Wave Response during Hydrostatic and Geostrophic Adjustment. Part ...

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Wave Response during Hydrostatic and Geostrophic Adjustment. Part II: Potential Vorticity Conservation and Energy Partitioning JEFFREY M. CHAGNON

AND

PETER R. BANNON

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 25 September 2003, in final form 16 August 2004) ABSTRACT This second part of a two-part study of the hydrostatic and geostrophic adjustment examines the potential vorticity and energetics of the acoustic waves, buoyancy waves, Lamb waves, and steady state that are generated following the prescribed injection of heat into an isothermal atmosphere at rest. The potential vorticity is only nonzero for the steady class and depends only on the spatial and time-integrated properties of the injection. The waves contain zero net potential vorticity, but undergo a time-dependent vorticity exchange involving latent and relative vorticities. The energy associated with a given injection may be partitioned distinctly among the various wave classes. The characteristics of this partitioning depend on the spatiotemporal detail of the injection, as well as whether the imbalance is generated by injection of heat, mass, or momentum. Spatially, waves of a scale similar to that of the injection are preferentially excited. Temporally, an extended duration injection preferentially filters high-frequency waves. An instantaneous injection, that is, the temporal Green’s function, contains the largest proportions of the high-frequency waves. The proportions of kinetic, available elastic, and available potential energies that are carried by the various waves are functions of the homogeneous system. For example, deep buoyancy waves of small horizontal scale primarily contain equal portions of available potential and vertical kinetic energy. The steady state contains more available potential energy than kinetic energy at small horizontal scale, and vice versa. These qualities of the wave energetics illustrate the mechanisms that characterize the physics of each wave class. The evolution and spectral partitioning of the energetics following localized warmings identical to those in Part I are presented in order to illustrate some of these basic properties of the energetics. For example, a heating lasting longer than a few minutes does not excite acoustic waves. However, Lamb waves of wide horizontal scale can be excited by a heating of several hours. The first buoyancy waves to be filtered by an extended duration heating are those of the deepest and narrowest structure that have a frequency approaching the buoyancy frequency. The energetics of the steady state depends only on the spatial and time-integrated properties of the warming. However, the energetics and transient evolution toward a given steady state depend on the temporal properties of the warming and may differ widely.

1. Introduction The first part of this study (Chagnon and Bannon 2005a, hereafter Part I) presented the solution to a linear compressible model of hydrostatic and geostrophic adjustment to prescribed injections of mass, momentum, and heat. The model solution was used to investigate the transient response of the statically stable large-scale environment to a rapidly produced, localized imbalance. The goal of this paper is to examine the potential vorticity and energy partitioning in order to present a more general picture depicting the relationship between the qualities of the imbalance generation mechanism and those of the response. Corresponding author address: Jeffrey M. Chagnon, Dept. of Meteorology, The Pennsylvania State University, University Park, PA 16802. E-mail: [email protected]

© 2005 American Meteorological Society

JAS3419

Potential vorticity (PV) and energy are useful measures for characterizing an evolving geophysical flow. Both are globally conserved when the flow is frictionless and adiabatic. Analysis of PV may directly yield the steady, hydrostatic, and geostrophic state toward which the perturbed flow evolves (e.g., Rossby 1938; Obukhov 1949; Schubert et al. 1980; Chagnon and Bannon 2001). Analysis of the energetics and its partitioning among various spectral and wave contributions conveniently establishes relationships between the nature of the imbalance and that of the response. Previous investigations of geostrophic adjustment have partitioned the energy associated with a given imbalance among the buoyancy waves and steady state (e.g., Veronis 1956; Vadas and Fritts 2001). Likewise, investigations of hydrostatic adjustment have partitioned the energy among acoustic waves and the steady state (Bannon

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1995; Sotack and Bannon 1999). The energy of the transients was inferred by taking the difference between the energy of the initial imbalance and that remaining in the steady state. Here, however, three distinct classes of transients are present: acoustic waves, buoyancy waves, and Lamb waves. The previous method of inferring the energy partitioning is therefore insufficient. We partition the energetics among the steady-state, acoustic wave, Lamb wave, and buoyancy wave classes. According to the analysis of Part I, each of these classes of transient plays a distinct role in the adjustment process. The complex transient dynamics of the adjustment and its dependence on the spatiotemporal detail of the injection may be conveniently summarized by analyzing this energy partitioning. Section 2 describes the conservation of PV in the model. Section 3 describes the conservation of energy and derives its partitioning. Section 4 clarifies the abstract presentation of sections 2 and 3 by demonstrating the energy partitioning following localized heating corresponding to the three cases of Part I. These cases consider warmings characteristic of the cumulus, the intermediate, and the mesoscale. Section 5 summarizes the characteristics of energy partitioning and suggests directions for future research. Dikiy (1969) suggests that free solutions of the type presented in Part I can be written as a sum of distinct contributions from the steady state, acoustic, and buoyancy waves. We present a rigorous analysis of this assertion for forced solutions in the appendix that includes the injection terms. Note that equations from Part I will be distinguished from those presented here by a subscript I. For example, Eq. (2.4I) refers to the equation labeled (2.4) in Part I. Furthermore, as in Part I, “mode” refers to a specific spatial basis function and “class” refers to a specific time dependence. For example, the solutions are written as a sum of horizontal and vertical orthogonal modes, each of which may be separated into three classes according to distinct time dependencies: acoustic wave, buoyancy wave, and steady state. The one exception is the Lamb mode that contains only an acoustic wave class (i.e., the Lamb wave) and a steadystate class.

FIG. 1. Schematic illustration of the PV balance. The PV is comprised of latent and relative vorticity. The latent vorticity is in turn comprised of a thermal stratification and an elastic contribution. Thick arrows denote generation mechanisms. Thin arrows denote positive conversion mechanisms.

The notation is traditional, and is defined in Part I. (For the moment, we relax the assumption of Part I of invariance in the y direction.) The PV (2.1) is essentially a linearized form of Ertel’s PV, up to a factor of the base state density. It resembles traditional forms of linearized quasigeostrophic PV that are partitioned between relative and “stretching” vorticities. Here, the compressibility of the model introduces an additional term. The latent (i.e., hidden or unrealized) vorticity is comprised of an elastic contribution ␨e proportional to the density perturbation ␳⬘ as well as a thermal stratification contribution ␨␪ proportional to the vertical gradient of the perturbation potential temperature ␪⬘. The elastic contribution is absent in the anelastic and Boussinesq approximations. The conservation of each contribution to the PV is described by ⭸␨r ⭸t ⭸␨e ⭸t ⭸␨␪ ⭸t

2. Potential vorticity conservation The potential vorticity q is a scalar quantity conserved by the compressible, homogeneous system (2.4I) that consists of relative ␨r and latent ␨l vorticity: q ⫽ ␨r ⫹ ␨l,

