## 1. Introduction

Rossby’s (1938) introduction of the adjustment problem used a momentum forcing to represent a sudden deposition of horizontal momentum into an infinite strip of the ocean by a surface wind stress. The Coriolis force associated with this flow would initially not be in balance with the pressure field and an adjustment would ensue. The original reduced-gravity shallow-water problem has been thoroughly examined [Cahn (1945); Mihaljan (1963); see Blumen (1972) for a review]. Atmospheric applications include the momentum forcing due to gravity waves that is an important component of the general circulation of the middle atmosphere (e.g., Fritts 1993). The sources of the gravity waves include excitation by airflow over orography, tropospheric convective activity, and jet stream adjustment. Various mechanisms (e.g., critical layer absorption and wave breaking) lead to the deposition of the gravity wave momentum. Zhu and Holton (1987) describe the atmospheric response to such a horizontal momentum forcing using an anelastic hydrostatic model. More recently, Vadas and Fritts (2001) developed a nonhydrostatic Boussinesq model of this process. They note that vertical momentum forcing can be a source of initial imbalance but focus their attention in their mathematical solutions on zonal momentum forcing.

The role of a vertical momentum forcing has taken on new significance in light of the work of Fovell et al. (1992) and Lane et al. (2001). For example, the latter study analyzed the output of an anelastic numerical model of deep convection and interpreted the nonlinear advective terms as sources of gravity wave activity. They examined the stratospheric wave response to this source but did not consider the problem in the context of atmospheric adjustment, defined here as the response to both a geostrophic and hydrostatic imbalance and the adjustment to a final equilibrium state in hydrostatic and geostrophic balance.

Practical applications of the momentum adjustment problem include the initialization problem of hydrostatic models (Williamson and Dickinson 1972). Later studies (e.g., Weygandt et al. 1999) focus on initiating an updraft from radar data of the wind field into a forecast model. De Lima Nascimento and Droegemeier (2006) studied the impact of abruptly removing the horizontal and vertical momentum fields in a numerical model of a convective system. Their results indicate that the horizontal rather than the vertical fields are more important in maintaining the system. Fiedler (2002) has discussed the initialization of updrafts in compressible, nonhydrostatic models. Chagnon and Bannon (2005b) have tested the Fiedler proposal to eliminate acoustic waves by having the updraft be balanced by compensating horizontal inflows and outflows such that the anelastic continuity equation is satisfied. They find that this approach significantly reduces the amount of acoustic energy generated but does not entirely eliminate the acoustic response. Chagnon and Bannon address the atmospheric adjustment to instantaneous momentum forcing in a compressible, linear, analytic model with an isothermal atmosphere. The purpose of this study is to continue their analysis by applying momentum forcings in a nonlinear, numerical model. This study examines the atmospheric adjustment of three different atmospheres (isothermal, nonisothermal, and tropopause) in a nonlinear, compressible, two-dimensional model to four different momentum forcings (vertical, divergent horizontal, nondivergent horizontal, and transverse circulation) with a size characteristic of an isolated, deep, cumulus cloud.

This study parallels the procedures of Fanelli and Bannon (2005) and applies momentum forcings to the three model atmospheres used in their study. The analysis here will be through examination of the perturbation fields, the energetics, and the potential vorticity. Examination of the perturbation fields allows for the observation of waves induced by the initial momentum forcing. Examination of the potential vorticity allows for the observation of the reestablishment of an equilibrium state. Examination of the energetics allows for the determination of wave type, wave frequency, and final atmospheric state.

Section 2 describes the numerical model and the idealized atmospheres used. Section 3 describes both the traditional and available energetics analyzed in this study. Sections 4 and 5 present the atmospheric responses to divergent vertical and horizontal momentum forcing. Section 6 presents the responses to an imposed transverse circulation. Section 7 presents the responses to nondivergent momentum forcing. Section 8 presents the summarized results and conclusions. Here, the results from the vertical and horizontal momentum forcings shall be compared and contrasted.

## 2. Numerical model

### a. Governing equations

*f*plane. The governing equations in Cartesian coordinates arewhere

*f*is the Coriolis parameter,

*g*is the acceleration due to gravity,

**∇**is the two-dimensional del operator (in the

*x*and

*z*directions), and

**k**is the unit vector in the

*z*direction. The variables

**u**,

*ρ*,

*p, θ,*and

*T*are the three-dimensional velocity, density, pressure, potential temperature, and temperature. All flow fields are assumed to be independent of the

*y*direction. The constants

*R*,

*c*, and

_{p}*c*are the ideal gas constant and the specific heats of dry air at constant pressure and constant volume. The nondimensional pressure is defined aswhere

_{υ}*p*

_{00}= 1000 hPa. Equations (2.1)–(2.5) are the momentum equation, pressure equation, continuity equation, thermodynamic energy equation, and the equation of state, respectively.

