Michael Frewer

No new scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence

Keywords: Statistical Physics, Turbulence, Symmetry Analysis, Lie-Groups, Scaling, Closure Problem

. No novel analytical achievements are made by Sadeghi & Oberlack, as the title of their study misleadingly wants to suggest. In fact, the already established self-similar scaling laws obtained by Bahri through simple dimensional analysis are already more general in the application than the ones obtained by Sadeghi & Oberlack through an overly complicated and therefore unnecessarily performed Lie-group symmetry analysis. The claim that it has the virtue of not being an ad-hoc method is not true. Because, instead of using an a priori set of scales as the classical method, the Lie-group method has to make use of an a priori set of symmetries, namely to select the correct relevant symmetries from an infinite and thus unclosed set. For example, a nonphysical scaling symmetry is selected which in the course of the analysis has to be discarded since it is not compatible with the data simulated. Hence, the Lie-group symmetry method in turbulence is just another common trial-and-error method and not a first-principle method that can bypass the closure problem.

Proof that the scaling laws in [1] are not new

To have an overview of what will be proven in this section, here a brief summary:

1. The first key result (4.24) derived in [1, Sadeghi & Oberlack (2020)] corresponds exactly to the already published result (4.20) in [2, Bahri (2016)], or alternatively to the result (2.14) in [3, Bahri et al. (2015)].

2. The second key result (4.25) in [1, Sadeghi & Oberlack (2020)] corresponds exactly to the result (5.10) in [2, Bahri (2016)]. Note that this result was not published in [3, Bahri et al. (2015)] and thus can only be found in [2, Bahri (2016)], but which is not mentioned or cited in [1, Sadeghi & Oberlack (2020)].

3. The third result (4.26) in [1, Sadeghi & Oberlack (2020)] also constitutes no new result as claimed, since it is just a direct consequence of Bahri's scaling laws when applied correctly to the underlying equations. The same is true of their results (5.7)-(5.10), which again are just a direct consequence of Bahri's results. In particular, the last result (5.10) was already established back in the 1990s by George & Speziale et al.

4.

While [1, Sadeghi & Oberlack (2020)] considers in their scaling only the case of zero thermal diffusivity (α = 0), Bahri et al. considered the case of non-zero thermal diffusivity (α = 0). Hence, one could assert that the scaling laws (4.24)-(4.26) derived in [1, Sadeghi & Oberlack (2020)] are indeed different and thus can be claimed as novel. But, as will be proven below, such an assertion is invalid. The self-similarity results of Bahri et al. include and clearly go beyond those of Sadeghi & Oberlack.

1.1. Proof that (4.24) and (4.25) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] are not new

The two key results, the temporal scaling law (4.24) and (4.25) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] θ 2 ∝ (t + t 0 ) 2n+m , u 2 θ ∝ (t + t 0 ) 2n+m-1 , (

where t 0 , n and m are arbitrary constants, correspond exactly to the already published scaling laws (2.14) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], and to (4.20) and (5.10) in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF], respectively

θ 2 = C 2 [ℓ θ ] 2C 1 α/A θ -1 , vθ = C 2 α β [ℓ θ ] 2C 1 α/A θ -3 , (1.2)
where the scalar (temperature) length scale ℓ θ is given by (2.10) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], or alternatively by (4.16) in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF], as

ℓ 2 θ = ℓ 2 0θ + A θ (t -t 0 ), (1.3)
where ℓ 0θ , A θ and t 0 are arbitrary constants. To note is the typo in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF], where the exponent for the scaling law (5.10) misses an additional -2, which can be easily seen when deriving (5.10) from the correctly printed relations (5.9) and (5.6).

Proof: By re-defining the constant virtual time origin t 0 in (1.3) as

A θ t ′ 0 := def ℓ 2 0θ -A θ t 0 , ( 1.4)
we can equivalently rewrite (1.3) as

ℓ 2 θ = A θ (t + t ′ 0 ), (1.5)
and therefore both scaling laws in (1.2) as

θ 2 = C 2 [ℓ 2 θ ] C 1 α/A θ -1/2 , vθ = C 2 α β [ℓ 2 θ ] C 1 α/A θ -1/2-1 = C 2 [A θ (t + t ′ 0 )] C 1 α/A θ -1/2 , = C 2 α β [A θ (t + t ′ 0 )] C 1 α/A θ -1/2-1 ∝ (t + t ′ 0 ) C 1 α/A θ -1/2 , ∝ (t + t ′ 0 ) C 1 α/A θ -1/2-1 . (1.6)
If we now re-define the constant exponent in terms of the parameters used in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] as

C 1 α/A θ -1/2 =: def 2n + m, (1.7)
then we get

θ 2 ∝ (t + t ′ 0 ) 2n+m , vθ ∝ (t + t ′ 0 ) 2n+m-1 , (1.8)
which are exactly the two "new" scaling laws (1.1) derived in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], where v = u 2 is the cross stream velocity.

Two things should be noted here: First, Bahri et al. derive their scaling laws in spectral space, while Sadeghi & Oberlack in physical space. But this difference is not the issue here, especially since the scaling laws are purely temporal and thus for 1-point quantities exactly the same in spectral as in physical space. The issue here is that Sadeghi & Oberlack claim these laws as "new", which is not true. Second, the parameter m in (1.1) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] results from a nonphysical scaling symmetry which the authors used. Their own analysis, however, later revealed this nonphysical feature when they tried to ensure compatibility with the simulated data, with the result that this symmetry, and thus its associated parameter m, must be set to zero (see the last sentence on p. 18 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]). Therefore, exactly as in (1.6), the scaling (1.8) due to m = 0 only exhibits one essential scaling exponent, and not two.

1.2. Proof that the self-similar results in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF][START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] are more general than in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] One could make the assertion that the proof in the previous section is inappropriate for the case studied in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], for the reason that therein the case of zero thermal diffusivity (α = 0) was considered, while Bahri et al. considered the case of non-zero thermal diffusivity (α = 0). Hence, one could assert that the two scaling laws (4.24) and (4.25) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] are indeed different and thus can be claimed as novel. But, as will be proven now, such an assertion is not valid.

