1. Introduction
a. Background
Horizontal density and temperature gradients are very small in the Tropics. The well-known consequence of this is that, on large scales, the dominant balance in the temperature equation is between heating and vertical advection of potential temperature. A tropical heat source causes vertical motion and an associated horizontal flow but only small temperature changes. A number of studies have neglected horizontal temperature variations as a simplifying assumption in idealized models of the tropical climate (e.g., Neelin and Held 1987; Pierrehumbert 1995; Miller 1997; Zeng 1998; Zeng and Neelin 1999; Raymond 2000; Clement and Seager 1999; Sobel and Bretherton 2000, hereafter SB00). SB00 suggest that this assumption may be able to serve as the foundation of balanced dynamical models for the study of tropical dynamics that explicitly include diabatic processes. In such models, both the “dry” aspects of the fluid dynamics and the diabatic processes would be constrained by the assumption of nearly horizontally uniform free tropospheric temperature. Radiative transfer becomes governed essentially by tropospheric humidity, cloudiness, and surface properties when the free tropospheric temperature profile is held fixed. Convection becomes governed by near-surface temperature and humidity (since these largely control the convective available potential energy and convective inhibition), and also by free tropospheric relative humidity, since this governs the effectiveness with which entrainment of environmental air into clouds can suppress convection. In such models, then, moisture becomes an important dynamical variable. The humidity of the atmosphere at a given horizontal location—leaving aside for the moment questions about vertical structure—can have a strong effect on the diabatic heating at that location and, hence, on the vertical velocity and horizontal divergence. In the presence of background absolute vorticity, this divergence in turn drives a large-scale rotational flow through vortex stretching and horizontal advection of absolute vorticity.
Though atmospheric water vapor is not conserved in the presence of precipitation and surface evaporation, the horizontal advection of water vapor does have an influence on its large-scale horizontal distribution and thus on the distribution of deep convection. Chen et al. (1996) noted that the occurrence of deep convection during the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) field experiment appeared to be linked to the direction of the low-level zonal wind more than to its speed. The surface wind speed would control convection if surface fluxes were the factor of primary importance, as in some quasi-equilibrium theories (e.g., Emanuel et al. 1994, and references therein). On the other hand, Chen et al. (1996) suggested that in the TOGA COARE intensive flux array, westerlies carry relatively moist air compared to easterlies, so that the asymmetry could be partly explainable through the advection of water vapor. Spectacular examples of the same process came in the same experiment when the advection of very dry subtropical air from the north was observed to suppress convection (Parsons et al. 1994; Numaguti et al. 1995; Yoneyama and Fujitani 1995; Yoneyama and Parsons 1999). Evidence of a more general, statistical nature for the dynamical importance of free tropospheric humidity was presented by Sherwood (1999). From a modeling perspective, the importance of horizontal water vapor advection is shown explicitly in the simulations of SB00 using modified versions of the models of Neelin and Zeng (2000, hereafter NZ00; see also Zeng et al. 2000) and Rennó et al. (1994a,b).
b. Outline
In this study, we first elucidate (section 2a) the basic scaling under what we call, for obvious reasons, the weak temperature gradient (WTG) approximation. In this approximation, horizontal temperature gradients are neglected at leading order. Small temperature perturbations can be obtained at next order from a balance requirement on the leading order flow. For maximum simplicity, the scaling analysis is performed using a shallow water system driven by a given mass source, analogous to a heating. The basic dynamics of this model has, in essence, been presented by Held and Hoskins (1985; hereafter HH85); similar ideas can also be found in the discussions of Sardeshmukh and Hoskins (1987, 1988). Compared to HH85, we formulate the problem somewhat differently, allow for nonlinearity, present an explicit solution with emphasis on the local tropical response (as opposed to teleconnections to midlatitudes), and address some additional details.