### b. Momentum forcings

*τ*is the duration of the forcing;

*δU*is its amplitude (m s

^{−1}); and the functions

*F*,

*G*, and

*H*describe its temporal, vertical, and horizontal structure; and

**is a general unit vector. The forcing structure in the horizontal isand that in the vertical isHere**

*û**a*is the horizontal width of the forcing,

*h*is the height of the forcing above the surface, and

*d*is the vertical depth of the forcing. The cosine-squared horizontal and vertical structures prevent infinite divergences from forming at the boundaries of the forced region. The forcing is applied over a finite duration with the temporal structureThe factor of 2/

*τ*in (2.7) ensures that the integral over time

*t*from

*t*= 0 to

*t*=

*τ*yields a net forcing of

*δU G*(

*z*)

*H*(

*x*). The sine-squared temporal structure is chosen because it produces smooth fields and both it and its first derivative are continuous functions of time.

Figure 1a shows the temporal structure of the forcing for duration *τ* = 5 min. The forcing is turned on slowly, reaching a maximum at 150 s when it begins to decrease until it is turned off at 300 s. This structure helps to avoid exciting sound waves of unnaturally large amplitude as predicted by Fiedler (2002) due to the addition of a wind field that does not closely satisfy the continuity equation. By slowly injecting the forcing, these sound waves are avoided as opposed to a sudden forcing, wherein most of the injected energy goes into sound waves, leaving little energy to force the nonacoustic adjustment.

A variety of momentum forcings are examined including vertical, horizontally divergent, and horizontally nondivergent forcings. The general form of this forcing is displayed in Fig. 1b. In the divergent, nondivergent, and vertical momentum cases, the forcing has an amplitude of 10 m s^{−1}, a half-width *a* = 10 km, and a vertical depth *d* = 11 km, extending from 0 km above the lower boundary (*h* = 0 km) to 11 km. Due to the symmetry of the vertical momentum case, only one-half of the domain is included. In addition to these momentum forcings, we also use a prescribed transverse circulation satisfying the anelastic continuity equation. In this case, the forcing (Figs. 1c and 1d) extends from 0 to 11 km in the vertical and 0 to 30 km in the horizontal. This prescribed circulation resembles the circulation of a convective cell.

### c. Initial conditions

*p*

_{*}. The first is an isothermal hydrostatic atmosphere defined bywhere

*T*, and

_{*}, p_{*}, θ_{*}= T_{*}*ρ*

_{*}are constants satisfying the ideal gas law and Poisson’s relation at the surface

*z*= 0. Here

*H*is the density scale height and

_{s}= RT_{s}/g*κ = R/c*. The parameter settings are

_{p}*f*= 10

^{−4}s

^{−1},

*g*= 9.81 m s

^{−2},

*R*= 287 J kg

^{−1}K

^{−1},

*c*= 1004 J kg

_{p}^{−1}K

^{−1},

*c*= 717 J kg

_{υ}^{−1}K

^{−1},

*T*= 255 K,

_{*}*p*= 1000 hPa,

_{*}*ρ*= 1.37 kg m

_{*}^{−3}, and

*H*= 7460 m. The second atmosphere has a surface temperature

_{s}*T*of 288 K and a constant lapse rate of 6.5 K km

_{*}^{−1}. The third is similar to the second but has an isothermal stratosphere above 11 km.

### d. Numerical techniques

Bryan and Fritsch (2002) give a thorough description of the numerical model. The model is integrated with third-order Runge–Kutta time differencing and fifth-order spatial derivatives for the advection terms that are written as the sum of a flux form term and a divergence term. This approach improves the conservation of the variable being advected in the numerical model.

The two-dimensional numerical domain is 15 km in height *z* and either 200 or 400 km in horizontal distance *x*. The resolution is 200 m in *x* and *z* with a time step of 0.30 s (Figs. 4 through 18). (For the energetics given in Table 2, the vertical momentum forcing cases have resolutions different from those given above. The time step for the nonisothermal atmospheres is 0.05 s with 50-m resolution in *z* and 100-m resolution in *x*. For the isothermal atmosphere, the time step is 0.01 s with 25-m resolution in *z* and 100-m resolution in *x.* This change in resolution was necessary due to the stronger atmospheric response to the vertical forcing and only had effects on the total traditional energy without changing the relative partitioning into its constituent parts.) Although the time splitting feature of the model gives comparable results to those presented here, that feature (along with the associated divergence damping) is not utilized in this study. The boundary conditions are flat, rigid horizontal boundaries at *z* = 0 and 15 km where the vertical velocity is set to zero. The effect of the upper boundary condition was tested by including a Rayleigh damping layer (Durran and Klemp 1983) in the upper 3 km of the domain to damp gravity waves. The results (not shown) produced distortions of the flow above 12 km but did not alter the response below the layer. It did cause spurious increases and decreases in the total energy. To conserve energy accurately, no numerical sponges are employed in the results presented here. Thus the methodology is identical to that of the companion study by Fanelli and Bannon (2005). In the vertical momentum and transverse circulation cases, a mirror boundary condition is applied at the origin (*x* = 0 km) because the thermodynamic variables and the vertical velocity are symmetric about the origin and the horizontal velocity is asymmetric. At *x* = 200 km, an open boundary condition is employed following Durran and Klemp (1983). The horizontal momentum cases follow approximately the same basis except the horizontal domain is 400 km, and both lateral boundaries at *x* = −200 and *x* = 200 km are open boundaries. In all simulations, all derivatives in the *y* direction vanish.

