## 1. Introduction

Many atmospheric phenomena important for the low-frequency variability (periods longer than 10 days) are characterized by planetary spatial scales (i.e., scales on the order of Earth's radius ≈ 6 × 10^{3} km). Such phenomena are the orographically and thermally induced quasi-stationary Rossby waves, eddy-driven teleconnection patterns such as the northern and southern annular modes (NAM and SAM), Pacific–North America pattern (PNA) or the North Atlantic Oscillation (NAO), large-scale ultralong persistent blockings, and the polar/subtropical jet. On the other hand, the synoptic eddies, which are responsible for the variability with periods of 2–6 days, have as a characteristic length scale the internal Rossby deformation radius (Pedlosky 1987), which is around 1000 km. Although the spatial and temporal separation between the planetary- and synoptic-scale atmospheric motion is not so pronounced as in the ocean, the different scales are evident in spectral analyses of tropospheric data: observations (e.g., Blackmon 1976; Fraedrich and Böttger 1978; Fraedrich and Kietzig 1983) as well as simulations (e.g., Gall 1976; Hayashi and Golder 1977) show the presence of isolated peaks in the wavenumber–frequency domain. Figure 1 [taken from Fraedrich and Böttger (1978)] displays three such peaks in the spectrum of the meridional geostrophic wind. There is a maximum associated with the quasi-stationary Rossby waves with zonal wavenumber *k* = 1–4 and with periods larger than 20 days. The other two maxima at *k* = 5–6 and at *k* = 7–8 result from the synoptic waves. These are eastward-propagating long and short waves associated with different background stratifications (Fraedrich and Böttger 1978). The overall picture of three maxima persists during the different seasons for the Northern Hemisphere, indicating a separation between the planetary and synoptic scales. However, the interactions between the two scales are of great relevance to the atmospheric dynamics as stressed in many studies (e.g., Hoskins et al. 1983).

The fast growth of computational resources in atmospheric sciences over the last decades leads to a huge increase of complexity in atmospheric models. This becomes apparent if one considers the development of the comprehensive general circulation models (GCMs). However, as simulations with those models become closer and closer to observations, the interpretation of the results with regard to improving the models as well as the understanding of the climate system becomes more difficult. This stresses the importance of simplified models, utilized for studies of many aspects of the circulation, such as instability, wave propagation, and interaction. As discussed by Held (2005), the development of a hierarchy of reduced models provides a tool for systematic improvement of the comprehensive models and thereby contributes to our understanding of the climate system.

One prominent example (if not the most prominent) of simplified model equations is the quasigeostrophic (QG) theory (Charney 1948), which describes the baroclinic generation and evolution of the synoptic-scale eddies. This theory is derived under the assumption of a horizontally uniform background stratification and small variations of the Coriolis parameter (Pedlosky 1987)—an assumption that is often violated if one considers motions on a planetary scale (such as the phenomena mentioned in the first paragraph). Reduced model equations, which do not make use of the latter assumption and model planetary-scale motions, are the planetary geostrophic equations (PGEs; Robinson and Stommel 1959; Welander 1959; Phillips 1963). The PGEs describe balanced dynamics; however, the relative vorticity advection is absent in the potential vorticity (PV) equation (the consequences of which we will discuss later). Much effort has been made to develop simplified models valid for the planetary and synoptic scale and for the interactions between the two scales. Pedlosky (1984) proposed a two-scale model for the ocean circulation, where the dynamics on the large scale is governed by the PGEs and the dynamics on the small scale by a modified QG equation, which is influenced by the large scale. Mak (1991) incorporated in the QG model effects due to spherical geometry of Earth by considering higher-order terms. Vallis (1996) introduced the geostrophic PV model, which can be reduced to QG or to PG model by imposing an appropriate scaling. Numerical simulations (Mundt et al. 1997) with the geostrophic PV model showed that this model improves the circulation patterns over the latter classical models. Luo developed multiscale models for planetary–synoptic interaction and applied them for studies of blockings (Luo et al. 2001; Luo 2005) and NAO dynamics (Luo et al. 2007).

