vertical rise of a buoyant plume: some elementary comments on closure problems and dissipative...

Atmmphrric Environmrnt Vol. IO. pp. 665-670. Pergamon Press 1976. Printed in Great Britain.

LETTER TO THE EDITORS

VERTICAL RISE OF A BUOYANT PLUME: SOME ELEMENTARY

COMMENTS ON CLOSURE PROBLEMS AND DISSIPATIVE PROCESSES

(First received 10 March 1975 and in final form 30 January 1976)

Much has recently been written about buoyant plumes; reviews of various aspects have been presented by Briggs (1969), Scorer (1970), Morton (1971), Fay (1973) and Csanady (1973). The purpose of this note is to try to resolve a few questions about buoyant plume modeling that may also arise in other students’ minds as the rather substantial literature is studied. Specifically, the topics to be addressed here are the different meanings of the closure problem and the influence of dissipative phenomena. It also may be of some interest to see the governing equations developed directly from the conservation laws of mass, momentum and energy, without the Boussinesq approximation and without relying on pseudo conservation “laws” of volume, buoyancy, kinetic energy or heat. Because of both the pedagogical tenor of this note and the crudeness of the model (see later) only a simple application will be considered, i.e. to the rise of a vertical plume.

GOVERNING EQUATIONS AND THEIR CLOSURE

The governing equations for plume rise are the conservation laws for mass, momentum and energy. The usual fluid equations can be transformed into the desired form for a buoyant plume by integrating the equations over the cross section of the plume, defining suitable average plume properties and identifying the flux of mass, momentum and energy through the plume boundaries. This is done in Briggs (1970). However, it is simpler and perhaps more desirable to rederive the conservation equations using a control volume of height AZ and area equal to the cross sectional area of the plume (see Fig. 1). Then using the notation shown in Fig. 1 it is easy to obtain, e.g. the continuity equation

;(pPwR’) = 2p,u,R,

pi jpe

:P ‘e %P Te ___-_

i

-__ t Y

e

RlZl

Fig. 1. Notation: p = pressure, p = density, T = temperature, w = updraft velocity, R = plume radius, o, = entrainment velocity; subscripts i = internal, e = external or en-

vironment, p = plume.

where U, is known as the entrainment velocity. Throughout this note steady state conditions are assumed as well as “top-hat” profiles for the plume variables (or it can be considered that we will deal only with cross-sectional aver- ages of the plume variables).

Before writing the momentum and energy equations it is important to notice that even if the entrainment velocity, drag coefficient, etc., were known still the equations will not be closed: there are 5 dependent variables (pi, pp, T,, w and R) and only 4 equations (conservation of mass, vertical momentum and energy and the equation of state). Two more equations, for the components of the horizontal momentum, are available but they introduce two more unknowns, viz. the components of the horizontal velocity. This might be called the first closure problem. It arises from the implicit assumption that a separate plume can be identified from the rest of the fluid which thereby introduces the unknown, R. Otherwise one could in principle solve the full six fluid equations (counting the vector momentum equation as three equations) for the six unknowns, p, p. 7: v and from the result define a suitable plume radius. However the full set of equations can not be solved except numerically (Fox, 1970; Hanna, 1972).

To proceed analytically a new assumption is necessary. Scorer (1970) assumes R = yz and obtains the proportiona- lity constant using experimental data but Csanady (1973, p. 185) criticizes this assumption. The implicit closure assumption used by Morton et al. (1956) appears to be to ignore the variation of pp except in the buoyancy term (Boussinesq approximation) and to assume that the buoyancy flux is constant. Here, as in Briggs (1970) but without assuming that plume rise is isentropic and without using the Boussinesq approximation, it will be assumed that the horizontal gradient of pressure is negligible. Physi- cally this appears to be a reasonable approximation since the plume’s “boundary” is free to adjust to the ambient pressure. It is a free surface. Since this assumption closes the equations, it defines a plume boundary and it might be appropriate in what follows to refer to the resulting plume radius as the “free-surface” radius. However, this terminology will not be pursued.