## 3. Energetics

**u**, would be defined asFor this case, the total kinetic energy produced by the forcing, KE

_{f}*, can be defined bywhere the integral is over the range of the forcing. This KE*

_{f}*is the amount of energy the atmosphere would have if it responded to the momentum forcing directly without any subsequent adjustment. However, this energy is not the kinetic energy seen by an adjusting atmosphere. The kinetic energy actually generated by the forcing, KE*

_{f}_{gen}, is defined bywhere

**u**is the actual wind field at any time

*t*and

*ρ*is the density at a particular point at time

*t*. By Le Chatelier’s principle, KE

*will always be greater than KE*

_{f}_{gen}, but as

**u**approaches

**u**

_{f}the difference between KE

*and KE*

_{f}_{gen}may become infinitesimal (e.g., the nondivergent horizontal momentum case of section 7).

### a. Traditional energetics

**u**·

**u**/2, ie =

*c*, and pe =

_{υ}T*gz*are the specific kinetic, internal, and potential energies, respectively. The inviscid equation for the time rate of change of traditional energy isThe forcing initially increases the kinetic energy. This kinetic energy is then converted into potential or internal energy.

### b. Eulerian available energetics

*r*(e.g., Δ

*p = p − p*). The three terms on the right-hand side represent the departure in specific enthalpy

_{r}*h*=

*c*, the departure in specific entropy

_{p}T*s = c*ln

_{p}*θ*, and the departure in pressure, respectively. Here

*T*and

_{r}*p*are the temperature and pressure of an isothermal reference state and

_{r}*α*= 1/

*ρ*is the specific volume. The form of (3.6) allows the partitioning of the available energy into available elastic and available potential contributions, such thatThe Eulerian available potential energy represents the energy associated with the departure of the flow’s entropy from the reference entropy and iswhere the enthalpy is a function of potential temperature and nondimensional pressure. The Eulerian available elastic energy represents the energy associated with the departure of the flow’s pressure from the reference pressure and is

**∇**· (

*p*

**u**) and by the momentum forcing, but the available total energy changes due to the working by the perturbation pressure

*p′ = Δp*and by the momentum forcing.

Figure 3 gives a schematic showing the generation of and conversion between these available energy terms. Available potential energy is converted into kinetic energy during the descent of relatively dense air. This conversion occurs mainly in the buoyancy waves. Available elastic energy is converted into available potential energy during ascent in regions of relatively low pressure. This conversion occurs mainly in vertically propagating acoustic modes. Kinetic and available elastic energy each have a sink due to the redistribution of energy within the domain. In a closed system, Eulerian available energy is conserved in the absence of an external forcing.

## 4. Vertical momentum forcing

*δW*= 10 m s

^{−1}.

### a. Atmospheric adjustment in an isothermal base state

The response to the forcing (4.1) at 10 min in the isothermal atmosphere (2.11) is depicted in Fig. 4. Vertically trapped (i.e., standing) acoustic and buoyancy modes are excited, but no strong Lamb waves are present. These acoustic modes propagate horizontally and are seen in Fig. 4 beyond 100 km from the origin and have extremely small amplitudes (e.g., 0.3 hPa or less) with a wavelength of about 20 km. The slower moving buoyancy modes in the vertical velocity field with vertical wavenumber *n* = 1 (Chagnon and Bannon 2005a) have amplitudes up to 40 hPa. The perturbations in the potential temperature (Fig. 4c) result from the advection of low-level potential temperature upward through the forced updraft and the corresponding subsidence and advection of high-level potential temperature downward.

**is the absolute vorticity. The first term on the right-hand side corresponds to the advection of potential vorticity and the second term corresponds to its generation by the momentum sources. For the two-dimensional forcing used here, the generation term in the equation reduces toBecause of this result, the nondivergent horizontal momentum forcing,**

*ω*_{a}**v̇**, is the only forcing that can generate potential vorticity. In all other cases, the potential vorticity is neither created nor destroyed. Instead, the perturbations in the potential vorticity field observed in Fig. 5 initially result from the vertical advection of the base-state potential vorticity field that increases with height due to both the decrease of density and the increase in the potential temperature gradient with height. The upward vertical motion in the forced region advects lower values of potential vorticity aloft, resulting in the negative perturbation throughout the forced region (Fig. 5a). The corresponding subsidence outside the forced region, which is weaker due to its covering a greater area, advects higher values of potential vorticity toward the surface, resulting in the area of positive perturbation outside the forced region. In Fig. 5b, these perturbations extend from the lower to the upper boundary. Their wavelength of 20 km is due to the size of the forcing whose half-width is 10 km in the horizontal. Notice the change in tilt over time of these perturbations, going from forward leaning (in Fig. 5a) to rearward leaning (Fig. 5b) within the 20-min run time. This change in tilt with time shows that these perturbations are also being advected by the horizontal wind field. Although there are perturbations of potential vorticity within the domain, the total mass and potential vorticity remain constant (not shown).

