## 1. Introduction

Dynamic processes play a crucial role in the middle atmosphere. Circulation in the middle atmosphere transports minor species such as ozone, affecting chemical and radiative processes, and modifies the temperature structure through adiabatic heating/cooling associated with vertical branches. Middle-atmosphere circulation and transient dynamical phenomena such as stratospheric sudden warming (SSW) also affect tropospheric dynamics (Baldwin and Dunkerton 2001; Kodera and Eguchi 2009; Kohma and Sato 2013; Hirano et al. 2016). Lagrangian-mean middle-atmospheric circulation is mainly driven by the remote redistribution of momentum by atmospheric waves. Most previous studies have examined the middle atmosphere in terms of zonal-mean features. An approximate form of Lagrangian-mean circulation was derived as the residual circulation of transformed Eulerian-mean (TEM) equations by Andrews and McIntyre (1976). Wave forcing is described as the Eliassen–Palm (EP) flux divergence ∇ ⋅ **F** in the TEM equations. As the first approximation for the large-scale circulation at midlatitudes, the nonlinear terms are neglected to examine zonal-mean fields. Using these approximations, the equation of motion for the zonal direction is composed of the following three terms: the acceleration of zonal wind, the Coriolis term, and wave forcing. When only wave forcing is given, the other two terms cannot be determined from the zonal momentum equation alone. However, it is necessary to determine how wave forcing distributes to zonal wind acceleration and Coriolis torque, as the latter relates meridional circulation to zonal wave forcing.

*ψ*is the streamfunction of meridional circulation,

*ϕ*is latitude,

*z*is altitude,

*ρ*

_{0}is basic density,

*a*is the radius of the earth,

*ϕ*,

*z*) is angular momentum. The variables denoted by overbar are zonal-mean ones. The term downward control principle was coined based on the fact that the mean vertical velocity at a vertical level

*z*is determined solely by the vertical integral of wave forcing between

*z*and the top of the atmosphere. The DC principle is advantageous in that the contributions of each wave forcing to circulation can be evaluated, as (1) is a linear equation for wave forcing

However, the DC principle has some limitations. First, it uses the steady-state assumption. Generally speaking, wave forcing is not constant, and thus, the *u*_{t} term is not negligible. Extratropical planetary wave forcing is sometimes larger instantaneously than its climatological value and causes circulation change. The time scale of this variation of the wave forcing and mean circulation is one or two weeks. Such a strong wave-forcing event occurs several times in a given winter (Randel et al. 2002). The out-of-phase relationship of polar warming and tropical cooling resulting from such rapidly varying wave-forcing events have been illustrated through the satellite observations (Fritz and Soules 1970). In addition, the group velocities of gravity waves are generally much faster than those of Rossby waves. Thus, the wave forcing caused by a burst of gravity waves may vary more rapidly than that by a Rossby wave one. The response to such a transient wave forcing needs to be investigated to understand the dynamics of the atmospheric circulation.

Responses to unsteady wave forcing have also been investigated in several previous studies. Garcia (1987) theoretically investigated responses to oscillating forcing. When the frequency of forcing is sufficiently low, the forced circulation accords with the steady DC state. Hereafter, the steady state and the steady circulation obtained by the DC principle are referred to as the steady DC state and the steady DC circulation, respectively. When the frequency of forcing is sufficiently high, the response occurs independently of the degree of relaxation through what is referred to as “the adiabatic regime” response (Dickinson 1968). Holton et al. (1995) investigated the difference between adiabatic and steady responses using an idealized numerical model. The steady response, when the frequency of forcing is sufficiently low, accords to the steady DC state. When forcing oscillates rapidly, the meridional circulation does not reach the ground like the steady-state solution but is observed in the surrounding region of the wave forcing. Haynes et al. (1991) theoretically examined the formation of meridional circulation as a response to unsteady step function wave forcing using quasigeostrophic equations. The response is described in terms of the initial time evolution value and final steady state. The final steady state is the form of circulation that is identical to the steady DC state. In contrast, initial time evolution is described as a form of localized balanced circulation that varies in strength. The initial circulation has a vertically aligned two-celled structure that gradually extends downward after forcing occurs, which describes the formation of the steady DC circulation. Semeniuk and Shepherd (2002) investigated a case in which the radiative relaxation rate was not uniform temporally and spatially (while DC principle uses the uniform damping rate), and the response was found to be modulated by the spatial and temporal variability of relaxation.

