## 1. Introduction

Our ability to model the atmosphere is hindered by our inability to accurately represent all scales of motion simultaneously. Consequently, we are often required to make assumptions about those processes that occur on scales outside of the range of principal interest. We therefore distinguish those processes that are *external* from those that are *internal* to the system under consideration. Generally, the external conditions are those properties of the physical system that must be prescribed or constrained artificially in some manner, whereas the internal processes are supported and predicted by the model. For example, in Rossby’s (1938) classic geostrophic adjustment problem, the external process is the instantaneous acceleration of a strip of ocean, incorporated through the initial condition. The internal processes include the rotating shallow-water gravity waves and the potential vorticity (PV) conserving final state. In this example, the internal processes are the homogeneous solutions of the system of partial differential equations used to model the phenomenon of geostrophic adjustment in the ocean.

To develop a model to study how isolated convection interacts with its environment—an issue that has theoretical value, as well as practical significance to the problems of data assimilation and parameterization—some assumptions must be made about the manner in which the convection is generated and how it is perceived by the surrounding environment. The typical investigative approach toward modeling convective adjustment is to 1) select an appropriate numerical or analytical model, 2) externally introduce a perturbation or injection in order to encourage the development of convection or to represent the convection itself, and 3) examine the internal response of the model to the externally imposed imbalance. The intent of this paper is to examine how the second step in this approach may influence the characteristics of the internal response within the model.

From the perspective of the larger-scale environment, isolated convection takes place rapidly and locally and is a source of imbalance. How should this imbalance be represented in a model of the larger-scale environment? This question is the starting point for most convective parameterizations. The convection transports mass vertically as well as horizontally, and may therefore be modeled as an external source/sink of mass (e.g., Shutts and Gray 1994). Convection also generates regions of heating and cooling associated with condensation and evaporation, and may therefore be modeled alternatively as a heat source (e.g., Yanai et al. 1973; Lin and Smith 1986). Additionally, the rapid vertical accelerations associated with the convective updrafts are likely sources of buoyancy waves (e.g., Lane et al. 2001). One may therefore also be required to represent the convection as a transient source of vertical momentum. Because the origin of the imbalance associated with convection is somewhat ambiguous, a sensitivity study, such as the one presented here, is practical. The present goal is not to suggest how convection should be parameterized, but to seek to identify characteristics of the dynamic response by the larger-scale environment to the possible injection types that may be selected by a given parameterization. Furthermore, the analysis is not restricted to convection-like sources.

To examine these issues, a simple analytical approach is adopted. The model and its solutions are presented in Chagnon and Bannon (2005a, b, hereafter CB05) in order to study hydrostatic and geostrophic adjustment. Although the model’s physics are a simple approximation to those incorporated by the dynamic cores of sophisticated modern numerical models, the linear model of CB05 has the advantage of being analytically tractable and able to facilitate a careful sensitivity study. The model is compressible and is subjected to external, time-dependent injections of mass, momentum, and heat. The internal response to these external injections involves acoustic waves, buoyancy waves, Lamb waves, and a PV-conserving steady state. CB05 performed experiments to examine the response to heat injections of varying spatial and temporal scales. This paper extends the two part study of CB05 in order to examine how the type of injection affects the qualities of the adjustment. Section 2 reviews the essential features of the model and its solution, including the properties of the energy partitioning. Section 3 compares the response to a mass injection (e.g., the rapid vertical redistribution of mass that takes place inside a cumulus cloud) to that following a heat injection that both generate an equivalent distribution of PV. This compressible analysis differs from the Boussinesq studies of Raymond (1983) and Shutts and Gray (1994) who found that heat and mass injections have equivalent effects on the flow. Section 4 compares the response to external injections of divergent vertical and horizontal momentum. Section 5 examines the response to convection-like injection of circulation, including an isolated updraft with and without compensating horizontally divergent motion. This section is relevant to the initialization of updrafts in compressible numerical weather prediction models. Section 6 examines the adjustment to injection of nondivergent horizontal velocity that generates inertia-gravity waves following the deflection of the current by the Coriolis force (i.e., a problem analogous to Rossby’s adjustment problem). Section 7 summarizes and discusses the results.