The efficiency of the forcing in terms of the traditional energy, *η*, is defined as *η* ≡ TE/KE* _{f}* and the efficiency for the available energetics is

*η*= AE/KE

_{a}*. For any forcing,*

_{f}*η*and

*η*are the same as long as no energy leaves the domain.

_{a}Figure 6a depicts the change in total traditional energy over time. The atmosphere is forced with a KE* _{f}* of 0.5167 GJ m

^{−1}, but the TE for this case is only 0.0745 GJ m

^{−1}at 5 min. Thus the efficiency of the conversion of kinetic energy to traditional energy is therefore 14.4%. This low efficiency is due to the advection of air through the forcing region that keeps the air from accelerating to the maximum speed possible. The effect of this advection can be seen in Fig. 6c, where the updraft speed at the center of the forcing does not approach the maximum forcing of 10 m s

^{−1}. Note the increase of the PE during the time the forcing is on with a corresponding decrease in the IE. This behavior arises because the air is being forced upward. As the air rises, the potential energy increases. But this rising air also expands, which leads to adiabatic cooling of the parcel and a decrease in its internal energy. This difference in the PE and IE decreases as the forcing is shut off.

After the forcing is shut off, there is an oscillation between the IE and PE (Fig. 6a) with a period of about 90 s that is due to the *n* = 1 acoustic modes with zonal wavenumbers near zero. There is a slight drop in the IE at ∼12 min, which corresponds to the time when horizontally propagating acoustic waves exit the domain with a TE change of 0.18 GJ m^{−1}.

Figure 6b shows the available energetics. Here the KE is the same as in the traditional energetics. The variation between the APE and KE in Fig. 6b oscillates with a period of about 3 min and is due to the propagation of buoyancy waves with decreasing amplitude over time. The buoyancy oscillation period for the model atmosphere is about 6 min, but due to the quadratic nature of both the APE and KE, these oscillate at half the buoyancy period. The AE at 5 min is 0.0728 GJ m^{−1}, giving an efficiency of 14.1%. The difference between *η* and *η _{a}* seen here is due to truncation error. The AE decreases slightly (7.75 × 10

^{−6}GJ m

^{−1}), with the departure of acoustic waves starting around 12 min.

Figure 6c shows the change in vertical velocity with time at the center of the perturbation. Here, the collapse of the updraft starts at 3 min (just after the time of the greatest forcing). This collapse continues into a complete reversal of direction at 5 min, right after the forcing is shut off. The vertical velocity then continues to vary at the buoyancy oscillation period.

### b. Atmospheric adjustment in nonisothermal base states

In the nonisothermal atmosphere (Fig. 7), the forcing excites weak dipole Lamb wave packets as well as buoyancy waves and vertically trapped acoustic waves like those in the isothermal case (Fig. 4). The Lamb waves propagate horizontally away from the forced area at the speed of sound. They extend from the surface to the top of the domain. This dipole packet is shown in the pressure perturbation field (Fig. 7a) between *x* = 100 and *x* = 200 km and consists of a compression pulse followed by an expansion pulse. Buoyancy waves are also excited. Acoustic waves with a wavelength of 40 km are seen in the vertical velocity (Fig. 7b) and potential temperature (Fig. 7c) fields beyond 80 km from the origin. These waves have twice the wavelength of the isothermal case because of the difference in static stability between the nonisothermal and isothermal cases. Statically stable air responds more quickly to an upward perturbation of the density field than the nonisothermal cases, resulting in a faster oscillation between positive and negative modes.

Figure 8 shows the domain change in mass and potential vorticity (per unit length in *y*) from their initial values of 1.8067 × 10^{9} kg m^{−1} and 1.3316 × 10^{3} K m^{2} kg^{−1} s^{−1}. Note the spike in both mass and potential vorticity between 12 and 14 min. The Lamb packet gives rise to this spike as the compression pulse first leaves the domain through the open boundary, followed by the expansion pulse. In contrast, linear theory predicts no potential vorticity transport by the Lamb waves.

The traditional energy (Fig. 9a) oscillates between the potential and the internal energies with a period of 96 s. This oscillation is caused by the *n* = 1 acoustic modes with large horizontal wavelengths, as in the isothermal case. Here, the departing Lamb packet can be seen leaving the domain at about 12 min as an increase of 2.7 GJ m^{−1} and subsequent decrease in the TE, PE, and IE.

