This thesis work is an investigation of the interaction between large-scale eddies and the mean climate in a multi-level quasi-geostrophic beta-plane channel model. This study focuses on two aspects of this interaction. These aspects are the maintenance of the temperature structure near the lower boundary and the homogenization of potential vorticity in the mid-troposphere.

a. Dynamics at the lower boundary

The first question that is addressed in this process model study is what are the processes and feedbacks that are responsible for the observed temperature structure and eddy fluxes near the lower boundary? This question is motivated by the observation that heat fluxes in GCM's tend to be larger than observed heat fluxes even though the modeled temperature gradients are close to observations ( Stone and Risbey 1990; Gleckler et al. 1995). It has been proposed that this is due to inadequate representations of the processes and feedbacks that are responsible for the maintenance of the vertical temperature structure near the lower boundary (Stone and Nemet 1996). Three main studies are highlighted in this investigation of the maintenance of the temperature structure near the lower boundary. They are the impact of vertical resolution in the boundary layer, the importance of the feedback between vertical eddy heat fluxes and the static stability and thirdly, the sensitivity of the temperature structure to changes in the surface drag and eddy diffusion coefficients that define the boundary layer.

Two-level quasi-geostrophic and primitive equation models have been used to investigate the nonlinear equilibration of a baroclinically unstable basic state in a host of studies ( Held and Suarez 1978; Stone and Branscome 1991; Zhou and Stone 1992a; Zhou and Stone 1992b; Cehelsky and Tung 1991). These models of baroclinic instability generally assume that large scale waves, which are resolved by the two-level geometry, are primarily responsible for the modification of the mean temperature structure. This conclusion has typically been based on the scaling analysis of Held (1978a) which shows that the shallow waves are relatively inefficient in transporting heat.

Can two-level models, some of which are mentioned above, simulate the eddy-mean flow interaction in the troposphere? Observed annual mean midlatitude sensible heat fluxes in both hemispheres peak at 850 mb, become small in the mid troposphere and have a secondary maximum in the upper troposphere (Trenberth 1992). Therefore, modification of climate forcing in limited regions of the troposphere may have different eddy responses. This problem has been investigated in the context of greenhouse warming, where an increased concentration of greenhouse gases leads to smaller lower tropospheric temperature gradients but larger tropospheric gradients than does the present climate (Manabe and Wetherald 1975), by studies which used multi-level quasi-geostrophic models, such as Held and O'Brien (1992) and Pavan (1996). These studies found that the dynamics in a multi-level quasi-geostrophic model are more sensitive to changes in the lower tropospheric temperature gradients than to changes in the upper tropospheric gradients. It is hypothesized that this is due to the importance of shallow eddies which are more sensitive to changes in the vertical profile of the temperature structure than the deeper eddies which are only sensitive to mean tropospheric fields. These studies support the claim that two-level model studies can simulate eddy-mean flow interaction in a vertically averaged sense, but they also suggest that any simulation of the observed troposphere must be able to resolve the shallow waves that modify the vertical distribution of the eddy and mean fields. Therefore, it may be necessary to resolve these shallow eddies in order to simulate the eddy response to more realistic changes in climate forcing.

The concentration of sensible poleward heat fluxes above the planetary boundary layer in observations suggests that processes near the Earth's surface inhibit eddy heat transport in the regions where the source of available potential energy for the growth of the wave is a maximum. The dominant process that is responsible for the reduction in the eddy heat transports at the surface is surface friction. The role of surface friction in modifying the characteristics of the baroclinic waves has been studied by Valdes and Hoskins (1988) and Lin and Pierrehumbert (1988). These studies used linearized quasi-geostrophic and primitive equation models to demonstrate that Ekman pumping damps out the shallow waves at the top of the boundary layer, leaving the large scale waves to modify the baroclinically unstable basic state. This result has been disputed by Farrell (1985) who argues that, using realistic values of the eddy diffusion coefficient, Ekman damping damps out all normal mode baroclinic waves. Farrell argues that it is non normal modes that are responsible for the eddy heat transport that is observed in the midlatitude lower troposphere. Whether or not Ekman pumping completely stabilizes the normal mode baroclinic waves in the troposphere, it is clear that the results are very sensitive to the parameters that are used to define the boundary layer.

