Is buoyancy a relative quantity?

by

Charles A. Doswell III

Cooperative Institute for Mesoscale Meteorological Studies

Norman, Oklahoma

and

Paul M. Markowski

Department of Meteorology, Pennsylvania State University

University Park, Pennsylvania

 

Submitted as a Note to:

Journal of the Atmospheric Sciences

October 2001

_____________________

Corresponding author address: Dr. Charles A. Doswell III, Cooperative Institute for Mesoscale Meteorological Studies, 100 East Boyd Street, Room 1110, Norman, OK 73019. Email: cdoswell@hoth.gcn.ou.edu

NOTICE: This manuscript has been submitted to Journal of the Atmospheric Sciences. See the AMS policy statement regarding copyright. There may be differences between the paper as it appears here and final version, owing to revisions suggested by the reviewers.


ABSTRACT

 

The development of parcel buoyancy theory is reviewed, noting the assumptions and approximations that can be associated with this derivation. The view of buoyancy that results is associated with the density difference between a parcel and its environment; that is, parcel buoyancy is defined relative to its environment. Textbook presentations of buoyancy often involve a linearization of the vertical momentum equation about a hydrostatic base state, in order to provide a simple treatment of pure parcel theory.. This linearization is unnecessary to develop buoyancy, and it is specification of the base state that is of some concern. As knowledge of the immediate environment of convective storms increases, it is becoming increasingly difficult to be confident in our determination of this base state, especially in diagnosis of observations. Since many numerical simulation models associated with deep convection also employ this traditional formulation of buoyancy relative to the environment, its impact on the simulations is reviewed and, unless the effects of the vertical perturbation pressure gradients are ignored, it is shown the simulations are not actually dependent on the base state. However, the use of the traditional formulation might lead to erroneous interpretations of the model results. The classical treatment of buoyancy is not necessarily incorrect, but it is usually incomplete and may be misleading.


1. Introduction

The classical notion of buoyancy, as developed in standard textbooks (e.g., Emanuel 1994, §1.2) is actually a relative quantity. That is, buoyancy is defined in terms of the density difference between an air parcel and its "environment," where that environment is defined in terms of a base state that is in hydrostatic balance. This traditional concept of buoyancy is the result of decomposing pressure and density into hydrostatic and nonhydrostatic components, detailed in section 2. The textbook development of buoyancy is built around 1- dimensional pure parcel theory, often involving a linearization of the vertical momentum equation that also ignores perturbation pressure contributions to the force balance This approach is correct so far as it goes, but it is incomplete and may be misleading. Within actual deep convective clouds that incorporate many parcels, embedded within a distinctly nonhomogeneous environment, it's not clear just how applicable all these simplifications associated with pure parcel theory really are. This issue is discussed in section 3. This leads us to question the value of treating buoyancy in these relative terms. Our conclusions are presented in section 4.

 

2. Development of buoyancy in relative terms

The starting point for consideration of buoyancy is the vertical momentum equation in height (z) coordinates:

, (1)

where w is the vertical component of motion, is density, p is pressure, g is the acceleration due to gravity, is the angular rate of the Earth's rotation, is the latitude, u is the zonal component of motion, and Fz is the vertical component of any external forces (e.g., viscosity). It is common to ignore the last two terms and we shall do so, here. For a base state of pure hydrostatic balance, the lhs of (1) is zero, resulting in the well-known, simple relationship:

, (2)

where the overbar denotes the hydrostatic base state. It is customary to consider some sort of departure (or, perturbation) from the base state to be the only source for contributions to vertical acceleration.

The next step is to decompose the pressure and density into the hydrostatic basic state and a perturbation, denoted by :

(3)

Therefore, ignoring the last two terms of (1) and using (3), it can be seen that

. (4)

Making use of (2) in (4) results in

(5)

Term i on the rhs of (5) is that associated with the vertical gradient of the perturbation pressure, and term ii is that traditionally associated with buoyancy.

For textbook treatments of parcel theory (e.g, Holton 1992; Dutton 1976, or Emanuel 1994), the next step may involve a replacement of in the denominator of both terms in (5) with ; this involves making either an anelastic [] or Boussinesq [] approximation and linearizing the equation by treating the perturbation as if it is small compared to the basic state, neglecting high order terms. The formulation winds up looking identical to (5) only with replacing . Also common in textbook developments of parcel theory is the neglect of term i in (5) [e.g., as in Holton (1992, p. 54) or Dutton (1976, p. 70), or Emanuel (1994, p. 6)], attributing the effect of buoyancy only to term ii.