In a previous paper (Dolaptchiev and Klein 2009, hereafter DK) we presented reduced model equations valid for one particular regime of planetary-scale atmospheric motions; we refer to this regime as the planetary regime (PR). In the PR we consider the planetary horizontal scales and a corresponding advective time scale of 7 days (see Fig. 2). The PR model includes the PGEs and a novel evolution equation for the barotropic flow. As discussed in DK, in applications to the atmosphere of the PGEs the barotropic flow has to be specified because there is no advection of relative vorticity in the PGEs. The novel evolution equation in the PR provides a prognostic alternative relative to temperature-based diagnostic closures for the barotropic flow adopted in reduced-complexity planetary models (Petoukhov et al. 2000). The PR model takes into account large variations of the background stratification and of the Coriolis parameter, but it does not describe the synoptic eddies. This limitation motivates an extension of the validity region of the single-scale PR model to the synoptic spatial and temporal scales (see Fig. 2). In this paper we apply the same asymptotic approach from DK, but utilizing now a two-scale expansion resolving both the planetary and the synoptic scales. In doing so we can take into account in a systematic manner the interactions between the planetary and the synoptic scales with particular attention paid to the barotropic component of the background flow. Part of the derived model equations can be regarded as the anelastic analogon of Pedlosky's two-scale model for the large-scale ocean circulation (Pedlosky 1984). But whereas the latter model describes only interaction from the planetary to the synoptic scale, in the present model there is an additional planetary-scale evolution equation for the vertically averaged pressure that provides a reverse interaction (from the synoptic- to the planetary-scale dynamics) in the form of momentum fluxes due to the synoptic-scale velocity field. This type of feedback on the planetary scale differs from the one recently proposed by Grooms et al. (2011), where the PGEs are influenced by the synoptic scale through eddy buoyancy fluxes. We have to point out that momentum fluxes due to synoptic eddies are commonly considered as an interaction mechanism acting on atmospheric planetary-scale barotropic flow (e.g., Luo 2005; Luo et al. 2007)—however, to our knowledge, not in the context of PGEs.

The outline of this paper is as follows. In section 2 we briefly discuss the asymptotic method applied for the derivation of the two-scale model. Key steps in the derivation are presented in section 3. The asymptotic model equations are summarized and discussed in section 4. We compare the results from the asymptotic analysis with numerical simulations with a primitive equations model in section 5. A concluding discussion can be found in section 6.

## 2. Asymptotic approach for the derivation of reduced models for the planetary and synoptic scales

### a. Asymptotic representation of the governing equations

We utilize the multiple scales asymptotic method of Klein (2000, 2004, 2008). It has been applied in the development of reduced models—for example, for the tropical dynamics (Majda and Klein 2003), deep mesoscale convection (Klein and Majda 2006), moist boundary layer dynamics (Owinoh et al. 2011), and concentrated atmospheric vortices (Päschke et al. 2012).

*p*

_{ref}= 10

^{5}kg m

^{−1}s

^{−2}, the air density

*ρ*

_{ref}= 1.25 kg m

^{−3}, and a characteristic flow velocity

*u*

_{ref}= 10 m s

^{−1}. The above quantities define further the scale height

*h*

_{sc}=

*p*

_{ref}/

*g*/

*ρ*

_{ref}≈ 10 km (

*g*= 9.81 m s

^{−2}is the gravity acceleration) and its time scale

*t*

_{ref}=

*h*

_{sc}/

*u*

_{ref}≈ 20 min. We introduce a small parameter

*ɛ*as the cubic root of atmosphere's global aspect ratio

*a** ≈ 6 × 10

^{3}km is Earth's radius and Ω ≈ 7 × 10

^{−5}s

^{−1}Earth's rotation frequency. With these estimates we find

*ε*~ 1/8, …, 1/6, and henceforth consider asymptotic limits for

*ε*≪ 1. Next, the nondimensional Mach, Froude, and Rossby numbers in the governing equations are expressed in terms of

*ɛ*, which is referred to as a distinguished limit. The coupling of all characteristic numbers in terms of only one small parameter is motivated by the fact that asymptotic expansions of simple systems (such as the linear damped oscillator) give nonunique results if multiple independent parameters are used. With the present specific coupling, a variety of classical models can be rederived; see also Klein (2008) for further discussion of the distinguished limit. An alternative interpretation of

*ɛ*to the one in (1) is that

*ɛ*equals the Rossby number for the synoptic scales (see Fig. 2 for the synoptic length scaling). Introducing the distinguished limit, the nondimensional governing equations in spherical coordinates take the form

*λ*,

*φ*, and

*z*are longitude, latitude, and altitude. The nondimensional variables

*p*,

*ρ*, and

*θ*denote pressure, density, and potential temperature;

*u*,

*υ*, and

*w*are the zonal, meridional, and vertical velocity components;

*S*

_{u}_{,υ,w}and

*S*represent momentum and diabatic source terms;

_{θ}*γ*is the isentropic exponent; and the operator

*d*/

*dt*is given by

*R*=

*a*+

*ɛ*

^{3}

*z*(

*a*is of order 1 constant). We want to stress that the reference quantities used for the nondimensionalization, although valid for a variety of flow regimes, might not be characteristic for the particular regime of interest (e.g., the scale height is not an appropriate horizontal scale for the description of planetary- and synoptic-scale atmospheric motions). Within the asymptotic approach a particular regime of interest can be studied, if rescaled coordinates together with an asymptotic series expansion of the dependent variables are introduced based on physical arguments and intuition. The scaling and the asymptotic expansion should reflect the relevant physical processes in the flow regime of interest.

