Supplemental data for this article is available online at

The role of planetary-scale zonally-asymmetric thermal forcing on large-scale atmospheric dynamics is crucial for understanding low-frequency phenomena in the atmosphere. Despite its paramount importance, good theoretical foundation for the understanding is still lacking. Here, we address this issue by providing a general framework for including planetary-scale thermal forcing in large-scale atmospheric dynamics studies. This is accomplished by identifying two distinct geostrophic motions of horizontal length scale _{D}

The effects of zonally-varying thermodynamic or orographic forcing on large-scale atmospheric motions have been a main topic of investigation in atmospheric dynamics studies. This is because of the forcing’s possible relationship to storm tracks (Chang et al.,

Zonally-

Traditionally, zonally-symmetric states have been used to understand large-scale, quasi-geostrophic (QG) dynamics at the mid-latitude. In particular, linearized QG equations have mainly been used for baroclinic instability, addressing the linear growth rate of large-scale atmospheric waves and associated poleward heat transfer (e.g. Charney,

Broadly, the large-scale, synoptic motion is driven by the temperature contrast between low and high latitudes, due to the latitudinal imbalance of radiative-convective thermal forcing. Hence, an often-used balanced initial condition in numerical simulations is one obtained by combining thermal wind balance and meridional radiative-convective equilibrium temperature distribution:
_{E}

The contribution of eddies to the shaping of large-scale atmospheric circulation is well understood in the zonally-symmetric case (see, e.g. Schneider,

Consider now the planetary scale, where

Recently, careful studies of the relationship between planetary and synoptic scales in PG motions have been undertaken by Dolaptchiev and Klein (

The basic outline of this paper is as follows. In

Large-scale atmospheric motions are characterized by

a value which is close to the observed value. Taking

giving
_{D}_{D}

As we shall show, the introduction of _{s}

Hence,

Hence, letting

Then, using the above scaling of

The construction of leading order equations can be carried out based on the relationship between the two small parameters, _{0} a representative mid-latitude. In QG scaling, since

Similarly, the non-dimensional momentum equation for the

For the vertical momentum equation, we have

Here, it can be readily seen that the leading order equation contains

The last is the heat equation, which could also be non-dimensionalized based on

Consider the case, _{D}_{0} and

Using the observed values for the Earth’s atmosphere,

giving

At

Combining the two horizontal momentum equations and the continuity equation, we obtain

The vertical velocity _{1} can be replaced by the leading-order variables from the equation. This leads to

Now consider the case, _{D}_{0} is no longer zero. Thus, the leading-order equations become

In

There are two different geostrophic motions in the large-scale atmosphere, which suggests that the large-scale atmospheric motions are shaped by the mutual interactions of the two scales. The two length scales _{D}

Under geostrophic balance, the pressure can be expanded as
_{D}_{L}_{L}_{L}

Hence, the horizontal momentum equations are as follows:
_{0}. Note that the ‘effect of

The continuity equation at the planetary scale should be considered carefully. The beta effect in the planetary scale is considered as a term in the leading order equation. The consideration of the beta effect will be shown in the planetary equation separately from the quasi-geostrophic equation. The continuity equation is as follows:

Finally, the heat equation is as follows:
_{L}

The planetary variables are not expanded because the first order is enough to close the equations in this case. The equations for

Note that the geostrophic and hydrostatic balances are satisfied for both scales.

The continuity equation at

The heat equation to this order is

Therefore, to this order, the planetary scale part is closed. However, the QG equations are not closed: higher order must be considered to close the QG equations.

The

These equations lead to the evolution equation for the relative vorticity,

Now, the equations for the two scales are closed. The two scales equally contribute to balance the heat flux with the forcing _{L}

The leading equations in the multi-scale analysis are as follows:

Decoupled from the QG scale, the planetary scale motion is in balance with the forcing _{L}_{L}_{L}

In terms of the planetary scale, we can time- and zonal-average (denoted by the overbar and bracket,

The planetary scale temperature Θ_{L}_{L}_{L}

We can view the interaction from the QG point of view, where the planetary scale provides a mean balanced state and the QG-scale eddies grow by baroclinic instability. The basic relationship in the planetary scale is given by _{L}

This equation is identical with the one used in the baroclinic instability problem with

As already shown, the planetary equations simplify to the local energy flux balance and the geostrophic and hydrostatic balances, when _{L}_{L}

The main evolution equation is the heat equation, which indicates that the difference between the forcing and the mean horizontal heat and vorticity convergence in QG scales is in balance with planetary-scale advection.