## 2. Model and solutions

The physical model is a compressible, rotating, inviscid, ideal gas on an *f* plane subject to gravity −*g***z** where *g* is the acceleration due to gravity and **z** is the unit normal vector in the vertical direction. The governing equations are linearized about a resting, hydrostatic, stratified, isothermal basic state. Apart from the prescribed heat injection, the flow is adiabatic. The fields are assumed to be invariant in the *y* direction so that divergent horizontal motions are isolated in the *x* direction. Consequently, an injection of *y* momentum generates horizontally divergent motions only after being deflected by the Coriolis force. Furthermore, an injection of *x* momentum is unable to generate a nontrivial hydrostatic and geostrophic steady state because a balancing mass gradient in the *y* direction is not permitted. Here, the essential characteristics of the model and solution are described, and the reader is encouraged to refer to the original papers if more detail is desired. The same set of values for the model parameters as was used in CB05 are also used here, that is, *T*_{*} = 255 K, *p*_{*} = 1000 mb, *ρ*_{*} = 1.37 kg m^{−3}, *N _{s}* = 0.0194 s

^{−1},

*H*= 7.48 km,

_{s}*c*= 320 m s

_{s}^{−1},

*f*= 10

^{−4}s

^{−1}, where the subscript

*s*denotes a basic-state quantity, and the subscript * denotes a basic-state quantity evaluated at the surface

*z*= 0. All symbols have their standard meaning, with

*N*being the buoyancy frequency,

_{s}*H*the density-scale height,

_{s}*c*the speed of sound, and

_{s}*f*the Coriolis parameter.

**contains the time-dependent pressure, potential temperature, and velocity components, 𝗔 is the spatial-operator matrix, and**

*χ***F**is the injection vector. For example, this injection vector may be the external representation of the localized convection in the larger-scale environment. Equation (2.1) is solved by 1) representing the horizontal dependence in the

*x*direction by a complex Fourier transform whose wavenumber parameter is denoted by

*k*; 2) representing the vertical dependence by a set of basis functions whose wavenumber is denoted by

*m*and that satisfies the rigid lower boundary condition at

*z*= 0; 3) finding the temporal Green’s function corresponding to instantaneous injections of mass, momentum, or heat; and 4) convolving this solution with general time-dependent injections to find the general solution. The solutions have the general form:where

*χ*is one of the fields,

_{i}*n*is the index for the vertical basis functions,

*f*(

^{i}_{n}*z*) is the appropriate vertical basis function corresponding to the field

*χ*,

_{i}*ω*is the buoyancy wave frequency,

_{b}*ω*is the acoustic wave frequency, and the time-dependent coefficients depend on the qualities of the external injection. All quantities depend on the Fourier transform coefficient

_{a}*k*that is not explicitly represented in (2.2). The representation (2.2) implies that the solution may be written as a sum of spatial

*modes*, each of which has a time dependence composed of three wave

*classes:*steady (PV conserving), acoustic wave, and buoyancy wave. The coefficient

*a*of the steady class is indeed constant after the injection is turned off. The

_{n}*n*= 0 mode is the

*Lamb mode*, whose pressure decays exponentially with increasing height

*z*and has only a steady and acoustic wave class, the latter being the

*Lamb wave*. The horizontally propagating Lamb mode is hydrostatic in the vertical and does not contribute to the potential temperature or vertical velocity fields. It should be noted that the upper boundary at

*z*= 2

*D*= 30 km is assumed to be rigid (i.e.,

*w*= 0 at

*z*= 2

*D*). This condition affords the construction of a spectrum of vertical basis functions that are discrete in wavenumber

*m*, and the domain depth 2

*D*is sufficiently large with respect to the density-scale height such that the characteristics of the steady-state solution are indistinguishable from those in a semi-infinite atmosphere. In the limit where the upper boundary is arbitrarily high, the spectrum maintains its basic qualities, but the vertical wavenumber is continuous.

*a*, and

*b*denote the PV-conserving, acoustic wave, and buoyancy wave classes, respectively. Once again, the dependence on horizontal wavenumber

*k*is implicit. During the application of the injection, each of the spectral contributions to the energetics is time dependent. Following the injection, the contributions are constant. CB05 examined the dependence of the ratios of the spectral contributions on the spatial and temporal characteristics of a heat injection, and established several basic qualities of this partitioning:

- An extended duration injection filters waves relative to the temporal Green’s function solution. When the duration of the injection exceeds the period of the wave, then the wave is not excited significantly. This rule applies for each injection type. The PV-conserving class (
*ω*= 0) is unaffected by the temporal details of the injection, and depends only on the time-integrated characteristics of the injection. The acoustic waves are not generated significantly by injections of duration exceeding the acoustic cutoff period of 292 s. Lamb waves and buoyancy waves have a minimum frequency of*f*, the Coriolis parameter, and may therefore be excited by injections whose duration is several hours. - An injection with given spatial characteristics will tend to excite waves of similar scale. The exact projection depends on the type of injection. Because waves of a given spatial scale have fixed acoustic and buoyancy wave frequencies that are given by the dispersion relation, the exact energy partitioning, in light of the first quality, cannot be determined by considering the spatial and temporal spectra independently.