First of all, Bahri et al. did not rule out the case α = 0. Their study is valid for all α ∈ R, and thus includes the special case α = 0 which Sadeghi & Oberlack considered. To demonstrate this, let's look more closely, for example,1 at the scaling law for the temperature variance (2.14) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF]

θ 2 = C 2 [ℓ θ ] 2C 1 α/A θ -1 , (1.9)
which, as was proven before in Sec. 1.1, can be rewritten as

θ 2 ∝ (t + t ′ 0 ) 2n+m , ( 1.10)
where

ℓ 2 θ = ℓ 2 0θ + A θ (t -t 0 ), A θ t ′ 0 := def ℓ 2 0θ -A θ t 0 , C 1 α/A θ -1/2 =: def 2n + m, (1.11) with C 1 , C 2 , A θ , ℓ 2 0θ
, and t 0 being arbitrary constants. Now, if we naively put α = 0 in (1.9), it will result in a parameter-independent scaling exponent, which, according to (1.11), will imply the restriction 2n + m = -1/2 on the Sadeghi-Oberlackparameters (n, m) -a restriction, however, which is not present in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], regardless of whether m is zero or not. Naively one can thus conclude that the scaling for α = 0 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is more general than in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF]. But this is a fallacy! There are two ways how to prove that [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is not more general for α = 0 than [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], where both proofs make use of the fact that the scaling can be made independent of α. Too see that, we have to go back to the derivation of (1.9) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF]. The central terms therein are (2.2)-(2.4) and (2.12). Proof A (see below) directly puts α = 0 in (2.2) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], while Proof B separates out α in (2.2) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] by not specifying a particular value for it, thus showing that the scaling laws of Bahri et al. are valid for all α ∈ R, including the specific case α = 0.

Hence, the self-similarity results of Proof A: Starting off with (2.2) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] and directly putting α to zero, we get:

[ Ės,θ ]f 1 + E s,θ lθ ℓ θ f ′ 1 γ = -[βE s,vθ ]f 2 -[T s,θ ]g 1 . (1.12)
Dividing this whole equation by E s,θ /ℓ2 θ , we then get

Ės,θ ℓ 2 θ E s,θ f 1 + ℓ θ lθ f ′ 1 γ = -β E s,vθ E s,θ ℓ 2 θ f 2 - T s,θ E s,θ ℓ 2 θ g 1 , (1.13)
which, if self-similarity should hold, leads to the following conditions

Ės,θ ℓ 2 θ E s,θ = ǫ 0 1 , (1.14) ℓ θ lθ = ǫ 0 2 , (1.15) β E s,vθ E s,θ ℓ 2 θ = ǫ 0 3 , (1.16) T s,θ E s,θ ℓ 2 θ = ǫ 0 4 , (1.17)
where ǫ 0 1 , ǫ 0 2 , ǫ 0 3 , and ǫ 0 4 are arbitrary constants without any restrictions, with the 0-index indicating that α = 0. The above conditions (1.14)-(1.17) are the same as (2.3)-(2.6) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], yet, only independent of α. Hence, when renaming the first two ǫ-constants2

ǫ 0 1 = C 0 1 , ǫ 0 2 = A 0 θ /2, (1.18)
as in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], but now indicating that α = 0, then solving for (1.14) and (1.15), and finally using the relation (2.12) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], we end up with the same scaling law (2.14) as before,

θ 2 = C 2 [ℓ θ ] 2C 0 1 /A 0 θ -1 , (1.19)
but, now, only independent of α. Result (1.19) shows that for α = 0 the scaling exponent is indeed parameter-dependent, thus implying the identity C 0 1 /A 0 θ -1/2 = 2n + m without any restrictions on the Sadeghi-Oberlack-parameters (n, m).

Proof B: We start again with (2.2) in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], but now we do not specify any value for α, but leave it arbitrary, i.e., α ∈ R (including thus also the case α = 0):

[ Ės,θ ]f 1 + E s,θ lθ ℓ θ f ′ 1 γ = -[βE s,vθ ]f 2 -[T s,θ ]g 1 -α E s,θ ℓ 2 θ 2γ 2 f 1 , α ∈ R. (1.20)
Dividing this whole equation again by E s,θ /ℓ 2 θ , we then get

Ės,θ ℓ 2 θ E s,θ f 1 + ℓ θ lθ f ′ 1 γ = -β E s,vθ E s,θ ℓ 2 θ f 2 - T s,θ E s,θ ℓ 2 θ g 1 -α • 2γ 2 f 1 , α ∈ R, (1.21)
which, if self-similarity should hold, leads to the following conditions

Ės,θ ℓ 2 θ E s,θ = ǫ α 1 , (1.22) ℓ θ lθ = ǫ α 2 , (1.23) β E s,vθ E s,θ ℓ 2 θ = ǫ α 3 , (1.24) T s,θ E s,θ ℓ 2 θ = ǫ α 4 , (1.25)
where ǫ α 1 , ǫ α 2 , ǫ α 3 , and ǫ α 4 are arbitrary constants without any restrictions, with the α-index now indicating that α can take any value in the defining equation (1.21), including the value α = 0. If we now rename the first two ǫ-constants again as before in (1.18), but now indicating these as being constants for arbitrary α,

ǫ α 1 = C α 1 , ǫ α 2 = A α θ /2, (1.26)
and then solving the corresponding equations (1.22)-(1.23), we end up again with the very same scaling law as given in (1. [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration[END_REF]), yet, now for any value of α:

θ 2 = C 2 [ℓ θ ] 2C α 1 /A α θ -1 , α ∈ R. ( 1
∂θ ∂t + u 2 Γ + • • • = 0, (1.29)
where the dots indicate the remaining terms of the equations, and where Γ is the constant mean scalar gradient. Multiplying (1.28) by θ, (1.29) by u 2 , and then adding both equations, we get the resulting equation

∂u 2 θ ∂t + u 2 2 Γ + • • • = 0, (1.30)
which then in statistically averaged form reads

∂u 2 θ ∂t + Γ • u 2 2 + • • • = 0. (1.31)
Now, when u 2 θ evolves according to Bahri's scaling law (1.8), then the evolution of the transverse fluctuation variance u 2 2 is dictated by the above equation as

u 2 2 ∝ ∂ ∂t u 2 θ ∝ d dt (t + t ′ 0 ) 2n+m-1 ∝ (t + t ′ 0 ) 2n+m-2 , (1.32)
which exactly is the result (4.26) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF].

