Yue Wang

G Micchi

F Pistolesi
email: fabio.pistolesi@u-bordeaux.fr

Sensitivity of the mixing-current technique for the detection of mechanical motion in the coherent tunnelling regime

Keywords: NEMS, quantum dot, carbon nanotube, mesoscopic physics

In a recent publication we have studied theoretically the sensitivity of the mixing-current technique to detect nanomechanical motion by coupling the oscillator to a single-electron transistor in the incoherent tunnelling regime: Γ k B T, where Γ is the tunnelling rate, T is the electronic temperature, and k B are the Planck and the Boltzmann constant, respectively. In this work we consider the same problem when the detection device is a quantum dot in the coherent tunnelling regime ( Γ k B T). In order to reach the best sensitivity we find that one should enter the strong coupling regime, as described in the recent publication (Micchi et al 2015 Phys. Rev. Lett. 115 206802) where a mechanical bistability is described. In this regime the electronic detection device strongly modifies the effective potential of the oscillator and the non-linearities determine the form of the displacement fluctuation spectrum. We find theoretical upper bounds to the sensitivity for the detection of the oscillation amplitude of the oscillator. It turns out that it is convenient to work as close as possible to the bistability.

Introduction

For more than a decade nano-electromechanical systems have been investigated for their potential as efficient and ultrasensitive devices. A particularly succesful system is the suspended carbon nanotube. Letting a current through it allows for the detection of its displacement and, if needed, for its actuation. Recent advances with carbon nanotubes allowed for a record sensitivity in mass [1][2][3] and force [4,5] sensing. These devices are also promising to explore the regime of strong coupling between the mechanical oscillator and the electronic degrees of freedom. Several theoretical predictions have been formulated [6][7][8][9][10] suggesting the possibility of blockade of the current [11][12][13][14], cooling [15][16][17], heating [18], shuttling [19][20][21][22], phonon lasing [23], noise squeezing [24], or unusual mechanical [25,26] and magnetic response [27,28]. Detection of mechanical motion is still one of the most difficult tasks. Several detection methods exploit the high sensitivity of single-electron transistors to a variation of the gate charge. By coupling capacitively the oscillator to the gate of the Single Electron Transistor (SET) it is possible to detect the motion of the oscillator with a high accuracy [29]. One widely used method of detection is the mixing-current technique. Since the single-electron transistor has a very low band pass, one injects two slightly detuned signals in the device, one through the gate and the other through the bias voltage. The non-linear electronic response generates a current oscillating at a frequency equal to the difference of the two driving frequencies. One can show that the amplitude of this low frequency current has a component proportional to the amplitude of the mechanical oscillation. To our knowledge, for nanomechanical resonators this method was implemented in metallic SET by the group of Cleland back in 2003 [30]. It was later adapted to the detection of carbon nanotube by the group of McEuen [31]. This method is particularly efficient for carbon nanotubes, since their small size makes the detection through other methods, like optical ones [32], very difficult. The mixing-current technique has been used for instance in the observation of the first single-electron backaction effects in carbon nanotubes [33,34], ultrasensitive mass detection [2,3], the detection of the charge response function in quantum dots [35], the detection of magnetic molecules [36,37], and the observation of decoherence of mechanical motion [38]. The same method can also be implemented by frequency modulation [39].

In a recent publication [40] we investigated the maximal sensitivity that can be reached with this method when the electronic detection device is a single-electron transistor in the incoherent transport (also called orthodox Coulomb blockade) regime. This is the regime where the decoherence effect induced by the temperature T allows one to treat each hopping event between the metallic lead and the central island (see figure 1) independently. On the other side, for Γ k B T, one can show that the hopping between the left and right metal can happen coherently (here Γ is the tunnelling rate or also the energy width of the electron level on the island, k B the Boltzmann constant and in the following we will set the reduced Planck constant = 1). In this paper we consider the sensitivity of the device when the electronic transport is realized in this regime, known as the coherent regime. We will model the electronic system as in [25], that is relevant for the description of recent experiments [41,42]. This system has been shown to be particularly interesting in the strong coupling limit [25,26]. Defining F 0 the additional force acting on the oscillator when an electron is added to the suspended island and k the mechanical spring constant, it has been shown in [25] that the system undergoes a mechanical bistability at P = F 2 0 /k = πΓ with an expected universal quality factor Q of the order of 1.71. Since in our previous publication [40] we found that best sensitivities are obtained for large values of the coupling constant it is interesting to consider this model and study the condition for an optimal detection of the oscillation amplitude with the mixing-current technique. The main question is how the bistability and the unusual fluctuations of the oscillator may influence the sensitivity. We will find that in most cases increasing the coupling allows one to reach a better sensitivity and thus the ultimate limit is the onset of the bistability. Once in the bistable regime both the progressive reduction of the current due to the establishement of the current blockade and a strong telegraph noise due to the hopping between the two stable minima reduce the efficacity of the detection device.

The paper is structured as follows: section 2 gives an introduction to the model describing a single-level quantum dot coupled to a mechanical oscillator. In section 3 the basic equations for the sensitivity in the case of a quantum dot are derived following [40]. Section 4 considers the weak coupling limit where the width of the mechanical response function is controlled by the dissipation induced by the electronic degrees of freedom. Section 5 considers the effect of non-linearities induced in the effective mechanical potential. Analytical expressions for the sensitivity are given in this section. Section 6 presents numerical results in the strong coupling regime. Finally, section 7 gives our conclusions.

Model

We consider electronic transport through a single electronic level quantum dot capacitively coupled to a mechanical oscillator, as is the case, for instance, in state of the art experiments with carbon nanotubes [42]. The system can be described by the following Hamiltonian:

H = H L + H R + H T + d (x)d † d + p 2 2m + kx 2 2 . (1)
The first three terms describe the leads and their coupling to the electronic level:

H α = k (ε αk -µ α )c † αk c αk with α = L(R)
for the left (right) lead, H T = k t αk c † αk d, with c and ε αk the destruction operator and the energy of the electrons in the leads, respectively, and µ α the chemical potential. From these quantities one can define the lead's tunneling rate Γ α = πt 2 α ρ α with ρ α the density of the states and the single-level width Γ = Γ L + Γ R . For simplicity in the following we choose Γ L = Γ R = Γ/2. The last two terms of equation ( 1) describe a single mechanical mode of displacement x, momentum p, mass m, and resonating frequency ω 2 0 = k/m. The coupling between the electronic and the mechanical degree of freedom is encoded in the fourth term, that gives the energy of the single electronic level d (x). It depends on the displacement of the oscillator, that we assume coupled to the electronic level through the modulation of a gate capacitance C g (x):

d (x) = d0 -eC g (x)V g /C Σ .
(

) 2
We defined C Σ = C L + C R + C g the sum of the three capacitances associated to the three leads (see figure 1). The bias voltage satisfies the relation V = (µ R -µ L )/e. We will work in the typical regime of most experiments: ω 0 Γ and ω 0 eV or k B T. This allows one to use the Born-Oppenheimer approximation and to treat the mechanical mode as a classical degree of freedom [11,14,43,44]. The effect of the mechanical part on the decoherence of the electronic part is negligible, since the oscillator is very slow. By expanding the x-dependence we obtain: where 0 = d0 -eC g (0)V g /C Σ and F 0 = eC g (0)V g /C Σ . A brief discussion on the derivation of equations ( 2) and ( 3) is given in appendix A.

In a previous publication [40] we considered the sensitivity of this detection device in the incoherent tunnelling regime:

k B T Γ.
In the present paper we consider the opposite case of coherent tunnelling regime k B T Γ. For the description of the dynamics of the device we follow [25,26]. Since the oscillator is slow one can calculate the position-depenent current for given value of x [26]:

I(x) = e 2 dω 2π τ (z ω )[ f L (ω) -f R (ω)], (4)
where

τ (z) = 1 1 + z 2 (5)
is the energy dependent electronic transmission factor through the quantum dot, z ω = (ω -0 + F 0 x)/Γ, and f α (ω) = (1 + e (ω-µα)/T ) -1 is the Fermi distribution of lead α. This expression depends on the position of the oscillator, and in order to obtain the measured value one should average over the position of the oscillator with its statistical distribution. We are interested in the low voltage limit eV k B T . In this case τ is a smooth function of ω with respect to the Fermi distributions and we can approximate

f L (ω) -f R (ω) = δ(ω -µ)eV , where for V → 0 we defined µ = µ L = µ R . Equation (4) sim- plifies to I = e 2 V 2π τ (z x ) (6)
where

z x = (µ -0 + F 0 x)/Γ. ( 7
)
The displacement dynamics of the mechanical mode can be described by a Langevin equation:

m[ẍ + γ(x)ẋ + ω 2 0 x] = ξ(t) + F e (x). (8)
In equation ( 8) the quantity F e (x) ≡ F 0 n d ( 0 -F 0 x) is the average force acting on the oscillator. It is simply proportional to the average occupation of the dot n d ≡ d † d . The fluctuating part of the force generated by the electrons jumping in and out of the dot is modeled by the stochastic force ξ(t) that is assumed to have gaussian fluctuations with ξ(t)ξ(t ) = D(x)δ(t -t ) on a time scale longer than Γ -1 . Also originating from the fluctuation of the charge on the dot γ(x) is the dissipative coefficient. Both γ and D can be related to the spectrum of charge fluctuation on the dot

S nn (t) = n d (t)n d (0) : D(x) = F 2 0 S nn (x, ω = 0), γ(x) = -(F 2 0 /m)(∂S nn /∂ω)(x, ω) | ω=0 .
The explicit expressions for γ, F e and D are given in [26]:

n d = ∞ -∞ dω 2πΓ ( f L + f R )τ , (9)
S nn | ω=0 = α,β +∞ -∞ dω 2πΓ 2 f α (1 -f β )τ 2 , (10)
and

dS nn dω ω=0 = α,β +∞ -∞ dω 2πΓ 2 f α τ [ f β τ -(1 -f β )τ ] . (11)
From equation ( 8) one can derive a Fokker-Planck equation for the probability distribution P(x, p, t) that the oscillator is at position x with momentum p at time t:

∂ t P = p m ∂ x P -F∂ p P + γ∂ p ( pP) + D 2 ∂ 2 p (P) (12)
where F(x) = -kx + F e (x).

