B B Augstein

C Figueira

Morisson Faria

Influence of asymmetry and nodal structures on high-harmonic generation in heteronuclear molecules

Keywords: numbers: 33, 80, Rv, 42, 65, Ky

The relation between high-harmonic spectra and the geometry of the molecular orbital in position and momentum space is investigated. In particular we choose two isoelectronic pairs of homonuclear and heteronuclear molecules, such that the highest occupied molecular orbital of the former exhibits at least one nodal plane. The imprint of such planes is a strong suppression in the harmonic spectra, for particular alignment angles. We are able to identify two distinct types of nodal structures. If, for homonuclear molecules, the nodal planes are determined by the atomic wavefunctions only, the angle for which the yield is suppressed will remain the same for both types of molecules. In contrast, if they are determined by the linear combination of atomic orbitals at different centers of the molecule, there will be a shift in the angle at which the suppression occurs and a distortion in the nodal structure for the heteronuclear molecule, with respect to its homonuclear counterpart. This shows that, in principle, molecular imaging, in which homonuclear molecule is used as a reference while observing the wavefunction distortions in its heteronuclear counterpart, is possible.

Introduction

High harmonic generation, first discovered in the late 1980s [1], is a process which occurs when atoms or molecules are exposed to intense laser fields, and results in the emission of high frequency coherent radiation. It can be readily understood in terms of the three step model, [2,3], where an electron is ionized from the atom or molecule, propagates in the laser field, and recombines with its parent ion, upon which high harmonics are released.

Currently, there is much interest in this process due to the possibility of tomographic orbital reconstruction [4,5,6], and the study of quantum interference effects [7,8,9]. In particular quantum-interference minima and maxima due to highharmonic emission at spatially separated centers, and high-harmonic suppression due to the presence of nodal planes in the orbital wavefunctions, have been identified and discussed theoretically. Such studies have been extensively performed in diatomic molecules [6,7,8,9,10,[START_REF] De | [END_REF]20]. In many of these studies so far, homonuclear molecules with definite orbital symmetry, either gerade or ungerade, have mostly been used, whereas heteronuclear molecules driven by strong fields are now starting to attract attention [11,12,13]. The orbital wavefunctions in such molecules do not have a definite orbital symmetry. Furthermore, heteronuclear molecules also contain a static dipole moment. These attributes will affect the harmonic spectra and the observables related to other strong field phenomena. Hence, a legitimate question to ask is whether two-center interference patterns, or the imprint of nodal surfaces in general, in form of high-harmonic suppression, can still be observed in the spectra. This is specially interesting, since, experimentally, in the past few years, it has become possible to orient the permanent dipole moments of heteronuclear molecules with regard to the polarization direction of the driving field [11].

In this work, we compare isoelectronic homonuclear and heteronuclear molecules, with particular emphasis on nodal planes in the position space wavefunction and twocenter interference effects. We assume the Born-Oppenheimer approximation to be valid, which should hold for large enough molecules [9]. Specifically, we take the pairs Be 2 and LiB, and O 2 and NF. The homonuclear molecule in the former and the latter pair have a σ u and π g highest occupied molecular orbital (HOMO), respectively. Such orbitals contain nodal planes which are altered when a heteronuclear counterpart is chosen. In our investigations, we employ the single-active electron and the strong-field approximation and assume that the HOMO is the only active orbital. The influence of multiple orbitals has been discussed elsewhere [6,[START_REF] De | [END_REF]16].

This work is organized as follows. In Sec. 2 the molecular orbital construction is briefly stated, and the three step model (Sec. 2.1), saddle-point equations (Sec. 2.2) and interference conditions (Sec. 2.3) are discussed. In Sec. 3, we present our results. This section is split into two main parts. First, in Sec. 3.1, we address the molecular orbitals obtained by us, with a particular emphasis on their nodal structures, or the absence thereof. Subsequently, in Sec. 3.2, we discuss how such characteristics manifest themselves in the spectra. Finally, in Sec. 4, we state the main conclusions to be drawn from this work. Unless otherwise stated, we use atomic units throughout.

Model

Below, we will discuss the transition amplitudes employed in this work, for homonuclear and heteronuclear molecules. The molecular orbitals are calculated using the Linear Combination of Atomic Orbitals (LCAO) approximation, along with the Born Oppenheimer approximation. Under these assumptions, the molecular orbital wavefunction is given by

Ψ(r) = α c (L) α Φ (L) α (r + R/2) + (-1) lα-mα+λα c (R) α Φ (R) α (r -R/2) ( 1
)
where R is the internuclear separation, l α is the orbital quantum number and m α is the magnetic quantum number. We assume that the ions are positioned along the z axis, i.e., that R = Rê z . The coefficients c (ξ) α form the linear combination of atomic orbitals and are extracted from the quantum chemistry code GAMESS-UK [START_REF] Bush | GAMESS-UK is a package of ab initio programs[END_REF]. The indices (L) and (R) refer to the left or to the right ion, respectively, which is a distinction required for heteronuclear molecules, whereas for homonuclear molecules, c

(L) α = c (R) α = c α and Φ (R) α = Φ (L) α = Φ α .
The internuclear axis is taken to be along the z direction in the molecular fixed frame, and the direction of the laser polarization defines the z axis in the lab frame. The parameter λ α determines the orbital symmetry of the molecular orbital, with λ α =|m α | and λ α =|m α |+1 for homonuclear molecules of gerade and ungerade symmetry, respectively. Throughout, we have employed Gaussian type orbitals with a real 6-31 basis set composed of only s and p states. We find that polarized basis sets make little difference to the orbital wavefunctions. For more details on these wavefunctions we refer to our previous work [START_REF] De | [END_REF].

Transition amplitude

The transition amplitude for high harmonic generation, within the framework of of the strong field approximation (SFA) [2], reads as,

b Ω = -i ∞ -∞ dt t -∞ dt d 3 ka * rec (k + A(t))a ion (k + A(t )) exp[iS(t, t , Ω, k)] + c.c., (2)
with the action

S(t, t , Ω, k) = - 1 2 t t [k + A(τ )] 2 dτ -E 0 (t -t ) + Ωt (3)
where

a rec (k + A(t)) = k + A(t)| d • e x |Ψ and a ion (k + A(t )) = k + A(t )| E(t ) • re x |Ψ
, are the recombination and ionization dipole matrix elements, respectively. The terms d, e x and Ω denote the dipole operator and the laser-polarization vector and the harmonic frequency, respectively. The ionization potential of the highest occupied molecular orbital, denoted by |Ψ , is given by E 0 . The length gauge is used throughout as the interference minima in the harmonic spectra vanish in the velocity gauge [17,18], and the length form of the dipole operator is used as it accounts for the presence of the static dipole moment ‡. In the present work, we are neglecting the timedependent Stark shifts in the ionization potential. Recently, a modified SFA including ‡ In the length form, the dipole operator reads d = r. Recently, we have shown that if a Gaussian basis set is employed in the modeling of the molecular orbitals, the velocity form of the dipole operator leads to a vanishing static dipole moment in heteronuclear molecule. This problem is absent if the length form is taken. For details see [20].

such shifts has been developed and applied to heteronuclear molecules [12,13]. As a first approximation, as long as the static dipole moments are small, such Stark shifts can be neglected. Within the present framework, one may either incorporate the structure of the molecule in the dipole matrix elements, or in the action. If the former procedure is followed, all structural information about the molecular orbital is contained in the recombination matrix element, a rec (k + A(t)), which is given by

a rec (k) = 1 (2π) 3/2 d 3 rr • êz exp[-ik • r]Ψ(r), (4)
where Ψ(r) is defined in Eq. ( 1). This is the component of i∂ k Ψ(k) along the laser-field polarization, where

ψ(k) = α exp[ik• R 2 ]c (L) α Φ (L) α (k) ( 5 ) + (-1) lα-mα+λα exp[-ik• R 2 ]c (R) α Φ (R) α (k),
denotes the momentum-space wavefunction. This matrix element gives rise to any two center interference which occurs in the harmonic spectra. The ionization matrix element, a ion (k + A(t )), on the other hand, will mainly determine whether tunnel ionization will be suppressed or enhanced due to the shape of a particular orbital. Therefore it directly affects the overall harmonic yield, but is not responsible for any interference patterns. Both matrix elements are vanishing for nodes in the molecular orbitals.

