A two-dimensional, second-order model for turbulent flow in the upper atmosphere

Mathematics and Computers in Simulation 28 (1986) 243-260 North-Holland

243

A TWO-DIMENSIONAL, SECOND-ORDER MODEL FOR TURBULENT FLOW IN THE UPPER ATMOSPHERE

Enrico SCIUBBA

Dipartimento di Meccanica e Aeronautica, Universita’ di Roma “La Sapienza”, Rome, Italy

Steven C. GONZALES

Naval Civil Engineering Laboratory, Port Hueneme, CA, U.S.A.

Richard L. PESKIN

Department of Geophysical Fluid-Dynamics, Rutgers University, New Brunswick, NJ, U.S.A.

ABSTRACT

A second order closure model for the atmospheric, two-dimensional, steady, turbulent flow in a zonal channel for the B-plane is developed, and the non-linear system of equations resulting from the model is solved using a standard iterative technique. An important feature of the model is that it attempts to simulate thermal effects in the energy- and enstrophy budgets by introducing a fictitious buoyancy-like forcing term, which corresponds to a baroclinic force field of intensity F(T) acting southwards in the g-plane. From the original governing equations a full set of relations for the Reynolds stresses is developed, in addition to one equation each for the mean momentum- and the energy balance [in the energy equation, a Boussinesq approximation is employed). The governing equations are properly scaled from synoptic observations to closely resemble those of the large eddy dynamics. Then, an invariant modelling technique is used, to obtain a non-linear system for the Reynolds’- and the thermal stresses. Since the source of energy required to maintain the flow is in the mean temperature difference between two limiting latitude circles, the [negative] temperature gradient is assumed from empirical data corresponding to a stable environment. The model parameters are reduced to standard Reynolds-, Rossby-, Richardson- and Prandtl numbers. A two-step double iterative procedure is employed to achieve convergence while maintaining physically sound profiles. The model shows a very high sensitivity to the turbulent kinetic energy profile, and the predicted energy conversion mechanisms, in particular the meridional eddy momentum flux, the eddy potential energy, and the temperature variances, are in good agreement with the available experimental data. In particular, the model reproduces reasonably well the general dynamics of synoptic flows, including the reverse energy cascade and the direct enst.rophy cascade. Further possible developments and feasible research topics in this area are briefly discussed.

0378-4754/86/$3.50 0 1986, IMACS/Elsevier Science Publishers B.V. (North-Holland)

244 E. Sciubba et al. / A 2D Znd-order model for turbulent flow in the upper atmosphere

1 . INTRODUCTION

The numerical prediction of the general circulation in the atmosphere has received considerable attention over the past several years. The atmosphere itself can be regarded as a “thin layer” when compared with the radial dimension of the Earth: if not for disturbances created by the Earth’s rotation and by the differential heating between latitudes, it would essentially exist as a quiescent, homogeneously stratified layer of almost constant thickness, where the isobaric surfaces would be concentric spheres. Air motion, in the form of a meridional circulation, is induced by the differential heating between latitudes, and an energetically much stronger, zonal circulation is produced by the Earth’s rotation. Because of the steady motion of the Earth, this zonal circulation is considered approximately symmetrical.

Perturbations introduced at the Earth’s surface (mountain ranges, land/sea boundaries) interrupt the symmetrical nature of the zonal circulation: these small disturbances cause the zonal velocity component to change from latitude to latitude, thus introducing shear layers, which are the direct cause of cyclonic and anticyclonic eddies.

The first example of a numerical attempt to weather prediction is the famous “Richardson’s model” in the mid twenties”“, but the first real forecast was performed much more recently, in a paper of Charney, Fjijrtijft and von Neumann in which the (parabolic] vorticity equation was solved on a two-dimensional grid.

In the numerical simulation developed in this work, a general circulation model, first proposed by Tennekes 1181 will be studied. A two-dimensional, non-divergent turbulent flow field driven by a fictitious, horizontally oriented “buoyant” (baroclinic) forcing term is used to couple the thermal- and dynamic effects of the upper atmosphere. The zonal temperature gradient accounts for vortex amplification in developing disturbances. The “buoyant” forcing represents an approximation to vortex stretching and baroclinic effects. The surface friction is parameterized by use of a linear drag term, so that the sole source of energy is the temperature differential at the boundaries.

