Growing interfacially coupled oscillations of the ocean–atmosphere

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Abstract

A stability analysis of the coupled ocean–atmosphere is presented which shows that the potential energy (PE) of the upper layer of the ocean is available to generate coupled growing planetary waves. An independent analysis suggests that the growth of these waves would be maintained in the presence of oceanic friction. The growing waves are a consequence of relaxing the rigid lid approximation on the ocean, thus allowing an upward transfer of energy across the sea surface. Using a two and a half layer model consisting of an atmospheric planetary boundary layer, coupled with a two layer ocean comprising an active upper layer and a lower layer in which the velocity perturbation is vanishingly small, it is shown that coupled unstable waves are generated, which extract PE from the main thermocline. The instability analysis is an extension of earlier work [Tellus 44A (1992) 67], which considered the coupled instability of an atmospheric planetary boundary layer coupled with an oceanic mixed layer, in which unstable waves were generated which extract PE from the seasonal thermocline. The unstable wave is an atmospheric divergent barotropic Rossby wave, which is steered by the zonal wind velocity, and has a wavelength of about 6000 km, and propagates eastward at the speed of the deep ocean current. It is argued that this instability, which has a multidecadal growth time constant, may be generated in the Southern Ocean, and that its properties are similar to observations of the Antarctic Circumpolar Wave (ACW).

Introduction

The aim of this paper is to show that it is plausible for the coupled ocean–atmosphere to generate long period oscillations, which can be sustained in the presence of oceanic friction, and which have as their energy source, the release of potential energy (PE) from the ocean (Bye, 2003). The rate of supply of energy to the growing waves can be expressed in terms of the rate of working of the interfacial stress at the sea surface (which is the correlation between the atmospheric pressure and the sea surface slope). In the main body of the paper (Section 3), an instability analysis is presented which shows how this interfacial stress may be generated by the baroclinicity of the ocean. In essence, the results follow from the relaxation of the rigid lid approximation (which precludes the interfacial stress) in the instability analysis, thus allowing an upward flow of energy from the ocean to the atmosphere. The flow of energy downwards out of the ocean through form stress on the sea bottom is well known, and is of particular importance in deeply penetrating currents such as the Antarctic Circumpolar Current. The interfacial stress discussed in this paper arises from the same dynamics applied to the water column, in which the negative integrated pressure gradient is expressed, using Liebnitz’s rule, in the form,−Hη∂p∂xdz=−∂x−Hηpdz+pa∂η∂x+pB∂H∂xwhere 0x is towards the east, 0y is towards the north, z is vertically upwards, p is pressure, pa and pB are respectively the atmospheric and bottom pressure, H is the undisturbed ocean depth, and η is the surface displacement.

In Eq. (1), the last two terms are respectively the interfacial stress (at the sea surface) and the form stress (at the sea bottom). In most analyses, the former term is usually neglected owing to its small magnitude in comparison with the latter term, and also in comparison with the surface shear stress due to vertical frictional adjustment. In concept, however, the interfacial stress, which can operate on any scale, is a component of the total surface shear stress. For example, it is highly significant in the generation of wind waves. We show below that it can also act on a large scale, applicable to climate variability. In Section 2, we provide an independent analysis, which suggests that the unstable waves may continue to grow in the presence of oceanic friction.

The instability processes which act on the general circulations of the atmosphere and ocean separately are well known (Gill, 1982). Baroclinic instability in the atmosphere gives rise to growing Rossby waves which evolve into our weather systems (Eady, 1949), whereas in the ocean, a similar mechanism generates mesoscale ocean eddies on a time scale of months (Holland, 1978).

The joint instability of the coupled ocean–atmosphere in which the PE of the mean flow is converted into the kinetic energy (KE) of growing waves however has received scant attention. Our intention is to derive some basic properties of this mode of interaction for the atmosphere and the ocean. The analysis is an extension of earlier work for a barotropic atmosphere coupled to a barotropic ocean over a meridional sloping bottom (Bye, 1992), in which the ocean component is replaced by a one and a half layer ocean model supported on a constant deep zonal flow.

