G G Vilenski

E R Johnson
email: e.johnson@ucl.ac.uk

J G Esler

O J Rump

J Sommeria

Orographically generated nonlinear waves in rotating and nonrotating two-layer flow

Keywords: nonlinear waves, orography, two-layer rotating flow, forced rotating BDA equation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Orographically generated nonlinear waves in rotating and non-rotating two-layer flow 1. Introduction

Among the many observations of cloud patterns near orography, some stand out as showing stationary disturbances upwind of the orography. Johnson & Vilenski (2004) (simply 'JV' here) reproduce a NASA photograph of Guadalupe Island, Baja California taken during the Gemini-V flight and described by [START_REF] Stevenson | The 200-mile fishline[END_REF]. A low layer of stratocumulus cloud is moving at 6-10 knots past the island whose peaks reach 4500 ft and thereby project through, and interfere with, the cloud layer. [START_REF] Stevenson | The 200-mile fishline[END_REF] notes that a 'shock or bow' wave spreads from the north end of the island, 'similar to waves formed by a ship moving through water', and further observes that as these cloud features were photographed during four Gemini missions they must be considered climatic features of the Guadalupe marine atmosphere, whose fluid dynamic details must be investigated for proper analyses of atmospheric and ocean flows. [START_REF] Burk | The dynamics of wave clouds upwind of coastal orography[END_REF] describe similar wave clouds extending away from the Monterey-Big Sur coastline of central California, noting that it is the orographic forcing and stationary appearance in geostationary operational environment satellite (GOES) image loops that set these waves apart from propagating wave phenomena. They present a schematic description based on isolated single attached and detached shocks forced by obstacles in supercritical flows. [START_REF] Tjernstrom | Simulations of supercritical flow around points and capes in a coastal atmosphere[END_REF] and [START_REF] Soderberg | Supercritical channel flow in the coastal atmospheric boundary layer: idealized numerical simulations[END_REF] also discuss flow in the atmospheric boundary layer around points and capes in terms of the criticality of the flow, following [START_REF] Samelson | Supercritical marine-layer flow along a smoothly varying coastline[END_REF] and [START_REF] Samelson | The horizontal momentum balance in the marine atmospheric boundary-layer during CODE-2[END_REF] who treat the shallow marine atmospheric boundary layer under a low-level inversion along the Californian coast as supercritical flow with a sidewall. [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] model these phenomena theoretically in the context of shallow water flow over isolated orography, noting that a shallow water model can capture such atmospheric flow phenomena as long wakes in the lee of obstacles, quasi-steady vortices and vortex shedding, and observing that with realistic parameters a single-layer model as a representation of stratified air flow has even made some reasonable quantitative predictions of vortex shedding period and wake length [START_REF] Scha ¨r | Shallow-water flow past isolated topography. 2. Transition to vortex shedding[END_REF]. Because of the direct analogy between the shallow water equations and the compressible two-dimensional Euler equations, these analyses lead to flow patterns with isolated attached and detached shocks as in the schematic descriptions of [START_REF] Burk | The dynamics of wave clouds upwind of coastal orography[END_REF] and [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF]. However, the GOES observations and nonhydrostatic numerical computations in [START_REF] Burk | The dynamics of wave clouds upwind of coastal orography[END_REF] show a wave pattern that does not necessarily consist of an isolated cloud line but can consist of an initial line followed by a set of parallel lines of cloud. Figure 1 shows a closer view of Guadalupe Island from a more recent (11 July 2001) NASA photograph. The parabolic bow wave is again clearly visible, bearing out [START_REF] Stevenson | The 200-mile fishline[END_REF] observation that it is a climatic feature, but behind the initial bow wave a series of alternate crests and troughs also appears. JV note that extending the shallow water equations to include weak nonhydrostatic effects causes the single isolated shocks of the ship-wave analogies of [START_REF] Stevenson | The 200-mile fishline[END_REF], [START_REF] Burk | The dynamics of wave clouds upwind of coastal orography[END_REF] and [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] to split into multiple parallel crests similar to those observed. JV consider near-critical flow, where the oncoming flow speed is close to the dominant free-wave speed of the flow, in the limit where wave steepening due to weak nonlinearity is balanced by weak dispersion due to non-hydrostatic effects. For one-dimensional flow over ridges this leads to the forced Korteweg-de Vries (fKdV) equation [START_REF] Grimshaw | Resonant flow of a stratified fluid over topography[END_REF]) and in two-dimensional flow over isolated orography to the forced two-dimensional KdV or forced Kadomtsev-Petviashvili (fKP) equation [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF][START_REF] Akylas | Three-dimensional long water-wave phenomena[END_REF]. Solutions reproduce many of the features seen in numerical integrations of the fully nonlinear shallow-water equations of [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] and also give predictions of the drag exerted on the flow by the orography, perhaps one of the most important physical quantities required from local modelling to parameterize orography in large-scale models. [START_REF] Li | Synthetic aperture radar observation of the sea surface imprints of upstream atmospheric solitons generated by flow impeded by an island[END_REF] report upstream propagating solitary waves identified both from synthetic aperture radar and moderate resolution imaging spectroradiometer images near St Lawrence island in the Bering Sea. To model these observations numerically they treat the waves as one-dimensional and approximate the density profile as a two-layer fluid. The governing equation then becomes the forced extended KdV (feKdV) equation where, for sufficiently large amplitude waves or particular ratios of densities and depths, cubic nonlinearities can contribute to the motion in addition to the usual quadratic nonlinearities of the KdV [START_REF] Melville | Transcritical two-layer flow over topography[END_REF][START_REF] Grimshaw | Transcritical flow of a stratified fluid: The forced extended Korteweg-de Vries model[END_REF]. [START_REF] Johnson | Two-dimensional leaps in near-critical flow over isolated orography[END_REF] note the modification to these flow patterns when the forcing is more localized so the flow is weakly two-dimensional, governed by the forced extended KP (feKP) equation. Similar solitary waves advancing away from the front of an obstacle have been generated and observed in the laboratory by [START_REF] Maxworthy | The generation and propagation of internal gravity-waves in a rotating fluid[END_REF]. In both the cases described by [START_REF] Maxworthy | The generation and propagation of internal gravity-waves in a rotating fluid[END_REF] and [START_REF] Li | Synthetic aperture radar observation of the sea surface imprints of upstream atmospheric solitons generated by flow impeded by an island[END_REF], the flows are significantly subcritical and waves advance unsteadily upstream. The unsteady development of forced twodimensional solitary waves is described in [START_REF] Lee | Upstream-advancing waves generated by 3-dimensional moving disturbances[END_REF] and [START_REF] Li | Three-dimensional nonlinear solitary waves in shallow water generated by an advancing disturbance[END_REF]. It is the purpose of this paper to investigate experimentally, and model simply, nearly steady near-critical flow patterns, both weakly subcritical and weakly supercritical, to provide a possible mechanism for the observed steady flow patterns noted earlier and shown in figure 1.