Proof that (5.7)-(5.10) in [1] are not new

Also the 2-point results (5.7)-(5.10) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] do not constitute any new results, since also those are a direct consequence of Bahri's 1-point results (1.8). To demonstrate this, we take, for example, the considered 2-point equation (3.18) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] (the proof for R θθ and R 2θ is analogous) where we make the ansatz of a self-similar spatial variable r := r/ξ(t). Inserting this ansatz into (1.33), we get:

∂R 22 ∂t + • • • = 0. ( 1
0 = du 2 2 dt • R22 + u 2 2 • ∂ R22 ∂t + • • • ∝ (2n + m -2)(t + t ′ 0 ) 2n+m-3 • R22 + (t + t ′ 0 ) 2n+m-2 ∂r k ∂t ∂ R22 ∂r k + • • • = (2n + m -2)(t + t ′ 0 ) 2n+m-3 • R22 -(t + t ′ 0 ) 2n+m-2 r k ξ 2 dξ dt ∂ R22 ∂r k + • • • = (2n + m -2)(t + t ′ 0 ) 2n+m-3 • R22 - (t + t ′ 0 ) 2n+m-2 ξ dξ dt • rk ∂ R22 ∂r k + • • • (1.35)
which, in order to achieve a t-independent invariant equation, implies the relation

dξ dt = γ • ξ t + t ′ 0 , (1.36)
where γ is some arbitrary proportionality constant. Hence, we get the solution

ξ ∝ (t + t ′ 0 ) γ , (1.37)
which then yields the self-similar variable (5.10) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]

r = r (t + t ′ 0 ) n , (1.38)
if γ is particularly specified as n, as was done in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] in choosing a particular symmetry configuration.

Hence, only by using the results of Bahri et al. [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF][START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] and simple dimensional analysis, we yield the very same results as Sadeghi & Oberlack in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], thus showing that their results are not novel as claimed.

Further remarks and points for correction in [1]

1. The structure of Fig. 1 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is a repetition of Fig. 4.1 in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF] and Fig. 1 in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], only displayed for new DNS data. However, since [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] in particular is not mentioned or cited at this position or in the corresponding text on p. 18,3 the reader gets again the misleading impression that this figure and its power-law fits are unique to [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]. The reader is not made aware of the fact that Bahri et al. achieved power-law fits of similar quality first, and also that they were the first to recognize the fact that the scalar variance θ 2 is not decaying but growing in time.

2.

In [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] it is stated that "the Lie symmetry yields the interrelation among the various scaling exponents, as it leads to a connection between the temporal scaling of the velocity and scalar variance" [p. 15]. This result, however, is not a merit of Lie symmetry groups. C. Bahri got this result first without using Lie-groups, and in the process also obtained more information than Liegroups could give. Her result (5.10) in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF] for the temporal scaling of the scalar (heat) flux vθ leads to the conclusion "that the heat flux is governed only by parameters related to the temperature field. Thus, no information is needed about the velocity field to determine the heat flux in this particular flow configuration. This is of major importance to the turbulence community, and particularly the atmospheric sciences where the scalar flux is of main interest" [p. 78]. This important information cannot be obtained by the Lie-group method as performed by Sadeghi & Oberlack in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], which clearly is a serious drawback compared to the successful method used by Bahri et al.

3.

In [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] it is said that in " [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] it has been suggested that the TPC [2-point correlation] may not scale on the integral length scale, but instead on the Kolmogorov length scale, the Taylor microscale or rather the scalar Taylor microscale" [p. 20]. This statement is misleading, since Bahri et al. [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] considers 2-point correlations in spectral space and not in physical space as Sadeghi & Oberlack in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]. In clear contrast to 1-point quantities, the 2-point quantities for the considered flow configuration may show a different sensitivity on the scaling behaviour in spectral space than in physical space. Furthermore, Bahri et al. did not consider the velocity integral scale in their analysis, as it was done in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], but instead considered only the scalar integral scale at that point. Hence, the 2-point correlation analysis in [START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] cannot be compared with the one done in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF].

4.

In Fig. 4 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] it is not clear why for the velocity correlations only R22 is displayed. What about the other 2-point velocity correlations R11 , R33 , R12 , etc.? Were they just omitted because they maybe don't show such a good collapse when using the velocity integral length scale?

5.

The term 2ΓR 2θ in (3.13) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is not correct, since R 2θ = R θ2 . It either has to be replaced by Γ(R 2θ (r) + R θ2 (r)), or by Γ(R 2θ (r) + R 2θ (-r)).

6.

The last remark is on the usefulness of a Lie-group symmetry analysis in turbulence as it is carried out in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]. As was proven in the previous sections, a Lie-group symmetry analysis is only overly complicated and thus unnecessary for the flow configuration considered in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], since a classical selfsimilarity analysis already suffices to yield the same results. The classical approach in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF][START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] even led to more general results than the ones obtained in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]. But also for other turbulent flow configurations, the Lie-group symmetry method is of no significant analytical relevance as long as the equations are not modelled and remain unclosed [START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF]. The problem here is that we have turbulence, which unfortunately comes along with unclosed statistical equations. Hence, the set of symmetries is also unclosed, which means that if modelling of the statistical equations is not considered as in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], then (nearly) any symmetry can be generated, and thus also (nearly) any desirable scaling law. The simple reason for this is that at each order of the infinite hierarchy almost any change due to a variable transformation can always be balanced or compensated by an unclosed term at the next higher order. Ultimately this means that the choice of an invariance is made by the user and not dictated by theory, simply because one has an infinite set of invariant possibilities to choose from when performing a full and correct Lie-group symmetry analysis for unclosed equations, as explicitly shown and discussed e.g. in [START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF][START_REF] Frewer | A critical examination of the conformal invariance in the statistical equations of 2D turbulent scalar fields[END_REF][START_REF] Frewer | On new scalings in a temporally evolving turbulent plane jet using a different and physical choice of equivalence transformations[END_REF][START_REF] Frewer | On the use of applying Lie-group symmetry analysis to turbulent channel flow with streamwise rotation[END_REF][START_REF] Frewer | Is the log-law a first principle result from Lie-group invariance analysis?[END_REF]. A crucial information which is not shared with the reader in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]. Instead, as a deflection of the issue, false claims are made regarding the use of Lie-group symmetry method in turbulence, thus raising false hopes, like the following statement on p. 3 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF]:

"Despite the fact that the classical self-similarity to construct scaling parameters for the passive scalars has been known for a long time, there are still several open challenges. First and foremost, the classical self-similarity hypothesis can only be carried out when using an a priori set of similarity scales for all of the statistical moments in the transport equations. Therefore, it is one of the outstanding goals to obtain the similarity scales based on a more general analytical approach rather than in an ad hoc manner. In addition, the interaction between the velocity and passive scalar (temperature) fields becomes important in several cases, for example, when a constant mean temperature gradient is present. However, the lack of a unified approach that can predict the direct link between the scaling laws for the passive scalar and velocity moments is noted in the literature. Therefore, it is the main aim of this paper to use a more general technique, which is known as Lie symmetry analysis, to formally derive the scaling laws and similarity variables for a passive scalar flow advected by a turbulent flow."