In the following set of experiments, injections of mass, heat, and momentum are specified and the above qualities are demonstrated and extended.

## 3. PV-equivalent injections of mass and heat

^{−1}) that is characterized by a half-width

*a*, a depth

*d*, an elevation above the lower boundary

*d*, a duration

_{g}*τ*, and a time-integrated amplitude Δ

*θ*. The specific form of the warming iswhere

*H**(

*z, z*

_{1},

*z*

_{2}) =

*H*(

*z*–

*z*

_{1}) –

*H*(

*z*–

*z*

_{2}) is the top-hat function between

*z*=

*z*

_{1}and

*z*=

*z*

_{2}, where

*H*is the Heaviside step function. Such a localized warming, contoured in Fig. 1a, is similar to those that have been used by previous authors to examine convective adjustment.

The warming (3.1) generates a PV anomaly that is positive below the heating and negative above. Following the injection, PV is conserved. This time-integrated distribution of PV determines the qualities of the steady adjusted hydrostatic and geostrophic state toward which the atmosphere evolves. Therefore, PV-equivalent injections generate identical steady states. However, the transient evolution toward that steady state depends not only on the time-integrated structure of the injection, but also on its specific temporal details.

*θ*and Δ

*ζ*. The geometry of (3.2) (Fig. 1b) is a vertical source/sink dipole representing the transport of mass from lower levels to higher levels accomplished by a convecting cloud. The consideration of such an injection is conceptually appropriate in the context of this linear model, in which a rapid, small-scale physical process whose actual dynamics are highly nonlinear must be represented as a prescribed forcing. However, it should be noted that the representation of such convective mass fluxes in modern numerical models is only implicit. Even in those parameterizations characterized as “mass flux schemes” (e.g., Gregory and Rowntree 1990) the convective effects are explicitly incorporated in the larger-scale driving model through forcing in the heat and moisture equations.

Figures 2b,d present the acoustic response in the pressure field following an impulsive mass injection whose amplitude Δ*ρ* = 2.36 g m^{−3} corresponds to a heating of amplitude Δ*θ* = 0.1 K. Figures 2a,c present the response following the equivalent heating. The maximum pressure perturbation in Figs. 2b,d is 465 Pa; that is considerably larger than the 31 Pa generated by a heating of amplitude Δ*θ* = 0.1. The magnitude of the pressure perturbation following the mass injection is therefore an order of magnitude larger than that of the equivalent heat injection. Such an extraordinary perturbation suggests that an instantaneous mass injection on this scale is severely unphysical. A mass injection representing the vertical redistribution within a convecting cloud should have a time-scale characteristic of the advection by the updraft. The instantaneous mass injection is therefore a severe approximation of this process.

Figure 2 also demonstrates a qualitative difference between the acoustic response following the mass and heat injections. Four distinct vertically propagating wave fronts emerge from the perturbed region (Fig. 2b) following a mass injection, rather than the two observed following the heating (Fig. 2a). For the mass injection, expansion occurs in the upper source region while contraction occurs in the lower sink region, each of which generates pairs of upward- and downward-propagating wave groups. The contraction in the lower sink region initially generates a negative pressure perturbation on the lower boundary (see Fig. 2b). A larger positive perturbation next emerges on the lower boundary as the downward-propagating front originating from the source region reflects off the boundary. In contrast to the behavior during the acoustic regime, the subsequent buoyancy wave response (not shown) following the mass injection is qualitatively indistinguishable from that following the equivalent heat injection.

To emphasize that the difference in the response to mass and heat injections occurs primarily during the acoustic stage of adjustment, Fig. 3 presents the energy partitioning as a function of duration *τ*. The space between the curves depicts the fraction of the total available energy that is contained in each class of time dependence. For each duration, the total energy is partitioned among the PV, acoustic, buoyancy, and Lamb waves. As mentioned above, the steady-state response to an injection of heat as in Fig. 1a is equivalent to that generated by the mass source/sink dipole in Fig. 1b. The energy partitioning for these PV-equivalent injections is quite different for short duration. When *τ* < 10^{2} s (the approximate duration of acoustic wave attrition), the response to a mass injection is dominated by acoustic waves. Because injections of short and long duration generate the same steady state with the same energy, the total energy associated with a short duration injection is much larger than that associated with a long duration injection. For example, the instantaneous injection of heat produces approximately 5.8 times the total energy of a 10 000-s injection. In addition, the total energy of a short duration mass injection is much larger than that of an equivalent heat injection of similar duration. For example, an instantaneous injection of mass produces approximately 33.2 times the total energy of an injection of heat. For longer duration injections *τ* > 10^{2} s, the energy partitioning is very similar between the two cases. On such time scales the waves that are not filtered by the long duration injection are primarily responsible for achieving a horizontal (geostrophic) adjustment that is largely insensitive to the manner by which the PV has been introduced.