### b. Coordinates scaling for the two-scale planetary regime

The coordinates resolving the planetary and synoptic time and spatial scales are summarized in Table 1. The planetary coordinates *λ _{P}* and

*φ*are suitable for the description of horizontal variations on the order of

_{P}*a**. The corresponding planetary advective time scale is about 7 days and is resolved by

*t*. The synoptic-scale variables

_{P}*λ*,

_{S}*φ*, and

_{S}*t*describe motions with characteristic length scales of 1000 km (~

_{S}*ɛa**) and with a time scale of about 1 day. For

*z*no scaling is required. Since this coordinate was nondimensionalized using

*h*

_{sc}, it describes motions spreading through the full depth of the troposphere. A more detailed discussion of the scaling can be found in DK; the validity range of the two-scale planetary regime is sketched in Fig. 2.

Scaling for the planetary and synoptic coordinates.

### c. Sublinear growth condition

*U*

^{(i)}grows slower than linearly in any of the synoptic coordinates. This requirement is known as the sublinear growth condition. Suppose that

*X*denotes one of the synoptic coordinates

_{S}*λ*,

_{S}*φ*, or

_{S}*t*, and

_{S}*X*denotes the corresponding planetary coordinate

_{P}*λ*,

_{P}*φ*, or

_{P}*t*. Since we have

_{P}*X*=

_{S}*X*/

_{P}*ɛ*, we can formulate the sublinear growth condition for the coordinate

*X*as

_{S}*X*are held fixed with respect to

_{S}*ɛ*in the limit process. An immediate consequence from the last constraint is the disappearing of averages over

*X*of terms, which can be represented as derivatives with respect to

_{S}*X*. In particular, we have

_{S}*L*is some characteristic averaging scale for the coordinate

_{S}*X*. Equation (11) implies that in the asymptotic analyses the synoptic-scale divergence of a flux has no effect on the planetary-scale dynamics when the synoptic-scale averaging in (12) is applied. This follows directly from the sublinear growth condition in (10) and we will make extensive use of it in the derivation of the reduced model equations.

_{S}### d. Assumptions for the background stratification

^{−2}) s

^{−1}, which in nondimensional form implies a horizontally uniform

*ɛ*

^{2}) background potential temperature (Majda and Klein 2003). Similarly, as in DK, we allow here

*ɛ*

^{2}) variations on the planetary scales of the background potential temperature distribution. To remain consistent with the assumptions in the QG theory, we consider an order of magnitude smaller variations on the synoptic spatial and temporal scales—namely,

*ɛ*

^{3}). Thus the expansion for the potential temperature takes the form

## 3. Derivation of the planetary regime with synoptic-scale interactions

### a. Asymptotic expansion

#### 1) Notation

**u**=

**e**

_{λ}

*u*+

**e**

_{φ}

*υ*and

**e**

_{λ},

**e**

_{φ}, and

**e**

_{r}denote the unit vectors in spherical coordinates. Note that we do not need to make the traditional

*β*-plane approximation for the Coriolis parameter

*f*since its full variations are resolved by

*φ*.

_{P}#### 2) Key steps of the expansion

We substitute the ansatz (9) into the governing equations (2)–(7) and collect terms of the same order in *ɛ*. Following DK, we assume a radiative heating rate of about 1 K day^{−1}; this implies for the diabatic source term *S _{θ}* ~

*ɛ*

^{5}). The magnitude of the friction source terms is estimated as

*S*

_{u}_{,υ}~

*ɛ*

^{2}) if a relaxation time scale for the frictional effects of about 1 day is assumed. Source terms of this strength will induce leading-order synoptic tendencies in the momentum equation.

##### (i) Vertical momentum balance

*ɛ*

^{4}). If we make use of the ideal gas law in (7) and of the Newtonian limit [which states that

*γ*− 1 =

*ɛ*) as

*ɛ*→ 0], we obtain from the leading two orders hydrostatic balance [see Klein and Majda (2006) and DK for details]:

*p*

^{(0)}=

*ρ*

^{(0)}= exp(−

*z*) and

*p*

^{(1)}=

*ρ*

^{(1)}= 0. The next two orders of hydrostatic balance can be expressed as

*π*

^{(i)}=

*p*

^{(i)}/

*ρ*

^{(0)}.

##### (ii) Horizontal momentum balance

*π*

^{(2)}, consistent with the assumption (13) on Θ

^{(2)}. Further, the leading-order synoptic-scale horizontal pressure fluctuations are assumed an order of

*ɛ*smaller and are modeled by

*π*

^{(3)}. If we allow for a dependence of

*π*

^{(2)}on the synoptic scales, the horizontal pressure gradient

**∇**

_{S}

*π*

^{(2)}will appear in the

**u**

^{(0)}in the current asymptotic expansion in (9) describes dimensional velocities of the order

*u*

_{ref}]. Thus, the synoptic-scale variations of

*π*

^{(2)}imply unrealistic large synoptic-scale velocities; such variations are inconsistent with the QG scaling and will not be considered here. With the consideration above, together with the result

*w*

^{(0)}= 0 from section 3a(2)(iii) below, we obtain that

**u**

^{(0)}is geostrophically balanced with respect to the pressure gradient on the synoptic and on the planetary scale,

*π*

^{(2)}and

*f*do not depend on the synoptic scales, (22) implies that the synoptic-scale divergence of

**u**

^{(0)}disappears,

**u**

^{(0)}on the synoptic time scale appears in the next order equation,

*ɛ*

^{3}) momentum equation,

**u**

^{(1)}

**u**

^{(1)}by

**u**

^{(0)}(

**u**

^{(0)}·

**∇**

_{S}

**u**

^{(1)}).