The purpose of the rest of the paper is to illustrate some types of large-scale dynamical behavior, which we hypothesize can result from horizontal moisture advection in the Tropics. In order to understand these effects as clearly as possible we use a model framework with a number of strong simplifications. The first set of simplifications can be viewed as being embodied in the single-mode vertical structure and simple convective parameterization, as in the derivation of the Quasi-equilibrium Tropical Circulation Model (QTCM) developed at the University of California, Los Angeles (Neelin and Yu 1994; Yu and Neelin 1994; Neelin 1997; Neelin and Zeng 2000; Zeng et al. 2000), which we take as our starting point. We then make a number of further assumptions, perhaps the most restrictive of which is the use of an f plane to eliminate Rossby waves. This is of course unrealistic, but it is done in order to isolate the dynamics of interest. In particular, in the presence of a background horizontal gradient of moisture, modes exist that are somewhat reminiscent of Rossby waves. Use of an f plane brings these modes into better relief. The addition of the β effect is considered briefly, but a thorough analysis of its role in this system is deferred.
2. Weak temperature gradient scaling in the shallow water system
a. Scaling
The shallow water system can describe, with proper interpretation of the variables, the dynamics of a stratified compressible atmosphere, if the vertical structure of the motion is taken fixed. The most common set of assumptions used for this is that of linear normal mode theory (e.g., Matsuno 1966; Lindzen 1967; Gill 1980; among others), but somewhat different approximations can lead to a fundamentally similar result, as in the QTCM equations. Due to the multiplicity of interpretations, we use a “raw” shallow water system in this section to illustrate the fundamental scaling.
1) Basic scaling assumption
(i) Linear limit
The linear limit follows by putting Ro = 0. Then, in the weakly damped, low-frequency limit (1/(fT) ≪ 1, α/f ≪ 1), no term in the vorticity equation can balance the divergence. Therefore, we have to rescale ζ → ζ(fT). To get a balance in the divergence equation, we also need to rescale η → η(fT). This imples that the WTG approximation is now valid if Bu ≪ 1.
b. Resulting system
c. Comment on the diabatic scaling
The model of HH85 differs in a number of particulars from that presented above, most importantly in excluding nonlinearity, in explicitly including the β effect, and in using separate meridional and zonal scales. Nonetheless, their model has the same essential dynamical structure as that above, one which is somewhat counterintuitive if one is accustomed to thinking about balanced flows in midlatitudes. In HH85 (and the present work), the heating gives the divergence, which gives the vorticity, which gives the temperature. HH85 commented rather little on the dynamics of their model beyond making that basic point. Some further details warrant discussion in our view. For the most part the same comments would apply to HH85's model.
d. Dynamical properties
1) Balance
The model derived above is balanced, containing no explicit gravity waves. This is immediately evident from (10), which has no terms containing η. Further insight is gained into the nature of the balance by inspecting the divergence equation (12). The dynamical time derivative in this equation has been eliminated by using (10); ∂δ/∂t has been replaced by ∂Q/∂t, which here is considered given. Note that if T is small compared to f−1, consistency cannot be maintained, in that the tendency term in (9) will be larger than all the others. Physically, a heating that varies too rapidly must drive gravity waves. Hence our balanced scaling requires, unsurprisingly, that the heating must not vary on a timescale that is short compared to f−1.
2) Adiabatic limit, energy, and potential vorticity
WTG is an inherently nonconservative model, in that the scaling analysis used to derive it assumes that the heating is important at leading order. Because of this, it may be somewhat misleading to consider the properties of the equations in the adiabatic limit (Q = 0). However, it is worth doing so in order to place WTG in the context of other balance models.
When Q = 0, the WTG equations become formally identical to those for strictly 2D incompressible flow; that is, the system obeys pure barotropic vorticity dynamics. As a consequence, if the system is also inviscid (α = 0),
absolute vorticity is conserved on trajectories,
the global integral of the kinetic energy is conserved, and
volume is conserved.
When Q ≠ 0, absolute vorticity is no longer conserved. Neither is potential vorticity [(ζ + f)/h]; in fact, because in both leading order equations [(10) and (11)] h is approximated by H, WTG cannot distinguish between absolute and potential vorticity.