To see that neglecting term i in (5) results in an incomplete description of buoyancy, imagine the following thought experiment. At t = 0, a uniform density perturbation is introduced instantaneously within some finite volume embedded in a horizontally uniform, hydrostatically-balanced base state "environment" initially at rest. The presence of a density perturbation implies a change in the pressure, even when there is no flow yet in response to the resulting acceleration. That is, the perturbation pressure is , where is the contribution due to buoyancy, arising from the air within the perturbation volume having a different density than the hydrostatically-balanced base state, and is the dynamic contribution to perturbation pressure arising from flow field differences created by the perturbation. In this rather unphysical thought experiment, = 0 at t = 0 because there is initially no flow, but there must still be an acceleration. Note that cannot be in hydrostatic balance; if it were, by definition, there would be no acceleration. If no acceleration is associated with the introduction of the perturbation at t = 0, there could never be any motion [see the comments by Gordon (1981) on Schlesinger (1980)].1


1 Observe that for a finite volume, there are also lateral boundaries to that volume, and along these lateral boundaries, there are horizontal perturbation pressure gradient forces acting to accelerate the horizontal winds. The aspect ratio (i.e., the height relative to the width) of the perturbation volume becomes an important issue when buoyancy is considered in its complete, 3-dimensional form, as discussed in Das (1979).

Term ii on the rhs of (5) is not yet in its typical form, however. The Equation of State is simply

, (6)

where is the virtual temperature, and R is the gas constant for dry air. For the hydrostatic part,

,

and from (6), it can be shown that

.

It is traditional at this point to neglect the contribution of pressure perturbations to the density, such that

(7)

Therefore, if we ignore the first term on the rhs of (5) and substitute from (7), using the fact that , we find

. (8)

where B is a commonly-used form of the expression for the vertical acceleration due to "buoyancy" as the difference between the parcel temperature and that of a hydrostatic basic state.

This expression can be developed further to consider the notion of conditional instability in terms of lapse rate (Schultz et al. 2000; Bryan and Fritsch 2000). This is not the direction we wish to pursue herein; rather, we want to consider some issues associated with defining the basic state.

It should be noted that if the so-called Exner function

, (9)

where and cp is the specific heat of air at constant pressure, is employed as the pressure variable, then it is possible to derive a simple perturbation buoyancy formulation that does not ignore the pressure fluctuations; see Hane et al. (1981; p. 566) for this development.

 

3. Definition of the basic state

The use of a separate, single term (B) for "buoyancy" is traditionally derived from the simplified treatment of pure parcel theory and involves the use of a base state temperature profile. Note that a linearization of Eqn. (5) is not necessary, although it is a common approach in textbook explanations of parcel theory. Changing the base state alters the partitioning of the unbalanced vertical pressure gradient force between terms i and ii in (5). That is, the base state defines the temperature differences that result in both the traditional buoyancy term and the buoyancy contribution to the vertical perturbation pressure.2 We will return to this topic, but first we consider some difficulties associated with defining the base state.


2 Note that it is possible to define a base state in which the nonkinetic energy (i.e., the sum of potential, internal, and latent energies) is minimal after mass rearrangements (see Emanuel 1994; §6.6). Emanuel shows that when considering the available potential energy of the whole system (not just that of the parcel), the actual amount of energy available can be substantially less than that indicated by parcel CAPE. This is related to a similar concept developed by Wang and Randall (1996).

a. The preconvective environment

As discussed in Brooks et al. (1994), the problem of defining a reference profile (i.e., a "proximity" environmental sounding) to determine buoyancy diagnostically within the real atmosphere is not necessarily a simple one. There are at least two major difficulties. The first challenge is that the observations of the state of the atmosphere are finite in number and density, and measurements are not without error. Essentially, most of what we think we know about the vertical structure of temperature and dewpoint comes from balloon-borne sounding instruments. Given the practical realities of such measurements, it is likely that there is considerable unobserved spatial and temporal variability in atmospheric structure, prior to the initiation of deep moist convection. There are several recent indications of this (e.g., Markowski et al. 1998) associated with the 1994/1995 field campaigns of the VORTEX project (Rasmussen et al. 1994). Even the special field measurements can only hint at this unsampled variability; for parameters such as convective available potential energy (CAPE, or relative buoyancy) or vertical wind shear-related parameters, the variability can be quite large (Brooks et. al. 1996).

