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Journal of Marine Research, 61, 703–706, 2003 Reply to: “Comments on ‘A generic length-scale equation for geophysical turbulence models’ ” by L. Kantha and S. Carniel by Lars Umlauf1 ,2 and Hans Burchard3 1. Introduction Recently, Kantha and Carniel (2003) commented on some earlier work of Kantha (2003) and Umlauf and Burchard (2003) on the formulation of a generalized length-scale equation in Reynolds stress models for geophysical ows. With the present short note, we respond to their major criticism of the Umlauf and Burchard (2003) generic two-equation model which uses transport equations for the turbulent kinetic energy k and a second variable of the form k ml n with real exponents m and n, where l is the integral length scale of turbulence. While the major criticism of this generic model has been that the variable c } k ml n may be physically not sound in the general case, we demonstrate here that the transport equation for c for any real m and n can be transformed into a well-known equation for the dissipation rate e with some extra terms appearing, but with no non-integer powers of physical quantities. 2. Transformations and diffusion terms It is easy to show that for homogeneous ows, all models discussed within the present context are mathematically identical, apart from some differences in the empirical parameters (see also Baumert and Peters, 2000). Therefore, a discussion about the differences of two-equation turbulence models is only helpful if the role of the vertical turbulent transport is considered. The traditional model for turbulent transport of k and the length scale determining variable is down-gradient diffusion. Even though the limits of this approximation have been amply demonstrated, down-gradient diffusion is implemented in virtually all two- 1. Laboratoire d’hydraulique environnementale (LHE), Faculte ENAC, Ecole Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland. 2. Present address: Physical Oceanography and Instrumentation, Baltic Sea Research Institute Warnemuende, P.O. Box 301161, D-18112 Rostock-Warnemuende, Germany. email: [email protected] 3. Physical Oceanography and Instrumentation, Baltic Sea Research Institute Warnemuende, P.O. Box 301161, D-18112 Rostock-Warnemuende, Germany. 703 Journal of Marine Research 704 [61, 5 equation models, mainly because it yields useful results in the logarithmic region near walls and near the edge of free shear- ows. Though simple, down-gradient diffusion of turbulence is a model with a clear physical interpretation. Much less clear, however, is the question for which length scale determining quantity down-gradient diffusion should be postulated: is it more reasonable to claim pure down-gradient diffusion for e, for v, for kl, or for any other generalized length scale? Physically, it is dif cult to argue in favor of pure down-gradient diffusion for any particular variable, so that it seems more reasonable to evaluate this question in terms of the mathematical and numerical properties of the resulting models. The goal of Umlauf and Burchard (2003) was to formulate a model for a quantity c } k ml n that exhibits controllable properties and is mathematically and numerically optimal under certain constraints essential for turbulent oceanic ows. In general, these constraints can lead to exponents m and n being non-integers, a fact that has been criticized by Kantha and Carniel (2003). They argued that most non-integer values of the exponents are unphysical and suggested instead in their equation (11) the addition of an extra pseudodiffusion term to the generic equation of Umlauf and Burchard (2003). We want to point out the following regarding their line of argumentation: First, for every real m and n, an exact transport equation for c can be derived from the Navier-Stokes equations. Thus, c is always a valid physical (i.e. measurable) quantity. Second, the additional pseudo-diffusion term suggested in (11) of Kantha and Carniel (2003) is completely ad-hoc. No physical argument leads to its form, and no physical interpretation can be attributed to it. Third, there is a clear interrelation between the values of the exponents m and n and the appearance of extra diffusion terms in transformed length-scale equations: any equation for c with pure down-gradient diffusion can be transformed into an equation for a physically more intuitive quantity (like e, v, kt, or any other) with some extra transformation terms showing up. These terms are similar to the ad-hoc term suggested by Kantha and Carniel (2003). If, however, the transport equation for c with arbitrary m and n can always be mapped into a transport equation for a physically more intuitive quantity, the discussion about the physical permissibility of real m and n is rather academic. As an example, we demonstrate this last point for the transformation of the c-equation (1) of Kantha and Carniel (2003) into the e-equation. Using the cascading relation e } k 3/ 2l 2 1 the generic equation of Umlauf and Burchard (2003) can be transformed, similarly to the way outlined in Kantha and Carniel (2003), into an equation for the dissipation rate. From Eq. (1) and (2) of Kantha and Carniel (2003) it follows then S D eÇ m 3 $k 1 $ c 1 5 1 2 1 ~c e1P 1 c e3G 2 c e2e!, e n n c k 2 k (1) where the model coef cients obey the relationships c ei 5 S D m 2 c ci 3 1 , n 2 i 5 1, 2, 3. (2) Umlauf & Burchard: Reply to Kantha & Carniel 2003] 705 The sum of the rst two terms on the right-hand side of (1) becomes S D S D S 9 1 m 3 m 3 $ k 1 $c 1 nt 1 2 5 e9 1 1 n n c e sc k n 2 k 2 2 S DS DS S nt D D 9 1 1 2 k9 sk sc D S D m 3 nt k92 nt e92 nt k9 e9 3 3 1 m1 n21 n , 2 2 ~n 1 1! 