Before applying analyses like these to planetary flows it is necessary to estimate the effect of the Earth's rotation on the flow. [START_REF] Ostrovsky | Non-linear internal waves in a rotating ocean[END_REF] considers weak rotation in unforced one-dimensional flows and this, together with later work on one-dimensional flows and some two-dimensional unforced flowsgoverned by the rotating KP (rKP) equation-are described in detail in the reviews of [START_REF] Akylas | Three-dimensional long water-wave phenomena[END_REF] and Grimshaw et al. (1998b). In particular, Grimshaw (1998b) point out that rotational effects are important when they become comparable to non-hydrostatic dispersion and that in oceanic flows this occurs for motions with scales of the order of a few hundreds of kilometres in the open ocean and only a few tens of kilometres in shallow seas. Vilenski & Johnson (2004) consider an obstacle advancing at approximately critical speed into an otherwise quiescent rotating fluid, a set-up chosen for the possibility of verifying predicted flow fields against relatively straightforward towing tank experiments and because the natural division of non-rotating flows into subcritical, critical and supercritical flows remains. The flow is governed by the forced rKP (frKP) equation. They describe weakly nonlinear supercritical solutions showing how steady oblique solitary waves, unattenuated in non-rotating flow, decay with distance from the obstacle and how they can be interpreted in terms of the temporal decay of solitary waves in the one-dimensional problem described by Grimshaw et al. (1998a). These effects appear here in both the experimental and model results.

The fKP equation describes not only the vertical displacement of the freesurface of a near-critical shallow flow but also, as shown for the fKdV by [START_REF] Grimshaw | Resonant flow of a stratified fluid over topography[END_REF], the horizontal and temporal variation of the amplitude of a near-critical vertical mode in a continuously stratified flow of finite depth. [START_REF] Grimshaw | Resonant flow of a stratified fluid over topography[END_REF] show that, for oncoming flows with speeds near that of the fundamental linear mode, the vertical structure of the mode influences the governing equation only through the coefficients in the equation, determined by integrals of the vertical structure. Computed patterns, like those in JV, thus depend only weakly on the details of the vertical structure of the linear wave. This weak dependence of the horizontal patterns on vertical linear wave structure offers a possible explanation of the qualitative accuracy of the description of near-critical flows by the shallow water equations even when the vertical structure of the fundamental linear wave differs from the two-layer reduced gravity model. Such differences in vertical wave structure have been remarked on in models of coastally trapped waves on a inversion within uniformly stratified layers by [START_REF] Samelson | The vertical structure of linear coastal-trapped disturbances[END_REF] and [START_REF] Durran | Small-amplitude coastally trapped disturbances and the reduced-gravity shallow-water approximation[END_REF], although the vertical step orography in their examples leads to singular pressure fields (Schmidt & Johnson 1993b) and differences near continuous orography (following, for example, Schmidt & Johnson 1993a) could be smaller. [START_REF] Rottman | Solitary waves in the atmosphere[END_REF] review weakly nonlinear theory for the propagation of one-dimensional solitary waves in the atmosphere. They distinguish between two typical scenarios. In the first scenario the waves propagate within a strongly stratified waveguide with relatively weak stratification elsewhere. [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] note that in the marine atmosphere the top of the boundary layer often acts as this waveguide. In the second scenario the stratification is relatively uniform throughout most of the depth of the fluid, and in the atmospheric case the waves then occupy most of the troposphere with the tropopause and strongly stable stratosphere acting as an upper boundary, to give an effectively finitedepth flow. The first scenario leads to Benjamin-Davis-Acrivos (BDA) [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF][START_REF] Davis | Solitary internal waves in deep water[END_REF][START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF]) theory and the second to KdV theory. Hence [START_REF] Rottman | Solitary waves in the atmosphere[END_REF] note that in the atmosphere BDA theory applies mostly to low-level solitary waves, where waves propagate in a low-level stable layer topped by a very weakly stratified layer, whereas the KdV theory generally applies to deeper waves that occupy most of the troposphere.

They point out that the atmosphere often produces layers of nearly neutrally stratified air that are sufficiently deep to provide effective trapping of solitary waves (the BDA case), but rarely produces a sufficiently large increase in buoyancy frequency to make a good reflector and effectively cap a stratified layer (the KdV case). They suggest that the KdV may, however, be relevant when there is sufficient vertical shear in the oncoming flow to reflect vertically propagating internal waves. As the lower layer in the experiments described below is deep and neutrally stable and the effective oncoming flow is unsheared, the experiments relate most closely to the BDA case and are modelled below in terms of a forced, rotating, weakly two-dimensional BDA (fr2dBDA) equation.

Section 2 describes the experimental set-up of a two-layer fluid in a large rotating tank where a thin upper layer occupies 10% of the total fluid depth. Section 3 interprets the experimental observations of interface displacement in terms of solutions the fr2dBDA equation and §4 gives brief conclusions.

Experimental set-up

The experiments were performed in the large rotating tank at the LEGI-Coriolis facility in Grenoble, the same facility as used in the experiments of [START_REF] Maxworthy | The generation and propagation of internal gravity-waves in a rotating fluid[END_REF]. The present set of experiments differed from those of [START_REF] Maxworthy | The generation and propagation of internal gravity-waves in a rotating fluid[END_REF] in that here the obstacle (O in figures 2 and 3) was mounted on a carriage C whose speed could be accurately controlled so the obstacle could be moved along the water surface at constant speed. In the experiments of [START_REF] Maxworthy | The generation and propagation of internal gravity-waves in a rotating fluid[END_REF], the obstacle was bottom-mounted and moved only a fixed distance before being brought to rest; so interfacial solitary waves propagated away in front of the obstacle. One of the great advantages of the Coriolis set-up is the large width of the tank. The wavefield forced by obstacles towed at near-critical speeds spreads laterally over significant distances and in narrower domains wave reflection from sidewalls soon dominates the entire wavefield [START_REF] Ertekin | Waves caused by moving disturbance in a shallow channel of finite width[END_REF]). In the present experiments wall reflections do not (figure 4a). The aim was to give a system to model, when inverted, an atmospheric flow consisting of a thin lower layer below a much deeper upper layer, following [START_REF] Rottman | Solitary waves in the atmosphere[END_REF] and [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] who note that the marine atmosphere typically has a two-layer stratification profile, with the marine atmospheric boundary layer forming a lower layer of 1-2 km beneath free tropospheric air of depth 8-10 km. The value of inverting the experimental system is twofold: first, no problems arise due to bottom boundary layer effects at the obstacle and, second, the upper layer can be readily refreshed after each experiment with fresh water. The density difference was 2.5%, corresponding to interfacial long wavespeeds c 0 Z(g 0 H 1 ) 1/2 of 10 cm s K1 . The obstacle was moved across a diameter of the tank at a constant speed U in the range 5-30 cm s K1 , allowing the transition from subcritical to supercritical regimes to be explored in all cases. At these speeds the Froude number associated with the free surface is extremely small and consequently the upper surface acts as a rigid lid. During the experiments no surface waves were observed. In addition to non-rotating runs, the Coriolis facility allowed experiments with tank rotation periods of TZ60, 90 and 120 s.

