Reply to comments by K. Shafer Smith

Reply to comments by K. Shafer Smith Reply to Comments by K. Shafer Smith By 1Ka Kit Tung Department of Applied Mathematics University of Washington, Seattle, Washington 1 Corresponding author address: K. K. Tung, Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195-2420. E-mail: tung@amath.washington.edu 1 1. Introduction We welcome the opportunity to offer further explanations of our model published previously in this journal (Tung and Orlando, 2003; hereafter T&O). We thank Dr. Smith (Smith (2003); hereafter S) for providing numerical illustration of some subtle points concerning the effect of small-scale dissipation on the shape of the energy spectrum in two-dimensional turbulence. His analytic derivation of the dissipation scales in the presence of double cascades of energy and enstrophy is, however, problematic. In any case his analytic prediction of the transition scale in two-dimensional turbulence does not apply to the case of two-level model considered by T&O. Nevertheless, some issues raised are common to both 2D and quasi-geostrophic (QG) turbulence, and these will be discussed first here. As pointed out recently by Tran and Bowman (2003), and also by Tran and Shepherd (2002), the classical configuration of a dual cascade of Kraichnan (1967), Leith (1968) and Batchelor (1969), with a –5/3 spectral slope in the energy spectrum upscale of injection and –3 slope downscale of injection, is an idealization that is probably not achievable in any finite-domain numerical simulation or in nature. The KLB theory envisages an infinite domain in which a pure inverse energy cascade exists on the long-wave side of the scale of injection, transferring all of the injected energy to larger and larger scales without being damped. On the short-wave side of injection, there exists a pure direct cascade of enstrophy, transferring all of the injected energy downscale, dissipated by molecular viscosity even in the limit of vanishingly small viscosity coefficient. In finite domains of numerical simulations discussed here and in the 2 atmosphere, Ekman damping provides the physical infra-red sink of energy. This sink is ―imperfect‖ in the sense that not all of the injected energy can be absorbed at the large- scale end of the spectrum, and part of the injected energy is diverted downscale. In the presence of finite viscosity there is a downscale flux of energy on the short-wave side of injection, whose presence was demonstrated in the numerical simulation of T&O and in atmospheric observations. Gkioulekas and Tung (2004a) show that in general the dual cascade should be a dual mixed cascade: In a finite domain, there should be a double downscale cascade of both energy and enstrophy on the short-wave side of injection. In the limit of large separation of dissipation scales and the scale of injection, the downscale enstrophy flux is the leading cascade and the downscale energy flux is the subleading cascade. In this limit the contribution of the subleading cascade will be hidden, consistent with the prediction of S. In the atmosphere, there is less than one decade of separation between the energy injection scale and the largest scale permitted by the domain geometry. Furthermore the Ekman damping is not concentrated on the largest scales only. This physical situation leads to a configuration that is very different from the ideal dual pure cascades in the KLB theory. Nevertheless, diagnostics of atmospheric data (see Boer and Shepherd, 1993; Straus and Ditlevsen, 1999) show that most of the injected enstrophy still goes downscale and most of the injected energy goes upscale. There are, in these observational analyses, also a small fraction of enstrophy that goes upscale and a small fraction of energy that is diverted downscale. The numerical 2simulation of T&O reproduces this observed energy and vorticity budget quite well. On 2 For some reason, in the numerical model of Smith (2003), although a large fraction of injected energy goes upscale, only about half of the injected enstrophy goes downscale. In particular, of the total 16 units of enstrophy injected, 8-9 units go downscale and 3 units go upscale (see S’s Figure 1a). At least 4 units of injected enstrophy are unaccounted for. They are probably dissipated at the forcing scale. 