Andreea Grigoriu
email: grigoriu@ljll.univ-paris-diderot.fr

Herschel Rabitz
email: hrabitz@princeton.edu

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

H Rabitz ‡

Controllability Analysis of Quantum Systems Immersed within an Engineered Environment

Keywords: quantum control, controllability of non-Markovian equations, Lie group controllability, controllability in a non-compact Lie group AMS subject classifications. 81Q93 , 93B05

The problem of controllability of quantum systems interacting with an engineered environment, whose dynamics are described by a non-Markowian master equation is addressed. The manipulations of the dynamics is realized with both a laser field and a tailored non-equilibrium, and generally time-dependent, state of the surrounding environment. Lie algebra theory is used to characterize the structures of the reachable state sets and to prove controllability. The theoretical results are supported by examples.

1. Introduction. Since the first successful laboratory experiments obtained in the 1990s [START_REF] Assion | Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses[END_REF][START_REF] Judson | Teaching lasers to control molecules[END_REF], the control of quantum systems using laser fields has been subject to significant developments ( [START_REF] Assion | Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses[END_REF][START_REF] Brumer | Coherent chemistry: Controlling chemical reactions with lasers[END_REF][START_REF] Judson | Teaching lasers to control molecules[END_REF][START_REF] Levis | Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses[END_REF][START_REF] Weinacht | Controlling the shape of a quantum wavefunction[END_REF] etc.). An important part of the associated theoretical work has been devoted to the investigation of closed quantum systems with unitary dynamics. However, realistic physical situations include circumstances where the quantum system is not isolated, but interacting with an environment (e.g., a molecule in a solvent). For these type of systems, also called open quantum systems, one of the main characteristics is that the dynamics is non-unitary. Generic open systems may be difficult to control because the environment can contain irreversible dynamics that fights against control mechanisms. We consider in this work a particular situation when the environment can be engineered i.e. its characteristics chosen at will. The resulting circumstance is not a general open quantum system but a large closed system containing a small part is of interest to the experimenter. The mathematical formulation will be identical to that used to describe an open system except that the environment is considered controllable.

For weak system-environment coupling the dynamics of the system is described by a Markovian equation involving the reduced density matrix ρ d dt ρ(t) = Lρ(t) (1.1)

with the generator L in Lindblad form (see [START_REF] Breuer | The theory of open quantum systems[END_REF][START_REF] Gorini | Properties of Quantum Markovian Equations[END_REF][START_REF] Gorini | Completely positive dynamical semigroups of N-level systems[END_REF]) :

Lρ = -i[H, ρ] + i γ i (L i ρL * i - 1 2 L * i L i ρ - 1 2 ρL i L * i ), (1.2)
where the first term represents the unitary part of the system dynamics, with H the Hamiltonian of the system. The second term describes the dissipative dynamics, also referred as the dissipator, dependent upon the Lindblad operators L i . The time independent coefficients γ i represent coupling to the environment, functioning as the relaxation rates for different decay modes of the open system. In the following for any X, we denote by X * its adjoint.

For cases of strong system-environment coupling, structured and finite reservoirs, and/or low temperatures one often encounters non-Markovian processes, implying that the dynamics is governed by significant memory effects. A systematic approach to non-Markovian dynamics is provided 2 by projection operator techniques [START_REF] Haake | Statistical treatment of open systems[END_REF]18,[START_REF] Zwanzing | Experimental control of the transition from markovian to non-markovian dynamics of open quantum systems[END_REF]. These techniques can lead to integro-differential equations for the reduced density matrix, through application of the Nakajima-Zwanzig projection operator technique. If a time-convolution free projection operator technique [START_REF] Arimitsu | Expansion formulas in nonequilibrium statistical mechanics[END_REF][START_REF] Breuer | Genuine quantum trajectories for non-Markovian processes[END_REF][START_REF] Royer | Combining projection superoperators and cumulant expansions in open quantum dynamics with initial correlations and fluctuating hamiltonians and environments[END_REF]] is applied, one can deduce an approximate or even exact non-Markovian, first-order differential equation for the open system reduced density matrix ρ:

d dt ρ(t) = K(t)ρ(t). (1.
3)

The non-Markovian character of this equation is reflected by the fact that its generator K depends explicitly on time and does not have the Lindblad form above.

If we require that the density matrix be Hermitian and its trace constant, the most general form of the generator K(t) is given by [START_REF] Breuer | Genuine quantum trajectories for non-Markovian processes[END_REF][START_REF] Gorini | Completely positive dynamical semigroups of N-level systems[END_REF] :

K(t)ρ = -i[H(t), ρ] + i γ i (t)(A i (t)ρA * i (t) - 1 2 A i (t) * A i (t)ρ - 1 2 ρA i A * i ), (1.4)
where the Hamiltonian H(t), the operators A i (t) and the relaxation rates γ i (t) depend on time. If the relaxation rates γ i (t) are positive functions, the generator K(t) is in Lindblad form for each fixed t 0. Such processes may be called time-dependent Markovian although the corresponding dynamical map do not lead to a quantum dynamical semigroup [START_REF] Breuer | Measure for the degree of non-Markovian behavior of quantum processes in open systems[END_REF]. However if the relaxation rates become negative in certain time intervals (see examples of negative rates γ i in [START_REF] Breuer | Genuine quantum trajectories for non-Markovian processes[END_REF][START_REF] Breuer | Measure for the degree of non-Markovian behavior of quantum processes in open systems[END_REF]), the master equation in no longer in Lindblad form and the generator is not completely positive. Therefore non-Markovian character is also related to the emergence of negative rates.