Interface displacements were measured by a set of 20 acoustic probes (P in figure 2) spaced 20 cm apart along a spar transverse to and to one side of the obstacle motion. The probes consisted of an ultrasonic pulse emitter (e in figure 2), positioned above the interface, and a reflector (a small metallic plate, r in figure 2), positioned below it. The piezo-electric emitter was also used as a receiver, and the return time of the ultrasonic pulses measured. As the speed of sound is different in both layers, this return time varies linearly with interface displacement. The precision of measurement was 0.1 mm. The probe calibration was checked before each experiment by applying a vertical translation of 1 cm to the whole spar of probes, using a stepping motor. Similar interfacial probes have been used previously by [START_REF] Ramirez | Generation of internal waves over a shelf[END_REF]. Figure 4b gives time-series from a typical experimental run. Runs were performed in both directions to verify that the flow was symmetric about the centreline. Individual runs with the same parameters showed a high degree of reproducibility and runs with the same parameters in opposite directions showed good agreement: at the rotation rates here the observed wavefields away from the direct wake region were symmetric about the centreline. Thus the interface elevations in figure 5 are shown with data reflected about the centreline to represent the whole wavefield. The absence of data in the band near the centreline is due to the inner limit placed on the probe line to allow the obstacle to pass without obstruction. The gap between the innermost probe and the edge of the widest obstacle was approximately 9 cm. Time-series of interface displacements from the probes like those in figure 4b were converted into the spatial plots of figure 5 by assuming that the flow pattern was stationary relative to the obstacle. The validity of this assumption is discussed briefly in §4.

Typical runs in the tank, of diameter 13 m, took approximately 2 min. Data from the probes were stored in raw form, a simple program converted this to displacements in centimetres and a further program plotted these in the form of a frame of figure 5. Thus within a few minutes of the end of a run it was possible to examine the generated wavefield. In practice, it was necessary to wait about 20 min for the wavefield to decay after each run and so it was possible to perform over 18 runs a day with a given stratification and rotation speed while varying obstacle height and towing speed. The disruption to the basic stratification was minimal over one day but over a three-day period the interface thickened altering the interfacial wavespeed and making it more difficult to estimate the fractional depth of the upper layer occupied by the obstacle. At this stage the tank was drained, the two layers mixed, sufficient salt added to return the density to 2.5% above freshwater, this mixed fluid used to form the lower layer and freshwater added to form a new upper layer.

Various obstacle shapes were used in the experiments, including axisymmetric Gaussian obstacles of different volumes and the flat obstacle of figure 3. It was found that the wavefield generated by symmetric obstacles was determined almost entirely by the volume of the obstacle and was uninfluenced by the obstacle profile. Observations are thus reported below solely for the flat obstacle.

Interpretation of the experimental results

(a ) A model equation for forced, rotating, near-critical internal waves

As the lower layer is neutrally stratified and 10 times deeper than the upper layer, it is treated here as effectively infinitely deep so the appropriate model equation can be expected to belong to the BDA family. Following the derivation in [START_REF] Esler | Evolution-equations for weakly nonlinear, long internal waves in a rotating fluid[END_REF] and [START_REF] Johnson | A modern introduction to the mathematical theory of water waves[END_REF], and including modifications for twolayer flow (e.g. [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF], and for rotation [START_REF] Ostrovsky | Non-linear internal waves in a rotating ocean[END_REF], gives the forced, rotating, weakly two-dimensional BDA equation (fr2dBDA),

2h 0zt C 2Gh 0zz C 3 2 ðh 2 0 Þ zz C H½h 0 zzz Kn 2 h 0 C A 2 h 0YY ZKMB zz ; ð3:1Þ
for the evolution of the leading order interface displacement h 0 . Here H is the Hilbert transform

H½f ðxÞ Z 1 p PV ð N K N f ðx 0 Þ x 0 Kx dx 0 ;
where PV denotes the Cauchy principal value of the integral. The onedimensional form of equation (3.1) has been used previously by [START_REF] Matsuno | Forced Benjamin-Ono-equation and its application to soliton dynamics[END_REF] to describe the collision of a soliton with a ridge. The derivation of equation (3.1) follows from expansion in the small parameter dZH 1 /L/1, and it follows that zZd(xKUt)/H 1 is a spatial coordinate fixed in the frame of the obstacle which is moving at speed U, Y Z d 3=2 Ay=H 1 is a stretched y-coordinate and tZd 2 c 0 t/H 1 is a 'slow' time variable. At leading order h 0 is related to the true interface displacement h via hZdh 0 and B is a nondimensionalized obstacle height with a maximum value of unity. There are four non-dimensional parameters G, A, n and M associated with equation (3.1). They can be identified with criticality (G!0 indicates a super-critical obstacle tow speed), obstacle anisotropy, rotation and nonlinearity, respectively. These nondimensional parameters are directly related to the experimental parameters as follows: Recall that h is the maximum height of the obstacle; L y , the obstacle length scale in the y-direction; c 0 Z(g 0 H 1 ) 1/2 , the speed of long interfacial gravity waves, with g 0 the reduced gravity; and f, the Coriolis parameter. A balance of the final two terms on the left side of equation (3.1) shows that solutions can be expected to decay away from the axis over distances of the order of the Rossby radius, c 0 /f. It is noted below that this decay is more evident in the experimental observations that in either the linear or nonlinear solutions of equation (3.1).

G Z c 0 KU c 0 d K1 ; A Z L L y d K1=2 ; n Z fH 1 c 0 d K3=2 ; M Z h H 1 d K2 : ð3

(b ) Numerical calculations

In order to differentiate between the separate roles of criticality, rotation and nonlinearity in determining the wave pattern generated by the obstacle, equation (3.1) was integrated numerically on a workstation. Equation (3.1) is a model equation for the experimental problem and the results should be interpreted as such, rather than as a direct numerical simulations of the experiments. Nevertheless, in order to ensure that the parameter regimes most relevant to the experiments are explored, the model equation parameters G, A, n and M are calculated in each case from the corresponding experimental parameters using equation (3.2).

Equation (3.1) was solved in a doubly periodic domain, with dimensions 120L!120L y in terms of the obstacle length scales L and L y . The integrations continue until the interface height is found to be steady in a subdomain of dimensions 40L!40L y centred on the obstacle. The discretization is pseudospectral, and a resolution of 512!512 Fourier modes was found to be adequate to resolve the wave patterns in the parameter regime of interest. (Higher resolution was found necessary when nonlinearity dominated over dispersion, i.e. at large M, small jnj, jGj.) Time-stepping in the model proceeds according to the predictor-corrector method with time-step DtZ0.1.

The obstacle shape B is taken in the numerical integrations to be the Gaussian

Bðz; Y Þ Z expfKðz 2 C Y 2 Þg: ð3:3Þ
This shape is numerically convenient as it has little signature at the very high wavenumbers that can affect the time-stepping in the numerical model but are absent from the experimental observations due to weak viscous effects. The length scales of the obstacle are chosen as LZ30 cm, and L y Z60 cm to give the same volume and aspect ratio as the experimental obstacle. In all the numerical integrations described Dr/rZ2.5%, and H 1 Z6 cm, so that AZ 1:12 and, for the purposes of determining the other non-dimensional parameters using equation (3.2), dZ1/5 and c 0 Z12 cm s K1 . The parameters G, n and M vary as the tow speed U, the rotation rate f and the obstacle height h, respectively, are varied. To compare with the nonlinear solutions, linear calculations of the steady-state wake pattern were also obtained. For each Fourier component, equation (3.1) was solved with the nonlinear terms absent, and with the operator v/vt replaced by a scalar factor 3Z0.04. The use of a positive scalar factor ensures the correct causality in the solution for the obstacle wake [START_REF] Lighthill | Group velocity[END_REF], and the limit 3/0 corresponds to the exact linear solution in an infinite domain. In practice a finite value of 3 is necessary to restrict the scale of the wake to within that of the finite domain.