From ignoring the horizontal pressure gradient it follows that pt = p* = p and the vertical gradient of the pressure within the plume is the same as for the environmental air. This is taken to be given by the hydrostatic equation

dp -= dz

- PeS.

With the pressure now specified and with the equation of state

p = pp.&T P (3)

where d is the gas constant for the plume air, then the three conservation equations for the five plume variables, p, pp, T,, w and R, appear to be closed (see later). We now turn to the momentum and energy equations.

The z-component of the momentum equation states that the net rate of outflow of momentum from the control volume, viz. [d(p,~R’w*)/dz]A; is equal to the sum of the

665

666 Letter to the editors

vertical forces. These forces include: the weight, pplrR*Az; the resultant of the pressure forces,

Z pnR2(; - pxR jr+li + 2nRARp = -nR’(dp/dz)Az;

and the drag force, -(1/2)C,p,w22~RAz, where C, is an (unknown) drag coefficient. Thus, using (2) and assuming that the entrained air possesses no initial vertical velocity (which seems to be a fairly accurate assumption, see Slinn, 1972) there results the momentum equation

WZ w’R*) = gR*[p, - pP] - : R2pp I. (4)

The energy equation is just a statement of the first law of thermodynamics and requires for steady state conditions that the energy outflow from the control volume, p,nRZwA~(1/2w2 + c,,T,) at z + AZ, where c, is the specific heat at constant volume and per unit mass, is equal to the inflow plus the heat added within the control volume less the net work done by the fluid. The inflow of energy is p,nR2wAt(l/2w2 + c,T,) + p,ZxRAzu,AtE, all evaluated at z where E, is the energy of the entrained air (assumed to be known if v, is known). The heat addition is

OAtnR’Az - (C,/R)c,(T, - 7Jp,R*wAtAz,

where Q (assumed known) is the rate of heat addition per unit volume from processes such as water vapor condensation, and CH is an (unknown) heat transfer coefficient. The additional variables in the heat transfer term are intro- duced solely for convenience and to nondimensionalize CH. The net work done by the fluid in the control volume is against gravity, p,gnR*AzAt, against the pressure forces, [d(pnR%At)/dz]Az and against the drag force, (1/2)C,p,w22nRAzwAr. Thus the energy equation is

W2 2+gz+c,TP = OR’ + ZRp,v,E,

CH - %cP(TP - T,)p,R’w - 2 c;;p,Rzw (5)

where the work done against the pressure forces has been included in the enthalpy term, h = c,T = c,T + p/p.

In the above five equations there are five unknowns and three unspecified quantities (u,, CD, and C,). Environmen- tal conditions, E, and Q are assumed known. That some variables are unspecified might be referred to as the second closure problem. It is similar to and in fact can be derived directly from the usual closure difficulties with turbulent flows. This is discussed in detail by Fox (1970) and Morton (1971) where the relation between, e.g. C, and the correla- tion of velocity fluctuations is formulated. Here this second closure problem will be overcome by selecting simple expressions for these otherwise unspecified quantities. In particular it is noted that in most previous attacks on the buoyant plume problem, heat transfer and drag have been ignored and Taylor’s entrainment assumption (Taylor, 1945) U, = c(w has been used. The objective of the remainder of this note is to study the consequences of this versus alternative assumptions.

ISENTROPIC CASE

The simplest case to study is when the entrainment, drag, heat transfer and heat generation are all specified to be zero. This highly idealized case of isentropic rise of a vertical plume is now investigated to assist in later identifying the influence of dissipative processes. In this simple case the continuity and energy equations can be integrated to

p,wR’ = c, = plowoR; (6)

W* f gz + cpTp

4 T

=~~=~+gz,+c,T,,. (7)

The momentum equation can be converted to

ldw’ _f T, , I (0,

UPDRAFT VELOCITY, W

0 10 20 30 40 50 lm I ml) r \ 1 1

PLUME TEMPERATURE, Tp

250

7

c

260 270 280 290 300 I Kb I

I

-2 -1 0 1 2 3

NORMALIZED PLUME WIDTH, R/R0

Fig. 2. Plots of equations (11, 14 and 19) for the isentropic rise of a buoyant plume in an isothermal atmosphere:

T,, = 275 K, T,, = 300 K, w0 = 10 m s-‘.

