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## Abstract

In this paper the linear equatorial ocean response to stress forcing is analyzed in terms of vertically propagating waves. A new projection onto the meridional eigenfunctions of the pressure equation is derived for a single Fourier wave component. The projection demonstrates that the solution is regular and not singular at the inertial latitudes, and is more convenient to use than the corresponding projection onto the meridional velocity equation. The wavenumber spectrum from the resulting forced vertical structure equation is found for four different choices of the vertical profile for the body force. The spectrum is shown to be insensitive to the particular profile chosen. The projection is then used to study the effects of forcing and linear damping on the vertical propagation of space-time transformed energy in three wave modes: the Kelvin, first Rossby and mixed Rossby-gravity waves. When the buoyancy frequency is constant, the energy decay is exponential in depth with the coefficient proportional to the damping magnitude. Finally it is shown that linear damping effects are very different on each vertically propagating or vertically standing wave. Thus, it is fallacious to make deductions about meridional phase changes in the total solution to a general forced problem from the phase changes of each wave component.

## Abstract

In this paper the linear equatorial ocean response to stress forcing is analyzed in terms of vertically propagating waves. A new projection onto the meridional eigenfunctions of the pressure equation is derived for a single Fourier wave component. The projection demonstrates that the solution is regular and not singular at the inertial latitudes, and is more convenient to use than the corresponding projection onto the meridional velocity equation. The wavenumber spectrum from the resulting forced vertical structure equation is found for four different choices of the vertical profile for the body force. The spectrum is shown to be insensitive to the particular profile chosen. The projection is then used to study the effects of forcing and linear damping on the vertical propagation of space-time transformed energy in three wave modes: the Kelvin, first Rossby and mixed Rossby-gravity waves. When the buoyancy frequency is constant, the energy decay is exponential in depth with the coefficient proportional to the damping magnitude. Finally it is shown that linear damping effects are very different on each vertically propagating or vertically standing wave. Thus, it is fallacious to make deductions about meridional phase changes in the total solution to a general forced problem from the phase changes of each wave component.

## Abstract

A new type of standing equatorial wave mode is described that exists in the semi-infinite ocean 0 â©½ *x* â©½ *L*, âˆ’âˆž â©½ *y* â©½ âˆž. It consists of a *finite* sum of the meridionally trapped equatorial waves in an infinite *x* domain. The new mode is thus itself equatorially trapped and requires no energy sources or sinks at |*y*| = âˆž. However, it exists only for a discrete, countable set of pairs of values of the frequency Ï‰ and the ocean zonal width *L*. Previously described standing modes exist for any ocean width, but are *infinite* sums of trapped equatorial waves and require a continuous energy source in the west at |*y*| = âˆž to balance the continuous energy sink in the east at |*y*| = âˆž. Several examples of the new type of standing mode are given, and it is shown that as the standing mode period becomes very long, so the zonal scale becomes very short. The effect on the standing modes of bounding the basin meridionally is also described; energy is recycled round the basin by boundary-trapped Kelvin waves along the zonal walls. The amount of energy recycled in the new type of standing mode, however, is exponentially small compared to that recycled in the previously described standing modes.

## Abstract

A new type of standing equatorial wave mode is described that exists in the semi-infinite ocean 0 â©½ *x* â©½ *L*, âˆ’âˆž â©½ *y* â©½ âˆž. It consists of a *finite* sum of the meridionally trapped equatorial waves in an infinite *x* domain. The new mode is thus itself equatorially trapped and requires no energy sources or sinks at |*y*| = âˆž. However, it exists only for a discrete, countable set of pairs of values of the frequency Ï‰ and the ocean zonal width *L*. Previously described standing modes exist for any ocean width, but are *infinite* sums of trapped equatorial waves and require a continuous energy source in the west at |*y*| = âˆž to balance the continuous energy sink in the east at |*y*| = âˆž. Several examples of the new type of standing mode are given, and it is shown that as the standing mode period becomes very long, so the zonal scale becomes very short. The effect on the standing modes of bounding the basin meridionally is also described; energy is recycled round the basin by boundary-trapped Kelvin waves along the zonal walls. The amount of energy recycled in the new type of standing mode, however, is exponentially small compared to that recycled in the previously described standing modes.

## Abstract

Vertically propagating linear wave calculations using realistic equatorial buoyancy profiles are presented which show the percentage of the downward surface energy flux that reaches the deep equatorial oceans. The percentages vary widely depending upon the buoyancy profile and the equivalent depth but can be as low as 10% on average for equivalent depths between 1 cm and 1 m if the thermocline is sharp. This means that models with constant or weak thermocline buoyancy profiles, which allow all or most downward surface energy flux to reach the deep ocean, are very unrealistic in this respect. Another conclusion is that the observed, very low-frequency, small vertical-scale deep jets cannot be explained by linear wave theory as caused by surface forcing. It is also shown that a WKB analysis of observations can be misleading even if applied to a single vertically propagating wave in a region that excludes the main thermocline. Implications are that comparing estimates of the equivalent depth from the mixed Rossby-gravity wave dispersion relation and a WKB analysis is of little value because the error bars on both estimates are large, and that WKB estimates of downward vertical energy flux into the deep ocean can also be misleading.