##### (iii) Continuity equation

*w*

^{(0)}=

*w*

^{(1)}=

*w*

^{(2)}= 0 (see DK for details). The

*ɛ*

^{3}-order equation reads

**u**

^{(1)}(interpreted in the classical QG theory as the divergence due to the ageostrophic velocities) appears in the same order as the planetary-scale divergence of the leading order wind field

**u**

^{(0)}. Making use of (23), the next two orders in the continuity equation take the form

##### (iv) Potential temperature equation

^{(2)}is interpreted as a horizontally uniform background temperature distribution and all terms involving it, except the stratification term, are set to zero. Here we consider the variations on the planetary spatial and temporal scales of Θ

^{(2)}and their influence on the synoptic-scale dynamics of Θ

^{(3)}.

### b. Vorticity equation for the two-scale PR

**e**

_{λ}component of (24) and

**e**

_{φ}component of (24), we obtain

**u**

^{(0)}

### c. Averaging over the synoptic scales

**u**

^{(1)}. They can be eliminated in a way similar to that encountered in the classical QG theory; see also Pedlosky (1984). This leads to two-scale PR model equations describing the planetary- and the synoptic-scale dynamics; the model is summarized in the next section. A key step in the derivation is to split all variables into a synoptic-scale average and a deviation from this average. In the case of the variable

*π*

^{(3)}we obtain

## 4. Summary and discussion of the two-scale PR model

Using a two-scale asymptotic ansatz, we extended in a systematic way the region of validity of the planetary-scale model from DK to the synoptic spatial and temporal scales. The model presented here relies on the assumption that the variations of the background stratification are comparable in magnitude with those adopted in the classical QG theory. The model equations are summarized below; for convenience of notation the superscripts indicating asymptotic expansion orders are dropped.

- Planetary-scale model:
- Synoptic-scale model:

Equations (40)–(43) describe the planetary-scale dynamics and (44) and (45) describe the synoptic-scale dynamics. The model equations include two advection equations in (40) and (44) for a PV-type quantity and an evolution equation for the barotropic component of the background pressure in (41), derived after applying the sublinear growth condition.

If we leave the planetary-scale dependence of the variables out, (40) and (41) reduce trivially and the underlined terms in (44) vanish. In this case (44) is the classical PV equation from the QG theory. On the other hand, if we assume that the variables do not depend on the synoptic scales, the interaction terms in (41) and (42) vanish and (40) remains unchanged: thus we have the single-scale planetary model from DK.

In the general case, when both synoptic and planetary scales are included, (40), (41), and (44) provide the planetary-scale structure of Θ, *p*. The variable Θ characterizes the background stratification. But whereas in the classical QG model a horizontally uniform stratification is assumed, here it is governed by the evolution equation (40). Another difference to the QG theory is that we do not utilize a *β*-plane approximation in the derivation of the synoptic-scale model in (44). In the last model, variation of the Coriolis parameter *f* (as well as *β*) on a planetary length scale are allowed. Equations (40) and (43) constitute the PGEs. As discussed in DK, they do not represent a closed system, since a boundary condition for the surface pressure, or equivalently for the vertically averaged (barotropic component) pressure, is required. The latter is determined by the planetary barotropic vorticity equation (41). It is shown (see appendix B) that as in the single-scale PR the barotropic component of the background pressure

The two underlined terms in (44) describe interactions between the planetary and the synoptic scales, or more precisely the influence of the planetary-scale variations of the background pressure/temperature distribution on the synoptic-scale PV field. The first term can be interpreted as the advection of synoptic-scale PV by the planetary-scale velocity field and the second as the interaction of synoptic velocities with PV gradient afforded by the planetary-scale field. It is important to note that the latter PV includes only a stretching vorticity part, since the contribution from the relative vorticity (due to planetary-scale velocity field) is an order of magnitude smaller in the asymptotic analysis. We observe that the barotropic model of Luo (2005) contains such additional interaction term [see (2b) in Luo (2005)]. We speculate that this results from the fact that the latter author starts his asymptotic analysis from the equivalent barotropic vorticity equation, which itself is derived under the quasigeostrophic scaling. We observe further that the model for the synoptic dynamics in (44) reduces to the model of Pedlosky (1984) if we set *ρ*_{0} to one and consider plane geometry.