3 the short-wave side of injection, the direct enstrophy cascade is the leading cascade, and the direct energy cascade is the subleading cascade. The presence of the subleading energy cascade can be detected from the energy flux, which would be positive. This is seen in both T&O and in S. Whether the subleading cascade will manifest itself in the energy spectrum in the inertial range depends on the treatment of the small-scale dissipation, which is the subject of the exchange here. The numerical calculations of S actually provide support to some major points of our theory: We are glad to see that even in his two-dimensional model, which is quite different from that of T&O, the energy flux is downscale on the short-wave side of injection, as we predicted for either QG (Quasi-Geostrophic) or 2D (two-dimensional) turbulence in the presence of a small-scale sink of energy. This is in contrast to the prediction of the traditional KLB theory. When a downscale enstrophy flux , and a 3downscale energy flux , are simultaneously present in the same inertial range on the short-wave side of injection, T&O found the energy spectrum E(k) to possess a –3 spectral slope transitioning to a –5/3 slope, with the transition wavenumber given 4approximately by 1/2 k ~ (,,,，.(1) t 3 The notation may be confusing: , is actually the portion of the total energy injection rate that is diverted downscale. It is the same as , in other publications. uv4 Eq. (1) is an empirical result, inferred from the numerical output of T&O (their Figures 4 and 7). 4 , : by controlling the There are at least two ways for a numerical model to determine total energy dissipation rate, as T&O has done, or to hold the hyperviscosity coefficient , constant, and let the model determine the total dissipation rate. In the second approach ,, and hence k , then vary as resolution changes. S gives a more detailed discussion of this t second approach and its sensitivity to resolution. This topic was not explicitly discussed in T&O, and we welcome the Comment by S. The discussion in S is useful in reconciling the different numerical results obtained previously, mainly along the second approach. An important point we wish to emphasize to the readers is that the second approach was not the one adopted by T&O. T&O were trying to model the situation in the upper troposphere where there is a finite sink of energy at the small-scale end of the spectrum, and there are some recent measurements of the magnitude of such a sink. In the Abstract and in the text, T&O made it clear that the model proposed has two important ingredients: (i) a means for energy and enstrophy injection at the intermediate (synoptic) scales, which is accomplished with the use of a two-level QG model, and (ii) ―a small sink of energy at the small scales due to subgrid hyperdiffusion; this attempts to model the small-scale sink not resolved by the two-level QG model‖. Unlike the surface-QG model, or a more general QG model with evolving temperature on the top and bottom boundaries, a two-level model does not by itself provide the sink in (ii) as the viscous coefficient becomes vanishingly small. This absence of ―anomalous energy dissipation‖ is a property it shares with 2D turbulence in the KLB limit (see discussions in Tung and Orlando, 2003b), and has to be remedied with a finite subgrid parameterization. As is the case with most 5 parameterization of subgrid processes in atmospheric models, the hyperdiffusion used in T&O is resolution-dependent; its coefficient needs to be adjusted as resolution is increased so as to maintain a finite sink of energy, as discussed in T&O. S is correct in pointing this out, although this property has already been emphasized in T&O. It is obvious that after we introduced such a sink into the model it would not make much sense to then choose the model parameters and resolution so that the model approaches the KLB limit of vanishing small-scale dissipation rate. The numerical experiments shown in S are all of the type where the magnitude of the sink of energy (the total energy dissipation rate) is not controlled to model this observed atmospheric feature. Consequently its magnitudes vary wildly from case to case. In some of his cases, the energy sink is too weak relative to its enstrophy sink, when compared to that observed in the atmosphere, for the transition wavenumber to occur in the resolved 5 The magnitude of the sink in the model determines the range in