In similar studies Peskin and Sciubba 1131 and Sciubba 1161 int.egrated the equation of motion proposed by Tennekes: by ignoring the sink term in the turbulent kinetic energy equation they were able to obtain a closed-form solution, which allowed for adjustment of model parameters. Once the model parameters were adjusted, the simulat.ion continued without restrictions (including the sink term in the energy equation]. Their results for the eddy momentum flux, poleward eddy flux, and temperature variance were not well correlated with empirical data. In a more recent paper, Peskin, Kowalski and Gonzales 1141 employed a numerical pseudo-spectral method 1121 to study a numerical, two-dimensional turbulent model of the general circulation: the governing equations were

(‘)Richardson, in his classical book “Weather Prediction by Numerical Process” 119271, estimated that a working force of 64000 “computers” l=persons able to conmpute) would have been required to “keep up” with weather forecasting on a global basis. His results [Ap at two gridpoints) were off of one order of magnitude. He blamed the poor initial data; today we know that the numerical technique he employed (centered time, centered-space discretization for the - parabolic - vorticity equation] is intrinsically unstable: but he - or his “computers” - could not work out a sufficient number of time steps to see the solution actually blow up.

E. Sciubba et al. / A 2D Znd-order model for turbulent flow in the upper atmosphere 245

again the original Tennekes equations for the stream function and the energy, formulated in the Fourier space. Their simulation provides a foundation to consider additional studies in two-dimensional turbulence and relate the physical concepts to the general circulation. Their results for the total kinetic energy, velocity- and temperature-velocity variances well represent the accepted empirical data. However, results for the temperature autocorrelation and meridional momentum flux did not represent empirical data very well. The synoptic scales of motion in their study were given for only one set of parameters, indicating need to carry out a parametric study. The present work is the sequel to the above mentioned studies, and is concerned with providing an optimized set of parameters with a minimal computational time. This optimized set of parameters can be employed in refining the momentun- and energy balances of the general circulation.

Reynolds decomposition for the system of equations for the buoyantly forced, two-dimensional non-divergent flow with surface friction is used to derive the flux maintenance equations. The terms involving triple-and pressure correlations [i.e., flux divergence and return-to-isotrophy term) are modelled by use of a scaled invariant modelling [81. A second-order closure application of invariant modelling has been successful in three-dimensional turbulent simulation of atmospheric and vort.ex flows 121, growth and decay of turbulence in the atmosphere [31 and variable density t.urbulent flows 171. The velocity-, temperature- and diffusion scales introduced by invariant modelling

for dimensional consistency are of the order of the “zonal channel” width [lo6 ml and constitute the “scalar parameters” of the model.

The solution to the modelled system of equations is obtained here with an iterative procedure, employing standard central finite difference discretization for t.he governing equation, with forward- and backward differences used at the boundaries. The various constants are parametrized by use of the commonly accepted synoptic scales of the upper atmosphere. Particular attention was focused on the turbulent length scales introduced by invariant modelling and on a constant [ol,) which represents the amplification rate of baroclinic waves. The model parameters were adjusted to match atmospheric radiosonde data from Oort and Rasmussen 1101. After the properly adjusted parameters are obtained, an experimental simulation of the general circulation is studied.

The numerical results reasonably well predict the eddy kinetic energy and temperature variance; the poleward eddy flux is overpredicted in the mid- to upper latitudes but is still within the range of generally accepted data; the model predicts a weak southern transport of eddy momentum in the tropics and a weak northern transport of eddy momentum in the upper latitudes. The salient conclusions are: the use of invariant modelling is proper for a two-dimensional, quasi-barotropic model, the energetics of the simulation adjust themselves correctly to realistic atmospheric conditions, and the simplicity of the mathematical- and computational model makes its use very convenient as a parameter-tuning tool.

2 . THE GOVERNING EQUATIONS: TENNEKES’ MODEL

In 1977, Tennekes proposed a 2-D atmospheric model, artificially “forced”, in which the forcing term is a - fictitious - “buoyancy”-force acting towards the Equator in the B-plane (figure 1). This buoyancy is the energetic replacement for the “random stirring” previously used by Lilly [3] to simulate an enstrophy source; it does not feed external energy into the flow but its use requires a diabatic heat source. At the same time, it provides the necessary link between the velocity [and vorticity-1 and temperature fields. To formulate his model, Tennekes considered a p-plane synoptic flow, and explicitly


ignored viscous diffusion terms in the equation of motion. Also, he ignored thermal diffusion, and used a “temperature advection” equation instead of the complete thermodynamic energy equation. Some other interactions are not explicitly treated, i.e., there must be some “friction” term to allow for the enstrophy cascade. The equations of motion include the Coriolis term. In addition, a force field is present, of intensity I, acting southwards (the y-direction]. The system of equations to be solved in this model is:

where R is the diabatic source-term, p is a y-averaged density, and nii and

D? are Rayleigh-type linear friction terms (that is, “surface friction” terms). It

is evident that a 2-D model should have the following characteristics:

a) allow vorticity amplification in developing disturbances [in order to simulate

cyclonic areas): this property is the so-called ability to simulate baroclinic

instability;

bl lead to the conversion of eddy kinetic energy to eddy “potential“ [=position) energy;

c I produce a reverse energy cascade; and

dl take into account the poleward heat flux.