Gutowski et al. (1989) studied the role of surface friction and surface heat fluxes in modifying the eddy-mean flow interaction. Their study showed that there was an important negative feedback between the surface heat fluxes and the vertical eddy heat fluxes. When the surface heat fluxes were removed, the vertical eddy heat fluxes caused large increases in the static stability at the lower boundary. Surface heat fluxes counteract this large increase in the static stability by downgradient mixing of potential temperature. The reduction of the static stability in the boundary layer causes the boundary layer to be more baroclinically unstable which causes the large scale eddies to transport more heat poleward, reducing the meridional temperature gradients. In order to understand the dynamics that are responsible for the observed distribution of heat fluxes and mean temperatures, it is necessary to study the feedbacks between these processes and the equilibrated state that is determined by them.

These feedbacks, which are neglected in classic quasi-geostrophic models, are also important in the sensitivity of the temperature structure to changes in climate forcing. Studies of baroclinic adjustment which include variable static stability have demonstrated that the negative feedback between the vertical eddy heat flux and the static stability plays a significant role in reducing the sensitivity of the model's climate to changes in external forcing. In models with fixed static stability, the equilibrium shear is found to be linearly proportional to the static stability and relatively insensitive to the changes in external forcing ( Stone and Branscome 1991; Zhou and Stone 1992a). In two-level models that include a variable static stability, the sensitivity of the climate to changes in external forcing was reduced (Held and Suarez 1978; Held 1978a; Zhou and Stone 1992b), demonstrating the significance of the feedbacks between the dynamical heat fluxes in stabilizing the climate, as suggested by Stone (1972).

Increasing vertical resolution may modify the characteristics of the baroclinic waves by resolving the potential vorticity gradients more accurately. Previous studies of the barotropic point jet, which is mathematically homomorphic to the linearized Boussinesq Charney problem ( Lindzen et al. 1983) , have shown that the stability characteristics of a baroclinically unstable flow are a function of the vertical resolution. An increase in the vertical resolution resolves the delta function jump in the potential vorticity at the lower boundary of the model, making the model more baroclinically unstable. A decrease in the vertical resolution decreases the magnitude of the jump in the potential vorticity at the lower boundary, stabilizing the model. It is not clear how this result translates to the full three-dimensional quasi-geostrophic model, but it suggests that two-level models may not be simulating the degree of the instability in the troposphere accurately. Therefore, an increase in the vertical resolution may have a significant impact on the stability characteristics of the baroclinic waves.

b. Homogenization of potential vorticity

The second question that is addressed in this study is what is the extent of the pv homogenization in a multi-level quasi-geostrophic beta plane channel model and how does this relate to current theories about the nonlinear equilibration of the observed mid-latitude troposphere? This question is motivated by the study of Lindzen (1993) which hypothesized that a possible simplistic equilibrated state which describes the observed mid-latitude troposphere is a state where the pv gradients in the mid troposphere have been homogenized such that waves propagating along the tropopause and the surface of the earth can no longer interact.

Adiabatic and inviscid process model studies have demonstrated that linear adjustment theory predicts the equilibrated state of an initially unstable flow, but these model studies have had little success in simulating the observed climate. Model studies that include simple representations of diabatic heating and surface friction have been relatively successful in simulating the observed large scale dynamics (Held and Suarez 1978; Stone and Branscome 1991). These two-level model studies attempt to relate the climate of a forced/dissipated model to the climate predicted by linear/inviscid/adiabatic criteria. It is not clear that this is justified.