## 4. Injections of horizontally and vertically divergent momentum

Small-scale sources of force imbalances may be represented as injections of momentum. To examine the characteristics of the larger-scale response to such injections, this section compares injection of *x* and *z* momentum with geometry identical to the heating (3.1) with a velocity amplitude of 1 m s^{−1}. This pair of injections makes for an interesting comparison because neither is capable of generating a steady state under the assumption of homogeneity in the *y* direction. The difference between these injections therefore highlights many differences between the transient qualities of adjustment in the vertical and horizontal directions. Section 6 examines the response to injection of *y* momentum independently.

The differences during the acoustic adjustment are demonstrated by comparing the evolution of the pressure field following impulsive injections of depth *d* = 5 km and half-width *a* = 2 km. Figure 4 presents the evolution of the pressure field during the acoustic response to the injection of *x* momentum. The pressure field is dominated by a positive (negative) eastward (westward) propagating Lamb wave. This signal is associated with the initial pattern of horizontal convergence generated by the injection. The convergence (divergence) induced upstream (downstream) of the *x*-momentum injection initially generates an elevated positive (negative) pressure perturbation. The subsequent increase (decrease) in the total column mass is accompanied by a positive (negative) perturbation along the lower boundary that propagates away horizontally as the Lamb wave of an approximate amplitude of 50 Pa.

In comparison, the response to the injection of *z* momentum (Fig. 5) is dominated by vertically propagating acoustic waves. The amplitude of the fronts is similar to those generated by the injection of *x* momentum. Once again, the waves originate from the regions of convergence and divergence. The imposed updraft initially generates a large positive (negative) pressure perturbation in the region of upper (lower) level convergence (divergence).

The differences in the buoyancy adjustment are demonstrated by comparing the evolution of the potential temperature field following the impulsive injections at much later times. The buoyancy wave response to the injection of *x* momentum (Fig. 6) is dominated by a horizontal propagation of energy, as was the case during the acoustic response. The phase of these waves is primarily directed vertically because, unlike the acoustic waves, the phase and group velocity of the buoyancy waves are orthogonal. Because the evolution of the potential temperature field depends on the vertical advection of the base-state potential temperature, it follows that in Fig. 6a the initial negative perturbation in the upper-right and lower-left quadrants and positive perturbation in the lower-right and upper-left quadrants indicate rising and sinking motions, respectively. Without the addition of heat, the perturbations in potential temperature are entirely attributed to advection of the base state. The buoyancy waves originate in the region of vertical divergence (convergence) located upstream (downstream) of the initial current that develops in response to the initial horizontal convergence (divergence). In contrast, the response to the injection of *z* momentum (Fig. 7) is dominated by upward-propagating waves. The phase of such waves is primarily oriented horizontally. The waves originate in the initial updraft that induces the initial negative potential temperature perturbation in the updraft core.

The sensitivity of the response to an increased duration injection is also of interest. Section 2 reviewed the general effect of increasing the duration of injection: Waves whose period exceeds the duration of injection are not generated significantly. For buoyancy waves, the vertically propagating nonhydrostatic modes have the shortest relative periods (approaching the limit 2*π*/*N _{s}*). Because the response to the injection of

*z*momentum is primarily vertical and involves these nonhydrostatic modes, increasing the duration of the

*z*-momentum injection effectively suppresses the transient response. The response following an injection of 20-min duration is so weak that it is not shown. However, the response to the injection of

*x*momentum is not as severely affected by the injection duration. In fact, the response to an injection of 20-min duration with

*a*= 25 km and

*d*= 5 km (Fig. 8) is quite similar to that generated by the impulsive injection. Furthermore, the maximum amplitude of the transients following an injection of

*x*momentum of 20-min duration are |Δ

*x*| = 931 m, |Δ

*z*| = 12.3 m, and |Δ

*T*| = 0.26 K, where Δ

*x*and Δ

*z*are the total particle displacements in the horizontal and vertical directions, respectively. In contrast, those following the injection of

*z*momentum of 20 min are considerably smaller (i.e., |Δ

*x*| = 15 m, |Δ

*z*| = 1.5 m, and |Δ

*T*| = 0.04 K).