Since it does not have a prognostic equation for h, a prognostic equation for potential energy cannot be formed under WTG. We can say that potential energy is created by the heating but is immediately converted to kinetic energy. Given these limitations, we can predict confidently (despite the fact that the analysis here is restricted to the shallow water system) that a continuously stratified version of WTG would not support baroclinic instability. This may limit the applicability of WTG to the study of easterly waves in Africa, and perhaps other regions as well (though over the tropical Pacific, e.g., we are not aware of observational evidence for the importance of baroclinic instability).
3) Role of momentum damping
The rotational flow is affected by the Rayleigh friction in the above system, but the divergent flow, being diagnosed directly from continuity, is not. In cases in which the rotational and divergent flows are of the same order (the most prominent exception being the weakly damped, low-frequency, linear limit), this appears to be a formal inconsistency, though not an important one in our view. It results from the fact that the divergent flow has been calculated from what ultimately is a thermodynamic equation rather than a momentum equation. The friction term does appear in the divergence equation as a consequence of this assumption, but this equation has become a diagnostic for η. Hence the temperature is affected by friction although the divergent flow is not. This situation is independent of our choice of a Rayleigh friction and would be true for any momentum forcing. The inconsistency presumably will not cause large errors if the damping is small. On the other hand, if α ≫ f, then we expect that damping, rather than rotation, will allow significant η gradients to be sustained, and the scaling will fail even for Bu ≪ 1. WTG is fundamentally a nearly inviscid scaling.
4) Mass conservation
The truncation of the continuity equation (which plays the role of the thermodynamic energy equation) means that the global conservation properties of the exact equation are not inherently satisfied unless Q happens to integrate to zero over the domain. However, any residual in this integral implies a secular trend in the global mean temperature (represented by h), so the apparent problem can be remedied by carrying a separate equation for this global mean. In the axisymmetric solution below, we simply impose a Q whose global area integral vanishes. In the context of a more physical model, in which diabatic processes are explicitly parameterized, appropriate ways of dealing with this issue are discussed by SB00.
5) Scaling under β
We have assumed an f plane, but the scaling does not require this. The analysis can be put on an equatorial β plane, f = βy, so that in terms of the scale of variations of f we have f ∼ βL. The scaling in the nonlinear regime (U/L ∼ f) then requires U ∼ βL2, so L in this regime is a Rhines scale (Rhines 1975).
e. Axisymmetric solution for top-hat forcing
Since ur is proportional to r in the heating region r < 1, it is readily apparent from (19) that a potential singularity for ζ could result at the origin. To avoid this, we take ζ to be a constant in that region and obtain ζ(r) = −(1 + α)−1(≡ζ1) for r < 1. Obviously, in the outermost region ζ = 0. In the intermediate region 1 < r < R integration of the first-order ODE (19) yields a single integration constant. Hence ζ cannot be made continuous at both r = 1 and r = R. We choose to enforce continuity of ζ at r = 1. This leaves us with a small discontinuity at r = R, whose only damage is to produce a tiny kink in the azimuthal velocity uθ.
The actual analytical expression for η(r) is rather cumbersome and thus not particularly illuminating. In the heating region r < 1, however, it is easily seen that since both ur and uθ are proportional to r, the integrand in (24) is also linear in r and thus η is a a simple quadratic, as Fig. 1d attests.
We reiterate that to obtain this WTG solution, terms were dropped only in the continuity (temperature) equation (this is what allows us to get δ immediately from Q) while all other equations were solved exactly.
It should also be noted that there is a fundamental asymmetry in the problem due to the sign of f. We solved the above problem for positive Q with no restriction on the value of α. However, if the sign of Q is reversed, there is no physically reasonable steady solution for α < 1. In this case vortex stretching can increase the vorticity without bound, while for positive Q the same process reduces the absolute vorticity but cannot bring it below zero (taking f > 0) since vortex stretching or compression cannot occur where there is no absolute vorticity. If instead of Rayleigh friction the dissipation is made a stronger function of wind speed, such as a quadratic as in bulk aerodynamic surface drag formulas, then stable steady solutions are possible for negative Q and arbitrarily small damping coefficients, though they may involve very strong circulations. On a β plane, weakly damped or even inviscid solutions can be expected to exist for localized forcing of either sign, since energy will be radiated away from the forcing region by Rossby waves.