If (7) is differentiated and dw’/dz is eliminated between the result and (8) then there results

(9)

which states that the plume’s lapse rate, yp, is slightly larger than the dry adiababic lapse rate yd = g/c,.

For simple environmental temperature profiles, an explicit expression for T, can be found from the general solution to (9), viz

(10)

For example, for an isothermal atmosphere, T, = T,, then (10) yields

Tp = T,, exp - ‘v} i eo

If the environmental temperature profile can be fit over restricted height intervals z, < z I zi+, with a series of straight lines, i.e.

T, = T,i - Y& - zi)t (12)

where yei is the environmental temperature’s lapse rate in the ith interval, then (10) yields

(13)

Letter to the editors

UPDRAFT VELOCITY, W (m I ‘I)

DRY BUOYANT PLUME

T - 303 K

- I SO&RMAL ATMOSPHERE

Te - 215 K . Tee

---- WEAK INVERSION

Te - Ten + c PZ P

I I 1 I

0 1 2 3 4 5n 6 1 NORoWiLIZED PLUME CROSS SECTION, lR/RiIL

Fig. 3. A comparison of the isentropic rise of a buoyant plume in an isothermal atmosphere (qO = 275 K) with its rise against the temperature inversion r, = T,, + Ydz.

where Si = Yd/Yei. The updraft velocity can now be found simply by

substituting these solutions for TP into the energy equation. Thus for an isothermal atmosphere the result is

w2 w; -= - - Y(Z - z0) + c,T,o 2 2

[l - exp{ -“br “)}I.

(14)

For the case of non-zero but constant environmental lapse rates as in (12) the result is

w2 w? _=’ 2 2

-8(r-z,)+cJPi[l -(;I]. (15)

These results yield that the updraft will continue to increase until T, = T, which, of course, can also be seen from the original momentum equation.

To find the radius of the plume the plume’s density or the pressure are needed. From the hydrostatic equation and for an isothermal atmosphere

Y 9(z - zo) p=poexp - -~

Y-l c,T,o ’ (16)

where Y = c,/c, is the ratio of specific heats. For constant, non-zero, environmental lapse rates

T .‘d./(.‘- I)

P = Pi jGi C !

. (17)

Now pp can be found from the equation of state and when the result is substituted into the continuity equation, there results

or, for an isothermal atmosphere

667

(19)

where w is given in (14 and 15). These results for the isentropic rise of a dry plume are

illustrated in Figs. 2 and 3. Figure 2 is for the case of an isothermal atmosphere and in Fig. 3 a comparison is made between the plume’s behavior in an isothermal atmosphere with its behavior in a weak (but extensive) temperature inversion.

There are a number of interesting and informative fea- tures of these results. First it is interesting to see that the familiar “mushroom” shape is predicted. This follows even though entrainment has been neglected; it is required for mass conservation. The temperature of the plume falls steadily and even falls below the environmental temperature, until finally the kinetic energy is exhausted. Actually, though, it is clear that our approximations fail as w + 0. In reality there is another term in the energy equation, namely s*/2, where s is the horizontal speed. Thus as w-+0, the details of the results are not reliable.