It is clear from its description that the Tennekes’ mo’del satisfies bl and dl; the scope

of the original Tennekes’ paper - and, to a larger measure. of this research - is to

ascertain if and t,o what extent points a) and c) are satisfied as well.

Applying the standard Reynolds’ decomposition I? = F +

through (41, and averaging to introduce the Reynolds’ stresses after some manipulations:

f’l to equations

equationsm one obtai!:

1 I one equation for the zonal momentum:

u, q I ap g’ K a --T -- 3Xj - - P aXi

+ f Cijk ui + T” B - _z Di - z u; uj 0 h 1

(51

E. Sciubba et al. / A 20 Znd-order model for turbulent flow in the upper atmosphere 241

2) three equations for the zonal- (i=j=l], meridional- (i=j=21 eddy momentum variance and

for the eddy momentum flux Ii=l, j=2I:

3 I Two equation for the zonal- (K-11 and meridional- (K=21 temperature t

41 One equation for the temperature variance:

---- -_.-^ , , -- a$ B, +$ = - 2 8’ UI axy - + tipa 8% t B a 2 ijq + 2@ ii!!2 iE!l 1 J J

ax2 1

ax, axj

3. THE MODELLING TECJ-INIQUE AND RESULTING EQUATIONS

ranspo rt term:

(6a,b.c.l

Vabl

(81

At the present time, ther are several generally accepted models for second-order closure of the equations of motion 158,151. There is obviously an ample degree of arbitrariness in selecting a model, and, as it is often said, a good model is one which gives reasonable results, thus justifying a posteriori the initial - somewhat arbitrary - choices. Still, some general principles can be formulated which the model should obey, thus increasing the degree of rationality of the whole process. In particular:

al the model must be frame-invariant;

bl it must be also Galilean-invariant;

c 1 the modelled terms must have all the tensor properties and all the symmetries of the terms which they replace;

dl the modelled equations must satisfy all the conservational laws which the original set of equations satisfies;

248 E. Sciubba et al. / A 20 Znd-order model for turbulent flow in the upper atmosphere

e 1 the modelled terms must have the same dimensions as the terms that they replace; and

f 1 once points a) through e) above are satisfied, the modelled term should be so chosen to be the simplest possible from both a mathematical and physical point of view.

In addition, the model should be consistent with the physical properties of the motion being considered [ 171.

An important simplification IS achieved is only three independent length scales are postulated: one [X,1 for the velocity fluctuations, another (X21 for the temperature fluctuations, and a third IX,1 for the turbulent diffusion. This is not an entirely

arbitrary assumption: on one hand, it. is realistic that terms like __.-.-_ u’ u; 1 >

and R/BX J

liii-q scale wt th the same length scale. On the

other hand, one can always take int.o account minor differences between these length scales by using a numerical choice of the modelled be non-dimensionalized with the characteristic defined as follows:

coefficient of Olll in the relevant modelled equation. With this terms, and going hack to 2-D equations, the governing equations may in terms of the mean and turbulent velocity and temperature along length of the flow field. Thus the non-dimensional variables are

Non-dimensionalization of the governing equations requires two representations of the Reynolds number and a turbulent Rossby number. &, the “surface exchange” coefficient

takes the place of the kinematic viscosity. Re, and Re, are of order 10,000 and the

turbulent Rossby number is of order 0.01: i.e., the validity of the geostrophic approximation in this model is within 1X.

The dimensionless governing equations ar.ec2’:

Continuity: -__I_ --

‘2’Note that some numerical coefficients have been introduced to allow for model parameter adjustment. Initially, the numerical values of these coefficients have been derived from the existing literature; but later, some of them have been modified (see Tables I and II for the actual values used in the final numerical experiment).

E. Sciubba et al. / A 20 Znd-order model for turbulent flow in the upper atmosphere

Eddy momentum flux: _... __

Zonal eddy flux: --- c

249

I101

(111

(151

(16)


Meridional eddy flux: -... ---_..-- -... -.-.-

T x, a 0 = - Re, FF v’ $ 8 + 3 Re, -C ay 0

_ Rt! L-_Q m- + Re eo7_ LZ___-

T x, ,T N, q B R - h” V’ 0’ +

Re.. _.__._ - R$ u’ O’ T

Temperat.ure variance: . -... ~.._..~.__. _._ -.