his numerical model. magnitude of the energy flux—in fact the ensemble average of one is equal to that of the other at statistical equilibrium----as the energy transferred downscale by nonlinear wave-wave interaction is dissipated by the sink. [There are exceptions to this, of course, when dissipation occurs at the forcing scale. See Tran and Shepherd (2002).] Using model data provided to us by Smith (personal communication), the downscale energy flux in his 5 In the classical KLB theory, as the energy dissipation rate is taken asymptotically to zero, , approaches zero, and k approaches infinity. Only the –3 slope is then seen in the energy spectrum, as the transition to tthe –5/3 slope now disappears. Of course in numerical models such as that in S one cannot have infinite wavenumbers. The next best thing is then to have k occurring beyond the inertial range, in the dissipation t range. This is a requirement that a finite numerical model must obey in order for it to simulate the infinite-domain, infinite-resolution, nearly-inviscid result of the classical KLB theory. Only then would the -simulated spectrum be independent of the energy dissipation rate in the numerical model and achieve the k 3 shape predicted by the KLB theory for vanishing energy dissipation rate, and remain so as resolution is further increased. But this is not T&O’s model for the atmosphere. 6 ,,, , ,,, x,, in his nondimensional unit, and is equal to the total energy model Case C is ,,dissipation rate of ,,, x,, for this case, as expected. The enstrophy flux for Case C is ,, ？ , ，It does not change much for all three runs, also to be expected.) These numbers imply a transition wavenumber of 1/2 1/22 k ~ (,,,，~ (9/0.7) 10~400, t which is beyond the inertial range in his model. It is instead in the dissipation range, where E(k) does not possess a scaling law. That is why the transition scale disappears in Case C despite the fact that there is both a positive , and a positive , . Although Case C has the same dissipation coefficient , as Case B, the energy dissipation rate is different because the integral of the energy spectrum E(k) in the dissipation range, which yields the rate of energy dissipation, is different. It is thus not surprising that those features which are sensitive to the dissipation rate, such as k , are different in the two cases. t Although S does not have a still higher resolution run to show that the results of Case C will not be further changed with higher resolution, it is reasonable to accept his assumption that this is true. The energy spectrum has already dropped to zero exponentially at the small scales at the resolution of Case C that resolving even smaller scales will not make much of a difference to the energy spectrum at larger scales. This is probably the physical interpretation of S’s ―hyperviscous Kolmogorov (HVK) scale‖: When the resolution exceeds that implied by this scale, the energy dissipation rate will be independent of resolution even as the hyperviscosity coefficient is held fixed. In 7 that limit however, it often turns out that the resulting dissipation rate is too small as compared to the atmospheric value, although there may conceivably be a way to find a remedy to this problem. That small limit, however, is useful if one’s goal is to simulate the KLB dual cascade as closely as possible in a finite domain (see footnote 5). 2. Remarks on the analytic formulae The main result, derived in Eq. (2.7) of S, is a statement that the transition from the –3 to the –5/3 range should occur where the dissipation range starts if the Kolmogorov scale is ―resolved‖. There are inconsistencies in the derivation leading to that formula, and the derived formula is not useful as a prediction of the transition scale, except in the KLB limit. Section 2 of S gives a derivation of an approximate analytic formula for a Kolmogorov type ―dissipation scale‖, modified for hyperviscosity (called HVK scale). For energy dissipation, k is defined to be the wavenumber where the downscale energy flux is (1- ,) ,, ~90% of its value , in the inertial subrange, based on the following integral (see Eq.