3. Balanced waves on an f plane with a background horizontal moisture gradient
In this section we add a dynamical moisture equation to the shallow water WTG system. The heating is taken proportional to the moisture field, coupling the moisture equation to the others. The system is derived in appendix A from the QTCM equations of NZ00, and our notation retains some features of the QTCM's (resulting in some changes in variable names and new constant coefficients relative to the shallow water system of section 2). We also linearize the system to facilitate analysis of its wave propagation mechanisms.
The QTCM framework is useful for this purpose in several ways. The QTCM derivation itself (found in NZ00) illustrates specifically and systematically the degree and kind of physical assumptions that are required in order to obtain a system of this simplicity from the full atmospheric physics and dynamics. The numerical implementation of the QTCM (NZ00; Zeng et al. 2000) demonstrates the climate that these equations, in their full nonlinear form, can produce, providing a valuable link between climate modeling and theory. Finally, the QTCM framework provides a basis for estimating the parameters that appear in the equations below, and some version of which should appear in any analogous system (i.e., a reduction of the primitive equations that includes highly truncated vertical structure and an advected moisture variable).
However, the system below can also be derived by assumptions about the vertical truncation and physical parameterizations different in detail, if not in basic physical interpretation, from those used to derive the QTCM. The proportionality of the heating to the moisture can be viewed in this general context as resulting from linearization about a typical moisture profile of another convective scheme than the simplified Betts–Miller scheme used in the QTCM.
a. Basic state
b. The moist linear WTG model
All notation is defined in the appendix (which simply restates material from NZ00), but we briefly introduce here a few significant features unfamiliar from section 2. Dependent variables, without overbars, represent perturbation quantities; these are δ, ζ, T, and q. The flow fields now explicitly represent a baroclinic mode whose sign represents the velocity in the upper troposphere; the flow in the lower troposphere has the opposite sign. Consistent with section 2, ζ and δ are the vorticity and divergence of this baroclinic flow field. Here η is now explicitly relabeled as the temperature T; and κ = R/Cp, the ratio of the gas constant for air to the heat capacity of air at constant pressure. The mean depth H becomes the mean dry static stability MS. Ignoring these relabelings and new constant coefficients, (25)–(27) are equivalent to linearized versions of (10)–(12).
Here τc is a convective timescale, typically taken on the order of hours. It can be argued that the moisture field cannot react to convection nearly as quickly as the buoyancy field can and, thus, that the use of identical timescales in the moisture and temperature equations is inappropriate. For simplicity we have left this feature of the Betts–Miller scheme as is. However, the linear modes we obtain below are sensitive to τc, and so the possibility that a timescale on the order of days, rather than hours, could enter should be kept in mind.
Apart from the moisture equation (28), the above system (25)–(27) is isomorphic to the linear shallow water WTG system forced by a mass source given by a linearization of (10)–(12), with the mass source now controlled by the moisture. With the addition of (28), our linear model contains two true dynamical equations, for moisture and baroclinic vorticity. Correspondingly, in general we expect the dispersion relation to have two sets of solutions, or “modes.”
c. Results
1) Irrotational modes
(i) The case βq = 0
Here B has units of inverse time. Generally we expect
(ii) The case βq ≠ 0
2) Rotational modes
One branch of (39) is a rotational counterpart of that discussed in the preceding section; one easily sees that for sufficiently large ω and k = 0, (39) becomes (33). This mode is damped on the timescale B−1. For reasonable parameters, this mode has frequencies high enough that the basic WTG scaling becomes questionable [section 2d(1), and below], although this depends on βq among other parameters and so is not a general conclusion. For these reasons, this fast mode may not be physically significant, but this is not entirely clear, as it depends (among other things) on the effective B−1 that would obtain in a model with a model with a separate, large timescale for moisture. If desired, this mode can be eliminated from the equations at the outset by dropping the tendency term in (28). This renders the moisture a diagnostic variable, proportional to the meridional velocity υ.