The Charney-Stern criterion for instability is a necessary condition for instability but it is not sufficient. A basic state which has a change of sign in the meridional gradient of potential vorticity may be stable. A linear stability analysis of the equilibrated state proposed by Lindzen (1993) would show that the baroclinic normal modes have been stabilized but this state would still satisfy the Charney-Stern criteria necessary condition for instability.

The equilibrated state proposed by Lindzen (1993) is consistent with the observational studies of Sun and Lindzen (1994), Morgan (1994), and Fullmer (1982b) which have shown that the potential vorticity gradients in the extratropics are significantly less than $\beta$, the meridional gradient of planetary vorticity, in the mid troposphere but large in the planetary boundary layer and a maximum at the tropopause. These studies make it clear that the observed extratropical temperature structure is determined, in part, by the mixing of potential vorticity along isentropic surfaces.

In order to resolve this state it is necessary to be able to decouple the dynamics between the surface, the mid troposphere and the tropopause. A two-level model couples the dynamics in the lower troposphere with the upper troposphere. This causes the dynamics at the jet and the dynamics in the boundary layer to interact in a very artificial way. If this equilibrated state approximates the dynamics in the atmosphere then the equilibrated state of the two-level model will have very little to do with the observed climate. If the dynamics in the observed climate are roughly approximated by the state hypothesized by Lindzen (1993), a model with more than two levels would be necessary to resolve it.

There have been many studies done concerning the homogenization of pv in subtropical ocean gyres. Fundamental studies by Rhines and Young, RY from here on, (Rhines and Young 1982a,b; Young and Rhines 1982) have demonstrated that there is very efficient homogenization of pv in layers that are not directly in contact with the wind-driven Ekman pumping at the ocean's surface. These studies were done using simple quasi-geostrophic layer models which neglected cross interface motion. Neglecting cross interface motion means that diabatic processes, that change the density in a layer, are neglected. In models of the atmosphere, neglecting these processes result in a climate which bears very little resemblance to observations (Gutowski et al. 1989). Therefore, model's which neglect these processes may not be very useful in understanding the wave-mean flow interaction in the atmosphere

In subtropical ocean gyres, geostrophic streamfunctions which emanate from the eastern boundary have streamfunctions which are equal to zero, due to the Sverdrup streamfunction which vanishes at the easten boundary. Therefore, in order for there to be any circulation along a geostrophic contour, the contour must be disconnected from the eastern boundary. Rhines and Young chose an Ekman pumping velocity such that this condition is satisfied. The Ekman pumping velocity needed to close the geostophic contours is equivalent to the condition that the progress of a westward propagating baroclinic Rossby wave, with a characteristic horizontal velocity which exceeds $\beta L_d^2$, has been arrested (Pedlosky 1996), where $L_d$ is Rossby deformation radius. In the RY studies, pv which is enclosed by a closed geostrophic contour, in layers below the directly forced upper layer, become well-mixed as long as the dissipation is sufficiently small. After the pv becomes homogenized, the fluxes which were reponsible for the mixing, decay away.

This scenario of mixing in a subtropical ocean gyre is significantly different from mixing in the atmosphere. In the atmosphere, closed geostrophic contours are a product of wave-mean flow interaction and are not directly forced independent of the eddies. Also, heating and friction in the atmosphere are continually acting to destabilize the eddies. Therefore, the efficient mixing of pv, that was found in subtropical ocean gyres, would not be expected in the atmosphere. In the atmosphere, wave-mean flow interaction would be expected to occur in regions of strong damping ( McIntyre and Norton 1990) and near the critical level ( Stewartson 1978, Warn and Warn 1978, Killworth and McIntyre 1985). In the equilibrated state, eddy forcings would be expected to continually act against the forcings of external heating and dissipation in life cycles of 'saturation-propagation-saturation' (Edmon et al. 1980, Hoskins 1983, Hoskins et al. 1985, Held and Hoskins 1985, Randel and Held 1991).