Mixing current technique

In order to detect the displacement of the oscillator a widely used technique exploits the non-linear mixing of the current when two tones at ω 1 and ω 2 are injected in the gate and bias voltage, respectively:

V g = V g0 + V g1 cos(ω 1 t), V = V 0 + V 1 cos(ω 2 t). ( 13
)
Both frequencies are very close to ω 0 , thus the same argument of slow variation of d is valid also for the external driving.

We recently investigated in details [40] the response function λ of this technique:

I mx = λx m , (14)
where x m is the amplitude of oscillation of the mechanical degree of freedom: x(t) = x m cos(ω 1 t + φ) (assuming a linear mechanical response), and I mx is the low frequency (ω 1 -ω 2 ) mixing current signal. By expanding the gate and bias voltage dependence of the average current for ω 1 , ω 2 Γ one obtains

λ = 1 4e ∂ 2 I ∂n g ∂V dC g dx V g0 V 1 , (15)
with n g = C g V g /e. This quantity measures the sensitivity of the mixing current signal with respect to the amplitude of oscillation x m . As discussed in [40] the sensitivity of this measurement is limited by intrinsic electronic current noise (S shot I ) and by the noise induced in the current by the fluctuation of the oscillator due to the electronic device. This last contrib ution is called back-action noise (S ba I ). It is convenient to define the total added noise as follows [45]:

S add x = S mx λ 2 = S shot I + S ba I 4λ 2 . ( 16
)
In order to obtain the sensitivity of the device in the present paper we thus calculate the two contributions and look for the conditions that minimize their sum.

Intrinsic electronic current noise

The current fluctuates due to the discrete nature of the charge and due to the thermal fluctuations. For simplicity and coherence with the notation of [40] we use the notation S shot I , even if in practice in the following we will consider only the case of dominant thermal fluctuations. For a singlechannel two-terminal conductor the noise power spectrum

(S I (ω) = 2 dte iωt I(t)I(0) ) of the current fluctuations for eV, k B T Γ reads [46] S shot I = e 2 π 2k B Tτ 2 + eV coth eV 2k B T τ (1 -τ ) . (17)
In the regime of interest for most experiments in nanomechanical systems, eV k B T , we are left with the equilibrium contribution:

S shot I (ω) = 2k B Te 2 π τ (z). ( 18
)
Here τ (z) should be averaged over P(x, p, t), but for weak coupling the probability has a sharp peak at the equilibrium position x e of the oscillator, and one can simply use this value of x in the definition of z = z e .

Back-action current noise

The coupling of the mechanical oscillator to the electronic detector induces fluctuations even in the quasi-equilibrium limit of eV k B T . The fluctuation of the charge on the quantum dot induces a stochastic force that puts the oscillator in motion. The displacement of the oscillator affects the effective energy level position through equation ( 3). This generates fluctuations of the current and, therefore, on the mixing current signal I mx . We call these fluctuations backaction current noise. In the particular case of vanishing DC bias voltage (V 0 ) the expression of the back-action noise reads [40]:

S ba I = 8λ 2 S x (ω 2 ). ( 19
)
The factor 8 is due to the definition of the current noise and to the averages entering the mixing current detection. The frequency ω 2 enters equation (19) due to the correlation of the n g oscillations entering the second order expansion of the expression of the current (for more details see [40]). Once inserted into the definition of S add it will simply give the intrinsic fluctuation of the oscillator generated by the electrons. In order to evaluate it we need to obtain S x (ω 2 ), where typically ω 2 is very close to ω 0 .

Weak coupling regime

Let us begin to evaluate the behavior of the device in the weak coupling regime, whose region of validity will be defined more precisely by equation ( 48) in the following. The spectrum S x can be obtained by solving the Langevin equation ( 8) by Fourier transform:

S x (ω) = x(ω)x(-ω) = F 2 0 S n (ω) m 2 |ω 2 0 -ω 2 -iγ(x e )ω| 2 . ( 20
)
The charge noise is given by equation (10). In the limit

eV k B T one has α,β f α (1 -f β ) ≈ k B Tδ(ω -µ) giving S n (ω = 0) = 2k B T πΓ 2 τ 2 (z e ), (21)
where we used S n (ω) ≈ S n (0) for |ω| Γ, since due to the fast electronic response S n (ω) has a flat spectrum up to Γ ω 0 . Assuming the worst case where

|ω 2 -ω 0 | ω 0 /Q, with Q = ω 0 /γ we have S x (ω 2 ) = Q 2 F 2 0 S n k 2 . ( 22
)
A convenient way of rewriting it is:

kS x (ω 2 ) = 2Q 2 τ 2 (z e ) π P k B T Γ 2 (23)
where we introduced the coupling constant energy scale P = F 2 0 /k . We calculate the response function from equation ( 6):

λ = e 2 NV 1 4πL
E C Γ τ (z e ) ≈ F 0 eV 1 Γ eτ (z e ) 8π , (24)
where τ = dτ /dz, L = C g /C g , N the number of electrons on the dot ≈ C g V g /e, E C = e 2 /2C Σ the Coulomb energy of the island (see also appendix A). Substituting the expressions for the shot noise equation (18) and the back-action noise equation ( 19) into equation ( 16) we obtain:

kS add x = k B T Γ α ba Q 2 P Γ + α shot Γ eV 1 2 Γ P ( 25
)
with the numerical factors α ba = 4τ 2 (z)/π and α shot = 32πτ (z)/(τ (z)) 2 . One can in principle minimize this expression as a function of z and P in order to find the best operational point. (Note that a change in 0 implies a nonlinear change in z e since x e depends also on 0 , as we will see in more details later.) In practice, it is preferable to maximize first the response function λ in order to have a strong signal, and subsequently to look for the optimal value of the coupling constant that minimizes the noise. The response function λ is maximized for z e = 1/ √ 3 where τ = 3/4 and τ = -3 √ 3/8 ≈ -0.65. This gives α ba ≈ 0.716, α shot ≈ 178. 42. Minimizing now with respect to P we find

opt P = α shot α ba 1/2 Γ 2 QeV 1 , (26)
with a minimum noise of

kS opt x = 2 k B T Γ α ba α shot 1/2 Q Γ eV 1 . ( 27
)
Since the previous calculation is performed in the limit eV 1 k B T Γ, the minimum value of the ratio Γ/eV 1 is 10. This gives for the optimal value of the coupling

opt P Γ ≈ 157.8 Q . ( 28
)
and the minimum of the added noise

kS opt x = 226.1 Qk B T Γ . ( 29
)
One can find the value of Q from equation ( 11):

Q = πΓ 2 P ω 0 τ 2 (z e ) , (30)
that gives

kS x (ω 2 ≈ ω 0 ) = 2πk B T P Γ 2 ω 2 0 τ 2 (z e ) . ( 31
)
In the spirit of the paper of finding the best possible sensitivity, we consider only the effect of the dissipation that cannot be eliminated, since it is intrinsic to the detection method. Neglecting the 0 dependence of Q and substituting the value equation (30) into equation ( 29), quite surprisingly, one finds that both the shot and the back-action noise decrease as 1/ P :

kS add x = k B T P 4π τ 2 Γ ω 0 + 32πτ τ 2 . ( 32
)
However, the analysis so far is only valid for small values of P as we have made a weak-coupling approximation. We will see in the next section that the precise condition for that is given by equation (48). For typical values of the other param eters that we use in the following for the numerical calculations (ω 0 /Γ = 10 -3 , eV 1 /Γ = 0.1, and k B T/Γ = 10 -2 ) one finds a high value of the added noise of the order of kS add x ≈ 10 7 . Increasing the coupling reduces this value, but forces us to consider the effect of non-linearities.

Non-linear regime

When the coupling or the temperature increases, the oscillation amplitude at resonance induced by the coupling to the electrons increases. One cannot neglect anymore the nonlinear part of the effective force generated by the electrons on the oscillator. By evaluating explicitly n d for k B T Γ and for eV = 0 one obtains:

F(x) = -kx + F 0 1 2 + 1 π arctan µ 0 -0 + F 0 x Γ . (33)
The electronic force modifies significantly the equilibrium position of the oscillator that is no more in x = 0. Once we know the equilibrium position we can calculate the renormalized spring constant:

k = - dF dx xe = k - F 2 0 πΓ τ (z e ). (34)
It is convenient now to define new variables:

˜ 0 = 0 -µ 0 -P /2 Γ , and x = kx/F 0 -1/2. ( 35
)
We now write the equilibrium equation for the oscillator, that is F(x e ) = 0. In terms of the newly introduced variables we have:

-

x e + 1 π arctan(˜ 0 -πx e ˜ P ) = 0, (36)
where we have introduced also ˜ P = P /(πΓ). It is convenient to use again the variable z = ˜ 0 -πx˜ P and write the equlibrium equation as

˜ 0 = z e -˜ P arctan z e . ( 37
)
The function ˜ 0 (z e ), that is plotted in figure 2, can be regarded as the solution to the equilibrium equation. One clearly sees that for ˜ P > 1 there are three solutions, two stable and one unstable. We can find the condition on ˜ 0 for the appearance of the bistability by looking at the value of the maximum of the function ˜ 0 (z). This gives d˜ 0 /dz = 0 that has two solutions

z ± = ± ˜ P -1 (38)
for

˜ P > 1.
From these values we can find the two functions

˜ ± = z ± -˜ P arctan(z ± ) = ±[z + -˜ P arctan(z + )]. (39)
The system has only a single stable solution for |˜ 0 | > ˜ + , and is instead bistable otherwise (see figure 3). The change of the stable points has an important consequence on the resonating frequency of the oscillator, that is renor malized (see equation (34)). Defining ω m = k /m one has

ω 2 m /ω 2 0 = 1 -˜ P τ (z e ). (40)
For ˜ 0 = 0 if ˜ P < 1 the only stable solution is z = 0, while for ˜ P > 1 the value z = 0 is no more stable, and two new minima appear for z = 0. The behavior of the system in this case has been discussed in details in [25] where analytical expressions for S x in the case z e = 0 have been derived. However, note that when the stable solution is at z e = 0 then τ (z = 0) = 0, and thus the response function vanishes (λ = 0). For the purpose of using the device as a displacement detection one needs to work at ˜ 0 = 0.