Further, once w+ 0, it is clear that the plume will begin to accelerate downward because 7” < T,. It is expected that a solution to the set of equations can be found to describe the subsequent evolution of the plume but this has not been explored quantitatively. Qualitatively, the motion is one of damped oscillation which can be seen by rewriting the momentum equation in the form

2

i+&i2+&z=g c2

L 1 p-1 ) (20)

P e P e C,Te

where w = i = dz/dt and t (time) appears in the para- metric representation of the w(z) curve. The undamped frequency of this oscillation in an isothermal atmosphere is w2 = g2/c,Teo which is essentially the Brunt-VHisPla frequency, or the natural period is

2a - r = - \L/ cpTeo

9

which for T’, = 275 K is about 5.6 min. Vertical motion finally stops at the elevation where TP = T,, which for the case shown in Fig. 2 is at about 2.44 km.

ESTIMATE OF THE INFLUENCE OF DRAG

At first encounter most of the results found above seem to be physically reasonable except for the magnitude of the updraft velocity. It seems quite improbable that, even with an initial temperature excess as large as 25 K, updrafts in excess of 40 m S-I would be attained. Indeed if this were so there would be even less justification for ignoring drag. In this section an attempt will be made to estimate the influence of drag, continuing to ignore entrainment, and heat generation and transfer.

In this case the continuity equation can still be integrated to yield.

The momentum equation can be written as

(23)

The energy equation becomes

2c, wz = -R1. (24)

If dw2/dz is now eliminated between (23 and 24) then there results, as before,

_dTp- TP dz

= Yp = Yd F’ r


Thus the plume’s temperature profile does not change because of drag and the results found in the previous section for T, can be used without modification.

The updraft velocity can now be found from (23 or 24) provided C, is specified. We are not prepared to attempt to derive C, at the present time and instead rationalize as follows. Since R was seen to change but little over most of the plume rise (at least when drag was ignored) and since for large Reynolds numbers, CD for bluff bodies becomes independent of the Reynolds number (and, there- fore, of w) but especially since what is sought is only an indication of the influence of drag, then 2CJR in (23) is taken to be a constant inverse “momentum relaxation” length:

I, = g D

Then upon substituting (26) into (23) and using TP for an isothermal atmosphere as found in the previous section there results

Reasonable estimates for 1, and T,,Jyd suggest that (l/1,) % (yd/TpD). Then for z < Tpo/yd, (27) can be approximated by

+&[$- lJ[l -exp{ejJ. (28)

A similar result can be obtained when the environmental lapse rate is a non-zero constant. However, recognizing the approximations in the formulation, it is suggested that an approximate solution will be consistent. To obtain one it is noted that the momentum equation (23) predicts a relatively rapid relaxation (the exp(-z/E,,,) term) from any initial velocity to a “steady-state” velocity, obtained by putting the inertial term in the momentum equation (i.e. dw’/dz) equal to zero. Taking dw’/dz = 0 implies that there is a balance between buoyancy and drag forces. Thus it is proposed that an acceptable approximate description of the updraft velocity is

d w; -=-exp -’ 2 2 ( ! 1,

+ d,,,(~ - l)[l - exp(- t)), (29

or, in cases where the initial velocity is generated by buoyancy (for example, probably the exit velocity from a natural draft cooling tower) just

L \‘e i

In Fig. 4 a comparison IS made among (14, 27, 29 and 30) for an isothermal atmosphere. We have chosen the order of magnitude “guesstimate” CD = 0(10-i) and used I, = 300 m. In Fig. 4 the substantial influence (the assumed) drag has on the updraft velocity is obvious. Further, given that the drag coefficient is not known, nor is the initial velocity of the plume, there seems to be little point in using the “exact” solution (27) or even the approximate solution (29) unless a specific initial velocity must be fit. Instead, in what follows the concept implicit in (30) will be used; that is, that the buoyancy and (unknown) drag forces are always in balance.

I SOTHERMAL ATMOSPHERE

Tea - 275 K

.-- T --A__ PO

- 300K

-1

‘\ \

I

/

I’\ NO DRAG

/

/

INCLUDING DRAG:

“EXACT” SOLUTION APPROXIMATE SOLN. _ FREE INITIAL CONDS.