Or, more compact in their abstract: "It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis."

Considering the proofs in Sec. 1, which demonstrate that a simple dimensional analysis is already fully sufficient to confront the current scaling problem, as correctly derived by Bahri et al. in [START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF][START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF], and the fact that the set of symmetries in turbulence is unclosed, it is easy to unmask the above claims by Sadeghi & Oberlack as false. It is not true, that the Lie-group symmetry method in turbulence is free of assumptions. It is an ad-hoc method too, not in the same but in a similar way as the classical self-similarity method: Instead of using an a priori set of scales, the Lie-group method has to make use of an a priori set of symmetries, namely to select the correct relevant symmetries from an infinite (unclosed) set. In other words, the particular selection of the chosen symmetries (4.7) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is a plain assumption, because when performing a full and correct Lie-group symmetry analysis on the considered (unclosed) equations (3.16)-(3.18), one gets an infinite set of possible symmetries, 4 which is not mentioned in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF].

Another problematic issue not mentioned in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] is the fact that due to the arbitrariness involved when making a particular choice from an infinite (unclosed) set of symmetries, there is the high chance that one will select a nonphysical symmetry which is not reflected by experiment or numerical simulation. This clearly is the case for the chosen statistical scaling symmetry (4.6) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], with the group parameter a s , and later m (4.20), which clearly is nonphysical -see e.g. [START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF][START_REF] Frewer | Comment on 'Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow[END_REF][START_REF] Frewer | Comment on "Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence[END_REF][START_REF] Khujadze | Revisiting the Lie-group symmetry method for turbulent channel flow with wall transpiration[END_REF][START_REF] Frewer | A note on the notion "statistical symmetry[END_REF][START_REF] Frewer | Objections to a Reply of Oberlack et al[END_REF][START_REF] Frewer | Comment on "Statistical symmetries of the Lundgren-Monin-Novikov hierarchy[END_REF][START_REF] Frewer | On the physical inconsistency of a new statistical scaling symmetry in incompressible Navier-Stokes turbulence[END_REF].

Although this nonphysical symmetry (4.6) later had to be put to zero in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] once a specific initial condition with a prescribed energy spectrum was chosen for the simulation (see p. 18 in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] where the case m = 0 is discussed), its nonphysical feature is still downplayed as being only a symmetry where "We do not have an indication from the DNS data that the statistical symmetry (4.6a-n) connected to m plays a role here. Hence, in the following we set m = 0" [p. 18].

Despite all these signs and warnings, this nonphysical symmetry (4.6) keeps being propagated in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] as a scaling symmetry that plays "a key role for the understanding of moment scalings in wall-bounded shear flows" [p. 13], and that "this symmetry is a measure of intermittency" [p. 8]. In particular the latter statement about intermittency,5 here the authors should ask themselves why this symmetry was forced to zero in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] for a flow configuration that exhibits intermittency, regardless of the initial condition considered. In other words, if this scaling symmetry (4.6) truly is a measure of intermittency, then the parameter m for the currently considered flow configuration should be non-zero, or? Independent of this apparent conflict, the connection to intermittency of this scaling symmetry (4.6) is not true and has been clearly refuted several times by now within different statistical frameworks -see e.g. [START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF][START_REF] Frewer | Comment on 'Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow[END_REF][START_REF] Frewer | Comment on "Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence[END_REF][START_REF] Khujadze | Revisiting the Lie-group symmetry method for turbulent channel flow with wall transpiration[END_REF][START_REF] Frewer | A note on the notion "statistical symmetry[END_REF][START_REF] Frewer | Objections to a Reply of Oberlack et al[END_REF][START_REF] Frewer | Comment on "Statistical symmetries of the Lundgren-Monin-Novikov hierarchy[END_REF][START_REF] Frewer | On the physical inconsistency of a new statistical scaling symmetry in incompressible Navier-Stokes turbulence[END_REF]. Also from a pure phenomenological viewpoint, it is abundantly clear that intermittency is a symmetry-breaking phenomenon and not a symmetry-existing or symmetrypreserving one. Even if we would wrongly assume this to be the case, intermittency is definitely not described or featured by any global scaling symmetry, particularly not by the one given by (4.6) in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], which actually just mimics the standard scaling of a linear system when applied to an unclosed nonlinear system. Obviously, such a symmetry is only a mathematical artefact of the unclosed system itself and therefore indeed nonphysical, independent of the fact, of course, that this symmetry also violates the classical principle of cause and effect [START_REF] Frewer | Comment on 'Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow[END_REF][START_REF] Frewer | A note on the notion "statistical symmetry[END_REF][START_REF] Frewer | Comment on "Statistical symmetries of the Lundgren-Monin-Novikov hierarchy[END_REF].

Finally it needs to be noted that in various other publications by Oberlack et al., regarding Liegroup symmetries and turbulence, a second nonphysical symmetry is constantly included, the nonphysical statistical translation symmetry, which led to serious errors, e.g. in [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration[END_REF] and [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation[END_REF], which then led to the forced Corrigenda [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration -CORRIGENDUM[END_REF] and [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation -CORRIGENDUM[END_REF], respectively. Although [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration -CORRIGENDUM[END_REF] is a Corrigendum, it is still seriously in error -for a detailed discussion on this matter, see Appendix D in [START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF]. Also in [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation -CORRIGENDUM[END_REF], although a correction is given, it does not provide an explanation or a correction to the originally published Fig. 7, a figure in [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation[END_REF] that still cannot be reproduced from the data provided. Instead, a completely new figure based on new results is presented and therefore unrelated to the original one. Hence, in both cases [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration -CORRIGENDUM[END_REF] and [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation -CORRIGENDUM[END_REF], the same central question: How and with what tools did the authors manage to do the original Fig. 9(a) in [START_REF] Avsarkisov | New scaling laws for turbulent Poiseuille flow with wall transpiration[END_REF] and Fig. 7 in [START_REF] Sadeghi | On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation[END_REF]? Nevertheless, although this nonphysical translation symmetry is not included in [START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF], it can, however, be found again in several recent studies by Oberlack et al.