It is important to note that from the equations describing the planetary-scale dynamics only (41), but not (40), contains a feedback from the synoptic scale [see the underlined terms in (42)]. We consider the first underlined term in (42): after applying the chain rule, it will give rise in (41) to a term of the form *u _{S}*. Such fluxes will directly affect the barotropic component of the background pressure. The background temperature Θ, on the other hand, will be influenced only indirectly by the synoptic scales through the barotropic part of the flow: changes in the background pressure imply changes in the planetary-scale wind

**u**

_{P}and hence the temperature advection in (40) will be affected. Such type of feedback mechanism from the synoptic to the planetary scale is absent in the Pedlosky (1984) model and differs from the one proposed by Grooms et al. (2011). The latter author shows that for some anisotropic regimes (requiring either an anisotropy in the large-scale spatial coordinates or anisotropy in the large- and small-scale velocity fields) the planetary-scale motion can be influenced by the synoptic scale at leading order through eddy buoyancy fluxes. This does not contradict our results, since the barotropic component of the flow was omitted in the analysis of Grooms et al. (2011) and the PR is not characterized by an anisotropy. Further, we observe that (41) does not contain vertical advection and twisting terms. This is in accordance with budget analysis of low-frequency life cycle studies (Cai and van den Dool 1994; Feldstein 1998, 2002), which found that the corresponding terms are small and spatially incoherent. We note that terms multiplied by

*β*in (41) result from the advection of planetary vorticity by the ageostrophic flow (see appendix B). This is consistent with the analysis of Cai and van den Dool (1994): they found that such an advection is important for the very longest low-frequency wave.

## 5. Balances on the planetary and synoptic scales in numerical experiments

In this section we address the question how closely the reduced planetary–synoptic asymptotic model captures the dynamics of a more complete fluid–dynamical model of the atmosphere. For that purpose we perform simulations with a model based on the primitive equations (PEs). Since the PEs are derived from the full compressible-flow equations by assuming only a small aspect ratio of the vertical to horizontal length scale and the traditional approximation, these equations are much more comprehensive than the asymptotic model and apply to a wider range of scales. From the simulations with the PEs model we study the balances in the vorticity transport on the planetary and synoptic scale and compare them with the reduced asymptotic equations.

### a. Model description

*ζ*denotes the relative vorticity,

*f*is the planetary vorticity,

**u**is the horizontal velocity vector,

*ω*is the vertical velocity, and

*R*is the residuum due to errors in the interpolation of the fields from

*σ*to pressure levels (the PUMA model equations use a

*σ*vertical coordinate). Further, we have the friction relaxation time scale

*τ*and the hyperdiffusion coefficient

_{f}*K*. All model variables are nondimensionalized using Ω and

*a**.

We performed simulations with realistic orography as well as with an aquaplanet as lower-boundary condition. The model was run at a T21 horizontal resolution, with 10 vertical *σ* levels and with a time step of 30 min. For the analysis an output over 11 yr with 12-h time increment was used, the first 1 yr is ignored so as to not misinterpret any spinup effects. We used the default value of 70 K for the equator-to-pole temperature difference in the restoration temperature profile and the seasonal cycle in the model was switched off.

The inspection of the orography run shows that PUMA is able to produce key features of the atmospheric circulation reasonably well for a simplified atmospheric model. At midlatitudes a pronounced wavenumber-6, 7 structure with a period of about 8 days is visible over most of the simulation time. This wave implies a characteristic length scale of about 2000 km for the individual synoptic eddies; its time period is overestimated compared with the real atmosphere where the maximum of the synoptic activity lies around 4 days (Fig. 1). The time-mean 500-hPa geopotential height shows that the model reproduces the trough over eastern Asia, but it shifts the trough over Canada to Greenland. The weak trough over western Asia is absent in the model but a weak minimum over the Aleutian Islands is visible. In the real atmosphere the depression over these islands is confined to the lower troposphere only. An explanation of these discrepancies can be the absence of land–sea thermal forcing in the model.

### b. The two-scale PR in simulations

#### 1) Synoptic-scale dynamics

The power spectral density of various terms in the model equation (48) as a function of zonal wavenumber and frequency is presented in Figs. 3 and 4. From the plots it is visible that the terms *βυ* show two pronounced maxima. The first maxima is at zonal wavenumber *k* = 6 [*k* = 6, 7 for *k* = 5 and period between 9 and 10 days. This structure resembles the two peaks associated with synoptic activity found in observational data; see Fig. 1. Further, in the power density of *βυ* there is a hint of an isolated maximum at *k* = 1, 2 and frequency close to zero. This maximum results from quasi-stationary Rossby waves forced by orography because it is absent in the aquaplanet simulation.

To compare the magnitude of different terms in the vorticity balance on the synoptic scale, we computed the cumulative spectral density (sum over spectral density for some wavenumber/period interval) for zonal wavenumbers 4 ≤ *k* ≤ 8 and periods 7 ≤ *T* ≤ 10 days. The results for three different pressure levels are shown in Table 2. Overall, it can be stated that first the terms *f***∇** · **u** in Figs. 3 and 4), the resulting variations are one to two orders smaller than those of the individual terms, implying that they nearly balance. Both results are consistent with the leading-order vorticity balance in the asymptotic analysis (49), which states that on the synoptic scales the leading order in the expansion for the wind is divergence free.