Weakly non-linear regime

Here we follow [25] to derive the analyical form of S x (ω) for ˜ 0 = 0 (and thus z e = 0) when the quadratic part of the oscillator remains dominant. We begin by expanding the potential around the equilibrium position x e :

U(x) = U(x e ) - n=2,4 1 n! ∂ (n-1) F ∂x (n-1) xe (x -x e ) n + . . . . ( 41
)
Figure 2. ˜ 0 as function as z for ˜ P = 2 and ˜ P = 0.9. The appearance of the two extrema signals the appearance of the bistability.

This gives for ∆U = U(x) -U(x e ) in terms of y = (xx e )F 0 /Γ ∆U(y)/Γ = ay 2 + by 3 + cy 4 (42)

with a = (1 -τ ˜ P )/(2π˜ P ), b = -τ /3!π, and c = -τ /4!π.

All these functions have to be evaluated at z equilibrium. In order to find the effect of the non-linearities on the spectrum a crucial quantity is the energy dependence of the resonating frequency

2π ω E = 2m xR xL dx [2m(E -∆U(x))] 1/2 , (43)
where x L,R are defined by the equation ∆U(x L,R ) = E , with x L < x e < x R . The integral can be performed treating perturbatively (see appendix B) the cubic and quartic terms leading to

ω E = ω m + ω E E + . . . . (44)
Specifically we have:

ω E ≡ dω E dE (0) = C ω m Γ ˜ 2 P (1 -τ ˜ P ) 2 (45)
with

C(z, ˜ P ) = π 2 3c - 15b 2 4a . ( 46
)
The spectrum has the same form found in [25] supplemental materials:

S x (ω) = π(ω -ω m ) mω m ω E 2 k B T e -(ω-ωm)/ω E (0)kBT . ( 47
)
One relevant difference is that ω E can be either positive or negative: the cubic term induces a reduction of the frequency, while the quartic one an increase. The spectrum equation (47) has a maximum for ω = ω m + ω E k B T and a full width at half height of ∆ω = ∆ 2 |ω E (0)|k B T , with ∆ 2 ≈ 2.446.

We can first compare these expressions with the purely dissipative calculation of the previous section and find for which value of the coupling constant the width induced by the dissipation (γ) is of the same order of the width induced by the non-linearities (∆ω):

˜ P ≈ ω 0 k B T τ 2 ∆ 2 |C| 1, (48)
where the last inequality comes from the hypothesis of classical oscillator ω 0 /k B T 1. A change in the equilibrium position modifies the numerical factor C, but has no dramatic effect. This is also the condition of validity of the weak coupling approximations.

In order to obtain the sensitivity of the device in the weakly non-linear regime we need the maximum of S x (in the usual pessimistic assumption that ω 2 is closer to ω 1 than the width of the resonance):

kS max x = πe -1 (1 -˜ P τ )|ω E | . ( 49
)
Note that the maximum of the spectrum does not depend on the temperature. This may seem surprising at a first glance, but actually follows from the fact that the integral of the spectrum is dominated by the quadratic contribution that is proportional to the temperature (equipartition theorem), but now the width of the distribution also is proportional to T, thus the only way to keep the normalization is that the maximum does not depend on T.

We can now obtain the form of the added noise:

kS add x = 2kS max x + 8π k B T P τ τ 2 Γ eV 1 2 . (50)
Explicitly:

kS add x = 2π eC Γ ω 0 (1 -˜ P τ ) 1/2 ˜ 2 P + 8π 2 k B T Γ 1 ˜ P τ τ 2 Γ eV 1 2 .
(51) We set z = 1/ √ 3 that maximizes λ. We then find that both terms in equation (51) are monotonically decreasing as a function of ˜ P . This is normal for the second term, but unusual for the first one that encodes the back-action of the detection system. The minimum of the added noise is thus obtained at the maximum value of ˜ P for which the above expressions are still valid.

In order to find the validity region we evaluate the amplitude of the fluctuation of the variable y. Neglecting the quartic and cubic terms one finds from the equipartition theorem y 2 = k B T/2aΓ. The conditions on the smallness of the cubic and quartic terms read respectively a b y 2 1/2 and a c y 2 . Explicitly for the cubic term:

k B T Γ (3!) 2 2πτ 2 (1 -τ ˜ P ) 3 ˜ 3 P (52)
and for the quartic term

k B T Γ 1 12π 3 |τ | (1 -τ ˜ P ) 2 ˜ 2 P . (53)
If we choose, as usual, the value z = 1/ √ 3 to maximize λ, the quartic term of the expansion vanishes, leaving only the cubic part. We see that the condition given by equation (52) can always be satisfied since even for ˜ P = 1 it reads k B T/Γ 4/3π ≈ 0.42. Note that this is not possible for the case z = 0, for which the cubic term vanishes and the quadratic condition becomes k B T/Γ

(1 -˜ P ) 2 /˜ 2 P 4!π 3 that vanishes at ˜ P = 1. For this value there is a crossover to a purely quartic behaviour of the oscillator. For z = 0 the crossover between the quadratic and cubic correction happens for

1 = k B T Γ ˜ P 1 -˜ P τ 2 τ 2 π 16 , (54)
that defines a small region around the line z = 0 and ˜ P < 1.

Coming back to the expression for S add

x we set ˜ P = 1 and z = 1/ √ 3 to obtain the optimal value:

kS opt x = 3.4 Γ ω 0 + 140 Γ eV 1 2 k B T Γ . ( 55
)
We see that the last term is essentially the same that we calculated in the weak-coupling regime, where the dissipation determines the width. Here we simply could push the expression to its limit by calculating the first part that includes the non-linear contribution. The first term is large, due to the small value of ω 0 , and independent of the temperature (provided it is finite). The reason has been given above and the consequences are that for

k B T Γ < 0.024 eV 1 Γ 2 ≈ 2.4 10 -4 (56)
the current noise becomes negligible with respect to the intrinsic mechanical fluctuations. Using the same parameters as before (ω 0 /Γ = 10 -3 , eV 1 /Γ = 0.1, and k B T/Γ = 10 -2 ) we find an optimal value of kS opt x ≈ 3400, in good agreement with the numerical results of the next section.

Numerical evaluation of the fluctuations

In the previous section we have evaluated the added noise and the response function in different regimes performing some approximations on the evaluation of the stochastic fluctuations. For instance we evaluated the response function by setting z at the equilibrium value. A definition that takes completely into account the fluctuations reads instead:

λ = F 0 eV 1 4πΓ L dxdpP st (x, p)τ (z x ).
(57) In order to find the stationary P st (x, p) that solves the Fokker-Planck equation we discretize the equation and solve numerically the associated linear problem. Defining the Fokker-Planck operator as L such that

∂ t P = LP (58)
one can also find an explicit expression for the displacement spectrum: [14] S x (ω) = -2Tr x L

ω 2 + L2 xP st , (59)
where all the terms with a hat in equation ( 59) are super-operators acting in the space of probability, P st is a vector solution of LP st = 0 and x(t) = x(t) -x (see [14] for more details).

Using this approach we can calculate explicitly the average of λ. In the following all the figures are plotted for the following values of the parameters: ω 0 /Γ = 10 -3 , eV 1 /Γ = 0.1, and k B T/Γ = 10 -2 . We show in figure 4 the average of τ, that is the only contribution to λ that depends on the fluctuations. Also the low frequency (ω Γ) shot noise has to be averaged over the stationary distribution probability:

S shot I = 2k B Te 2 π dxdp P(x, p, t)τ (x). (60)
From these two expressions we can study the dependence of S shot I /λ 2 as a function of ˜ 0 for different values of the coupling constant. We show the result in figure 5. One sees that a higher coupling means a higher sensitivity. The main origin of the ˜ 0 dependence remains that given by the average of τ , with a minimum that moves at lower values ˜ 0 simply by the relation between z and ˜ 0 for given P (see figure 3 and equation ( 37)).

Finally performing the calculation of the spectrum we show in figure 6 the dependence of the maximum of the spectrum compared with the analytical approximations for the case ˜ 0 = 0 and as a function of ˜ P . This figure shows the nice agreement between the weakly non-linear approximation, equation (49), with the numerics till ˜ P ≈ 0.7, as expected from equation (53). For a very small value of the coupling constant ˜ P = 0.1 we believe that our numerical calculation is not sufficiently accurate to correctly reproduce the value of the maximum of the spectrum, since the peak is very narrow. The weak coupling (dissipation dominated) analytical result equation ( 31) is shown dashed. The two analytic approximations cross at a value of P /Γ that agrees with equation (48), ensuring their validity. The two curves disagree at smaller values of the coupling constant since we neglected the The inset shows the position of the minimum of τ obtained by the numerical evaluation (stars) and from the analytical expression (line). In this figure and in the following ones the other parameters are ω 0 /Γ = 10 -3 , eV 1 /Γ = 0.1, and k B T/Γ = 10 -2 . dissipation in the calculation that considers the weakly nonlinear terms.

We can now come to the numerical evaluation of the added noise equation (16) for two different values of the coupling constant: ˜ P = 0.8 and 1. The result is shown in figure 7 as a function of ˜ 0 . If we compare these results with those shown in figure 4, for instance for ˜ P = 1, we see that the position of the minimum moved at slightly lower values of ˜ 0 , but in particular the absolute value of the fluctuation has increased by more than a decade. This is due to the contribution of the fluctuations induced by the thermal motion in the non-linear regime.