  .33) Now, since we are on a search for temporal scaling laws, we redefine the 2-point function R 22 by factoring out the corresponding 1-point temporal scaling law (1.32) of Bahri et al.: R 22 (t, r) =:

  Bahri et al. go beyond those of Sadeghi & Oberlack. In other words, not the results of Sadeghi & Oberlack, but ultimately the results of Bahri et al. are the more general ones.

  .27)Result(1.27) shows that the scaling structure obtained in(1.19) is valid not only for α = 0, but for all α ∈ R. Hence, the scaling result obtained by Bahri et al. is more general than the result obtained by Sadeghi & Oberlack, who considered only the case α = 0.

	1.3. Proof that (4.26) in [1] is not new
	Also result (4.26) in [1] is not a new result as claimed, but only a consequence of Bahri's results (1.8)
	when correctly applied to the underlying equations. The relevant equations are the equation for the
	cross stream velocity (3.5) in [1]
	∂u 2 ∂t	+ • • • = 0,	(1.28)
	and the passive scalar (temperature) equation (3.4)

* Email address for correspondence: frewer.science@gmail.com

All arguments and proofs brought forward here for the scaling law of the temperature variance θ

, equally applies to Bahri's scaling law of the scalar flux vθ in[START_REF] Bahri | Fundamentals and scaling of passive scalar fields in isotropic turbulence[END_REF].

The renaming of ǫ2 in[START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] above (2.10) has a typo, which easily can be verified when integrating (2.4) to (2.10) in[START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF]: Instead of 'ǫ2 = αA θ ' it should read 'ǫ2 = αA θ /2'.

To note is that although[START_REF] Bahri | Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient[END_REF] gets cited in[START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] for other findings and discussions, it is not cited correctly at the key positions in[START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF].

To note is that the implied linearity of the equations (3.16)-(3.18) in[START_REF] Sadeghi | New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence[END_REF] still amplifies the closure problem when correctly applying a Lie-group invariance analysis to it. It is the unpleasant effect of the linear superposition principle which adds an additional infinite dimension to the already infinite dimensional Lie-algebra of invariant transformations that already results from the unclosedness of the equations themselves.

To note is that the group Oberlack et al. use the term "intermittency" for the nonphysical scaling symmetry interchangeably either for external (large-scale) or internal (small-scale) intermittency, e.g. as done for jet flow in[START_REF] Sadeghi | New symmetry-induced scaling laws of passive scalar transport in turbulent plane jets[END_REF], or for the transition to turbulence in[START_REF] Wac | Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence[END_REF], or for the center region of fully developed channel flow in[START_REF] Oberlack | Turbulence statistics of arbitrary moments of wall-bounded shear flows: A symmetry approach[END_REF]. For the latter case, see also the discussion in[START_REF] Frewer | A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach[END_REF].


Dr Renaud Colson

Learning from Elsewhere: from cross-cultural explanations to transnational prescription in criminal justice?

Comparing criminal justice systems in a transnational era

We are living at a time in which increasingly concerns about crime and security are shared across national borders and our responses are becoming transnational: this is evident in intensifying international efforts at crime-control co-operation and the dissemination of particular approaches to crime prevention, criminal justice and the rule of law. These developments are being implemented partly through international law (especially international criminal law and international human rights law) and partly through wider and more diverse forms of supranational influence and intergovernmental cooperation. There is now a substantial grey literature emerging from international institutions such as the United Nations, the European Union and the Council of Europe producing comparative data and research and seeking to promote particular views of good practice in relation to a variety of issues around crime, security and justice. 1 In this context, it is hardly surprising that the comparative study of criminal justice practice and policy and its transfer across national boundaries is beginning to emerge as a new field of research both in criminology and criminal justice. 2 One can distinguish two linked research questions within this field. The first is the more descriptive or analytical. How are concepts of good or best practice in security and criminal justice being constructed and circulated transnationally? How do transfers take place, through what channels and with what effects (good and bad)? Here the task is to chart what is actually happening in the development of transnational prescription in criminal justice practice and policy. The second research question is more directly normative with clear prescriptive implications. To what extent is it desirable to seek to identify transnational good or best practice? What are the difficulties and challenges in doing so validly? If transfers of policy and practice can sometimes be desirable, how should they be carried out in order to promote the exchange of good practice (as opposed to bad)?

What both descriptive and prescriptive research questions have in common is that they require an engagement with the underlying challenge of cross-cultural diversity in social and legal norms. If the definition and operation of concepts and practices in security and justice varies according to cultural contexts, then this raises questions both about the feasibility of effective transfer and its desirability. In turn, this then raises questions about the role of comparative researchers and cross-cultural research in informing good learning practices and criticizing bad ones. We may anticipate that, in so far as it is governments that are shaping transfers, they are likely to be driven by the immediacy of their own instrumental and political goals. Established in particular legal jurisdictions and shaped by the cultures of particular nations, they may struggle to come to terms with the significance of cross-cultural diversity. What part can or should cross-cultural research play in defining good transnational practice and assessing the consequences, both good and bad, of transfer? What kinds of comparative research might be appropriate to the task? There are several distinct strands to the published academic research that relates to these questions. First there is a substantial body of conceptually sophisticated and methodologically reflective research in politics and international relations on policy transfer studies. It even has its own acronym (PTS). But as yet, very little of this has been applied in the criminal justice and criminological field. 3 Secondly there is a burgeoning literature in comparative law which uses the concepts of legal transplant and legal adaptation to examine the transfer of legal rules and practices. But again, there has been only a limited focus on criminal justice. 4 There is a more substantial literature on broader questions of comparative criminal justice; but policy transfer is often not a direct focus and such work is often seen as in need of greater conceptual and methodological reflection. 5 In recent years we have seen the rise of a diverse academic and policy literature stretching across criminology, politics and international relations which treats a very wide range of transnational issues of public safety and social order as questions of 'security.' This includes a concern to define 'legitimate security' and to address normative questions about the 'good' in security. 6 Yet the approach has traditionally been based more on notions of security as a general human need with a dominant universal logic rather than something differently experienced, felt and managed by different groups. 7 The examination of the influence of cultural differences on the lived experience of security is as yet underdeveloped. 8 Finally, there are substantial specialist legal literatures on international and European criminal law and the development of European criminal justice. But the general drive of much of the research is towards developing coherent general normative principles to organise these different fields rather than examining their effects and implications for diverse legal cultures.9