10 20 30 40 50 M)

UPDRAFT VELOCITY, W tm s ‘*l

Fig. 4. A comparison among the predictions of equations (14, 27, 29 and 30) to show the influence of drag on the updraft velocity of a buoyant plume in an isothermal atmosphere. The “exact” solution is the result (27). For the approximate solution it is assumed that the inertial term in the momentum equation is negligible but the initial velocity w0 = 10 m s -’ is forced. The initial condition is

removed for the third curve.

ESTIMATE OF THE INFLUENCE OF HEAT TRANSFER

The original 5 equations contained 5 unknowns and 3 unspecified quantities (u,,. Cr, and C,,). Some of the consequences of taking all the unspecified quantities to be zero (isentropic case) and the case with only Co # 0 were seen in the above. Now consider the case with both CD # 0 and C, # 0 to see some of the consequences of heat transfer. In this case, the continuity equation can still be integrated to

p,wR’ = c,. (31)

The momentum equation does not contain any explicit dependence on heat transfer and is as before

Finally, if(31) is used in the energy equation,

1 2

=-;; In

- 2 c&T, - T,), (33)

where, again, I,,, = R/2C,. Rigorously the analysis can not proceed any farther

since the heat transfer coefficient is unknown. However, we rationalize as before and take

RIG, = I,. (34)

Letter to the editors 669

where I, is another unknown, “temperature relaxation”, constant length. If dw’/dz is now eliminated between (32 and 33)

(35)

The solution to (35), e.g. for an isothermal atmosphere, is

T, = T,,exp[- T,) + Te,,ill - exp[-- ;,)I, (36)

where

1 1 T=T+$. (37)

t eo Solutions for other simple environmental temperature profiles can be found similarly. To find the updraft velocity in this case, the momentum equation is returned to and it is again assumed that a balance has been established between drag and buoyancy:

wz T -=gl, q-1.

L 1 (38)

2 e

These results will be illustrated later for the case [compare (26) with (34)] I, = 21, which follows if C, = C,,.

ESTIMATE OF THE INFLUENCE OF ENTRAINMENT

If V, # 0 (but, for the moment, C, = 0 = C,) then the continuity equation cannot be integrated. It is

; (ppwRz) = Zp,u,R. (39)

If this is used in the momentum equation, it becomes

T VW - 225,

T, R (40)

Similarly for the energy equation, if the total energy of the entrained air is E, = (u:/2) + cpTe + gz, then

2 0, Tp zz- R ; r [c,(T, - T,) + 8~’ - $)I. (41)

e _

As yet a, has not been specified. If use is made of Tay- lor’s assumption that a, is proportional to the updraft velocity, v, = co-v the above momentum and energy equations become almost identical to the similar equations used earlier when heat transfer and drag were included. In particular the momentum equation, (40), is identical if in (32)

Cn = 2aT,/T, (42)

and the energy equation is essentially the same if in (33)

CH = 2aT,IT, = CD (43)

Consequently, given the crudeness of the specifications of v,, Cr, and Cm it seems inconsistent to modify the momentum and energy equations any further because of entrainment, and we propose to keep in them only the drag and heat transfer terms.

I I , I, I1 10 15 a a mr -1

UPDRAFT VELOCITY. W

However, the continuity equation is significantly altered. If v, = aw then (42) in (39) yields

I I I I I I I I

215 280 285 2% 295 300 K

PLUME TEMPERATURE, Tp

f (ppwR2) = f p,wR’. Vn

(44)

In (44) we might take 21, = I, as was suggested earlier, but instead it seems more appropriate to call 21, a third characteristic “entrainment” length 1,. Then (44) yields

p,wR’ = (p,,~R*bqW,)

Fig. 5. An illustration of the influences of: entrainment on the continuity equation, heat transfer on the energy equation, and drag on the momentum equation for a buoyant plume in an isothermal atmosphere. The (arbitrary) choices of the relaxation lengths are 1, = 900 m, 1, = 600 m, I, = 300 m. To determine the updraft velocity, inertial acceleration was ignored and no initial velocity

(45) was forced. T,, = 275 K, T,, = 300 K.

which means that the plume’s radius increases at a more rapid rate because of entrainment.