## Abstract

Vertically propagating linear wave calculations using realistic equatorial buoyancy profiles are presented which show the percentage of the downward surface energy flux that reaches the deep equatorial oceans. The percentages vary widely depending upon the buoyancy profile and the equivalent depth but can be as low as 10% on average for equivalent depths between 1 cm and 1 m if the thermocline is sharp. This means that models with constant or weak thermocline buoyancy profiles, which allow all or most downward surface energy flux to reach the deep ocean, are very unrealistic in this respect. Another conclusion is that the observed, very low-frequency, small vertical-scale deep jets cannot be explained by linear wave theory as caused by surface forcing. It is also shown that a WKB analysis of observations can be misleading even if applied to a single vertically propagating wave in a region that excludes the main thermocline. Implications are that comparing estimates of the equivalent depth from the mixed Rossby-gravity wave dispersion relation and a WKB analysis is of little value because the error bars on both estimates are large, and that WKB estimates of downward vertical energy flux into the deep ocean can also be misleading.

## Abstract

The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.

## Abstract

The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.

## Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskinsâ€™ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

## Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskinsâ€™ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

## Abstract

A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9â€“16 of the simulation there is a dominant oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.

## Abstract

A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9â€“16 of the simulation there is a dominant oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.

## Abstract

An accurate diagnosis of ocean heat content (OHC) is essential for interpreting climate variability and change, as evidenced for example by the broad range of hypotheses that exists for explaining the recent hiatus in global mean surface warming. Potential insights are explored here by examining relationships between OHC and sea surface height (SSH) in observations and two recently available large ensembles of climate model simulations from the mid-twentieth century to 2100. It is found that in decadal-length observations and a model control simulation with constant forcing, strong ties between OHC and SSH exist, with little temporal or spatial complexity. Agreement is particularly strong on monthly to interannual time scales. In contrast, in forced transient warming simulations, important dependencies in the relationship exist as a function of region and time scale. Near Antarctica, low-frequency SSH variability is driven mainly by changes in the circumpolar current associated with intensified surface winds, leading to correlations between OHC and SSH that are weak and sometimes negative. In subtropical regions, and near other coastal boundaries, negative correlations are also evident on long time scales and are associated with the accumulated effects of changes in the water cycle and ocean dynamics that underlie complexity in the OHC relationship to SSH. Low-frequency variability in observations is found to exhibit similar negative correlations. Combined with altimeter data, these results provide evidence that SSH increases in the Indian and western Pacific Oceans during the hiatus are suggestive of substantial OHC increases. Methods for developing the applicability of altimetry as a constraint on OHC more generally are also discussed.

## Abstract

An accurate diagnosis of ocean heat content (OHC) is essential for interpreting climate variability and change, as evidenced for example by the broad range of hypotheses that exists for explaining the recent hiatus in global mean surface warming. Potential insights are explored here by examining relationships between OHC and sea surface height (SSH) in observations and two recently available large ensembles of climate model simulations from the mid-twentieth century to 2100. It is found that in decadal-length observations and a model control simulation with constant forcing, strong ties between OHC and SSH exist, with little temporal or spatial complexity. Agreement is particularly strong on monthly to interannual time scales. In contrast, in forced transient warming simulations, important dependencies in the relationship exist as a function of region and time scale. Near Antarctica, low-frequency SSH variability is driven mainly by changes in the circumpolar current associated with intensified surface winds, leading to correlations between OHC and SSH that are weak and sometimes negative. In subtropical regions, and near other coastal boundaries, negative correlations are also evident on long time scales and are associated with the accumulated effects of changes in the water cycle and ocean dynamics that underlie complexity in the OHC relationship to SSH. Low-frequency variability in observations is found to exhibit similar negative correlations. Combined with altimeter data, these results provide evidence that SSH increases in the Indian and western Pacific Oceans during the hiatus are suggestive of substantial OHC increases. Methods for developing the applicability of altimetry as a constraint on OHC more generally are also discussed.

## Abstract

Results from two perturbation experiments using the Community Climate System Model version 4 where the Southern Hemisphere zonal wind stress is increased are described. It is shown that the ocean response is in accord with experiments using much-higher-resolution ocean models that do not use an eddy parameterization. The key to obtaining an appropriate response in the coarse-resolution climate model is to specify a variable coefficient in the Gent and McWilliams eddy parameterization, rather than a constant value. This result contrasts with several recent papers that have suggested that coarse-resolution climate models cannot obtain an appropriate response.

## Abstract

Results from two perturbation experiments using the Community Climate System Model version 4 where the Southern Hemisphere zonal wind stress is increased are described. It is shown that the ocean response is in accord with experiments using much-higher-resolution ocean models that do not use an eddy parameterization. The key to obtaining an appropriate response in the coarse-resolution climate model is to specify a variable coefficient in the Gent and McWilliams eddy parameterization, rather than a constant value. This result contrasts with several recent papers that have suggested that coarse-resolution climate models cannot obtain an appropriate response.

## Abstract

The NCAR Climate System Model, version one, is described. The spinup procedure prior to a fully coupled integration is discussed. The fully coupled model has been run for 300 yr with no surface flux corrections in momentum, heat, or freshwater. There is virtually no trend in the surface temperatures over the 300 yr, although there are significant trends in other model fields, especially in the deep ocean. The reasons for the successful integration with no surface temperature trend are discussed.

## Abstract

The NCAR Climate System Model, version one, is described. The spinup procedure prior to a fully coupled integration is discussed. The fully coupled model has been run for 300 yr with no surface flux corrections in momentum, heat, or freshwater. There is virtually no trend in the surface temperatures over the 300 yr, although there are significant trends in other model fields, especially in the deep ocean. The reasons for the successful integration with no surface temperature trend are discussed.