## 1. Introduction

The interaction between jet variability and eddies is a long-studied topic, but the interaction is not yet understood well enough to identify causal mechanisms for variability or sources of systematic errors in models. There are well-developed theoretical frameworks for the zonally homogeneous case (e.g., annular-mode variability); however, zonally asymmetric analyses including planetary-scale interactions are more complicated, and only partial theories for this case exist (Hoskins et al. 1983; Plumb 1985, 1986). Yet longitudinal variations and synoptic–planetary-scale interactions are important for the location and strength of the storm tracks and blocking episodes (Hoskins et al. 1983; Luo 2005; Simpson et al. 2014). These phenomena strongly affect the regional climate and its climate change. As the dynamical aspects of climate are not yet well understood, there is low confidence in circulation patterns simulated by global and regional models and their response to climate change (Shepherd 2014).

An important aspect of wave–mean flow interaction concerns barotropic and baroclinic processes and their links through eddy momentum and heat fluxes. It has recently been shown from observations for the southern and northern annular modes in Thompson and Woodworth (2014) and Thompson and Li (2015) that the zonal mean flow is affected only by momentum fluxes and not by heat fluxes, while the opposite is true for a so-called baroclinic annular mode (BAM) that is based on eddy kinetic energy (EKE). This decoupling goes against the usual transformed Eulerian mean (TEM) perspective, first introduced by Andrews and McIntyre (1976), within which both heat and momentum fluxes affect the zonal-mean-flow tendency through the Eliassen–Palm (EP) flux divergence. The decoupling was further investigated in Thompson and Barnes (2014), who found an oscillating relationship between EKE and heat flux with time periods of 20–30 days. A similar relationship was found between wave activity and heat flux in Wang and Nakamura (2015, 2016).

To derive a theoretical framework for understanding planetary–synoptic-scale interactions and the apparent decoupling of the baroclinic and barotropic parts of the flow, we use multiscale asymptotic methods as introduced in Dolaptchiev and Klein (2009, 2013, hereafter DK09 and DK13, respectively). This approach is taken as such methods provide a self-consistent (albeit idealized) framework for studying interactions between processes on different length and time scales, starting from a minimal set of assumptions. While the derived theory using these methods may not be quantitatively accurate for the atmosphere, it can still provide qualitative value, especially when trying to determine the causal relationships that are so elusive in standard budget calculations. This is analogous to the use of the quasigeostrophic approximation, which provides a clear qualitative picture of the large-scale flow and both planetary- and synoptic-scale eddies; however, for accurate representation of the flow (e.g., in weather prediction), the primitive equations are used. Therefore, the aim of this work is to find a theoretical framework by which to better understand the emergent properties of observations and model behavior rather than developing a predictive theory.

DK13 used a separation of length scales in the meridional and zonal directions, with an isotropic scaling for the synoptic scales, as well as a temporal scale separation between the synoptic and planetary waves. Isotropic scaling for the synoptic scales is standard in quasigeostrophic (QG) theory (Pedlosky 1987), and a meridional scale separation has been argued to be a useful and physically realizable idealization of baroclinic instability (Haidvogel and Held 1980). These assumptions allowed DK13 to study planetary- and synoptic-scale interactions. However, they did not derive a wave activity equation or develop explicit equations for the interaction with a zonal mean flow. These aspects are the focus of this paper. For simplicity, we derive the asymptotic equations for the case of small-amplitude eddies evolving in the presence of a zonal mean flow, which is an important special case of the DK13 framework. As well as giving a theoretical description for the interaction of a zonal mean flow with planetary- and synoptic-scale waves, this setting also allows a study of the link between baroclinic and barotropic processes and the relative importance of planetary- and synoptic-scale waves for these processes.

The outline of the paper is as follows. Section 2 gives the equations and assumptions used to derive the potential vorticity (section 3), wave activity and mean-flow equations (section 4), and the angular momentum budget for the zonal mean flow (section 5). The momentum, continuity, thermodynamic, and vorticity equations at different asymptotic orders, which are needed for the derivations, are given in appendix A. Further details on the derivations of the mean-flow and angular momentum equations and the nonacceleration theorem are given in appendixes B–D. The zonally homogeneous case with weak planetary-scale waves is discussed in section 6, and conclusions are given in section 7.