Figure 9b shows a much larger response in the AEE (about 0.1 GJ m^{−1}) than in the isothermal case (Fig. 6b), though the response is negative. This response is due to the lower static stability in the nonisothermal atmosphere and results from the greater lifting and expansion of the air parcels. The oscillation seen in the APE and AEE continues as these energies convert between each other. The frequency of this oscillation is the same as that in Fig. 9a between the PE and IE but is different from the observed variation in Fig. 6. There, the APE and KE oscillate with twice the period of the PE and IE and the oscillation is caused by buoyancy waves. Here the APE and AEE oscillate with the same frequency as the IE and PE and this oscillation is caused by acoustic waves. The APE also oscillates with the KE, though at a slower rate due to buoyancy waves propagating through the domain. Throughout these conversions, the AE remains constant after the forcing is shut off. The only variation in AE appears when the dipole Lamb wave leaves the domain, where there is a slight increase in the AE corresponding to the expansion pulse (0.018 GJ m^{−1}) followed by a decrease in AE as the compression pulse leaves the domain.

Figure 9c illustrates the differences between the isothermal and nonisothermal atmospheres. The nonisothermal atmosphere cases have a greater vertical velocity and a slower oscillation between positive and negative phases. In the isothermal atmosphere, the vertical velocity peaks near 1 m s^{−1} at 3 min, while in the nonisothermal atmospheres the vertical velocity maximum is nearly 2 m s^{−1} and the peak is near 4 min. The negative response is slower and stronger in the nonisothermal atmosphere. In the isothermal atmosphere, the minimum vertical velocity is stronger than the initial maximum (−1.5 m s^{−1}) and occurs at 6 min. The minimum in the nonisothermal atmosphere occurs near 9 min and has the same magnitude (nearly 2 m s^{−1}) as the initial maximum. The period of the oscillation in the isothermal atmosphere is 3 min, while the nonisothermal atmospheres oscillate at 5 min. These features are a consequence of the difference in static stability.

## 5. Divergent horizontal momentum forcing

*δU*= 10 m s

^{−1},

**i**is the unit vector in the

*x*direction (east), and the domain is extended to −200 km ≤

*x*≤ 200 km with the perturbation centered at the origin (

*x*= 0 km) with open boundaries at both the east and west edges of the domain.

### a. Atmospheric adjustment in an isothermal base state

The divergent horizontal momentum forcing (5.1) generates acoustic, buoyancy, and Lamb waves (Fig. 10). The Lamb waves form a compression wave packet and an expansion wave packet moving away from the initial perturbation in opposite directions. These packets are centered near *x* = ±145 km from the initial perturbation in Fig. 10a. The Lamb waves here are stronger than in the vertical momentum cases because the horizontal momentum forcing projects more strongly onto them than the vertical momentum forcing. The slower buoyancy waves corresponding to the *n* = 1 and *n* = 2 modes lie closer to the origin. There are still some weak acoustic modes, which can be seen in the perturbation vertical velocity and potential temperature fields (Figs. 10b and 10c).

Figure 11 shows the energetics of this case. The KE_{f} is 1.0334 GJ m^{−1} that results in a TE of 0.48 GJ m^{−1} at 5 min with an efficiency of 46.3%. This efficiency is greater than in the vertical momentum forcing case. Figure 11a shows that the vertical motion is not a major player. This result can be inferred from the very small increases in the PE and IE. In cases where the vertical motion plays an important role, such as during the injection of vertical momentum, a strong oscillation between the PE and the IE is present (see Fig. 6a). Note the strong decrease in the IE as the Lamb wave exits the domain at *t* = 12 to 14 min. There is also a weaker decrease in the PE, which leads to a 7.6 GJ m^{−1} decrease occurring between 10 and 14 min in the total energy perturbation. This change in the IE and PE shows that injecting horizontal momentum into the domain leaves the domain cooler and with lower mass after the packets leave. The kinetic energy (see Fig. 11b) changes very little at this point, showing that the exiting wave packets do not have strong wind speeds associated with them.

Figure 11b shows a weak, decaying buoyancy oscillation between the APE and the KE. It also shows that the AEE has a large effect on the total energy unlike the vertical momentum cases. The Lamb wave packets dominate the AEE that begins to decrease as the packets leave the domain between 12 and 14 min. There is no drop in the APE as the packets leave the domain because this energy is associated totally with the buoyancy waves. Due to the weak oscillation in the APE, strong buoyancy waves are not expected to form in this system. The kinetic energy drops slightly as the packets leave the domain. The AE has an efficiency of 46.8%, giving a total energy of 0.48 GJ m^{−1} after the forcing is turned off. The AE then drops approximately 0.1 GJ m^{−1} as the packets leave the domain.

Figure 11c shows the horizontal velocity in the *x* direction at the center of the forcing. Note that the maximum speed here is greater than in the vertical forcing case (Fig. 6c). This difference is due to the vertical forcing having to overcome the static stability, whereas here the forcing is in the direction of the atmospheric strata. The atmospheric response in the opposite direction near 7 min is much slower and lower in relative magnitude than the vertical forcing case. Also, the horizontal velocity at the center is very nearly zero at 10 min, while the vertical case had a nonzero amplitude even at 20 min into the run. This difference shows the effect of buoyancy fluctuations on the continuation of the atmospheric motion.