共2.1兲

where

␨r ⬅

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⭸␷⬘ ⭸x

⫺

⭸u⬘ ⭸y

⫽ ⫺f ⫽f

冉

⫽⫺

␨l ⫽ ␨e ⫹ ␨␪ ⬅ ⫺f

冉 冊

f ⭸ ␳sg␪⬘ ␳⬘ ⫹ . ␳s ␳s ⭸z Ns2␪s

共2.2兲

⭸u⬘ ⭸x

⭸u⬘ ⭸x

⫹

⫹

⭸␷⬘ ⭸y

⭸␷⬘ ⭸y

冊

冊 冉 ⫹

⫹

⭸␷˙ ⭸x

⫺

⭸u˙ ⭸y

冊

,

f ⭸ ␳˙ 共␳ w⬘兲 ⫺ f , ␳s ⭸z s ␳s

冉 冊

f ⭸ ␳s g⌰ f ⭸ . 共␳sw⬘兲 ⫹ ␳s ⭸z ␳s ⭸z Ns2␪s

共2.3a兲 共2.3b兲 共2.3c兲

The sum of (2.3a)–(2.3c) implies that in the absence of external injection the PV is conserved locally: ⳵q/⳵t ⫽ 0. In the presence of external injection, the PV satisfies ⭸q ⭸t

,

冉

⫽

⭸␷˙ ⭸x

⫺

⭸u˙ ⭸y

⫺f

冉 冊

f ⭸ ␳sg⌰ ␳˙ ⫹ . ␳s ␳s ⭸z Ns2␪s

共2.4兲

Figure 1 schematically depicts the conservation of PV given by (2.3). Latent vorticity may be generated by the injection of heat or mass, and relative vorticity may be generated by the injection of horizontal momentum with vorticity. Following the initial generation, vorticity

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is converted among the various contributions. For example, consider an injection of heat. Initially, the injection of heat decreases/increases the thermal stratification above/below the level of maximum heating. Consequently, there is negative/positive latent vorticity in the region above/below the maximum heating. Below the maximum heating, the upward motion can result in either 1) a decrease in the perturbation density or 2) horizontal convergence. In the first case, some of the positive latent thermal-stratification vorticity is converted into positive latent elastic vorticity. In the second case, there is a conversion of positive latent vorticity to positive relative vorticity required by Kelvin’s theorem. We therefore expect an injection of heat to induce a cyclone below the level of maximum heating whose strength is mitigated in part by the compressibility of the fluid. Similarly, we expect an anticyclone to develop above the level of maximum heating. Such characteristics are observed in the steady state (e.g., Chagnon and Bannon 2001). Figure 1 does not by itself imply a causality relationship between the various injections, conversion mechanisms, and latent and relative vorticity. Causality is inferred by examining the complete dynamics of the system. Equation (2.4) implies that the steady state depends only on the time integral of the injection and not its temporal details. The energetics and the transient response, however, do depend on the temporal details of the injection. In section 5 we compare the energy of PV-equivalent warmings in order to demonstrate the extent to which they induce equivalent adjustment processes. Equation (2.4) also implies that injections of different type may generate equivalent distributions of potential vorticity and therefore the same steady state. In a subsequent article (Chagnon and Bannon 2005b), we examine injections of different type (e.g., mass versus heat) that generate equivalent distribution of potential vorticity. Applying now the assumption made in Part I that the model is homogeneous in the y direction, Eq. (2.4) implies that an injection of either x momentum or z momentum does not generate any perturbation PV and therefore does not induce a perturbation steady state. An injection of x momentum and the subsequent deflection of this current by Coriolis forces may not produce a pressure gradient in the y direction and there is no mechanism for the retention of a horizontal mass gradient in the asymptotic steady state. Similarly, an injection of z momentum does not generate a response involving permanent changes to the fluid center of gravity. Furthermore, the global PV (defined as the mass integral of the PV 兰V ␳sq dV) remains constant if the injection mechanism on the right-hand side of (2.4) satisfies several necessary conditions: an injection of heat must be zero on the upper and lower boundaries; an injection of mass must conserve the total mass; and an injection of momentum must not induce a net torque about the vertical direction (an injection of momentum

VOLUME 62

satisfying a periodic lateral boundary condition satisfies this criterion). To examine the partitioning of the PV among the various classes of solution, we write the PV in terms of the field variable transformation introduced in Part I as Q ⬅ ␳s1Ⲑ2q ⫽

⳵V ⭸x

⫹

f ␬ ⲐHs

冉 冊 ⳵

f

P,

共2.5兲

f 共m2 ⫹ ⌫2兲 f Sn ⫺ 2 Pn, ␬ ⲐHs cs

共2.6兲

⭸z

⫺⌫ S⫺

cs2

which may be written in spectral space as Qn ⫽ ikVn ⫺

where Q is expanded about the vertical basis function fn, m ⫽ n ␲ /D is the vertical wavenumber of mode n in an atmosphere of depth D, k is the horizontal wavenumber, ⌫ ⫽ 3/14Hs is a compressible vertical decay scale, and Hs is the density scale height. As in Part I, Pn, Sn, Un, Vn, and Wn denote the projection of the massweighted pressure, potential temperature, and velocity components onto the nth vertical basis function. Following the convention used in Part I, the subscript k denoting the Fourier transform of the horizontal structure is implied but omitted and solutions are assumed homogeneous in the y direction. It may be readily verified that any transient solution (2.19I) of the homogenous system (i.e., with nonzero frequency ␻i ⫽ 0) contains no PV, Qn ⫽ 0. (This result is not generally the case for models in which the basic state contains nonzero gradients of PV.) The perturbation in PV introduced by the injection mechanism is therefore comprised only of the hydrostatic and geostrophic (i.e., steady state) class, whereas the buoyancy wave, acoustic wave, and Lamb wave classes contain no PV. That is, the waves may contain portions of relative and latent vorticity that are equal in magnitude but opposite in sign. For the steady-state class, which we can refer to interchangeably as the PV-conserving class, we may write (2.6) in terms of the steady contribution to the pressure field, Qn ⫽ Qn,steady ⫽⫺

f Ns2

冉

m2 ⫹ ⌫ 2 ⫹

Ns2 f2

k2 ⫹

Ns2 cs2

冊

Pn,steady.

共2.7兲

The solution of (2.7) is a generalization of the solution presented in Chagnon and Bannon (2001) in order to investigate the properties of the hydrostatic and geostrophic steady state in response to an injection of heat.