These qualities of the transient response to divergent momentum injections are summarized by Fig. 9, which presents the partitioning of energy between acoustic, buoyancy, and Lamb waves, and the steady state following instantaneous injections of *x* and *z* momentum of depth *d* = 5 km. For each horizontal mode *k*, the energy in each transient class is normalized by the total energy residing in that wavenumber. The spectra in Fig. 9 are therefore referred to as the “horizontal white noise” spectra. The injections of *x* and *z* momentum generate no PV and no steady state. Thus the partitioning consists only of acoustic, buoyancy, and Lamb contributions. The left side of each panel represents large horizontal scales for which the buoyancy wave motions are primarily horizontal and the acoustic wave motions are primarily vertical. At the large horizontal scales on the left side of each panel, we observe that the response to *x* momentum involves mainly buoyancy waves, whereas that following the *z*-momentum injection involves mainly acoustic waves. Because the injection of *z* momentum is in the direction of the acoustic waves whose particle motions are naturally parallel to the injection, this *z*-momentum injection is an efficient generator of acoustic waves. In contrast, at these large horizontal scales, the injection of *x* momentum is in the direction of the buoyancy waves whose particle motions are naturally in the direction of forcing. At small horizontal scales, on the right side of these panels, the circumstance is reversed. Furthermore, because Lamb waves produce strictly horizontal particle motions, the injection of *x* momentum generates Lamb waves whereas the injection of *z* momentum cannot.

## 5. Injection of a convective circulation

The previous section compared the response to injection of vertically divergent and horizontally divergent momentum. The comparison treated each injection in isolation, and the response involved acoustic and buoyancy waves whose characteristics were very different depending on the direction of injection. In nature, vertical and horizontal motions can occur simultaneously. This section considers the response to an injection of a circulation resembling that accompanying convection.

A basic issue in the initialization of a numerical model is the representation of the velocity field associated with a cumulus cloud. For example, suppose that satellite infrared imagery indicates the existence of a cumulus cloud, but that detailed velocity data does not exist for this updraft. The associated velocity fields must therefore be assumed. The results of this section indicate that the prescription of a divergent vertical velocity field alone will generate acoustic waves. To avoid the generation of these spurious acoustic waves, Fiedler (2002) suggests compensating the updraft by a horizontally divergent momentum field that satisfies the anelastic continuity equation. Two questions arise in response to Fiedler’s suggestion: Will the injection of a two-dimensional velocity field that satisfies the anelastic continuity equation suppress the generation of acoustic waves? How will the buoyancy adjustment to the isolated updraft be effected by the imposition of the horizontally divergent momentum field?

*z*

_{*}=

*π*(

*z*–

*d*–

_{g}*d*/2)/

*d, a*is the half-width of the primary updraft,

*b*is horizontal displacement of the compensating downdrafts on the flanks of the updraft core,

*δ*(

*t*) is the Dirac delta function, and

*c*is a constant that must be fixed such that the horizontal integral across the injection (5.1) is zero so that all of the mass transported upward by the updraft core is transported downward on the flanks of the updraft. For a sufficiently large domain,

*c*= 1. The injection (5.1) is contoured in Fig. 10a, where

*a*= 2 km,

*b*= 6 km,

*d*= 1 km,

_{g}*d*= 5 km, and the amplitude is Δ

*W*= 1 m s

^{−1}(kg m

^{−3})

^{−1}. The vertical velocity injection is vertically divergent and will therefore excite an adjustment whose qualities resemble those presented earlier in this section, including the generation of vertically propagating acoustic waves. Fiedler (2002) suggests that these acoustic waves may be suppressed if one also specifies a horizontally divergent horizontal momentum injection that makes the full two-dimensional velocity field satisfy the anelastic continuity equation. Such a horizontal momentum injection is given byFigure 10b contours the horizontal momentum injection (5.2) that is characterized by regions of inflow and outflow at the base and top of the updrafts. The horizontally divergent injection (5.2) will excite an adjustment resembling that presented in section 4, including the generation of acoustic waves.