A typical phenomenon for which wave-forcing unsteadiness is critical is the stratospheric sudden warming (SSW). SSW is a phenomenon in which the temperature of the polar stratosphere increases by several tens of degrees Celsius over several days (Sherhag 1952). Zonal winds are usually eastward in the winter stratosphere, allowing planetary-scale Rossby waves to propagate upward (Charney and Drazin 1961). Westward forcing caused by Rossby wave breaking thus induces a poleward flow, and adiabatic heating associated with downwelling over the polar cap warms that region. When SSW occurs, wave-forcing time scales may be so short that behaviors of the resultant circulation differ from those expected by the DC principle. In fact, Matsuno (1971) theoretically showed that circulation accompanied by SSW includes a vertically aligned two-celled structure. The circulation causes not only polar stratospheric warming but also polar mesospheric cooling, midlatitude stratospheric cooling, and midlatitude mesospheric warming. This quadrupole temperature pattern was validated through a numerical model (Matsuno 1971) and through satellite observations (Labitzke 1972).

Hitchcock and Shepherd (2013) investigated a recovery response of the Arctic polar vortex occurring after SSW events using a similar framework as that employed by Haynes et al. (1991). Most previous studies of transient circulation responses have used the quasigeostrophic (QG) equations, which apply an assumption whereby zonal wind and temperature satisfy thermal wind balance relations. When the time scale of forcing is sufficiently short, however, the resulting response can include an unbalanced component. In such cases, the response should be described through a primitive equation. In fact, Zhu and Holton (1987) examined gravity wave generation as a response to unsteady wave forcing in the primitive equation. It is thus necessary to treat the time tendencies of meridional circulation explicitly (which is ignored in QG equations) to investigate responses to switch-on forcing, which occurs at various frequencies.

The dynamics of primitive equations can be divided into two variables (Saujani and Shepherd 2006). One component is geostrophic motion and is described by a slow variable. The other component is described by two fast variables, which include a gravity wave solution in their linearized equations. However, when nonlinear terms and/or external forcing exist, fast variables include slowly varying components, which describe ageostrophic dynamics referred to as slaved components (Leith 1980; Yasuda et al. 2015).

It is also worth noting that Zhu and Holton (1987) examined the evolution of fast variables to study the secondary generation of gravity waves, but they did not examine slave components corresponding to circulation induced by wave drag. The secondary generation of the gravity wave is also confirmed in numerical simulation. A compressive numerical model of Satomura and Sato (1999) reproduces small-scale gravity waves generated by the breaking of mountain waves, which are similar to the observations (Sato and Hirota 1988).

The purpose of this study is to theoretically examine the response of zonal-mean meridional circulation to unsteady wave forcings by using primitive equations, which allows us to treat the time evolution of fast variables. Two issues are highlighted. One pertains to the formation process of meridional circulation in terms of structures and time scales. The other concerns how the wave forcing distributes the zonal flow acceleration term and Coriolis term in the zonal momentum equation. This study is structured as follows: In section 2, descriptions of the governing equations and analysis method applied are given. In section 3, responses to unsteady wave forcing are calculated from Green’s function. In section 4, the time scale of circulation formation and relation to forcing time scales are discussed. In section 5, the distribution of wave forcing to the zonal flow acceleration term and Coriolis term are examined. A summary and concluding remarks are presented in section 6.

## 2. The governing equations and method

*f*plane are used as the governing equation (Vallis 2006):where

*p*′ is normalized pressure, which is the pressure deviation divided by density

*ρ*

_{0};

*b*is buoyancy;

*f*≡ 2Ω sin

*ϕ*

_{0}is the Coriolis parameter; and

*N*is the buoyancy frequency. Material derivative is defined asand

*R*

_{0}is the Rossby number. The

*f*plane is used (i.e.,

*β*effects are neglected), as a Rossby wave–type response is not expected in zonal-mean equations. Linear relaxation terms with the same relaxation factor

*κ*are included in the momentum equations [(3) and (4)] and thermodynamic equation [(6)]. Three external forcings (zonal forcing

*X*, meridional forcing

*Y*, and diabatic heating

*L*

_{F}, and

*H*

_{F}, dimensionless parameters are

*t** =

*ft*,

*x**=

*xL*

_{F},

*y**=

*yL*

_{F}, and

*z**=

*zL*

_{F}, yieldingThese equations have dimensionless numbers, including a Rossby number

