Vertical mixing of passive scalars owing to breaking gravity waves

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Dynamics of Atmospheres and Oceans 23 (1996) 371-378

Vertical mixing of passive scalars owing to breaking gravity waves V. Schilling, D. Etling * Institut fiir Meteorologie und Klimatologie, Universitiit Hannover, Herrenhiiuser Str. 2, 30419 Hannover, Germany

Received 1 July 1994;revised 31 January 1995;accepted 24 February 1995

Abstract In this paper, some aspects of dispersion of air pollutants as emitted from aircraft in the lower stratosphere have been investigated. As this part of the atmosphere is always stably stratified, mixing as a result of small-scale turbulence is very slow. Instead, effective vertical mixing can be provided by breaking gravity waves. We have examined the mixing properties of those events by means of a numerical model, which simulates the wave development as well as the dispersion of passive trace substances. From these simulations, an effective diffusion coefficient for the entire event of a breaking gravity wave Of about 0.7 m 2 s- a was calculated.

1. Introduction Large parts of geophysical flow systems such as the atmosphere and ocean are stably stratified with moderate or weak turbulent activities. Mixing of passive scalars (e.g. air pollutants) is rather slow under these circumstances, except for some singular events such as Kelvin-Helmholtz instabilities or breaking gravity waves. This is especially true for the lower stratosphere, where there is little or no turbulence under normal circumstances. Concerning the problem of vertical diffusion of passive scalars, this has become of some concern because of the possible influence of high-altitude aircraft traffic on climate (e.g. as a result of the greenhouse effect or an ozone hole). Because pollutants from aircraft emissions are set free in the form of narrow line sources (of a few metres width), it is of

* Corresponding author. 0377-0265/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-0265(95)00429-7

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interest in modelling chemical reactions in global climate models to determine how fast these trace substances are distributed vertically. On the smallest scales, this process is usually due to small-scale turbulence in the atmosphere. However, as the lower stratosphere is very stably stratified, no detectable permanent turbulence level is found in these regions. It has therefore long been suggested that turbulence is created by shear instabilities (Kelvin-Helmholtz instabilities) or breaking gravity waves (e.g. Fritts and Rastogi, 1985) within limited local and spatial scales. The question then arises of the net effect from these events for the dispersion of passive scalars, if averaged over space and time as suitable for large-scale numerical models of atmospheric chemistry (e.g. Crutzen and Briihl, 1990). In this paper we describe some numerical simulations on the dispersion of passive scalars owing to breaking gravity waves. This extends earlier work on the mixing of air pollutants by Kelvin-Helmholtz instabilities (Schilling and Janssen, 1992).

2. The flow model Numerical simulations of gravity waves were performed with a two-dimensional model based on the anelastic version of the Boussinesq approximation subject to the usual Reynolds averaging. Equations of motion, continuity equation and heat transfer equation can be written as O~i Oui --+Uk--= ~t Oxk ~uip

10p Po ~xi

+

0 --

+

O[OUiOUk~

g ~ o (~i3 ~Xk K ~

= o

[ ) -+ OXk

3X i

(1)

(2)

Ox i

--

i~t

+ uk

i~xk

-- - - K h - -

i~xk

(3)

Oxk

In (1)-(3) u i is the mean velocity vector, ~J the potential temperature and pressure. K m and K h are the eddy viscosity and eddy heat diffusivity, respectively. Both are related via a Prandtl number by K h = K m / P r . The eddy viscosity is obtained from the Prandtl-Komogorov relation

(4)

K,. = o d E ~/2

where E = ~/2/2 is the turbulent kinetic energy and l is a mixing length. The latter is related to the grid size A = (Ax Az) 1/2 by •

l = min(A,O.76N-lE1/2), t = A, N 2<~0

N: > 0 (5)

V. Schilling,D. Etling/ Dynamicsof Atmospheres and Oceans23 (1996)371-378

373

The turbulent kinetic energy is obtained from the usual equation

~E OE [ ~i ~lk~ ~li - - + U k - - =Kin [ ) -+ --Kh N 2 + ot oxk oxk

~

~E

E 3/2 +c,-t

(6)

In (5) and (6), N is the local Brunt-V~iis~il~i frequency;

(7) Eqs. (1), (3) and (6) have been solved numerically on a finite difference grid. Details on the numerical methods have been given elsewhere (Schilling and Janssen, 1992). The boundary and initial conditions will be given in Section 4.