(2.1) in S): kv,skE(k)dk,, , ,, , ，,， ,kf Similarly, an enstrophy HVK wavenumber, k, is defined from: ,, k,,，2skE(k)dk,, , ,, , ，,， ,kf 8 To evaluate the above integrals, a form for the energy spectrum E(k) needs to be assumed. S appears to have used 2/3 -5/3, k , C=6, (4) E(k) ~ C11 in the first integral, but 2/3 –3E(k) ~ C, k , C=1.3, (5) 22 in the second integral, and assumed the two C’s to be the same. This procedure is inconsistent since in the definition of the dissipation scales, (2) and (3), E(k) should be the same. Although S is following the procedure introduced by Frisch (1995), this problem does not arise in the case of 3D turbulence considered by Frisch. In 3D turbulence there is only a downscale energy flux, and (4) would be the appropriate energy spectrum to use in (2). In this way S derives an energy dissipation wavenumber, k his ～,, Eq. (2.2), and an enstrophy dissipation number, k his Eq. (2.3): ,,～ 31/(3s-2)3/(3s-2)k = a (,,, ) , where a = (,(3s-2)/(6C)) , (6) ,,1 31/3s1/sk = b (,,, ) , where b = (s,/(2C)) . (7) ,,2 If one tries to be more consistent and uses the same E(k) in the integrals, the energy dissipation scale would become dependent on the enstrophy flux as well as on the energy flux. There does not appear to be a way to disentangle the two scales if one uses S’s integral definition for the dissipation scales (2) and (3). In the following, we will proceed with the rest of S’s argument using (6) and (7). Combining (6) and (7), we get: 2,,, , , k, (8) ,, 333swhere , , ，，s – 2/3)/s)(C/C) (k , k，. 21,,,, (9) 9 , by unity to arrive at his conclusion that the S approximates the proportionality constant transition wavenumber (1) is the same as the Kolmogorov dissipation wavenumber: 22 k= ,,, , k (10) t,, We see from (9) that this parameter depends on the order of hyperdiffusion sensitively for low s, and for high s, it depends critically on how close to unity the ratio of the two dissipation numbers is. Furthermore the ratio of the two universal constants in , is 3(C/C) = 0.010, which S set to one because it was raised to a very small power in a and 21 b. Note that however, in arriving at (8), a and b are themselves raised to a very high power. For s=18: For molecular viscosity s=2, and , ,,,,,,, , 542 , , ,,,,,？ (k , k， k, for s=18 . ,,,,,, S hypothesizes that the two dissipation scales are exactly the same, claiming that it is ―qualitatively obvious‖. While qualitatively it may be true, it is the exact quantitative values that are important, because the ratio is raised to a very high power, 54. The reason S gives for assuming that the two dissipative scales are the same is based on a local 2argument: the wavenumber for which E(k) or kE(k) begins to feel the effect of viscosity should be the same. While this is true, the dissipation wavenumbers are instead defined by S by the integrals (2) and (3). There is no indication that these two scales, namely (6) and (7), defined this way, should be identical: As we mentioned in the Introduction, the , in Eq. (6) is the amount of energy flux diverted downscale from the imperfect infra-red sink and consequently it depends critically on the nature of the infra-red energy sink and the separation of that sink from the injection scale. On the other hand, the downscale 10 ,, being the dominant (leading) cascade, is much less sensitive to the enstrophy flux 6conditions on the infrared end (Danilov, 2004). Since the enstrophy dissipation wavenumber is independent of , ～ while that for energy is dependent on , , there is no a priori reason to expect that the two dissipation wavenumbers should be the same regardless of the condition in the infrared end. Let us for a moment suppose that they happen to be approximately the same but not identical. If (k , k， , ,,？？～ then (k , ,,,,,, 54k ，=0.58 ,, and (8) becomes 2,,, , 0.0052 k (11) ,, Using the above formula instead of (10) would have led to a completely different conclusion concerning the nature of the transition wavenumber in (1). Gkioulekas and Tung (2004a) provided a different theoretical framework, which allowed them to bypass the difficulties of S’s derivation mentioned above. In Gkioulekas and result, that the transition scale Tung (2004b) they managed to show that S’s main coincides with the dissipation scale, is correct in the KLB limit of 2D turbulence. 3. Resolving the Kolmogorov scale What is the meaning of ―resolving‖ the Kolmogorov scale? It does not mean that the dissipation scale is not resolved if the model’s truncation scale is larger than this Kolmogorov scale. S acknowledges this point, and so there is no disagreement here. Whether or not the energy dissipation scale in the model is actually resolved can be 6 This may not be true in the model of S. See footnote 2. 