Functionalism and its critiques

In order to develop conceptual reflection on the terms and conditions of sound comparison and its implications for policy transfer in the specific contexts of criminology and criminal justice, an obvious starting point is mainstream comparative law. In the last thirty years, the rise of globalization and the harmonization of legal fields as diverse as commercial law, company law, constitutional law or human rights law have stimulated academic research on the methods of legal comparison. The dominant approach relies on the idea that only laws which fulfil the same function can be compared. According to the so-called functional method, the goal of comparison is to understand how different societies solve similar problems in diverse ways. This in turn opens the door to legal progress because the contrast between various solutions helps to identify best practices and to circulate them.

At first sight, the identification of functional equivalents in different legal systems seems to provide an appropriate tool to overcome the descriptive and normative challenges facing comparatists in the field of criminal justice. Not only does it provide a framework to identify what is worth comparing (different legal rules and normative practices deployed to solve similar criminal problems and analogous security issues), but it also allows us to assess these norms and possibly promote or criticise their diffusion depending on how they fulfil their goal. A good example is provided by the way the development of negotiated justice and pleabargaining in different states is described as a response to a common problem, namely an increasing burden on criminal procedures due to an increasing crime rate. 10 In the light of such a diagnosis, it then becomes possible to compare the various national manifestations of pleabargaining in order to determine how each of them meet this goal, what overall impact it has on the functioning of the criminal justice system, and evaluate whether it should be extended further or applied in other contexts. 11 Although dominant in comparative scholarship, the functional approach has been under fierce attack. The critique is manifold and bears both on the method itself (criticised as relying on a simplistic and instrumental view of legal technique) and its goals (said to be practice-oriented, lacking neutrality and blind to cultural diversity). Some critiques claim that the mainstream approach precludes a sophisticated understanding of the relationship between norms and their functions and is heavily tilted towards hegemonic harmonization, not to say unification, of laws. Ultimately, its most radical opponents reject the quest for similarities at the core of the functionalist approach and claim that the legal comparisons should essentially focus on differences instead of pseudoconvergences. 12 This critique seems all the more powerful in the field of criminal justice, a domain where distinctive legal characteristics are deeply entrenched. Rooted in history and tradition, criminal law -especially in its procedural dimension -should not be viewed as a mere set of technical arrangements. It emerges within a historical context and it articulates local political and cultural constructs which seem to determine its development as much as its explicit function. As a result, the quest for functional equivalence to explain the circulation of criminal models may prove problematic as it overlooks other fundamental determinants in the transfer of norms and institutions. Thus, in understanding a practice like negotiated justice, we may need to think more about the effects of legal and political cultures to explain the ways in which the apparent homogenization of legal practices may conceal continued differences or even the development of unpredictable consequences. 13 This in turn may lead us to be very cautious with, or even to abstain from, any normative statement on the desirability of legal transfer. Should comparatists resign themselves to oversimplification in order to shape the increasing normative exchanges which characterize criminal justice systems? Or should they sacrifice policy prescription on the altar of cultural complexity? At first sight, the fierceness of the ongoing methodological dispute and the depth of the epistemological discrepancy between the functionalist and the hermeneutic perspectives seem to rule out any reconciliation of the two points of view. Yet compromise should not be seen as precluded. Although functional and interpretive approaches are based on very different epistemological starting points, both stress the need for a contextual analysis of law.

Comparison as Translation

Translation studies can provide a useful guide to the researcher involved in the comparison and the circulation of norms and practices between criminal justice systems. Language and law are analogous in many ways. Both are systems of rules (linguistic or legal) which evolve in particular social, economic and political contexts. Both are the result of historical processes shaped by interaction with other linguistic and legal systems, and these interactions often reflect hegemonic power relationships. Moreover, comparative law and translation studies have much in common. 14 Comparative lawyers are often obviously confronted with the issue of translation when working on foreign laws. But beyond this visible connection, which simply flows from different legal systems being in different languages, there are other similarities between the two disciplines. Indeed, both provide methods to mediate meanings and identify correspondence between distinct semiotic universes, and both engage in debates on the value of these methods and their results.

The first lesson that researchers involved in the comparison of criminal laws can learn from translation studies is that mediating the exact same information from one language into another is impossible. While the translator should strive to say 'almost the same thing', [START_REF] Ecco | Dire quasi la stessa cosa. Esperienze di traduzione[END_REF] it can never achieve an 'ultimate' translation in the target language without a loss. Because languages do not signify identically, all translation necessarily implies some sort of distortion. In an influential essay, [START_REF] Benjamin | Die Aufgabe des Übersetzers[END_REF] Walter Benjamin illustrated this phenomenon using a simple example: while the word German Brot and the French word pain refer to essentially the same bread, these two words are not interchangeable as they mean something different to a German, who will probably have in mind Vollkornbrot, and to Frenchman, who will most likely be thinking of a baguette. Similarly, the comparatist lawyer willing to describe foreign rules and legal concepts can hardly claim to fully render their meaning to its audience. [START_REF] Glanert | Translation Matters[END_REF] American plea bargaining, German Absprachen, French comparution immediate sur reconnaissance préalable de culpabilité, and Italian pattegiamento may well all be referred to as guilty pleas, but they are not interchangeable. On the contrary, it could be argued that each of these concepts excludes the others. They have different meanings. Each of them refers to a distinct set of rules and practices, all deeply culturally bound and reflecting particular legal mentalities which are incommensurable.