3. The dispersion model Dispersion of a passive scalar within gravity waves owing to mean flow and turbulence is simulated by means of a Lagrangian particle model (e.g. Legg and Raupach, 1982). Several thousands of weightless particles are set free at the source and their trajectories are followed by x i ( t q - A t ) = x i ( t ) + [ui(/) + u ' i ( t ) ] h t

(8)

The turbulent velocity fluctuations u'i are obtained from (9)

u'i( t ) =RLU'i( t -- A t ) + u ; ( t)

where RL is the Lagrangian autocorrelation given by (10)

R L = exp( - A t / z L )

u~* is a random velocity, which can be obtained from a stochastic process via the Monte-Carlo method: u" = (1 w* = (1

n2

xl/2

--r~L) "2"/2 --/~L)

~ 002 Or.A+ ( 1 - - R L ) r L "~x

v,

~tr2

(11)

OrwA"t" (1 --RL)T L OZ

In (11) X is a random number with normal distribution and or. and crw are the velocity variances. These are obtained from the turbulent kinetic energy by ~ru = 1.0E 1/2,

0% = 0.4E 1/z

(12)

The Lagrangian time scale r L needed in (10) and (11) is given by "cL = K r a / E 1/2

(13)

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V. Schilling, D. Etling / Dynamics of Atmospheres and Oceans 23 (1996) 371-378

By solving Eqs. (1)-(13) numerically it is possible to obtain particle positions for every timestep At of the flow development. Concentrations can then be obtained by counting the number of particles contained in a box of the numerical grid.

4. Simulation of breaking gravity waves The aim of our study was to investigate the mixing of passive scalars owing to breaking gravity waves within the lower stratosphere. As initial condition we choose a stably stratified shear flow with stratification N = 8 × 10 -3 s-1 = constant (OO/Oz = 1 K per 100 m) and mean velocity u 0 according to u0(z) = - z ~ U t a n h [ ( z - zc ) / H ]

(14)

with H = 1000 m, z,. = 25 km (critical level) and AU = + 20 ms-1. The computational domain was taken between 10 and 30 km height in the vertical and 16 km in the horizontal direction. The gravity waves were initiated at the lower domain level (z = 10 km). This simplifies the physical observation that gravity waves in the stratosphere are excited by gravity waves starting in the troposphere. Following Fritts (1985), a single wave was prescribed at the lower boundary by a vertical velocity

w( x,t ) = Wo[sink( x - ct ) ] g( t )

(15)

where g ( t ) is an initialization function

g(t) = sin2(27rt/T), g(t) =1,

t <~ T

t>T

The wave amplitude was increased in time according to (15) until t = T (usually T = 300 s) and was then kept to constant at its final value w 0 = 1 ms-1. Wavelength L = k / 2 r r was varied between 2 and 8 km and phase speed c between 0 and 16.7 m s-~. An example of numerical simulations for the case L = 8 km and c = 0 m s - ~ is given in Fig. 1. About 30 min after the wave was excited at the lower boundary (at z = 10 km), a large amplitude wave can be seen at mid-level (about 22 kin) which breaks subsequently (t = 40 m i n - 1 h). One should note that wave breaking occurs already at z = 20 km, which is below the critical level at z = 25 km. Hence, in this case, wave breaking is not due to wave-critical level interaction, but is due to self-induction of a local critical level by tilting of phase lines owing to wind shear (Koop, 1981; Weinstock, 1982). The wave breaking event has a dramatic effect on the dispersion of passive trace particles, which are injected into the gravity wave at time t = 30 min at position z = 20 km. Although the wave is starting to break at time 40 rain, the particle cloud is stretched initially more or less in the horizontal direction as a result of shear effects. However, after t = 50 min, vertical transport starts to be very fast and the particle cloud is distributed vertically over about 6 km within a few minutes. This behaviour is presented more clearly in Fig. 2, where the cloud