If the latter procedure is followed, this will lead to the coherent sum

M = µ,ν M µν (6)
over four distinct possible processes, where

M µν = -i ∞ -∞ dt t -∞ dt d 3 kJ µν (k, t, t ) exp[iS µν (t, t , Ω, k)] + c.c.. (7)
The indices µ, ν refer to the left or right ion in the molecule, as L or R. Each of these terms will exhibit a similar form to Eq. ( 2), with S(t, t , Ω, k) replaced by the modified actions S µν (t, t , Ω, k), where µ and ν indicate the ion with which the electron recombines and from which it is released, respectively. The prefactor a * rec (k + A(t))a ion (k + A(t )) will be replaced by

J µν (k, t, t ) = α,β Ξ µ,ν c * (µ) α c (ν) β ∂ k(t) Φ * (µ) α (k(t))∂ k(t ) Φ (ν) β (k(t )), (8)
where k(τ ) = k + A(τ ), with τ = t, t , and, once more, the indices µ, ν refer to the left or right ion. The coefficients Ξ µ,ν are introduced to account for the parity of the atomic wave functions composing the orbitals, such that Ξ

L,L = 1, Ξ R,L = (-1) lα-mα+λα , Ξ L,R = (-1) l β -m β +λ β and Ξ R,R = Ξ L,R Ξ R,L .
The modified actions are defined as

S νν (t, t , Ω, k) = S(t, t , Ω, k) ± [A(t) -A(t )] • R/2 ( 9 )
forµ = ν, where the negative and positive signs apply to the left and right ion, respectively, and

S µν (t, t , Ω, k) = S(t, t , Ω, k) ± k • R ± [A(t) + A(t )] • R/2 (10)
for µ = ν. In Eq. ( 10), the positive and negative signs correspond to S RL and S LR , respectively. We will show below that the action (9) leads to the so-called "direct harmonics", for which the active electron leaves and recombines with the same ion, while Eq. ( 10) gives the so-called "exchange harmonics", for which the electron tunnels from one ion and recombines with the other. Within the strong-field approximation, such harmonics have been first identified in [22]. For systematic studies of the exchange harmonics using saddle-point methods and a linear combination of 1s orbitals see [23,18].

Saddle-point equations

To solve the HHG transition amplitude (2) we use the saddle-point approximation [19], i.e., we seek values for t, t and k such that the action is stationary. This results in saddle-point equations which can be related to the orbits of an electron in an external laser field.

If one considers the transition amplitude (2), i.e., if one incorporates the influence of the molecule in the prefactor, then the saddle-point equations ∂S(t, t , Ω, k)/∂t = ∂S(t, t , Ω, k)/∂t = 0 and ∂S(t, t , Ω, k)/∂k = 0 lead to

[k + A(t )] 2 2 + E 0 = 0, (11) t
t dτ [k + A(τ )] = 0. (12)
and

[k + A(t)] 2 2 + E 0 = Ω. ( 13
)
These equations have the following physical interpretation. Initially the electron is tunnel ionized from the binding potential. This results in Eq. (11), which gives the conservation of energy. Note that this equation has only complex solutions, as a consequence of the fact that tunneling has no classical counterpart. Secondly, Eq. ( 12) results in the electron momentum in the continuum being constrained such that the electron returns to its place of origin, which we take to be the geometric center of the molecule. Finally, Eq. ( 13) represents conservation of energy at recombination. The return condition (12) guarantees that the momentum k and the external field are collinear. Hence, for a linearly polarized field the angle between the intermediate momentum k and the internuclear axis R is equal to the orientation angle θ L between the molecule and the field.

If, on the other hand, one incorporates the molecular structure in the action, it will be necessary to distinguish between the direct and exchange harmonics. The former will lead to the saddle-point equations ∂S νν (t, t , Ω, k)/∂t = ∂S νν (t, t , Ω, k)/∂t = 0 and ∂S νν (t, t , Ω, k)/∂k = 0, where S νν (t, t , Ω, k) is given by Eq. ( 9). The latter can be obtained from the derivatives of S νµ (t, t , Ω, k), µ = ν, i.e., Eq. ( 10), with regard to t, t and k.

Tunnel ionization and recombination for both the direct and the exchange harmonics will be given by the saddle-point equations

[k + A(t )] 2 2 + E 0 ± E(t ) • R/2 = 0 (14)
and

[k + A(t)] 2 2 + E 0 ± E(t) • R/2 = Ω. ( 15
)
The terms proportional to the field E(τ ), τ = t, t , and to the internuclear distance R can be interpreted as potential-energy shifts, which appear due to the fact that the ions the molecule are displaced with regard to the origin of the coordinate system. These potential-energy shifts appear when the length-gauge formulation of the SFA is considered, and have raised a great deal of debate (for details see [23,18]). For the direct harmonics generated at the left and the right ions (ν = L and ν = R), one must take the negative and positive signs for the potential-energy shifts in Eqs. ( 14) and ( 15), respectively. In contrast, for the exchange harmonics, the potential-energy shifts in such equations exhibit opposite signs. Specifically, ionization from the left ion and recombination with the right ion will lead to negative sign in Eq. ( 14) and positive sign in Eq. ( 15), while these signs will be reversed if the electron is freed from the right ion and recombines with the left ion.

The return condition will remain the same as in the modified-prefactor case, with the difference that now the electron is returning to each ion, instead of to the geometric center of the molecule. For the exchange harmonics, the return condition will read

- t t dτ [k + A(τ )] ± R = 0, (16)
where the positive and negative signs correspond to S RL (t, t , Ω, k) and S LR (t, t , Ω, k), respectively. In the former case, the electron is released in the continuum from the left ion and recombines with the right ion, while in the latter case, ionization from the right ion and recombination with the left ion takes place. Eq. ( 6) also shows that the intermediate electron momentum is no longer collinear with regard to the external laser field. This has been pointed out recently in [24] as leading to elliptically polarized harmonics in molecules, from a linearly polarized driving field.

In the particular studies performed in this work, we are not considering the periodic contributions to high-order harmonic generation over many cycles. These would lead to sharp harmonic peaks, which would make the suppressions studied in this work more difficult to single out.

Interference Condition

Maxima and minima in the momentum-space wavefunction may be determined by writing the exponents in Eq. ( 5) in terms of trigonometric functions. In this case, one obtains

ψ(k) = α C (α) + cos k • R 2 + iC (α) -sin k • R 2 , (17)
with

C (α) ± = (-1) lα-mα+λα c (R) α Φ (R) α (k) ± c (L) α Φ (L) α (k). ( 18
)
Calling ϑ = arctan[iC

(α) + /C (α)
-],we find

ψ(k) = α C (α) + 2 -C (α) - 2 sin[ϑ + k • R/2]. (19)
Eq. ( 19) exhibits minima for ϑ + k • R/2 = nπ. Note, however, that the coefficients C (α) ± defined in Eq. ( 18) depend on the wavefunctions at the left and right ions. Since these wavefunctions themselves depend on the momentum k, one expects the two-center patterns to be blurred for heteronuclear molecules. In contrast, for homonuclear molecules, c

(L) α = c (R) α and Φ (L) α (k) = Φ (R)
α (k). This implies that the momentum dependence in the argument ϑ cancels out and that the interference condition in Refs. [8,[START_REF] De | [END_REF] is recovered. In this case, sharp interference fringes are expected to be present.

The above-stated condition does not only lead to interference fringes in the boundstate momentum wavefunctions but also in the high-harmonic spectra. This is due to the fact that the dipole matrix elements depend on the wavefunctions (5). In this latter case, k → k + A(t) in Eqs. ( 5)- (19).

For a linearly polarized monochromatic field of frequency ω, using the saddlepoint equation ( 15), the generalized interference condition ( 19) may be expressed in terms of the harmonic order n as

n = E 0 ω + 2(κπ -ϑ) 2 ωR 2 cos 2 θ L , (20)
where E 0 is the absolute value of the bound-state energy in question, κ is an integer number, θ L is the orientation angle, R is the internuclear distance and ϑ is defined above. This interference condition has been first derived in [21] and has been used by us in previous work [START_REF] De | [END_REF]. In the specific case of orbitals of π g type, which are constructed entirely from ptype atomic orbitals such that l α = 1, are of gerade symmetry such that λ α = 0, and are made from a homonuclear species such that Φ (R)

α (k) = Φ (L) α (k) and c (R) α = c (L)
α , one finds that ϑ = 0. This implies that for the zeroth order interference, where κ = 0, the second term of (20) will vanish leaving the angular independent term

n = E 0 ω . (21)
This minimum, however, would occur exactly at the ionization threshold, for which the present framework is expected to break down. Hence, it will not be discussed here.

Results

Position and momentum space wavefunctions

In this section, we will display the specific molecular orbitals employed by us when calculating the results that follow. We choose two pairs of molecules composed of a homonuclear molecule and an isoelectronic heteronuclear molecule. The first pair, Be 2 and LiB, possess eight electrons and a σ highest occupied molecular orbital, while the second pair, O 2 and NF, have sixteen electrons and a π highest occupied molecular orbital.

In Fig. 1, we exhibit the HOMO for homonuclear molecules, Be the z-axis. The position-space wavefunctions of Be 2 and LiB are of σ type with Be 2 having ungerade symmetry, σ u , and a nodal plane along the x-axis at z = 0. For LiB, there is a bias towards the Boron atom. One can also clearly see the contribution of the p type atomic orbital which is introduced by the Boron atom. This gives rise to a nodal structure of conical type, whose apex is displaced from z = 0 towards positive values of the z coordinate. The position-space wavefunctions of O 2 and NF are of π type with O 2 having gerade symmetry, π g , and a nodal plane along the x-axis at z = 0 and the z-axis at x = 0. NF shows a bias towards the Nitrogen atom, so that the nodal plane along z = 0 in Fig. 1.(c) was distorted into a curved surface (see Fig. 1.(d)). In contrast, the plane along the x axis is left unaffected. In summary, Figs. 1.(c) and(d) show that the distortions of the electron charge cloud along the internuclear axis removes any nodal planes perpendicular to it for heteronuclear molecules. This implies that the probability density at the geometric center of the molecule is no longer vanishing, and that high-harmonic generation is no longer completely suppressed when what was a nodal plane in the homononuclear case is parallel to the laser-field polarization. This is particularly true if one assumes that the active electron returns to the geometric center of the molecule, as implicit in Eq. (12).