The obstacle used in the experiments had a width of the order of the total fluid depth and so, to determine the importance of the particular model equation chosen, integrations were also performed using the frKP equation following Vilenski & Johnson (2004) and the feKP equation following [START_REF] Johnson | Two-dimensional leaps in near-critical flow over isolated orography[END_REF], using linear wave parameters from both the two-layer model and computed from the measured experimental density profiles. With parameter values derived from the experiments, all models showed the same qualitative behaviour, and so only integrations of the fr2dBDA are described below.

(c ) Numerical solutions and interpretation

Figure 6 shows the steady state solutions of the non-rotating (nZ0) fr2dBDA equations with hZ3 cm (MZ12.5) at three different criticalities GZ0.208, K0.625 and K1.458 corresponding to towing speeds UZ11.5, 13.5 and 15.5 cm s K1 . The left panels show the linear solutions and the right panels the nonlinear solutions, allowing the effects of nonlinearity to be easily identified. At the subcritical towing speed UZ11.5 cm s K1 , nonlinearity generates two weak bow waves up to 30L ahead of the obstacle, with the wave pattern behind the obstacle modified only slightly from its linear counterpart. At weakly supercritical towing speeds (not shown) the two bow waves are stronger and form closer to the obstacle. With increasing supercriticality the inner bow wave decays, its curvature increases, and it eventually merges with the dispersive wave pattern. By moderate supercriticality (UZ13.5 cm s K1 panel) there is, therefore, only a single, large-amplitude bow wave ahead of the obstacle. With further increases in criticality this solitary bow wave begins to decay and also merges with the dispersive wave pattern until by UZ15.5 cm s K1 the V-wave pattern (e.g. [START_REF] Jiang | V-waves, bow shocks, and wakes in supercritical hydrostatic flow[END_REF] strongly resembles its linear counterpart.

The numerical results of figure 6 can be compared directly with the experimental results shown in figure 5: both show similar developments in the forced wave pattern once allowance is made for the difference in the critical speed c 0 between the numerical results and the experiments caused inter alia by the diffusion acting on the stratification profile in the experiments. The two weak bow waves of the UZ7.5 cm s K1 experiment are stronger and closer to the obstacle in the UZ10.0 cm s K1 experiment, whereas in the UZ12.5 cm s K1 experiment only a single bow wave with much greater curvature remains distinct from the dispersive wave pattern behind the obstacle.

Figure 7 illustrates how increasing nonlinearity affects the wave pattern. The hZ3 cm (MZ12.5), UZ13.5 cm s K1 numerical solution (left panel) should be compared with its hZ5 cm (MZ20.83) counterpart (right panel). Numerical integrations of the fKP equation in JV similarly show that increased nonlinearity, or equivalently reduced dispersion, leads to the formation of multiple solitary bow waves ahead of the obstacle. This transition between different solitary wave regimes has profound implications for the drag exerted by the obstacle on the flow (JV). Examining figure 7 and series of similar numerical solutions (not shown) shows that the extent to which the solitary wave pattern extends ahead of the obstacle depends only weakly on nonlinearity, but depends sensitively on the criticality, as can be seen in figure 6.

Figure 8 shows linear and nonlinear rotating steady states for the numerical integrations with fZ0.105 s K1 (so nZ0.587), corresponding to a Rossby radius of c 0 /fZ1.3 m. To emphasize the nonlinearity of the wave pattern topographic forcing has been chosen to be strong with hZ6 cm so MZ25. Because of the increased nonlinearity, and the tendency of the nonlinear solutions in the rotating regime to form extremely sharp-crested waves (e.g. [START_REF] Shrira | Propagation of long nonlinear waves in a layer of rotating fluid[END_REF][START_REF] Shrira | On long strongly nonlinear waves in a rotating ocean[END_REF]Grimshaw et al. 1998b;[START_REF] Esler | Evolution-equations for weakly nonlinear, long internal waves in a rotating fluid[END_REF], it was found necessary to apply either a weak diffusion (ZKkP 2 h 0z , where P 2 fZf zz Cf YY ) or hyper-diffusion (Zk H P 4 h 0z ) to the right-hand side of equation (3.1) in order to maintain numerical stability. Very similar wave patterns were obtained with either choice, although the diffusion operator had a stronger damping effect on the wavetrain far behind the obstacle. Figure 8 was generated using hyper-diffusion with k H Z5.4!10 K4 .

The subcritical (UZ11.5 cm s K1 ) and moderately supercritical (UZ13.5 cm s K1 ) numerical solutions show that the main effect of nonlinearity in rotating flow in this parameter regime occurs in the wake behind the obstacle. The bow wave is relatively unmodified by nonlinearity, but the trough following the obstacle (marked as T in figure 5) is broadened and deepened and is followed by either one (for UZ13.5 cm s K1 ) or two (for UZ11.5 cm s K1 ) sharp-crested, large-amplitude solitary waves, which do not appear in the linear solutions. The crests of the solitary waves can be seen to merge with those of the dispersive wavetrain away from the centre axis. The dispersive wavetrain itself is modified by nonlinearity, as the waves have sharp crests and relatively broad troughs.

Interpreting the rotating, experimental results on the basis of figure 8 shows clearly, for UZ10 cm s K1 , a bow wave (with crest C 1 in figures 4b and 5) and a steep, large amplitude wave (with crest C 2 in figures 4b and 5) following the broad deep trough after the obstacle (marked as T in figure 5). As in the numerical solutions wave C 2 is a significant distance behind the obstacle (over 2 m) and maintains its amplitude towards the centre axis. Similarly to Karpan (1967) for one -layer non-rotating flow, it can be shown that the far-field weakly nonlinear supercritical steady two-dimensional rotating flow is governed by the corresponding one-dimensional unsteady equation provided the crossstream coordinate of the two-dimensional equation is identified with the slow time of the one-dimensional equation. In Vilenski & Johnson (2004) this allows solutions of the frKP equation to be interpreted in terms of the Ostrovsky (rotating KdV) equation [START_REF] Ostrovsky | Non-linear internal waves in a rotating ocean[END_REF]. Here a similar interpretation is possible and the cross-stream development of the interface displacement observations of figure 4b is closely modelled by the temporal development of a solution of the one-dimensional unforced rotating BDA equation.

To our knowledge this is this first time this type of topographically forced nonlinear solitary wave has been recorded experimentally in a rotating fluid system. By contrast, it is debatable whether there is a clear signature of nonlinearity in the supercritical flow (UZ12.5 cm s K1 ) or indeed, with the exception of the multiple bow waves, in the subcritical flow (UZ7.5 cm s K1 ).

Discussion and conclusions

Experimental observations have been presented of the wavefield forced by a finite height obstacle travelling at near-critical speed through a two-layer rotating and non-rotating fluid in a tank sufficiently large to allow the lateral development of the pattern to be observed without interference from wall reflections. Comparisons with numerical solutions of the model fr2dBDA equation show that close to criticality bow waves appear in the experimental flows that are absent from linear theory but modelled by the weakly nonlinear equation. Further from criticality the difference between the linear and nonlinear flows disappears and the experimental results are well described by linear theory. Rotation is shown to introduce sharp crested highly nonlinear waves in the wake behind a near-critical obstacle. Recent exact solutions for fully nonlinear transcritical flow over a one-dimensional parabolic ridge in rotating shallow -500 500 1000 1500 0 distance from obstacle (cm) h = 5 cm, U= 13.5 cm s -1 , t =2 h = 5 cm, U=13.5 cms -1 , t=4 h=5 cm, U=13.5 cms -1 , t=8 water (Esler et al. in press) typically show a shock downstream of the ridge followed by a wavetrain of finite-amplitude Poincare waves of limiting amplitude [START_REF] Shrira | Propagation of long nonlinear waves in a layer of rotating fluid[END_REF][START_REF] Shrira | On long strongly nonlinear waves in a rotating ocean[END_REF]Grimshaw et al. 1998b). Numerical integrations (not shown here) of a fully nonlinear, finite-volume, shallow water model show shocks appearing behind obstacles in non-dispersive rotating flows, providing further evidence of the nonlinearity of the sharp-crested waves of figure 5. The experiments of figure 5 also show that rotation confines the wake to within a distance of order the internal Rossby radius, c 0 /f for critical speed c 0 and Coriolis parameter f, of the axis. More rapid rotation confines the wake more closely to the axis and experiments for rotation periods of TZ60 s and TZ90 s (so fZ0.06, 0.08 s K1 ), giving Rossby radii of 50 and 75 cm, gave a clearly visible wake with little signature away from the axis and so not shown here.