## 2. The multiscale asymptotic model

### a. Nondimensional compressible flow equations

*ε*

^{1}(DK09). To obtain the nondimensional equations, the DK09 and DK13 scaling parameters

^{2}are used, based on the assumption that the waves are not propagating faster than the speed of sound. In this process, the following nondimensional numbers appear (DK09): Rossby

^{3}(

*ε*according to

*S*denotes source–sink terms (

*p*is nondimensional pressure,

*θ*is nondimensional potential temperature,

*ρ*is nondimensional density,

*z*is altitude from the ground,

*ϕ*is latitude, λ is longitude,

*t*is time, all parameters are nondimensional, andNote that the shallow-atmosphere limit

*R*, the material derivative, (2), involves horizontal advection terms

### b. Assumptions for multiscale asymptotic methods

To derive the multiscale asymptotic version of the equations, some assumptions must be made. In particular, we assume small-amplitude eddies in the presence of a zonal mean flow. This approximation is made in order to gain qualitative insight into the behavior of the system and to allow connection with previous theories of wave–mean flow interaction. This can be considered a special case of DK13, with the eddies (but not the zonal mean flow) scaled down by one order of *ε*. The assumptions for the scale separation between the synoptic, planetary, and mean flow in time, height, latitude, and longitude are given in Table 1 (following DK13), where the subscripts *m*, *p*, and *s* represent mean, planetary, and synoptic scales, respectively. Note that

The assumptions for the scale separations between planetary (*p*), synoptic (*s*), and zonal mean flow (*m*).

^{−1}; DK09), which would require a different treatment. Note that here the leading-order variation in potential temperature

*z*, not only on

*z*, which is the case for the static stability parameter in QG theory.

To have a well-defined asymptotic expansion, (3), the sublinear growth condition (DK13) is required. This means that variables at any order grow more slowly than linearly in any of the synoptic coordinates, which effectively means that any averaging over the synoptic scales

The full set of equations at different asymptotic orders using the assumptions from this section is given in appendix A. This includes the momentum, thermodynamic, and continuity equations, thermal wind, hydrostatic balance, and the vorticity equation. These equations are used in the following sections to derive potential vorticity, wave activity, and mean-flow equations.

## 3. Potential vorticity equation

*f*do not depend on

The background PV gradient

The planetary-scale PV equation, (10), also resembles the QG PV equation; however, the planetary-scale PV, (8b), only includes the stretching term (again because of the planetary scaling we chose). Note that the planetary- and synoptic-scale PV equations are independent of each other in this small-amplitude limit, which implies no direct interaction between planetary and synoptic scales—their interaction only occurs via source–sink terms, the mean flow, or at higher order. This independence is not present in DK13’s finite-amplitude theory where the synoptic- and planetary-scale waves interact at leading order.

This analysis suggests that the QG approximation can be used locally for both planetary- and synoptic-scale PV. Note, however, that this is only true in this small-amplitude case (in the finite-amplitude theory of DK13, this approach is not applicable for the planetary scales).

The PV equation in (11) is closely related to the Ertel PV equation. However, it includes vertical advection, which is problematic with respect to obtaining a QG wave activity equation. As shown in (7), we can eliminate the vertical advection term by including it in the stretching term of the synoptic- or planetary-scale PV. This is similar to the classical QG approximation of Charney and Stern (1962), in which they point out that the QG PV equation is the QG approximation to the PV equation; however, the QG PV is not the QG approximation to the Ertel PV (because the QG PV equation only includes horizontal advection). Notice that in (11), there is also the mean-flow PV, whereas (7) only has the background PV gradient that came from this mean-flow PV (but not via the direct meridional derivative of

## 4. Wave activity equation and the equations for the mean flow

### a. Wave activity equation

Note how the planetary-scale EP flux does not have a meridional component (no momentum flux) and that the synoptic-scale EP flux closely resembles Plumb (1985)’s total flux

We can also relate these expressions to Hoskins et al. (1983)’s **E** vector, where the difference is in the zonal component of the **E** vector, which lacks the wave activity advection