Just how might we characterize the "environment" of the storms, given this reality? Existing observations suggest that assuming horizontal homogeneity could be a gross oversimplification. If the observed environment exhibits a large amount of variability, just what does parcel buoyancy relative to some hypothetical homogeneous, fixed basic state actually represent?

 

b. The environment after storms begin

The other challenge arises once deep convection commences. After deep moist convection is initiated, major changes to its surroundings can extend several km (Brooks et al., 1994; Weisman et al. 1998) from the convective vertical motions. Those alterations to the environment can be quite large, especially in the volatile parameters related to vertical wind shear, but also in thermodynamic profiles. In the real atmosphere, of course, convection must have some impact on its environment. That is, deep moist convection is a response to the conditions that gave rise to convection. If there were no impact on those conditions, once deep convection began, it could go on indefinitely. Bjerknes (1938) certainly was among the first to consider this effect, developing the so-called slice method (see Emanuel 1994; pp. 175-178) as an attempt to address it. However, our knowledge of the magnitude and possible impact of these changes to the local environment in the vicinity of deep moist convection is limited. Once deep moist convection is underway, it is constantly altering its own immediate "environment," so assuming that a single, fixed reference state that can be used to define buoyancy seems unlikely.

 

c. Numerical simulations

A question of some import follows from the traditional form of the buoyancy process [i.e., Eqn (8)]. If the forces on a parcel inside a convective cloud truly depend on the temperature difference between the parcel and some "external" reference state profile, how does the parcel within that cloud "know" (or "feel") its temperature relative to this reference state? Parcels inside clouds are presumably surrounded by other parcels with similar thermodynamic characteristics. Note that Bryan and Fritsch (2000) raised some similar issues in a different context. The only parcels that can "know" anything about the temperature difference between themselves and their "environment" would be those on the outer periphery of the convective cloud, where lateral entrainment becomes a complicating issue. A 1-dimensional theory of buoyancy cannot legitimately address the issue of the environment outside a cloud except through the artifice of the basic state.

The assumption when using Eqn. (8) clearly is that the vertical acceleration on a buoyant parcel starting from rest is due to the relative buoyancy defined by the difference between the basic state and the parcel. The environmental buoyancy is presumed to be essentially zero (i.e., the basic state is hydrostatic). The very term CAPE implies that what is being considered is that part of the potential energy that is "available"; that is, assuming that the reference state has no available potential energy.

Although it might well be true that the basic state contributes little, if any, potential energy for sustaining deep moist convection, our concern is associated with defining the basic state. Determination of buoyancy in (8) depends on the choice of the basic state profile. However, that buoyancy would somehow depend on the choice of a base state is unacceptable, in physical terms. Given the rather arbitrary nature of the base state, the vertical acceleration associated with buoyancy cannot be subject to arbitrary change.

Equation (5), which does include the impact of perturbation pressure, has been used extensively in numerical models (although in different forms), many of which define buoyancy in just this way (e.g., Clark 1973; Soong and Ogura 1973; Schlesinger 1975; Klemp and Wilhelmson 1978; Leslie and Smith 1978; Dudhia 1993; Wicker and Wilhelmson 1995). These model formulations do not ignore term i, of course. If we rewrite (5) as follows:

, (10)

this rearranged form of the equation highlights the physically distinct contributions to the vertical accelerations. Term i of (10) is due to the dynamic perturbation pressure (a function of the flow field), which is clearly independent of the choice of a thermodynamic reference state. Term ii combines the traditional buoyancy term with the buoyant contribution to the perturbation pressure gradient force. The apparent dilemma of buoyancy depending on the choice of a base state is resolved by observing that changing the base state leaves the sum of the two parts of term ii unchanged. That is, shifting the "B-term" upward or downward by changing the basic state results in an equal and opposing change to the vertical perturbation pressure gradient forces. The net force associated with both buoyant contributions is actually independent of the basic state when both contributions to buoyancy are included. A special case, showing this compensation between the two parts of term ii in (10) is developed in the Appendix. Ignoring term i prevents a proper compensation for changes to the basic state from occurring and so Eqn (8) remains erroneously dependent on the basic state.