A balanced field with a given thermal forcing is given by the planetary geostrophic motion without the influence of the QG eddies. The equations for the balanced field are as follows:

The linearized PV equation for the QG scale with the given balanced field is

This equation represents a generalized baroclinic instability problem with a zonally asymmetric basic state. With the given forcing _{L}

The implication of PG equations can be elucidated by showing how to construct a balanced field with a non-zonal thermal forcing. With a given large-scale heat flux, a solution to the PG equations without the contribution of QG scales satisfies both the geostrophic and the hydrostatic balances. Such a solution could be used as an initial state for numerical simulations, for example, to study the development of QG eddies through baroclinic instability. Moreover, a long-term simulation would lead to a statistically-stationary state showing a balance between planetary-scale advection and eddy convergence in the QG scale. The construction of a balanced field from the PG equations with a given thermal forcing would reveal the relationship between thermal forcing and statistics of QG-scale eddies.

The PG equations mainly describe two major physical processes – the Sverdrup relationship and the forced heat equation:

The latter equation simply shows the local temporal change and the large-scale advection of potential temperature due to the planetary scale forcing. The interesting part is the Sverdrup relationship, which comes from the incompressible continuity in spherical geometry. It shows the relationship between vertical mass flux and meridional mass flux. The relationship could be represented in terms of the mass fluxes via vertical column integral,

An important explicit example of non-zonal balanced flow is
_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}

The velocity, temperature and forcing associated with the pressure field, _{L}_{L}_{L}

Another example describes a balanced state where zonal-asymmetry, originated mainly from surface thermal forcing, disappears with height. This is closer to a realistic atmosphere. The example is

This is constructed by the given pressure (or streamline) _{L}_{L}_{L}

Same as in

_{L}

The above two examples are the solutions of the planetary geostrophic motion without the influence of QG scales. The balanced state is the energy source for the growth of QG eddies through the baroclinic instability. Thus, identifying a balanced state is crucial for a physical interpretation of the statistics of geostrophic eddies in the atmosphere. For example, in the Northern hemisphere, land–sea contrast is one of the major factors controlling the spatial distribution of geostrophic eddies. Hence, it should be possible to initialize a numerical calculation of a balanced state with the thermodynamic forcing induced by the land–sea contrast, which could reveal the role of zonal asymmetry from the land–sea contrast on the baroclinic growth of synoptic waves and its feedback to the balanced state.

Large-scale atmospheric dynamics is controlled by processes of different scales and the interactions between them. Unravelling the interactions – and ultimately understanding the dynamics in full – requires a systematic approach. In this paper, we have used an asymptotic approach to identify two separate scales in large-scale atmospheric dynamics and to understand the mutual interactions between them. Multi-scale analysis, in particular, is used in spatial and temporal domains to elucidate the interactions.

The two scales are distinguished by noting how the thermodynamic variables are scaled by the Rossby number _{0}. Two important parameters in this work,

In this work, we have argued that the variation and structure of the large-scale atmospheric motion can be viewed in terms of the interaction of two geostrophic motions. One motion, the PG motion, can be interpreted as a balanced field directly forced by large-scale radiative-convective imbalance, as just noted, and the other motion, aptly descried by the familiar QG vorticity equation, governs the life cycle of synoptic waves that grow via the energy extracted from the balanced field.

The interaction is systematically revealed in a multi-scale analysis, in which the PG motion provides the mean field in the QG vorticity equation. The mean field provides the background condition, for example, for the growth of QG perturbations. It is important to note that baroclinic instability problems studies thus far have mainly employed zonally-symmetric flow with vertical shear represented often with a simple linear dependence with height. However, such a setup is not realistic. Our approach affords a generalisation of the traditional baroclinic instability study (cf.,

Another significant perspective addressed through the multi-scale analysis is the construction of a statistically-stationary large-scale flow – i.e. PG motion forced by both the large-scale radiative imbalance and the QG eddies. Averaging the QG variables in time domain shows the contribution of the QG motion to the slow time evolution of the PG motion. The time-averaged horizontal heat flux convergence and vertically-integrated relative vorticity flux convergence act as forcing in the planetary heat equation. Then, the PG motion is balanced by the radiative imbalance and the heat and vorticity flux convergences. The mutual interaction of the two scales discussed in this work extended the study by Dolaptchiev and Klein (

W.M. acknowledges a Herchel-Smith postdoctoral fellowship. This work was initiated at the 2015 Geophysical Fluid Dynamics Summer Study Program ‘Stochastic Processes in Atmospheric & Oceanic Dynamics’ at the Woods Hole Oceanographic Institution, which is supported by the National Science Foundation and the Office of Naval Research. W.M also acknowledges the support of Swedish Research Council grant no. 638-2013-9243. J.Y-K.C. acknowledges the hospitality of the Kavli Institute for Theoretical Physics, Santa Barbara and the Department of Astrophysical Sciences, Princeton University, where some of this work was completed. We are grateful to Joe Pedlosky for very helpful discussions and comments.

No potential conflict of interest was reported by the authors.