Nonetheless, the synoptic-scale EP flux is similar to the QG form of EP flux (e.g., Edmon et al. 1980), especially if zonally averaged. The planetary-scale wave activity implies that the momentum fluxes and hence barotropic processes at those scales are less important than heat fluxes and baroclinic processes. Also, this emphasizes the fact that planetary and synoptic scales do not interact directly but rather through other processes (source–sink terms or the mean flow) as the two wave activity equations are at different orders and have no “cross” terms. The wave activity equations are at different orders as the planetary and synoptic PV equations, (10) and (9), are multiplied by

*s*) in (12) and over planetary scales (

*p*) in (13) giveswhere

### b. Barotropic equation

*z*(denoted by the overline and

*z*). This yields the momentum equation in (B6) (see appendix B for more details), which can be used to derive the barotropic pressure equation, taking

*f*, and recalling (A4):withwhere the double-underlined terms represent eddy forcing of the mean flow,

This equation implies that although both the synoptic- and planetary-scale momentum fluxes affect the barotropic part of the mean flow, only the planetary-scale heat fluxes

The zonal-mean-flow equations at different orders can be further written in TEM form (Andrews and McIntyre 1976; Edmon et al. 1980), from which a nonacceleration theorem can be derived using the wave activity equations. This is addressed in appendix D. Note that an evolution equation for

### c. Baroclinic equation

## 5. Angular momentum conservation

Apart from the mean-flow equations (baroclinic and barotropic) and the eddy equations (wave activity), angular momentum conservation provides additional information about the transfer of angular momentum between Earth and the atmosphere, which has implications for the surface easterlies in the tropics and westerlies in the midlatitudes (e.g., Holton 2004). Hence, it is important to show that such a budget can be found also in the asymptotic model.

*a*is the radius of Earth, Ω is Earth’s rotation rate,

*ϕ*is meridional coordinate,

*u*is zonal velocity, and

*M*is angular momentum per unit mass.

^{4}

*M*in (22) gives the nondimensional angular momentum equationusing

*M*, (22). Here,

*M*is conserved [using the fifth-order momentum equation, (A8)] in the absence of source–sink terms and orography, yieldingwhere

Note that the surface pressure tendency

## 6. The zonally homogeneous case

These findings may help explain the empirical results of Thompson and Woodworth (2014), who found that the barotropic and baroclinic parts of the Southern Hemisphere (SH) flow variability were decoupled, with the barotropic part of the flow [characterized by the southern annular mode (SAM), based on zonal-mean zonal wind] being only affected by the momentum fluxes and the baroclinic part of the flow (characterized by the BAM, based on EKE) being only affected by the heat fluxes. We assume here that the wave activity is closely linked to EKE. Indeed, Wang and Nakamura (2015, 2016) found that wave activity during the SH summer is only affected by the heat fluxes under an average over a few latitudinal bands (approximately 10°), giving an equation similar to (27a). Here, we put this view into a self-consistent mathematical perspective.

*l*is meridional wavenumber,

*T*is temperature, the square brackets represent the zonal mean, and the asterisk represents perturbations therefrom. This relation is not present here because of the chosen scaling and the averaging over synoptic scales. Equation (18) does in fact have the heat fluxes, acting on synoptic scales, which because of the sublinear growth condition (DK13) disappear in (27c), as mentioned above.

Pfeffer (1987, 1992) argued that heat fluxes (vertical EP fluxes) grow in the part of the domain with low stratification parameter *S*. Pfeffer’s *S* can be related to *ε* as

Zurita-Gotor (2017) showed further that there is a low-frequency suppression of heat fluxes (at periods longer than 20–30 days) and concluded that, at longer time scales (considered here), the meridional circulation and diabatic processes are more important for the baroclinicity than the synoptic-scale heat fluxes [consistent with (27c)].

## 7. Conclusions

In this paper, we have provided a theoretical framework for planetary–synoptic zonal-mean-flow interactions in the small-amplitude limit with a scale separation in the meridional direction, as well as in the zonal direction, between planetary and synoptic scales. Thus, the synoptic-scale eddies are assumed to be isotropic (which is the case also in QG theory). These assumptions allow us to derive strong results, for example, a lack of direct interaction between the planetary and synoptic waves and a lack of a direct link between the baroclinic and barotropic components of the flow when only synoptic-scale fluxes are considered.