Given the diversity of models that describe the quantum system-environment interaction, the problem manipulating them becomes difficult to address, especially since environment is usually composed of a large number of degrees of freedom which are difficult to control. Despite that, research on control of such systems is strongly motivated by a large number of applications including quantum computing [START_REF] Chuang | Quantum computers, factoring, and decoherence[END_REF], laser cooling, quantum reservoir engineering, management of decoherence, chemical reactions and energy transfer in molecules [START_REF] Rabitz | Control of quantum dynamics: Concepts, procedures and future prospects[END_REF].

Optimal control by a tailored non-equilibrium, and generally time-dependent, state of the surrounding environment has been addressed in the literature (see [START_REF] Pechen | Teaching the environment to control quantum systems[END_REF] for more details). The master equation for the system interacting with an environment, characterized by its distribution function n k (t) is given by

d dt ρ(t) = -i[H, ρ(t)] + ω γ ω (n k (t))(A ω ρA * ω - 1 2 A * ω A ω ρ - 1 2 ρA * ω A ω ), (1.5)
where the coefficients γ ω (t) 0 determine the transitions rates between energy levels with transition frequencies ω.

Control by lasers fields normally affects the system through Hamiltonian evolution and transforms pure states into pure states, while control by the environment (i.e., control by γ(t)) affects a system through dissipative dynamics and can be used to steer the system from a pure or a mixed state into mixed and in some cases pure states (a familiar example is the cooling of a thermalized quantum system, which requires coupling to a reservoir). This type of incoherent control by the environment may be combined with coherent fields to allow for simultaneous control through both the Hamiltonian and dissipative parts of the system dynamics.

The equation for a system that simultaneously interacts with an electromagnetic field (t) and an environment described by a function γ(t) 0 generally has the form [START_REF] Pechen | Teaching the environment to control quantum systems[END_REF]:

d dt ρ(t) = -i[H 0 + H ef f + (t)H 1 , ρ(t)] + i γ i (t)(A i ρA * i - 1 2 A * i A i ρ - 1 2 ρA * i A i ), (1.6
) where H 0 is the Hamiltonian describing the coherent part of the dynamics, H ef f is the effective Hamiltonian describing the unitary part of the system-environment interaction and H 1 is the dipole moment operator describing the interaction between the system and the field (t).

Although the notion of simultaneous control by both a laser and by the environment has been mentioned in the literature [START_REF] Pechen | Teaching the environment to control quantum systems[END_REF], a theoretical analysis of the controllability has not been investigated yet. The goal of this paper is to propose such an analysis in a generic situation. A fundamental question involving the system above is whether for any pair of states ρ i and ρ f , both positive semidefinite, Hermitian, tr(ρ i ) = tr(ρ f ) > 0, a control ( (t), γ(t)) exists such that ρ(0) = ρ i and ρ(T ) = ρ f for some T > 0. If the answer is positive the system is called controllable. Since γ(t) may also take negative values, the present work addresses the controllability analysis for situations when both (t) and γ(t) are arbitrary real numbers. Under this hypothesis the property of complete positivity of the density matrix is lost; moreover with dynamics being non-Markovian. Note that if physical considerations require γ(t) to have further specific properties the results of the analysis will not apply, but a practical proposal is to impose the constraints in the numerical simulations by penalization, leading to an optimal control framework.

Thus we suppose that the system follows a first-order differential equation (1.6) for the reduced density matrix ρ. The generator for the above equation is time dependent and has the same form as K(t), previously defined by equation (1.4). Our main goal is to analyze the controllability (as defined above) of the equation (1.6).

Another motivation for this analysis is given by systems that are not controllable in the isolated setting (γ(t) = 0); one direction on addressing controllability can rely on the use of specially tailored environments, characterized by γ(t), applied through the dissipative part of the dynamics. It should be noted, however, that the assumption γ(t) ∈ R does not imply that the propagation of an arbitrary positive matrix at time t with the help of the master equation (1.6) necessarily leads to a positive matrix for future times. It is important to mention that although we want to introduce results as close as possible to physical circumstances, the complexity of the theoretical problems addressed in the framework of controllability of quantum systems interacting with an environment is a complex topic for which we have a partial analysis.

The balance of the paper is as follows: in Section 2 we introduce the general framework, the main notation together with a short review of Lie algebra controllability results previously introduced in the literature. In Section 3 we characterize the reachable sets and we prove two controllability results. The last section is dedicated to numerical computations.

2. The problem. We consider a quantum system interacting with an environment. The system is manipulated by both a laser field of intensity (t) and a tailored non-equilibrium, and generally time-dependent rate, γ(t) depending on the state of the surrounding environment; the dynamics is described by equation (1.6). We suppose that the quantum dynamics takes place in a finite dimensional space (e.g., either intrinsically so or because a suitable large basis set approximation has been chosen). Thus, ρ(t) is a N × N complex matrix for some integer N > 0, H 0 , H ef f and H 1 are real N -dimensional symmetric matrices (thus Hermitian), A i is an arbitrary N -dimensional matrix and , γ are real time dependent functions.

In the following, we study the controllability (see the definition above) of the N -dimensional equation (1.6). Without loss of generality we consider throughout the paper only the circumstance when all γ i are null except γ 1 . This assumption is not restrictive since the same general results can be obtained for the case when more then one γ is non-null. Since the effective Hamiltonian H ef f is assumed to commute with H 0 we can redefine H 0 to be H 0 + H ef f and omit H ef f in the sequel. Therefore, under the above hypothesis equation (1.6) becomes:

d dt ρ = -i[H 0 , ρ] -i (t)[H 1 , ρ] + γ(t)(AρA * - 1 2 ρA * A - 1 2 A * Aρ). (2.1)
In the following, for simplicity, we use the notation K for the generator of the equation (2.1)

K(t)ρ = -i[H 0 , ρ] -i (t)[H 1 , ρ] + γ(t)(AρA * - 1 2 ρA * A - 1 2 A * Aρ).
(2.2) systems (2.1) with γ positive and constant. The problem of kinematic state controllability (KSC) of open quantum systems whose dynamics are represented by Kraus maps has been considered in [START_REF] Wu | Controllability of open quantum systems with Kraus-map dynamics[END_REF].

