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## Abstract

It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.

## Abstract

It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.

## 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 ⩽ *x* ⩽ *L*, −∞ ⩽ *y* ⩽ ∞. 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*| = ∞. However, it exists only for a discrete, countable set of pairs of values of the frequency ω 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*| = ∞ to balance the continuous energy sink in the east at |*y*| = ∞. 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 ⩽ *x* ⩽ *L*, −∞ ⩽ *y* ⩽ ∞. 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*| = ∞. However, it exists only for a discrete, countable set of pairs of values of the frequency ω 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*| = ∞ to balance the continuous energy sink in the east at |*y*| = ∞. 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 Hoskins’ (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 Hoskins’ (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

The equilibrium climate sensitivity of a climate model is usually defined as the globally averaged equilibrium surface temperature response to a doubling of carbon dioxide. This is virtually always estimated in a version with a slab model for the upper ocean. The question is whether this estimate is accurate for the full climate model version, which includes a full-depth ocean component. This question has been answered for the low-resolution version of the Community Climate System Model, version 3 (CCSM3). The answer is that the equilibrium climate sensitivity using the full-depth ocean model is 0.14°C higher than that using the slab ocean model, which is a small increase. In addition, these sensitivity estimates have a standard deviation of nearly 0.1°C because of interannual variability. These results indicate that the standard practice of using a slab ocean model does give a good estimate of the equilibrium climate sensitivity of the full CCSM3. Another question addressed is whether the effective climate sensitivity is an accurate estimate of the equilibrium climate sensitivity. Again the answer is yes, provided that at least 150 yr of data from the doubled carbon dioxide run are used.

## Abstract

The equilibrium climate sensitivity of a climate model is usually defined as the globally averaged equilibrium surface temperature response to a doubling of carbon dioxide. This is virtually always estimated in a version with a slab model for the upper ocean. The question is whether this estimate is accurate for the full climate model version, which includes a full-depth ocean component. This question has been answered for the low-resolution version of the Community Climate System Model, version 3 (CCSM3). The answer is that the equilibrium climate sensitivity using the full-depth ocean model is 0.14°C higher than that using the slab ocean model, which is a small increase. In addition, these sensitivity estimates have a standard deviation of nearly 0.1°C because of interannual variability. These results indicate that the standard practice of using a slab ocean model does give a good estimate of the equilibrium climate sensitivity of the full CCSM3. Another question addressed is whether the effective climate sensitivity is an accurate estimate of the equilibrium climate sensitivity. Again the answer is yes, provided that at least 150 yr of data from the doubled carbon dioxide run are used.

## 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 9–16 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 9–16 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.