We have found that both momentum-space wavefunctions, whose absolute values are presented in Fig. 2 are symmetric with respect to (p x , p z ) → (-p x , -p z ) in comparison to their position-space counterparts. Despite that, the momentumspace wavefunctions for homonuclear and heteronuclear molecules exhibit different properties, especially if a real basis set is used in the construction of the positionspace wavefunctions ψ(r). They come from the properties of Fourier transforms and will determine whether a particular node will be blurred in the heteronuclear case. An even wavefunction in position space will lead to a real wavefunction in momentum space, as the Fourier transform of a real and even function should be real. For the same reason, if the position-space wavefunction is odd, the corresponding momentum-space wavefunction will be pure imaginary. Neither property will hold if the position-space wavefunction does not possess a well-defined symmetry. In this case, its momentumspace counterpart will be complex.

We will now have a closer look at the nodes in the momentum-space wavefunctions, whose absolute values are depicted in Fig. 2. For Be 2 , the momentumspace wavefunction of the σ u HOMO is pure imaginary, and exhibits a clear node along the p x -axis (see Fig. 2.(a) for |ψ(k)|). This node is caused by the linear combination of atomic orbitals giving rise to the σ u orbital. The above-mentioned node is lost in the absolute value of the HOMO momentum space wavefunction of LiB, as shown in Fig. 2.(b), even though a suppression along the p x axis is still present. This feature is blurred by the real part of ψ(k), which does not have this node, and is of comparable magnitude. We have also verified that this node is still present in Im[ψ(k)] (not shown). Hence, the loss of symmetry in position space is what leads to the loss of the clear nodal plane in momentum space.

In comparing O 2 and NF momentum space wavefunctions, one can see that the features are quite similar for one of the nodal planes. Specifically, the well defined nodal plane along the p x -axis observed for O 2 has been slightly blurred for NF (see Figs. 2.(c) and (d), respectively). Note once more that an even and real position-space wavefunction, such as the HOMO in O 2 , leads to a real momentum-space wavefunction. This is not the case for NF, and the non-vanishing imaginary part of the momentum space wavefunction gives rise to the blurring. Its imaginary part, however, is much smaller than its real part. This implies that the NF molecule is quite close to having gerade type symmetry, and therefore one would expect similar features in the position and momentum space wavefunctions as compared to those of O 2 .

In contrast, the other nodal plane, along the z-axis, remains completely unaffected by the presence of the static dipole moment in NF. Indeed, this nodal plane is just as pronounced in both molecules, irrespective of being homonuclear or heteronuclear. Mathematically, this is because one can define two types of nodal structure in position space: one of which is due to the characteristic of the p-type atomic orbitals, and therefore disappear at x = 0, irrespective of the contraction coefficients and exponential coefficients, and the other of which is due to the linear combination of atomic orbitals. The latter results in a cancelation for a homonuclear molecule, and hence a nodal plane. This cancelation does not occur for a heteronuclear molecule. This argument can be mapped into the momentum space using Fourier transforms (see Eq. ( 5)). The nodal plane along the p z -axis is of the first type, and is therefore present for both molecules, and the nodal plane along the p x -axis is of the second type and not present for NF.

Harmonic Spectra

We now present the harmonic spectra computed in a linearly polarized monochromatic wave of frequency ω and amplitude ωA 0 , for all molecules discussed previously. Unless otherwise stated, we will consider that the electron leaves and returns to the geometric center of the molecule, i.e., we will consider the action (3) and the saddle-point equations ( 11)- (13). These results are displayed in Fig. 3. It can be seen that all the spectra contain a series of cuts, i.e., a complete suppression of the yield for particular angles. Most notably in the case of O 2 these occur at orientation angles θ L =0,90,180,270 and 360 degrees (see Fig. 3.(a)) §. By observing the position space wavefunction in Fig. 1.(c) and the absolute value of the momentum space wavefunction in Fig. 2.(c) this can immediately be attributed to the nodal planes of the molecular wavefunction, which occur at these angles, and therefore prohibit ionization or recombination. The situation in the case of NF, however, is modified, as shown in Fig. 3.(b) for different orientation angles. The harmonic signal completely vanishes at orientation angles θ L =0,180 and 360 degrees, because at these angles the nodal planes in the highest occupied molecular orbital occur due to the nature of the p-type atomic orbitals. However, due to the polar nature of these heteronuclear molecules, and a linear combination of atomic orbitals which no longer cancels out, the nodal planes at orientation angles θ L =90 and 270 degrees are no longer present.

In fact, several differences with regard to the homonuclear case are observed. Firstly, the signal is no longer completely suppressed, but there is a distinct minimum which occurs at orientation angles (θ L = 123 and 238 degrees). There is also a second minimum, but not a cut, beyond harmonic n=35, which occurs at the orientation angles θ L =90 and 270 degrees, that is, where the nodes were in the homonuclear molecule. We attribute this to the remnants of the nodal plane in its heteronuclear counterpart.

Inspection of the position and momentum space wavefunction suggests that the polar nature of the heteronuclear molecules deforms the nodal plane such that, although there is no longer a node, there is a suppression which occurs at the angle to which the plane has been deformed. Hence, the probability density associated to the wavefunction is small, but nonvanishing. This leads to a minimum, but not a complete suppression in the spectrum. As the molecule is rotated the new minimum in the wavefunction is first experienced after θ L =90 degrees but then occurs at the same angle before θ L =270 degrees. Therefore the spectra has reflectional symmetry about an orientation angle of 180 degrees.

Note that, for O 2 , the two-center interference maxima and minima predicted by Eq. ( 21) are located either beyond the cutoff or at the ionization threshold. Therefore, they do not occur in the parameter range of interest. This also holds for its heteronuclear counterpart. Now observing the spectra for Be 2 , displayed in Fig. 3.(c), one may identify nodes at θ L =90 and 270 degrees, as one would expect by observing the position and momentum space wavefunctions. In the spectra of the heteronuclear counterpart LiB, presented in Fig. 3.(d), these nodes have been replaced by minima at θ L =45 and 315 § Note that orientation and alignment express different physical concepts. This is particularly relevant for heteronuclear molecules: whislt the permanent dipole moment of such molecules may be oriented parallel or antiparallel to the laser field polarization, the molecular axis may be aligned parallel to the laser field. This would mean that a parallel alignment alignment angle would be a coherent superposition of the parallel and antiparallel orientation. For homonuclear molecules, due to their inversion symmetry, in practice, parallel orientation means parallel alignment. degrees, for the same reasons as described above.

The spectra for Be 2 also exhibit two-center interference minima, corresponding to κ = 0, κ = 1 and κ = 2 in Eq. ( 20), whose energy position depends strongly on the orientation angle. These are shifted considerably in the heteronuclear counterpart LiB. This is because the Be 2 molecular wavefunction is constructed almost entirely of s-type atomic orbitals, whereas in the case of LiB the Boron atom introduces p-type atomic orbitals into the molecular wavefunction and hence changes the interference condition. Note that, with regard to Be 2 , there is a blurring in the two-center patterns for LiB. This is expected according to the discussion in Sec. 2.3. This is shown in more detail in Fig. 4, in which the individual contributions of the s and p states to the spectra of Be 2 and LiB are presented (upper and lower panels, respectively). For these individual contributions, there is also a considerably simpler interference condition for homonuclear molecules, which can be obtained from Eq. ( 20) by setting either C + or C -equal to zero. Specifically the s-state and p-state contributions for an ungerade orbital exhibit minima at the harmonic frequencies

Ω (u) s = E 0 + 2κ 2 π 2 R 2 cos 2 θ L (22)
and

Ω (u) p = E 0 + (2κ + 1) 2 π 2 2R 2 cos 2 θ L , (23)
respectively, where κ is a integer. In Fig. 4.(a), we may identify the minima for κ = 1 and κ = 2 according to Eq. ( 22), while in Fig. 4.(b) the minima corresponding to κ = 0 and κ = 1 can be easily found. If the s p mixing is considered, the full expression ( 20) must be used. The contributions from p-type orbitals to the harmonic spectrum of Be 2 are orders of magnitude smaller than the contribution of p-type orbitals in LiB. Therefore the overall spectrum of Be 2 is dominated mostly by the s-type atomic orbitals. In contrast, in LiB, both types of orbitals lead to comparable contributions to the spectra, and Influence of asymmetry and nodal structures on high-harmonic generation in heteronuclear molecules13 the s p mixing will influence the energy position of the overall LCAO patterns. Apart from that, the ionization potentials and internuclear separation of both molecules are slightly different. This will influence the interference condition (20) further.

The effects of s p mixing are also displayed in Fig. 5.(a) and Fig. 5.(b) for a fixed mid-plateau harmonic. The figure clearly shows that where one observes that, for Be 2 , the s-type orbitals dominate and the main effect of the p-type orbitals is to introduce a small shift in the interference minima. On the other hand, for LiB, the overall maxima and minima are considerably altered by s p mixing. In the figure, one can also see that the nodal planes occurring at orientation angles θ L =90 and 270 degrees in Be 2 , are shifted by the contribution of the s-type orbitals in LiB.