The behavior of a Markovian master equation for a N -level quantum mechanics system driven by a coherent control field (t) is analyzed [START_REF] Altafini | Controllability properties for finite dimensional quantum Markovian master equations[END_REF][START_REF]Coherent control of open quantum dynamical systems[END_REF]. It is shown that the system can be accessible, but neither small-time controllable nor controllable in finite time.

A deterministic two stage method for engineering arbitrary pure and mixed states for quantum systems subject to the Markovian evolution is proposed in [START_REF] Pechen | Engineering arbitrary pure and mixed quantum states[END_REF]. In the first stage, the system evolves under the action of a suitable optimal incoherent control. In the second stage, the system evolves under the action of a suitable coherent laser control. But in that work the focus in on the control of the dissipative regime i.e. the control by the coherent part (the laser of intensity (t) is already supposed complete), while here we on the contrary want to use γ(t) to bring supplementary control in addition to that of (t).

In this paper we use the Lie algebraic approach (cf. [START_REF] Altafini | Controllability of quantum mechanical systems by root space decomposition of su(n)[END_REF][START_REF] Dahleh | Controllability of molecular systems[END_REF][START_REF] Rabitz | Wavefunction controllability in quantum systems[END_REF]) to analyze the controllability of quantum systems those dynamics is described by equation (2.1). An advantage of this technique is that it also provides an explicit controllability criteria, which can be verified using numerical computations.

As will be seen in the sequel, controllability analysis of systems defined by (2.1) must deal with the loss of compactness that arises from the non-Hermitian nature of the generator K defined by (2.2). Another problem is that we do not know a priori the structure of the reachable states, identifying the Lie group remains a task to be investigated.

Background on controllability on Lie groups.

Let us consider a connected but not necessarily compact Lie group G with Lie algebra L(G) and control system

dX dt (t) = X 0 (X(t)) + m i=1 u i (t)X i (X(t)), (2.3)
where X 0 and X i are right-invariant vector fields on G. where we denote by e the identity of the Lie group G; thus, describing the set R t (e)Y allows for completely describing all the other reachable sets. When the final time is not specified, we will use

R(Y ) = ∪ t 0 R t (Y ). (2.6)
We take the admissible controls u i (t) to be the set of all locally bounded and measurable functions.

Consider L to be the Lie algebra generated by X 0 , X 1 , ..., X m and S its corresponding Lie group (Lie subgroup of G). We do not assume that S is compact.

The results proved below build on the following reformulation of a result in [START_REF] Jurdjevic | Control systems on Lie groups[END_REF][Thm 6.6] (to which we refer for further details):

Theorem 2.1 [START_REF] Jurdjevic | Control systems on Lie groups[END_REF]. If there exists a constant control u = (u 1 , ..., u m ) and a sequence of positive numbers {t n } with t n δ > 0, for some δ, with the property that lim n→∞ X(t n , u, e) exists and belongs to S (the closure is relative to S ) then R(e) = S. 3. Controllability. We define the operators H 0 , H 1 and T as follows:

H 0 : ρ -→ -i[H 0 , ρ] H 1 : ρ -→ -i[H 1 , ρ] T : ρ -→ AρA * - 1 2 ρA * A - 1 2 A * Aρ, (3.1)
and rewrite equation (2.1) as:

d dt ρ = H 0 ρ + (t)H 1 ρ + γ(t)T ρ. (3.2)
We introduce the sets of matrices:

Γ N = {Z ∈ C N ×N |Z = Z * }, Γ 0 N = {Z ∈ C N ×N |Z = Z * , tr(Z) = 0}, (3.3)
and recall that Γ N (respectively Γ 0 N ) has dimension N 2 (respectively N 2 -1) when seen as a vector space over R. Note that, denoting by ad x the map y → ad x (y) = [x, y] (the adjoint representation), we also have

H 0 = ad -iH0 , H 1 = ad -iH1 . (3.4)
Also, when

A = A * then T = -1 2 ad A • ad A . A simple computation indicates that for any Z ∈ Γ N : H 0 (Z), H 1 (Z), T (Z) ∈ Γ N .
Moreover all three operators are linear. Thus H 0 , H 1 , T ∈ Lin(Γ N , Γ N ), the space of linear operators from Γ N to Γ N (this also results from Eqn (3.4)). Also note that for any matrix Z:

tr(H 0 Z) = tr(-i[H 0 , Z]) = 0 tr(H 1 Z) = tr(-i[H 1 , Z]) = 0 tr(T Z) = tr(AZA * - 1 2 ZA * A - 1 2 A * AZ) = 0.
Thus we also have H 0 , H 1 , T ∈ Lin(Γ 0 N , Γ 0 N ). In particular the trace of ρ will not change during the evolution. Moreover as we will see below, when A is Hermitian another conservation law exists: if ρ(0) = λI for some λ ∈ R then ρ(t) = λI for all t ≥ 0. This motivates the following definition: Definition 3.1. The evolution (3.2) is density matrix controllable if for any positive semidefinite Hermitian matrices ρ i and ρ f with tr(ρ i ) = tr(ρ f ) > 0 there exists a time t ≥ 0 and locally bounded measurable controls (•), γ(•) such that the solution of the evolution equation (2.1) starting at 0 from ρ i reaches ρ f at time t. When A = A * we suppose

ρ i = tr(ρi) N Id, ρ f = tr(ρ f )
N Id. The standard Lie group setting for investigating the control of an isolated quantum system works with the Lie group of unitary (or special unitary) matrices U (N ) (or SU (N )). Here, on the contrary, we work on the group of one-to-one transformations of Γ N to Γ N (or the same for Γ 0 N ). As a numerical illustration of the difference, the dimension of U (N ) as a real Lie group is N 2 while the dimension of the one-to-one transformations from Γ N to itself is N 4 .