### b. Atmospheric adjustment in nonisothermal base states

The nonisothermal cases (not shown) differ slightly from the isothermal case. The Lamb packets are stronger and fewer buoyancy modes are excited. In the constant lapse rate case, the buoyancy waves propagate slower and are located within 50 km of the initial perturbation after 10 min. These correspond to the *n* = 3 buoyancy wave. Unlike the isothermal case, the *n* = 1 and *n* = 2 buoyancy waves are not excited in either nonisothermal case because of the low static stability in these cases. The energy is trapped within the strata of the isothermal atmosphere, which has a much stronger response to vertical motion than the nonisothermal cases. The tropopause case follows the nonisothermal case with minor differences in the fields and energetics due to the isothermal layer aloft.

## 6. Transverse circulation forcing

**u̇**is defined such thatis satisfied. Here

*δW*= 10 m s

^{−1}, and

*L*= 0 is the distance between the edge of the updraft and the edge of the downdraft. Figure 1c depicts the vertical portion of the forcing and Fig. 1d the horizontal portion. Because of the two-dimensional structure of the forcing and its symmetry in

*x*, the model domain is taken to be confined to the positive

*x*direction (0 ≤

*x*≤ 200 km).

### a. Atmospheric adjustment in an isothermal base state

For comparison, the forcing is run with only the horizontal forcing, with only the vertical forcing, and with the full forcing. Note the difference in the vertical velocity at the center of the perturbation among the three cases (Fig. 12a). In each case, the vertical velocity comes to a maximum slightly after 2.5 min (the time of maximum forcing) and then responds in a buoyancy oscillation. In the vertically forced case, the response is a negative vertical velocity that has nearly the magnitude of the original perturbation. In the horizontally forced case, the vertical velocity is much stronger (*w _{c}* > 4 m s

^{−1}) and is offset from the time of the strongest forcing and actually occurs at 3 min. The negative response to this strong upward velocity is less than half as strong (

*w*∼ .5 m s

_{c}^{−1}) and occurs later in the run than in the vertically forced cased (nearly 8 min). The full circulation case has a maximum velocity of about 5.5 m s

^{−1}. However, the negative response is much weaker (

*w*∼ 2.5 m s

_{c}^{−1}) and comes later than in the vertical velocity case (∼7 min). This result shows that the full circulation is actually dominated by the horizontal forcing.

The different forcings also create different perturbation fields. When only the vertical portions are forced (Figs. 13b,d,f), the circulation decays rapidly after the maximum forcing at 2.5 min and is unable to be observed in the perturbation fields when the forcing is turned off at 5 min (Fig. 13f). The vertical-only forcing does not force a strong, corresponding, supportive horizontal velocity circulation (Fig. 13d), which prevents the divergence (convergence) of air out of (into) the outflow (inflow) area. This lack of horizontal mass removal initially leads to a pressure gradient force that acts downward (upward) in the updraft (downdraft) region. This pressure gradient force partially counteracts the forcing and stops the circulation. In contrast, the horizontal-only forcing creates a circulation that lasts longer than the one created with the vertical-only forcing (Figs. 13a,c,e). The horizontal velocity perturbation (Fig. 13c) closely resembles the spatial structure of the forcing shown in Fig. 1d. The corresponding vertical velocity structure (Fig. 13e) is much stronger than the vertical velocity seen in Fig. 13f. The corresponding pressure perturbation (Fig. 13a) shows that this horizontal circulation produces a strong pressure gradient that is resolved through vertical motion flowing from high to low pressure areas as it was in the vertical circulation case. This response is different, however, in that the pressure gradient acts to continue the circulation, not destroy it. This result agrees with the analysis of de Lima Nascimento and Droegemeier (2006), where the forcing from only a vertical velocity perturbation was unable to recreate the expected circulation.This response is due to the strong vertical pressure gradient set up by the horizontally forced circulation (Fig. 13a), which is much weaker in the vertically forced circulation (Fig. 13b).

In the fully forced circulation case, the perturbation continues the longest period of time into the run, maintaining the circulation for a short time after the forcing has ended (Fig. 14). Notice the weak perturbations in the potential temperature (Fig. 14d) that are of opposite sign to the vertical velocity (Fig. 14c). This result demonstrates that the circulation does not continue because there is no thermodynamic forcing to sustain it. This transverse circulation is dominated by the horizontal forcing (Fig. 14b). Notice that the *u*′ field closely resembles the forcing shown in Fig. 1d, but the *w*′ field does not resemble that shown in Fig. 1c. Also, note the difference in the *θ*′ and *w*′ fields. This result is as expected from the comparison between the vertical and horizontal circulations previously discussed. The perturbation potential temperature works against the vertical velocity, suppressing the vertical legs of the circulation, and the induced circulation does not last long after the forcing is turned off.

The fully forced transverse circulation injects 5.908 GJ m^{−1} into the domain. The total traditional energy at 5 min (Fig. 15a) is 5.468 GJ m^{−1}, which gives an efficiency of 92.5%. Unlike the vertical perturbation scenarios (Fig. 6), there is very little oscillation between energy types within the system, again suggesting a dominant contribution from the horizontal forcing. There is, however, a much sharper response to the horizontally forced Lamb packets. Note the much faster recovery to almost the initial atmospheric state after the packets leave the domain at around 12 min than in the divergent horizontal momentum forcing case (Fig. 11a). Therefore, the atmosphere is cooler and less massive (not shown) than the initial state.