In the results discussed so far, we have assumed that the electron left and returned to the geometric center of the molecule. In other words, all the molecular structure has been incorporated into the prefactor. However, a legitimate question would be how the present results would change if instead, the action contained the molecular structure. In the latter case, there will be four different geometric paths along which the electron may return and the so called " exchange harmonics", in which one electron leaves from one ion and recombines with the other, are expected to play a role.

In Fig. 6, we exhibit the intensity of a plateau harmonic for O 2 and NF computed with both methodologies. As previously discussed, both molecules exhibit two types of nodal structures, which are either determined by either the LCAO or the properties of the atomic wavefunctions only.

For O 2 (Fig. 6.(a)), we find that the yield computed using both methods is very similar, with a strong suppression for both types of nodal planes. In fact, the only differences encountered are small changes in the overall yield. In contrast, for NF (Fig. 6.(b)), the suppression due to the nodal structure related to the LCAO is considerably reduced. This can be understood as follows. In the modified-prefactor Influence of asymmetry and nodal structures on high-harmonic generation in heteronuclear molecules14 computation, the main emphasis is on the shape of the molecular wavefunction. For the heteronuclear case, this wavefunction has been deformed such that it is no longer vanishing at the geometric center of the molecule. Hence, the nodal structure is still present, even though it is observed for a different angle. However, the electron is always returning to the geometric center of the molecule. In contrast, in the modified-action picture, one may view the suppression at the LCAO nodes as the quantum interference due to recombination at spatially separated centers. Hence, the molecule behaves as the microscopic counterpart of a double slit experiment. If the centers in the molecule are identical, then the electron orbits will interfere as if they were leaving from identical "slits". Thus, sharp two-center interference patterns are expected. If, however, the molecule is heteronuclear, this implies that one of such "slits" is larger than the other, i.e., that the contributions from one center of the molecule will dominate. Hence, the nodes caused by the quantum interference of different geometrical paths will be less distinct.

Conclusions

In this work, we investigated the dependence of the high-order harmonic spectra on the orientation angle between the diatomic molecules and the laser-field polarization, for isoelectronic pairs consisting of a homonuclear and a heteronuclear molecule. We employed a single active electron approximation, using the HOMO as the active orbital, within the strong-field approximation.

Our studies lead us to conclude that there are two types of nodal structures in the bound-state orbitals of molecules, which will lead to a strong suppression in the harmonic yield when such structures are parallel to the laser-field polarization. The first type is caused by the nodes in the atomic orbitals at each ion, and is present at the same location for homonuclear molecules and their heteronuclear counterparts. The second type is due to the sum or subtraction of atomic orbitals at different centers within the linear combination of atomic orbital (LCAO) approximation. Both types of minima are present for homonuclear molecules. For their heteronuclear counterparts, however, the asymmetry of the molecule eliminates the latter nodes. This implies that the imprints of the first type of nodes in the harmonic spectra are common to isoelectronic homonuclear and heteronuclear molecules, and could in principle be observed in both cases.

In general, the asymmetry also leads to some blurring in the interference patterns determined by the LCAO at different centers. Furthermore, depending on the molecule, s p mixing will be different. This will lead to shifts in the energy positions of the two-center patterns. On a more technical level, we have also been able to map the symmetry or asymmetry of the molecular orbitals in position space to properties of their momentum-space counterparts. Apart from that, if one considers that the electron travels different paths in the molecule, returning to the right or left ions instead of to its geometric center, additional blurring will be observed for the twocenter nodes in the heteronuclear case.

The above-stated findings show that nodal structures in a heteronuclear molecule can be related to those in an isoelectronic homonuclear molecule. Hence, in principle, the latter can be used as a reference point in order to understand the behavior of the former, by using high-order harmonic generation. The shifts in the nodal planes present for homonuclear molecules due to the distortions in the wavefunctions, or the absence thereof, can be mapped into features in the high-harmonic spectra. This is a very good example of how symmetry breaking in a molecule reflects itself in the harmonic spectra, may shed some light in the imaging of heteronuclear molecules. Another example of this are the even harmonics, which are present in oriented heteronuclear molecules (for a recent example see [12]).

Finally, we would like to comment on the fact that, throughout this work, the Stark shifts in the ionization potentials of the molecules have been neglected in the strong-field approximation. The first-order shifts lead to binding energies which depend on the alignment angle. In fact, if the static dipole moment is oriented parallel or antiparallel to the laser-field polarization, the bound-state energies will decrease, or increase, respectively. This would imply a shift in the cutoff towards higher energies in the antiparallel case. The second-order shifts are much smaller and do not depend on the alignment angle. For homonuclear molecules, the first-order Stark shift is vanishing, so that orientation effects can be neglected. For heteronuclear molecules such as NF, for which the static dipole moment is small, the Stark shift is expected to be negligible. For LiB, however, the static dipole moment is much larger. Hence, further distortions in the spectra due to the above-mentioned effect are expected .

2 and O 2 ,

 2 and heteronuclear molecules, LiB and NF, as obtained with GAMESS-UK [Figs. 1.(a),(c),(b) and (d) respectively]. The internuclear axis is chosen to be along

Figure 1 .

 1 Figure 1. Highest occupied molecular orbitals in position space for the homonuclear molecules Be 2 and O 2 [panels a) and c)], and their heteronuclear isoelectronic counterparts LiB and NF [panels b) and d)]. The geometric center of the molecules is located at (x, z) = (0, 0). For the heteronuclear molecules, B and N are located at positive values of the z coordinate as indicated on the right-hand side of the figure. The orbital symmetries in panels a), b), c) and d) are 2σu, 4σ, 1πg and 2π, respectively, and the bond lengths are R (Be 2 ) = 4.642 a.u., R (LiB) = 4.642 a.u., R (O 2 ) = 2.280 a.u., and R (NF) = 2.485 a.u. For the heteronuclear molecules, the static dipole moments point towards the positive values of the z coordinate as shown by the arrows on the right-hand side of the figure. The static dipole moments computed considering only the charge cloud of the HOMO read d (HOMO) LiB = 0.946132531 a.u. and d (HOMO) NF = 0.11647725 a.u. for LiB and NF, respectively, while the total static dipole moments of both molecules are d LiB = 1.9895 a.u. and d NF = 0.0407 a.u.

Figure 2 .

 2 Figure 2. Absolute values of the highest occupied molecular orbitals in momentum space for the homonuclear molecule Be 2 and O 2 [panels a) and c), respectively] and their heteronuclear counterparts LiB and NF [panels b) and d), respectively].

  at i on Angl e θ L (degrees) at i on Angl e θ L (degrees)

Figure 3 .

 3 Figure 3. Harmonic spectra as a function of orientation angle for the molecule a) O 2 of ionization potential 0.2446 a.u., b) NF of ionization potential 0.2246 a.u., c) Be 2 of ionization potential 0.2390 a.u., and d) LiB of ionization potential 0.1942 a.u.. A linearly polarized laser field of frequency ω=0.057 a.u. and intensity I = 4 × 10 14 W/cm 2 is used. The spectra have been computed using the length gauge and employing the dipole operator in the length form.

Figure 4 .

 4 Figure 4. Harmonic spectra as a function of orientation angle for a) the s-type orbitals of the Be 2 HOMO, b) the p-type orbitals of the Be 2 HOMO, c) the s-type orbitals of the LiB HOMO, d) the p-type orbitals of the LiB HOMO, for the same parameters as in the previous figure.

14 O 10 HFigure 5 .

 14105 Figure 5. Contributions of the s, p and all atomic orbitals to the yield of the 25 th harmonic of a) Be 2 and b) LiB as functions of the orientation angle θ L . All parameters are the same as in the previous figure.

Figure 6 .

 6 Figure 6. The harmonic yield of a) O 2 and b) NF for the 25 th harmonic vs the orientation angle, computed using either a modified prefactor and single-atom saddle-point equations, or a modified action (black and red lines, respectively). All parameters are the same as in the previous figure.

Acknowledgments

We thank H. J. J. van Dam, P. J. Durham, P. Sherwood and J. Tennyson for very useful discussions. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC), and by the Daresbury Laboratory.

For orientation angles 0 < θ L < π/2 and 3π/2 < θ L < 2π the Stark shift will lower the potential barrier and cause a decrease in the high-harmonic cutoff, while for π/2 < θ L < 3π/2 the opposite will occur and the cutoff will increase. For perpendicular orientations, there is no Stark shift.


Fabien Racape

Simon Lefort

Edouard Francois

Marie Babel

Olivier Deforges

ADAPTIVE PIXEL/PATCH-BASED SYNTHESIS FOR TEXTURE COMPRESSION

Keywords: texture, synthesis, characterization

This paper presents an adaptive scheme for synthesizing missing textured regions. In synthesis-based compression approaches, large textures are removed at encoder side and filled in at decoder side. This work proposes a synthesizer in which both complementary pixel-based and patch-based approaches are used. According to results shown by synthesis algorithms, patch-based and pixel-based approaches are efficient with different kinds of texture. Two algorithms are adapted to the region synthesis context: the sample patch is built from surrounding texture and potential anchor blocks inside the removed region; the scan order depends on a confidence map in order to take advantage of the designed patch; DCT-descriptors provide the minimum size of matching window which is a required parameter for both synthesizers. Finally an "a posteriori" gradient based assessor enables the synthesizer to switch between algorithms.

