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## Abstract

A Pacific Ocean–global atmosphere general circulation model is used to simulate the climatic mean state and variability in the Tropics, up to interannual timescales. For this model no long-term trend in climate occurs, but there are systematic differences between the model mean state and observations: in particular, the east equatorial Pacific sea surface temperature is too high by several degrees. Along the equator the seasonal variability in sea surface temperature is good although some features of the seasonal cycle are unrealistic: for example, the east Pacific convergence zone crosses the equator twice a year, residing in the summer hemisphere.

Despite some deficiencies in the simulation of the mean state, there is substantial interannual variability, with irregular oscillations dominated by a 2-yr cycle. A principal oscillation pattern analysis shows that the interannual anomalies are typically generated in the west Pacific and move eastward along the equator, with closely connected oceanic and atmospheric components. The patterns are similar to those associated with observed El Niño events. Rainfall anomalies associated with the model El Niño events also have several realistic features.

Idealized seasonal prediction experiments were made by slightly perturbing the atmospheric component: three 6-month hindcasts were thus made for each of several start times spread through an El Niño cycle. Predictability of central Pacific sea surface temperature anomalies was best for hindcasts starting near a warm El Niño peak. Generally, hindcasts starting in September and December were more accurate, with less spread, than those starting in March and June. The behavior and predictability of seasonal rainfall in several regions was also analyzed. For example, a warm model El Niño produces enhanced rainfall in the central equatorial Pacific and reduced rainfall in the Indian region, which is reproduced consistently in the hindcasts.

The model also shows variability on shorter timescales, and an example is presented of a spontaneous westerly wind burst in the west Pacific and its oceanic impact.

## Abstract

A Pacific Ocean–global atmosphere general circulation model is used to simulate the climatic mean state and variability in the Tropics, up to interannual timescales. For this model no long-term trend in climate occurs, but there are systematic differences between the model mean state and observations: in particular, the east equatorial Pacific sea surface temperature is too high by several degrees. Along the equator the seasonal variability in sea surface temperature is good although some features of the seasonal cycle are unrealistic: for example, the east Pacific convergence zone crosses the equator twice a year, residing in the summer hemisphere.

Despite some deficiencies in the simulation of the mean state, there is substantial interannual variability, with irregular oscillations dominated by a 2-yr cycle. A principal oscillation pattern analysis shows that the interannual anomalies are typically generated in the west Pacific and move eastward along the equator, with closely connected oceanic and atmospheric components. The patterns are similar to those associated with observed El Niño events. Rainfall anomalies associated with the model El Niño events also have several realistic features.

Idealized seasonal prediction experiments were made by slightly perturbing the atmospheric component: three 6-month hindcasts were thus made for each of several start times spread through an El Niño cycle. Predictability of central Pacific sea surface temperature anomalies was best for hindcasts starting near a warm El Niño peak. Generally, hindcasts starting in September and December were more accurate, with less spread, than those starting in March and June. The behavior and predictability of seasonal rainfall in several regions was also analyzed. For example, a warm model El Niño produces enhanced rainfall in the central equatorial Pacific and reduced rainfall in the Indian region, which is reproduced consistently in the hindcasts.

The model also shows variability on shorter timescales, and an example is presented of a spontaneous westerly wind burst in the west Pacific and its oceanic impact.