11 revealed more accurately with a model diagnostic of its energetics. For the results shown in T&O, several diagnostics are used to ensure that the small-scale dissipation sink is resolved by the resolution adopted. Figure 5 of T&O shows the rate of energy dissipation by the hyperviscosity for each wavenumber in the 129-km resolution run. It is clear from this figure that for wavenumbers higher than 180, the subgrid diffusion is the dominant term---with a very rapid dissipation rate of ~ 1/(0.6 days)--- and it balances the energy transferred downscale by the wave-wave interaction. Therefore the energy dissipation was adequately resolved in T&O to their satisfaction. (The reduced black and white figure in the printed version of the journal is admittedly difficult to read. The curve we are referring to is the light dotted one which veers downward from the horizontal axis near wavenumber 180.) S asserts, without giving any justification, that ― if this HVK scale is not resolved by the model, i.e., if the hyperviscous coefficient is too small for the given enstrophy flux, then enstrophy will build up at the small (sic) wavenumber end of the spectrum. While any finite hyperviscosity will ultimately dissipate the enstrophy, the small (sic) wavnumber spectrum will be altered by the constipation of enstrophy if the HVK scale is not resolved‖. It is not clear what reference state S is referring to: ―Constipated‖ as compared to what? ―The spectrum altered‖ from what? In view of what is said in the Introduction and in footnote 5 it is reasonable to suggest that the reference state S has in mind, though never acknowledged in his Comment, is the KLB limit. 12 A further confusion arises when one examines more carefully the truncation scale adopted by T&O. One finds, surprisingly, that the Kolmogorov dissipation scale is ―resolved‖ in that model, though barely so. In his estimates, S uses order-of-magnitude figures in a formula with various inaccuracies (e.g. setting a=1 and b=1 in (6) and (7)), while comparisons are made with the truncation wavenumbers at the second significant figure. The case picked by S happens to have the least meridional resolution (with 1/8 the number as in the zonal direction); all other cases shown in T&O’s Figures 3 and 4 have higher meridional resolution (with ? the number of wavenumbers in the zonal direction). Yet they all exhibit the same behavior of spectral shape, showing insensitivity to meridional resolution as long as the number of wavenumbers used in the meridional direction equals or exceeds 1/8 of that in the zonal direction (1/7.7 is the ratio of the meridional width to the zonal length of the channel in T&O). For this case, S concludes that the zonal direction is ―resolved‖ while the meridional direction is not. In view of the fact that the adopted resolution, in terms of grid distance, is about the same in either direction, it is more reasonable to conclude that either both are not resolved or both are resolved. Our re-calculation shows that the latter is the case. Then why does the transition scale for this case in T&O not obey Eq.(10) predicted by S? Why does the spectrum found not the same as in the KLB limit? These confusing paradoxes are probably a result of the inaccurate prediction of Eq. (10)—it could just as well be Eq. (11). An interesting question arises. If the case of T&O picked by S to examine satisfies S’s criterion, though barely, of having the Kolmogorov dissipation scale resolved, does it 13 mean that its transition scale will become independent of resolution for higher resolution? There are other higher resolution runs presented in T&O. The case highlighted in their Figure 3 has more than double the meridional resolution and about 30% higher zonal resolution. The transition scale found is moved only very slightly to higher wavenumber. However in that case the shift was attributed by T&O to the parameterization scheme, which lowers the hyperviscosity coefficient slightly as resolution increases. It appears that if that coefficient were held fixed as resolution was increased, the transition scale would not have changed. Although this fact cannot be ascertained without further numerical experiment, it does suggest that a useful application of the concept of the Kolmogorov