3. Energy partitioning a. Basic properties Chagnon and Bannon (2001) presented a discussion of the available energetics of this linear model, including the mechanisms of conversion between kinetic

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(KE), available potential (APE), and available elastic (AEE) energies. The purpose of this section is to extend that analysis by presenting the partitioning of the energetics into distinct contributions from each of the spatial modes, which in turn are comprised of distinct contributions from each class of transient. This partitioning is not always possible [e.g., in the case of certain sheared flows (Held 1985)] and must therefore be demonstrated explicitly (see the appendix). Furthermore, the partitioning of each class of transient among KE, APE, and AEE is fixed for a given spatial mode. This latter partitioning is a property of the homogeneous system and is presented in some detail at the end of this section. The energy density is 1 TE ⫽ 2

冉

U*U ⫹ V*V ⫹ W*W ⫹

g2 Ns2

S*S ⫹

1 cs2

冊

P*P , 共3.1兲

where a superscript * denotes a complex conjugate. Equation (3.1), expressed in terms of the field variable transformation, is the traditional available energy density. The first three terms comprise the KE, the fourth is the APE, and the last is the AEE. The energy density is governed by ⭸ ⭸t

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˙ ⫹ V*V ˙ ⫹ W*W ˙ 共TE兲 ⫹ ⵜ · 共PU兲 ⫽ U*U

ˆ ⫽ TE ˆPV ⫹ TE ˆa ⫹ TE ˆb. TE n n n n

g2 Ns2

S*S˙ ⫹

1 cs2

P*P˙ ,

ˆPV ⫹ r共␻ , t兲TE ˆa ⫹ r 共␻ , t兲TE ˆb, TEn共t兲 ⫽ r 共0, t兲TE n a n b n 共3.6兲 where the filtering function r depends on the frequency of the contribution and is given by r 共␻, t兲 ⫽

冋冕

t

e⫺i␻t⬘␴t共t⬘兲 dt⬘

D共m2 ⫹ ⌫ 2兲 Un*Un ⫹ Vn*Vn ⫹ 共m2 ⫹ ⌫ 2兲 2

冉

Wn*Wn ⫹

g2 Ns2

Sn*Sn

冊

⫹

1 cs2

Pn*Pn

册

共3.3兲

that depends on the horizontal wavenumber k and vertical wavenumber m of the mode. Each mode is, in turn, comprised of acoustic wave, buoyancy wave, and PVconserving classes. The appendix demonstrates that there is no interaction among these distinct classes even in the presence of active injections. The spectral energy density may therefore be partitioned among the classes: TEn ⫽ TEnPV ⫹ TEna ⫹ TEnb.

册冋 冕

t

册

ei␻t⬘␴t共t⬘兲 dt⬘ .

0

共3.7兲

共3.2兲

where ⵜ ⬅ (⳵/⳵x, ⳵/⳵y, ⳵/⳵z) and U ⬅ (U, V, W). The global integral of the divergence of the energy flux vector PU is zero for periodic or closed rigid boundaries. Thus, in the absence of an injection, the energy is conserved globally. Because the spatial modes are orthogonal, we may consider the conservation of the spectral energy density

冋

共3.5兲

We may write the time-dependent spectral energy density generally in terms of the spectral energy density of the Green’s function

0

⫹

TEn ⫽

Following the shutoff of the injection mechanisms, the distinct acoustic wave (TEan), buoyancy wave (TEbn), and PV-conserving (TEPV n ) contributions to the spectral energy density are time independent. Each of these contributions consists of KE, APE, and AEE that undergo time-dependent conversions. [Note that the Lamb mode (n ⫽ 0) contains only a steady class and acoustic wave class, which is the Lamb wave.] The spectral energy density excited by an injection of arbitrary temporal structure may be generalized in terms of that of the temporal Green’s function solution (i.e., the response to an instantaneous injection function). Let the spectral energy density of the temporal Green’s function be given by

共3.4兲

and ␴t(t) is a function describing the time dependence of the injection function. Through the representation of the energy partitioning (3.6) we may investigate separately the effects of the spatial and temporal characteristics of the injection on the partitioning. The temporal Green’s function energy partitioning contains the dependence of the spatial characteristics as well as the type of injection. The filtering function (3.7) incorporates all of the time-dependent effects. The result of (3.6) and (3.7) is a generalization of the buoyancy wave filtering demonstrated by Vadas and Fritts (2001). Consider some examples of the filtering function. The first example is that of the delta function ␴t(t) ⫽ ␦(t), for which the filtering function is r(␻, t) ⫽ 1. In this case, the partitioning is that of the temporal Green’s function. Another example is for a top-hat injection of duration ␶, ␴t(t) ⫽ [H(t) ⫺ H(t ⫺ ␶)]/␶, where H is the Heaviside step function. The filtering function for this injection is

r 共␻, t兲 ⫽

1 共␻␶Ⲑ2兲2

冦

sin2 2

sin

冉冊 冉 冊

␻t , 2

t⬍␶

␻␶ , 2

tⱖ␶

.

共3.8兲

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ergy because acoustic waves have frequencies greater than the acoustic cutoff frequency NA ⫽ 0.0214 s⫺1 (and hence periods ⬎294 s). The sine squared in time provides a smoother filter with less energy remaining in the higher-order harmonics.

b. Energy ratios It is useful to examine some characteristics of the energy partitioning of the homogeneous solutions. The energetics may be decomposed into its various spatial contributions. Equation (3.4) asserts that these may be partitioned among the time-dependent classes (i.e., acoustic, buoyancy, and PV-conserving classes). Each of those classes may in turn be partitioned among KE, APE, and AEE. For a given mode and wave class, these relative proportions of the time-averaged KE, APE, and AEE (i.e., average over an integer multiple of wave periods) are fixed. (Note that at any instant in time there is an exchange between the energy terms, e.g., the KE may be zero at one instant and nonzero at some later instant at the expense of APE.) The ratios of these time-averaged energies may be derived from the dispersion relation (3.6I), the form of the spectral energy terms (3.3), and the form of the eigenvectors (3.7I). One finds APE:AEE ⫽

FIG. 2. (a) Top-hat (solid) and sine-squared (dashed) injection functions ␴t (t) and (b) their corresponding filtering functions for t ⬎ ␶ plotted as a function of the injection duration parameter ␶ times the frequency ␻ of the mode.

An injection that is somewhat smoother in time such as the sine-squared injection ␴t(t) ⫽ 2[H(t) ⫺ H(t ⫺ ␶)]sin2(␲ t/␶)/␶ has a filtering function given by r 共␻, t兲 ⫽

1 共␻␶Ⲑ2兲2

冤

×

冉 冊 冉 冊 2␲ ␶

␻2 ⫺

2␲ ␶

冥冦 2

2

2

sin2 sin2

冉冊 冉 冊

␻t , 2

t⬍␶

␻␶ , 2

tⱖ␶

.

共3.9兲 Figure 2 presents the filtering functions (3.8) and (3.9) for t ⬎ ␶. Waves are filtered relative to the Green’s function solution. The filtering is an increasing function of the wave frequency and of the injection duration. For example, an injection of duration exceeding several minutes should not generate much acoustic wave en-

KEz:APE ⫽

KEx:APE ⫽

cs2共m2 ⫹ ⌫ 2兲Ns2 共Ns2 ⫺ ␻2兲2

␻2 Ns2

冉

共3.10a兲

,

共3.10b兲

,

Ns2 ⫺ ␻2 f 2 ⫺ ␻2

冊冉 2

␻2 ⫹ f 2 Ns2

冊冉

k2 m2 ⫹ ⌫2

冊

,

共3.10c兲 where an overbar denotes a time average computed over a period of the wave, and the KE is separated into horizontal (KEx) and vertical (KEz) contributions. Note that the horizontal component KEx contains contributions from both the x and y components of velocity, u and ␷. The overbar denoting the temporal average is omitted from the following figures and discussion. Figure 3 presents this partitioning for the PVconserving, acoustic wave, and buoyancy wave classes. The scale dependence of the energy ratios may be explained in terms of the physical mechanisms that are at work in each wave class. Consider first the PV class (Fig. 3a), for which ␻ ⫽ 0. Equation (3.10a) indicates that the ratio of AEE to APE depends only on the vertical scale of the wave. Indeed, the dashed curves in Fig. 3a are independent of horizontal wavenumber k, and they demonstrate that the ratio AEE:APE is larger for the deep mode. The explanation for this behavior requires the consideration of hydrostatic balance. The