We can conclude that the best sensitivity of the device is found for the largest coupling available before the bistability:

˜ P = 1.
One could in principle explore also the bistable region, but the presence of two minima that are quasi degenerate for ˜ 0 1 introduces an additional source of low-frequency noise. As discussed in [25], a strong peak at low frequency due to hopping of the system between the two stable positions appears at the transition. The noise persists for a large region after ˜ P = 1, increasing of 6 orders of magnitude between ˜ P = 1 and ˜ P = 1.2. A second limitation of the bistable region is the strong reduction of the current. The two stable points for large value of the coupling correspond to the empty and full dot. This is particularly clear for 0 = 0. From equation (37) and ˜ 1 one finds

z e ≈ ± π 2 ˜ P (61)
leading to a transparency of the junction of the order of τ ∼ (π˜ P /2) -2 1. The current through the device is thus very weak.

Conclusions

In this work we have theoretically investigated the sensitivity of the mixing current technique when the oscillator is measured by coupling it to a quantum dot in the coherent tunnelling regime. We found that a stronger coupling helps to obtain better sensitivities, mainly because it increases the value of the response function λ, therefore leading to the reduction of the intrinsic electronic current noise once referred back to the displacement. In principle the best coupling value is determined by a compromise between the back-action noise and the electronic noise, but in our case we have shown that the system undergoes a bistability before reaching this ideal value. We thus considered in detail the behaviour of the device close to the bistability region and found the best sensitivity that can be achieved with the device, before entering the bistable regions. We found that in the weak coupling regime the displacement fluctuation has a standard Lorentzian form with a width controlled by the electronically-induced dissipation. In this limit the best sensitivity that can be obtained is given by equation (32). Its validity is constrained by the condition in equation (48) on the coupling constant, thus limiting its scope to a very weak-coupling regime and relatively low value of sensitivity. For larger coupling constant we were able to obtain analytically the form of the displacement spectrum and thus to obtain the sensitivity of the device along the line in the ˜ 0 -˜ P plane defined by z = 1/ √ 3 where λ is maximal. The analytical approach is based on a weak non-linear expansion, that on the line z = 1/ √ 3 holds all the way till the critical value ˜ P = 1. The best sensitivity is given by equation (51) or for ˜ P = 1 by equation (55). We then performed numerical calculations of λ and of the S add x that allowed us to validate the findings of the previously described analytical results and to observe the small deviations. A comparison with the results obtained in [40] for the single electronic level in the incoherent regime can be interesting. There it was found that the optimal value for kS x = 203Q(E C /Γ 0 ), where E C is the Coulomb energy. The Coulomb energy does not enter in our case, it is typically of the same order of Γ 0 or larger. One thus finds that the coherent transport regime should allow better sensitivity. A last comment is in order: we did not investigate the bistable region in detail, since the telegraph noise and the reduction of the current appear to seriously deteriorate the quality of the detectors.

In conclusion, this study indicates clearly that even in the presence of non-linear fluctuations close to the bistability, one finds that increasing the coupling always improves the sensitivity of the device for the detection of the amplitude of oscillation.

In particular in order to evaluate I 1 + I 2 we need the sum y L + y R = -b Ẽ/a 2 . We finally obtain:

I 1 + I 2 = π a 1/2 - Ẽ 4πa 5/2 C (B.7)
with C given by equation (46) in the main text.

ORCID iDs

F Pistolesi https://orcid.org/0000-0002-5897-0347

Figure 1 .

 1 Figure 1. Schematics of the detection device.

Figure 3 .

 3 Figure 3. Stability diagram as a function of ˜ 0 and ˜ P . The region between + and -is bistable. The dashed lines give the values of ˜ 0 for which z = ±1/ √ 3 (see equation (37)) and thus τ is maximum.

Figure 4 .

 4 Figure 4. τ as function of ˜ 0 for different values of the coupling constant ˜ P = 0.2, 0.6, 1.The inset shows the position of the minimum of τ obtained by the numerical evaluation (stars) and from the analytical expression (line). In this figure and in the following ones the other parameters are ω 0 /Γ = 10 -3 , eV 1 /Γ = 0.1, and k B T/Γ = 10 -2 .

Figure 5 .

 5 Figure 5. The value of kS shot I /λ 2 as a function of ˜ 0 for P = 0.2, 0.6, 1.

Figure 6 .

 6 Figure 6. maximum of the displacement spectrum (stars), S max xx , as function of P compared to the two analytical expressions valid in the weak coupling regime (dashed) and in the weakly nonlinear regime (full line).

Figure 7 .

 7 Figure 7. The added noise as a function of ˜ 0 for ˜ P = 0.8 and P = 1.

d (x) = 0 -F 0 x, (3)

Acknowledgments

YW thanks the China Scholarship Council for financial support. FP thanks the Région Nouvelle Aquitaine for financial support.

Appendix A. Single level energy and force

In this appendix we give a short derivation of the gate voltage dependence and of the additional force acting on the oscillator when an electron is added. For more details on the electrostatic model we refer the reader to the appendix of [40]. In order to find the gate voltage dependence of the energy level we simply calculate the difference in the electrostatic energy and electronic energy when one electron is added to a quantum dot where a charge Q is already present:

Using the expressions for the electrostatic energy one finds

with V L , V R , and V g the voltage applied to the left, right, and gate electrode. The first two terms represent the contribution of the local electrostatic energy, the third one is the contribution of the sources, and the last one is the electronic level energy. (We consider spinless electrons.) In order to reach the strong coupling regime one will typically work in the limit of Q e. Moreover for the same reasons typically

. With these approximations one obtains the result given by equation (2).

The variation of the force acting on the oscillator can be obtained by calculating the derivative of ∆E:

With the usual conditions Q e, V V g , and Q = -eN ≈ -C g V g one obtains

where E C = e 2 /2C Σ is the Coulomb energy and L = C g /C g is the length scale of the induced coupling.

Appendix B. Derivation of dω E /dE

The resonating frequency is given by equation ( 43) that can be recast in the form

with

and I 2 the same integral taken between 0 and y R . Here y L,R = (x L,R -x e )F 0 /Γ and Ẽ = E/Γ. By definition Ẽ = ay 2 L,R + by 3 L,R + cy 4 L,R , we can thus substitute this expression in the integrals. We then introduce ξ = y/y L,R . This gives:

3) This form is particularly convenient to perform the expansion for y L,R 1. In this limit the cubic and quartic terms can be treated as a perturbation of the quadratic term. At order y 2 L,R one has:

(B.4) All the integrals can be performed giving:


H Kellay

Hydrodynamics experiments with soap films and soap bubbles: A short review of recent experiments

In this short review, I focus on recent experiments that benefit from the advantages offered by the twodimensionality of the flow in suspended thin liquid films to reconsider hydrodynamics problems which have resisted a full understanding. The first problem discussed here concerns friction drag in channel flows. The use of turbulent channel flows, using thin liquid films, allows measurements of friction drag as well as mean velocity profiles for flows with different spectral exponents. Is there a link between the spectral properties of the turbulence and the mean velocity profiles or the frictional drag? This is the first question considered. The second issue examined considers the long time dynamics of large scale vortices. These are obtained in half bubbles rendered "turbulent" through thermal convection. These vortices, which live in a quasi two-dimensional environment, have a long time dynamics where their vorticity goes through what seems to be a well-defined cycle with generic features.

INTRODUCTION

Since the seminal work of Kraichnan [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF] and Batchelor [START_REF] Batchelor | Computation of the energy spectrum in homogeneous two-dimensional turbulence[END_REF] elucidating the role of spatial dimension on the properties of turbulence, the search was on for finding experimental systems to test and explore these properties. Experimental tests of the phenomenology of turbulence uncovered by this work had become a subject of interest and curiosity: Not one but two inertial ranges, new scaling laws, and a different direction for the energy cascade. Different systems have been studied since by different groups. [START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF][START_REF] Tabeling | Two dimensional turbulence: A physicist approach[END_REF][START_REF] Boffetta | Two dimensional turbulence[END_REF] Experiments have tested the predictions of this work through measurements of scaling laws for different quantities including energy and enstrophy spectra as well as the different moments of velocity or vorticity differences across different scales. [START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF][START_REF] Tabeling | Two dimensional turbulence: A physicist approach[END_REF][START_REF] Boffetta | Two dimensional turbulence[END_REF] Experiments did not stop at those tests and used quasi two-dimensional systems to study scalar dispersion and fluctuations, the dynamics and properties of large scale vortices, the interaction of a flow with different structures, etc. [START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF][START_REF] Tabeling | Two dimensional turbulence: A physicist approach[END_REF][START_REF] Boffetta | Two dimensional turbulence[END_REF] Work using electromagnetically driven flows in thin layers of fluid, flows driven by the excitation of surface waves, or flows in suspended liquid films had set the tone for such investigations and numerous studies have been carried out to examine issues as diverse as mixing, the role of shear on flows of thin liquid films, and the fluctuations of energy injection. Now original work is being carried out in different groups around the globe using such systems or variants of these. [START_REF] Xia | Upscale energy transfer in thick turbulent fluid layers[END_REF][START_REF] Francois | Inverse energy cascade and emergence of large coherent vortices in turbulence driven by Faraday waves[END_REF][START_REF] Liu | Janus spectra in two-dimensional flows[END_REF] Here, in this short review, I will only focus on some very recent work carried out in collaboration with different groups whereby the twodimensional nature of the flows in suspended liquid films has been used to study the dynamics of large vortical structures or to bring new insight into friction drag. It is my purpose here to bring to the attention of the reader that new insight into some "old" issues can be obtained in quasi 2D systems.