dissipation scale is in predicting resolution independence for fixed hyperviscosity coefficient. The secondary prediction of the location of the transition scale with respect to the dissipation scale is not useful in its present form. 4. Remarks on modeling philosophy In classical fluid mechanics, the viscous term is due to the physical process of molecular diffusion, and the governing partial differential equation, the Navier-Stokes equation, describes the physical situation down to all macro-scales. Since the viscous coefficient is a physical quantity in the Navier-Stokes equation, it makes physical sense to hold it constant as one increases resolution. Two-dimensional turbulence simulation with the second-order viscous diffusion has been problematic however, due to the demanding requirement on numerical resolution. A recent successful simulation by Lindborg and Avelius (2000) of the KLB limit uses high-order hyperdiffusion, as S is doing in his note. Hyperdiffusion does not have the same physical basis as molecular diffusion. Whether or 14 not this numerical device is ―physical‖ depends on what physical process we want it to model. Holding its (artificial) coefficient constant as resolution increases is not necessarily the physical thing to do. Let us reiterate what T&O are trying to model physically: (i) There is evidence that the energy flux over the mesoscales is downscale. Cho et al -62-3, ~+10 ms, for turbulence in the free (2003) give a magnitude of the order of troposphere. At statistical equilibrium this downscale energy flux must be dissipated by a -62-3small-scale sink of energy, with the rate of dissipation , =,~10 ms. This rate is very D small (compared to the large-scale sink), but not negligible. (ii) The small-scale dissipation may occur at scales not correctly described by the QG theory, so the common mathematical approach of considering dissipation at the infinite wavenumber limit is irrelevant for our model. T&O lists a number of mechanisms, such as frontal genesis and gravity wave radiation, which are known to occur in the atmosphere as sinks for QG energy transferred to the small scales by nonlinear wave-wave interactions, and which are known to be not describable by the QG equations. The existence of a non-zero sink of energy at the small scales is modeled in T&O by a hyperdiffusion of order s, the order is chosen so as to leave as much of the resolved scale uncontaminated by this viscosity as feasible. The rate of the dissipation rate can be calculated diagnostically---that is, after the fact--- as: s+2 2s ,, ,,, ,k|,,，k)| dk=2,, ,k|,，k， dk D 15 The coefficient , is varied and the dissipation rate calculated. The dissipation rate is generally smaller the smaller the coefficient ,～ but larger , sometimes would actually yield a smaller ,by having the effect of driving E(k) down precipitously in the D dissipation range. Controlling ,is not an easy task because E(k) cannot be specified a D priori. T&O developed an algorithm for controlling the dissipation rate and obtained a ange of such values. The algorithm, which is explained in T&O, involves adopting a , r that adjusts with resolution such that the last resolved scale has a dissipation rate due to hyperdiffusion that is ten times that of the Ekman damping rate. This algorithm is not perfect in its ability to hold the total dissipation rate constant as resolution is changed, but because the it is able to hold it to the same order of magnitude, which is all that is needed observed rate is known in order-of-magnitude only. The range for the runs shown in Figure 3 in T&O is, in descending order of resolution, -62-3, ~ 0.5-1.1 x10 ms, D which is very close to the observed range of dissipation rates in the free troposphere. This then determines the downscale energy flux -62-3, ~0.5-1.1 x10 ms. There perhaps may exist a better numerical filter, which removes at the truncation scale a -62-3fixed amount of energy per second, say 0.5 x10 ms, independent of the particular truncation scale at which it is applied. Then the simulated energy spectrum would be 16 independent of resolution at scales larger than the truncation scale. We have not found such an ideal filter, but the device with hyperdiffusion comes quite close in achieving our modeling goal. 5. Conclusion. We thank Smith (2003) for his comments, which help contrast two different approaches taken in treating