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FIG. 3. Ratios of KE to AEE or APE (solid lines) and APE to AEE or its inverse (dashed lines) for the (a) PV-conserving, (b) buoyancy wave, and (c) acoustic wave classes as a function of horizontal wavenumber k. The first column gives the ratios for a deep mode (m ⫽ 1/20 km) and the second for a shallow mode (m ⫽ 1/1 km).

hydrostatic pressure perturbation is related to the potential temperature perturbation via

冋

⳵ ⭸z

⫺

␬ Hs

册冉 冊 p⬘ ␳s

⫽

g␪⬘ . ␪s

共3.11兲

The left side of (3.11) is proportional to the square root of AEE, and the right side is proportional to the square root of APE. The derivative operator on the left side of (3.11) implies that for a deep mode a larger pressure perturbation (and hence AEE) is required to maintain this balance. Now consider the ratio of KE to APE in the steady state (note that KEz ⫽ 0). Equation (3.10c) with ␻ ⫽ 0 implies that the ratio is given by a modal Burger number, k2Ns2/(m2 ⫹ ⌫ 2)f 2. Let us first examine the deep mode in the left panel of Fig. 3a. The ratio KE:APE is a decreasing function of increasing horizontal scale. Consider the geostrophic balance of the steady state. A given geostrophic wind must be balanced by a horizontal pressure gradient that is proportional to a horizontal gradient of perturbation potential temperature. When the horizontal scale is large, a larger perturbation potential temperature is required to maintain this balance, which thus requires a relatively

larger value of APE. For the shallow mode, the dependence of the ratio KE:APE on horizontal scale is the same as for the deep mode, except that the curve is shifted toward larger horizontal wavenumber k. Next, we consider the energy partitioning for the buoyancy waves (Fig. 3b) whose frequencies lie between the inertial and buoyancy frequencies (Fig. 2I). Recall that the direction of the particle displacements may be determined by the ratio of the horizontal and vertical wavenumbers. Let us first consider the deep mode case displayed in the left panel of Fig. 3b whose vertical wavenumber m is fixed. For the wide scales (on the left side of the panel), the waves’ particle displacements are nearly horizontal, the frequencies approach the inertial frequency, and the motions are approximately hydrostatic. Then, from (3.10), the waves are rich in horizontal kinetic energy but poor in vertical kinetic energy relative to the elastic and potential energies. The two available energies behave in a manner similar to that for the PV class discussed above. At these wide scales, the mechanism for conversion from KEx to APE is via horizontal mass convergence. This mass convergence is proportional to the scale of the gradient of the current. The ratio KEx:APE is

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TABLE 1. Fraction of total energy residing in the various wave classes and the total energy following the heat injections of Part I. Cumulus scale (␶ → 0)

Cumulus scale (␶ ⫽ 4 min)

Intermediate scale

Mesoscale

0.25 0.05 0.68 0.02 8.43

0.00 0.01 0.95 0.04 4.67

0.00 0.03 0.74 0.23 155

0.00 0.01 0.46 0.53 572

Acoustic Lamb Buoyancy Steady Total energy (MJ m⫺1)

therefore an increasing function of k. For the narrow scales (on the right side of the panel), the displacements are nearly vertical, the frequencies approach the buoyancy frequency, and the motions approach pure buoyancy oscillations. Then, from (3.10), the waves are rich in vertical kinetic energy and poor in both horizontal kinetic energy and elastic energy relative to the potential energy. The waves convert equal potions of APE and KEz. Similar considerations apply for the shallow mode on the right panel of Fig. 3b where the wide-scale behavior dominates and the transition to the buoyancy limit is shifted toward larger horizontal wavenumbers. The energy partitioning for the acoustic waves (Fig. 3c) behaves in the opposite sense as that of the buoyancy waves. Let us first consider the deep mode case displayed in the left panel of Fig. 3c whose vertical wavenumber m is fixed. For the wide scales (on the left side of the panel), the waves particle displacements are nearly vertical and the frequencies are relatively constant (Fig. 2I) but always greater than the acoustic cutoff frequency that is greater than the buoyancy frequency. Then, from (3.10), the waves are rich in vertical kinetic energy but poor in horizontal kinetic energy relative to the elastic and potential energies. The vertical kinetic energy is always greater than the potential energy. For the narrow scales (on the right side of the panel), the displacements are nearly horizontal, the frequencies approach the acoustic frequency, and the motions approach pure acoustic oscillations. Then, from (3.10), the waves are rich in horizontal kinetic energy and poor in both vertical kinetic energy and potential energy relative to the elastic energy. The waves convert equal potions of AEE and KE. Again, similar considerations apply for the shallow mode on the right panel of Fig. 3c where the wide-scale behavior dominates and the transition to the horizontally propagating modes is shifted toward larger horizontal wavenumbers. For the Lamb mode (not shown in Fig. 3), the partitioning consists of only AEE and KEx. For the acoustic class of the Lamb mode, that is, the Lamb wave, the time-averaged proportions of these energies is KEx:AEE ⫽ 1 ⫹

2f 2 cs2k2

.

共3.12兲

At small horizontal scale, the Lamb wave contains approximately equal portions of AEE and KEx. At large

horizontal scale, the Lamb wave contains a much larger portion of KE. For the steady class of the Lamb mode: KEx:AEE ⫽

cs2k2 f2

,

共3.13兲

and the spectral partitioning is reversed. As in Fig. 3a, the steady state contains a significant relative proportion of KEx only at the narrowest horizontal scales. At such scales, only a small change in the horizontal distribution of mass is required to balance an injected current. The above presentation of the partitioning of the energy in each class among KE, APE, and AEE is useful for explaining the partitioning of total energy associated with a given injection function among the classes of time dependence. The next section demonstrates this utility.

4. Energetics of adjustment to heat injection This section clarifies the abstract presentation of the previous sections by demonstrating the energy partitioning following localized warming of the form used in the experiments of Part I. The localized warming, given by (4.1I), is characterized by a half-width a, a depth d, and a duration ␶. These parameters were fixed to represent warmings on the cumulus scale (a ⫽ 1 km, d ⫽ 5 km, ␶ → 0), intermediate scale (a ⫽ 25 km, d ⫽ 9 km, ␶ ⫽ 20 min), and the mesoscale (a ⫽ 100 km, d ⫽ 9 km, ␶ ⫽ 2 h) that were elevated a height dg ⫽ 1 km above the rigid lower boundary and had an amplitude ⌬␪ ⫽ 1 K. Table 1 presents the total energy partitioning among the various classes of transient corresponding to these experiments. For the cumulus-scale heating structure, we consider two durations: the instantaneous heating that was used in Part I, and a 4-min heating that will be used later in section 4a. Consider first the fraction of energy in the steady state. This fraction increases as the spatial scale of the warming increases, which is consistent with the spectra presented in Chagnon and Bannon (2001) wherein the energy was partitioned between the waves and the KE, AEE, APE of the steady state. Acoustic wave energy was generated appreciably by only the instantaneous cumulus-scale warming. The filtering function presented in Fig. 2 suggests that the rapid nature of this warming permitted the generation