The presentation of the experimental results as a fixed wave pattern travelling with the obstacle assumes that the flow has had sufficient time to become steady before the waves arrive at the probe spar. The obstacle reaches the spar in the middle of the tank in non-dimensional time in the range tZ3.8-6.4, although waves forming the outer edges of the pattern can take twice as long to arrive. Some indication of the extent to which this is sufficient for nonlinear bow waves to develop may be gauged from figure 9, which shows the temporal development of the bow waves in the case of the non-rotating multiple bow wave numerical solution with hZ5 cm, UZ13.5 cm s K1 . The steady state for this experiment is shown in figure 7 (right panel). Figure 9 shows the wavefield at tZ2, 4 and 8. At tZ2 a single bow wave has separated from the dispersive wake, while by tZ4 a second bow wave has begun to separate. It is not until tZ8, however, that the bow wave pattern begins to resemble the final state shown in figure 7 (right panel). It is clear, therefore, that although significant development of nonlinear bow waves has taken place by the time the wave pattern passes the probe spar in the experiments, it is by no means certain that the bow wave pattern has attained its steady state. Experiments intended to detect multiple bow wave patterns would thus need careful design to ensure that t is sufficiently large when measurements are taken.

Figure 1 .

 1 Figure 1. A wavefield above Guadalupe Island, Baja California with crests (solid lines) and troughs (dashed lines) indicated.

Figure 2 .

 2 Figure 2. The experimental set-up. (a) The top view. (b) A vertical cut. The obstacle O is moved along the upper layer by the carriage C which rides along rigid rails R sufficiently far above the surface to clear the spar supporting the interfacial probes P. Each probe consists of an ultrasonic emitter/receiver e above the interface and a reflector r below the interface.

Figure 5 .

 5 Figure 4. (a) A typical density profile showing the particularly sharp interface of the rotating experiments. (b)The time-series of interface displacements from the innermost five probes (in order from the closest at the bottom, with successive traces displaced upwards by 4 cm to avoid overlap) for figure5for rotating flow with towing speed UZ10 cm s K1 . The crest of the bow wave is marked C 1 and the more prominent sharp-crested lee wave is marked C 2 .

Figure 6 .

 6 Figure6. Linear and nonlinear non-rotating (nZ0) f2dBDA steady states corresponding to obstacle height hZ3 cm (MZ12.5) and towing speeds UZ11.5, 13.5 and 15.5 cm s K1 (GZ0.208,K0.625, K1.458).

Figure 7 .

 7 Figure 7. Non-rotating f2dBDA steady states for UZ13.5 (GZK0.625) at nonlinearities corresponding to: Left panel: hZ3 cm (MZ12.5). Right panel: hZ5 cm (MZ20.83).

Figure 9 .

 9 Figure 9. Snapshots at tZ2, 4 and 8 illustrating the temporal development of the non-rotating f2dBDA numerical solution when hZ5 cm and UZ13.5 cm s K1 .

This work was funded by the UK Natural Environment Research Council Grant NER/A/S/2000/01323 and under the European Union contract HPRI-2001-CT-00168, 'Access to Research Infrastructures' of the program 'Improving Human Potential'. The authors are grateful to Henri Didelle and Samuel Viboud for their expert technical assistance.

As in the corresponding experiments, C 1 is the bow wave; T, the long, deep trough; and C 2 , the sharp-crested lee wave in near-critical rotating flow.


Adrien J Göttle

Fabienne Alary

Martial Boggio-Pasqua

Isabelle M Dixon

Jean-Louis Heully

Azadeh Bahreman

Sven H C Askes

Sylvestre Bonnet

Pivotal Role of a Pentacoordinate 3MC State on the Photocleavage Efficiency of a Thioether Ligand in Ruthenium(II) Complexes: A Theoretical Mechanistic Study

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The photochemistry of ruthenium complexes has attracted much attention due to their potential as light-activatable anticancer agents, emissive probes, or photoactivable synthons and prodrugs. 1 Under visible light irradiation, some ruthenium complexes are found to undergo photochemical substitution processes, as shown in Scheme 1, in Indeed, the presence of such bulky ligands is usually acknowledged to decrease the ligand field by decreasing the overlap between the Ru d orbitals and the pyridine nitrogen lone pairs.

In a low ligand field the 3 MC states are more accessible, and several experimental reports demonstrate that in such complexes the efficiency of ligand photosubstitution is indeed enhanced. 3

From a mechanistic point of view it is usually postulated, considering the limited experimental proofs for the existence of intermediates with an increased or reduced coordination number (hepta-or pentacoordinated intermediates), 5 that the ground-state substitution of monodentate ligands L by solvent molecules operates by an interchange mechanism in which the weakening of the initial Ru-L bond is more or less concerted with the formation of the Ru-solvent bond. 6

The aim of the present investigation is first to see whether the classical picture of the photoreactivity of polypyridyl ruthenium complexes is consistent with the topology of the triplet excited state potential energy surface ( 3 PES) for this family of complexes. The second objective is an attempt to rationalize the much higher photoreactivity of [Ru dmbpy] 2+ (φ = 0.30), in comparison to that of [Ru biq] 2+ or [Ru dcbpy] 2+ (φ = 0.12 and 0.13, respectively).

The article is organized as follows: first, the ground states of the four complexes are studied in order to detect a possible steric hindrance effect on the geometries and on the molecular orbital diagrams. Then, we discuss the TD-DFT absorption spectra of the four complexes and we assign the main electronic transitions. Finally, the lowest triplet PES with characteristic connecting points for the four species are reported to discuss the formation and the fate of intermediates involved in the photochemical cleavage of the Hmte ligand upon irradiation (Scheme 1). Although the calculation of the absorption spectra of such complexes is routine, the computation of the excited state relaxation pathways and their interpretation remain challenging. All of this information contributes to shedding light on the difference in photoreactivity of the four complexes.

Computational methods

All of the calculations were performed using the Gaussian 09 7 and Orca 3.0 8 program packages. DFT calculations were carried out using the hybrid-type Perdew-Burke-Ernzerhof exchange correlation functional 9 (PBE0) together with Grimme's D3(BJ) 10 model to add molecular mechanics damped dispersion. We used a triple-ζ quality basis set, including one polarization function, for C, N, O, S, and Cl atoms (Ahlrichs TZVP) 11 and a double-ζ quality basis set also including one polarization function for hydrogen atoms (Ahlrichs SVP). For the ruthenium atom, a Stuttgart relativistic small-core effective potential with its associated polarized basis set including two f and one g polarization functions was used. 12 The choice of PBE0 and of the basis set is based on previous works performed on [Ru(bpy) 2 (DMSO) 2 ] 2+ complexes. 13 Solvent effects (water) were taken into account in geometry optimization calculations using the polarized continuum model 14 (PCM) as implemented in G09, and all stationary points were characterized as minima or transition states (TSs) using analytical harmonic vibrational frequency calculations. Unrestricted DFT calculations were performed to optimize structures on the lowest triplet PES.