## Abstract

A number of linear models of the steady-state response of the tropical atmosphere to sea surface temperature (SST) anomalies have been proposed, all based on the shallow-water equations. Despite their formal similarity, the various models have very different physical interpretations and suggest widely varying values for key parameters, including the mechanical damping rate (or coefficient of Rayleigh friction), the strength of the coupling to SST, and the “effective stability” of the lower troposphere. In order to place empirical constraints on these coefficients, the linear momentum equations are inverted to obtain the scalar forcing fields *P*′(interpreted as vertically integrated boundary-layer pressure anomalies) that best reproduce observed surface wind anomalies through the period 1984–90. This gives an optimum value for the mechanical damping rate, independent of the coupling parameterization. Direct optimization of a fully linear Gill-type model of ocean-atmosphere coupling reveals that the problem of identifying optimum values for the other two parameters (coupling and stability) is degenerate; if one parameter is fixed, the other is well constrained by the data, but if both are allowed to vary, the cost function (rms error in the model output winds) has no well-defined minimum. Models of this type also suggest a simple relationship between *P*′ and anomalies of SST and divergence, however. A significant but strikingly different relationship is found between these three quantities derived from the observations. If uniform, linear coupling is assumed, this result suggests that the large-scale response of the tropical atmosphere to SST anomalies is consistent with a moderately moist-unstable boundary layer, with the stability of the response being maintained by turbulent diffusive processes. This may be parameterized most simply by introducing biharmonic diffusion on Pinto the “thermodynamic” equation of a Gill-type model. Simple forms of nonlinear coupling both to SST and to divergence are also investigated. Although the possibility that nonlinear effects are important cannot be excluded, no evidence is found to suggest that either of two widely used nonlinear coupling parameterizations represent an improvement on a fully linear scheme.

## Abstract

A number of linear models of the steady-state response of the tropical atmosphere to sea surface temperature (SST) anomalies have been proposed, all based on the shallow-water equations. Despite their formal similarity, the various models have very different physical interpretations and suggest widely varying values for key parameters, including the mechanical damping rate (or coefficient of Rayleigh friction), the strength of the coupling to SST, and the “effective stability” of the lower troposphere. In order to place empirical constraints on these coefficients, the linear momentum equations are inverted to obtain the scalar forcing fields *P*′(interpreted as vertically integrated boundary-layer pressure anomalies) that best reproduce observed surface wind anomalies through the period 1984–90. This gives an optimum value for the mechanical damping rate, independent of the coupling parameterization. Direct optimization of a fully linear Gill-type model of ocean-atmosphere coupling reveals that the problem of identifying optimum values for the other two parameters (coupling and stability) is degenerate; if one parameter is fixed, the other is well constrained by the data, but if both are allowed to vary, the cost function (rms error in the model output winds) has no well-defined minimum. Models of this type also suggest a simple relationship between *P*′ and anomalies of SST and divergence, however. A significant but strikingly different relationship is found between these three quantities derived from the observations. If uniform, linear coupling is assumed, this result suggests that the large-scale response of the tropical atmosphere to SST anomalies is consistent with a moderately moist-unstable boundary layer, with the stability of the response being maintained by turbulent diffusive processes. This may be parameterized most simply by introducing biharmonic diffusion on Pinto the “thermodynamic” equation of a Gill-type model. Simple forms of nonlinear coupling both to SST and to divergence are also investigated. Although the possibility that nonlinear effects are important cannot be excluded, no evidence is found to suggest that either of two widely used nonlinear coupling parameterizations represent an improvement on a fully linear scheme.

## Abstract

A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve potential vorticity *q* when unforced. Regions of closed *q* contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of closed height contours.)

Motion is wavelike for very weak disturbances (maximum particle speed *Û*; ≪ long planetary wave speed *ĉ*). The height field disperses like a group of linear Rossby waves, and tracers have small, oscillatory (mainly north-south) displacements, with very little scatter.

When *Û*≈*ĉ*, the planetary *q* field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent “self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good indicator of tracer movement.)

When *Ü*≫*ĉ*, the *q* field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient *q* (and fluid elements) around itself. This wrapping effect leads to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the center-of-mass anomaly (for a buoyancy-generated disturbance).

Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed *ĉ*, whereas a cyclone travels slower than *ĉ*. The predictions are confirmed by the numerical experiments.

Some estimates of mixing based on tracer separation are given.

## Abstract

A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve potential vorticity *q* when unforced. Regions of closed *q* contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of closed height contours.)

Motion is wavelike for very weak disturbances (maximum particle speed *Û*; ≪ long planetary wave speed *ĉ*). The height field disperses like a group of linear Rossby waves, and tracers have small, oscillatory (mainly north-south) displacements, with very little scatter.

When *Û*≈*ĉ*, the planetary *q* field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent “self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good indicator of tracer movement.)