*R*

_{0}=

*U*/

*fL*

_{F}, a Burger number

*B*

_{u}= (

*NH*

_{F}/

*fL*

_{F})

^{2}, and a modified Ekman number

*κ** =

*κ*/

*f*. Hereafter, asterisks are omitted. As we are interested in large-scale atmospheric responses to forcing, it is assumed that the Rossby number is sufficiently small. Thus, the governing equations become linear as follows:As in Saujani and Shepherd (2006), three variables are used—linearized potential vorticity

*q*, horizontal divergence

*δ*, and ageostrophic vorticity

*γ*:The governing equations are transformed into the equations using

*q*,

*δ*, and

*γ*:whereandThe equation of the slow variable

*q*[(22)] is obtained by cross differentiation of (14) and (15) and using the definition (19). The equations of fast variables [(23) and (24)] are obtained by the time differentiation of (20) and (21). The linear PV

*q*is referred to as a slow variable because it describes geostrophic flows with small tendencies. Horizontal divergence

*δ*and ageostrophic vorticity

*γ*are fast variables that include zonally symmetric gravity waves. The dispersion relation of the gravity wave solution iswhere

*ω*is an angular frequency and

*l*and

*m*are meridional and vertical wavenumbers, respectively. Note that the fast variables also include a slaved component with small tendencies when nonlinear terms and/or external forcing terms are present (Leith 1980).

To investigate these linear equations for a given forcing, Green’s function method is useful. Green’s function has two advantages. The first relates to its low calculation cost: once Green’s function is obtained, the response to any forcing can be determined by convolution alone. The other relates to the treatment of boundary condition: the boundary condition under which the response becomes zero at infinity can be used easily when applying Green’s function. This condition is applied to all boundaries (i.e., upper, lower, and meridional boundaries) in the present study.

*δ*and

*γ*) obeys the following equation:where

*D*(

*s*) is the delta function for the independent variable

*s*. To apply Green’s function, the Laplace transform is used following Zhu and Holton (1987). The Laplace transform of Green’s function for (30) isHere, the boundary condition is used so that the response is zero at infinity. On the other hand, Green’s function of the slow variable

*q*fulfillsand the solution isThis formula for

*G*

_{q}indicates that the spatial pattern of slow variable does not change over time, although the amplitude can change.

## 3. Results

*X*, meridional forcing

*Y*, diabatic heating

*X*(

*y*,

*z*,

*t*) =

*F*

_{X}(

*y*,

*z*)

*F*

_{T}(

*t*) The spatial structure

*F*

_{X}(

*y*,

*z*) of

*X*takes a Gaussian shape vertically and a shape with considerable kurtosis in the latitudinal direction:

where *L*_{F} is the latitudinal (i.e., horizontal) scale, *c* and *d* are parameters, and (*y*_{0}, *z*_{0}) is the central latitude and height of the forcing (Fig. 1a).The aspect ratio of the forcing *α* is defined as the ratio of vertical to the horizontal length scales of the wave forcing. Now, the aspect ratio of the forcing (*α* = *NH*_{F}/*fL*_{F}) is taken to be one for the normalization. Dependence of the response on *α* will be investigated in section 4. A Gaussian spatial pattern applies for diabatic heating *Y* as is shown in Fig. 1b. Three forms of time dependence are considered:

- (i)
(ii) and (iii)

*H*(

*t*), which changes in a discontinuous manner from 0 to 1 at

*t*= 0; note that the Laplace transform of the step function is

*s*:

*t*) function is dependent on the forcing time scale

*T*

_{F}as follows:

These forcings are given at a latitude of 30°. The time scale of the relaxation is taken as 1/*κ* = 200, which corresponds to 32 days (1/*κ* in dimensional form equals 1/(*κ***f*) = 2.76 × 10^{6} s). This parameter setting is somewhat different from the real middle atmosphere where the thermal damping time scale is much faster and the mechanical damping one is much slower. However, the initial response of the fast variable does not depend on the strength of the relaxation because the time scale of the relaxation is determined by the slow variable. Thus, the dependence of the relaxation can be neglected as long as the initial response is considered. Our interest is mainly the initial transient response of the fast variables. The relation between our result and Haynes et al. (1991) is discussed later in this section and in section 6. Forcing responses are calculated in the dimensionless form, but results are shown in the dimensional form to facilitate comparisons with the real atmosphere.