We investigate controllability results in two distinct situations: when the matrix A is Hermitian or not.

Remark 3.1. The particular case of A being Hermitian corresponds to unital operator T , i.e. T I = 0, which is a situation frequently addressed in quantum information processing. N that contains the identity operator. The canonical notation of this group is GL + (Γ 0 N ). As we do not impose any particular structure on Γ 0 N except that of vector space, Γ 0 N is isomorphic to R N 2 -1 ; thus the Lie group GL + (Γ 0 N ) is isomorphic to GL + (N 2 -1), the Lie group of invertible matrices of dimension (N 2 -1) × (N 2 -1) with positive compact. We will denote by Lie(G 1 ) the Lie algebra of G 1 which is therefore Lin(Γ 0 N , Γ 0 N ). Denote by H G1 0 the element of Lie(G 1 ) that is constructed from H 0 (and the same for H 1 and T ). We associate to the evolution equation (3.2) the following evolution equation on the group G 1 :

d dt X(t) = H G1 0 + (t)H G1 1 + γ(t)T G1 X(t), X(t = 0) = X 0 .
(3.5)

We will also write X(t; , γ; X 0 ) when we will need to make explicit the dependence on the parameters; of course X(t; , γ; X 0 ) belongs to the Lie group G 1 . Then by definition ρ(t) = X(t; , γ; e)ρ(0) (here e is the identity of the Lie group G 1 ). Theorem 3.2. If the Lie algebra Lie{H G1 0 , H G1 1 , T G1 } ⊂ Lie(G 1 ) generated by {H G1 0 , H G1 1 , T G1 } has dimension (N 2 -1) 2 (as a vector space over the real numbers), then the system (3.2) is density matrix controllable.

Proof. Without loss of generality we can suppose that tr(H 0 ) = 0 and tr(H 1 ) = 0. As a side remark, note that since A is Hermitian we can show by computation that T is a Hermitian operator from Γ 0 N to itself i.e., T * = T . However H 0 and H 1 are skew-Hermitian as operators from Γ 0 N to itself. We consider the following change of variables:

ρ(t) = ρ(t) - tr(ρ(0)) N I, (3.6)
which implies that ρ(0) ∈ Γ 0 N . However this does not imply directly that ρ(t) ∈ Γ 0 N for all t > 0. To prove this one recalls that A is Hermitian and H 0 , H 1 , T ∈ Lin(Γ 0 N , Γ 0 N ); after invoking the Frobenius theorem for the equation:

ρ = -i[H, ρ] + T (ρ), (3.7)
one obtains that ρ(t) ∈ Γ 0 N for all t ≥ 0. From the above relations we conclude that H 0 (ρ), H 1 (ρ) and T (ρ) belong to Lin(Γ 0 N , Γ 0 N ). The central question is to characterize R(e) for the system (3.5). We will use Theorem 2.1

with G = G 1 . Then L = Lie iH G1 0 , iH G1 1 , T G1 is the Lie algebra generated by H G1 0 , H G1 1 and T G1 . Recall that L ⊂ Lie(G 1 ) and dim R (Lie(G 1 )) = dim R (Lin(Γ 0 N , Γ 0 N )) = (N 2 -
1) 2 (dimension as vector space over R). By hypothesis dim R (L) = (N 2 -1) 2 thus L = Lie(G 1 ) and S = G 1 . Take now = 0,γ = 0, then equation (3.5) becomes:

d dt X(t) = H G1 0 X(t), X(t = 0) = e, (3.8)
with solution X(t) = e tH G 1 0 ; take a sequence of positive numbers {t n } with t n δ > 0 for some δ (strictly positive, arbitrary but fixed); we need to prove that lim n→∞ e tnH G 1 0 ∈ S.

(3.9)

But, H G1 0 is a skew-Hermitian map, which means that e tnH G 1 0 belongs to the group SO(Γ 0 N ) of the special orthogonal transformations of Γ 0 N ; it is known that SO(Γ 0 N ) is the maximal compact subgroup of G 1 = GL + (Γ 0 N ). By compactness we obtain that up to extracting a subsequence lim n→∞ e tnH G 1 0 exists and belongs to S. Therefore R(e) = S. In particular X can reach any one-to-one transformation from It should be noted that in particular this set contains matrices that are not necessarily positive semidefinite. This means that the control, even between two states that are positive semidefinite may also use intermediary states ρ(t) that are not physical. Should such a circumstance appear in practice, a numerical algorithm that penalizes, e.g. the presence of non-physical states, may be used. The result above indicates that an initial guess that offers perfect control but with a possible non-physical trajectory, can always be subject to such an algorithm.

Remark 3.4. A straightforward extension is to consider the circumstance when several nonnull controls γ 2 ,... are present; the theoretical results can be proved in the same manner.

Remark 3.5. In this paper two controllability results are proved for γ(t) ∈ R but not necessarily positive [START_REF] Breuer | Genuine quantum trajectories for non-Markovian processes[END_REF]. However physical considerations may impose γ to be positive (see [START_REF] Breuer | The theory of open quantum systems[END_REF] for more details). Previous controllability results cannot be used due to the loss of compactness. Therefore the controllability analysis of the system defined by (2.1), for γ(t) ≥ 0, remains for now a conjecture and leads to the question: are Theorems 3.2 and 3.3 true if γ(t) takes only positive values ? 4. Application. In the following we illustrate the theoretical results introduced in the above section. For this purpose consider two finite-dimensional systems defined by

3. 1 .

 1 Situation I: Hermitian operator A. Let us now consider the connected Lie group G 1 of one-to-one linear transformations of Γ 0

Acknowledgments. The authors are grateful to Alireza Shabani for useful discussions and comments.