INTRODUCTION

In State-of-the-art compression schemes, such as H.264/AVC, pixel-wise redundancies are reduced by using predictions and transform domain operations. However, spatio-temporal approaches, exploiting redundancy based on the Mean Squared Error (MSE) criterion, are not able to take visual redundancy into account. Detailed textures may seem near stationary to the Human Visual System, but totally irregular according to MSE criterion. Meanwhile, texture synthesis algorithms [START_REF] Wei | Fast texture synthesis using treestructured vector quantization[END_REF][START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF] have shown great results. The purpose of new coding schemes is to detect regions where exact positions of texture patterns are irrelevant to the human eye. The whole region is not encoded since only a few patterns are sufficient to synthesize a satisfactory region. An interesting framework has been designed in [START_REF] Zhu | Video coding with spatio-temporal texture synthesis[END_REF] where some 8x8 blocks are removed at the encoder side and synthesized by the decoder. Texture synthesis is performed using the algorithm presented in [START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF], however using 8x8 blocks offer poor overlap for the patch-based technique. The work presented in [START_REF] Ndjiki-Nya | Generic and robust video coding with texture analysis and synthesis[END_REF] proposes a closed-loop analysis-synthesis approach. As in [START_REF] Zhu | Video coding with spatio-temporal texture synthesis[END_REF], groups of pictures (GOP) are considered for a spatio-temporal scheme.

Each potential region from a GOP is both analyzed and synthesized, using side information from the texture analyzer. A first synthesizer is designed for Rigid textures with global motion, which has a great similarity to global motion compensation (GMC). Another synthesizer, inspired by the patch-based approach developed in [START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF], processes Non-rigid textures with local and global deformations. The scheme has recently been upgraded in [START_REF] Ndjiki-Nya | Optimization of video synthesis by means of cost-guided multimodal photometric correction[END_REF] with photometric corrections, using Poisson editing and covariant cloning. However, Y. Liu [START_REF] Liu | The promise and the perils of near-regular texture[END_REF] pointed out that synthesis algorithms work more efficiently with different kinds of textures and addressed the specific case of near regular textures.

This paper presents a framework using both complementary pixel-based and patch-based approaches. The scheme includes a texture characterization step, based on DCT-domain descriptors, that outputs the approximated feature size of texture patterns to be synthesized. The two following texture synthesis algorithms have inspired this work: the basic pixelbased approach developed in [START_REF] Wei | Fast texture synthesis using treestructured vector quantization[END_REF] and the algorithm presented in [START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF] for the patch-based synthesis.

This paper is organized as follows. In section 2 is introduced the texture analysis at encoder side. In sections 3 and 4 are respectively described pixel-based and patch-based chosen algorithms. Section 5 finally presents some experimental results and the switching method between both approaches.

TEXTURE ANALYSIS

This section focuses on the analysis at the encoder side which aims at characterizing texture patterns and designing the removed regions and sample patches for synthesis. Note that the segmentation has been performed by the region extraction scheme presented in [START_REF] Deforges | Color lar codec: a color image representation and compression scheme based in local resolution adjustment and self-extracting region representation[END_REF].

Texture characterization

This part of the scheme outputs the required parameters in order to optimize the synthesizers. Chosen descriptors are derived from those described in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF] which are computed from a Fourier transform. Since the characterization requires little sizes of block, i.e. inferior to 32 pixels wide, DCT domain has been preferred. Indeed, Fourier transform outputs descriptors a) square patch b) chosen patch with a lower resolution in frequency, which is a problem to describe a signal from a small block. Like in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], descriptors are computed from concentric circles, except that they are integrated on a quarter of circle. Their computation follows

D DCT (λ) = π/2 θ=0 |C (λ, θ))| 2 dθ (1)
where C corresponds to the DCT coefficient at position (λcosθ, λsinθ). The DC coefficient is not taken into account in order to be invariant to mean luminance variations, so the descriptors vector has the size of DCT block minus one coefficient. According to [START_REF] Wei | Fast texture synthesis using treestructured vector quantization[END_REF][START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF] and our experiments, the size of neighborhoods has to be greater than the texture pattern, in order to produce visually good results. By experimenting a large set of arbitrary chosen texture patches, we noticed that if the first coefficient (low frequency) is greater than any others in the descriptors vector, the block size can be considered smaller than the pattern. Indeed, the first coefficient of each block corresponds to a single variation of luminance over the block used for DCT computation. The output size parameter corresponds to the minimum size NxN of DCT block that provides a non-monotone decreasing vector, which is given by

N out = min N B N ×N DCT |D(1) = max λ=[2;N -1] {D(λ)} (2)
where B is a DCT block of size NxN. Thus, an approximate size is given by testing descriptor sizes from 4x4 to 32x32 pixels. Fig. 4. Pixel-based synthesis order.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

Texture patch design

Texture synthesizers from the literature process with a rectangular input patch. Its relevance is essential to ensure an efficient synthesis in this particular context. One notices that cropping a patch close to the texture region to be synthesized can lead to inconsistencies between candidate pixels and previously reconstructed borders. Figure 1 a) shows a case in which there is a variation of luminance, leading to a failing synthesis. We propose to define the patch as the surrounding MBs of the region to be synthesized. This source patch for is depicted in figure 1 b).

Encoding strategic anchor blocks

Large textured regions require anchor preserved blocks in order to prevent artefacts. After experiments, it has been decided to encode some anchor macroblocks (MB) at strategic locations depicted in figure 2 since the region size exceeds a fix number of MBs in width or height. Coupled with a confidence-based synthesis order described in next section, they prevent visible seams at synthesis junctions. The next two sections describe the synthesis tools for filling in these regions, using information provided by descriptors.

PIXEL BASED TEXTURE SYNTHESIS

The pixel-based synthesizer derives from the basic scheme presented in [START_REF] Wei | Fast texture synthesis using treestructured vector quantization[END_REF]. This algorithm relies on the matching between current pixels' neighborhood and those in a texture patch, using the Sum of Squares Error (SSE) as criterion. In order to improve the research process, the exact neighborhood matching described in [START_REF] Sabha | Texture synthesis using exact neighborhood matching[END_REF] is used. The following describes the adaptation to the context of removed regions.

Adaptive neighborhood size using texture characterization

Texture characterization enables the synthesizer to get an approximate neighborhood. In order to refine the neighborhood sizes for matching, a set of neighborhood sizes are tested, for example 7x7 and 9x9 if the best descriptor is 8x8 large. The chosen distance minimizes the norm distance

D m,n (p, c) = Nm,n k=0 (p k -c k ) 2 N m,n (3)
where N m,n is the number of pixels contained in the neighborhood of current pixel, m and n respectively representing its height and width. c k and p k denote the kth pixel's luminance in the current neighborhood and the considered one in the patch respectively. Considering various neighborhood sizes enables to better catch texture patterns size and shape.

Confidence-based synthesis order

The goal is here to exploit available whether previously decoded or synthesized pixels which are, at this point, the only confident data. A confidence map has been designed to synthesize the most reliable pixel at each time. The map ing law is depicted in figure 3, where initial computing (a) from previously decoded pixels is represented on the left side and the update during synthesis on the right side. The neighborhood, used for texture synthesis, serves for computing a map on which previously decoded pixels are initialized with a unique confidence value. Thus, confidence in pixels to be synthesized is computed following:

w(i, j) = 1 N 2 N/2 k=-N/2 N/2 l=-N/2 w(i + k, j + l). (4)
The map is then updated (b) by assigning the previously computed w(i, j) on its location.

PATCH BASED TEXTURE SYNTHESIS

The patch based method used can be decomposed into two main steps. The synthesis order, depicted in figure 5, is based on the confidence map described in section 3.2

Finding the best matching patch

Each patch added to the output texture is chosen by comparing the already known parts, depicted in figure 6 a), with all the possible positions in the input patch. The SSE criterion is used to find the best matching area. That is, the patch size is determined by previously described DCT-descriptors.

Cutting the patch for a seam-less texture

Most of the time, the added patch does not match exactly the already synthesized texture, then a second step consists on cutting the patch with the texture using a graphcut method like described in [START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF]. Costs between overlapping patches is computed from the extended version in [START_REF] Kwatra | Graphcut textures: image and video texture synthesis using graph cuts[END_REF] using gradients. Figure 6 presents a typical case due to surrounding blocks and synthesis order. The next section provides some results and proposes a switch method to optimize the synthesis.

COMPARING AND SWITCHING ALGORITHMS

This section aims at comparing and presenting how to choose the best algorithm for each region.

Results and comparison

One has now to determine whether to use patch-or pixelbased technique depending on textures' characteristics. Then

Switching between algorithms

Since the patch-based approach produces visible artefacts when failing, an a posteriori method is proposed. Three gradient-based criteria are proposed in order to quantify the amount of created artefacts. ∆Gl denotes the gradient's differences at cut locations beten synthesized and original images, while ∆Gh and ∆Gv represent horizontal and vertical mean Sobel gradients' differencies, again between output and input textures. After experiments on a large set of textures, maximum thresholds {5; 5; 20} have been fixed for ∆Gh, ∆Gv and ∆Gl respectively to determine acceptable synthesis. Gl characterizes the located seams to determine if the cuts are hidden enough, while Gh and Gv aim at pointing out such oriented edges, highly detectable by the Human Visual System, like in figure 10 c). As the texture is expected synthesizable by characterization step, the pixel-based synthesizer is chosen for high gradients.