### a. Steady-state solutions

*G*can be obtained from the following equation:The

_{s}*G*

_{δs}(

*y*,

*z*) for fast variables:On the other hand, the steady-state Green’s function for linearized PV becomes

*X*is determined. Here,

*Y*and

*δ*in the meridional section determined using Green’s function (38) is shown in Fig. 2a. Figure 2a shows that meridional circulation occurs in a two-celled vertical structure (Fig. 2c). The structure of

*q*obtained based on (39) or directly calculated from (40) is shown in Fig. 2b. Physical parameters such as zonal wind

*u*, buoyancy deviation

*b*, meridional wind

*υ*, and vertical wind

*w*are obtained using the solutions of

*δ*and

*q*, and corresponding results are shown in Fig. 3.

This two-celled steady solution is different from the result of DC principle (Haynes et al. 1991). This difference is caused mainly by the presence of dynamical relaxation given in the present study. The DC principle specifies no mechanical relaxation other than boundary layer friction, considering that radiative relaxation is a major dissipation process in the middle atmosphere. On the other hand, in this study, the mechanical damping rate has the same value as the thermal one. Thus, the steady state of the present study may be an unrealistic situation but can be regarded as the “semi”-steady state corresponding to the initial response of Haynes et al. (1991). Detailed discussion will be made in section 6.

*X*=

*Y*= 0 isThe spatial pattern of diabatic heating

*δ*and

*q*are shown in Figs. 4a and 4b, respectively. Physical variables

*b*,

*u*,

*υ*, and

*w*obtained through the inversion of fast and slow variables are shown in Figs. 4c–f. Meridional circulation involves an upward branch in the heating region and two downward branches to the south and north of the heating areas.

*Y*differs from responses to zonal forcing

*X*and diabatic heating

*Y*, as is shown in Fig. 1b. The governing equation with

*X*=

*γ*only. In other words,

*q*and

*δ*are zero. Thus, no geostrophic motion or meridional circulation occurs. The response only involves ageostrophic horizontal winds. When

*κ*≪ 1, (48) becomesThe structures of

*γ*calculated from (49) and of

*u*,

*p*, and

*b*are shown in Fig. 5. Note that the meridional equation of motion isIn the forcing region,

*fu*and ∂

*p*/∂

*y*have the same sign. Interestingly, however, in some regions, the two terms have opposite signs, indicating that these two terms satisfy the relation of geostrophic balance. This result shows that ageostrophic vorticity contains a flow component of geostrophic balance, which should be distinguished from QG flow.

As previously mentioned, the inclusion of both dynamical and thermal relaxation in the present study is not a realistic condition for the steady state. However, the steady-state solution gives an insight into the semi-steady condition as discussed in section 5. The relation between our results and those by the DC principle will be examined in section 6.

### b. Time-varying forcing

*X*. Responses for meridional forcing

*Y*and diabatic heating

*H*(

*t*) is considered. First, the response of

*q*is examined. Note again that Green’s function for this case is described by (33), which is a monotonically decreasing function in time. Thus, Green’s function is obtained from the integral of (33) in time as

Second, the response of horizontal divergence *δ* is investigated. The Laplace transform of Green’s function is described by (31). The Laplace inverse transformation in time is performed using the fast Fourier transform. Horizontal divergence *δ* obtained in this way is illustrated in the meridional–height cross section of Fig. 6. Immediately after forcing is introduced at *t* = 0, a response appears around northern and southern edges of the forcing. This response spreads latitudinally with time. It is evident that the strength of the response at *t* = 6 h (Fig. 6f) is almost equal to the steady state (see Fig. 2). However, there is a notable difference in the zero-line structures shown in Fig. 6f and of the steady state. The structure of horizontal divergence *δ* at a later time is shown in Fig. 7. The wavelike structure has a large amplitude whose magnitude is roughly the same as that of the circulation for *t* = 1 and *t* = 2 days (Figs. 7a and 7b). Both the structure and strength of *δ* for *t* = 3 and *t* = 4 days (Figs. 7c and 7d) are similar to those of the steady state, although a wavelike structure, which has low frequency, is observed above and below the forcing. This result suggests that the initial response of circulation described by the QG equations takes a few days to develop. The waves that initially appear are likely gravity waves judging from the phase structure and propagation direction (not shown). These gravity waves must have zonally symmetric structure (zonal wavenumber *k* = 0) and displacements only in the latitude and altitude directions. To investigate wave characteristics, anomalies from the steady state were obtained (Fig. 8). Gravity wave–like patterns are clearly shown. The phase line of disturbances at a later time is more horizontal, meaning that disturbances of lower frequencies and smaller horizontal wavenumbers have larger amplitude in later time. The frequency of this disturbance is nearly the same as the inertial frequency (Figs. 8e–g). We note that these gravity waves have quite large meridional scales comparable to those of meridional circulation.