FRICTION DRAG IN TWO-DIMENSIONAL CHANNELS

One such issue that has attracted the attention of experimentalists including my group, that of W. I. Goldburg in the US, the group of P. Chakraborty and G. Gioa in Japan, and that of N. Goldenfled in the US is the use of two-dimensional flows to study friction drag in 2D channels. The basic question can be stated simply: does friction drag in 2D show similar phenomenology as in 3D channels where much is known or are there fundamental differences between the two? As it turns out, the 2D case with its two different inertial ranges allows testing theories of friction drag in depth. Let me first introduce friction drag and how 2D flows may bring some insight into this problem which is related to how small scale properties and macroscopic properties of turbulent flows are linked.

Indeed, there are gaps in our fundamental understanding of turbulent flows in channels or pipes. Two aspects of such flows have been the subjects of extensive, but separate, research efforts: their macroscopic properties and their turbulent spectrum. The macroscopic properties of a turbulent flow are the properties of interest for practical and engineering purposes such as the frictional drag experienced by a flow past a wall [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Schlichting | Boundary-Layer Theory[END_REF] and the mean velocity profile of the flow in a pipeline. [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Schlichting | Boundary-Layer Theory[END_REF] The turbulent energy spectrum of the velocity fluctuations on the other hand is representative of the smallscale statistics of the flow as noted by Kolmogorov several decades ago. [START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Kolmogorov | Local structure of turbulence in incompressible fluid at a very high Reynolds number[END_REF][START_REF] Obukhov | Energy distribution in the spectrum of turbulent flow[END_REF][START_REF] Sreenivasan | Fluid turbulence[END_REF] The link between the macroscopic properties and the turbulent spectrum has however remained unexplored. Only recently has experimental and theoretical work begun to focus on this issue and some of these results suggest that this link has important repercussions.

Interest and measurements of turbulent drag in pipes started almost a century ago. In the 1930s, Nikuradse carried out measurements of the friction factor of a turbulent pipe flow as a function of the Reynolds number Re = UR/ν 15 (here U is the mean velocity of the fluid, R is the radius of the pipe, and ν is the kinematic viscosity of the fluid) and as a function of r/R, the scale of the roughness of the pipe wall r normalized by the pipe radius R. [START_REF] Nikuradse | Stromungsgesetze in rauhen rohren[END_REF] This data set is usually shown in the form of plots of the so-called friction factor f (which is a non-dimensional measure of the pressure drop in pipes: f ∼ R∆P/ρU 2 L where ∆P is the pressure drop over a length L, ρ is the density of the fluid, and U is the mean velocity in the pipe) as a function of Re for different values of r/R. These data are shown in Fig. 1 where the asymptotic empirical scaling of Blasius, f ∼ Re 1/4 for smooth pipes, and Strickler, f ∼ (r/R) 1/3 for rough pipes, is clearly visible.

A possible link between the small-scale properties of turbulence and the macroscopic response was originally based on these data and comes from two distinct studies. One development is the roughness-induced criticality theory of Goldenfeld, [START_REF] Goldenfeld | Roughness-induced critical phenomena in a turbulent flow[END_REF] which derives the general form of the scaling law f (Re, r/R): to satisfy the asymptotic scalings f ∼ Re 1/4 and f ∼ (r/R) 1/3 , the friction factor must conform to the scaling law f (Re, r/R) = Re 1/4 g(Re 3/4 , r/R). This scaling law imposes that roughness must play an important role in any description of the turbulent wall bounded flows. The other development is the momentum-transfer theory of Gioia and Chakraborty, [START_REF] Gioia | Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory[END_REF] which derives an expression for f (Re, r/R). This expression is consistent with the scaling law stated above and is a functional of the turbulent energy spectrum E(k). This latter development is the first attempt to give a physical argument explaining Nikuradse's data set. [START_REF] Allen | Turbulent flow in smooth and rough pipes[END_REF] This momentum-transfer theory also yields the exponents of the asymptotic scalings of Blasius and Strickler as functions of the inertial-range exponent of the turbulent spectrum in 3D pipe flows (the spectral exponent of Kolmogorov, 5/3). The momentum transfer theory is especially useful because it can make precise predictions for macroscopic observables, such as the friction factor or the mean velocity profile, and relate their functional forms to the energy spectra.

These recent developments can, in particular, be generalized to arbitrary spectra. For a spectrum of arbitrary inertialrange exponent α [recall that α is defined so that E(k) ∼ k α within the inertial range], momentum-transfer theory yields the generalized Blasius scaling [START_REF] Goldenfeld | Roughness-induced critical phenomena in a turbulent flow[END_REF][START_REF] Gioia | Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory[END_REF][START_REF] Allen | Turbulent flow in smooth and rough pipes[END_REF][START_REF] Gioia | Rough-pipe flows and the existence of fully developed turbulence[END_REF] f ∼ Re (1-α)/(1+α) (1) and the generalized Strickler scaling [START_REF] Goldenfeld | Roughness-induced critical phenomena in a turbulent flow[END_REF][START_REF] Gioia | Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory[END_REF][START_REF] Allen | Turbulent flow in smooth and rough pipes[END_REF][START_REF] Gioia | Rough-pipe flows and the existence of fully developed turbulence[END_REF] f ∼ (r/R) (α-1)/2 .

(2)

These predictions explicitly use the value of the spectral exponent α and therefore can take into account differences in the statistical properties of the turbulence such as those which arise when going from 3D to 2D turbulence. A crucial aspect of this theory is that the wall stress is dominated by the characteristic velocity at the dissipation scale for smooth walls and the scale of the roughness for rough walls. The above relations 1 and 2 are then easily derived from the scaling of the characteristic velocity versus length scale and the scaling of the dissipation scale versus Re. Since the scaling of the characteristic velocities versus scale and that of the dissipation versus Re depend on α, an explicit dependence on this parameter arises in the scaling of f.

A direct test of this powerful theoretical development would need to use different values of α but this is not possible in three-dimensional turbulent flows since the value of α = 5/3, the Kolmogorov value, is observed for most known turbulent flows. A way to change α is to use two-dimensional flows. [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Batchelor | Computation of the energy spectrum in homogeneous two-dimensional turbulence[END_REF][START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF][START_REF] Tabeling | Two dimensional turbulence: A physicist approach[END_REF][START_REF] Boffetta | Two dimensional turbulence[END_REF] We have recently performed experimental measurements of the friction factor in two-dimensional soap-film flows (where the two walls are smooth and the injection of vorticity is carried out using an array of several small cylinders) that exhibit the enstrophy cascade (a type of spectrum that does not exist in 3D flows, and for which α = 3) and verified that on smooth walls f ∼ Re 1/2 , in accord with (1) for α = 3. [START_REF] Tran | Macroscopic effects of the spectral structure in turbulent flows[END_REF] Note that the generalized Blasius scaling of Eq. ( 1) and these experimental results cannot be explained using the classical theory of frictional drag formulated by Prandtl. [START_REF] Schlichting | Boundary-Layer Theory[END_REF][START_REF] Tran | Macroscopic effects of the spectral structure in turbulent flows[END_REF] Numerous variants and alternatives to the theory have been proposed. [START_REF] Schlichting | Boundary-Layer Theory[END_REF][START_REF] George | Is there a universal log law for turbulent wall-bounded flows?[END_REF][START_REF] Barenblatt | A mathematical model for the scaling of turbulence[END_REF] But the original theory of Prandtl and its variants are based on dimensional analysis and similarity assumptions, with no explicit reference to the turbulent spectrum.

In a second experiment, we have used a different way to generate turbulence in a soap film channel by using one rough and one smooth wall along with two different injection schemes (1 rod or an array of rods to generate turbulence) as illustrated in Fig. 2. This type of channel allows obtaining different spectral exponents. [START_REF] Tran | Macroscopic effects of the spectral structure in turbulent flows[END_REF] While the injection scheme using several small rods produces an enstrophy cascade type flow with an exponent α = 3. The spectral exponent α in the second configuration (using 1 rod) is close to 5/3 as expected when an inverse cascade of energy occurs in two-dimensional flows. [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF] The exact reasons for obtaining an exponent of 5/3 following the introduction of a rough wall and an injection of vorticity using a single large rod are not clear but it has been argued 4 that wall roughness changes decaying turbulence into forced turbulence. The wall roughness forces the turbulence at small scales by injecting small scale eddies produced near the rough wall into the main flow favoring an inverse cascade. It is possible that using multiple rods with vortices, the size of the small rods renders observing the inverse cascade scaling more difficult. Nevertheless, the results for the energy spectra shown in Fig. 3 clearly point to two different scaling exponents. This new channel allows obtaining values of α of 3 or 5/3 depending on the injection scheme used: an array of cylinders as in Fig. 2(d) produces α = 3 while a single rod [Fig. 2(b)] produces α = 5/3. This channel, therefore, allows testing the Blasius generalized scaling for two different spectral exponents α by simply switching from a rod to an array of cylinders. We recall in Fig. 2 (right panel) that the friction factor is obtained from measuring the shear rate near the smooth wall in the viscous sub-layer where the velocity varies linearly with distance from the smooth wall. In this sub-layer, the Reynolds stresses are estimated to be negligible so the wall stress is given directly by the shear stress near the wall: the wall shear rate multiplied by the viscosity as indicated in the inset of Fig. 2 (right panel). Figure 3 shows measurements of the friction factor f versus Re for the two different channels. The friction factor scaling versus Re clearly depends on the value of α: as this value changes from 3 to 5/3, the scaling of the friction factor changes from Re 1/2 to Re 1/4 . This variation is in agreement with the expected variation from the generalized Blasius scaling.

The two experiments outlined above show the power of prediction of the simple spectral link theory discussed above, as it is capable of distinguishing between turbulent flows that differ in their spectral content. Considering the importance of linking the macroscopic properties to the fluctuations and of predicting friction drag for a variety of situations, it is essential to put this link to stringent tests beyond the two tests discussed above. Experiments are ongoing to examine the roughness dependence, for example, or the role of polymers on the friction drag. As mentioned above, direct predictions for the roughness dependence of the friction factor are at hand and need to be tested explicitly. Further, the use of polymers reduces drag and even if a large literature is devoted to this problem, no consensus exists as to its origin. Both issues can benefit from the ease of experimentation offered by these films and from the two-dimensionality of the flow which allows measuring all components of the velocity, the vorticity, and the local thickness of the film.