CONCLUSION AND FUTURE WORK

This paper presents an intra-frame compression framework using two kinds of texture synthesis, according to their favorite types of texture. The switch between algorithms is done a posteriori since it is based on patch-based synthesis artefacts measurement. Ongoing research focuses on the descriptorbased segmentation as well as the characterization process of detected textures in order to improve synthesizers' adaptation.

Fig. 1 .

 1 Fig. 1. Texture patch design.

Fig. 2 .

 2 Fig. 2. Anchor blocks. Up: synthesis of an empty region. Bottom: synthesis with anchor blocks.

Fig. 3 .

 3 Fig. 3. Confidence map order. Confidences w a , w b and w c are computed for center pixels from equation 4 using neighboring confidences. At initialization w a = 0.64 for instance. After synthesizing pixel a, w b is updated by assigning w 1 a .

Fig. 5 .Fig. 6 .

 56 Fig. 5. Patch-based synthesis order.

Fig. 7 .

 7 Fig. 7. Results on the Coastguard.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Results on the Morocco Trial sequences.

Fig. 10 .

 10 Fig. 10. Resulting gradients for the 4 textures on the right.


Christopher Davies

Christian Thomas

Instabilities and transition in three-dimensional flows with rotation LOCAL AND GLOBAL STABILITY OF ROTATING DISK BOUNDARY LAYERS

The von Kármán boundary-layer flow above a rotating disk can be shown to be absolutely unstable, by deploying a local linear analysis where the base flow is artificially simplified by taking it to be homogeneous along the radial direction [START_REF] Lingwood | Absolute Instability of the Boundary-Layer on a Rotating-Disk[END_REF]. Making use of the same approximation to represent the base flow, absolute instability has also been identified for the whole family of rotating disk boundary layers that, through the variation of the Rossby number, continuously connect the Kármán flow to the Bödewadt flow [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. Moreover, absolute instability still persists if suction is introduced at the disk surface [START_REF] Lingwood | On the effects of suction and injection on the absolute instability of the rotating-disk boundary layers[END_REF] or there is an applied axial magnetic field [START_REF] Jasmine | Convective and absolute instability in the incompressible boundary layer on a rotating disk in the presence of a uniform magnetic field[END_REF], though its onset is postponed to higher Reynolds numbers. Thus absolute instability is pervasive for rotating disk boundary layers. However, we will illustrate how the results obtained using a localized analysis for the homogenized version of the base flow cannot just be taken, in an immediate manner, to predict the correct global behavior for the genuine radially inhomogeneous base flow. The relationship between the local and global stability properties turns out be rather more complex and interesting than many researchers might have anticipated.

For the case of the von Kármán boundary-layer flow, direct numerical simulations of linearized disturbance development in the genuine flow showed that the predicted absolute instability does not in fact give rise to any unstable linear global modes [START_REF] Davies | Global behaviour corresponding to the absolute instability of the rotatingdisc boundary layer[END_REF]. This is despite the fact that the temporal growth rates for the absolute instability display a marked increase with the radial distance from the rotation axis [START_REF] Davies | Global stability of the rotating disc boundary layer[END_REF]. The apparent disparity between the radially increasing strength of the absolute instability and the absence of any global instability can be understood by considering analogous behaviour in impulse solutions of the linearized complex Ginzburg-Landau equation [START_REF] Hunt | Instability of flows in spatially developing media[END_REF]. These solutions show that detuning, arising from the radial variation of the temporal frequency of the absolute instability, may be enough to globally stabilize disturbances. Depending on the precise balance between the radial increase in the growth rates and the corresponding shifts in the frequencies, it is possible for an absolutely unstable flow to remain globally stable.

Globally stable behaviour was found to persist in numerical simulations where mass injection was introduced at the rotating disk surface, modifying the base flow. More interestingly, it was discovered that globally unstable behaviour is promoted when suction is applied [START_REF] Thomas | The effects of mass transfer on the global stability of the rotating-disk boundary layer[END_REF]. This contrasts with the locally stabilizing influence of suction that had previously been identified, using the approximation of a radially homogenized base flow. For the inhomogeneous flow, impulsively excited disturbances display an increasingly rapid temporal growth at the radial position of the impulse, albeit without any selection of a dominant frequency, as would be more usual for an unstable global mode. A similar form of global destabilization, overcoming locally stabilizing effects, was reproduced when an axial magnetic field was applied.

We will discuss numerical simulation results to illustrate the new type of instability, and show how it can be successfully described using impulse solutions of the linearized Ginzburg-Landau equation. More generally, we will review the subtle connections that can persist between the local and global linear behavior, for a range of the rotating disk boundary layer flows that have been mentioned above. For example, the figure depicts the space-time development of a disturbance that was generated in a rotating disk boundary layer with Rossby number Ro = 0.4. Results for both the artificially homogenized version of the flow and the genuine flow are shown. The flow parameters were chosen so that, for the homogenized version of the base flow, the disturbance is at the margin of absolutely instability. In marked contrast to what might have been predicted by a naive use of local stability results, it may be very clearly seen that there is a strong form of global stability in the genuine flow. The disturbance is not even convectively unstable, although it does display transient growth.

The previously unsuspected complexity of the relationships between the local and global stability properties of linearized disturbances in rotating disk boundary layers can be expected to have important consequences for any nonlinear developments. It is conceivable that, in some instances, the character of the linearly unstable disturbance may be such as to discount the possibility of any analytical approach to describing the emergence of nonlinearity that, in the interests of tractability, tries to postpone the incorporation of the effects of the inhomogeneity of the base flow. For the case of the rotating disk boundary layer with suction, for example, destabilization due to the radial inhomogeneity may need to be accounted for right from the very outset of the analysis, irrespective of the smallness of the initial disturbance amplitude.

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

Figure 1 :

 1 Figure 1: Spatio-temporal development of the azimuthal vorticity at the disk surface for an impulsively excited disturbance with Ro = 0.4 and azimuthal mode number n = 20. The disturbance was excited at r e = Re/Ro = 313. (a) Homogenized flow with Re = 125 (b) Radially inhomogeneous flow.


Shintaro Imayama

P Henrik Alfredsson

R J Lingwood

Instabilities and transition in three-dimensional flows with rotation AN EXPERIMENTAL STUDY OF LAMINAR-TURBULENT TRANSITION OF A ROTATING-DISK FLOW

INTRODUCTION

The main objective of this experimental study is to measure the laminar-turbulent transitional region of the rotating-disk boundary layer. In 1921, von [START_REF] Kármán | Über laminare und turbulente Reibung[END_REF] derived the exact similarity solution of the Navier-Stokes equations for the basic laminar flow on an infinite rotating disk. Typically 28 to 32 stationary vortices are observed within the convectively unstable region in unforced experiments (e.g. [START_REF] Gregory | On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk[END_REF]), which are excited by unavoidable roughnesses fixed at the disk surface, but the flow is unstable to travelling modes as well. Lingwood (1995a) suggested that instability on the rotating disk is not only of convective nature but also that the flow becomes absolute unstable at R=510 (R = r * (Ω * /ν * ) 1/2 , where r * is the radius of the disk at the measurement position, Ω * is the rotational speed of the disk, ν * is the kinematic viscosity of the flow and * means dimensional value). In the present work we map the flow regimes from the stable laminar flow region, through the unstable and transitional regions, into the fully turbulent region. The measurements were performed on a rotating disk shown in Figure 1. The apparatus is in principle the same as used by [START_REF] Lingwood | An experimental study of absolute instability of the rotating-disk boundary-layer flow[END_REF] and has been lent by Cambridge University to KTH, where it has undergone certain modifications. The disk is now made of glass and has a diameter of 474 mm and a thickness of 24 mm and is mounted on the original aluminium-alloy disk. The aluminium-alloy disk is connected to DC-servo motor through an iron disk and a vertical shaft. The air bearing mounted around the shaft is operated with high pressure air to achieve low noise and vibration levels. The surface of the glass disk is polished and the roughness is within 1 µm and the imbalance within 10 µm at the edge of the glass disk. To attenuate the effect of the disk edge, a stationary wood plate is fixed around the glass disk. A single hot-wire probe, operated by a constant temperature anemometer (CTA) with overheat ratio 0.8, is used for the velocity measurement. The sensitive area is made of platinum, with a diameter of 5 µm and a length of 1 mm. The signal from the CTA is digitized using a computer with a 16 bit A/D converter and is sampled at a rate of 720 data points per revolution. The hotwire is mounted on a two-axis traverse system. One of the traverses moves in the horizontal (radial) direction, and the other traverse is mounted on the horizontal traverse at a 45 • inclination to not disturb the axial flow which approaches the rotating disk from above.