We derived planetary- and synoptic-scale PV equations, (10) and (9), and equations for the eddies [wave activity equations, (14) and (15)]; the barotropic part of the zonal mean flow, (17); and the baroclinic part of the zonal mean flow, (19). A crucial step in deriving these equations was finding a form of the PV equation that eliminated the effect of vertical advection. The synoptic-scale PV then resembled QG PV, and the planetary PV resembled that of planetary geostrophy, that is, with only stretching vorticity representing PV on planetary scales (e.g., Phillips 1963). These equations provide an alternative view to the conventional transformed Eulerian mean (TEM) framework [first introduced in Andrews and McIntyre (1976)], which combines all components into two equations that are linked through the Eliassen–Palm flux.

The background PV gradient, (8c), that emerged from the equations lacks the relative vorticity term as in planetary geostrophy (Phillips 1963), implying the dominance of baroclinic processes for eddy generation. Thus, this PV gradient resembles that of Charney’s baroclinic instability model (e.g., Hoskins and James 2014) but is more general as it includes variations in static stability in both the vertical and meridional directions. The latter should be stressed, as this is the main difference to QG dynamics in this model.

In terms of the baroclinic life cycle (Simmons and Hoskins 1978), the barotropic pressure equation, (17), would be relevant in the breaking region of the storm track, and the baroclinic equation, (19), would be more relevant in the source region. We also showed that only the planetary-scale heat fluxes affect the baroclinicity, (19); that both planetary and synoptic-scale momentum fluxes, as well as planetary-scale heat fluxes, affect the barotropic zonal mean flow, (17); and that the planetary waves and synoptic-scale eddies only interact via the zonal mean flow or the source–sink terms or at higher-order approximations. Since both the barotropic [(17)] and baroclinic [(19)] parts of the zonal mean flow are affected by the planetary-scale heat fluxes, the latter could provide a link between upstream and downstream development of storm tracks. The barotropic equation, (17), was also directly linked to the angular momentum equation, (26), which has not been noted in previous work. This linkage revealed the importance of planetary-scale heat fluxes (via meridional mass transport) for the angular momentum budget (Haynes and Shepherd 1989).

The importance of planetary-scale waves was also noted in Kaspi and Schneider (2011, 2013), who found that the termination of storm tracks downstream is related to stationary waves and the baroclinicity associated with them. Stationary waves are especially important locally in contributing to heat fluxes, which enhance temperature gradients upstream and reduce them downstream.

When considering only the synoptic-scale eddies (when planetary-scale eddies are weak, as, for example, in aquaplanet simulations or in the Southern Hemisphere), we find that under synoptic-scale averaging the barotropic zonal mean flow, (27b), is only affected by the momentum fluxes, the baroclinicity, (27c), is only affected by the source–sink terms, and wave activity, (27a), is only related to heat fluxes (as in Thompson and Woodworth 2014). This suggests that the baroclinicity is primarily diabatically driven. Understanding the decoupling of the baroclinic and barotropic parts of the flow (in the case of weak planetary-scale waves) is addressed in a companion study (Boljka et al. 2018), where it is shown that at time scales longer than synoptic the EKE is only affected by the heat fluxes and not momentum fluxes, confirming relation (27a).

Along with helping to understand a variety of previous results in the literature, one potential use of the theory presented here is to help understand the barotropic response to climate change, which is fundamentally thermally driven. In general, we need a better understanding of the interaction between the baroclinic and barotropic parts of the flow, where planetary-scale heat fluxes and diabatic processes may play an important role.

This theoretical framework could be extended by allowing finite-amplitude eddies (as in DK13) and by relaxing the assumption of a separation of scales in latitude (e.g., Dolaptchiev 2008).

## Acknowledgments

This work was funded by the European Research Council (Advanced Grant ACRCC, “Understanding the atmospheric circulation response to climate change” Project 339390). We thank the three anonymous reviewers for their comments, which helped improve the original manuscript. We acknowledge Mike Blackburn, Rupert Klein, Stamen Dolaptchiev, Alan Plumb, and Brian Hoskins for helpful discussions.