## 4. Time scale of the circulation formation

### a. Theoretical prediction

As stated above, the steady-state response involves vertically two-cell aligned circulation. The purpose of this subsection is to theoretically investigate the time scale of circulation formation after the forcing is introduced. In this section, dimensionless variables are denoted by asterisks.

*B*

_{u}and the strength of linear relaxation

*κ**. A condition in which the time scale of the relaxation is much longer than the inertial period is considered. Thus, the governing equation for the fast variable

*δ** isNote again that

*κ** is negligible for our discussion of fast variables, and hence, only dependence on the parameter

*B*

_{u}is considered.

The dependence of the response can also be determined through a dimension analysis. Parameters determining this response include the *N*, *f*, horizontal scale *L*_{F}, and vertical scale of the forcing. In a Boussinesq system, there are no other specific length scales. Thus, responses are determined based on the horizontal–vertical length ratio of the forcing *H*_{F}/*L*_{F} when *f* and *N* are constant. The vertical length scale of the dimensionless form becomes *N*/*f*)*H*_{F}, meaning that the change in *f* and *N* is regarded as the change in the modified vertical scale. Thus, the time scale is dependent only on the normalized aspect ratio (hereafter referred to as the aspect ratio) *α* = *NH*_{F}/*fL*_{F}, which is the square root of the Burger number. Note that the aspect ratio of the forcing so far is taken to be one for the normalization. Here, the dependence of the time scale on *α* is examined.

*t** =

*ft*, the time scale of circulation formation should beThus, the time scale of circulation formation can be written using an undetermined function Ψ(⋅) as follows:This equation indicates that

*T*is a function of aspect ratio

*α*only.

*T*

_{g}, as the large-scale gravity waves of large amplitudes dominate as the transient response (Fig. 7). This time scale

*T*

_{g}is written as the ratio of the latitudinal scale of forcing (

*L*

_{F}) to the latitudinal group velocity of gravity waves |

*c*

_{gy}|:We use the dispersion relation of the hydrostatic inertia–gravity wave (29),In addition, from the wavenumbers

*l*= 2π/

*L*

_{F}and

*m*= 2π/

*H*

_{F}of most dominant gravity waves, we obtainwhereThe fact that is proportional to

*f*

^{−1}is consistent with our theoretical expectation (54). The function Ψ

_{g}can also be obtained by using

*T*

_{g}=

*H*

_{F}/|

*c*

_{gy}|.

### b. Numerical calculation

*δ*into the following two components—a circulation component

*A*(

*t*)

*δ*

_{s}(

*x*) and remaining component

*δ*′:where

*δ*

_{s}(

*x*) is the spatial structure of steady-state horizontal divergence. The magnitude of the circulation component

*A*(

*t*) is determined to minimize the spatial integralThis is equivalent to the following condition:

The responses for *α* = 0.2, 1, and 5 are shown in Figs. 9a–c. It is evident that *A*(*t*) oscillates and decays. The oscillation frequency is almost the same as the near-inertial frequency. This result reflects the fact that near-inertial oscillation slowly decays and that the spatial structure of near-inertial oscillation resembles that of steady-state circulation. Figures 9a–c also show that for larger *α*, that is, a longer vertical length of forcing *H*_{F} at a given *L*_{F}, *A*(*t*) decreases faster, and thus, circulation formation is also faster. The envelop of the difference between the circulation component *A*(*t*) including near-inertial oscillations and the steady-state value |*A*(*t*) − 1| for *α* = 0.2, 0.33, 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, and 5.0 are shown in Fig. 9d. Responses to the forcings with larger *α* values approach the steady state faster, consistent with the theoretical prediction (58). The numerically obtained formation time scale *T*_{B} is defined as the time it takes for |*A*(*t*) − 1| to become smaller than a particular value of *B*. By taking *B* = 0.15, the *α* dependence of the *T*_{0.15}(*α*) is shown in Fig. 9e. To illustrate a comparison to the theoretical prediction, the *α* dependence of theoretically predicted *T*_{g} is denoted by the solid line. It is evident that the *α* dependence of *T*_{0.15}(*α*) coincides well with that of *T*_{g}(*α*).