V. Schilling, D. Etling / Dynamics of Atmospheres and Oceans 23 (1996) 371-378

375

0

16

v eq

30

v tq

Rt'I

4

8

12

16 0

x (kin)

4

8

12

x (k~)

Fig. 1. Structure of a gravity wave with L = 8 km and c = 0 ms- i for various times after excitation at the lower boundary (z = 10 km). Shown are lines of constant potential temperature O and a cloud of particles set free at t = 30 min, z = 20 km and x = 1 km.

d e v e l o p m e n t w i t h o u t gravity wave contours is shown in a s u b s e q u e n t extrapolation o f the x direction (time is n o w c o u n t e d from the injection of the particle cloud, i.e. after the gravity wave field has d e v e l o p e d for 30 rain).

5. Effective diffusivity in breaking gravity waves T h e vertical mixing o f passive scalars o w i n g to breaking gravity waves (Figs. 1 and 2) c a n n o t be regarded as turbulent diffusion in the usual sense, b e c a u s e mixing is d u e to the special e v e n t o f wave breaking. H o w e v e r , for large-scale m o d e l s o f

V. Schilling, D. Etling / Dynamics of Atmospheres and Oceans 23 (1996) ~71-378

376

3O 26°

t,i

~i 18= 140 rri£11 10

I

I

I

I

3O 26' 22-

H 2 0 ~rLin

tO

I

3026-

t~

22iS'

60 rr~ir~ I0

I

I

I

I

3O

"7 .i

t,,l

261

'

22iS.:, 14-

.==....

.;....

•

';.,-.-.... .-~:';:-

,; , . . I s

;

:

"::'.

.

... •

" ~"

r .'"..

.'~

90 m~,r~ I0

x (~m) Fig. 2. Dispersion of trace particles within the breaking gravity wave shown in Fig. 1 for various times after the particles have been injected at position x = 1 km and z = 20 km. (At time of injection, the gravity wave has already developed for 30 min; see Fig. 1).

the stratosphere it is necessary to parameterize diffusion by an effective subgrid diffusion coefficient. Hence, it may be useful to define a diffusion coefficient for the total event o f a breaking gravity wave. This is, o f course, not an easy task, and it is not clear from the outset which m e t h o d will provide reasonable estimates.

I~. Schilling, D. Etling / Dynamics of Atmospheres and Oceans 23 (1996) 371-378

377

OZ 321-

2b

~'0

6'0

8b

160

time{rain) Fig. 3. Temporal development of the spatial averaged effective vertical diffusivity/~c for the case of the breaking gravity wave in Fig. 1.

For the case of mixing in Kelvin-Helmholtz waves, several methods of evaluating an effective diffusion coefficient have been tested (Schilling and Janssen, 1992). The estimate from a diffusion equation averaged over horizontal planes seemed to yield the most plausible results. If we denote the horizontally averaged concentration by 5, the one-dimensional diffusion equation can be written as

a~

a

a~

a t = 0z K¢ 0z

(16)

Here Kc(z) is an eddy diffusivity which can be evaluated from (16) by inserting concentrations c(z, t) as obtained from simulations with the Lagrangian model. For the case presented in Figs. 1 and 2, the temporal development for the vertical mean of Kc

t~c(t)= fz~ tS~kmK~( z , t ) d e

(17)

is shown in Fig. 3. Vertical mixing seems to be most effective in the first half-hour of the breaking gravity wave. If we take the mean over the whole event of say 90 min duration, we obtain for the vertically and temporal averaged effective diffusivity