When *Ü*≫*ĉ*, the *q* field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient *q* (and fluid elements) around itself. This wrapping effect leads to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the center-of-mass anomaly (for a buoyancy-generated disturbance).

Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed *ĉ*, whereas a cyclone travels slower than *ĉ*. The predictions are confirmed by the numerical experiments.

Some estimates of mixing based on tracer separation are given.

## Abstract

The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region, with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.

## Abstract

The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region, with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.

## Abstract

When forecasting sea surface temperature (SST) in the Equatorial Pacific on a timescale of several seasons, most prediction schemes have a spring barrier; that is, they have skill scores that are substantially lower when predicting northern spring and summer conditions compared to autumn and winter. This feature is investigated by examining predictions during the 1970s and the 1980s, using a dynamic ocean model of intermediate complexity coupled to a statistical atmosphere. Results show that predictions initialized during the 1970s exhibit the typical prominent skill decay in spring, whereas the seasonal dependence in those predictions initialized during the 1980s is rather small. Similar changes in seasonal dependence are also found in predictions based on simple persistence of observed SST anomalies.

This decadal change in the spring barrier is related to decadal variations found in the seasonal phase locking of the SST anomalies, which is largely determined by the timing of El Niño events. The spring barrier was strong in the 1970s, when El Niño was strongly phaselocked to the annual cycle. An analysis of observed SST anomalies from 1900 to 1990 shows several changes in behavior on a decadal scale, with the largest change being from the 1970s to the 1980s.

The seasonal dependence of model heat content predictions is investigated and found to be similar to that for SST, except that it shows a winter barrier rather than the spring barrier evident in SST.

## Abstract

When forecasting sea surface temperature (SST) in the Equatorial Pacific on a timescale of several seasons, most prediction schemes have a spring barrier; that is, they have skill scores that are substantially lower when predicting northern spring and summer conditions compared to autumn and winter. This feature is investigated by examining predictions during the 1970s and the 1980s, using a dynamic ocean model of intermediate complexity coupled to a statistical atmosphere. Results show that predictions initialized during the 1970s exhibit the typical prominent skill decay in spring, whereas the seasonal dependence in those predictions initialized during the 1980s is rather small. Similar changes in seasonal dependence are also found in predictions based on simple persistence of observed SST anomalies.

This decadal change in the spring barrier is related to decadal variations found in the seasonal phase locking of the SST anomalies, which is largely determined by the timing of El Niño events. The spring barrier was strong in the 1970s, when El Niño was strongly phaselocked to the annual cycle. An analysis of observed SST anomalies from 1900 to 1990 shows several changes in behavior on a decadal scale, with the largest change being from the 1970s to the 1980s.

The seasonal dependence of model heat content predictions is investigated and found to be similar to that for SST, except that it shows a winter barrier rather than the spring barrier evident in SST.

## Abstract

Many features of the El Niño–Southern Oscillation (ENSO) phenomenon have been successfully simulated by coupled models during the last decade; however, some fundamental differences in model behavior remain. They can be classified into two categories according to whether the oscillation is self-sustained within the Pacific sector or whether some external impacts are needed to maintain the oscillation. In the first category, the delayed oscillator scenario describes ENSO as an oscillation generated and maintained by the coupled instability and oceanic waves, without the need for any external impacts. In the second category, the system has two steady states of equilibrium and an external forcing is needed to move the system from one state to another. Recent observational analyses suggest possible interactions or connections between external influences and ENSO variability.

The effects of external impacts on ENSO variability are investigated here by using a simple coupled ocean–atmosphere model. The impacts considered are wind-stress anomalies associated with the seasonal monsoonal cycle, and the tropospheric quasi-biennial oscillation in the Indian and western Pacific region. It was found that 1) the external impact plays an important role in triggering ENSO variability when the coupled system in the Pacific could not support the oscillation by itself, 2) the impact regulates the original self-sustained oscillation to a seasonally phase-locked time evolution; and 3) the periods of the resulting oscillations could be three times that of the external forcing, a result of the interaction between the external forcing and the coupled system in the Pacific.