### c. The dependence of circulation formation on the time scale of forcing

*α*= 1. To examine the dependence of the time scale of circulation formation on the forcing time scale

*T*

_{F}, Green’s function method is used once again. The time evolution of the forcing

*t*) and magnitude of circulation

*A*(

*t*) obtained using (61) are shown in Figs. 10a and 10b for

*T*

_{F}= 0.1 and

*T*

_{F}= 1.5 days. When

*T*

_{F}is small,

*A*(

*t*) (i.e., the response does not follow the forcing); instead, it shows an oscillatory behavior whose frequency is near the inertial frequency. On the other hand, when the forcing changes slowly, (i.e., for larger

*T*

_{F}), the response follows the forcing. To quantitatively investigate this feature, the dependence of circulation formation on

*T*

_{F}is determined. Figure 10c presents a function

*M*(

*T*

_{F}) showing the maximum difference

*A*(

*t*) from the magnitude of forcing

*t*) as a function of

*T*

_{F}. It is clear that the response is slaved to the forcing when

*T*

_{F}is longer than 1 day (i.e., the inertial period). Thus, it can be concluded that the response to a slowly varying wave forcing such as Rossby wave forcing would be slaved to the forcing, whereas an abrupt forcing such as that caused by a gravity wave burst would produce zonally symmetric gravity wave radiation.

## 5. Relative magnitudes of the *u*_{t} and −*fυ* that balance the zonal forcing in a semisteady state

*u*

_{t}and Coriolis torque −

*fυ*that arise as the response to a given wave forcing are investigated when the forcing is only

*X*. In section 4, the time scale of circulation formation was estimated. After

*t*≥

*T*

_{g}(i.e., several days) for the forcing with the step function, the fast variable roughly accords with that of the steady state. On the other hand, the time scale of the slow variable

*κ*

^{−1}is usually much longer than that of fast variables in the stratosphere. Thus, even when meridional circulation described by fast variables becomes steady state, the slow variable may not yet reach the steady state. Moreover, the relaxation term can be ignored when fast variables reach the steady state. This point can be understood as follows. The effect of the relaxation, which is proportional to the strength of the geostrophic wind, is not large as long as the characteristic time scale of the fast transient response such as gravity waves is shorter than 1/

*κ*. In other words, in the time-dependent momentum equation,the term

*κu*plays a negligible role as long as ∂

*u*/∂

*t*≪

*κu*, which is satisfied during

*t*< 1/

*κ*. Thus, from here, we consider a case in which the fast variable become a steady state while the relaxation term of the slow variable remained sufficiently small. This is realized when

*T*

_{g}<

*t*< 1/

*κ*. Because the fast variables are steady, and the slow variable is not steady, this state is regarded as a “semi”-steady state. In the semisteady state, relaxation terms are negligible, and thus, the Coriolis torque and the adiabatic heating are balanced with the acceleration terms, which is the same situation of the initial response of Haynes et al. (1991).

*u*

_{t}and −

*fυ*are obtained from (65) and (66),

*X*can be obtained within a limit of

*κ*≪

*f*as follows:where

*α*=

*NH*

_{F}/

*fL*

_{F}as before and [⋅] denotes here the magnitude of the term. This result shows that when the aspect ratio

*α*is larger (i.e., when the vertical scale of the forcing is larger), meridional wind is weakened, and zonal wind acceleration is longer. The expression of the strength of two terms is the special case of (14) of Garcia (1987) with a condition where the frequency of the forcing is zero (i.e., steady) and the strengths of mechanical and thermal relaxations are the same. His equation uses a simple typical wavelength for each of the meridional and vertical scales. However, in general, wave forcing is composed of wavelengths distributed over a wide range. It is interesting to compare the simple theoretical expectation with solutions numerically obtained using the Green’s function for the more realistic forcing with a shape of (34) (Fig. 1a).

