Nonturbulent dispersion processes in complex terrain

965

NONTURBULENT DISPERSION PROCESSES IN COMPLEX TERRAIN* The paper by Fosbcrg et 01. examines the importance of llow divergence or convergence on pollutant concentration calculations. An exact solution to the advcction-diffusion equation under certain idealized conditions is presented from a paper by Roberts, Roberts’ solution is identical to the standard point source Gaussian equation. modified by a term containing the v&city divergence V. 1’. All equation numbers used in the present discussion paper will refer to equations written in the original paper. with the cxceptton of Equation (2a) defined below. Let us make . distinction bctwecn ‘total divergence’ defined by

(2) and “surface

divergence”

defined

by

Let us now examine the conditions under which the total divergence will affect the concentration solution as given by Roberts’ equation (3). The validity of the solution requires that U and d remain reasonably constant over the downwind distance x. Assuming the flow to be steady. we can use the total mass conservation equation to express the divergence as V.F=

P U dp p dx for the quasi one-dimensional flow that is under consideration. It can be seen that neglecting V* V is equivalent to neglecting compressibility effects. The conditions during which compressibility becomes important are for sizeable velocity, temperature or pressure changes. Let us suppose that these changes occur over a topographic feature of horizontal extent x. For an isentropic velocity change. a text on compressible flow will show that dp __=

W From a brief search of the more easily available references. it seems as if the authors may have misinterpreted the literature. They state that “airflow patterns.. are characterized by significantly large values of divergence” referring to Defant (1951). In Defant’s article, convergence and divergence are mentioned only twice. In both instances. they refer to the compressing and spreading of streamlines in land-sea breezes or over complex terrain. However, divergence of the streamlines is not equivalent to divergencc of the velocity vector field. The former refers to the behaviour of a velocrty component whereas the latter refers to the net sum of the components. For an incompressible fluid. it is necessary that the net or total divergence is zero (V. li = 0). However. the fluid may have diverging streamlines as described by. for example. r’i. & >tl (:).‘ r:Y in which case the u component is decelerating. Secondly. the authors state that mountain valley winds “generate substantial divergence”. based on a paper by Fosherg (1969). This is quite unusual since the mass continuity equation in the latter paper on two-dimensional flow over a mountain explicitly describes the divergence as being zero. i.e. \‘. v=

0

The flow problem was solved numerically. Since no finite difference scheme can satisfy the flow equations exactly. the result was that small non-zero values occurred when \;’ C’ was evaluated. It appears to be a numerical error rather than divergence that the authors are quoting. Thirdly. the authors state that “estimates of mesoscalc divcrgencc (Schaefer. 1973) showed maximum values of 10 ‘+s ’ over moderate terrain.. .“. However. Schaefer’s paper dots not give values of total divergence. but only surface divcrgencc as defined by Equation (2a) above. Fourthly. the authors tind that divergence has “occasronal extreme values of IO 4s-‘. (Anderson. 1971)“. But the foundation of Anderson’s wind field analysis is that the lotal divergence is zero. From this he estimates the surface divergence and the horizontal wind field. l Fosbcrg M. A.. Fox D. G.. Howard E. A. and Cohen J. I). (1976) Am~o.sphrric E~~rrronmmt IO, 1053-1055.

_%!

_.&&!!j u

P

where M is the flow Mach number (the ratio of fluid speed to sound speed). Using the above equations. the velocity divergence is approximated by

Since most wind speeds are much less than the speed of sound (344mss’ at 20°C). the Mach number M is very small. There are cases of exceptionally strong katabatic winds. which can blow steadily for several days, draining down mountain slopes from a glacier. If forced through a converging channel, they are further intensified. For a steady wind with a mean speed of 50 m s- I, not v,arying by more than 50% over the distance .x. the standard Gaussian solution is altered by a factor of -

w

6,x T

u

(

exp(TO.01)

z

= I 5 0.01

>

where the choice of sign depends on whether the velocity change is positive or negative. In examining the effect of a temperature or pressure change, and using the equation of state. p = pRT. dp

dp

dT

P

P

T

Assuming a finite pressure change Ap/p to be of the same order or smaller than the temperature change AT/T. the velocity divergence is approximated by &=V.ii:

u AT .-, x T

Typical temperature changes in air passing over a city or from land to water or from sunny to shady surfaces of a hill arc about IO’C. For such a change. the standard Gaussian solution is altered by a factor of cxp