The new model, which uses the one and a half layer baroclinic structure of the oceanic instability analysis, predicts that the removal of the rigid lid at the sea surface gives rise to an additional suite of unstable baroclinic modes, which are coupled with the atmosphere. These modes are of much longer wavelength than the mesoscale eddies of the rigid lid analysis. The results of this model are presented in Section 3.

The earlier model predicted the existence of a suite of unstable oceanic barotropic modes, which are also coupled with the atmosphere. These modes are of somewhat longer wavelength than in the barotropic spectrum, which is produced by the progression of the synoptic systems. The results of this model are summarized in Section 4.

The essential feature of both models is an upward transfer of energy from the ocean to the atmosphere, released from either the main or the seasonal thermocline. The two models, which, for brevity, are called respectively, the baroclinic model and the mixed layer model, are compared in Section 5, where possible applications of the analysis to the climate system are also considered.

Section snippets

Interfacially coupled zonal ocean–atmosphere oscillations

In order to introduce the concept that the interfacial stress at the sea surface can sustain oscillations which have as their energy source, the release of PE from the ocean, we will consider a solution of the shallow water equations applied to a constant depth ocean in which the motion is forced by a zonally propagating atmospheric pressure disturbance. The governing equations are,∂u∂t−fv=−1ρ∂pa∂x−g∂η∂x−Ru∂v∂t+fu=−1ρ∂pa∂y−g∂η∂y−Rv∂η∂t+H∂u∂x+∂v∂y=0in which (u, v) are the components of velocity

The two and a half layer baroclinic instability of the coupled ocean–atmosphere (the baroclinic model)

Consider the two and a half layer coupled ocean–atmosphere shown in Fig. 1, and suppose that there exist constant zonal velocities, Ui, in the three layers (i=1,3), and that infinitesimal perturbations of meridional geostrophic velocity, vi(i=1,2) can occur in the atmospheric and oceanic upper layers (the two active layers) together with infinitesimal displacements of the layer surfaces, such that the streamfunctions for the two active layers are,ψi=−Ui(y−y0)+φieik(x−Ct)where φi are the

The baroclinic instability for a one layer atmosphere coupled with a topographic ocean (the mixed layer model)

The corresponding results for the topographic ocean, coupled with a one layer atmosphere (Fig. 2), in which the displacement of the oceanic lower layer is,η3=s(y−y0)where s is the meridional slope of the interface, were originally presented in an extended analysis in Bye (1992). Alternatively, they can be derived using the methods for the baroclinic model in Section 3, with,β2=β+f0sH2andγ=1For this model, the potential vorticities of the two layers reduce to the approximate expressions,P1∼βandP2

Application to the climate system

The important question is whether the instabilities discussed in 3 The two and a half layer baroclinic instability of the coupled ocean–atmosphere (the baroclinic model), 4 The baroclinic instability for a one layer atmosphere coupled with a topographic ocean (the mixed layer model) play any role in the coupled ocean–atmosphere dynamics. Table 1 shows the predicted properties of the unstable waves for the two models, for H1=1 km (which is representative of the depth of the atmospheric planetary

Conclusion

The main thrust of our analysis has been to demonstrate that an upward transfer of energy (apart from heat energy) between the ocean and the atmosphere can occur through the rate of working of the interfacial stress at the sea surface. We have presented a simple analysis which shows this upward energy transfer is generated through the instability of the coupled ocean–atmosphere, which is a generic process, akin to the baroclinic instability of the atmosphere and ocean separately. The inviscid

Acknowledgements

The paper was completed during a Fellowship at the Hanse Institute for Advanced Study in Delmenhorst, Germany in July and August 2003. The comments of a reviewer are also gratefully acknowledged.

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