*X*] and [−

*fυ*], respectively, are evaluated aswhere, as the latitude

**y**

_{0}at which this integral is performed, the central latitude of forcing is used. The interval of the integral used to calculate [−

*fυ*] and [

*X*] is the region where −

*fυ*< 0 and

*X*< 0, respectively. Figure 12 shows the theoretical and numerical results for the magnitude of acceleration and Coriolis torque as a function of

*α*. The broken curve stands for the

*α*dependence of the relative strength of the Coriolis torque theoretically calculated by (71). It is seen that the Coriolis torque is weaker for larger

*α*. The dots of Fig. 12 stand for the numerically calculated

*α*dependence. These two

*α*dependences are qualitatively similar; however, the difference becomes larger when

*α*is larger than 1. This is attributable to the fact that the numerical calculation includes various meridional and vertical wavenumbers in the forcing, whereas only one wavenumber is considered in the theoretical investigation. This is an advantage of our method providing the meridional circulation formed by the wave forcing having wavelength in a wide range as in the real atmosphere. When

*α*> 3 (i.e., vertically long wave forcing), (71) shows the

*fυ*magnitude quite small, which is less than 1/10 of the wave forcing. In contrast, the numerical result suggests that |

*fυ*| should be significantly large even for such large

*α*, which is 4 times larger than that from (71) for

*α*= 5. This large discrepancy is due to the fact that the wave forcing given in the present study is composed of a multitude of wavenumber components and relating minor wavenumber components that largely affect the ratio of [−

*fυ*] to [

*X*].

*p*

_{z}=

*b*, the strengths of the two terms on the left-hand side of (75) are obtained as follows:It is interesting that the distribution of the forcing to tendency and meridional circulation for diabatic heating

*X*. The results of the numerical calculation using the Green’s function method are shown in Fig. 13.

## 6. Relations with the previous studies

In this study, the strength of dynamical relaxation (*κ*_{F}) is taken as the same as that of thermodynamic relaxation (*κ*_{T}). When two relaxation coefficients *κ*_{F} and *κ*_{T} differ, however, steady-state circulation will also differ from vertically aligned two-celled circulation patterns shown in this study. However, it is expected that the evolution of the circulation until the relaxation plays an important role (*t* < 1/*κ*) does not depend on relative strengths of mechanical and thermal damping rates, because the relaxation term is proportional to the strength of the wave-induced geostrophic flow (the deviation of the geostrophic flow from the balanced background flow), which is sufficiently small initially. Thus, as long as the degree of relaxation is sufficiently small at an early stage after the forcing is given, the formation process of two-celled circulation is the same as that shown through the present study. This two-celled circulation will gradually change to a steady state with the time scale of relaxation depending on the strength of *κ*_{F} and *κ*_{T}. Therefore, the steady-state meridional circulation calculated based on the DC principle in Haynes et al. (1991) differs from the vertically aligned two-celled circulation values obtained through the present study. The difference is attributed to the fact that there is no mechanical (frictional) relaxation in the DC principle. However, it should be noted that the response in the semisteady state of our study (*T*_{g} < *t* < 1/*κ*) and the initial response in Haynes et al. (1991) have similarities. In the semisteady state of the present study, the gravity waves disappear and quasi-steady meridional circulation remains. As the geostrophic wind is quite weak, the relaxation terms are negligible. Thus, the acceleration term and Coriolis torque are balanced with the wave forcing in the semisteady state. On the other hand, Haynes et al. (1991) uses QG equations. Thus, there are no gravity waves. Soon after the forcing is on, the geostrophic wind component in the transient response is considered to be quite small. These features are the same as those in the semisteady state of the present study. The differences between the solution by Haynes et al. (1991) and ours appear after the condition of the semisteady state. For the case of the present study, the meridional circulation does not change, and the geostrophic wind becomes strong and approaches asymptotically to an unrealistic steady state. On the other hand, because only the thermal relaxation exists, the response in Haynes et al. (1991) changes from the initial semisteady state to the realistic steady DC state. The initial transient behavior including radiation of gravity waves investigated in section 4 cannot be investigated in QG equations. However, it is possible that such a transient response including gravity waves occurs in the real atmosphere as a transient response before reaching the semisteady state. The evolution of the circulation caused by wave forcing and the relation between the semisteady state of the present study and the steady DC state is schematically illustrated in Fig. 14. This figure shows the relation of evolutions of geostrophic flow (slow variable *q* or *u* and *T*) and meridional circulation (fast variable *δ* and *γ* or *υ* and *w*) examined by the present study and by Haynes et al. (1991). The white arrows in Fig. 14 stand for the time formation process of our study. The fast variables change rapidly and reach the steady state on the time scale of gravity wave propagation, while the slow variable changes much more slowly on the time scale of mechanical and thermal relaxation. The transient circulation response to the steady DC state is denoted by colored arrows in Fig. 14 The meridional circulation initially has a two-celled structure, but the structure as well as the magnitude gradually changes as the relaxation becomes larger depending on the respective relaxation coefficient. Thus, the steady meridional circulation obtained by the DC principle without mechanical relaxation and with strong thermal relaxation is much different from the two-celled circulation that initially forms. For the formation of steady DC meridional circulation, interaction between the slow variable and fast variables caused by the relaxation is important.

