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## 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

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

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

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

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

Ocean heat uptake and the thermohaline circulation are analyzed in present-day control, 1% increasing CO_{2}, and doubled CO_{2} runs of the Community Climate System Model, version 2 (CCSM2). It is concluded that the observed 40-yr trend in the global heat content to 300 m, found by Levitus et al., is somewhat larger than the natural variability in the CCSM2 control run. The observed 40-yr trend in the global heat content down to a depth of 3 km is much closer to trends found in the control run and is not so clearly separated from the natural model variability. It is estimated that, in a 0.7% increasing CO_{2} scenario that approximates the effect of increasing greenhouse gases between 1958 and 1998, the CCSM2 40-yr trend in the global heat content to 300 m is about the same as the observed value. This gives support for the CCSM2 climate sensitivity, which is 2.2°C.

Both the maximum of the meridional overturning streamfunction and the vertical flow across 1-km depth between 60° and 65°N decrease monotonically during the 1% CO_{2} run. However, the reductions are quite modest, being 3 and 2 Sv, respectively, when CO_{2} has quadrupled. The reason for this is that the surface potential density in the northern North Atlantic decreases steadily throughout the 1% CO_{2} run. In the latter part of the doubled CO_{2} run, the meridional overturning streamfunction recovers in strength back toward its value in the control run, but the deep-water formation rate across 1-km depth between 60° and 65°N remains at 85% of the control run value. The maximum northward heat transport at 22°N is governed by the maximum of the overturning, but the transport poleward of 62°N appears to be independent of the deep-water formation rate.

## Abstract

Ocean heat uptake and the thermohaline circulation are analyzed in present-day control, 1% increasing CO_{2}, and doubled CO_{2} runs of the Community Climate System Model, version 2 (CCSM2). It is concluded that the observed 40-yr trend in the global heat content to 300 m, found by Levitus et al., is somewhat larger than the natural variability in the CCSM2 control run. The observed 40-yr trend in the global heat content down to a depth of 3 km is much closer to trends found in the control run and is not so clearly separated from the natural model variability. It is estimated that, in a 0.7% increasing CO_{2} scenario that approximates the effect of increasing greenhouse gases between 1958 and 1998, the CCSM2 40-yr trend in the global heat content to 300 m is about the same as the observed value. This gives support for the CCSM2 climate sensitivity, which is 2.2°C.

Both the maximum of the meridional overturning streamfunction and the vertical flow across 1-km depth between 60° and 65°N decrease monotonically during the 1% CO_{2} run. However, the reductions are quite modest, being 3 and 2 Sv, respectively, when CO_{2} has quadrupled. The reason for this is that the surface potential density in the northern North Atlantic decreases steadily throughout the 1% CO_{2} run. In the latter part of the doubled CO_{2} run, the meridional overturning streamfunction recovers in strength back toward its value in the control run, but the deep-water formation rate across 1-km depth between 60° and 65°N remains at 85% of the control run value. The maximum northward heat transport at 22°N is governed by the maximum of the overturning, but the transport poleward of 62°N appears to be independent of the deep-water formation rate.

## 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 trapped equatorial standing modes described theoretically by Gent (1979) are reproduced in a single vertical-mode numerical ocean model. integrations are carried out in domains whose longitudinal extents are characteristic of the widths of the Atlantic and Pacific Oceans, as well as in a narrow ocean in which the simplest possible standing mode can exist. The modes are shown to be very insensitive to small changes in basin width and to the inclusion of friction, and somewhat sensitive to the inclusion of the nonlinear terms and to rotation of the rectangular basin relative to the equator. Moreover, they arise spontaneously from simple atmospheric forcing or from random initial conditions. Typically, 20–40% of the energy input to the equatorial ocean remains trapped in a number of distinct standing modes after about nine years of integration time. The wider the ocean domain, the more energy remains trapped near the equator. These unexpected results have important implications for equatorial ocean dynamics and tropical air-sea interaction.

## Abstract

The trapped equatorial standing modes described theoretically by Gent (1979) are reproduced in a single vertical-mode numerical ocean model. integrations are carried out in domains whose longitudinal extents are characteristic of the widths of the Atlantic and Pacific Oceans, as well as in a narrow ocean in which the simplest possible standing mode can exist. The modes are shown to be very insensitive to small changes in basin width and to the inclusion of friction, and somewhat sensitive to the inclusion of the nonlinear terms and to rotation of the rectangular basin relative to the equator. Moreover, they arise spontaneously from simple atmospheric forcing or from random initial conditions. Typically, 20–40% of the energy input to the equatorial ocean remains trapped in a number of distinct standing modes after about nine years of integration time. The wider the ocean domain, the more energy remains trapped near the equator. These unexpected results have important implications for equatorial ocean dynamics and tropical air-sea interaction.