T, -;.-_; an 0 = 2 Ke, Pr ;, V 0 2~ t Re, Pr :z $ Q _a_ Q---F +

0 ay 3 7

+ _L ayL

8’ 0’ ---- _ 2 _G1 3-;8; 1;

I171

Four non-dimensional length scales and one non-dimensional temperature scale are introduced:

x __I _ L

.!2 turbulent temperature length scale I. -- characteristic length scale

x 2 _ I.

h L -:

In addition to the above

turbulent velocity length scale _..~~~~~~~~ristic_i~~~~~-~c~i e.-

turbulent diffusion length scale charactertstic length scale

depth of atmosphere ___-- &m%c-Eiii th s c a 1 e

fluctuation temperature scale _._.._-___-_.-_._.._- mean temperature scale

ratios the dlmensionless term:

I28a,b,c,d,e)

Ke, N,. $ I

in equations 23), 241, 26) and 27) represents the baroclinic source term contribution to the flow field. A linear stability analysis [I81 provides an amplification parameter for very short waves:

(191


By nondimensionalizing c& , the effects of unstable baroclinic waves may be parametrized by the following:

Ke, N, $- 0

where:

120)

N, directly depends on the mean temperature gradient for growing disturbances. If the mean temperature gradient has a fairly constant value the atmosphere is quasi-barotropic and computational srability IS assured unless perturbations, resulting from local disturbances entering the zonal channel from the boundaries. cause a zonal temperature inversion resulting in a locally positive temperature gradient in the S-N direction.

4 NUMERICAL PROCEDURF:

The governing system of equations are second order non-linear partial differential equations 01 eight unknowns: mean velocity, ti, mean temperature, T. zonal velocity variance, U’ U’, meridional velocity variance, V’ V’, tonal eddy f 111x, II’ 0’. meridional eddy flux, V’ El’. temperature -___ variance. 8’ tl’, and momentum eddy flux, U’ V’. The numerical procedure employed IS a standard iteration technique which numerically reduces the partial non-linear equations to ordinary differential equations. As an example consider the final form of the meridional eddy flux equation [equation 16) abovel:

The left hand side is a second order ordinary differential equation in V’ 0’. The right hand side is assumed known from the prior iteration [with the initial value provided by either radiosonde data or a mathematical function). The left hand side is central differenced for the first and second derivative. The solution vector is the updated meridional eddy flux and is held constant until the (K+ll-th iteration.

The procedure used was to separate the unknowns assuming that the equations derived from _- ---- _.___- --___ the Reynolds stress equation, U’ U’, V’ V’, and U’ V’ can be treated separately from the unknowns derived from the thermal stress equation, __--_- -I_.-- 8’ U’ , 8’ V’. and temperature variance 8' 8'. This may be thought of as solving for the energy equation then using the updated temperature field for the momentum equation when solving the Navier-Stokes equation for a buoyant fluid. In order to achieve convergence for the simulation it was necessary to treat the mean temperature and mean velocity profiles as “data” throughout the iteration procedure. The mean temperature and mean velocity profiles are available from radiosonde data. Of primary importance is the calculation of the correlation of the dynamics of the second order terms and the energetics of the general circulation.


The system of equations involves six unknowns with the turbulent kinetic energy being ___-_. --- solved from U’ U’ and V’ V’. The starting point for the iteration procedure is t.o assume initial profiles close to the stability range for which solutions are valid. This was accomplished by assuming the radiosonde data from Oort and Rasmussen 119711 as initial input profiles. The convergence criterion was chosen consistent wit.h the numerical procedure, i.e., the solutions for the zonal and meridional eddy fluxes and temperature variance were checked so that the variation from the K-th to the (Ktll-th iteration was within a specified tolerance.

A one step iteration solution did not achieve convergence. Convergence was obtained by solving equations 251, 27) and 261, zonal eddy flux, temperature variance, and meridional eddy flux respectively. The zonal momentum variance, 221, meridional momentum variance, 231, and the eddy momentum flux, 24), were solved after introduction of the “new” solutions until convergence was obtained. Each solution is updated and appropriately substituted for the next iteration.

Slip boundary conditions were employed throughout the entire simulation. Standard forward- [southern boundary) and backward- [northern boundary) difference were used.

5. RESUL.TS AND DISCUSSION

A selection of proper time- and length scales was necessary before programming the governing equations and determining the required physical quantities. An order-of-magnitude analysis of the governing equations and known synoptic scales of motion help simplify the choice of several parameters.