TD-PBE0 calculations were performed to compute the first 20 singlet vertical electronic excitations in water using the conductor-like screening model (COSMO). 15 These calculations were performed on the ground state (GS) optimized structures in water. Plots of the intensity vs excitation energy were obtained with the orca_mapspc program that applies a Gaussiantype line shape function to the calculated transition with a defined full width at half-maximum of 1200 cm -1 . The nature of the transitions was determined using natural transition orbitals. 16 Computations of the enthalpic and entropic corrections at 298 K allow the calculation of the activation energies of the photodissociation processes in water. In order to describe deactivation funnels along this pathway, we searched for minimum energy crossing points (MECPs) between the ground and lowest triplet states with Orca. Triplet excited states and MECPs were optimized with PBE0-D3 in the solvent phase.

Optimization of the minimum energy path and calculation of the TSs connecting the two 3 MC states, found in the case of [Ru biq] 2+ and [Ru dcbpy] 2+ , have been performed with PBE0-D3 by using the nudged elastic band (NEB) 17 method associated with the STRING 18 module implemented in the NWChem 19 package.

Results and discussion

Ground state geometrical and electronic structures. The structures of the fully optimized GS structures of the four complexes are shown in Figure 1 along with the numbering of some key atoms. The main structural parameters of the studied complexes in their GS and in their lowest triplet states ( 3 MLCT and 3 MCs) are gathered in Table 1. Further information on the optimized structures can be found in Tables S1-S5, S7, S8, S10-S12, S14-S16, and S18 in the Supporting Information. In the unhindered complex [Ru bpy] 2+ , the terpyridine and the bidentate ligands are planar and lie perpendicular to each other. In contrast, in the three other complexes the bulkiness of the substituted bidentate ligand prevents this optimal geometry.

The distortion of the octahedral geometry around the Ru atom is highlighted by the angle defined by the mean planes of the tpy and N-N ligand. This angle is 89.7° in the unhindered The computed metal-ligand bond lengths are reported in Table 1 together with angles that are relevant for the discussion (see Figure 1 show σ-antibonding interactions between the d orbital of the metal and the S and N5 atoms.

3 MC states are the result of the occupation of these antibonding orbitals. 21

Adding substituents at the bpy ligand lowers the dσ* energy, confirming the influence of steric effects on the ligand field. This effect is tempered in the case of [Ru dmbpy] 2+ by the σ donor effects of the methyl group, which provoke a slight destabilization of this antibonding orbital.

As for the geometrical parameters, dcbpy and dmbpy induce very similar electronic effects. (ii) two different types of connecting points: i.e., TSs between 3 MLCT/ 3 MC and MECPs between 3 MC and 1 GS.

The geometries of the stationary points and their relative free energies are reported in Tables 1 and3, respectively. Figure 4 represents the free energy pathway connecting 3 MLCT and 3 MC states for the four complexes. 3 MLCT states. The geometries of the lowest MLCT states for the four complexes closely resemble those of their respective GS geometry. In the four species, the SOMO-1, namely the hole, is essentially localized on the Ru atom, while the SOMO, namely the electron, is localized on tpy for [Ru bpy] 2+ , [Ru dmbpy] 2+ , and [Ru dcbpy] 2+ and on biq in [Ru biq] 2+ , as could be predicted from the orbital ordering of the ground state (Figure 2).

3 MC states. The commonly accepted model for the photophysical cascade resulting in ligand dissociation ends by the population of a potentially reactive 3 MC state. 2c For two complexes we found that two distinct 3 MC minima are able to play a major role in ligand photocleavage.

It is noteworthy that both structures correspond to the same electronic state.

These two optimized 3 MC geometries result from the same excitation of a Ru (4d) electron to a dσ* MO that is antibonding between the metal and the departing Hmte ligand and the nitrogen atom in a trans position (N5). A closer look at the geometries of these optimized 3 MC states, displayed in Tables 1 and3, reveals that two different structural isomers of the 3 MC state have been characterized, sharing the same electronic occupancy. The first isomer, referred to as 3 MC hexa , is characterized by a geometry where the Ru-S bond elongates to ca.

2.9 Å while the Ru-N5 elongates to ca. 2.4 Å. A slight distortion evidenced by the angle between the mean planes of the tpy and N-N ligands is still present with respect to GS, as shown in but not for [Ru bpy] 2+ . From an energetic point of view, the binding energy was calculated between the metallic moiety [Ru(tpy)(N-N)] 2+ and the Hmte ligand corrected for intramolecular basis set superposition error. 23 The Hmte ligand interacts with the rest of the complex more strongly in the 3 MC hexa state (mean value of ca. 12 kcal mol -1 ) than in the 3 MC penta (mean value of ca. 7 kcal mol -1 ). Photorelease of the monodentate ligand is thus facilitated in the 3 MC penta state. As detailed later, the existence and population of the 3 MC penta state clearly facilitates photocleavage. Connecting points. For the four complexes, the geometries of the TS between the 3 MLCT and 3 MC states (TS-3 MLCT/ 3 MC) were optimized, as well as that of the MECPs along the photodeactivation coordinate (MECP-3 MC/ 1 GS, denoted 1/3 MECP). Geometries, electronic structures, and energetic data of these states are detailed in Tables S6, S9, S13, and S19-S22.

The four TSs connecting the 3 MLCT and the 3 MC minima are easily accessible from the 3 MLCT state, since the highest Gibbs activation energy was found to be 8.8 kcal mol -1 (for the bpy ligand). The Ru-S bond initiates its elongation, up to 0.4 Å, in these TS structures.

The spin density on ruthenium in the TS states has values intermediate between the usual ones expected for a 3 MLCT state (∼0.8) and a 3 MC state (∼1.8). Note that both the Ru-S elongation and the spin density values are consistent with the Hammond postulate. 24 Large

Ru-S elongation (0.402 Å) and high spin density (1.30) are consistent with the endergonic profile of [Ru bpy] 2+ . In contrast, a small Ru-S elongation (0.128 Å) and a lower spin density on ruthenium of the [Ru dcbpy] 2+ complex (1.16) are consistent with the most exergonic profile of the series. These particular geometries are the key points upon which we will build our description of the photorelease of the Hmte ligand.

Table 3. Theoretical mechanistic scheme. A schematic, but realistic, overall energy profile for the photosubstitution process is shown in Figure 5. In this sketch we adopt the terminology proposed by Morokuma et al. 2d To formalize the change in reaction coordinate together with the chemical system during the photosubstitution process, we distinguish three areas: i.e., different reaction coordinates are successively followed along this profile. In the first area, called the "transition process", the process is initiated; it is the MLCT state domain. The second area is where the photocleavage step occurs; it is the region of the 3 MC states. The last area (out of the scope of this work) terminates the process by forming the aqua product. Upon irradiation, the population of the 1 MLCT manifold occurs, followed by ultrafast ICs and ISCs that lead to the population of 3 MLCT states on the lowest 3 PES. From there, overcoming the TS-3 MLCT/ 3 MC state allows the complex to access the 3 MC region, where the departure of the Hmte ligand can occur. Similarly, a vast 3 MC region had been identified in Ru sulfoxide complexes, in that case leading to linkage isomerization. 22a,b

N-N

From our results, three situations can be distinguished that illustrate the different behaviors of the different complexes. They are schematized in Figure 5 by inserts (a)- (c). Insert (a): from the 3 MC hexa state, the 1/3 MECP is easily accessible; the system will be able to surmount this minor barrier and will relax to the ground state by nonradiative decay and without ligand substitution (route 1). The proximity and accessibility of the 1/3 MECP structure from the 3 MC hexa minimum guarantees a high photostability for these complexes: i.e., low photosubstitution quantum yields.