EXPERIMENTAL CONDITIONS

The hot-wire orientation makes it mainly sensitive to the circumferential velocity component (i.e. the axis of the sensing wire is aligned with the radial direction) and although the flow is threedimensional this component is much larger compared to the radial and axial components. The calibration of the hot-wire is carried out on the rotating disk using the theoretical laminar mean profile. The hot-wire was calibrated using a modified King's law to extend the low velocity range (Johansson & Alfredsson,1982). The measurements were carried out at a fixed radius position of 198 mm, varying the Reynolds number with the rotational speed.

RESULTS AND DISCUSSIONS

Measured circumferential mean velocity profiles spanning the range R = 430 -630 are shown in Figure 2 . The abscissa axis is the non-dimensional circumferential velocity (V = V * /(r * Ω * ), where V * is the circumferential velocity). The vertical axis is the non-dimensional height from the wall (z = z * (Ω * /ν * ) 1/2 , where z * is the wall normal position of hot wire). At Reynolds number R from 430 to 510, the experimental data correspond to theoretical laminar profile shown as a solid line, except in the very low velocity region where effects of heat convection on the hot-wire are observed. At R = 550, deviations of the experimental data from the theoretical EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation laminar profile can be observed. Lingwood (1995a) suggested that the absolute instability appears at R = 510 triggering nonlinear effects that one would expect to generate Reynolds stresses thereby distorting the laminar velocity profile. As the Reynolds number increases, the velocity profile changes completely. At R = 630 the boundary-layer thickness has become much larger and the velocity gradient has increased close to the wall.

Figure 3 shows the non-dimensional intensity of the circumferential velocity fluctuation (v rms = v * rms /r * Ω * , where v * rms is the root mean square of the circumferential velocity) against the wall normal position. At R = 510, the peak appears at z = 1.3. At R = 550, the rms has a higher value not only at z = 1.3 but also z = 4. At R = 630, fluctuations develop over the entire boundary-layer height. Figure 4 shows the spectra at each Reynolds number measured at z = 1.3. At around ω * /Ω * = 30, peaks start to appear and the intensity increases with Reynolds number. This result corresponds to the previous study that between 28 and 32 stationary vortices are observed once the individual patterns (generated by individual fixed roughnesses) have merged and filled the whole circumference of the disk. Above R = 510 harmonics of the basic frequency starts to appear, but at R = 630 the spectrum becomes flat without any distinct peaks characteristic of fully-developed turbulence.

The talk will include a further investigation of the transition region including intermittency and higher moments, as well as details of the turbulent flow itself.

This project is supported by the Swedish Research Council. The first author acknowledges support from the Scandinavia-Japan Sasakawa Foundation. We would also like to acknowledge the help of the late Dr Tim Nickels who was instrumental in arranging the loan of the rotating disk equipment from Cambridge University to KTH.

Figure 1 :

 1 Figure 1: The experimental setup of the rotating disk.

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Mean velocity profiles in the range R = 430 -630, spanning the stable laminar region, the unstable, transitional and turbulent regions.


INSTABILITY OF STATIONARY STREAMWISE VORTICES EMBEDDED IN A SWEPT-WING BOUNDARY LAYER

Viktor KOZLOV 1 , Valery CHERNORAY, 2 Alexander DOVGAL 1 & Lennart L ÖFDAHL 2 1 Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, Russia.

2 Chalmers University of Technology, Göteborg, Sweden.

Motivation

Research results on transition to turbulence in boundary layers testify to a pronounced contribution of streamwise vortices and streaky structures to the laminar flow breakdown. Modulating the base flow, they generate local velocity gradients with high-frequency instabilities starting to grow. The same applies to a swept-wing flow which is unstable to stationary crossflow vortices, the latter being subject to amplifying secondary oscillations.

A number of experimental, theoretical and numerical studies has been performed on this topic indicating that the instability phenomenon under the consideration is of much importance during the boundary-layer transition, e.g., [START_REF] Nitschke-Kowsky | Instability and transition of a three-dimensional boundary layer on a swept flat plate[END_REF][START_REF] White | Secondary instability of crossflow vortices[END_REF][START_REF] Fischer | Primary and secondary stability analysis of a three-dimensional boundary-layer flow[END_REF][START_REF] Malik | Secondary instability of crossflow vortices and swept-wing boundary-layer transition[END_REF][START_REF] Högberg | Secondary instability of crossflow vortices in Falkner-Skan-Cooke boundary layers[END_REF]; for an overview of the problem see [START_REF] Saric | Stability and transition of three-dimensional boundary layers[END_REF].

In the present work we deal with the secondary disturbances of a swept-wing flow perturbed by the stationary vortices at their different transverse spacing. Most of all we are interested in the effect of the spatial arrangement of the vortices on the high-frequency boundary-layer oscillations. To make it clear, wind-tunnel experiments were performed focusing on the laminar flow disturbances excited in a controlled manner.

Experimental set-up

The experimental runs were carried out at the Department of Thermo and Fluid Dynamics, Chalmers University of Technology. A wing model with a C-16 profile was tested at a 45 degree sweep angle. The stationary vortices were generated in the laminar boundary layer by an elongated and a circular roughness elements placed one by one on the wing surface. The first of them produced at its tips a pair of isolated vortices of opposite rotation while the second element generated the vortices interacting with each other as is sketched in figure 1. The high-frequency harmonic perturbations evolving along the vortices were excited through a tiny hole on the wing surface behind the roughness elements. In each case, the secondary oscillations were excited underneath the primary vortices. The flow pattern was examined through hot-wire measurements with single and V-wire probes which were positioned by a computer controlled traversing mechanism. To observe the secondary perturbations, velocity measurements at each point in the spatial domain were phase-locked with the disturbances generator so that volumetric flow patterns were created. Some of the obtained results are illustrated here as iso-surface plots of the nonstationary velocity component taken at a fixed time moment.

Wind-tunnel data

In the flow disturbed by the elongated roughness element two vortices (Vortex 1 and Vortex 2) are generated originating at the left-hand and the right-hand sides of the element, respectively. Spaced in the transverse direction, they seem to develop without significant interaction with each other. When changing the elongated EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation roughness element to the circular one, the counter-rotating vortices interfere at diminution of their spacing, thus, it is expected that mean-velocity gradients generated by the vortices become smaller and the boundary layer is stabilized to the high-frequency disturbances.

Indeed, such an effect was observed comparing the secondary oscillations evolving along the isolated and interacting vortices, an example is given in figure 2. In both cases, the disturbances were excited at low levels of the periodic forcing so that linear oscillations were induced at the position of their generator. As is seen, they appear as oblique waves which are transported in narrow regions of the basic-flow stationary modulation. Riding on the isolated vortex the perturbations are well periodic in the major part of the measurement domain, while propagating away from the generator they amplify and finally result in chaotic motion. One observe that the periodicity of the oscillations is contaminated by small-scale irregularities in the aft part of the box shown at the left of figure 2 testifying to a broad spectrum of velocity fluctuations. Otherwise, at the vortex interaction, the high-frequency disturbances do not evolve into 3D irregular structures so that their coherence is found in the entire measurement domain. Our conclusion is that the behavior of secondary perturbations involved in the swept-wing transition is quite different depending on the transverse spacing of the primary basic-flow disturbances. That is, amplification of the secondary oscillations is obviously reduced when the isolated counter-rotating vortices start to interact, approaching each other. In this way, the transitional flow can be controlled by roughness elements of a proper configuration, modulating the primary boundary layer perturbations and their secondary instabilities.

The present contribution to EC525 was supported by the Russian Foundation for Basic Research (08-01-00027), the Ministry of Education and Science of the Russian Federation (RNP.2.1.2.541), and the government contract (14.740.11.0354).

Figure 1 :

 1 Figure 1: Experimental model and roughness elements generating the stationary vortices.

Figure 2 :

 2 Figure 2: Controlled perturbations of Vortex 1 (lef t) and of the interacting vortices (right). Chord Reynolds number is 390,000, excitation frequency is 210 Hz, amplitude levels shown in yellow and green are ±0.5% of U 0 (lef t) and ±0.1% of U 0 (right).


Lothar Rukes

Kilian Oberleithner

Christian Oliver Paschereit

Instabilities and transition in three-dimensional flows with rotation LOCAL LINEAR STABILITY ANALYSIS OF A TURBULENT SWIRLING JET UNDERGOING VORTEX BREAKDOWN

Introduction

Swirling jets are known to exhibit a variety of instability phenomena. A number of different stability theories are available today that can partly explain these phenomena. Of the many theories the linear theory applied to a laminar base flow has probably been studied the most. Although this theory is in a strict theoretical sense only valid for a base flow that is a steady solution of the Navier-Stokes-equations, it can be successfully applied to time-averaged turbulent flows [START_REF] Gaster | Large-scale structures in a forced turbulent mixing layer[END_REF]. The results of such studies are often in very good agreement with experiments. Oberleithner et al. [START_REF] Oberleithner | Self excited oscillations in swirling jets: Stability analysis and empirical mode construction[END_REF] found the wave-number and frequency of a global mode derived from the local linear theory to be in good agreement with the experimental observations. The goal of this study is to extend the results of [START_REF] Oberleithner | Self excited oscillations in swirling jets: Stability analysis and empirical mode construction[END_REF] by using the linear theory to examine the properties of a global mode in a swirling turbulent jet for different stages of the vortex breakdown. Properties of the global mode are obtained from the results of a local analysis, as shown in Gallaire et al. [START_REF] Gallaire | Spiral vortex breakdown as a global mode[END_REF]. Therefore, it is necessary to locate regions of absolute and convective instability in the flow domain. The analysis will be based on the time-averaged turbulent flow derived from PIV-measurements.