One interesting feature of (42) is its sign. If βq is positive, corresponding to decreasing humidity toward the Pole, the intrinsic phase velocity of the waves is eastward, the opposite of the standard Rossby wave. We may understand the propagation mechanism by considering the phase relations for a low-frequency wave that has the background moisture gradient drier toward the pole. A positive moisture anomaly in such a wave will be collocated with a positive low-level υ anomaly. This would seem to amplify the wave, but it does not since the moisture advection is balanced by “damping” at the rate B. The low-level vorticity maximum is approximately one quarter-cycle west of the moisture maximum, but the low-level convergence maximum, which is collocated with the moisture maximum, creates a maximum in the vorticity tendency there, causing eastward propagation.
A second interesting feature of (42) is the inverse dependence of the frequency on the background gross moist stability, in contrast to the role this parameter plays in theories of gravity waves coupled to convection (e.g., Emanuel et al. 1994). A third is that the convective timescale τc does not appear in (42).
3) Brief comment on the addition of the β effect
Given that the standard β effect would lead to westward propagation where our “βq effect” leads to eastward propagation, it is desirable to quantify the relative importance of these two effects. A fuller analysis of the β-plane case is under way and will be presented in the near future, including a thorough study of the stability properties and using, as is most appropriate, the equatorial β plane. For the moment we only briefly address the issue of most immediate relevance here, which is the direction of phase propagation for both β and βq nonzero.
4. Discussion
a. Comparison with other balance models
Both in dry and moist implementations, the WTG dynamics is balanced in a sense similar to that of other balance models, although the balance occurs fundamentally in the temperature rather than momentum equation. The model has no explicit gravity waves for the same reason that, for example, quasigeostrophy does not; they are assumed to be infinitely fast. The differences are that in WTG dynamics the adjustment process is assumed to occur in an atmosphere that is rotating weakly at most, and is assumed to be a response to a sustained heating rather than an initial imbalance, which may occur in either the mass or momentum fields. The adjustment process in our case might be considered “convective adjustment,” as described by Bretherton and Smolarkiewicz (1989), rather than geostrophic adjustment, but the essence of the process is the same.
In quasigeostrophic theory P is conserved following adiabatic inviscid geostrophic flow. In our analysis q is not generally conserved, except in the special case of no evaporation anomalies and the limit of vanishing gross moist stability; these are probably not generally good assumptions, but the existence of this limit at least points to the fact that horizontal advection (the dominant process in quasigeostrophic dynamics) plays a role in determining the q field.
b. Qualifications
The relevance of the dynamical modes derived here to the real atmosphere bears discussion. One may be tempted, as is always the case with models using any particular convective closure scheme, to question how strongly the results depend on the characteristics of that scheme. We cannot answer that question with precision, but at a qualitative level the main feature that is necessary for modes resembling those described here to exist is that convective heating, all else being equal, be an increasing function of the moisture content of the atmosphere. This is, presumably, a rather generic feature of all reasonable convective schemes, though its quantitative importance may vary from one scheme to the next.
Another limitation inherent in the moist model used here is the single moisture variable with a fixed vertical structure. It is natural, and is the case in the QTCM, that the vertical structure function for moisture projects fairly strongly on boundary layer moisture, which is directly related to convective available potential energy (CAPE) in virtually any formulation. The vertical structure function also projects on free tropospheric relative humidity. The latter is generally presumed to affect convection through entrainment of environmental air into clouds, with buoyancy being more strongly decreased the drier the environment. However, there is also evidence that air above the boundary layer can participate directly in convective circulations, particularly in active mesoscale convective systems (Kingsmill and Houze 1999a,b; see also Zipser et al. 1981; Jorgensen et al. 1991), suggesting that environmental humidity can affect the convection through mesoscale dynamics somewhat differently than the standard picture of entrainment into an isolated cloud. As the effect of environmental humidity at different levels on convection is not well quantified at present, the limitation to a single vertical degree of freedom, with attendant vagueness about which level is most important and how, seems appropriate. However, it does bear stating that free tropospheric humidity can generally be expected to be more strongly affected by horizontal advection than boundary layer humidity, since the latter is more directly controlled by relatively fast processes (Betts and Ridgway 1989).