## APPENDIX A

### The Multiscale Asymptotic Version of the Primitive Equations

Using the assumptions from section 2b, the momentum, thermodynamic, continuity, hydrostatic, and thermal wind balance equations at different orders

#### a. Hydrostatic balance

*p*and

*θ*as defined in (47) in (DK09):where

*θ*is taken (and this relationship will only be used in this case).

#### b. Momentum equations

Below is the list of all momentum equations up to fifth order. Note that we derive the PV and wave activity equations from the third-order momentum equation and we obtain a barotropic equation for the mean flow from the fifth-order momentum equation.

: Geostrophic balance for zonal mean wind:The subscript *m*refers to the mean flow, andis related to the zonal-mean zonal velocity. Note that . : Geostrophic balance for first-order wind (planetary- and synoptic-scale perturbations to the zonal mean):Here, (with subscripts *p*and*s*referring to planetary and synoptic waves, respectively), such thatand . : First nontrivial order, used to derive PV equations: : We require only the *u*-momentum equation:: Again we require only the *u*-momentum equation, used to derive the barotropic pressure equation (equation for the zonal-mean zonal flow):

*x*and

*y*coordinates, the tilde is used when

#### d. Thermodynamic (θ) equations

Below is the list of all needed thermodynamic equations. Note that all orders below

: : :

#### e. Continuity equations

This is the set of all continuity equations (also the trivial ones as they give us information about vertical velocities).

, , and : [here, note that from the thermodynamic equation, (A10), and that by definition]: : : :

*z*/

*a*come from corrections to the shallow-atmosphere approximation at higher orders. Note that these terms vanish in the zonal mean and/or synoptic-scale average.

#### f. Vorticity equation

*π*,

*ρ*,

*θ*) that do not depend on synoptic scales (up to third order) are zero. This yields (following DK13)where

## APPENDIX B

### Derivation of the Mean-Flow Equations

#### a. Barotropic equation

This section shows the steps in deriving the barotropic pressure equation—combining the correct thermodynamic, hydrostatic, thermal wind, momentum, and continuity equations (see appendix A) with the *z* (denoted with an overline). Note that the vertical mean assumes

#### b. Baroclinic equation

- Thermodynamic equations at
:where terms with *z/a*come from corrections to the shallow-atmosphere approximation. - Continuity equations at
: - Momentum equations at
:

*z*and using (B9a) and (B9b). This yieldswith

These equations are then used in section 4c to derive the final baroclinic equation for the mean flow, (18) and (19).

## APPENDIX C

### Derivation of the Angular Momentum Equation

This appendix shows the derivation of angular momentum conservation for the zonal-mean-flow

Deriving an angular momentum equation for the mean flow means that something that corresponds to the fifth-order momentum equation, (A8), must be used. This means that, for example, *Du*/*Dt* has to be fifth order, which overall makes the angular momentum equation, (23), a second-order equation; thus, the rest of the terms in the equation must follow that pattern.

*ε*everywhere):

*B*is an arbitrary scalar, and

^{C1}assuming no source–sink terms and assuming there is no orography (for simplicity) yields angular momentum conservation

*z*) and using Gauss's theorem again, which givesHere, the overbar denotes an average over

*f*, take

*f*. This yieldswhere source terms were omitted for simplicity, the left-hand side can be simplified towithand the last term in the equation can be simplified to

## APPENDIX D

### The Nonacceleration Theorem

*r*representing residual velocity and

*i*representing its order):which satisfy the continuity equations at different orders.

*u*-momentum equation, the

Note that (D7) and (D8) provide equations for zonal-mean-flow variations on shorter time scales (synoptic and planetary), which have dynamical importance for higher-frequency atmospheric flow (e.g., baroclinic life cycles or barotropic annular modes with time scales of 10 days or less). Upon averaging over these scales, the slower variations in the mean flow

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^{1}

The variable *ε* is defined as *g* is Earth’s gravitational acceleration; *ε* is a constant in the range from 1/8 to 1/6.

^{2}

We set pressure

^{3}

Note that the Rossby number (Ro) used in DK09 and DK13 is

^{C1}

Gauss's theorem generally states **G** is a three-dimensional vector, **n** is a normal vector on surface *S*, and *V* of interest. Note that in the case of