Kc°~ = ft =4°° o sI?,c(t')dt = 0.7 m2s -1

(18)

This would be the average value for a breaking gravity wave over a vertical volume depth of 10 km and a time scale of about 90 min. This value is comparable with estimates of Kc, ~ given by other workers (e.g. Woodman and Rastogi, 1984). More examples of breaking gravity waves for other combinations of wavelength and phase speed, and the dispersion of passive scalars within those wave events have been given by Schilling (1993). As can be seen in Figs. 1 and 2, vertical spreading of the passive tracer is mainly due to resolved scale motions of the breaking gravity wave. Hence within the Lagrangian dispersion model, diffusion owing to subgrid-scale turbulence (Eq. (11)) seems to be of minor importance for the whole mixing event. Therefore, the effective diffusion coefficient Kce~ as evaluated from Eqs. (16)-(18) will not

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V. Schilling, D. Etling/ Dynamics of Atmospheres and Oceans 23 (1996) 371-378

depend very much on the method used for the subgrid part of the Lagrangian dispersion model discussed in Section 3. The final stage of breaking gravity waves will lead to layers of small-scale turbulence, as is often observed in the lower stratosphere. This is clearly a three-dimensional process (e.g. Isler et al., 1994) which cannot be handled properly with a two-dimensional model as presented here. However, with regard to dispersion of trace substances, our results indicate (see, e.g. Fig. 3) that vertical mixing is most effective within the early stages of wave breaking. As this part of the whole process can be still regarded as dominated by two-dimensional motions the use of a two-dimensional numerical model seems to be justified for the purpose of this study.

Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft.

References Crutzen, P.J. and BriJhl, C., 1990. The atmospheric chemical effects of aircraft operations. In: U. Schumann (Editor), Air Traffic and the Environment. Springer, Berlin, pp. 96-106. Fritts, D.C., 1985. A numerical study of gravity wave saturation: nonlinear and multiple wave effects. J. Atmos. Sci., 42: 2043-2058. Fritts, D.C. and Rastogi, P.K., 1985. Convective and dynamical instabilities due to gravity motions in the lower and middle atmosphere: theory and observations. Radio Sci., 20: 1247-1277. Isler, J.R., Fritts, D.C., Andreassen, O. and Wasberg, C.E., 1994. Gravity wave breaking in two and three dimensions: 3. Vortex breakdown and transition to isotropy. J. Geophys. Res., 99: 8125-8137. Koop, C.G., 1981. A preliminary investigation of the interaction of internal gravity waves with a steady shearing motion. J. Fluid Mech., 113: 347-386. Legg, B.J. and Raupach, M.R., 1982. Markov-chain simulation of particle dispersion in inhomogeneous flows: the main drift velocity induced by a gradient in Eulerian velocity variance. Boundary-Layer Meteorol., 24: 3-13. Schilling, V.K., 1993. Effektive Diffusion yon Luftbeimengungen in der Strato-sph~ire verursacht dutch Kelvin-Helmholtz Instabilit~iten und brechende Schwerewellen. Bet. Inst. Meteorol. Klimatol. Univ. Hannover, 44:146 pp. Schilling, V.K. and Janssen, U., 1992. Particle dispersion due to dynamical instabilities in the lower stratosphere. Contrib. Atmos. Phys., 65: 259-273. Weinstock, J., 1982. Nonlinear theory of gravity waves: momentum deposition, generalized Rayleigh friction, and diffusion. J. Atmos. Sci., 39: 1698-1710. Woodman, R.F. and Rastogi, P.K., 1984. Evaluation of effective eddy diffusive coefficients using radar observations of turbulence in the stratosphere. Radio Sci., 19: 243-246.