-(

6,.x u >

: cxp( T 0.03) = I T 0.03

where again the choice of sign depends on whether the temperature change is positive or negative. For an adiabatic process. density and temperature changes arc related by dp_ = 2,5d? P

T

Discussions

966

givmg a slightly larger value for divergence, i.e..

which, since V *0 5 0, becomes

The largest adiabatic temperature change experienced by a wind is probably in a Fdhn. If it warms by 30°C in its descent over a distance x, the standard Gaussian concentration calculation is increased by a factor of

if 6 is defined as

exp

f&r - -_- 2 exp(+0.26) = 1.3. u> (

These examples represent typical as well as extreme meteorological conditions. If meteorological and surface conditions are such that Roberts’ equation is applicable (U and 6 are constant), then it seems that the effect of total divergence, or compressibility, can usually be neglected. In the case of extreme conditions in which total divergence is non-negligible. there is probably a much greater error due to uncertainties in U, ep and cr than in the omission of divergence effects. .4tmosphric

Environment

Service

C. S. MATTHIAS

Down.www

(d) We considered that the lead term in (d) is in fact the two-dimensional divergence Z/dy + &/t?z, that the c?ic/&x contribution is overwhelmed by the U transpoft as are the x diffusion terms. and that advection by the V field could be neglected. We did this to calculate the magnitude of the correction, since estimates of two-dimensional divergence were available. Because this last consideration is questionable. we can alter the form of our solution to include the advection effect. However. the main point of our paper is to illustrate that the contribution of the V field to cry and u, is through

and

On turio Cunada

AUTHORS

REPLY

We thank Matthias for commenting on our paper. His criticism provides an opportunity for us to clarify two key points that were not fully discussed in our manuscript. First, our intent was to explore the mechanism for systematic or organized effects of complex topography on the dispersion coefficients np and u,, and secondly to suggest that two-dimensional divergence might be used as an index of the organized flow effect. Clearly, as Matthias correctly states. the three-dimensional divergence must be zero or nearly so for any realistic tropospheric dispersion apphcations. Our notation in the original paper was misleading in this regard. To rectify this confusion, consider the following definition of a velocity field: V = 5 + V + V’ = (U + li + u’, t? + I:‘, i + w’) (a) where V = ensemble mean velocity without topography = (u. 0. 0). V = mean topographically induced variation from V, V’ = remaining small-scale variations from the ensemble mean. generally considered turbulent fluctuations. The presence of topography, for the purpose of this discussion, leads to the V field, which is not approximated as hpmogeneous and/or isotropic as are the V’ turbulence terms, hut represents a larger scale (in space) effect on plume dynamics. Clearly, when averages are taken over space scales much larger than the topographic variation, the contribution of V becomes less significant. Introduction of (a) into the conservation of mass expression from the original paper. after performing an ensemble average, yields: ;;rx + V*($)

+ V.(jO)=

V*(K.Vi)

(b)

which, as we showed in equation (3) of the original paper, can be formally incorporated into the conventional Gaussian solution. Equation (3) in the original paper provides a formal illustration that diffusion from point sources in complex terrain can be treated using a Gaussian model. But the conventional (I) and o, coefficients, designed to parameterize the turbulent diffusion prpcess, must be supplemented with contributions from the V field, the organized circulation patterns induced by the terrain. We imply that the following two alternatives are available for modeling point source dispersion in complex terrain: (1) Direct measurement or simulation through use of a meteorological wind field model to determine V + V, coupled with conventional Gaussian calculation or other ‘turbulent’ diffusion approximation, (2) Direct measurement of by and n,. which contain the dispersion on a +iciently large space scale to incorporate the organized V flow field created by the topographic forcing. It Seems necessary to reiterate that the V field is due to differential thermal convection from mountain slope surfaces, coupled with mechanical interactions of the terrain with the mean meteorological winds This suggests the terrain-augmented u’s will be strongly dependent on insolation (stability) and on terrain form. While the former effect has already been noted in the literature, the latter has prevented generalization of mountain dispersion data. We strongly support the need for more comprehensive field experimentation to develop terrain-influenced dispersion parameters. USDA Fores! Service Rocky Mountain Forest Range Experimenrul Fort Collins

CO 80521 U.S.A.

M. and

Station

A. FOSBERG D. G. Fox