In Fig. 5 these results are illustrated for the case of an isothermal atmosphere. The specific equations used are (45) with I, = 900 m, (36) with I, = 600 m and (38) with I, = 300 m. Upon comparing Figs. 4 and 5, the substantial difference caused by heat transfer is seen. In Fig. 4 it is seen that drag reduced the final plume rise by a factor of about 2 from the case with no drag (for our essentially arbitrary choice of 1,). Now, in Fig. 5, it is seen that heat transfer reduces the plume rise by about another factor

of 2. Also, the plume’s radius can be seen to be consider- ably larger than without entrainment, and increases almost linearly with height for a considerable portion of the rise. At this point it is interesting to recall that Scorer assumes that R = yz to close his equations.

ESTIMATE OF THE INFLUENCE

OF A HEAT SOURCE

So far in this report a cursory examination has been made of the consequences of drag, heat transfer and entrainment. This meets the objective of this report. How- ever before closing it may be of some interest to look at the consequences of heat production, Q. and the case of a moist plume may seem ideal. However it is not ideally simple because in this case a host of new unknowns enter

NORMALIZED PLUME RADIUS, R/R0

D 1 2 3 4 5 6 7


besides Q, namely: the mass concentration of dry air, pd; water vapor, pL’; cloud water, prw as well as rain water or even ice, and there are also the possibly-different veloci- ties for each constituent. Correspondingly there is a host of new equations, for example, continuity equations for each constituent.

Rather than delve into the full problem for a ,moist plume, it is proposed just to indicate the effect of Q. For example, suppose that the plume is saturated and that entrainment is negligible. Then the rate of heat production per unit volume Q, is just LP,,, where L is the latent heat of condensation and P,, is the rate of production of cloud water. Ignoring entrainment, conservation of water vapor requires

; (pdm,wR2) = -PcwRZ, (46)

where m = pr/pd is the mixing ratio. Upon substituting Q from (46) into the energy equation and taking pp G pd

(47)

If the Clausius-Clapeyron equation is now used to evaluate dm,/dz (e.g. see Haltiner and Martin, 1957) and if dw’/dz is eliminated between (47) and the momentum equation, then the plume’s lapse rate becomes

where dd and d, are the gas constants for dry air and water vapor, respectively, and Y, is the moist adiabatic lapse rate. Thus one consequence of the presence of water vapor (and Q) is to replace the dry adiabatic lapse rate, Y,,, in our previous results, by the moist adiabatic lapse rate Y,,,. Other changes needed to describe a moist plume, though, include accounting for the weight of the cloud water in the momentum equation and including the cloud water evaporation necessary to saturate the entrained air. Such effects are contained in the cumulus cloud models of, e.g. Simpson and Wiggert (1970) and have recently been used in a buoyant plume model by Hanna (1972).

An interesting case considered by Gilford (1967) is for a highly radioactive plume such as from a hypothetical worst-case reactor accident. In this case the heat is generated from radioactive decay within the plume. If it is assumed that drag and buoyancy are in balance, then ignoring entrainment and heat transfer it is seen that the energy equation yields

Yp = Yd - (Qlc,P,w). (49)

It is noted that the approximations leading to (49) have yielded yp = y,, if Q = 0. From (49) it is seen as expected and as with a moist plume, that the plume’s lapse rate is decreased if there is an internal heat source. As Gifford shows this leads to greater plume rise and consequently some reduction in the hazard.