Cumulative power spectral density for various terms in the vorticity equation (48) and at three different pressure levels. Shown is the sum over power density for zonal wavenumbers 4 ≤ *k* ≤ 8 and periods 7 ≤ *T* ≤ 10 days at 50°N and in units of Ω^{4}. The following abbreviation is used: *F*_{fr} for the Rayleigh friction and hyperdiffusion terms,

The next order vorticity balance (50) from the PR suggests that terms including vorticity tendency, relative vorticity advection, planetary vorticity advection, and horizontal divergence (multiplied by *f*) are next in importance in the vorticity transport. Indeed, Table 2 shows that *βυ*, *f***∇** · **u** are larger than terms involving advection by the vertical velocity *ω* or the dissipation term *F*_{fr}. However, the individual magnitudes of the first terms show variations within a wide range from 10^{−1} up to 10^{−3}. Clearly, *F*_{fr} is even one order smaller and at 300 and 500 hPa is comparable with *f***∇** · **u**.

Table 3 shows that the spectral properties discussed so far are observed at different latitudes as well.

#### 2) Planetary-scale dynamics

*f*

**∇**

_{P}·

**u**

^{(0)}and

*βυ*

^{(0)}. Thus, (50) can be used to study the leading-order vorticity balance on the planetary scale and the effect of the synoptic scales on that balance. We average (50) over the synoptic spatiotemporal scales in order to obtain the net influence on the planetary-scale motions. The resulting equation reads

*f*and

*ζ*flux in the PEs model. They suggest the following leading-order balance on planetary spatial and temporal scales: (i) the terms

*βυ*sum to zero up to next-order asymptotic corrections. Therefore, we consider the standard deviation of the terms

**∇**·

**u**

*ζ*and

**∇**·

**u**

*f*in the PEs simulation relative to the standard deviation of the individual terms entering in the definitions of

**∇**·

**u**

*ζ*and

**∇**·

**u**

*f*. To extract variations with particular zonal and meridional scale, we expand the data in spherical harmonics. Each harmonic has a total wavenumber

*n*and zonal wavenumber

*k*; the difference

*n*−

*k*defines the so-called meridional wavenumber and gives the number of nodes from pole to pole. Thus, modes with small

*n*and

*k*(

*n*≥

*k*) describe variations on planetary spatial scales in both zonal and meridional direction. On the other hand, if

*n*,

*k*, or both become larger, the corresponding spherical harmonic will capture synoptic spatial scales as well.

Figures 5a and 5b depict the normalized standard deviation of the spectral coefficients for **∇** · **u***ζ* as a function of *n*, where the normalization factor is given by the mean over the standard deviation of the terms *n* ≤ 3 at both pressure levels. However, the transition between a regime with compensation and noncompensation is smooth.

The averaging operator in (52) includes, in addition to a spatial averaging, an averaging over the synoptic time scales as well. Because of this we applied a low-pass filter to the data (Blackmon 1976), filtering out the synoptic time scales and all other time scales with periods smaller than 10 days. The results for the normalized standard deviation of **∇** · **u***ζ* are shown in Figs. 5c and 5d. The time filtering shifts the position of the maximum with roughly one total wavenumber to the left and reduces the standard deviation at higher *n*. However, for lower wavenumbers nearly no changes are observed compared to the unfiltered data, indicating that the large-scale spatial modes are dominated by long-period variations.

From (51) we expect that the divergence of planetary vorticity flux vanishes on the planetary spatial scales. This balance differs from the leading-order result on the synoptic scale (49), which states that *f* times the divergence of the wind vanishes. Figures 6a and 6b display the normalized standard deviation of **∇** · *f***u** and *f***∇** · **u** for different total wavenumbers *n*. The term **∇** · *f***u** has similar distribution as the term **∇** · **u***ζ* from Fig. 5: as *n* increases it increases monotonically up to a maximum and then declines; the smallest values correspond again to small *n*. The term *f***∇** · **u** has a different behavior: it decreases at the beginning until it saturates around some low, constant value. The saturation is reached around *n* = 5 and *n* = 6 for the 200- and 500-hPa pressure levels, respectively. At this wavenumber the synoptic-scale balance (49) is reached. From the graph of the **∇** · *f***u** term, it appears that the balance on the planetary scale (51) is satisfied for *n* = 1, 2 where the smallest values are reached and the curve is below the one for *f***∇** · **u**. As in the case of **∇** · **u***ζ*, the transition between the planetary and synoptic regime in **∇** · *f***u** and *f***∇** · **u** is smooth.

Figures 6c and 6d show that the application of a low-pass filter to the data does not change qualitatively the behavior of **∇** · *f***u** and *f***∇** · **u**. The results reported in this section were also observed in an aquaplanet simulation.