2 12 m1 n 2 sc k sc e sc k e 2 2 (3) with primes denoting derivation with respect to the vertical coordinate z. The rst term on the right-hand side of this equation is the traditional diffusion term of the e-equation. Due to the transformation it is now complemented by a number of new terms, of which only the second term on the right-hand side also has the character of a diffusion term. All other terms must be considered as additional production/destruction terms. Note that for c } e q, as suggested by Kantha and Carniel (2003), one has c } eq 3 n 5 2q, 3 m 5 q. 2 (4) With these values for m and n, (1) and (3) reduce to Eq. (9) of Kantha and Carniel (2003). From (1) and (3) the following points are evident. First, in homogeneous ows, all models are identical to the k-e model if parameters are converted according to (2). Second, for any real values of m and n, the equation for c can be transformed into an equation for the more intuitive quantity e. The role of m and n in the original equation is to create additional diffusion and production/destruction terms (similar to the term postulated by Kantha and Carniel, 2003) in the e-equation. In light of this, the distinction of Kantha and Carniel (2003) between physically sound and not sound models is completely arbitrary. It is interesting to note that Kantha (2003) did the inverse procedure. He transformed the standard k-e model of Rodi (1987) into an equation for c. As a consequence, for every c-equation derived in this manner, there exists a structurally much simpler e-equation which computes identical results. It is not evident why modelers should code a numerically much more demanding transport equation for c, when they can obtain the same result with the standard e-equation. Kantha (2003) also changed the parameters of this k-e model from the original values suggested by Rodi (1987), a procedure which rst requires testing against standard ow situations. Yet another approach has been suggested recently in the note by Kantha and Carniel (2003). They added to the c-equation suggested by Umlauf and Burchard (2003) ad-hoc an extra term and observed that this new equation can be transformed into an equation for c11 j containing only the classical down-gradient diffusion term. Then, however, one can always set cUB 5 c11 j, where cUB is the generic variable suggested by Umlauf and Burchard (2003), and the equation of Kantha and Carniel (2003) is seen to be a special case of the generic equation of Umlauf and Burchard (2003). In this context, we want to point out that the method of Kantha and Carniel (2003), even though it satis es the constraints imposed by Umlauf and Burchard (2003), it leads to different values for the model 706 Journal of Marine Research [61, 5 parameters. It has been, however, clearly demonstrated by the latter authors that any parameter set differing from their suggestion will fail in the self-similar mixing layer, an important ow in ocean modeling. 3. Conclusions Kantha (2003) has shown how the k-e model including vertical turbulent transport of k and e may be identically transformed into a generic model formulation similar to the model recently suggested by Umlauf and Burchard (2003), but with some additional forcing terms. This gives an interesting perspective on the k-e model, but might have little practical relevance, since the numerical calculation of the additional terms would probably consume a considerable amount of computational time without resulting in higher accuracy. In this note, we have gone the opposite way: we have identically transformed the generic two-equation model back to the k-e model, likewise containing some additionalproduction/ destruction terms. Also, this approach might be of little practical relevance, but it clearly demonstrates that the generic model with arbitrary real exponents, often criticized as unphysical, is identical to a well-accepted model type without odd powers of physical quantities. REFERENCES Baumert, H. and H. Peters. 2000. Second-moment closures and length scales for weakly strati ed turbulent shear ows. J. Geophys. Res., 105, 6453– 6468. Kantha, L. H. 2003. The length scale equation in turbulence models. Nonlinear Proc. in Geophys. (in press). Kantha, L. H. and S. Carniel. 2003. Comments on “A generic length-scale equation for geophysical turbulence models” by L. Umlauf and H. Burchard. J. Mar. Res., 61, 693–702. Rodi, W. 1987. Examples of calculation methods for ow and mixing in strati ed ows. J. Geophys. Res., 92, 5305–5328. Umlauf, L. and H. Burchard. 2003. A generic length-scale equation for geophysical turbulence models. J. Mar. Res., 61, 235–265. Received: 8 September, 2003.