The model parameters listed in Table II are held constant throughout the simulation [except as noted). The dynamics of the general circulation depend on several important fact.ors, two of them being the temperature distribution and advection of turbulent eddies. It was determined from the results of Kowalski, et al. (61 and Tennekes’ 1181 linear stability analysis that a specified temperature distribution was required to assure a proper energy closure. The justification of presenting results at two distinct pressure levels [i.e., altitudes) was to verify the effects of the temperature distribution on the buoyantly forced barotropic model.

The choice of invariant modeling to obtain a second order closure is strongly dependent on

the eddy kinetic energy. An eddy kinetic energy 1q21”’ closure check was necessary. Illustrated in Figure 5 is a comparison of the transient eddy kinetic energy simulation for two different pressure levels. Variations of modeled parameters will he discussed lat.er. For both levels the general trend is slightly underpredicted. The results indicate a continually increasing energy throughout the entire channel implying a northward transport. of eddy kinetic energy. The maximum is shifted to the north of the one shown by the empirical data. The eddy kinetic energy is calculated by summing the zonal and meridional variance as seen from Figures 6 and 7. The zonal variance solution underpredicts the observed data between 20 and 45 degree north. The results for the meridional variance are in relatively good agreement. The underprediction of the zonal variance is a direct result. of the poor prediction of the eddy momentum flux. As seen from Figure 8 the eddy momentum flux is relatively weak in the range indicated, thus forcing the solution of the zonal variance to be underestimated. The meridional variance is well predicted: since the turbulent Rossby number is within the geostrophic range of validity 10.011 and the turbulent Reynolds number is of order 10,000, the ratio


of these two non-dimensional parameters is of the same order of magnitude as the channel width. Considering the relatively weak eddy momentum flux, the terms multiplied are of the same order of magnitude as the turbulent velocity length scale. The simulation then accounts for the meridional wind to span the entire channel. A second reason for the good prediction of meridional variance is the Tennekes buoyancy parameter: the meridional transport of the wind component is accurately simulated with the parameterized “effective gravitational” term. This parameter as mentioned ,earlier simulates the amplification of baroclinic waves. For the zonal variance the underpredicted results are also related to the eddy momentum flux. With a relatively weak eddy momentum flux, in order to accurately simulate the zonal variance the mean shear of the flow field must be greater.

Referring to Figure 8, the numerical result for the eddy momentum flux is not in good agreement with observed data. The eddy momentum flux is required to maintain the mean zonal flow against frictional dissipation. The solutron indicated a weak southern transport of momentum at low-to-mid latitudes and weak northern transport at upper latitudes.

The local value of the eddy moment.um flux depends on the zonal shear, the bu.oyant forcing t.erm, and the anisotropy of t.he flow field. The flow is nearly isotropic everywhere in the channel. with the exception of the immediate proximity of the boundaries; the buoyant forcing term as defined above produced the correct meridional variance solution: it follows that the zonal shear field is inadequate to simulate the transport of westerly momentum. Since the conversion from eddy kinetic energy to zonal kinetic energy (reverse energy cascade) is given by [‘,I:

(211

one then expect.s this reverse cascade to be small - if present at all - in the channel [see Figure 121. It is here that the original assumption of flow two-dimensionality shows its limitations: like any 2-D model, Tennekes’ model is unable to successfully reproduce either the eddy-potential to eddy-kinetic or the eddy-kinetic to zonal-kinetic energy conversion. In the atmosphere, eddy momentum is NOT the balancing term hetween the mean zonal shear and the frictional losses: vertical transport of both eddy- and zonal momentum enter the picture as well. On a local scale, vortex stretching - denied in this 2-D model - is the main energy exchange mechanism. Furthermore, temperature gradients introduce real baroclinic effects whose amplitude vartes LOCALLY, thus also contributing to both the C(A,&l and the C&K,1 conversions. The fictitious “baroclinic forcing” Icy,) used in the present simulation is useful to achieve an approximate closure of the ove_rali energy balance, but it cannot be expected to take care of the energy exchange mecharnsms equally well.

The temperature variance solutron are illustrated in F’igure 9. The results are consistent wrth observed data such that the maximum variance occurs in the northernmost part of the channel. The results indicate overprediction in the tropics and underprediction in the ---- upper latitudes (0’ 0; is a measure of the total available eddy potential energy].