* This work was supported by "Agence Nationale de la Recherche" (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01, US-Department of Energy and Army Research Office.

Γ 0 N to itself. For any ρ i and ρ f with tr(ρ i ) = tr(ρ i ), ρ i -tr (ρi) N I = 0 and ρ f -tr(ρ f )

N I = 0 we can find a transformation to map the non-null vector ρ i -tr (ρi) N I ∈ Γ 0 N to the non-null vector ρ f -tr(ρ f ) N I = ρ f -tr (ρi) N I ∈ Γ 0 N i.e., we have controllability for ρ thus for ρ. Remark 3.2. Following the proof of Theorem 3.2 we can also conclude that the set of all density matrices reachable from ρ i (with ρ i = tr(ρi) N I ) with dynamics described by equation (3.2), for A being a Hermitian operator, is

(3.10)

Situation II: arbitrary operator A.

Let us now consider the connected Lie group G 2 of one-to-one linear transformations of Γ N that contains the identity operator and preserves the trace:

By construction of Γ N we only have a canonical linear space structure, thus Γ N can be identified with R N 2 ; and thus G 2 is isomorphic with

We know that G 2 is connected but not compact. We will denote by Lie(G 2 ) the Lie algebra of

Since tr(ρ) is a linear operation and if ρ is viewed as a vector in R N 2 then tr(ρ) =< α, ρ > with α a vector with "1" on the positions corresponding to ρ ii , i = 1, ..., N 2 and zero elsewhere. In this case the Lie algebra is thus isomorphic with the set (endowed with its canonical Lie algebra structure):

We have thus

Denote by H G2 0 the element of Lie(G 2 ) that is constructed from H 0 (and the same for H 1 and T ). We associate to the evolution equation (3.2) the following evolution equation on the group G 2 :

(3.15)

We will also write X(t; , γ; X 0 ) when we will need to make explicit the dependence on the parameters; of course X(t; , γ; X 0 ) belongs to the Lie group G 2 . Then by definition ρ(t) = X(t; , γ; e)ρ(0).

as a vector space over the real numbers) then the system (3.2) is density matrix controllable.

Proof. Without loss of generality we can suppose that tr(H 0 ) = 0 and tr(H 1 ) = 0. We did not yet prove that G 2 is indeed a Lie group: this results from Cartan's theorem that states that any closed subgroup of a Lie group is a Lie subgroup (we include G 2 in the group GL + (Γ N ) of 8 one-to-one transformations on Γ N of positive determinant). We will use the same line of proof as in Theorem 3.2 and invoke Theorem 2.1 (this time without any change of variables) for G = G 2 .

Then L = Lie iH G2 0 , iH G2 1 , T G2 is the Lie algebra generated by H G2 0 , H G2 1 and T G2 . Recall that L ⊂ Lie(G 2 ) and dim R (Lie(G 2 )) = (N 2 -1)N 2 (dimension as vector space over R). By hypothesis dim R (L) = (N 2 -1)N 2 thus L = Lie(G 2 ) and S = G 2 . Take now = 0,γ = 0, then equation (3.15) becomes:

with solution X(t) = e tH G 2 0 ; take a sequence of positive numbers {t n } with t n δ > 0 for some δ (strictly positive, arbitrary but fixed); we need to prove that lim n→∞ e tnH G 2 0 ∈ S.

(3.17)

But H G2 0 is a skew-Hermitian map which means that e tnH G 2 0 belongs to the group SO(Γ N ) of special orthogonal transformations of Γ N which is a subgroup of G 2 . Since SO(Γ N ) is compact it implies that up to extracting a subsequence lim n→∞ e tnH G 2 0 exists and is orthogonal. All orthogonal transformations e tnH G 2 0 preserve the trace thus lim n→∞ e tnH G 2 0 exists, is orthogonal and trace preserving and hence an element of S. Therefore R(e) = S. In particular X can reach any trace preserving one-to-one transformation from Γ N to itself. For any ρ i and ρ f with tr(ρ i ) = tr(ρ i ) we can find a trace preserving one-to-one transformation to map ρ i ∈ Γ N to ρ f ∈ Γ N i.e. we have controllability for ρ.

Remark 3.3. From the proof of Theorem 3.3 we obtain that the set of all density matrices reachable from ρ i with dynamics described by equation (3.2), for A arbitrary operator, is the set

(3.18) First note that the system defined by H 0 and H 1 alone (i.e., with A = 0) is not controllable as the dimension of the Lie algebra generated by -iH 0 and -iH 1 is 4, short of 3 2 -1 = 8 needed for controllability. Also note that for the system defined by (4.1) A is a symmetric matrix. We want to verify if systems (4.1) and (4.2) are controllable i.e., verify the hypotheses of Theorem 3.2 and Theorem 3.3.

To do so we choose a parameterization such that we can write (3.2) as a linear system

Numerically H0 , H1 , T are N 2 × N 2 dimensional matrices and ρ is a N 2 × 1 vector. For the Hamiltonian part of the dynamics this is known as the Liouville equation in the adjoint representation.

In order to analyze the controllability we need to numerically compute the dimension of the Lie algebra (as subalgebra of N 2 × N 2 matrices) generated by {i H0 , i H1 , T }, which we denote by Lie{i H0 , i H1 , T } and verify for A a Hermitian N -dimensional matrix if dim R (Lie{i H0 , i H1 , T }) = (N 2 -1) 2 or when A is an arbitrary if dim R (Lie{i H0 , i H1 , T }) = (N 2 -1)N 2 . For the system in (4.1) numerical computations give the result dim R (Lie{i H0 , i H1 , T }) = 64, (4.4) and for the system (4.2)

dim R (Lie{i H0 , i H1 , T }) = 72. (

We conclude that both systems are controllable.