A modified version of the delayed oscillator equation was used to examine further details of the interaction. It was found that the match of half of the period of the external forcing with the delay time of the reflected oceanic waves from the western boundary arriving at the air–sea interaction region to turn off an event is a key factor in determining how they interact. If the time-matching condition is satisfied, the oscillation period will be three times that of the forcing. It is also shown that wind stress associated with the quasi-biennial oscillation could influence significantly the original self-sustained oscillation in the Pacific, making the amplitude and interval between two successive warm or cold phases variable, as observed in ENSO events.

## Abstract

Many features of the El Niño–Southern Oscillation (ENSO) phenomenon have been successfully simulated by coupled models during the last decade; however, some fundamental differences in model behavior remain. They can be classified into two categories according to whether the oscillation is self-sustained within the Pacific sector or whether some external impacts are needed to maintain the oscillation. In the first category, the delayed oscillator scenario describes ENSO as an oscillation generated and maintained by the coupled instability and oceanic waves, without the need for any external impacts. In the second category, the system has two steady states of equilibrium and an external forcing is needed to move the system from one state to another. Recent observational analyses suggest possible interactions or connections between external influences and ENSO variability.

The effects of external impacts on ENSO variability are investigated here by using a simple coupled ocean–atmosphere model. The impacts considered are wind-stress anomalies associated with the seasonal monsoonal cycle, and the tropospheric quasi-biennial oscillation in the Indian and western Pacific region. It was found that 1) the external impact plays an important role in triggering ENSO variability when the coupled system in the Pacific could not support the oscillation by itself, 2) the impact regulates the original self-sustained oscillation to a seasonally phase-locked time evolution; and 3) the periods of the resulting oscillations could be three times that of the external forcing, a result of the interaction between the external forcing and the coupled system in the Pacific.

A modified version of the delayed oscillator equation was used to examine further details of the interaction. It was found that the match of half of the period of the external forcing with the delay time of the reflected oceanic waves from the western boundary arriving at the air–sea interaction region to turn off an event is a key factor in determining how they interact. If the time-matching condition is satisfied, the oscillation period will be three times that of the forcing. It is also shown that wind stress associated with the quasi-biennial oscillation could influence significantly the original self-sustained oscillation in the Pacific, making the amplitude and interval between two successive warm or cold phases variable, as observed in ENSO events.

## Abstract

The effects of viscosity and finite- differencing on free Kelvin waves in numerical models (which employ the Arakawa *B*- or *C*-grid difference schemes) are investigated using the *f*-plane shallow-water equations with offshore finite-difference grids, (assuming alongshore geostrophy). Three nondimensional parameters arise: Δ [=(offshore grid spacing)/(Rossby radius)], ε characterizes the offshore lateral viscous effect and α the combined vertical and alongshore viscous effect. This study is more relevant to *baroclinic* Kelvin waves which tend to suffer poor offshore resolution because of their small Rossby radii.

For inviscid models (ε = α = 0), as Δ increases (resolution worsens), the alongshore speed increases dramatically in the *B*-grid, but stays constant at the gravity wave speed in the *C*-grid. Models with damping only (α > 0, ε = 0) behave similarly. With lateral viscosity (ε > 0, α > 0), increasing ε decreases the speed in both the *B*- and *C*-grids—the drop in speed being less severe when the free-slip boundary condition is imposed instead of the no-slip one. As Δ increases, the speed declines in the *B*-grid, but in the *C*-grid, worsening resolution cancels the viscous slow-down, with speed rising to that when ε = 0.

Our theory predicts the alongshore phase speed, the temporal decay rate and the offshore structure for *B*- and *C*-grid models of given viscosity and grid-spacing and of given boundary conditions (e.g., no-slip or free-slip). The predictions are checked against observations from two- and three-dimensional model—including the Bryan-Cox model—with good agreement.