The second lesson that translations studies can provide to comparatist is that since there is no "true" translation, the translator is bound to betray somehow the original text and the host language, [START_REF] Ost | Law as Translation[END_REF] and he should accordingly choose an interpretative strategy. Does he want to convey the strangeness of the original text and communicate its cultural and linguistic particularities to a foreign readership at the risk of being cryptic and elusive? Or does he aim at a more transparent discourse in the host language at the risk of assimilation of the cultural other? Depending on its choice, the translator will stick to the word or to the spirit of the original text, to the letter or to the genius of the host language. This decision is sometimes framed in ethical terms, as a choice between "authenticity" and "ethnocentrism". [START_REF] Schleiermacher | On the Different Methods of Translating[END_REF] Yet, Julian Barnes has rightly pointed out that the translation that works best may sometimes be the one least faithful to the original. [START_REF]Translating Madame Bovary[END_REF] This "politics of translation" in relation to the work of translators mirrors the "politics of comparison" which is also at play, although usually implicitly, in the work of legal comparatists. [START_REF] Kennedy | The Methods and the Politics[END_REF] Indeed, similar issues arise in the comparative description and appreciation of foreign criminal justice systems.

In order to describe foreign rules and foreign institutions, the comparatist can try to do justice to their originality, to emphasize their otherness to the point of incomprehension, or he can choose to emphasize sameness over difference in order to make the foreign model understandable and possibly transferable. The choice between these two approaches has political consequences. The study of law, including comparative law, is not only a contemplative activity: part of its value derives from its ability to transform legal representations and legal practices. [START_REF] Muir-Watt | La fonction subversive du droit comparé[END_REF] These political stakes are especially high as comparative arguments are frequently used in the framing of national and transnational criminal policies. When involved in such enterprise, the comparatist cannot always take refuge in the ethereal debate on cultural appropriation and the authenticity of the comparative process. He necessarily has to take a stance, if only because the knowledge he produces is likely to have a governance effect, because cross-cultural comparisons play a crucial role in accelerating the circulation of policy ideas and legal models.

Like the translator, the comparatist should reflect on the purpose of his research to determine what comparative strategy to adopt. This is all the more necessary if he is engaged in a process of law reform, whether it be new procedural guarantees for individuals, increased crime control efficacy or the administrative modernisation of criminal justice. In that case, he should remember that any legal transfer from one setting to another is fraught with uncertainty. [START_REF] Nelken | Comparatists and Transferability[END_REF] All jurisdictions have an established framework for thinking and doing criminal justice (a criminal justice language) that imposes particular constraints but also offers specific opportunities. As a result, the most faithful transfer from one jurisdiction to the other may not be the most effective. Yet there may be occasions where showing one's fidelity to the original model may be a matter of important political symbolism and effectiveness in practice on the ground a lesser concern. 24

Article summaries

With this volume, we hope to place reflection on comparative methodology at the centre of scholarly debate on transnational harmonisation of criminal law. This Special Issue seeks to do this by asking notable cross-cultural researchers to reflect on their own experiences, positive and negative, in attempting in particular studies to define good transnational practice or to negotiate cross-cultural difference. In so doing, it seeks to develop insights into both of the research challenges with which we started: the descriptive analysis of mechanisms of international transfer of policy and practice and the prescriptive evaluation of the characteristics of good and bad transfers. All contributors engage with comparative practices and methodologies in the specific contexts of transnational criminal policy, and all discuss studies or initiatives that represent good practice as well as counter examples where good practice has been defined invalidly and/or unhelpfully.

The first article of the special issue provides a general perspective "on transnational policy flows in security and justice". It offers an overview of cross-national comparative research in relation to crime control and penal policy. Trevor Jones and Tim Newburn's state-of-the-art account of policy transfer studies shows that there is now a sufficient body of work on the subject to make some general observations on the increasing circulation of criminal policy models and criminological ideas around the globe. The article addresses empirical questions relating to how far and in what ways crime, security and justice policies are actually transferred across national boundaries, what happens to them in the process, and what factors might explain such phenomena. These include 'rational' attempts at policy learning from other jurisdictions, but also 'coercion' orchestrated by powerful 'hegemon' states and 'mimicry' whereby norms and policies are emulated for symbolic reasons rather than likely effectiveness. Jones and Newburn then proceed to reflect on the desirability of such cross-national exchange. Observing that much research in relation to policy transfer in crime control adopts a pessimistic position in the light of the 'punitive turn' that has emerged in many western countries, the authors discuss the normative principles that might be used to explore the possibilities for progressive policy transfer.

The identification of such progressive policy in order to promote their diffusion across borders is central to the construction of global social 24 See both Nelken and Jones and Newburn for examples in this Special Supplement indicators. David Nelken's article "on the significance of context in and for transnational criminal justice indicators" notes the growing interest of criminologists and comparative law scholars in such data. Drawing upon the example of the recent global targets set by the United Nations in the field of criminal justice, Nelken questions the way ideas that count as good practice are selected and how they are used to frame transnational comparisons. Global indicators which are used to evaluate local conditions in terms of overarching standards are neither purely descriptive nor politically neutral: they have knowledge and governance 'effects". Those engaged in exercises of transnational ranking and target setting may often be tempted to disregard contexts. But applying similar (criminal) standards to dissimilar places may have adverse consequences when pursuing projects of social improvement, such as the fight against corruption or gender violence. While Nelken does not rule out the possibility that, with proper design, global indicators might have the potential to make effective contributions, he warns against the dangers of simplistic comparison. Calling for more empirical studies on how global indicators accomplish commensuration between diverse settings, Nelken's article highlights the methodological difficulties attached to governance by numbers in transnational criminal policy, especially in the diffusion of best practices and ideal standards. Field's article, which focuses on the "comparative analysis of youth justice cultures", examines precisely some of the methodological challenges in using comparative research to inform policy transfer. It is rare to attempt the transposition of whole criminal justice systems from one jurisdiction to another: it is more common to seek to transfer particular elements of a system such as a penal measure or a legal role. But how can we predict the impact of a transferred element when it is placed within a different cultural context? Where criminal justice cultures are very different, their elements may not be directly comparable, or their significance may be conditioned by the system of which they are part. Drawing on a bilateral comparative study of youth justice in Italy and England and Wales conducted with Nelken, Field explains the difficulty of comparing particular pre-trial interventions in two jurisdictions where the relationship of pre-trial and trial phases and between civil and criminal intervention is very different. Arguing for the need to understand criminal justice practices in the light of the legal and political cultures within which they are imbedded, Field argues that learning from very different 'elsewheres' should be seen in terms of broadening the criminal justice imagination rather than furnishing particular solutions with predictable consequences. Renaud Colson's paper on transnational drug policy demonstrates that comparative law has the potential to expand the agenda of "thinkable" possibilities. The article explores the insights that may be gained from the vast amount of comparative data produced to monitor the implementation of the international drug control regime. At a time of growing doubt about the benefit of criminalisation of drug use, it provides a case study on the ways that epistemic communities may rely on comparative research to identify best practices and promote them as normative alternatives in the face of a long entrenched legal dogma. In order to explore these issues, the article looks at the UN drug control system from the perspective of comparative law. It shows how the concept of legal transplant provides a useful tool to understand the limits of transnational criminal law designed on a global scale to tackle the "drug problem". The article also identifies the various types of legal comparison that might contribute to addressing this failed transplant. It argues that while comparative analysis can certainly make substantial contributions to legal and policy knowledge, its ability to provide universal solutions applicable on a transnational basis should not be overrated. Even when grounded in sophisticated design, legal comparisons are always open to interpretation and can sometimes prove erroneous. In this respect, comparison of laws and regulatory measures on drugs is no substitute for rigorous policy evaluation. Special Issue concludes with two particular case-studies which consider the potential impact of comparative research on the building of transnational norms. In the first, Jackie Hodgson reflects on her recent experiences as a member of cross-cultural research teams. The teams were funded by the European Commission to do comparative empirical research aimed at informing the development of EU Directives on suspects' rights to legal advice in police custody. She notes that the legal form of transnational norms can make a significant difference to their effectiveness: the European Union's capacity for proactively legislating the particulars of suspects' rights has a potential to influence practice that goes beyond the more abstract and reactive interventions of the European Court of Human Rights. But in understanding the potential impact of Directives in different jurisdictions, what is essential is empirical research that goes beyond the comparison of formal legal texts to examine the variable ways that these may be translated into the everyday practices of law and the experiences of suspects in different jurisdictions. Comparative empirical research enables us to understand the different ways in which the professional organisation of criminal defence or the provision of legal aid may shape the real meaning of a right to custodial legal advice. It enables us to appreciate the impact of variable institutionalized relations between police, prosecutor, defence lawyer and judge. It provides us with a basis not just for advice on constructing legal norms but also on the training that may shape professional cultures. Critical to this is the kind of insight that can be generated by cross-cultural research teams.