Olivier Praud

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Adam M Fincham

Decaying grid turbulence in a rotating stratified fluid

Rotating grid turbulence experiments have been carried out in a stably stratified fluid for relatively large Reynolds numbers (mesh Reynolds numbers up to 18 000). Under the combined effects of rotation and stratification the flow degenerates into quasihorizontal motions. This regime is investigated using a scanning imaging velocimetry technique which provides time-resolved velocity fields in a volume. The most obvious effect of rotation is the inhibition of the kinetic energy decay, in agreement with the quasi-geostrophic model which predicts the absence of a direct energy cascade, as found in two-dimensional turbulence. In the regime of small Froude and Rossby numbers, the dynamics is found to be non-dissipative and associated with a symmetric and highly intermittent vertical vorticity field, that displays k -3 h energy spectra. For higher Rossby numbers, fundamental differences with the quasi-geostrophic model are found. A significant decay of kinetic energy, which does not depend on the stratification, is observed. Moreover, in this regime, although both cyclones and anticyclones are initially produced, the intense vortices are only cyclones. For late times the flow consists of an assembly of coherent interacting structures. Under the influence of both rotation and stratification, they take the form of lens-like eddies with aspect ratio proportional to f/N.

Introduction

Turbulence in the atmosphere and oceans is generally influenced by planetary rotation and density stratification. Both effects tend to produce highly anisotropic flow structures, with quasi-horizontal velocity fields. However, while rotation alone produces columnar eddies aligned with the axis of rotation, stratification produces layered eddies of short vertical extent. The turbulence dynamics under these two antagonistic effects is still not fully understood.

The present paper reports an experimental study of freely decaying grid turbulence in the presence of both stratification and rotation. The experiments were performed on the 'Coriolis' turntable in Grenoble, whose large scale (13 m in diameter) allows the production of turbulence dominated by stratification and rotation effects. Stratification is characterized by the Brunt-Väisäläf r e q u e n c yN and rotation by the Coriolis parameter f . The ratio f/N is varied in the range 0 <f/N<1.2, to study the competition between these two effects. The case with pure stratification (f = 0) has been reported earlier [START_REF] Praud | Decaying grid turbulence in a strongly stratified fluid[END_REF], and here we focus on the influence of rotation. In the non-rotating stratified regime, the flow splits into multiple layers with pancake eddies. [START_REF] Riley | Direct numerical simulation of homogeneous turbulence in density-stratified fluids[END_REF] and [START_REF] Lilly | Stratified turbulence and the meso-scale variability of the atmosphere[END_REF] have suggested that, in the limit of small Froude number (strong stratification), stably stratified turbulence could obey two-dimensional dynamics. However, numerical (see [START_REF] Herring | Numerical simulation in forced stably stratified turbulence[END_REF][START_REF] Smith | Generation of slow large scales in forced rotating stratified turbulence[END_REF] and experimental (see [START_REF] Fincham | Energy dissipation and vortex structure in freely decaying stratified grid turbulence[END_REF]Praud et al. 2005) studies have shown that the shear between layers increases with stratification, such that the global energy dissipation remains close to that of ordinary three-dimensional turbulence. This is in stark contrast with the dynamics of two-dimensional turbulence, where in the limit of large Reynolds numbers there is no energy dissipation.

When turbulence is subjected to rotation alone, energy transfer toward the small scales is severely reduced, so that inertial waves and two-dimensional eddies thus coexist for a long time. The tendency towards two-dimensionalization has been extensively studied in homogeneous isotropic turbulence using theoretical and numerical simulations, both DNS (direct numerical simulations) and LES (large-eddy simulations) (see [START_REF] Bardina | Effect of rotation on isotropic turbulence: computation and modelling[END_REF][START_REF] Bartello | Coherent structures in rotating three-dimensional turbulence[END_REF][START_REF] Cambon | Energy transfer in rotation turbulence[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scale in forced, rotating three-dimensional turbulence[END_REF][START_REF] Cambon | Advanced in wave turbulence: rapidely rotating flows[END_REF]). It has been found that nonlinear interactions between inertial waves, which initiate the transition from three-dimensional to two dimensional turbulence do not lead to a truly two-dimensional dynamics [START_REF] Cambon | Energy transfer in rotation turbulence[END_REF]. In particular, it does not exhibit the classical inverse energy cascade [START_REF] Smith | Transfer of energy to two-dimensional large scale in forced, rotating three-dimensional turbulence[END_REF]. When boundary effects constrain inertial waves, the tendency toward two dimensionalization is favoured. This is the case for a fluid confined between two plates perpendicular to the axis of rotation. Then the only source of energy decay is the Ekman friction at the bounding walls. The effect of vertical confinement was numerically investigated by [START_REF] Godeferd | Direct numerical simulatins of turbulence with confinement and rotation[END_REF].

In addition, rotating turbulence is associated with the emergence of persistent columnar vortices aligned with the rotation axis. Nevertheless, they should not be considered as exact two dimensional vortices. The creation of such organized structures has been observed in various experiments, first by de Verdiere (1980), and in numerical simulation of forced rotating turbulence [START_REF] Smith | Transfer of energy to two-dimensional large scale in forced, rotating three-dimensional turbulence[END_REF]). In the case of unforced initially three-dimensional isotropic rotating turbulence, the spontaneous generation of such vortices is enforced by the presence of solid boundaries which vertically confine the flow [START_REF] Godeferd | Direct numerical simulatins of turbulence with confinement and rotation[END_REF]. The Ekman pumping, induced by the presence of horizontal walls, appears to be essential in generating coherent vortices as these features are not obtained in the unforced homogeneous case.