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FIG. 4. Evolution of the horizontally integrated (a) AEE, and (b) KE, as a function of time t and height z in units of kJ m⫺2 following an instantaneous cumulus-scale heating. The contour interval is 0.02 up to 0.1, and 0.05 thereafter.

of acoustic waves, while even the heating of duration 4 min filtered most of the acoustic waves (the acoustic cutoff period is 2␲/NA ⬇ 294 s). In contrast, because Lamb waves exist at all frequencies greater than f, Lamb wave energy was generated by each warming. The fraction of buoyancy wave energy is maximized following the cumulus-scale heating of 4-min duration. This result should be interpreted carefully. The total energy generated by a warming of a given spatial structure is maximized in the temporal Green’s function solution (i.e., when the warming occurs instantly rather than gradually). The total buoyancy wave energy generated by the instantaneous heating is slightly larger than that following the 4-min heating because the 4-min heating approaches the duration of the maximum buoyancy period ␶b ⫽ 2␲/Ns ⬇ 324 s and therefore is a weak filter for the highest frequency buoyancy waves. The total energy is also an increasing function of the heating spatial scale. The decrease in the buoyancy wave energy fraction with increasing scale does not therefore imply a decrease in total buoyancy wave energy.

a. Spatial and temporal evolution of the wave energetics Before examining the energy spectra in detail, we analyze the transient evolution of energetics following the warmings applied in Part I. The analysis of Part I established that the response by the various wave classes depends on the scale and duration of warming. Here, we consider a cumulus-scale warming that generates an acoustic wave response within the first minute and a buoyancy wave response within the first fifteen minutes. We then examine the lower-frequency buoyancy wave response to a mesoscale warming, which involves a buoyancy wave response within the first few hours, and an inertial wave response within the first day following the warming. The vertical and horizontal propagation of energy during these response regimes are examined following each heating. The vertical

propagation of energy is demonstrated by presenting the evolution of the horizontally integrated energy density (i.e., integral over all horizontal wavenumbers) as a function of time t and height z. The horizontal propagation of energy is demonstrated by presenting the evolution of the vertically integrated energy density (i.e., integral over all vertical wavenumbers) as a function of time t and distance x. Consider the cumulus-scale heating with a Diracdelta function temporal dependence (␶ → 0) that generates a large acoustic response. To demonstrate the vertical propagation of energy during the acoustic wave regime, Fig. 4 presents the evolution of the horizontally integrated available elastic and kinetic energies in the first minute following the heating (cf. Fig. 5aI). Two distinct vertically propagating acoustic wave groups emerge from the top and bottom of the heated layer. The latter reflects off the lower boundary around t ⫽ 10 s. Comparison of the amplitude of these signals in Figs. 4a and 4b reveals that these waves contain approximately equal portions of AEE and KE, which is consistent with the energy ratio analysis presented in Fig. 3c. Because the acoustic waves do not carry a significant proportion of APE, we have not shown the evolution of APE in the first minute (i.e., the APE does not evolve significantly in the first minutes, as can be seen in Fig. 8I). The slope of the wave groups in Fig. 4 implies a propagation speed of approximately 320 m s⫺1, which is approximately the sound speed cs. Because the heating initially generates a pressure perturbation, but no motion, the initial vertical profile of horizontally integrated AEE (along the left side of Fig. 4a) resembles that of the heating, whereas the KE (left side of Fig. 4b) is initially zero. Examining the right side of Fig. 4a reveals that most of the initial profile of AEE is carried off by the acoustic waves. What remains in their wake, at t ⫽ 60 s, is a small but deep perturbation in AEE whose vertical structure may be explained by considering the horizontally propagating Lamb waves as

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FIG. 5. Evolution of the vertically integrated (a) AEE, and (b) KE, as a function of time t and distance x in units kJ m⫺2 following an instantaneous cumulus-scale heating. The contour interval is 0.02 up to 0.1, and 0.05 thereafter.

well as the acoustically adjusted state of Bannon (1995). In contrast, the right side of Fig. 4b indicates that the KE is increasing in magnitude in the wake of the acoustic waves at t ⫽ 60 s. This increase corresponds to the initial conversion of APE into upward motion in the heated region accomplished by the buoyancy waves. According to Fig. 3b, for large k these buoyancy waves primarily involve conversion between APE and KE (as will be demonstrated in Figs. 6 and 7). To demonstrate the horizontal propagation of energy during the acoustic regime, Fig. 5 presents the evolution of the vertically integrated AEE and KE in the first minute following the cumulus-scale heating (cf. Fig. 5bI). Within the first several seconds, a wave group emerges from the heated region near x ⫽ 0 and propagates outward at approximately the speed of sound cs (as implied by the slope of the group). This horizontally propagating Lamb wave contains approximately equal portions of AEE and KE, as could be predicted from (3.11). It is interesting to note that the Lamb wave leaves in its wake some AEE (⬍0.05 kJ m⫺2) corresponding to the horizontal expansion accomplished by the Lamb wave. Furthermore, at x ⫽ 0, the AEE exhibits a fluctuating signal within the first 30 s corresponding to the instances when the vertically propagating acoustic waves interfere and reflect off the rigid lower boundary. These interference/reflection events are primarily evident below z ⫽ 5 km in Fig. 4. The increase of KE at x ⫽ 0 and toward t ⫽ 60 s in Fig. 5b indicates the onset of the buoyancy wave motions in the core of the heated region, as was noted in Fig. 4b. It is difficult to isolate the energetics of the buoyancy wave response to the cumulus-scale heating. Although the acoustic waves dominate the AEE and the buoyancy waves dominate the APE, both wave classes contribute to the KE. To isolate the buoyancy wave response, we extended the duration of the cumulus-scale heating to 4 min. This heating duration should, according to Fig. 2, filter most acoustic waves while retaining most of the buoyancy wave energy because 4 min ex-

ceeds the period of most acoustic waves and is comparable to the acoustic cutoff period 2␲/NA ⬇ 294 s, but is less than the minimum buoyancy period ␶b ⫽ 2␲/Ns ⬇ 324 s. Table 1 indicates that the acoustic waves are effectively filtered while the buoyancy waves are marginally filtered. Figures 6 and 7 present the evolution of the horizontally integrated and vertically integrated APE and KE, respectively, during the buoyancy wave response to a cumulus-scale heating of duration ␶ ⫽ 4 min. The APE generated initially by the heating is converted by the buoyancy wave motions into KE. Figure 6 demonstrates that some of this energy is propagated upward into the region above the heated layer. At the midlevel of heating, z ⫽ 3.5 km, a high-frequency buoyancy wave is evident. Its period is approximately 3 min—half of that observed in the potential temperature field (see Fig. 8bI). This frequency doubling occurs because APE is proportional to the square of the potential temperature. The amplitude of this signal at the midlevel of heating decreases in time because some of its energy is propagated vertically. Because this signal is associated with the highest-frequency buoyancy waves whose period is approximately 324 s, one would expect that increasing the duration of heating beyond 4 min will filter this wave. The existence of these highfrequency buoyancy waves do not indicate a resonant behavior because, according to Fig. 2, any heating whose duration is less than 4 min will generate these wave appreciably, and, when the duration is exactly 324 s, the amplitude of this wave is less than when the duration is shorter. To identify resonant behavior, one must examine the relationship between the spatial structure of the forcing and the free oscillations as well as the temporal structure, which are not independent in the wave dispersion relation. Figure 7 demonstrates that some of the initial energy is carried by horizontally propagating buoyancy waves. Several distinct wave groups are evident in Fig. 7, the fastest of which is of the deepest vertical scale. The slope of the leading distinct group in Fig. 7 implies a