MEAN VELOCITY PROFILES IN 2D TURBULENT CHANNEL FLOWS

A related issue to friction drag concerns the functional form of the Mean Velocity Profiles (MVPs) in a turbulent channel or pipe flows. As is well known now in 3D turbulent pipe flows, the mean velocity profiles have universal features with a structure that is dependent on the distance from the wall. Three regions dominate the mean velocity profile: a viscous sublayer where the viscosity dominates the flow near the wall, a logarithmic layer farther out, and a buffer zone making the link between the viscous layer and the logarithmic zone. Again and considering the importance of the structure of the velocity profiles in setting the friction drag, we set out to determine whether such a universal structure has any relevance to 2D channel flows.

Turbulent flows over solid boundaries are ubiquitous, whether in natural or industrial settings. [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Schlichting | Boundary-Layer Theory[END_REF] For threedimensional (3D) wall-bounded turbulent flows, Prandtl, in 1925, postulated the dominant role of the wall shear stress (which slows down the flow near the wall due to the no slip boundary condition) to write down the law of the wall for the mean velocity profile (MVP). [START_REF] Pope | Turbulent Flows[END_REF] The near-wall region can be divided into three zones. Starting from zero velocity at the wall, the mean velocity linearly increases with distance from the wall in the viscous sublayer. In the buffer layer, this velocity continues to increase but at a smaller rate before changing into a logarithmic variation with increasing distance from the boundary. The logarithmic variation, known as the log law, was derived by von Karman in 1930 and is widely considered to be a fundamental result in the study of 3D turbulent wall bounded flows.

The law of the wall for the mean velocity profile, U(y) [where U(y) is the longitudinal velocity at a distance y from the wall, and denotes averaging over time] is generally written in terms of the wall variables, U + = U(y) / √ τ w /ρ and y + = y/(ν √ ρ/τ w ), where τ w is the wall shear stress, ρ is the density of the fluid, and ν its kinematic viscosity. This law is written as follows: U + = f(y + ), where the functional form of f remains unspecified. In the viscous sublayer, U + = y + . In the log layer U + = (1/κ) ln(y + ) + B, where κ is the von Karman constant and B is the offset of the log law. At high Reynolds numbers (Re), the log region spans most of the spatial domain of the law of the wall, and, as Prandtl demonstrated, for a given Re the log law can be used to predict the wall shear stress. [START_REF] Pope | Turbulent Flows[END_REF] Precise measurements of the two parameters, κ and B, are thus crucial to estimate the wall shear stress. In computing the frictional drag in pipe flows, or dissipation over different engines, these two parameters need to be known with high precision.

Numerous studies have shown that at high Re, these parameters are independent of Re, and that for different types of flows, the variation in κ is within 10% and the variation in B (for flows over smooth walls) is within 25%. [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Marusic | Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues[END_REF][START_REF] Smits | High-Reynolds number wall turbulence[END_REF] These observations indicate that the log law may well be a universal feature of 3D wall-bounded turbulent flows. [START_REF] Gioia | Spectral theory of the turbulent mean-velocity profile[END_REF] The derivation of the law of the wall and the log law is based primarily on dimensional analysis and similarity arguments. [START_REF] Pope | Turbulent Flows[END_REF] These laws are independent of the dimension of the flow and the underlying turbulent spectrum. While 3D flows allow for only one turbulent spectrum (the energy cascade), two-dimensional (2D) turbulent flows allow two different turbulent spectra (the enstrophy cascade and the inverse energy cascade) [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Batchelor | Computation of the energy spectrum in homogeneous two-dimensional turbulence[END_REF][START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Kellay | Two-dimensional turbulence: A review of some recent experiments[END_REF][START_REF] Tabeling | Two dimensional turbulence: A physicist approach[END_REF][START_REF] Boffetta | Two dimensional turbulence[END_REF] as we have seen above for channel flows. In contrast to 3D, there was no experimental study of the MVP in 2D or quasi-2D wall-bounded turbulent flows before. We therefore set out to study experimentally the near-wall MVP of quasi-2D turbulent channel flows. [START_REF] Samanta | Scaling of near-wall flows in quasi-two dimensional turbulent channels[END_REF] As in the previous friction drag measurements, the channels used allow for either energy cascade dominated flows or enstrophy cascade dominated flows with two different spectral exponents. Does the velocity profile depend on any way on the type of spectrum used? Are velocity profiles dependent on spatial dimension? These are the two questions we set out to answer.

To test the validity of the law of the wall and the log law, we measured MVPs for different Re and analyzed them in wall coordinates, U + vs. y + . In Fig. 4, we plot the MVPs for both turbulent spectra in semi-log coordinates. For each spectrum type, the MVPs corresponding to different Re collapse onto a universal curve in the near wall region confirming the validity of the law of the wall, U + = f(y + ). (Farther away, the MVPs for different Re deviate from the universal curve, entering the domain where this law no longer holds.) Examining this universal curve we note the following: in the viscous sublayer, U + = y + ; in the buffer layer, the curve tends toward the logarithmic region; and in the log layer (which in semilog coordinates is a straight line), U + = (1/κ) ln(y + ) + B, thus validating the log law. These results appear to suggest an interesting prospect: that the law of the wall and the log law are not only universal features of 3D wall-bounded turbulent flows but also of quasi-2D wall-bounded turbulent flows. There are, however, two differences concerning the parameters of the log law.

First, consider the von Karman constant, κ. To estimate the value of κ, we take each MVP (such as the ones shown in Fig. 4) and replot as y + dU + /dy + vs. y + , wherein we look for a plateau, which, according to the log law, corresponds to 1/κ. For each type of spectrum, we plot the values of κ vs. Re in Fig. 5. For flows with an inverse energy cascade, we find that κ is approximately constant, with a mean value of 0.14. For flows with an enstrophy cascade, we find that at large Re, κ is approximately constant, with a mean value of 0.13, and that at lower Re, κ assumes smaller values. Within experimental uncertainty, we conclude that at high Re, κ in quasi-2D flows is independent of the turbulent spectra. This value of κ is markedly different from that in 3D flows, where, to our knowledge, such small values of κ have not been observed. [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Marusic | Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues[END_REF][START_REF] Smits | High-Reynolds number wall turbulence[END_REF] Next, consider the offset of the log law, B. Limited by the experimental scatter, we estimate B from the MVPs shown in Fig. 4 using our above mentioned estimate of κ, we compute a best fit line for the log law, whose intercept at y + = 1 corresponds to the value of B (see the arrows in Fig. 4). In contrast to κ, B depends on the turbulent spectrum: for flows with an inverse energy cascade, B ∼ 20; for flows with an enstrophy cascade, B ∼ 7. Interestingly, this latter value is close to 3D channel flows where B = 5. [START_REF] Pope | Turbulent Flows[END_REF] Using quasi-2D turbulent flows in soap-film channels, we have tested that despite the differences with 3D wall-bounded flows in dimensionality and turbulent spectra, the near-wall MVP is composed of a viscous sublayer, a buffer layer, and a log layer, and obeys the law of the wall and the log law. The parameters of the log law are however very different: the von Karman constant κ is independent of the turbulent spectra and is about one-third of its 3D counterpart; the offset of the log law B depends on the turbulent spectra and for the enstrophy cascade is close to its 3D counterpart. These results, which question the classical approach, await theoretical explanation.

VORTICES IN SOAP BUBBLES

In the above two examples, the two-dimensional nature of the flows and their ability to show different spectral properties have allowed testing new theories of friction drag as well as examine the validity of the log law of the wall under hitherto unexplored conditions of dimensionality and spectral properties. In the following, we use the 2D nature of flows in thin liquid layers to examine features related to large scale vortical structures and their dynamics.

Interest and open questions

Vortices are prominent features of fluid flows. They span length scales ranging from an insect's size [START_REF] Hu | The hydrodynamics of water strider locomotion[END_REF] to planetary dimensions. [START_REF] Marcus | Numerical simulation of Jupiter's great red spot[END_REF] The understanding of the movement of these vortices, their structure, as well as their long time dynamics is important for different aspects whether at the level of small engines, natural turbulent flows, or planetary atmospheres. [START_REF] Green | Fluid Vortices[END_REF] Vortices are found in turbulent flows at all scales, in the wake of a bluff body, or in atmospheric flows whether on Earth or on other planets. [START_REF] Frisch | Turbulence: The Legacy of A. N. Kolmogorov[END_REF][START_REF] Marcus | Numerical simulation of Jupiter's great red spot[END_REF] In the latter case, the observed single vortices are giant and very long lived. [START_REF] Marcus | Numerical simulation of Jupiter's great red spot[END_REF][START_REF] Polvani | Simple dynamical models of Neptune's great dark spot[END_REF] These vortices may also be short lived and capable of long distance travelling such as tropical cyclones (TC) on Earth. [START_REF] Chan | Global Perspectives on Tropical Cyclones[END_REF][START_REF] Emanuel | Tropical cyclones[END_REF] The exact structure of giant vortices and notably that of the great red spot on Jupiter or the dark spot on Neptune is still being examined, but some of their features have been measured: The two great spots are elliptical, for example, [START_REF] Polvani | Simple dynamical models of Neptune's great dark spot[END_REF][START_REF] Liu | Flow structures of Jupiter's great red spot extracted by using optical flow method[END_REF][START_REF] Choi | Velocity and vorticity measurements of Jupiter's great red spot using automated cloud feature tracking[END_REF] and tropical cyclones can be modeled with some success using a modified version of an ideal vortex known as the Rankine vortex. [START_REF] Mallen | Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency[END_REF] Nevertheless, when it comes to predicting their trajectories (as for TCs on Earth), the evolution of their intensity, or their lifetime, vortices resist description and modeling mainly due to the fact that their interactions with the environment are complex. [START_REF] Emanuel | Tropical cyclones[END_REF][START_REF] Emanuel | Thermodynamic control of hurricane intensity[END_REF]39 Take, for example, the intensity of tropical cyclones (wind speed) and its temporal evolution. This is an issue that has been intensely studied over the past few decades with some success: thermodynamical models as well as simple coupled ocean atmosphere models seem to capture some of the essential features. [START_REF] Emanuel | Thermodynamic control of hurricane intensity[END_REF][START_REF] Ooyama | Numerical simulation of the life cycle of tropical cyclones[END_REF] However, its prediction remains a formidable task: several factors such as the interaction with the sea and the environment, the structure of the TC itself, as well as the role of rotation on the dynamics of such vortices 39,[START_REF] Smith | An investigation of rotational influences on tropical-cyclone size and intensity[END_REF] have to be considered. It is therefore highly desirable to have simple experimental systems where the study of the generic properties of these vortices can be carried out. [START_REF] Francois | Inverse energy cascade and emergence of large coherent vortices in turbulence driven by Faraday waves[END_REF][START_REF] Smith | An investigation of rotational influences on tropical-cyclone size and intensity[END_REF][START_REF] Sommeria | Laboratory model of a planetary eastward jet[END_REF][START_REF] Solomon | Shear flow instabilities and Rossby waves in barotropic flow in a rotating annulus[END_REF][START_REF] Seychelles | Thermal convection and emergence of isolated vortices in soap bubbles[END_REF][START_REF] Meuel | Hurricane track forecast cones from fluctuations[END_REF][START_REF] Turner | The carbonated-water tornado vortex[END_REF]