For large rotation rates (small Rossby numbers Ro), intense vortices of both signs emerge in a symmetric way [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF]. At moderate rotation rates (Ro ∼ O(1)), by contrast, axial coherence is observed preferentially for cyclonic vortices, for which local rotation adds to the background rotation, while the anticyclones tend to remain three dimensional or instable [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF][START_REF] Godeferd | Direct numerical simulatins of turbulence with confinement and rotation[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scale in forced, rotating three-dimensional turbulence[END_REF]. For instance, the experiments of [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF] have clearly shown the presence of intense cyclonic vortices in a background of weaker anticyclonic vortices.

The case of rapid rotation combined with a strong stratification is usually called geostrophic turbulence. It is the typical regime for meso-scale atmospheric and oceanic motion. An analytical theory has been proposed by [START_REF] Charney | Geostrophic turbulence[END_REF], using the quasigeostrophic model, valid in the limit of small Froude and Rossby numbers. Charney noted a formal analogy with two dimensional turbulence, and on the basis of inviscid conservation laws, he proposed the occurrence of an inverse cascade of total energy and a direct cascade of potential enstrophy (squared potential vorticity). He assumed three-dimensional isotropy of the energy and potential enstrophy spectra, once the vertical coordinate has been rescaled by N/f, according to the invariance of the quasi-geostrophic equations under vertical stretching with a corresponding change in N/f. [START_REF] Herring | Statistical theory of quasi-geostrophic turbulence[END_REF] largely confirmed Charney's conjectures in the framework of a statistical closure model. The dynamics is then completely dominated by the potential vorticity modes and exhibits the inverse cascade of energy and the forward cascade of potential enstrophy.

This quasi-geostrophic approximation, however, discards some physical effects, such as the interaction of turbulence with internal inertio-gravity waves. The various resonant interactions and energy transfer have been investigated by [START_REF] Bartello | Geostrophic adjustment and inverse cascades in rotating stratified turbulence[END_REF] and [START_REF] Smith | Generation of slow large scales in forced rotating stratified turbulence[END_REF]. The results of [START_REF] Smith | Generation of slow large scales in forced rotating stratified turbulence[END_REF] suggest that the quasi-geostrophic dynamics is valid for small Froude number (Fr ≪ 1) in the domain 1/2 6 f/N 6 2. A common result of the various studies is the general tendency of energy transfer to slow modes when the Rossby or Froude numbers are small. Therefore, under the joint effects of rotation and stratification, energy is protected from dissipation, as the direct energy cascade to small scales is severely reduced. Furthermore Ekman friction is limited by stratification to regions near the boundaries, with weak influence in the interior.

Ageostrophic effects such as wave-vortex interactions could however play a crucial role in rotating stratified turbulence [START_REF] Babin | On nonlinear baroclinics waves and adjustment of pancake dynamics[END_REF] suggested that these wave-vortex interactions contribute to the nonlinear adjustment to the geostrophic state. Asymmetry between cyclones and anticyclones is also due to effects beyond the quasi-geostrophic approximation (see [START_REF] Hakim | A new surface model for cyclone-antcyclone asymetry[END_REF]). These effects involve interactions of different space and time scales and may be difficult to capture in numerical computations. A recent review on the effect of stratification and rotation on turbulence flows can be found in [START_REF] Cambon | Turbulence and vortex structures in rotating and stratified flows[END_REF].

Because of practical constraints, few experimental studies of turbulence in the presence of stratification and rotation have been performed. Linden, Boubnov & Dalziel (1985) described the response of a rotating stratified fluid to forcing produced by horizontal sources and sinks. In these experiments, the flow appears to evolve with a balance between an inverse cascade produced by the two-dimensionality of the flow and a cascade due to baroclinic instability. The authors also noticed the development of an asymmetry between cyclones and anticyclones: for f/N >2 a predominance of cyclonic vorticity is observed.

The formation of coherent structures is a generic feature of the collapse of turbulence regions in rotating stratified fluids. They are lenticular with characteristic aspect ratios of order f/N, and the vertical and horizontal scales adjust, such that the effects of stratification and rotation balance. Baroclinic and internal barotropic instabilities constrain the vortex shape. This natural tendency to form coherent structures has been observed and extensively discussed by [START_REF] Mcwilliams | The emergence of isolated vortices in turbulent flow[END_REF][START_REF] Mcwilliams | Statistical properties of decaying geostrophic turbulence[END_REF][START_REF] Mcwilliams | The vortices of geostrophic turbulence[END_REF] and McWilliams, Weiss & Yavneh (1999). Numerical simulations have shown that the presence of such intermittent vortices could modify classical statistical predictions such as the enstrophy cascade process, leading to steeper energy spectra. For late times, the dynamics of the flow is controlled by the vortex interactions.

The present paper focuses on the combined influence of rotation and stratification on the dynamics of turbulence, including kinetic energy decay, three-dimensional structure of the flow and the emergence of coherent vortices. For small Rossby numbers, the results are found to be consistent with the quasi-geostrophic model, in particular the very weak energy decay, the emergence of coherent vortices which are geostrophically balanced and a strongly intermittent vertical vorticity field, statistically symmetric with respect to cyclones and anticyclones. For moderate rotation rates, departure from this model is observed: the flow is dominated by cyclonic vortices which appear stronger and more barotropic, whereas the anticyclones remain fragmented.

The paper is organized as follows. In § 2 the experimental apparatus and the measurement technique are described; the inhibition of the decay of kinetic energy is discussed in § 3. A general description of the flow, including the length-scale evolution and a discussion of the antagonistic effects of rotation and stratification is given in § 4. The emergence of coherent vortices and the intermittent and symmetric properties of the vorticity field are commented on in § 5. Finally the conclusion is presented in § 6.