Insert (b): the coexistence of two minima ( 3 MC hexa and 3 MC penta ) on the flat 3 MC surface opens two competing pathways. In addition to the possibility of following route 1 and restoring the initial complex, the 3 MC penta state can be easily reached from the 3 MC hexa state.

Indeed, the resemblance of the 3 MC hexa and 3 MC penta reflected in the very close energies between these two MC states results in a very small activation barrier: ca. Insert (c): Direct population of the 3 MC penta state by interconversion from the 3 MLCT state, which is a consequence of the absence of a 3 MC hexa local minimum, favors route 2 with relaxation of the 3 MC penta to the ground state of the aqua complex or to diffusional separation. 2e These complexes are ideal for ligand solvolysis and will show high photosubstitution quantum yields.

In the following discussion we confirm the well-accepted model where the accessibility and the population of a 3 MC state is a key factor to achieve photoinduced ligand exchange. From

Figure 4, we can see that the 3 MC states (MC hexa and/or MC penta ) for [Ru biq] 2+ , [Ru dcbpy] 2+ , and [Ru dmbpy] 2+ are lower in energy than the corresponding 3 MLCT states, consistent with the absence of luminescence for these three species. [Ru bpy] 2+ has its 3 MC hexa state above its 3 MLCT state, and these two states are separated by the largest free activation energy barrier (8.8 kcal mol -1 ). A small backward activation free energy barrier will favor the 3 MC hexa →

3 MLCT back-conversion. Such results are consistent with experimental observations indicating that [Ru bpy] 2+ is the only complex of the series which displays a weak luminescence at 298 K (see Figure S3 in the Supporting Information). The vertical emission energy calculated from the 3 MLCT state is in excellent agreement (630 nm) with the experimental emission wavelength (650 nm; see Figure S3). As illustrated in insert (a), our calculations on the lowest 3 PES of [Ru bpy] 2+ located only one 3 MC minimum. Close to this state, we have identified a highly accessible crossing point, 1/3 MECP, located only 2.8 kcal mol -1 above the 3 MC hexa minimum. The accessibility of this MECP, which provides a facile pathway back to the ground state, combined with the absence of 3 MC penta , prevents efficient photoreactions from occurring and explains the small experimental value found for the quantum yield of photo-substitution of the Hmte ligand by a water molecule (φ = 0.022). 3b

According to our analysis, the best schematic picture explaining the behavior of [Ru bpy] 2+ is that described by insert (a).

For [Ru dmbpy] 2+ the 3 MLCT state must overcome a small barrier of 5.3 kcal mol -1 to directly reach a 3 MC penta state, which is computed to be 5.2 kcal mol -1 lower in energy than the 3 MLCT state. Once this 3 MC penta state is populated, the nearby crossing point 1/3 MECP is easily accessible (0.9 kcal mol -1 ), that allows the system to deactivate into the ground state.

However, in this case the highly elongated Ru-S distance observed in the 3 MC penta state suggests a Ru-S bond cleavage pathway following a quasi-dissociative mechanism.

Meanwhile, the absence of an energy barrier on the flat landscape of the 3 PES must facilitate the decoordination of the ligand, which fits with the highest observed photosubstitution quantum yield in the series (φ = 0.30). Thus, according to this analysis, insert (c) best describes the photochemical behavior of [Ru dmbpy] 2+ . It is noteworthy that, to ensure the absence of 3 MC hexa geometry as a minimum in the case of the dmbpy ligand, we tried to reach the 3 MC hexa isomer by starting a geometry optimization from the 3 MC hexa state of the [Ru dcbpy] 2+ complex, replacing the chloride atoms by methyl groups. No true minimum was found, however: i.e., [Ru dmbpy] 2+ only displays the 3 MCpenta isomer.

Finally, [Ru dcbpy] 2+ and [Ru biq] 2+ behave similarly. For both complexes we were able to distinguish two local minima on the excited 3 PES, 3 MC hexa and 3 MC penta , with the 3 MC penta isomer being more stable than the usual 3 MC hexa isomer and with only a small barrier between these two structures. In our analysis the intermediate photostability measured for these two complexes is best illustrated by insert (b). Indeed, the population of the 3 MC hexa state may either lead to repopulation of GS via the 1/3 MECP (almost degenerate with 3 MC hexa , i.e., located at 0.3 and 0.9 kcal mol -1 ) following route 1 or undergo ligand substitution via the 3), are different enough to contradict these conclusions.

Our hypothesis is that these differences in photoreactivity are mainly influenced by the topology of the 3 MC surface and by the geometry of the photoreactive states. Following our calculations, we propose the following photosubstitution mechanism: (i) population of the 3 MC hexa state leads to a structure where the Ru-S distance is slightly elongated. By staying in the coordination sphere of the metal, the Hmte ligand prevents later access of a solvent molecule to a d orbital of the Ru atom. In addition, from this 3 MC hexa state, the nonreactive deactivation pathway to the GS is easy due to the accessible 1/3 MECP crossing point. If a 3 MC penta minimum is present on the 3 PES, the Hmte ligand undergoes a misalignment and the Ru-S bond distance is elongated to such an extent that the d orbital of Ru becomes available for the attack of a water molecule. It might be noted that in the 3 MC penta isomer the [Ru(tpy)(N-N)] 2+ moiety recovers a near-octahedral arrangement of the coordination sphere charac-terized by angles between the mean planes of the tpy and N-N ligands around 85°.

Thus, it appears that bulky bidentate ligands not only distort the geometry of the complex in the GS but also generate on the 3 PES the more reactive 3 MC penta isomer where the misaligned conformation favors ligand exchange. In [Ru dmbpy] 2+ the existence of a unique 3 MC penta local minimum with accessible d orbitals facilitates Hmte photodissociation and minimizes nonreactive deactivation pathways, which explains the very high experimental value of φ.

Conclusion

The results presented in this work show that discussing the energy level of the dσ* or "e g -like" molecular orbitals of ruthenium polypyridyl complexes in their ground state is not sufficient to predict the effectiveness with which crucial photoreactive 3 MC states are populated.

Instead, geometry optimization of these triplet states must be achieved, as it is the only way to will be to investigate the end of the process, i.e., the formation of the photoproduct (route 2 on Figure 5), in order to ultimately better design metal complexes suitable for medicinal applications. To reach this objective, a molecular dynamics study of the reactivity and of the photoreactivity of this series of complexes in aqueous solution is in progress.

Supporting information

Cartesian coordinates, energies, and orbitals of all the optimized stationary points, structures of the MECPs, MEPs, and the experimental emission spectrum of [Ru bpy] 2+ in water

Corresponding Author Fabienne Alary. E-mail fabienne.alary@irsamc.ups-tlse.fr

  Scheme 1. Photosubstitution of the Hmte ligand by an aqua ligand (top) and ligand structures (N-N) used in this work (bottom) a

Figure 1 .