small-scale dissipation of energy in numerical models. As pointed out by him, the approach taken by Tung and Orlando (2003a) of controlling the rate of energy dissipation by ―subgrid‖ diffusion is unconventional in 2D turbulence studies, but is motivated by an observed atmospheric feature. In traditional 2D turbulence alculations, the rate of small-scale dissipation is not controlled but is allowed to vary as c resolution is changed. This gives the appearance of sensitivity of the model results to numerical resolution. There does not appear to be a right or wrong approach; which approach one should adopt depends on one’s modeling goal. To reproduce the result of T&O in S’s calculation, after one finds a correct small-scale dissipation rate at one resolution, one must adjust the magnitude of the coefficient of hyperviscosity to maintain that dissipation rate as resolution is increased. As pointed out by S, the model proposed by T&O is a finite resolution, finite dissipation model of atmospheric turbulence. Extending the QG model to arbitrarily small scales is probably nothing more than a mathematical exercise. We suggest that hyperviscous Kolmogorov scale defined by S is probably useful in predicting resolution-independence for fixed hyperviscosity coefficient. Although 17 holding the coefficient fixed is not the approach taken by T&O, it may be useful to those who do numerical simulation of 2D turbulence using the classical approach. The secondary prediction of S that the transition scale should be located at the dissipation scale when the Kolmogorv scale is resolved is found to be highly inaccurate---the range of uncertainty is so large that that it does not preclude the case of the transition scale located in the middle of the inertial range--- and consequently the formula is not useful. Acknowledgment The work is supported in part by National Science Foundation, Division of Atmospheric Sciences, under grant ATM 01-32727. References: Batchelor, G.K, 1969. Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids, II, 233-239. Boer, G.J., and T.G. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci., 40, 164-184. Cho, J. Y. N., R. E. Newell, B. E. Anderson, J. D. W. Barrick, and K. L. Thornhill, 2003. Characterizations of tropospheric turbulence and stability layers from aircraft observations, J. Geophys. Res., 108, D20, 8784. 18 Danilov, S, 2004. Nonuniversal features of forced 2D turbulence in energy and enstrophy ranges, Discrete and Continuous Dynamical Systems B, to appear. st Frisch, U., 1995. Turbulence: The legacy of A. N. Kolmogorov. 1ed. Cambridge Press, 296pp. Gkioulekas, E. and K. K. Tung, 2004a. On the double cascades of energy and enstrophy in two-dimensional turbulence, Part 1, Theoretical formulation, Discrete and Continuous Dynamic Systems B, in press. Gkioulekas, E. and K. K. Tung, 2004b: On the double cascades of energy and enstrophy in two-dimensional turbulence, Part 2., Approach to the KLB limit, and interpretation of experimental evidence, DCDS B, in press Kraichnan, R.H., 1967. Inertialranges in two-dimensional turbulence. Phys. Fluids,10, 1417-1423. Leith, C.E., 1968. Diffusion approximation in two-dimensional turbulence. Phys. Fluids, 11, 671-673. Lily, D., 1989. Two-dimensional turbulence generated by energy sources at two scales. J. Atmos. Sci., 46, 2026-2030. 19 Lindborg, E. and K. Avelius, 2000. The kinetic energy spectrum of the two-dimensional enstrophy turbulence cascade. Phys. Fluids, 12, 945-947. Nastrom, G.D. and K.S. Gage, 1985. A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950-960. –3-5/3 ―The k and k energy spectrum of atmospheric Smith, K.S., 2003. Comment on turbulence: Quasigeostrophic two-level model simulation‖. J. Atmos. Sci., this issue. Straus, D.M., and P. Ditlevsen, 1999. Two-dimensional turbulence properties of the 51A, 749-772. ECMWF reanalyses. Tellus, Tran, C.V. and T.G. Shepherd, 2002. Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence. Physica D, 165, 199-212. Tran, C.V. and J.C. Bowman, 2003. On the dual cascade in two-dimensional turbulence, Physica D, 176, 242-255. –3-5/3Tung, K.K. and W.W. Orlando, 2003a. The k and k energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation. J. Atmos. Sci. 60, 824-835. Tung, K.K. and W.W. Orlando, 2003b. On the difference between 2D and QG turbulence, Discrete and Continuous Dynamical Systems B, 3, 145-162. 20 21