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FIG. 6. Evolution of the horizontally integrated (a) APE, and (b) KE, as a function of time t and height z in units of kJ m⫺2 following cumulus-scale heating of 4-min duration. The contour interval is 0.02 up to 0.1, and 0.1 thereafter.

propagation speed of approximately 33 m s⫺1 that is associated with the horizontal group velocity of the waves whose vertical wavelength is approximately equal to twice the heating depth. The next major group has an apparent propagation speed of approximately 14 m s⫺1 and is associated with the higher vertical wavenumber modes that accomplish the lower boundary rotors observed in Part I (see Fig. 61). The horizontally trapped wave that corresponds to the high-frequency oscillation at the midlevel of the heating in Fig. 6 is also evident near x ⫽ 0 in Fig. 7. This high-frequency buoyancy oscillation is not likely to be excited by any injection of duration exceeding the buoyancy period, nor by an injection whose vertical–horizontal aspect ratio of the injection is less than unity. Part I demonstrated that the response to the intermediate-scale and mesoscale heatings were somewhat similar, except that the mesoscale heating excited lowfrequency inertia-gravity waves, generated a larger steady state, and generally produced displacements of smaller vertical–horizontal aspect ratio. Therefore, we

will present only the evolution of the energy fields following the mesoscale heating (Figs. 8 and 9). In contrast to Fig. 6a, Fig. 8a demonstrates that the vertical propagation of horizontally averaged APE occurs rather smoothly, without distinct “wavelike” features. Furthermore, the KE (Fig. 8b) that emerges is displaced vertically relative to the APE. The heights of maximum KE correspond to the locations of the primary upper-/ lower-level outflow/inflow. These outflow/inflow regions are located where there are large horizontal pressure gradients, which are not necessarily located near the height of maximum perturbation potential temperature. The PV (2.5) is proportional to the vertical gradient of potential temperature. Also evident in Fig. 8a within the heated layer is a low-frequency gravity wave that generates a second maximum in APE at approximately t ⫽ 13 h. The corresponding wave in KE is displaced vertically and lags the signal in APE by several hours. The wave that is affected by the Coriolis force is superimposed on a nonzero profile that we identify as the PV-conserving steady state. Figure 9

FIG. 7. Evolution of the vertically integrated (a) APE, and (b) KE, as a function of time t and distance x in units of kJ m⫺2 following cumulus-scale heating of 4-min duration. The contour interval is 0.01 up to 0.06, and 0.03 thereafter.

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FIG. 8. Evolution of the horizontally integrated (a) APE, and (b) KE, as a function of time t and height z in units of kJ m⫺2 following mesoscale heating of 2-h duration. The contour interval is 2 up to 20, and 5 thereafter.

demonstrates that some of the initial vertically integrated energy is carried away from the heated core by horizontally propagating gravity waves. The leading group has a propagation speed of approximately 86 m s⫺1, and the subsequent groups have speeds of approximately 53 and 42 m s⫺1. As in Fig. 7, the leading wave speed corresponds to the horizontal group velocity of the wave whose vertical wavelength is approximately twice the heating depth, and the slower speeds correspond to successively shallower modes. These speeds exceed those of the narrower gravity waves generated by the cumulus-scale heating, which is consistent with the k dependence of the slope of the dispersion curves in Fig. 2I. A large proportion of energy remains near the heated region in the form of inertia-gravity waves whose horizontal group speed is relatively small (e.g., the slope of the dispersion curves for the buoyancy class in Fig. 2I asymptote to zero for small k).

b. Spectral energy partitioning We now present a more general picture of the relationship between the heating and the wave response.

Figure 10 presents a series of partitioned energy spectra that illustrate the effect of varying heating width, depth, and duration. These spectra correspond to a warming with horizontal structure that projects equally onto all horizontal modes. We therefore refer to these as horizontal “white noise” spectra, with a horizontal wavenumber plotted along the abscissa. At each horizontal wavenumber k, the total energy is partitioned among the acoustic wave, buoyancy wave, Lamb wave, and steady-state classes according to (3.4). Each of these contributions is in turn normalized by the total energy residing in that wavenumber. The resulting energy fractions are represented as the partitioned regions, indicated in Fig. 10a. The spectra in the left column correspond to a deep warming, while those on the right correspond to a shallow warming. The spectra in the top row correspond to a very rapid warming, while those in subsequent rows correspond to increasing heating duration. We discuss the contribution of each class in turn. First, consider the generation of acoustic wave energy. An instantaneous warming (Fig. 10a) maximizes the generation of acoustic wave energy. In the small k

FIG. 9. Evolution of the vertically integrated (a) APE, and (b) KE, as a function of time t and distance x in units of kJ m⫺2 following a mesoscale heating of 2-h duration. The contour interval is 0.01 up to 0.1, and 0.05 thereafter.

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FIG. 10. Energy spectra partitioned among acoustic waves (A), buoyancy waves (B), Lamb waves (L), and the steady state (S) [as shown in (a)] following a horizontal white noise heating of depth (left) d ⫽ 10 km and (right) d ⫽ 1 km of durations (a) instantaneous, (b) 1 min, (c) 20 min, (d) 1 h, (e) 2 h, (f) 6 h, and (g) 10 h. The regions between the various curves denote the fraction of energy residing in each class of transient. The dashed curves demark the upper bound on the Lamb wave region and the lower bound on the buoyancy wave region.

limit (i.e., the horizontally invariant limit) the partitioning is identical to that of the hydrostatic adjustment considered by Bannon (1995) in which acoustic waves accounted for a fraction cp/c␷ ⫽ 28.6% of the total energy, the remainder contained in the steady state. In the shallow heating case, the acoustic wave energy is scale insensitive over the range of scales presented. Extending the heating duration to 1 min (Fig. 10b) results in a significant suppression of the acoustic waves. Because

the acoustic wave frequency is a decreasing function of the wave depth (see Fig. 2I) the higher-frequency shallow acoustic waves are preferentially filtered. The acoustic wave frequency is also a decreasing function of wave width, which explains why acoustic waves at large k are also preferentially filtered. The Lamb wave, which has a deep exponentially decaying structure in the vertical, is generated more readily by a deeper heating (cf. left column to right

