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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.
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Journal of Marine Research
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[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 difŽ 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 coefŽ 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 Ž 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 Ž 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 Ž 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 satisŽ es the constraints
imposed by Umlauf and Burchard (2003), it leads to different values for the model
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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 stratiŽ 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 stratiŽ 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.