## 6. Conclusions and outlook

Using a two-scale asymptotic ansatz, we extended in a systematic way the region of validity of the planetary-scale model from DK to the synoptic spatial and temporal scales. The resulting multiscale model is summarized in (40)–(45). Already Mak (1991) incorporated in the QG model spherical geometry by considering higher-order terms, but his model is valid for motions characterized by length scales smaller than the planetary scale. The model presented here consists of two coupled parts—for the planetary and for the synoptic dynamics. This is different from the geostrophic potential vorticity model of Vallis (Vallis 1996; Mundt et al. 1997), which consists of a single PV equation valid on the planetary and on the synoptic scales. The latter model is derived by choosing an appropriate scaling, which allows both the limit for the QG model and the limit for the PGEs, whereas here we have applied a multiscale asymptotic derivation. The two-scale wave models of Luo (2005) and Luo et al. (2007) assume a scale separation between planetary and synoptic motion only in zonal direction; here, we considered a horizontally isotropic planetary scaling. A study with the asymptotic approach, as applied here, of anisotropic motions with planetary zonal scale, but meridionally confined to the synoptic scale, reveals a model that describes a coupling between the planetary evolution of the leading QG PV and the synoptic evolution of the first-order PV corrections from the QG^{+1} model of Muraki et al. (1999) [details of this regime can be found in Dolaptchiev (2009)]. The anisotropic multiscale ocean model of Grooms et al. (2011) is another example for an anisotropic scaling of the large-scale coordinates (here the planetary coordinates): the meridional coordinate in this model resolves a planetary length scale, whereas the large-scale zonal coordinate resolves a scale between the planetary and the synoptic spatial scales. In the context of the atmosphere, the external Rossby deformation radius (Obukhov scale) might be a natural choice for an intermediate large-scale length scale between the planetary and synoptic scales. Such scale is relevant for atmospheric blockings and within the present asymptotic approach it can be accessed in a systematic way.

Equations (40) and (44) represent the anelastic analogon of Pedlosky's two-scale model for the large-scale oceanic circulation (Pedlosky 1984). In his study Pedlosky (1984) applied an asymptotic expansion in two small parameters: one is the Rossby number and the other is the ratio between the synoptic and the planetary length scales. For the derivation of his model, he considered the case when the ratio between the two small parameters is of the order one. Expressing in terms of *ɛ* Pedlosky's expansion parameters for our setup, it can be shown that their ratio is again one, which means that we have considered the same distinguished limit. The analysis of Pedlosky starts from the incompressible equations on a plane; here, we study the compressible ones on a sphere. Nevertheless, the model PV transport equations have the same structure and are identical if we set *ρ*_{0} in (40) and (44) to one and neglect the effects due to the spherical geometry. A fundamental difference is the absence of a counterpart to the barotropic vorticity equation (41) in Pedlosky's model. In the ocean the barotropic component of the planetary-scale flow is determined, for example, by prescribing the surface wind or by including some additional friction in the leading-order momentum equation. This is not applicable to the atmosphere, since the surface winds should be a part of the solution and the frictional effects are much smaller than in the ocean.

The additional evolution equation for the barotropic component of the flow [see (41)] provides the only feedback from the synoptic-scale processes to the planetary-scale flow in the form of momentum fluxes. No such feedback is contained in Pedlosky's model. This type of feedback mechanism on the planetary scale differs from the one recently proposed by Grooms et al. (2011), where the planetary-scale motion is influenced by the synoptic scale through eddy buoyancy fluxes.

One possible application of the two-scale PR model presented here is its implementation in the atmospheric module of an earth system model of intermediate complexity (EMIC; Claussen et al. 2002). The Climate and Biosphere (CLIMBER) EMIC (Petoukhov et al. 2000) solves a type of the PGEs (40) and (43), but it uses a temperature-based diagnostic closure for the barotropic component of the flow. Here (41) represents a prognostic alternative, which may provide for more realistic increased large-scale, low-frequency variability in future implementations.

In EMICs the synoptic fluxes are often parameterized as a macroturbulent diffusion. In this context the model for the synoptic-scale dynamics (44) can be regarded as a higher-order closure. The solution of the additional evolution equation for the synoptic scales might be avoided by applying a stochastic mode reduction strategy (Majda et al. 2003; Franzke et al. 2005; Franzke and Majda 2006; Dolaptchiev et al. 2012). Using this strategy one can derive stochastic differential equations for some “slow” variables taking into account in a systematic manner the interactions from the “fast” variables. In the case of the two-scale PR model, we have a natural separation between fast (synoptic) and slow (planetary) modes. Thus one might apply a stochastic mode reduction procedure to the reduced two-scale model and derive a stochastic parameterization for the synoptic correlation terms in (41), which is consistent with the synoptic-scale model (44). An alternative approach avoiding synoptic-scale parameterization is followed by Luo (2005) and Luo et al. (2007) in studies of blockings and NAO dynamics. The latter phenomena are considered as nonlinear initial value problems of planetary-synoptic interactions; this allows one to assume a synoptic eddy forcing prior to the evolution of the planetary-scale motion.