*T*

_{DC}of the circulation forcing by the wave forcing, which is located in the height

*h*from ground, iswhere

*H*is the scale height and

*κ*

_{T}is the thermal damping rate, which is denoted as

*α*in their study, and

*n*th zonal-mean Hough mode. Using the typical values for the middle atmosphere,

*h*= 50 km,

*H*= 7 km,

*N*= 100

*f*,

*α*= 6378 km,

*κ*

_{T}= 1/5 days, and

*n*= 6; the time scale

*T*

_{DC}is 10 days, which is much longer than the time scale of the circulation formation described by the fast variable

*δ*.

It is worth noting that there is another difference originating from the treatment of density stratification between the two studies. In the Boussinesq approximation (i.e., an infinite scale height used in the present study), the meridional circulation in the semisteady state is completely symmetric in vertical, while the initial two-celled circulation in Haynes et al. (1991) is not symmetric but weaker in the upper cell.

## 7. Summary and concluding remarks

The formation of meridional circulation for given dynamic forcing and/or diabatic heating in a meridional cross section was examined using zonal-mean *f*-plane equations by applying a Green’s function. The variables are translated to linearized potential vorticity *q*, horizontal divergence *δ*, and ageostrophic vorticity *γ*. Green’s functions were analytically obtained for respective variables.

The steady solution of meridional circulation responding to constant zonal forcing is composed of two vertically aligned cells. For forcing taking the shape of a step function in time, large-scale gravity waves with a broader range of frequencies are radiated as a transient response. The frequencies of the gravity waves near the source region become lower with time, and a quasi-steady meridional circulation finally remains. The quasi-steady meridional circulation pattern accords well with the steady-state solution for a constant forcing. The time scale needed for meridional circulation formation depends on the aspect ratio of the wave-forcing structure as is consistent with a theoretical expectation of the dimensional analysis. In addition, we found that the group velocity of gravity waves and the spatial scale of forcing determine the time scale of circulation formation. A case in which forcing patterns gradually change over time is also examined. When the time scale of forcing change is longer than the inertial period, the response does not include gravity wave radiation but instead involves meridional circulation that changes slowly the following time-varying forcing. This suggests that the meridional circulation for slowly varying forcing always occurs in accordance with that estimated from the steady-state assumption. The distribution ratio of wave forcing to zonal wind acceleration and Coriolis torque is also investigated. The distribution ratio is determined based on the shape of wave forcing and is explained through a dimensional analysis. For forcing with a large aspect ratio (i.e., a long vertical scale), most wave forcing is distributed to the acceleration term, and the meridional circulation becomes quite weak.

From these theoretical investigations, the following conclusions can be drawn concerning circulation formation under unsteady forcing. For wave-forcing processes occurring over shorter time scales than the inertial period, the fast variables include radiating zonally symmetric gravity waves as an earlier transient response and reaches a semisteady state with a vertical two-celled structure over a time period determined by the group velocity of gravity waves and by the length of the forcing.

The conventionally used assumption in the quasigeostrophic framework is only appropriately applied when the forcing time scale is longer than the inertial period. On the other hand, the time scale of the slow variable, which is determined based on the degree of linear relaxation, is usually much longer than that of the fast variable in the stratosphere.

In this study, calculations were performed for a zonal-mean two-dimensional system. However, generally speaking, wave forcing is zonally nonuniform. Thus, the responses to the three-dimensional forcing are also important and will be investigated in our future studies.

The authors thank anonymous reviewers for their critical reading and constructive comments. One of the authors (Y. H.) thanks Yasuda Yuki for his help in the theoretical treatment of governing equations. All figures in this paper were drawn using the DCL library. This work was supported by JSPS KAKENHI Grant (A) 25247075 and by JST CREST (JPMJCR1663).

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