c. Accuracy
While the scaling analysis used to derive the WTG system can guide our expectations regarding the accuracy of WTG solutions compared to those of more fundamental equations, we have not addressed this here by direct comparison with more exact solutions. Work aimed at addressing this question is currently well underway in the context of classic models of tropical atmospheric dynamics, such as the Gill model (1980) and simple axisymmetric models of the Hadley circulation. The results so far are encouraging, and this work will be reported in due course. Further discussion is outside our present scope.
d. Possible observational relevance
The eastward-propagating moist mode derived in section 3c(2) was obtained as the result of a theoretical investigation, rather than being a conscious attempt to explain any particular observed phenomenon. Nonetheless, we present a couple of tentative speculations here on its possible relevance to the real atmosphere. Careful data analyses will be required to determine whether these speculations have merit.
One is tempted by the eastward propagation to suggest that our rotational moisture waves may have some relevance to the Madden–Julian oscillation (MJO), although the addition of the β effect complicates this matter, as discussed in section 3c(3). This application also seems inconsistent with the common interpretation of the MJO as having a fundamentally gravity or Kelvin wave–like aspect. Our mechanism may, however, be consistent with the recent work of Raymond (2001), who suggests, based on his numerical simulations, that the MJO may be more fundamentally rotational, and less Kelvin wave–like, than previous work has suggested. Raymond sees evidence in his simulations for a wave propagation mechanism very similar to that of the slow mode in section 3c(2).
Perhaps more likely than relevant to the MJO proper, our mechanism may be relevant to aspects of intraseasonal variability in Northern Hemisphere summer, when the centers of convective activity are displaced well off the equator and the associated flows are consequently more rotational in character. We are thinking here of the “convection seesaw” discussed by Zhu and Wang (1993), and to the related monsoon onset and break periods over the Asian continent and tropical western North Pacific discussed by Murakami and Matsumoto (1994). The relevance of both moisture advection and vorticity dynamics to the Asian monsoon onset is clearly suggested by the simulations of Xie and Saiki (1999).
Another possible application is to the propagation of African easterly wave disturbances across the Atlantic basin (e.g., Reed et al. 1988a,b). Given the rotational nature of these disturbances, it has not been obvious how they stay localized against Rossby wave dispersion. One possibility is that our moisture wave mechanism is operating on the moisture gradient between the air masses advected from the Sahara and the moister landmass to its south. This would reduce the “effective β” and hence inhibit dispersion. In this context it is worth pointing out that the horizontal advection of the background moisture gradient acts in our slow mode as a distinct effect from the WTG dynamics per se, and one that will also exist in a fully baroclinic system. One might, for that matter, also expect this effect in moist midlatitude disturbances in the vicinity of a strong horizontal moisture gradient, although WTG will not be appropriate for analyzing this situation.
5. Conclusions
We first derived the basic WTG scaling for dry shallow water dynamics subject to an imposed mass source, representing a heating. This results in a balanced system fundamentally similar to that derived by HH85. The fundamental balance assumption occurs in the continuity (temperature) equation and, in fact, no additional approximations are needed. Physically, this assumption is that horizontal temperature gradients are negligible despite spatial variations in diabatic heating. In the resulting system, the vorticity equation retains the only dynamical time derivative, and the divergence equation becomes a diagnostic equation for the height (temperature). The scaling requires that the Burger number be small, but the dynamics is fundamentally different from that obtained in this regime in the adiabatic case. The key feature in the WTG system is the a priori importance of diabatic heating (mass source Q), which can be expressed by defining a small nondimensional parameter containing a Q scale in the denominator.