Most cloud-scale models treat convection explicitly using expressions akin to (5) and employ a reference state for defining buoyancy. The subtraction of a reference state is done primarily to minimize truncation errors in the calculations. Cloud models have proven capable of simulating successfully many aspects of the behavior of deep moist convection, and we are not disputing the general validity of those results. We note, however, the initiation of a cloud-scale model with a "buoyant bubble" is analogous to our unphysical thought experiment, except that in such a simulation, the initiating "bubble" is 3-dimensional, thereby including all the implied complications beyond simple parcel theory associated with higher dimensionality. At the initial time, no known cloud model simulation that begins with a buoyant bubble is accounting for at t = 0; it appears that the models can adjust to this initial discrepancy within a few time steps, and show no ill affects thereafter (G. Bryan 2001, personal communication).

Many mesoscale numerical models alleviate any buoyancy created during the simulation through parameterized convection; recently, Bryan and Fritsch (2000) have shown that mesoscale models create resolvable-scale buoyancy, as well. It is also the case that most mesoscale models use the artifice of a reference state for defining buoyancy, and for the same reason: minimization of truncation error. As with cloud scale models, mesoscale models have achieved considerable success in helping to understand physical processes, so there is no cause to believe their treatment of buoyancy is essentially incorrect. Notably, mesoscale model simulations are not typically initiated with unphysical initial conditions.

In situations involving deep convection in a sheared environment, the effects of perturbation vertical pressure gradients in (5) simply cannot be ignored, as discussed in Rotunno and Klemp (1982), Weisman and Klemp (1982; 1984), inter alia . In fact, these authors have demonstrated that the first term in (5) can be as large or even larger than that due to "buoyancy" (i.e., B alone). Whereas the perceived contribution of "buoyancy" is typically limited to that of term ii in (5), a complete description of the contribution owing to buoyancy is the combination of terms embodied in term ii of (10).

 

d. Physical interpretation of buoyancy

Another potential problem with the traditional parcel theory of convection is the parcel itself: the notion of a parcel is another artifice that is open to question. As it stands, we have essentially no measurements of pressures within deep convective clouds, so it is unclear just what might be happening with regard to pressure fluctuations within them. This makes the traditional neglect of the importance of pressure values in density fluctuations at least subject to validation. If we somehow could obtain highly detailed measurements of all the basic atmospheric variables within deep convective clouds, what might we find? Most penetrations of deep convection have been in the Tropics (e.g., LeMone and Zipser 1980), and the within-cloud observations of temperature, humidity, and vertical motion contain a lot of variability. Is each bump and wiggle in those aircraft penetration observations a parcel? What pressure fluctuations are associated with the temperature variations? Do the corrugations and convolutions seen in cumulonimbus clouds (Fig. 1) constitute processes approximating true individual "parcels"? Just what might that say about a real cloud, versus the relatively smooth theoretical constructs we have used traditionally to understand deep convection? Most current numerical cloud models have horizontal grid spacings on the order of 1 km, which does not even come close to resolving the visible structures we see in actual deep convective clouds (cf. Fig. 1). Does a highly smoothed representation of that structure, eliminating most of the variability we see in real clouds, lead to misrepresentations of the dynamics? Given our lack of detailed observations, it's hard to know how to answer this.

 

4. Conclusions

Buoyancy is an unbalanced vertical pressure gradient force attributable to variations in density within the atmospheric column. The traditional use of a reference state and treatment of buoyancy as a relative quantity is simply an artifact of a simplified pedagogical treatment and a device to reduce truncation error in numerical simulations. Regrettably, this artifice tends to obfuscate the application and understanding of how buoyancy works. Although many generations of meteorologists have learned about convection in the traditional "parcel theory" approach that ignores the effects of buoyancy-associated perturbation pressure, this simplification limits our ability to conceptualize processes properly in deep, moist convection. Furthermore, although the application of Eqn. (5) in numerical models, in any of several different forms, does not result in any significant error, its use invites misinterpretation of model results since it seems to imply that "buoyancy" is wholly contained in B. As we have shown, B by itself depends on the reference state and must thereby be an incomplete description of buoyancy effects.

Moreover, it is no longer obvious that considering only the B-term is the best way to understand the physics of deep atmospheric convection in a diagnostic sense. When doing observational studies, there is no way to determine , and so any errors associated with choosing an inappropriate reference state for calculating B are uncompensated. Although the concept has a long history and is a foundation for the development of forecasting parameters such as CAPE, the use of the B-term alone results in buoyancy having a strong apparent dependence on the choice of a base state, which is not physically correct.