Figure 15b again shows the dominance of the horizontal component over the vertical component of the forcing. Here the AEE is nearly 16% of the total available energy of 5.4655 GJ m^{−1} at 5 min. This amount of AE again corresponds to an efficiency of 92.5%. Also, there is no oscillation between APE and KE, unlike the vertical forcing (Fig. 6), and there is little decrease in AEE as the packets leave the domain.

### b. Atmospheric adjustment in nonisothermal base states

Figure 12b shows the difference in the vertical velocities between the three atmospheres for the fully forced circulation. There is very little difference in the constant lapse rate and tropopause cases. The isothermal atmosphere has a lower maximum (near 5 m s^{−1}) that occurs sooner (3.5 min) into the run than in the nonisothermal atmospheres that have a maximum vertical velocity of about 7 m s^{−1} at 4 min. These results agree with the pattern seen in Fig. 9c with the vertical momentum forcing case. Comparison of the two figures indicates that the presence of the horizontal and downdraft forcings help to strengthen the updraft. However, the strength of the corresponding negative response (relative to the maximum updraft) is much lower and slower than in the vertical momentum cases. The response of both the isothermal and nonisothermal atmospheres is approximately the same at −2.5 m s^{−1} with the minimum seen at 7 and 10 min, respectively.

## 7. Nondivergent horizontal momentum forcing

*δV*= 10 m s

^{−1}and

**j**is the unit vector in the

*y*direction. Again, the domain is extended to −200 km ≤

*x*≤ 200 km. In contrast to (7.1), Rossby (1938) considered an instantaneous input of nondivergent momentum in a shallow-water model.

### a. Atmospheric adjustment in an isothermal base state

The nondivergent horizontal momentum forcing (7.1) generates weak acoustic, buoyancy, and Lamb waves. Figure 16 shows the response of the model isothermal atmosphere at 10 min. Note the leading edge of a weak Lamb mode located beyond 150 km from the pressure perturbation in Fig. 16a. The buoyancy waves are seen (Figs. 16b,c) within 100 km of the origin. These weak waves are the *n* = 1 and *n* = 2 buoyancy modes. The nondivergent horizontal wind field (Fig. 16d) has very high positive velocities (≃10 m s^{−1}) in the center of the perturbation and very weak negative velocities throughout the rest of the domain. This nondivergent meridional wind field excites a response in the zonal wind field as well (not shown). The maximum of this zonal response is much weaker than the meridional perturbation but does show an interesting separation into two maxima bracketing the center of the perturbation.

The potential vorticity perturbation (Fig. 17) forms a dipole pair that brackets the initial momentum perturbation and this perturbation remains constant after the initial perturbation of the system by the forcing at 5.8 × 10^{−5} and −5.8 × 10^{−5} K kg^{−1} m^{2} s^{−1} on the west and east side of the perturbation, respectively. This large amount of potential vorticity is expected because this is the only case where potential vorticity can be generated (4.3). The potential vorticity is not advected through the domain, either, as can be seen from the lack of change between Figs. 17a and 17b.

Almost all of the initial kinetic energy (KE* _{f}* = 1.0334 GJ m

^{−1}) injected into the domain remains kinetic energy (Fig. 18). The total traditional energy at 5 min is 1.0334 GJ m

^{−1}, which gives an efficiency of 100.0%. There is a small decrease in KE starting just before

*t*= 12 min. This decrease can be accounted for by the weak Lamb wave that begins to leave the domain at this time. It is seen as the weak, sudden, but continuous decrease in the IE and PE in Fig. 18a. This result shows that, even though the Lamb mode begins to leave the domain, it does not entirely leave the domain during the time interval of the study. This agrees with the assumptions from Fig. 16a, where the Lamb wave leading edge is seen moving through the domain, but there is no trailing edge.

Figure 18b shows the available energetics of the system. Note that here, too, the KE dominates the energetics. However, the APE and AEE are nonzero and show the presence of very weak Lamb and buoyancy modes. Because both the total available energy and the total traditional energy are dominated by the kinetic energy, they are equal at 5 min (1.0334 GJ m^{−1}) and their efficiency is the same (100.0%).

Figure 18c shows the meridional horizontal velocity at the center of the perturbation. Note that after the initial increase in the velocity caused by the forcing, the velocity remains constant. This lack of change in the meridional velocity after the forcing is turned off is the expected result for nondivergent momentum forcing at these short times before significant Coriolis deflection occurs.

### b. Atmospheric adjustment in nonisothermal base states

In the nonisothermal cases, the energetic response matches the isothermal case. There is, again, no change in the total potential vorticity nor is there a change in the perturbation potential vorticity dipole with time, showing that these cases may result in a steady state as well. This steady state excites much less perturbation potential vorticity than the isothermal case (9.7 × 10^{−6} and −9.7 × 10^{−6} K kg^{−1} m^{−2} s^{−1} on the west and east sides of the perturbation, respectively) because of the reduction in the static stability.