Figure 10 shows that the meridional transport of sensible heat overpredicts the observed data by a factor of 3 for both the 400 and 500 mb solutions. As mentioned earlier the forced buoyancy term (a,] is of extreme importance in this results, and this will be emphasized further in t.he following discussion. For the same reason stated above, namely the inability of any 2-D model to produce a realistic energy cascade, a tuning of o,


can only achieve a more realistic overall trend, but cannot reflect the local dynamics of the thermal transport in synoptic flows, which are essentially controlled by 3-D effects. Physically the conversion from zonal potential energy to eddy potential energy and the conversion process of eddy potential energy to eddy kinetic energy [91,

C. (K,, K,] = - 1 \” .@T $; dy ’ 0

are greater at the mid-to-upper latitudes. From Figure 10 it can be seen that the diabatlc heating is consistent with the dynamics of the general circulation [see Equation 211 and the decision to assume a fixed temperature distribution is justified for energy c 1 0 s u r e

S h o w n in Figure 11 a r e tt-I e numerical results for V’ 8’ for varying a m P 1 i f i C a t i n parameter, u,. All other parameters listed in Table II remain unchanged. The’

amplification rate is varied from o.sx10-5 to 1.0x10’5 s-1. The increments of cy,

are 0.5, 0.6, 0.7, 0.8, and 1.0X10-’ s-‘. The value suggest ed by Tennekes (19771, is 1.0X10 5 5“.

Not shown are the results for the eddy kinetic energy, zonal and meridional variances. The solutions for these variables are similar to the previous graphs. Therefore the eddy kinetic energy, zonal and meridional variances are insensitive to the parameter 01,. One would expect this to nccur for the znnal momentum variance but not for the meridional momentum variance. ‘Thus the simulation shows that energy is being redistributed within the channel. independently of 01, and any slight variation of the energy dlstrlhutlon IS due to the intrinsic transport processes of the simulation.

This verifies that. the energetics of the at,mosphere become dependent on t.he conversion mechanism of energies and not on the eddy available potential energy 0’ 0’ for an increasing amplification of barocllnic wa ves. Figure 11 provides an excellent example for the previous argument. The meridional eddy flux or transport of sensible heat becomes tntnlly dependent on the nmpllflcation parameter. There IS a strong southward transport of sensible heat and a relativrly weak northward transport in the tropic region. The simulation then suggests that the amplification of baroclinic waves controls the conversion process of zonal eddy available potential energy to eddy available potential energy. Figure 11 also shows why the amplification parameter chosen in Table II is

0.8x10-” s-1. For this value the simulation produces good agreement between the observed data and numerical closure of Ihe diabatic heating rate.

6 SUMMARY

Numerical simulation of a turbulent two-dimensional, nondivergent flow was undertaken using an invariant modelling technique for a second-order closure model; several turbulent. length scales were introduced along with the dimensional scaling based on the eddy kinetic energy. The important covarlances and variances used to describe the energetics of the atmosphere for. large eddy transport are analyzed herebelow.


From a numerical and mathematical point of view the eddy kinetic energy is the primary agent in obtaining a realistic closure. The importance of this term cannot be overetiphasized because the numerical closure is heavily dependent on it. The result for each numerical experiment is consistent with nost aspects of observed atmospheric data.

---.. The zonal momentum variance [u’ u’l is within the observed atmospheric data. The result is underpredicted by as much as 507. in the mid-latitudes and approaches atmospheric data in the upper latitudes. Uy inoking closely at the data it would be extremely difficult to simulate this variance at the mid-latitudes, because this simulation depends on smooth increasing and decreasing slopes. However, the results demonstrate the general trends: the primary feature is that the relative maximum is obtained.

The meridional momentum variance (v’ v’l is in good agreement with observed data. The simulation is able t.o predict well a steady increasing variance. The influence of the parameterized buoyant forcing term, g’ O’o/T,, is tested and the results indicate the coupled turbulent velocity and temperature fields are well represented by the parameter choice under discussion. As in the atmospheric data both the zonal and meridional variance are similar and peak at or near 60 degrees latitude. Neither an increase in turbulent Reynolds number nor an increase of t.he turbulent velocity length scale show any effect other than to increase the variance consistently throughout the channel (both results not shown here).