Experimental setup and data processing

The experimental apparatus used to generate the swirling jet is shown in fig. 1. The three-dimensional velocity-Figure 1: Experimental apparatus (all dimensions are in millimeters) [START_REF] Oberleithner | Three dimensional coherent structures of the swirling jet undergoing vortex breakdown: Stability analysis and empirical mode construction[END_REF]. The used coordinate system is indicated. The measurements presented here were obtained at a Reynolds number of around 21000. field was measured using a Stereo-PIV system. The laser sheet was introduced perpendicular to the nozzle in the x-z-plane, extending up to seven nozzle diameters in the x-and 3.5 nozzle diameters in the z-direction (positive and negative). Measurements were taken for swirl numbers between 0 and 1.2, covering flow configurations from a non-swirling-jet to a fully developed breakdown state. Especially in the swirl number range between 0.7 and 0.9 measurements were concentrated, because in this range the vortex-breakdown was observed to develop.

Numerical method

The numerical discretization method employed is based on a Chebyshev collocation approach [START_REF] Khorrami | Application of spectral collocation techniques to the stability of swirling flow[END_REF]. The stability of the flow is analyzed by solving the resulting matrix eigenvalue problem with MATLABs eig-, respectively eigs-routine. In order to find the saddle-points determining the type of the instability a method adapted from [START_REF] Rees | Hydrodynamic instability of confined jets & wakes & implications for gas turbine fuel injectors[END_REF] is used. This method is based on a truncated Taylor-series expansion around the saddle point. Because the truncated series can only be expected to converge in the vicinity of the saddle-point, good starting values are needed. These are obtained by constructing a global picture and the corresponding cusp map at two axial positions and identifying the relevant saddle points by human eye. From these two known saddle points the wave-number and frequency of the relevant saddle point at the next axial location can be extrapolated and the EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation truncated Taylor series in conjunction with the eigs-routine is used to converge to the exact wave-number and frequency of the saddle-point.

Results

An exemplary result of the analysis is given in fig. 2. The streamline-plot illustrates the analyzed flow and the bottom right plot provides a magnified view of the absolutely unstable region. The computational code used to produce this graph was used in [START_REF] Oberleithner | Self excited oscillations in swirling jets: Stability analysis and empirical mode construction[END_REF] as well, thus demonstrating that it produces correct predictions of the global modes frequency and wave-number. Furthermore, the existence of two regions of absolute instability seems plausible, since Gallaire et al. [START_REF] Gallaire | Spiral vortex breakdown as a global mode[END_REF] found a similar behavior in their study. The results of the stability analysis are expected to provide a better understanding of the mode selection of the naturally occurring global mode and its properties. Additionally, the results are supposed to shed light on the global modes interaction with the dynamics of the vortex breakdown. The swirling jet exhibits vortexbreakdown in this case. The azimuthal wave-number was set to one, which is the azimuthal wave-number of the naturally occurring mode, as it is observed from experimental data. Two regions of absolute instability are present (indicated by gray shading in the graph). One immediately downstream of the nozzle exit and the other one in the region of the breakdown bubble.

Figure 2 :

 2 Figure2: Wave-number k and frequency ω at the saddle point location for one measurement, plotted over the nondimensionalized axial coordinate (D denotes the nozzle diameter). The swirling jet exhibits vortexbreakdown in this case. The azimuthal wave-number was set to one, which is the azimuthal wave-number of the naturally occurring mode, as it is observed from experimental data. Two regions of absolute instability are present (indicated by gray shading in the graph). One immediately downstream of the nozzle exit and the other one in the region of the breakdown bubble.


Ubaid Qadri

Dhiren Mistry

Matthew Juniper

Instabilities and transition in three-dimensional flows with rotation SENSITIVITY ANALYSIS OF SPIRAL VORTEX BREAKDOWN

Introduction

Vortex breakdown occurs in some swirling flows, such as those in gas turbine combustion chambers. Ruith et al [START_REF] Ruith | Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation[END_REF] and Gallaire et al [START_REF] Gallaire | Spiral vortex breakdown as a global mode[END_REF] established that the initial breakdown is steady and axisymmetric but that an unsteady spiralling breakdown mode develops on top of this, due to a region of absolute instability.

In this paper, we investigate the linear stability of the steady axisymmetric solutions obtained by Vyazmina et al [START_REF] Vyazmina | The bifurcation structure of viscous steady axisymmetric vortex breakdown with open lateral boundaries[END_REF] for vortex breakdown in incompressible flows at Re = 200. We compare local and global stability analyses and relate the global behaviour of the flow to its local stability properties. Using results from the global stability analysis, we map the regions of the flow that are most sensitive to external forcing and internal feedback.

Methodology

We use a Low Mach Number (LMN) formulation of the Navier-Stokes (N-S) equations to obtain steady axisymmetric solutions in a cylindrical domain with open lateral and convective outlet boundaries. The Grabowski velocity profile [START_REF] Ruith | Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation[END_REF] is imposed at the inlet. We study the linear stability of these flows for a range of values of the swirl parameter.

In the global stability analysis, we superpose small perturbations of the form û(x, r) exp(imθ + σt), where m is the azimuthal wavenumber and σ is the complex eigenvalue. The direct and adjoint Linearized N-S (LN-S) equations are discretized with a compact finite difference scheme in space and a 4 th order Runge Kutta scheme in time. The resultant matrix eigenvalue problems are solved with the implicitly restarted Arnoldi algorithm. The regions that are most sensitive to external forcing and internal feedback are found from the direct and adjoint eigenmodes [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF].

In the local stability analysis we apply the WKBJ approximation (i.e. we assume a locally parallel base flow) and superpose small perturbations of the form û(r) exp(i(kx + mθ -ωt)), where k is the local complex wavenumber and ω is the local complex angular frequency. The absolute frequency, ω 0 , is calculated at each axial location. This distribution of ω 0 is interpolated with a Padé polynomial and then continued analytically into the complex x-plane. The linear global frequency, ω g (which is equivalent to iσ), and the wavemaker position are estimated from the position of the relevant saddle point of ω 0 in the complex x-plane. The response of each slice to the estimated global frequency, ω g , is then calculated and the WKBJ approximation is inverted in order to obtain the mode shape ũ(x, r).

Results

Figure 1 shows the base flow at a swirl parameter of Sw = 1.0, together with the absolute growth rate, the most unstable direct and adjoint global modes and the structural sensitivity maps for m = 1 helical perturbations. There is an axisymmetric breakdown bubble around x = 1, which causes a wake downstream. The global stability analysis predicts one unstable eigenmode, which has iσ = 1.1655 + 0.0352i. This matches the results from nonlinear DNS [START_REF] Ruith | Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation[END_REF], suggesting that linear wavefront theory is valid in this case. The direct global mode, which represents the flow structure that dominates in the long time limit, has maximum amplitude at x = 9, in the wake downstream of the bubble. The adjoint global mode, which represents the most unstable initial condition or the receptivity of the direct mode to external forcing, is localized between the inlet and the upstream edge of the bubble.

The direct and adjoint global modes are combined to give the structural sensitivity maps. These show where a modification in the linearized equations produces the greatest drift of the eigenvalue [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]. The first map shows the regions that have most influence on the growth rate of the direct global mode. The second map shows the regions where there is maximum coupling between the direct and adjoint velocity components, which indicates the core of the instability (Fig. 17 of [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]). We find that the instability core (as defined by [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]) is located in the upstream end of the breakdown bubble. By considering the global stability of the breakdown bubble and the wake separately, we find that the global frequency of the flow matches that of the bubble rather than that of the wake. We conclude that the wavemaker region is in the recirculation bubble, contrary to expectation. The local stability analysis predicts two regions of absolute instability: a small region corresponding to the bubble and a large region corresponding to the wake. We find that the instability core obtained from the global analysis corresponds to the region of absolute instability in the bubble. This supports the conclusion that the linear global mode is being driven by a wavemaker situated in the breakdown bubble.

For a given azimuthal wavenumber, m, more eigenmodes become unstable as the swirl increases. Furthermore, eigenmodes at higher m also become unstable as the swirl increases. The local analysis indicates that there are one or two valid saddle points (i.e. k + /k -pinch points) at each streamwise location. Broadly, these group into one saddle point that dominates in the bubble and another saddle point that dominates in the wake. As the swirl increases, the saddle point in the wake becomes more absolutely unstable.

We find that the local analysis is valid only for slowly developing, weakly non-parallel flows. Furthermore, it cannot conclusively identify the location of the wavemaker region in flows which have more than one region of absolute instability. [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]

EUROMECH

  Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation

Figure 1 :

 1 Figure 1: (a) streamlines and azimuthal velocity (colour) of the base flow for Sw = 1.0; (b) local absolute growth rate, ω 0,i ; (c) real component of the direct global mode for radial velocity; (d) real component of the adjoint global mode for radial velocity; (e) sensitivity of growth rate; (f) structural sensitivity as defined by[START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]