Experimental realization of flows in curved geometry and observations

In recent years, we have, in my group at Bordeaux, introduced a system where a half bubble heated from below is used to shed light on these problems. First of all, and to mimic atmospheric flows, the geometry needs to be spherical. Second, the FIG. 6. The bubble: (left) setup: a brass disk (1) with a circular groove (3) can be rotated using a continuous motor (6) connected to it by a shaft (5). This disk is heated by the proximity of a hollow annulus (2) connected to a water circulation bath. The bubble is blown using the soap solution in the groove (3). The inner side of the brass disk is covered by a Teflon coating (2 mm thick) to minimize the heating of the air inside the bubble. The temperature at the equator of the bubble is set by the temperature of the water bath. (Right) Images showing the detachment of thermal plumes from near the equator and rising towards the pole, images of the full bubble with a vortex being formed by a large thermal plume and a zoom on a vortex, the colors are interference colors of white light being reflected by the thin water layer constituting the bubble.

system should be amenable to simple diagnostics and measurements. The half bubble system allows these features: the geometry is naturally curved, and the thin liquid film nature of the bubble makes it relatively easy to visualize the flow and its prominent features. Further, the system is simple enough to introduce rotation effects and therefore mimic, even more realistically, the sought after flows. In order to produce flow in such a system, a simple scheme consists of heating the bubble at the equator. Direct visualizations using color CCD cameras or infrared cameras allow extracting the thickness field as well as the temperature field. In addition, and either through following thickness inhomogeneities or through the use of Laser Doppler velocimetry (after seeding the solution with micrometer size particles), information about the velocity field or the velocity fluctuations can be obtained.

The system used is a half bubble heated from below [START_REF] Seychelles | Thermal convection and emergence of isolated vortices in soap bubbles[END_REF][START_REF] Meuel | Hurricane track forecast cones from fluctuations[END_REF][START_REF] Meuel | Intensity of vortices from soap bubbles to hurricanes[END_REF][START_REF] Seychelles | From intermittent to nonintermittent behavior in two dimensional thermal convection in a soap bubble[END_REF] in a specially designed cell capable of rotating the bubble at different rates (see Fig. 6). Once formed, the bubble is subject to strong convection due to the heating at the base of the bubble. [START_REF] Seychelles | From intermittent to nonintermittent behavior in two dimensional thermal convection in a soap bubble[END_REF] Images of the bubble with well-defined thermal plumes are shown in Fig. 6. By the way, this convection and its statistical properties can be studied in detail both for the velocity fluctuations using laser Doppler velocimetry and through infrared imaging for the temperature fluctuations. [START_REF] Seychelles | From intermittent to nonintermittent behavior in two dimensional thermal convection in a soap bubble[END_REF] The most intriguing aspect is that after a short period, a large vortex may emerge. The emergence of vortices occurs both for bubbles not subjected to rotation and in bubbles subjected to rotation. The formation of these vortices occurs when a large plume, the result of the merging of a few smaller ones, rises to near the top of the bubble and forms a swirl as shown in Fig. 6. This is only one possible way to form these vortices and our study did not allow us to uncover all the possible ways leading to their formation. Once they form, the large vortices are followed using video imaging. A large vortex of this kind is visible in Fig. 6. The vortex has a well-defined center or "eye" and develops as a spiral structure of dimensions between 1 and 2 cm.

From velocity and vorticity profile measurements, it turns out that these vortices have profiles that are typical of a variety of known vortices. [START_REF] Green | Fluid Vortices[END_REF] The ones examined here can be reasonably approximated by a Gaussian vortex. [START_REF] Green | Fluid Vortices[END_REF] The azimuthal velocity profile of such a vortex obeys V(r) = Γ/(2πr)(1exp(-r 2 /4λ 2 )). Here Γ is the circulation, r is the distance from the center of the vortex, and λ is a characteristic length. From the azimuthal velocity, we obtain the vorticity profile as ω(r) = V(r)/r + ∂V(r)/∂r. The Gaussian functional shape seems to capture the essential features of this vortex.

Long time dynamics of vortices

Besides the characterization of the structure of the observed vortices, we have also made measurements of the azimuthal velocity of the vortex over longer periods of time spanning several turnover times of the vortex. A typical turnover time of these vortices is a fraction of a second (0.1 s). An example of the long time dynamics of the azimuthal velocity is displayed in Fig. 7. Note that a long time variation can be observed in these measurements. This long time variation shows periods of constant low velocity and vorticity followed by an intensification period with higher velocities and vorticities. The measured velocity is for the same distance from the center so variation in the velocity is not related to the probe particle meandering towards the center of the vortex or moving away from it. Changes in the velocity or vorticity are solely due to the change of the vortex intensity in this case. The vortex followed here ends up reducing its velocity in the end and practically disappears in the background flow. We are not aware of other measurements of such long time dynamics of vortex properties in other systems or in theoretical work.

A curious aspect of our observations is that during the intensification period, the trajectory of the center of the vortex shows a trochoidal like motion as seen in Fig. 7: the vortex center wobbles a few times around its mean position. The period of this oscillation of the center of the vortex is roughly one turnover time that is comparable to the oscillation period observed for some tropical cyclones. This is seen for different vortices with the period of trochoidal motion being roughly the same and close to 0.1 s. The link between this trochoidal motion and the intensity of the vortex in our case is not clear since this type of motion was observed both for the increasing intensity phase and during the decreasing intensity one. Different reasons have been proposed to explain this peculiar feature observed during the motion of some TC including instability of the core of the vortex or the existence of a double vortex structure giving rise to a periodic displacement of the vortex core with respect to its periphery. Our own observations do not allow a precise determination of the mechanisms at play here but our velocity measurements seem to exclude the double vortex hypothesis, and visualizations of the vortex do not seem to indicate large deformations of the core during this phase.

Let us come back to the intensification of vortices. In Fig. 8 we show several intensification events from the soap bubble as well as from numerical simulations of a half hemisphere heated at its equator and carried out by Bruneau and Fischer from the Institute of Mathematics in Bordeaux. [START_REF] Meuel | Intensity of vortices from soap bubbles to hurricanes[END_REF] Both FIG. 8. Superposition of intensification events from the bubble vortices and from the numerics: vortex intensity from five different intensification events at different frequencies (of 0, 0.2, and 0.6 Hz) versus time. Three intensification events from the numerics are also shown. The velocity axis has been normalized by the maximum velocity and the time has been normalized by a characteristic time τ and shifted so that the position of the maximum velocity is at zero. Adapted from Meuel et data sets, from five different intensification events in the soap bubble and three different events from the numerical simulations, show an intensification period where the velocity increases up to a maximum value we note V max followed by a decrease in intensity. The time scale in this figure has been shifted, so that V max occurs at time zero, and normalized by a characteristic time τ. The velocity was simply normalized by V max . An intriguing feature of this representation of the data is that a simple rescaling of the velocity and time axes leads to a collapse of all data sets onto a single universal curve suggesting that the intensification dynamics shows similar generic features for different vortices at different bubble rotation rates as well as for vortices from the numerical simulations. The time constant τ used for the rescaling of the data in this figure turns out to be roughly 0.07 s for the bubble vortices and for the vortices from the numerical simulations. This time scale is comparable to a turnover time. The mechanisms behind the intensification and decline of the vortices are however difficult to decipher.

The experimental system proposed here shows a variety of interesting features concerning the properties of vortices on the surface of the bubble: the existence of long lifetime vortices whose structure is well defined and whose long time dynamics shows intriguing properties such as intensification events and trochoidal motion. These features do not seem to have been observed before in other experimental systems so it is difficult to compare these observations to other laboratory experiments or to theoretical predictions.

Comparisons with tropical cyclones

While the vortices observed here are quasi twodimensional and therefore very different from natural giant vortices such as Tropical cyclones, some of the properties observed in our system (trochoidal motion and intensification) are also characteristic of Hurricanes and Typhoons. For the sake of comparison, we have examined whether such qualitative similarities can be made more quantitative. A particular feature we have examined in detail concerns the link between the intensification observed here and that of tropical cyclones. These are the only known vortices for which intensification as observed in our experiments is documented. In Fig. 9, we show the variation of the wind velocity V(t) versus time t for a few hurricanes (data obtained from Ref. 49). Note that the normalization used above is useful to superimpose intensification data from different hurricanes. This figure shows that this rescaling works reasonably well for this data set. The only difference is the time constant (given in the figure) which varies but hovers around 6 h.