The eddy momentum flux deserves considerable attention. The trends are opposite to observed data, showing southward transport of momentum in the tropics and northward transport in the upper latitudes. Through exhaustive efforts of testing the simulation the results still held. In this simulation the converslon mechanism from eddy kinetic to zonal kinetic energy (reverse energy cascade] is poorly predlcted. The forced buoyancy parameter and the transport of sensible heat are strong enough to maintain the energetics of the atmosphere, therefore a salient point to consider is that the model adjusts Itself to an overall energy balance: this suggests that one possible way of producing a more -_ realistic u’ v’-profile would be that of revising the mathematical form in which some of the terms in equation 6~) are modelled. Studies in this direction are underway. Once again, though, one must consider that the seemingly poor energetics of the model are caused by its intrinsic inability to simulate the conversion of eddy-potential energy to eddy-kinetic energy [C(A,,&jl as well as to model an effective reverse energy cascade

IW&,ll. This is a major problem, and though its effects can be mitigated by carefully tayloring the energy balance between the (pseudo-j baroclinic forcing term and the friction dissipation, poses an important question about the absolute validity of a “modified barotropic model” like Tennekes’ for the simulation of synoptic flows. In the light of the present results, as well as those of similar numerical experiments (Donaldson, Sullivan & Rosenbaum, 1972; Nielsen, Chen & Trebbia, 19811, one would conclude that the best application of Tennekes’ type models is as horizontal flow models in a multi-level simulation which includes at least a simple vertical hydrostatic equation. The possibility of developing a simple “three-layer” code is also being presently investigated by the Authors.

The temperature variance is a measure of the available eddy potential energy in the atmosphere. The general trend is an overprediction in the tropics and a slight underprediction in the upper latitudes. The results are in good agreement with observed data. For increased turbulent Reynolds number there is an increase In temperature variance indicating an increase the amount of eddy available potential energy in the atmosphere.


The transport of sensible heat follows the general trends of the atmospheric data. The results are overpredicted throughout the simulation indicating the conversion process from eddy available potential energy to eddy kinetic energy is over-estimated. However, this is expected because the conversion from eddy kinetic energy to zonal kinetic energy is minimal. Therefore an increase in transport of sensible heat is required in order to maintain the overall energy balance.

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J.G. Charney, K. Fjijrtijft and J. Van Neumann, Numerical Integration of the Barotropic Vorticity Equatio:~, Tellus, vol. 2, #4. C. dup. Donaldson, Calculation of Turbulent Shear Flows for Atmospheric and Vortex Moti,ons, AIAA .I., Vol. 10, al [19?2]. C. dup. Donaldson, R.D. Sullivan and H. Rosenbaum, A Theoretical Study of the Generation of Atmospheric Clear-Air Turbulence, AIAA J., Vol. 10, a2 (19723. S.C. Gonzales, A Two-Dimensional Second Order Closure Model of t.he Upper Atmosphere, Ph.D. Thesi?, Rutgers University [1984]. Y. Kaneda and D.C. Leslie, Tests of Subgrid Models in the Near-Wall Region Using Represent.ed Velocity Fields, J. Fluid Mech., a132 (19831. A.D. Kowalski and R.L. Peskin, Numerical Simulation of Relative Dispersion in Two-Dimensional, Decaying Turbulence, J. Fluid Mech., alO9 (19811. W.S. Llewellen, M.E. Teske and C. dup. Donaldson, Variable Density Flows Computed by a Second-Order Closure Description of Turbulence, AIAA J., Vol. 14, #5 (19761. G.L. Mellor and J.H. Herring, A Survey of the Mean Turbulent Field Closure Models, AIAA J., Vol. 14, ~5 [1976]. J.E. Nielsen, T.C. Chen and J.J. Tribbia, A Numerical Simulation of the Atmospheric General Circulation with a Barotropic Model, NSF Report for ATM-7951800 (19811. A.H. Oort and E.M. Rasmussen, Atmospheric Circulation Statistics, NOAA Professional Paper #5 (19711. S.A. Orszag, Design of Large Hydrodynamic Codes, Proc. 3rd ICASE Conf. on Scientific Computing (19761. S.A. Orszag, Spectral Methods for Problems in Complex Geometries, Proc. 3rd IMACS Int. Symp. on Advances in Computer Methods for P.D.E. (19791. R.L. Peskin and E. Sciubba, A Turbulent Model for Synoptic Mid-Latitude Flows in the Upper Atmosphere, Proc. 5th Int. IMACS Symp. on Advances in Computer Methods for P.D.E (19811. R.L. Peskin, A.D. Kowalski and S.C. Gonzales, Two-Dimensional Simulation of Large Scale Atmospheric Turbulence and Diffusion, 37th APS Conf. (19841. W.C. Reynolds, Computation of Turbulent Flows, Adv. Geophys. 18A (1976). E. Sciubba, A Two-Dimensional Model for Turbulent Flow in the Upper Atmosphere, Ph.D. Thesis, Rutgers IJniversi ty (1982). C.S. Speziale, Closure Models for Rotating Two-Dimensional Turbulence, Geoph. Astroph. Fl. Dyn., Vol. 23 [1983]. H. Tennekes, The General Circulation of Two-Dimensional Turbulent Flow on a B-plane, JAS, Vol. 34 (19771.