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column in Fig. 10). The Lamb wave is remarkably resilient to increasing heating duration. The minimum frequency for the Lamb wave at wide scales is f (see Fig. 2I), which corresponds to a period of approximately 17.4 h. It is therefore not surprising that Lamb wave energy may persist through a heating duration of 6 h (Fig. 10f) and that the narrower waves are preferentially filtered. Very little energy is projected onto the Lamb wave at the widest scales. The Lamb wave’s restoring mechanism is directed horizontally and is therefore limited when the heating is horizontally invariant. At such scales, energy is projected onto the steady-state class of the Lamb mode. The buoyancy wave energy generated by an instantaneous heating (Fig. 10a) is maximized at narrow horizontal scales and asymptotes to zero at wide scales. This spectral “decay” at wide scales is more severe when the heating is shallow. Buoyancy waves are excited in regions where there are horizontal gradients in the heating (for example, see the lowest vertical wavenumber buoyancy wave in Figs. 6I and 8I). The waves effectively accomplish a horizontal redistribution of the mass, momentum, and thermal fields. If the heating does not generate these horizontal gradients, then the atmosphere achieves a balanced state by undergoing a vertical expansion that may be accomplished by acoustic waves if the heating is sufficiently rapid. Buoyancy waves are therefore preferentially excited by a deep and narrow heating that generates large unbalanced horizontal gradients in the initial fields. The deepest and narrowest buoyancy waves are also of highest frequency. As the heating duration is increased, these deep and narrow waves are preferentially filtered, as is indicated by the emergence of a peak in the buoyancy wave spectrum that shifts toward wider scales. Those buoyancy waves that are generated by a heating of duration 6 h and longer (Figs. 10f,g) are sufficiently wide and shallow and have such a long period that they behave like inertia-gravity waves. It should be noted that the total energy residing in the steady-state class of a given spatial mode does not change as the heating duration is extended (assuming the net heating is fixed). The steady state may be obtained directly from the PV generated by the heating. The net PV generation depends only on the timeintegrated heating and not on its temporal detail [e.g., consider the time integral of the right side of (2.4)]. Therefore, in Fig. 10, where the steady-state fraction appears to increase as the heating duration is extended, the total energy generated by the heating is actually decreasing. A rapid heating is indeed potentially more energetic than a gradual heating. This feature is due to the fact that the energy generation on the right side of (3.2) is proportional to the product of the injection rate and the field perturbations that modulate the total input of energy. Therefore, if the spectra presented in the columns of Fig. 10 are excited by heatings that have the

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same time-integrated structure and amplitude, then Fig. 10 implies that the wave response to a given heating may differ significantly from the response to the corresponding time-integrated heating. The temporal detail of the heating is crucial for determining the characteristics of the wave spectrum.

5. Summary The first part of this study presented an analytic solution to a hydrostatic and geostrophic adjustment problem. The model was constructed under the assumption that hydrostatic and geostrophic imbalance may be generated rapidly and on small scales by such phenomena as convection, and that the response to these disturbances by the larger-scale environment is approximately linear. This notion has formed the basis for many investigations into the relationship between the source and wave spectrum as well as the function of waves in the adjustment process. The advantage of the present study is its generality; it considers the adjustment of a fully compressible, stratified, rotating atmosphere with a rigid lower boundary to injections of various spatial and temporal scale. The second part of this study examines the PV and energy partitioning of the transients generated during the adjustment. The PV is conserved and projected entirely onto the steady-state class. The other transients undergo a vorticity exchange, but the sum of the latent and relative vorticities (i.e., the PV) carried by these waves is zero. The vorticity exchange is similar to that in a quasigeostrophic model, except the latent vorticity is comprised of two parts: a thermal stratification or stretching vorticity and an elastic vorticity. When a column is stretched or compressed, some of the thermal stratification vorticity that might have been exchanged with the relative vorticity may be stored as elastic vorticity. The perturbation energy in this linear model projects distinctly onto the various spatial modes and wave classes. The various waves do not exchange energy with each other. This orthogonality of the energetics is a fortunate consequence of the simplicity of the base state in which there is no shear. We are also able to conveniently separate the effects of the temporal and spatial characteristics of the injection on the qualities of the partitioned spectrum. The effect of an extended duration injection is to filter waves relative to the temporal Green’s function. For example, if the wave period is much shorter than the duration of the injection, then the wave will not be excited significantly by the injection. 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) conclude that acoustic waves are not generated significantly by heating of duration exceeding 2 min. Addi-

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tionally, the comparison of the wave filtering function given by a top-hat injection to that of a sine-squared injection implies that an injection that is smoother in time will project less energy onto higher-frequency harmonics. That is, the energy spectrum is affected not only by the duration of the injection, but also by the higher-order temporal derivatives of the injection (e.g., see Part I, Fig. 15I and discussion thereof). Furthermore, we analyzed the time-averaged proportions of AEE, APE, and KE carried by the acoustic wave, Lamb wave, buoyancy wave, and steady-state (PV) classes. These ratios contain interesting information concerning the physical mechanism associated with the waves at various scales. For example, we considered buoyancy waves at wide horizontal scale. Such waves contain approximately horizontal particle motions and are able to convert KE to APE via horizontal mass convergence. As the horizontal scale of such waves is increased, the ability of a current of a given amplitude to generate APE is reduced. Therefore, the ratio KE:APE in these large-scale low-frequency buoyancy waves is an increasing function of horizontal scale. The qualities of the energetics are demonstrated in the context of the specific experiments of Part I that examined the adjustment to heating characteristic of the cumulus scale, an intermediate scale, and the mesoscale. The acoustic waves generated by a rapid heating contain approximately equal portions of AEE and KE that are propagated away from the heated region vertically. Similarly a Lamb wave propagates AEE and KE away from the heated region horizontally. The cumulus-scale heating also generates buoyancy waves. The highest-frequency buoyancy waves that contain approximately equal portions of KEz and APE propagate some of their energy vertically but remain horizontally trapped near the heated core. Some of the APE initially generated by the heating is propagated horizontally by buoyancy waves of deeper scale that correspond to the waves emerging from the heating region in Fig. 6I. The response to a mesoscale heating is qualitatively quite different from that to the cumulus-scale heating. Highfrequency acoustic waves and high-frequency horizontally trapped buoyancy waves are not generated. The vertical propagation of APE occurs smoothly, and the corresponding signal in KE is out of phase vertically to the APE. In particular, the KE is maximized near the top and bottom of the heated region where horizontal pressure gradients are largest. This signal reflects the establishment of the PV-conserving steady state consisting of an upper-level anticyclone and a lower-level cyclone. Superimposed on this signal is a low-frequency buoyancy wave that is deflected by the Coriolis force. Finally, the dependence of the adjustment on the width, depth, and duration of the heating is summarized by examining the spectral energy partitioning for the heating used in Part I. An instantaneous heating generates the largest proportion of wave energy. Increasing