## 8. Conclusions

This study compares and contrasts the responses of rotating, compressible atmospheres to different forms of momentum forcings. In general, the flow fields are consistent with those in Chagnon and Bannon (2005b), and like that linear study there is no evidence of wave breaking, in part because of the short duration of the numerical integrations and the upper boundary condition. Each momentum forcing produces an atmosphere altered from its initial state of rest. Only the nondivergent momentum forcing may result in a steady state, as can be seen from the constant perturbation potential vorticity (Fig. 17). Based on linear theory, the other cases should not result in such a steady state. In the vertical and divergent horizontal momentum forcing case, potential vorticity is neither created nor destroyed. Instead, the wind field advects potential vorticity around the domain without causing any change in the total potential vorticity (e.g., Fig. 5). In the fully forced transverse circulation case, no steady state is observed. This lack of a steady state is surprising because convective circulations, like the one studied here, are observed within the real atmosphere and are maintained for longer time periods. Here, the transverse circulation is lost soon after the forcing is shut off due to a lack of thermodynamic support for the circulation (cf. de Lima Nascimento and Droegemeier 2006). If there had been a potential temperature field that acted to increase vertical velocity, rather than one to suppress it (e.g., Fig. 14), the circulation would have increased in strength, rather than quickly decreasing. The implication of this finding is that the initialization of an updraft requires that there be thermodynamic support (e.g., positive buoyancy, pressure gradient forces) for the updraft to be sustained.

The forcings not only excite potential vorticity changes but atmospheric waves as well. Table 1 summarizes the Lamb wave response and the change of traditional and available energy as the Lamb pulse leaves the domain. Both divergent and nondivergent horizontal momentum forcings result in a monopole Lamb packet that carries a large amount of traditional energy (TE) with it but relatively little total available energy (AE). The energy changes are similar for all three atmospheres. In contrast the transverse circulation forcing excites a dipole Lamb wave packet of similar strength in the three atmospheres. Last, the vertical momentum forcing produces no Lamb wave in the isothermal case. The change in the TE and AE observed in this case are due solely to the acoustic modes leaving the domain. In the nonisothermal atmosphere the vertical momentum forcing projects onto the Lamb wave. This observed dominance of the Lamb wave in the traditional energetics for all momentum forcings agrees with the analysis of Fanelli and Bannon (2005) for the case of thermal forcing.

The efficiency of the conversion of KE* _{f}* to TE (Table 2) differs depending on the forcing but is similar for the three model atmospheres (within 5%). A high efficiency occurs when the wind vector

**u**approaches the applied wind vector

**u**

*. The most efficient conversion occurs in the nondivergent horizontal momentum case where the velocity at the center of the forcing remains constant after the forcing is shut off (see Fig. 18c). The next most efficient is the forced transverse circulation whose efficiencies range between 92% and 96% and results from the strong support the vertical and horizontal branches provide each other. The divergent horizontal momentum case maintains an efficiency of ∼50%. The least efficient forcing is the vertical momentum forcing. This small efficiency is caused by the forcing accelerating air parcels across the isentropes rather than parallel to the isentropes as in the horizontal momentum forcing cases. The efficiency of conversion of KE*

_{f}*to AE,*

_{f}*η*, is within 0.5% of

_{a}*η*in all cases (Table 2). This agreement between the efficiencies is expected, as stated in section 4.

The efficiency of a forcing increases monotonically as the duration of the forcing decreases. For example, for the horizontally divergent momentum forcing, the efficiency decreases to 39.7% for a duration of 10 min, while it increases to 63.7% for a duration of 2.5 min when compared with the 5-min duration, which gives an efficiency of 46.3% in the constant lapse rate atmosphere. Similarly for the vertical momentum forcing, the efficiency increases to 25.0% for a duration of 2.5 min from the 20.1% efficiency with 5-min duration.

Future research should include an investigation into the thermodynamic support necessary to maintain a transverse circulation. Also, moist processes and more realistic base-state atmospheres with mean flows should be considered.

## Acknowledgments

We thank Jeffrey M. Chagnon for helpful discussions and comments on a draft of the manuscript, and Paul F. Fanelli for his help with the numerical model. The National Science Foundation (NSF) under NSF Grants ATM-0215358 and -0539969 provided partial financial support.

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Type of Lamb wave generated for each atmosphere and forcing and the change in the total traditional and total available energy caused by the Lamb wave leaving the domain. The changes in TE and AE are calculated by subtracting the energy at 14 min from that at 10 min and taking the absolute value. The vertical momentum and transverse circulation changes are doubled to encompass the entire domain.

Total energetics in the different atmospheres for the different forcings at the time the forcing is shut off and their efficiencies compared with the total energy injected by the momentum forcing for each case. The vertical momentum forcing and the fully forced transverse circulation energies are doubled so that the domain energies are comparable with the horizontal momentum forcings.