We went a step further and superimposed data from other hurricanes alongside our data in Fig. 10. The curves from different tropical cyclones, from our vortices, and from our numerical simulations are superimposed in this representation suggesting that the variation of the velocity versus time is similar for such very different vortices spanning several orders of magnitude in length and time scales. In itself this result is perhaps not surprising since many different vortices may show an increase and a decrease in intensity. However, a notable feature is that the characteristic time τ needed to normalize the data turns out to be roughly constant and of order 6 h for TCs and 0.07 s for our vortices as shown in the inset. For the numerics, the value of τ is about 0.07 s for the three vortices shown in very good agreement with the experiments. Each system therefore seems to be characterized by a single mean time constant. While the exact meaning of this time constant is not clear at present, its order of magnitude points to roughly one turnover time for the experiments and the numerics, and roughly one turnover time for the tropical cyclones if the radius of hurricane force winds is considered. In order to test whether for tropical cyclones the time constant is robust and not just a coincidence, we have performed an analysis of a large ensemble of events. A time constant was obtained from superimposing all the data onto the same universal curve. The histogram of τ values obtained from an analysis of 171 TCs in the Atlantic and the Pacific basins is shown in Fig. 10. Note that the histogram is well defined suggesting that the value of τ has a mean of 6 h with a standard deviation of about 2 h. Furthermore, and in Fig. 10, we have added data from a compilation of tropical cyclone intensity variation with time (over 56 storms in the Atlantic and 73 storms in the Pacific) obtained from Ref. 50. The normalization as above of this data set using a time constant of 6 h works reasonably well.

The superposition of the data shown in Fig. 10 suggests that a simple relation for the variation of the wind speed versus time can be written: V(t) = V max (1 ± (tt max )/16τ) or equivalently ∂V(t)/∂t = ±V max /16τ for the increase and the decrease in intensity before and after t max . Note that there are deviations for t > t max (see Katrina case in Fig. 10) when the hurricane approaches landfall and the intensity decreases fast. From this relation, the maximum attainable wind speed V max is directly related to the rate of change of the velocity during the intensification period. Separate estimates of the velocity rate of change ∂V(t)/∂t and the maximum speed V max agree with this relation as seen in the inset of Fig. 10. Further, since the temporal variation of the velocity turns out to be roughly linear, the duration of the intensification period or t max can be determined as it lasts 16τ. Note that the mechanisms at play in the intensification of vortices here and of tropical cyclones are very different: our vortices are prone mostly to interactions with the background flow while tropical cyclones even though very extended in their lateral dimension have a full and complex three-dimensional structure which is essential to their dynamics. Nevertheless, the superposition of the data sets suggests that the dynamics of the intensification of vortices may have generic features that call for a full understanding.

CONCLUSION AND PERSPECTIVES

This short review brought together recent work on flows in thin liquid films, either in flat or curved geometries. In both cases, the quasi two-dimensional nature of the system allows to reconsider important fluid dynamics questions. Friction drag with different turbulent flows, and vortex properties in curved geometry have been revisited. These thin liquid films can be used to consider other problems such as the interaction of flows with different structures. Experiments on thin filaments, deformable structures, or flow induced oscillations have been examined. [START_REF] Childress | Fluid-structure interactions: Research in the courant institute's applied mathematics laboratory[END_REF][START_REF] Orchini | Galloping instability and control of a rigid pendulum in a flowing soap film[END_REF][START_REF] Lacis | Passive appendages generate drift through symmetry breaking[END_REF] One interesting development by my group in collaboration with an Italian and a Swedish team focused on flow induced symmetry breaking in the problem of flow structure interaction. [START_REF] Lacis | Passive appendages generate drift through symmetry breaking[END_REF] This development opens the way towards passive flow induced locomotion with implications for sedimentation and particle transport in turbulent flows. Other perspectives are open in the use of these films to study flows in porous like materials in the presence of numerous obstacles or flows with deformable walls. The half bubbles introduced here can be rotated in a simple setup opening the way to a full study of rotation effects on thermal convection or Marangoni convection in curved geometries. Our initial studies already indicate the suppression of turbulent fluctuations with the Rossby number and reduced heat flux. There are certainly other developments to come.

FIG. 1 .

 1 FIG. 1. Frictional drag f vs Re for pipe flows on walls of roughness r/R: Nikuradse's data. Note the empirical Blasius and Strickler scalings. Adapted with permission from G. Gioia and P. Chakraborty, "Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory," Phys. Rev. Lett. 96, 044502 (2006). Copyright 2006 American Physical Society.

FIG. 2 .

 2 FIG. 2. (Left) The new soap film channel: (a) the film (in gray) hangs between two steel blades (shown in black) that are suspended by fishing wires kept taut by weight W. One of the blades has a serrated edge (size of serrations 2 mm spaced by 4 mm). The soap solution is injected from an upper reservoir and collected at a bottom reservoir. The gravity driven flow reaches mean velocities, tuned by varying the injection flux, between 0.5 and 3 m/s. The frictional drag is measured only on the smooth side of the channel. (b) Turbulence is generated by a rod of 6 mm diameter resulting in a flow with a 5/3 exponent. The turbulence can be made visible at a particular instant by (c) interference fringes in white light. (d) Turbulence is generated with the teeth of a comb (diameter 1 mm and spacing of 3 mm) resulting in a flow with a 3 exponent. The turbulence can be made visible in (e) using interference fringes in white light. (Right) Velocity profile near the smooth wall: Lower inset shows a zoom on the viscous sub-layer where the velocity varies linearly with distance to the wall giving access to the shear rate at the wall. The upper inset shows the shear stress versus distance from the wall. Note that the shear stress increases as the wall is approached and shows a well-defined value near the wall. Adapted with permission from Kellay et al., "Testing a missing spectral link in turbulence," Phys. Rev. Lett. 109, 254502 (2012). Copyright 2012 American Physical Society.

FIG. 3 .

 3 FIG. 3. Log-log plots of the energy spectrum [longitudinal (solid line) and transverse (dashed line) components are shown] in the two channels. (a) A rod at the inlet while (b) a comb at the inlet. Spectra are shown at two distances in the channel: near the center y/w = 0.5 and near the wall at y/w = 0.125 (w is the width of the channel) showing the homogeneity of the turbulence in the transverse direction to the flow. (c) Friction factor variation versus the Reynolds number for the two flow configurations of Fig. 4. Note the difference in the exponent from 1/2 to 1/4 as predicted when the cascade switches from an enstrophy cascade to an energy cascade. (a) and (b) are reproduced with permission from Kellay et al., "Testing a missing spectral link in turbulence," Phys. Rev. Lett. 109, 254502 (2012). Copyright 2012 Americal Physical Society. (c) Part of the data in this figure is adapted from Ref. 24.

FIG. 4 .

 4 FIG. 4. Velocity profiles for the two different cascades (left: energy cascade and right: enstrophy cascade): wall units have been used here and data from different Reynolds numbers (given in the legends) are rescaled together. The data in this figure are adapted from Ref. 28.

FIG. 7 .

 7 FIG. 7. Long time of the azimuthal velocity and the vorticity of a single vortex, the upper inset shows the trajectory with visible trochoidal motion during the intensification phase marked by the red and black dots. The bottom inset shows the variation of the latitude and longitude of the vortex position versus time for the trajectory in the upper inset. Note the oscillations in the trajectory. Adapted from Meuel et al., "Intensity of vortices from soap bubbles to hurricanes," Sci. Rep. 3, 3455 (2013). Copyright 2013 Author(s), licensed under a Creative Commons Attribution 3.0 Unported License.

FIG. 9 .

 9 FIG. 9. Intensification for several hurricanes in the North Eastern Pacific. Here again the time and velocity axes have been normalized as in Fig. 8. The time constants are given along with the name of the TC in the figure. Adapted from Meuel et al., "Intensity of vortices from soap bubbles to hurricanes," Sci. Rep. 3, 3455 (2013). Copyright 2013 Author(s), licensed under a Creative Commons Attribution 3.0 Unported License.

FIG. 10 .

 10 FIG. 10. Hurricane intensity from five different hurricanes in the Atlantic as well as two compilations extracted from Ref. 50 for the Atlantic and Pacific oceans and vortex intensity from five different intensification events versus time. Three intensification events are from the numerics. The upper right figure shows the characteristic time versus maximum velocity for the events shown in the main figure and for a few additional hurricanes (Georges 1998, Bill 2009, Rita 2005, Katrina 2005, Beta 2005, Helene 2006, Paloma 2008, Dolly 2008, Omar 2008, Isaac 2006, Earl 2010). The bottom right figure shows the linear relation between intensification rate and maximum velocity for a dozen hurricanes (different from the ones used in the upper right figure and the main figure: Opal 1995, Andrew 1992, Hugo 1989, Dean 1989, Gilbert 1988, Gloria 1985, Camille 1969, Igor 2010, Gordon 2006, Florence 2006, Dean 2007, Chris 1994). The bottom figure shows a histogram of the time τ obtained from an analysis of 171 tropical cyclones in the Atlantic and the Pacific oceans. The mean value of τ is 6 h with a standard deviation of 1.8 h. Adapted from Meuel et al., "Intensity of vortices from soap bubbles to hurricanes," Sci. Rep. 3, 3455 (2013). Copyright 2013 Author(s), licensed under a Creative Commons Attribution 3.0 Unported License.

  al., "Intensity of vortices from soap bubbles to hurricanes," Sci. Rep. 3, 3455 (2013). Copyright 2013 Author(s), licensed under a Creative Commons Attribution 3.0 Unported License.

ACKNOWLEDGMENTS

This work has been carried out with several colleagues and students to whom I am indebted and thankful: W. I. Goldburg, X. L. Wu, N. Goldenfeld, P. Chakraborty, G. Gioia, C. H. Bruneau, P. Fischer, M. Bessafi, F. Ingremeau, T. Meuel, F. Seychelles, T. Tran, R. Cerbus, D. Samanta, S. Bagheri, and A. Mazzino.