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the duration of heating causes changes in the wave spectrum relative to that of the temporal Green’s function that may be understood vis-à-vis the dependence of the wave period on the wave spatial scale. Because the wave period and wave spatial scale are inextricably linked via the dispersion relation, the influence of heating width, depth, and duration on the characteristics of the spectra are mutually dependent. This is the basis for the warning promulgated by Holton et al. (2002) that the dominant waves generated by tropical heating may not be obtained unambiguously given the vertical structure of the heating alone. It is interesting to note that the steady state is unaffected by changes in the temporal structure of the heating. The PV, and hence the steady state, depends only on the time-integrated injection. Injections that are applied slowly so as to not generate acoustic waves must still be accompanied by the same net expansion or compression required by the given total input of potential vorticity. The current study emphasizes that the manner by which the steady state is achieved depends strongly on the duration of the injection. The interpretation is simple: A very rapid injection generates a severely unbalanced initial configuration. Such an unbalanced configuration requires the production of a large amount of wave energy in order to accomplish the adjustment. An injection that is more gradual generates a less severely unbalanced configuration requiring much less wave energy to accomplish the adjustment. From the perspective of the surrounding environment, the transient response to these injections is quite different. If a heating is applied slowly enough such that acoustic waves and nonhydrostatic buoyancy waves are not excited, then the air located above the heating will experience a slow vertical displacement toward an equilibrium position rather than oscillations about the equilibrium position. Because the model used in this study was linear, the results were insensitive to the amplitude of the heating. A typical mesoscale convective system may be accompanied by a heating that exceeds the 1-K amplitude prescribed in section 4a. The nonlinear effects associated with increasing the forcing amplitude as well as those that may be influenced by a less stable nonisothermal basic state must be examined in the context of a nonlinear model and will be the topic of a future investigation. In nature, the injections that generate imbalance may be comprised of either mass, momentum, and heat injections, depending on the physical circumstance. The present study is limited to consideration of only an injection of heat. Future research will address injection of mass and momentum. Acknowledgments. The insight of Professors Sukyoung Lee and John H. E. Clark as well as the three anonymous reviewers was invaluable to the preparation

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of this manuscript. Partial financial support was provided by the National Science Foundation under NSF Grants ATM-9521299, ATM-9820233, and ATM0215358.

where Nn is the domain integral of the nth vertical basis function squared and

APPENDIX Dn ⫽

Orthogonality of the Energetics Section 3 presents the characteristics of the energy which, as we assert, may be partitioned distinctly among the orthogonal spatial modes which in turn may be partitioned distinctly among the acoustic wave, buoyancy wave, and PV-conserving time-dependent classes. This appendix proves these assertions by exploiting the qualities of the solution presented in Part I. It is convenient to recall the form of the linear system of partial differential equations, ⭸␹ 共2.41) ⫹ A␹ ⫽ F, ⭸t where the solution state vector is ␹T ⫽ (U, V, W, S, P), ˙, V ˙, W ˙ , S˙ , P˙ ), and A is a the injection vector is F ⫽ (U spatial matrix operator. The energy density (3.1) may be written in terms of the solution vector ␹ 1 TE ⫽ ␹*TD␹, 2

冤 冥

D⫽

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

2

0

0

0

0

0

0

g

0

Ns2 0

2D

0

,

共A.2兲

⭸ ⭸t

冕 冕

⬁

0

⫺⬁

1

0

0

共m ⫹ ⌫ 兲 2

2

0

0

0

0

0

0

g

0

0

0

0

0

0

2

Ns2

共m2 ⫹ ⌫2兲

1

0

cs2

冥

.

共A.6兲

First, let us consider solutions of the homogeneous problem of the form (3.91) ␹n ⫽ En⌿n where En is the matrix of eigenvectors of An. The spectral energy density (A.5) may be written as 1 T M ⌿ N . TEn ⫽ ⌿* 2 n n n n

共A.7兲

The elements of the wave interaction matrix Mn are * E1j ⫹ E2i * E2j ⫹ 共m2 ⫹ ⌫2兲E3i * E3j Mij ⫽ E1i ⫹

g2共m2 ⫹ ⌫2兲 Ns2

* E4j ⫹ E4i

* E5j E5i cs2

Mii ⫽

1

k2共␻i2 ⫹ f 2兲

cs

共f 2 ⫺ ␻i2兲2

⫹ 2

⫹

共A.8兲

.

共m2 ⫹ ⌫2兲共␻i2Ns2兲 共Ns2 ⫺ ␻i2兲2

.

The spectral energy density may therefore be partitioned among the steady, acoustic, and buoyancy cona b tributions as in (3.4), TEn ⫽ TEPV n ⫹ TEn ⫹ TEn, where 共A.3兲

TEnPV ⫽ M11C 1*C1, TEna ⫽ M22C 2*C2 ⫹ M33C *3 C3,

冕 冕

0

共A.9兲

␹*TDA␹ dx dz ⫽ 0.

TE dx dz ⫽

0 0

0

cs2

The energy is therefore governed by 2D

0

1

⬁

⫺⬁

1

Using the dispersion relation, it may be shown that Mij ⫽ 0 when i ⫽ j. The wave interaction matrix is therefore diagonal, which implies that there is no interaction among the distinct time-dependent contributions to the energy. The diagonal elements of Mn are

and the sum of the conversion terms is zero,

冕 冕

冤

共A.1兲

where the diagonal matrix operator D is

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2D

⬁

0

⫺⬁

TEnb ⫽ M44C 4*C4 ⫹ M55C 5*C5,

␹*TDF dx dz. 共A.4兲

Because the spatial modes are orthogonal, we may consider the conservation of the spectral energy density (3.4): 1 T TEn ⫽ ␹* D ␹ N , 2 n n n n

共A.10兲

共A.5兲

and the Ci are the coefficients of the PV-conserving, acoustic wave, and buoyancy wave contributions defined in (3.9I). For solutions of the homogeneous problem, the distinct PV-conserving, acoustic wave, and buoyancy wave contributions to the energy are time independent. Next consider the energetics of solutions to the forced Eq. (3.111). Section 3b of Part I demonstrated the structure of these solutions to be of the general form ␹n(t) ⫽ EnGn(t) where the ith element of Gn is

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Gi ⫽ ci(t) exp(i␻it) with ci(t) defined in (3.151). Following the analysis of the energetics of the homogeneous solutions, we may also write the energy of the forced solution as 1 T M G N . TEn ⫽ G* 2 n n n n

共A.11兲

The spectral energy density of the forced solutions may therefore be partitioned in a manner similar to that of the homogeneous solutions. The spectral energy density of the temporal Green’s function, denoted in (3.5) ˆ ⫽ TE ˆPV ⫹ TE ˆa ⫹ TE ˆb, is given by as TE n n n n ˆPV ⫽ M C *C , TE n 11 1 1 ˆa ⫽ M C *C ⫹ M C *C , TE n 22 2 2 33 3 3

共A.12兲

ˆb ⫽ M C *C ⫹ M C *C , TE n 44 4 4 55 5 5 where the coefficients Ci are provided in (A.1I)–(A.5I). The representation (3.6) of time-dependent spectral energy density of the forced solution (A.11) in terms of the spectral energy density of the Green’s function and a filtering function r(␻, t) follows directly from the relationship between the temporal Green’s function and the general solution of the forced problem (3.11I) given by (3.12I) and (3.15I). REFERENCES Bannon, P. R., 1995: Hydrostatic adjustment: Lamb’s problem. J. Atmos. Sci., 52, 1743–1752.

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