The first question to consider is whether or not the total injection **u̇** = *u̇***x̂** + *ẇ***ẑ** will suppress the acoustic waves that are generated by each directional injection alone. First consider the response to the injection of vertical momentum (5.1) during the acoustic stage. Figure 11 plots the evolution of the vertical profiles of perturbation pressure and vertical velocity at the center of the updraft *x* = 0 during the first minute following the injection. Acoustic wave groups are generated in the divergent regions at the top and bottom of the updraft. The wave groups propagate upward and downward, respectively, at the speed of sound *c _{s}* = 321 m s

^{−1}, with the downward-propagating front reflecting off the lower boundary between

*t*= 10 and 25 s. In comparison, the response to the injection of horizontal momentum (5.2) during the acoustic stage (Fig. 11b) shows acoustic wave groups whose characteristics are strikingly similar to those generated by the vertical momentum injection (5.1), except that the sign of the perturbations is reversed. The characteristics of the vertical velocity field in the updraft core are indeed dissimilar between these two cases, which indicates that the buoyancy wave response may be different—an issue that is examined in more detail below. The extent to which the acoustic wave fronts are suppressed by the total two-dimensional injection is demonstrated by examining the evolution of the perturbation pressure and vertical velocity profiles in the first minute following the total nondivergent injection (Fig. 12), that is, the sum of the right and left columns in Fig. 11. The acoustic wave fronts are indeed significantly reduced in amplitude. However, the cancellation is not perfect, and although the waves that propagate vertically away from the updraft core are approximately filtered, there remains a significant negative pressure perturbation that is approximately confined to the updraft region and that evolves on this acoustic time scale. The vertical velocity perturbation that remains in the updraft core is much larger in amplitude than the perturbations remaining in the acoustic waves and the acoustic waves are therefore not evident in the vertical velocity field (Fig. 12b).

The updraft undergoes an adjustment on a longer time scale associated with the buoyancy waves. We now consider the effect of prescribing the compensating horizontal momentum injection on the adjustment of the updraft core. Figure 13 presents the evolution of the vertical profile of the perturbation potential temperature field at the updraft center *x* = 0 during the first 20 min following the injection. The potential temperature is presented because it effectively isolates the perturbations associated with buoyancy wave motions. In this linear adiabatic problem, the potential temperature perturbation is generated entirely through vertical advection of the basic-state potential temperature. The perturbation potential temperature is therefore proportional to the time-integrated vertical velocity (i.e., the net vertical displacement). Following the injection of the horizontal momentum (5.2) alone (Fig. 13a) there is a gradual upward displacement of the region where the updraft core exists in (5.1), with sinking occurring above and below this region. Vertically propagating gravity waves are also evident above the region containing the main signal. The buoyancy waves carry away most of this kinetic energy such that the potential temperature perturbation, and hence the updraft, is removed by *t* = 15 min. In contrast, the adjustment following the injection of vertical momentum (5.1) (Fig. 13b) shows that the updraft is removed very efficiently by vertically propagating gravity waves. The updraft core is entirely removed within the first 5 min. Figure 13c presents the adjustment following the total two-dimensional adjustment. It is clear that the initial updraft is supported much more efficiently when a horizontally divergent injection is prescribed in addition to the initial updraft. Figure 13d, which plots the evolution of potential temperature perturbation at a point in the center of the updraft (*x* = 0 km, *z* = 3.5 km), demonstrates that the horizontally divergent momentum injection encourages a much larger net displacement than the injection of vertical momentum alone. Therefore, although the initialization procedure effectively (but not completely) filters acoustic waves from the solution, the characteristics of the subsequent response during the buoyancy stage of adjustment are significantly affected.

## 6. Injection of horizontally nondivergent momentum

The injection of *y* momentum induces a response quite different from that induced by the divergent injections of *x* and *z* momentum of section 4 because the assumption of invariance in the *y* direction makes this injection nondivergent. The waves examined in section 4 were primarily generated in regions of convergence and divergence. Following an injection of *y* momentum, the slow deflection of the current by Coriolis forces produces regions of convergence and divergence. The transient response to an injection of *y* momentum is therefore expected to involve only the lowest-frequency waves.

Consider an injection of *y* momentum with geometry given by (3.1) of depth *d* = 5 km, elevation *d _{g}* = 1 km, half-width

*a*= 25 km, duration

*τ*= 20 min, and amplitude 1 m s

^{−1}. The initial imbalance associated with this injection resembles that of the classic Rossby adjustment problem. Figure 14 presents the evolution of the pressure field in the first 2 h following the injection. Although the amplitude of the pressure field is only O (1 Pa), there is a discernable Lamb wave signal. A temporal cross section of the pressure field along the lower boundary (Fig. 15) demonstrates the Lamb wave that induces a far-field change in pressure. By the end of the first hour (Fig. 14c), a high (low) pressure perturbation emerges on the right (left) periphery of the current. These localized perturbations indicate the convergence/divergence associated with the deflected current. The evolution of the potential temperature field (Fig. 16) demonstrates that vertical divergence (convergence) compensates the horizontal convergence (divergence) on the right (left) periphery of the current. Because the width of the current is very small compared to the Rossby radius of deformation

*L*=

_{R}*d N*/

_{s}*f*= 970 km, the current itself undergoes very little change. This dependence is illustrated in Table 1 that lists the amplitude of the steady-state current versus the half-width

*a*. A wider current induces a steady state of smaller amplitude. Conversely, as the width of the initial current is increased, the amplitude of the transients also increases.