The final article by John Jackson charts the transnational development of legal norms relating to security-cleared counsel (special advocates): these are lawyers acting in trials where evidential material is regarded as so sensitive to the interests of the state that their access to that material is conditional on not disclosing it to their 'client'. Like Hodgson, Jackson emphasizes the limits of the European Court of Human Rights' capacity to define best practice through judicial decision-making. It struggles to compare widely enough across different jurisdictions, to look deeply enough into the variable practices discovered and to pay close enough attention to the local context. Like Hodgson, Jackson argues for the potential of comparative empirical research to provide better insights: in this case, the challenge is to define procedures that best balance the needs of state security against suspects' rights of participation. Drawing on his own interview-based study of practice in Canada, the United States and the UK as well as previous comparative research by Cole and Vladek, he too suggests that what is needed is 'interpretivist' research that engages with the professional cultures and procedural traditions that underpin domestic and transnational practice.

See for a notable exception, T Jones and T Newburn, Policy Transfer and Criminal Justice: Exploring US Influence over British Crime Control Policy (2007). See Jones and Newburn, this volume for the authors' reflections on the significance of policy transfer studies to comparative criminal justice.

For a sample of studies, see D Nelken and J Feest (eds), Adapting legal cultures(2001). For a notable exception to the neglect of criminal justice, see M Langer (2004) 'From Legal Transplants to Legal Translations: The Globalisation of Plea Bargaining and the Americanization Thesis in Criminal Process'' 45 Harvard International Law Journal 1

David Nelken's work involves a continuing concern that those working in comparative criminal justice should be more reflexive in the development and use of concepts. See among others: Contrasting Criminal Justice : Getting From Here to There (2000) and Comparative Criminal Justice : Making Sense of Difference (2010).

A Crawford and S Hutchinson (2016) 'The Future(s) of Security Studies' 56(6) BJ Crim 1049, 1053. For a particular attempt to develop the concept of legitimate security see S Virta and M Branders (2016) 'Legitimate Security? Understanding the Contingencies of Security and Deliberation' 56(6) BJ Crim 1146

A Crawford and S Hutchinson (2016) 'The Future(s) of Security Studies' op.cit.,1053.

But see for an agenda: A Crawford and S Hutchinson (2016) 'Mapping the Contours of 'Everyday Security: Time, Space and Emotion' 56(6) BJ Crim 1184. There are important exceptions to this generalisation that offer pointers for future development. On urban security in Europe, see A Edwards and G Hughes (2013) 'Comparative European Criminology and the Question of Urban Security', 10(3) European Journal of Criminology 257 and the special issue that this introduces. On policing, see the Special Issue of European Journal of Policing Studies 2(1) Policing European Metropolises edited by A Edwards, P Ponsaers, A Recasens, I Brunet, A Verhage (2014), and E Devroe, A Edwards and P Ponsaers (eds), Policing European Metropolises: The Politics of Security in City-Regions (2017).

See nonetheless R. Colson and S. Field (eds), EU Criminal Justice and the Challenges of Diversity: Legal Cultures in the Area of Freedom, Security and Justice (2016)

M. Langer, " From Legal Transplants to Legal Translations", op. cit., 37.

M. Wade, "Why some old dogs must learn new tricks: Recognising the new in EU criminal justice?" in Colson and Field (2016), op.cit. n.000

P. Legrand, 'The Same and the Different', in Comparative Legal Studies: Traditions and Transitions, eds. P. Legrand & R. Munday (2003), 240.

E Grande, Legal Transplants and the Inoculation Effect: How American Criminal Procedure Has Affected Continental Europe (2016) American Journal of Comparative Law 64, M Langer, op.cit. n000

See, among others, V. Grosswald Curran, "Comparative Law and Language", in Oxford Handbook of Comparative Law, eds. M. Reimann & R. Zimmerman, (2006), chap. 20; C. Laske, "Translators and Legal Comparatists as Objective

This Special Supplement emerged from a workshop jointly organised by the Centre for Crime, Law and Justice and the Centre of Law and Society at Cardiff Law School in May 2017. The project benefited from the financial support of the Centre of Law and Society, the Cardiff University International Visiting Fellow scheme and the Max-Planck-Institut für ausländisches und internationales Privatrecht in Hamburg.