We then applied this scaling to a linear moist model on an f plane. The key new feature that is added to the shallow water dynamics here is an independent equation for conservation of water. The moisture in turn controls the convective heating and through this governs the balanced dynamics in a manner somewhat reminiscent of (though, in important ways, different than) potential vorticity in classic balance models such as quasigeostrophy. Linear analysis on simple, artificially constructed basic states brings out the existence of modes of variability due fundamentally to horizontal moisture advection to which, as far as we know, no analogs have been previously studied theoretically.
In the low-frequency limit with a background moisture gradient, a mode exists that is identical to the barotropic Rossby wave in mathematical form, though it relies on a different physical mechanism and has phase propagation toward the east. This mode has a significant growth rate over a finite range of (low) wavenumbers. We tentatively suggest that this mode may have some relevance to intraseasonal variability and easterly waves. An interesting problem that may complicate attempts to evaluate the validity of these suggestions is that the mode's phase speed in the presence of the β effect is ambiguous in both sign and magnitude for reasonable parameter choices, due to cancellation between the restoring effects of β and the moisture gradient.
Acknowledgments
An e-mail discussion with Dr. K. A. Emanuel helped to inspire the analysis in section 3. Drs. C. S. Bretherton (who suggested the name “weak temperature gradient approximation”), A. J. Majda, D. J. Raymond, and J.-I. Yano commented insightfully on the manuscript in advance of submission, leading to a number of improvements. We also thank M. A. Cane, N. Harnik, I. M. Held, D. J. Muraki, J. D. Neelin, R. A. Plumb, and B. Stevens for discussions. AHS acknowledges support from NSF Grant ATM-0096195, and from a David and Lucile Packard Foundation Fellowship for Science and Engineering. A critical portion of the work was done during a visit by AHS to Stockholm University in August 2000, sponsored by the International Meteorological Institute in Stockholm. LMP acknowledges support from NSF.
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APPENDIX
QTCM Equations as Basis for the Moist Model
The QTCM equations are derived and discussed at some length in Neelin (1997) and NZ00. We give only a minimal introduction to them here and refer the reader to these studies and the references therein for further information. Numerical solutions to the equations (or approximate versions of them) under various conditions are presented in Zeng et al. (2000), Lin et al. (2000), Chou et al. (2000), and SB00.
If we linearize the QTCM equations, make the WTG approximation, set f = constant, set all momentum transfer coefficients except ϵ1 (which is relabeled α in section 3b) to zero, ignore the barotropic mode, and drop all subscripts “1” on variables and constants in the baroclinic mode, we arrive at the linear model in section 3b.
The flow, computed using the WTG approximation, resulting from a simple axisymmetric heating Q with zero net integral. All variables are identically zero for r ≥ 10. The case illustrated here is for α = 1 and R = 10
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
The flow, computed using the WTG approximation, resulting from a simple axisymmetric heating Q with zero net integral. All variables are identically zero for r ≥ 10. The case illustrated here is for α = 1 and R = 10
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
The flow, computed using the WTG approximation, resulting from a simple axisymmetric heating Q with zero net integral. All variables are identically zero for r ≥ 10. The case illustrated here is for α = 1 and R = 10
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Real part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Real part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Real part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Imaginary part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Imaginary part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Imaginary part of the frequency for the rotational modes, as a function of meridional and zonal wavenumbers
Citation: Journal of the Atmospheric Sciences 58, 23; 10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2
Note that we assume only a single length scale, rather than two distinct scales, for the meridional and zonal directions. This is consistent with the fact that most calculations to follow will not explicitly include the β effect. However, our basic scaling analysis holds on a beta plane (and becomes very similar to that of HH85) if we take f ∼ βL. Our length scale must be defined by the length scale of Q, so for the present analysis we assume Q is not strongly asymmetric in x and y.
However, this value depends on the particular basis functions chosen for moisture and baroclinic velocity, and so cannot be regarded as precisely known.