E. Sciubba et al. / A 2D Znd-order model for turbulent flow in the upper atmosphere 257

LIST OF SYMBOLS

f

hp

*

i,j,k

KS 1.

P cl” R Re

t 1 11 , v

x,Y

a,

P

1,

x2

1,

Coriolis parameter Effective acceleration Scale height of the atmosphere Coordinate indeces Surface energy exchange coefficient Zonal channel width Pressure Turbulent kinetic energy Diabatic heating term Reynolds number Time Temperature Velocity components Cartesian coordinate on the g-plane

Baroclinic amplification factor

“beta”-plane coefficient Turbulent velocity length scale

Turbulent temperature length scale

Dissipation length scale

Suffixes and Other

s-1

m s2 m

m s2 m N mm2 m s2

OK s-’

s

*lx m s-l m

s-1

m-1 s-l

m

m

m

-. Mean

‘”

i,j,k

0

lnstantaneous component Fluct.uating component Vector components Standard, average

r---- --- -.__- __-l_---___ -.-__

_-Table__

A -2 z 0.25 L

x3 L = 0.50

I El’ 0 H = 0.01 1__.___-_~ _-:__” “-..--_-----_-_J_-__._--.__

A 2 I. = 0.1

H i = 0.01


FIGURE 1, THE PHYSICAL DOMAIN

DEGREES LRTITUOE

FIGURE 2, THE COMPUTATIONAL DOMAIN

FIGURE 3. tlELH VELOCITY PROFILE FOR 400 IWD Se8 118 PRESSURE LEVEL (OORT AHD RASHUSSEH, 1971)._

275.0 -

. . *. .N

2 --._

% 225.0 ,....I...*,

0.0 25.0 50.0 75.0

DEGREES LATITUDE

FIGURE 4. MERH TEMPERATURE PROFILE FOR 499 MD SD9 118 PRESSURE LEVEL (OORT RHO RRSWSSEHI 1971).

E. Sciubba et al. / A 20 Znd-order model for turbulent jlow in the upper atmosphere 259

25.0

? f

i.3 20.0

85 5 15.0 u I-

2 10.0

ii k 5.0 is

0.0 0.0 25.0 50.0 75.0

DEGREES LRTITUOE

FIGURE 5. TRMSIEHT EDDY KlHEfIC ENERGY FOR THE 408 IHD 588 tlB PRESSURE LEVELS.

$ 200.0 I: ac

2 100.0

3 s d 50.0

6 z Liz 0.0 ki

-f$ $ DATA m--w

0.0 25.0 50.0 75.0

DEGREES LRTITUOE

g 200.0 1 x

8 g 150.0

g >

f 100.0 l-

5 H

50.0

0.0 0

FIGURE 6.

20.0

.O 25.0 50.0 75.0

DEGREES LRTITUOE

ZOIML HDHEI~TUtl WRIAtdCE FOR THE 4BB AIdD SOB iI8 PRESSURE LEUELS.

: -*‘Kof% - ----5oot% kSENl bUATIoN

DEGREES LRTITUOE

FIGURE 7. MERIDIOHLL t9OllEl~TUH ‘JARIdtKE FOR THE 488 AND FIGURE 8. EDDY l4OHEHlUl4 FLUX FOR 588 HB PRESSURE LEUELS. PRESSURE LEVELS.

THE 409 AHD 580 HB


kSENI bUATlCt_ _ _ _ _

t- 0.0 La ” ” ” a B ’ * ” ” 0.0 25.0 50.0 75.0

DEGREES LFITITUDE

FlGUkE ?. TEtlPERIIlIJRE ULRIRNCE FDR THE 488 RHO 588 )(B PRESSURE LEVELS.

25.0 ?

2 20.0 INCREI’ITNTS OF-X 1O-5 l/S

5 ii 15.0

is 3 10.0 -1

g 5.0

ii

g 0.0 0 0 I:

-5.0 DEGREES LATITUOE

FIGURE 11. )IER1DlOlML EDDY FLUX FOR UARYIMC AHPLIfICATION Df BhROCLIWlC YAUEB.

25.0

c 5 20.0

& z 15.0 r

2 10.0 d s ii 5.0

iz r

0.0 0

-5.0 t DEGREES LATITUDE

FIGURE lo. MRIDIOHIL TRAHSPORT FOR THE 406 RHO 580 18 PRESSURE LEVELS.

FIGURE 12. bIMCNSIONLE.SS REVERSE ENERGY CASCADE

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A two-dimensional, second-order model for turbulent flow in the upper atmosphere