 1 Figure 1. Structure of [Ru bpy] 2+ with atom labeling and definition of the mean planes of the tpy and N-N ligands (biq ligand). H atoms have been removed for clarity.

  for atom numbering). The maximum discrepancy between computed and experimental bond lengths available for [Ru bpy] 2+3b and [Ru dcbpy] 2+3b is ∼0.01 Å for Ru-N bonds and 0.02 Å for Ru-S bonds. The introduction of bulky substituents on the bidentate ligand provokes an elongation of Ru-N4 and Ru-N5 bond lengths by 0.04-0.06 Å, which is in the range of previously reported values for related complexes. 2a,20 At this stage, it is thus impossible to differentiate the three N-N ligands in terms of geometrical parameters.

Table 1 .

 1 Selected bond lengths (Å) and angles (deg) for the ground state and for the lowest triplet states located trans to the thioether ligand. b Single-crystal X-ray data are provided in parentheses for [Ru bpy] 2+ 3b and [Ru dcbpy] 2+ .3b c This structure, called 3 MC penta for the sake of clarity (to distinguish it from 3 MC hexa ), can be viewed as a weakly bound complex.The molecular orbital (MO) diagrams that characterize the electronic structures of the four complexes in their ground state are reported in Figure2. For all of the complexes, the three highest occupied molecular orbitals (HOMOs) are metallic d orbitals with minor contributions of the ligands, while the lowest unoccupied molecular orbitals (LUMOs) are the π* orbitals of the polypyridyl ligand (tpy, N-N, or both). It is noteworthy that the localization of the LUMO reflects the π-accepting properties of the bidentate ligand. For [Ru bpy] 2+ the LUMO is only located on the tpy ligand and for [Ru dcbpy] 2+ and [Ru dmbpy] 2+ the LUMO is distributed on both polypyridyl ligands; the LUMO of [Ru biq] 2+ mainly resides on the biquinoline ligand. The large delocalized π system of the biq ligand is responsible for the intercalation of a lowlying π* energy level, which significantly decreases the HOMO-LUMO energy gap in comparison to the other complexes. Higher in the diagram, metal-centered vacant MOs (dσ*)

Figure 2 .

 2 Figure 2. Ground state MO diagrams illustrating the electronic structure of the investigated complexes. Energy levels related to metallic orbitals are displayed as solid lines, and those of ligand-based orbitals are shown as dashed lines. Only frontier MOs are drawn for clarity. The HOMO energy is set to 0.

Figure 3 .

 3 Figure 3. Simulated absorption spectra of the four complexes. Theoretical and experimental values of the absorption maxima are indicated in nanometers for comparison (λ theor /λ exp ).

Figure 4 .

 4 Figure 4. Schematic representation of the free energy profiles computed with DFT in water at 298 K for [Ru bpy] 2+ (in red), [Ru biq] 2+ (in blue), [Ru dmbpy] 2+ (in black), and [Ru dcbpy] 2+ (in green). The 3 MLCT states were deliberately placed at the same origin for clarity.

Table 2 .

 2 Comparative views of the GS, 3 MC hexa , and 3 MC penta states of [Ru(tpy)(N-N)(Hmte)] 2+ complexes along the N2-Ru axis a a Some atoms are hidden for clarity. Values indicate the Ru-S bond distance and the angle (in boldface) between the mean planes of tpy (horizontal) and the bidentate ligand (more or less vertical).

  energy (in kcal mol -1 ) of the stationary points involved along the photodissociation profiles calculated in water using COSMO with 3 MLCT states as reference. b Energy difference between 3 MC states and ground states. c Experimental photosubstitution quantum yield of [Ru(tpy)(N-N)(Hmte)] 2+ in water according to ref 3b.

Figure 5 .

 5 Figure 5. Schematic potential energy profiles including the lowest energy states involved in the proposed decoordination pathway. Red dots refer to optimized geometries (minima, TS, MECP). The transition process

  Figures S1 and S2 in the Supporting Information. The coordination of a water molecule is not addressed in this study, but our hypothesis is that the 3 MC penta could cross the ground state PES of the aqua product, enabling solvent substitution to yield [Ru(tpy)(N-N)(H2O)] 2+ (route 2 in Figure5) or to undergo diffusional separation. 2e Both routes are kinetically and

3

  photosubstitution quantum yields. Indeed, relative to [Ru bpy] 2+ , the three hindered complexes [Ru biq] 2+ , [Ru dcbpy] 2+ , and [Ru dmbpy] 2+ show more distorted GS geometries and lower energies for their reactive 3 MC states, which is consistent with the higher experimental photo-substitution quantum yields in water.However, in a more refined discussion, distorted geometries, as well as electronic and steric effects of the substituents on the bidentate ligand, are insufficient to explain the difference in photosubstitution quantum efficiency between [Ru biq] 2+ and [Ru dcbpy] 2+ on the one hand and [Ru dmbpy] 2+ on the other hand. Previous works of Takeuchi 25 on similar molecules and

  observe how relaxation from the Franck-Condon geometry may open new pathways on the triplet PES that explain the experimentally observed differences in photoreactivity.Different mechanistic schemes that allow the photocleavage of the Hmte ligand were identified. Along the reaction path, the intervention of a pentacoordinate 3 MC intermediate, the ability for the excited complexes to populate this photoreactive state, and nonradiative deactivation toward the GS govern the quantum yields of photosubstitution. The very low quantum yield of photosubstitution reported for [Ru bpy] 2+ (φ = 0.022) is explained by the absence of 3 MC penta and a predominant nonradiative deactivation path favored by an accessible 1/3 MECP from the 3 MC hexa . In contrast, for the highly photoreactive [Ru dmbpy] 2+ complex (φ = 0.30), the existence of only one type of photoreactive state, i.e., 3 MC penta , favors the formation of a photocleaved intermediate which can easily relax to accommodate a water solvent molecule. The intermediate photosubstitution quantum yields reported for [Ru biq] 2+ and [Ru dcbpy] 2+ (φ ≈ 0.12) result from a combination of the two previous competitive processes. This study highlights the processes that are involved in the first stages of the photosubstitution of the Hmte ligand by a solvent molecule. Our next goal

Table 2 .

 2 This 3 MC hexa state has been characterized for [Ru bpy] 2+ , [Ru biq] 2+ , and [Ru dcbpy] 2+ but not for [Ru dmbpy] 2+ . The second isomer, referred to as 3 MC

penta , 2e exhibits a much longer Ru-S distance, up to 4 Å, accompanied by a shortening of the trans Ru-N5 bond and by a planarization of the bidentate ligand, as testified by the values of mean plane angles which increase to reach 85.7° in the case of [Ru dcbpy] 2+ . These rearrangements clearly design these 3 MC penta states as ideal starting structures for the photoinduced Hmte departure. Similar structures have been previously proposed for the photorelease of a pyridine ligand. 2j These 3 MC penta states were found for [Ru biq] 2+ , [Ru dcbpy] 2+ , and [Ru dmbpy] 2+

Acknowledgements

This work was granted access to the HPC resources of the CALMIP supercomputing center under the allocation 2014-[1041] and 2015-[1133]. The NWO (The Netherlands Organization for Scientific Research) and the European Research Council are acknowledged for a VIDI and a Starting grant, respectively, to S.B.