The reduced barotropic vorticity equation has the potential to provide a diagnostic tool for studying planetary-scale low-frequency dynamics in GCM or in observations. A number of studies (Cai and van den Dool 1994; Feldstein 1998, 2002; Franzke 2002) on the life cycle of atmospheric low-frequency anomalies utilize budget analysis with the streamfunction tendency equation. In particular, with such an analysis the importance of different interaction terms (e.g., interactions with the time-mean flow or high- and low-frequency transients) can be assessed systematically. In this context, the asymptotic analysis presented here stresses the importance of the barotropic, zonally symmetric component of the flow for the low-frequency dynamics. Further, it identifies terms containing zonally and vertically averaged synoptic-scale momentum fluxes (or planetary meridional gradients of such fluxes) as relevant planetary–synoptic interactions. Those terms can be evaluated from observational data or GCM simulations and might be used as a diagnostic tool in interaction studies. Thus, the reduced planetary-scale barotropic vorticity equation provides an alternative framework to apply a budget analysis when the growth and decay of zonally symmetric anomalies with a planetary meridional scale (e.g., NAM and SAM) are investigated. Such a model might give new insights in the interactions between the different spatial scales. Those spatial interactions are studied in the literature (Cai and van den Dool 1994; Feldstein 1998, 2002; Franzke 2002) by splitting the flow into zonal average and its deviation, whereas in the present approach the planetary and synoptic scales are associated with different ranges in wavenumber space. Another application of the present model is to use it as a data-driven planetary-scale model in a way similar to Feldstein (2002). In such a model the synoptic fluxes are prescribed from GCM simulation or observation and the effect on the planetary-scale dynamics can be studied by solving the reduced model equations.

The analysis from section 5 of numerical simulations with a primitive equations model showed that the leading-order balances in the vorticity transport are consistent with the two-scale asymptotic model. In particular, we find that for modes with planetary spatial scales (modes corresponding to spherical harmonics with a total wavenumber ≤ 2) the horizontal fluxes of relative and planetary vorticity are nearly divergence free. However, the transition between a planetary and a synoptic regime is smooth in the primitive equations model. The comparison between the numerical experiments and the asymptotic models can be extended in the present framework by considering the thermodynamic equation or higher-order balances on the planetary and synoptic scales. The asymptotic analysis revealed that some higher-order terms involve corrections to the leading-order wind. These corrections can be calculated from the model output by considering the divergent part of the wind.

In the future we plan to solve the two-scale PR model numerically. This raises the question about the model behavior in the tropics where *f* tends to zero. If no frictional effects are considered, the geostrophically balanced leading-order wind has a singularity at the equator. However, the asymptotic analysis of Majda and Klein (2003) showed that the background temperature field in the tropics is horizontally uniform (also known as the weak temperature gradient approximation). This condition on the temperature implies a vanishing pressure gradient that compensates the growth due to *f*. In the case of the two-scale PR model, further analysis is required; this model should be matched in a systematic way to the intraseasonal planetary equatorial synoptic-scale model of Majda and Klein (2003).

## Acknowledgments

The authors thank the reviewers for their comments and suggestions, which helped to improve the draft version of the manuscript. S. D. is thankful to U. Achatz for useful discussions. This contribution is partially supported by Deutsche Forschungsgemeinschaft, Grant KL 611/14.

## APPENDIX A

### PV Formulation of the Two-Scale Model

*S*. In (A2), both the planetary and the synoptic scales are involved; we have reduced the unknown variables to two

_{pυ}*π*

^{(2)}(

*X*,

_{P}*z*) and

*π*

^{(3)}(

*X*,

_{S}*X*,

_{P}*z*), since

**u**

^{(0)}, Θ

^{(2)}, Θ

^{(3)}, and

*ζ*

^{(0)}can be expressed in terms of them; see (22), (20), (21), and (31). Next, we derive two separate equations for the unknowns; as usual in the multiple-scales asymptotic techniques, this is achieved by applying the sublinear growth condition [see also Pedlosky (1984)].

*S*and

_{pυ}*S*the deviations from this average. The advection terms on the rhs of (A3) can be written as the divergence of a flux

_{q}## APPENDIX B

### Evolution Equation for the Barotropic Component of the Pressure

As discussed in DK, the planetary-scale PV equation (A7) requires a closure for the vertically averaged pressure *p*^{(2)} (barotropic component). Here we derive an evolution equation for that component in the two-scale setup form section 2b. To see the net effect from the synoptic scales on the planetary-scale pressure distribution, we have to average first the asymptotic equations from section 3 over the synoptic variables.

#### a. Averaging over the synoptic scales

##### 1) Continuity equation

##### 2) Potential temperature equation

##### 3) Momentum equation

#### b. Derivation of the evolution equation for the planetary-scale barotropic pressure

*z*and

*λ*to obtain

_{P}*λ*and

_{P}*z*;

*ρ*

^{(2)}in (B10) can be expressed with the help of (B9) in terms of

*p*

^{(2)}only [see also (73)–(77) from DK]; thus, we have

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