## Abstract

The effects of viscosity and finite- differencing on free Kelvin waves in numerical models (which employ the Arakawa *B*- or *C*-grid difference schemes) are investigated using the *f*-plane shallow-water equations with offshore finite-difference grids, (assuming alongshore geostrophy). Three nondimensional parameters arise: Δ [=(offshore grid spacing)/(Rossby radius)], ε characterizes the offshore lateral viscous effect and α the combined vertical and alongshore viscous effect. This study is more relevant to *baroclinic* Kelvin waves which tend to suffer poor offshore resolution because of their small Rossby radii.

For inviscid models (ε = α = 0), as Δ increases (resolution worsens), the alongshore speed increases dramatically in the *B*-grid, but stays constant at the gravity wave speed in the *C*-grid. Models with damping only (α > 0, ε = 0) behave similarly. With lateral viscosity (ε > 0, α > 0), increasing ε decreases the speed in both the *B*- and *C*-grids—the drop in speed being less severe when the free-slip boundary condition is imposed instead of the no-slip one. As Δ increases, the speed declines in the *B*-grid, but in the *C*-grid, worsening resolution cancels the viscous slow-down, with speed rising to that when ε = 0.

Our theory predicts the alongshore phase speed, the temporal decay rate and the offshore structure for *B*- and *C*-grid models of given viscosity and grid-spacing and of given boundary conditions (e.g., no-slip or free-slip). The predictions are checked against observations from two- and three-dimensional model—including the Bryan-Cox model—with good agreement.

## Abstract

Free Kelvin wave solutions of the linear shallow-water equations are described, for an *f*-plane. Lateral and vertical viscous effects are represented by terms ν∇^{2}u and *d*u, respectively, where (*u*,*v*) is the (onshore, longshore) velocity. Both no-slip and free-slip boundary conditions are considered.

When ν = 0 and *d* = 0, the lognshore phase speed decreases as longshore wavelength increases. Decay time is independent of wavelength, so the shorter waves are more efficient at sending information alongshore. For ν = 0 and *d* = 0, speed still decreases with increasing wavelength, but ht longer waves decay more slowly, and the longshore decay distance is now largest for long waves. Several examples are given.

The wave properties are much less dependent on ν when free-slip rather than no-slip conditions are used.

The onshore velocity is nonzero when ν > 0. This property is used to estimate ν = 10^{3}–10^{4} m10^{2} s^{−1}, from previous observations of free baroclinic coastally-trapped waves off Peru.

Longshore geostrophy is a good approximation unless ν is large and wavelength is small. With longshore geostrophy wave properties can be found in terms of just two nondimensional parameters: ε, related to offshore viscous effects, and α, which combines vertical and alongshore viscous effects. Wave properties for a wide range of values of ε and α are given.

Effects of lateral and vertical diffusion can be added. With longshore geostrophy, the wave properties can be deduced by simply reinterpreting the parameter α.

## Abstract

Free Kelvin wave solutions of the linear shallow-water equations are described, for an *f*-plane. Lateral and vertical viscous effects are represented by terms ν∇^{2}u and *d*u, respectively, where (*u*,*v*) is the (onshore, longshore) velocity. Both no-slip and free-slip boundary conditions are considered.

When ν = 0 and *d* = 0, the lognshore phase speed decreases as longshore wavelength increases. Decay time is independent of wavelength, so the shorter waves are more efficient at sending information alongshore. For ν = 0 and *d* = 0, speed still decreases with increasing wavelength, but ht longer waves decay more slowly, and the longshore decay distance is now largest for long waves. Several examples are given.

The wave properties are much less dependent on ν when free-slip rather than no-slip conditions are used.

The onshore velocity is nonzero when ν > 0. This property is used to estimate ν = 10^{3}–10^{4} m10^{2} s^{−1}, from previous observations of free baroclinic coastally-trapped waves off Peru.

Longshore geostrophy is a good approximation unless ν is large and wavelength is small. With longshore geostrophy wave properties can be found in terms of just two nondimensional parameters: ε, related to offshore viscous effects, and α, which combines vertical and alongshore viscous effects. Wave properties for a wide range of values of ε and α are given.

Effects of lateral and vertical diffusion can be added. With longshore geostrophy, the wave properties can be deduced by simply reinterpreting the parameter α.