Figure 17 presents the white noise partitioning (as in Fig. 9) for an injection of *y* momentum. The first row corresponds to an injection of 1-h duration. The partitioning consists mainly of buoyancy waves dominant at the large horizontal scales in Figs. 17a,c, and PV is dominant at small scales. Acoustic waves are not generated significantly [maximum fraction <10^{−3}%). Let us consider the physical mechanism that underlies this behavior. Initially, all the fields except the *y* momentum are zero. CB05 demonstrated that the ratio of horizontal KE to APE required in the steady state is large on scales that are narrow compared to the modal Rossby radius of deformation *L _{R}* =

*N*/

_{s}*f m*. The initial distribution of KE therefore requires only a small change in APE to achieve the steady state. In other words, at small horizontal scale (compared to

*L*), the pressure field does not require significant adjustment to balance the initial velocity (i.e., only a small change in APE is required to balance the KE associated with current). The small change in the horizontal gradient of pressure may then be accomplished by waves of low energy relative to the kinetic energy of the current.

_{R}For the relatively deep injection in Figs. 17b,d, the steady state is more dominant at wider scales than for the shallow injection in Figs. 17a,c at similar horizontal scale. The deeper injection projects preferentially onto structures whose modal Rossby radius is relatively larger that therefore effectively shifts the partitioning toward wider scales.

Figures 17b,d present the partitioning following an injection of duration 12 h. Because the time scale associated with the preferred waves generated by the *y*-momentum injection is relatively long, the transient response is relatively insensitive to changing duration when the duration is less than several hours. Essentially, the response to the *y*-momentum injection is dominated by the PV-conserving contribution and the low-frequency inertia-gravity waves.

The adjustment to the injection of the *y* momentum indeed resembles that of the classic Rossby adjustment problem in which a slab-symmetric strip of ocean is suddenly accelerated. Veronis (1956) and Veronis and Stommel (1956) analyzed the partitioning of available energy between inertia-gravity waves and the geostrophic steady state in the Rossby adjustment problem (generalized to a two-layer ocean model accelerated gradually rather than instantly). The present analysis leads to conclusions that are qualitatively similar to those of these previous analyses, particularly with respect to the affect of forcing width and duration on the partitioning between gravity waves and the steady state. However, the present analysis may be distinguished from the previous analyses insofar as the context is a compressible, continuously stratified model atmosphere, in which Lamb and acoustic waves are excited, and whose available energetics include an elastic contribution.

## 7. Conclusions

Depending on the type of the imbalance generation mechanism (i.e., whether the imbalance is introduced through a prescribed injection of heat, mass, or momentum), the characteristics of the steady and transient responses and the wave spectrum may be very different. The waves that are generated by a localized imbalance play a significant role in the adjustment of the larger-scale toward a balanced state. Additionally, the waves generated by tropospheric sources are considered crucial to the dynamics of the stratosphere (e.g., Holton 1983). Therefore, in order to accurately model the interaction between the small-scale forcing and the larger-scale dynamics, it is necessary to know something about the injection type as well as the spatiotemporal details of the forcing. For example, it has been demonstrated that nonhydrostatic buoyancy waves dominate the wave spectrum at narrow horizontal scales when the injection is a rapid one of vertical momentum. Such waves are relatively more likely to propagate into the stratosphere. Lane et al. (2001) were able to attribute the primary origin of vertically propagating buoyancy waves in an anelastic numerical simulation of convection to forcings in the vertical momentum equation. The present analysis demonstrates that the potential for the updraft to generate these buoyancy waves is largest for the deepest updrafts. A rapid, deep injection of heat is also capable of generating these waves efficiently.

Wider, hydrostatic, rotating buoyancy waves are efficiently generated by the injection of nondivergent horizontal momentum with vorticity. These injections constitute the imbalance generation mechanism of the classic Rossby adjustment problem that is analogous to the injection of the *y*-momentum of the present problem. The injection of a divergent horizontal current generates a response in the along current direction. Such an injection therefore does not efficiently generate nonhydrostatic buoyancy waves whose group velocity is mainly directed vertically. However, if introduced rapidly and on small scales, such an injection efficiently generates horizontally propagating acoustic waves.

