the model (the entrainment rate and the boundary layer depth) are assumed to have uniform climatological mean values over the Pacific basin, optimal parameter values can be derived by matching the model winds to the climatology. Such a procedure leads to boundary layer depths between 300 and 400 m and entrainment rates slightly less than 1 cm s^{−1}. A somewhat more general model of the boundary layer winds (the so-called *K*-profile parameterization), is used to show that accounting for the vertical structure of the wind profiles yields somewhat larger optimal estimates of *w*
_{e} and *h.* Overall, the incorporation of the entrainment effect is critical, indicating that the acceleration of the near-surface winds by momentum mixing with the free atmosphere is a first-order effect that should not be neglected in simple models. In physical terms, this effect is one of resisting the turning of the winds.

## Abstract

Winds over the tropical Pacific are interpreted using mixed-layer theory. The theory—which posits that the surface winds can be derived in terms of a force balance among surface drag, pressure gradients, Coriolis forces, and the vertical mixing of momentum into the boundary layer (entrainment)—is very successful in predicting the seasonal climatology of the surface winds. The model is also used as a basis for interpreting previous results. In particular the model illustrates why studies that model the momentum flux divergence as a Rayleigh damping find optimal damping coefficients that are anisotropic. A linear variant of the model, which also incorporates entrainment but neglects the quadratic relation between the wind speed and the surface stress, is also found to predict the surface winds skillfully. In addition to improving the representation of the winds, it leads to realistic representations of the divergence of the vector wind. If the key parameters of the model (the entrainment rate and the boundary layer depth) are assumed to have uniform climatological mean values over the Pacific basin, optimal parameter values can be derived by matching the model winds to the climatology. Such a procedure leads to boundary layer depths between 300 and 400 m and entrainment rates slightly less than 1 cm s^{−1}. A somewhat more general model of the boundary layer winds (the so-called *K*-profile parameterization), is used to show that accounting for the vertical structure of the wind profiles yields somewhat larger optimal estimates of *w*
_{e} and *h.* Overall, the incorporation of the entrainment effect is critical, indicating that the acceleration of the near-surface winds by momentum mixing with the free atmosphere is a first-order effect that should not be neglected in simple models. In physical terms, this effect is one of resisting the turning of the winds.