As noted in Doswell and Rasmussen (1994), CAPE generally is not an accurate predictor of vertical motion in storms.3 Variability in the severe convective environment is being recognized as an important factor in the behavior of storms, so it is becoming increasingly clear that it is difficult to interpret CAPE estimates unambiguously in terms of vertical motion in deep convection, in part because it inevitably involves assumptions about the base state used in CAPE computations. CAPE can be viewed properly as a simple parameter summarizing certain aspects of the complex thermodynamic structure within the environment for deep convection, but cannot be used as an accurate estimator of the contribution to "buoyancy" in deep convective storms.


3 There are many factors besides the base state that can alter the accuracy of the vertical motion estimates derived from CAPE: the choice of which parcel to lift, the neglect of dynamic pressure perturbation effects in sheared environments, whether moist adiabatic ascent is pseudo-adiabatic or reversible moist adiabatic, the complex topic of entrainment, and the impact of freezing on the process, to name only some of them.

In particular, the association between CAPE and the occurrence of severe convection is open to question, in part because of the growing recognition of the importance of dynamic vertical pressure [i.e, in (10), above] gradient forces on updrafts. Specification of a proper "environmental" sounding has been recognized as a troubling issue for some time and is not physically relevant to parcel buoyancy, as we have tried to show. The atmosphere does not recognize the distinction we impose between a base state and a perturbation; this distinction has proven to be useful but it can be misleading in the way we have described.

In this short note, we have shown that a proper treatment of buoyancy cannot be limited to the traditional B-term, involving only the relative temperature difference between a parcel and an arbitrary base state. Vertical accelerations on a parcel are not dependent on the base state, as physical logic dictates. Parcels do not need to "know" their temperature relative to their environment; in the simplified, 1-dimensional treatment of convection via parcel theory, buoyancy is an unbalanced vertical pressure gradient force associated with density perturbations. In this classical theory of buoyancy, the base state is irrelevant because of the compensatory relationship between both terms associated with buoyancy. Therefore, the force balance in a 1-dimensional treatment is limited to what is going on within a vertical column of parcels, as it should be.

The pedagogical development of parcel theory needs modification to include this often-neglected aspect of the problem. Moreover, any diagnosis from model simulations (or from observational data) that considers the B-term to be a complete description of buoyancy effects is potentially misleading. We believe that the time has come to consider revising our traditional approach to understanding buoyancy in the context of deep convective storms.

 

Acknowledgments. We appreciate many helpful discussions on this general topic with Drs. Erik Rasmussen and Louis Wicker. Drs. David Schultz and Kerry Emanuel made several useful suggestions for improving the presentation. Mr. George Bryan, Dr. Peter Bannon, and Dr. Mike Fritsch also helped correct some errors within an early version of the manuscript ; George Bryan also contributed some valuable discussion and generously shared his simulation results with us.


APPENDIX

Compensation between the perturbation pressure and relative temperature

terms to produce independence from the base state

 

Consider the special case of an isothermal, hydrostatic atmosphere. The temperature is To, so the hydrostatic equation can be integrated for the pressure and density:

, (A1)

where po and are the surface pressure and density, respectively, and z is the height above the surface. Now consider an isothermal, hydrostatic base state, given by . For this base state, the pressure and density profiles are given by

, (A2)

Therefore, the perturbation pressure and density can be shown to be:

, (A3)

Now consider the two terms in part ii of Eqn. (10). The first is the perturbation pressure gradient term; differentiating the expression for in (A3) with respect to z yields

, (A4)

whereas the relative density term is given by

, (A5)

It is easy to see that the two separate contributions to buoyancy [i.e., Eqns. (A4) and (A5)] are equal in magnitude but of opposite sign, yielding a zero sum, which must be the case since both the base state and the actual atmosphere are hydrostatic. In this special case, the two terms are equal and opposite no matter what the choice of might be, including the trivial case where = 0.

Obviously, showing this in more general cases is considerably more complicated, but the principle will remain the same. Changes to the base state will always result in changes to the two parts of term ii in (10) that will be equal in magnitude and of opposite sign.


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FIGURE CAPTIONS

Figure 1. Cumulonimbus cloud during the afternoon of 18 May 1990, in the Texas Panhandle (© 1990 C. Doswell), showing the complex "bubbly" appearance typical of cumuliform clouds.