The rapid injection (or rapid redistribution) of mass is a very efficient generator of acoustic waves. The large pressure perturbations associated with such injections are directly removed by the acoustic waves. Similarly, a rapid heating generates a large pressure perturbation to which acoustic waves may respond. The significance of these sources in nature is much less than what one may conclude from the examination of the temporal Green’s function solution presented in Fig. 2. A restriction is placed on the ability of any injection to generate particular waves: the duration of the injection must be less than or equal to the period of the waves. Vadas and Fritts (2001) presented a similar conclusion with respect to Boussinesq buoyancy waves generated by time-dependent sources of momentum and heat. Similarly, Sotack and Bannon (1999) concluded that vertically propagating acoustic waves are not generated significantly by heating of duration exceeding 2 min.

The circulation associated with convection may consist of both updrafts and downdrafts and regions of inflow/outflow. It has been demonstrated that the response to an injection of vertical momentum alone is very different than the response to horizontal momentum alone. Additionally, the response to the injection of vertical momentum alone is different than the response to an injection of momentum that satisfies the anelastic continuity equation. Fiedler (2002) suggested, as a data initialization technique intended to filter acoustic waves, that the updraft associated with a cumulus cloud should be accompanied by horizontal motions such that the initial velocity field satisfies the anelastic continuity equation. This approach is successful at filtering most, but not all, of the acoustic waves that would be generated by the updraft alone. However, this approach also affects the buoyancy wave adjustment, encouraging a stronger updraft that does not collapse as quickly. The Fiedler (2002) initialization technique may therefore have the additional benefit of improving the ability of a numerical weather prediction model to support the existence of an initial updraft.

Whether it is most appropriate to model the imbalance associated with a region of convection as a heating, or as a momentum injection, or as a mass redistribution is ambiguous and returns us to the issue of imbalance origin, the discussion of which is admittedly speculative. In reality, convection can accomplish all of these injections and the process of convective adjustment has thus been modeled as a response to both heat and mass injection. In a Boussinesq model, Raymond (1983) and Shutts (1994) argue that the transient response to these injections should be precisely equivalent. A comparison of injections that generate equivalent distributions of PV provides insight into the compatibility of these studies. The PV associated with a warmed column of air is equivalent to that generated by a vertical mass source/sink dipole. However, a very rapid injection of mass generates a much larger acoustic response than does the heating. If the duration of the injection exceeds several minutes, then the response to the PV-equivalent heating and mass injections are very similar. It is therefore essential to carefully consider the time scale associated with the injection.

The results of the current study may be used to comment on the notion that the response to a given injection and the “averaged” injection are in some sense equivalent or asymptotic. The steady state corresponding to the PV introduced by a given injection is independent of the temporal characteristics of the injection. Furthermore, if one defines a reasonable spatial averaging operator, then it may be shown that the spatially averaged steady-state response to a given injection is equivalent to the steady-state response to the spatially averaged injection; that is, these averaging operators commute between the injection and steady state. However, this commutability does not hold for the transient response. It has been demonstrated that, by “averaging” the temporal structure of a given injection, one will preferentially filter high-frequency waves; by spatially averaging a given injection, one projects less energy onto finer-scale structures. As Holton et al. (2002) have argued, the mutual dependence of the spatial and temporal characteristics of the transients complicates matters. A temporal averaging operator acts, in some sense, as a spatial averaging operator, and vice versa. For applications in which the nature of the transients is of primary concern, these matters must be taken into consideration.

The model used in this study was two dimensional and linear. These assumptions do not severely limit the general relevance of the conclusions presented in this section, but a few caveats should nonetheless be noted. The assumption of horizontal homogeneity in one dimension restricts all divergent motions and the propagation of wave energy to a two-dimensional plane. As a consequence, the wave energy density along a traveling disturbance decays more slowly than in the general three-dimensional case. The slab-symmetric assumption conveniently afforded the isolation of divergent irrotational injections of momentum from nondivergent rotational injections of momentum. Linearized PV is generated only by an injection of momentum in the latter category. In the general nonlinear case, PV generation is more complicated. The generation mechanisms are modulated by the evolving flow and local generation can also arise from advection. Additional sources must be considered in a model that is moist and viscous. Furthermore, the lessons learned from examining a linear model must be applied cautiously to a circumstance in which internal instability may be released on a scale similar to that upon which the linear model has been applied.

## Acknowledgments

Partial financial support was provided by the National Science Foundation under NSF Grants ATM-9820233 and ATM-0215358. We thank David R. Stauffer for suggesting the relevance of this work to the initialization of updrafts in numerical weather prediction.

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Maximum amplitude of the steady-state current *υ _{f}* as a function of the half-width

*a*of the initial current following an injection of

*y*momentum of amplitude 1 m s

^{−1}.