CONCLUDING REMARKS

With reference to the introduction the following conclu- sions are drawn. First it appears that there are two separate closure problems for the buoyant plume equations. One is derived from the assumption that a separate plume exists, identifiable from the rest of the fluid medium. This introduces the new unknown, R. To overcome this closure problem it was assumed and it appears reasonable, physically, that the horizontal pressure gradient is negligible. However other assumptions can be and have been made. The second closure problem relates to the specification of the entrainment, drag and heat transfer and is similar to the familiar closure difficulties with the equations for turbulent flows. Here the consequences of a number of differ-

ent assumptions were examined. For example, for the sim-

plest case, with t’,, C, and Cn all set to zero, then it appeared that the (isentropic) vertical rise of a buoyant plume was too fast, too hot and too high. To modify this some drag, heat transfer and entrainment were added (albeit in an approximate manner by introducing constant relaxation lengths) and this led to obvious consequences. It might be of particular interest to recall that Taylor’s entrainment assumption introduces terms into the momentum and energy equations which are similar to drag and heat transfer terms but with quite specific expressions for the drag and heat transfer coefficients. It is clear that a major research task that should be undertaken is to obtain realistic specifications of these coefficients. In this regard it appears that most rapid progress could be made by in- terpreting field data for R, w, and Tp for days with negligible mean wind and measured T, and p.. Such work is now in progress at our laboratory and will be reported later by Wolf et al.

Ackyowlrdgemmts-Thanks are given to G. A. Briggs and G. Arnason for helpful comments on an earlier version

of this report. B. C. Scott and T. J. Bander helped the author recognize difficulties with other plume models. Financial support was obtained from Battelle Institute’s Physical Science Project and from USAEC Contract AT-(45-1)-l 830.

W. GEORGE N. SLINN Atmospheric Sciences Department, Battelle, Pacific Northwest Laboratories, Rich/and, WA 99352, U.S.A.

REFERENCES

Briggs G. A. (1969) Plume Rise, AEC Critical Review Series. TID-25075. available from NTIS Springfield. VA.

Briggs G. A. (1970) A simp/e model for bent-over Phmle rise. Ph.D. Dissertation, Pennsylvania State Univ. and contribution No. 39 from ATDL, NOAA, Oak Ridge, TN.

Csanady G. T. (1973) Turbulent Diffusion in the Environ- ment. Reidel, Boston, MA.

Fay J. A. (1973) Buoyant plumes and wakes. Ann. Rev. Fhtid Mech. (Edited by Van Dyke M. et al.) Annual Reviews Inc., Palo Alto, CA.

Fox D. G. (1970) Forced plume in a stably stratified fluid. J. geophys. Res. 75, 6818-6835.

Gifford F. A. (1967) The rise of strongly radioactive plumes, J. appl. Met. 6. 644649.

Haltiner G. J. and Martin F. L. (1957) Dynamical and Physical Meteorology. McGraw-Hill, New York.

Hanna S. R. (1972) Rise and condensation of large cooling tower plumes. J. appl. Met. 11, 7933799.

Morton B. R. (1971) The choice of conservation equations for plume models. J. geophys. Rex 76, 7409-7416.

Morton B. R.. Taylor G. I. and Turner J. S. (1956) Turbu- lent gravitational convection from maintained and in- stantaneous sources. Proc. R. Sot. (Lond.) A 23, l-23.

Scorer R. S. (1970) Plumes. jets and entrainment coefficients, notes taken by W. G. N. Slinn and B. Wilhelm- son, in The Microphysics and Dynamics of Convective Clouds-A Colloquium. NCAR-RN-52, National Center for Atmospheric Research, Boulder, CO.

Simpson J. and Wiggert V. (1970) Models of precipitating cumulus towers. Mon. Weather Rev. 97, 471-489.

Slinn W. G. N. (1972) Kinematically possible flows exterior to jets and plumes. PNL Ann. Rept. for 1971 to the USAEC DBER BNWL-1651 pt I, Battelle-Northwest, Richland, Wash. Available from NTIS, Springfield, VA.

Taylor G. I. (1945) Dynamics of a mass of hot gas rising in the air. USAEC Report MDDC-919 (LADC-276), Los Alamos Scientific Laboratories.

vertical rise of a buoyant plume: some elementary comments on closure problems and dissipative...

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