Experimental set-up and procedure

2.1. Experimental apparatus The experiments were performed in a closed rectangular channel (8 m × 4m) constructed on the 'Coriolis' rotating platform (a circular tank with diameter 13 m) see figure 1. The axis of rotation is vertical to within 3 × 10 -6 rad, and the angular velocity is constant within a precision of Ω/Ω < 10 -4 . The average water depth is 0.9 m (the variations due to the parabolic free surface are less than 2 cm). The whole tank is filled with water to avoid leaks from the channel. The tank is linearly stratified with common salt by diluting an ultrasonically maintained constant-head filling flow with a prescribed flux of brine provided by a computer-controlled volumetric pump. The water temperature is stabilized and controlled to better than 1 • C, such that density variations associated with temperature fluctuations can be neglected. The density stratification is measured before and after each experiment by a profiling conductivity probe that moves at 2 cm s -1 .

The turbulence is generated by towing a rake of vertical flat plates of width W = 10 cm and mesh spacing M = 45 cm, at a constant speed U along the full length of the channel. The rake is hung from a carriage that moves above the free surface and its displacement is driven through a computer-controlled DC motor, ensuring good repeatability of the experiments. The velocity of the rake increases linearly from 0t oU and, at the end of the motion, decreases linearly back to zero. The acceleration and deceleration are chosen to minimize the generation of internal gravity waves by the accumulation of the fluid between the rake and the endwalls of the channel. A set of vertical plates is used instead of a standard grid with the objective of minimizing internal wave generation and avoiding selection of a particular vertical length scale. The plates were chosen instead of rods, as they provide a well-defined separation point that is generally independent of the Reynolds number. When generating turbulence by means of this rake we introduce a maximum of kinetic energy into the horizontal modes which minimizes dissipation of energy due to the initial three-dimensional turbulence state. This helps to maintain the energy entering the rotating stratified turbulence state, which is then attained at higher Reynolds numbers.

A high-resolution scanning digital particle image velocimetry system was used in these experiments, as described by [START_REF] Fincham | Low cost, high resolution DPIV for measurement of turbulent fluid flow[END_REF] and [START_REF] Fincham | 3D measurement of vortex structures in stratified fluid flows[END_REF].

This system provides the horizontal velocity projection in a volume (2.48 m × 2.48 m × 0.5 m), with relative precision about 2 %. The spatial derivatives along x, y and z are obtained by differentiation of the velocity data. Turbulence statistics are obtained within this volume as a function of time, whose origin is chosen when the rake is located at the centre of the measurement area. The measurement volume is centred in the channel to avoid boundary effects. Figure 1 shows the experimental facility and the orientation of the measurement area. More details about the experimental techniques are provided in Praud et al. (2005) and Praud & Fincham (2005).

Scales and parameters

We denote by l h a horizontal length scale and by V a typical horizontal velocity. Since the motion quickly becomes quasi-horizontal, we choose l h /V for the advective time scale. The buoyancy time scale based on the mean stratification is 1/N where N = √ -(g/ρ 0 )(∂ρ/∂z) is the Brunt-Väisälä frequency. The ratio of these two time scales defines a horizontal Froude number which compares buoyancy effects to inertia and therefore measures the relative importance of stratification:

Fr h = V/l h N.
(2.1)

The relative importance of rotation is similarly characterized by the Rossby number:

Ro = V/l h f (2.2)
where f =2Ω is the Coriolis parameter (Ω is the background angular rotation rate); the vertical axis is chosen to be aligned with the axis of rotation and the mean density gradient, ∂ρ/∂z. The ratio f/N characterizes the relative importance of rotation and stratification.

From the main control parameters, which are the mesh size M, the velocity of the rake U , the Coriolis parameter f and the Brunt-Väisäläf r e q u e n c yN, we define the initial Froude number, Fr M , Rossby number, Ro M and Reynolds number Re M : where ν is the mean kinematic viscosity, i.e. the viscosity of the salt solution at mid-depth of the tank (viscosity varies by 5.6 % with depth in the most stratified case, with δρ/ρ =2.5 %).

Fr M = U MN ,R o M = U Mf , Re M = UM ν , (2.3) Exp U (cm s -1 ) Re M N (s -1 ) f (s -1 ) Fr M Ro M f/N ba 0.
= Re M u * L * h , Ro = Ro M u * /L * h and Fr = Fr M u * /L *
h ,whereu * and L * h are respectively the dimensionless turbulent velocity, given by the square root of the measured kinetic energy (see figure 4 below and the dimensionless horizontal integral scale given by (4.1) below. At t * = tU/M = 10, which correspond roughly to the beginning of the well-established turbulent decay, u ∼ 0.15U and L h ∼ 1.6M; thus, in comparison with definitions (2.3) the Reynolds number should typically be divided by 5, and the Froude and Rossby numbers by 10. As the velocity decreases and the length increases, the Reynolds, Froude and Rossby numbers decrease with time.

f/N = 1.2 f = 0 f/N = 0.15 f/N = 0.3 f/N = 0.6 Ro M -1 (rotation) Fr M -1 (stratification) f = N
Fi gure 2. Diagram of the different experiments as a function of their initial non-dimensional parameters, based on the grid mesh M and velocity U (the corresponding non-dimensional parameters based on the turbulent velocity and integral scale are about 10 times lower).

Since the Froude number is initially small for all the experiments, the flow constantly evolves in conditions of strong stratification (Fr h ≪ 1). In the absence of rotation, the vertical Froude number, based on a vertical length scale, can be initially of order 1 (even if Fr h is very small) as already observed by Praud et al. (2005), [START_REF] Godeferd | Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. part 2. large-scale and small-scale anisotropy[END_REF] and [START_REF] Billant | Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid[END_REF]. However, due to the decay of turbulence, the vertical Froude number rapidly decreases to values much smaller than unity. When rotation is present the ratio of the horizontal to the vertical Froude numbers, given by the ratio of the vertical to the horizontal length scales, is thus proportional to f/N. During this decay, the ratio f/N does not change, that is, the evolution of each experiment takes place along a line of constant f/N (see figure 2) with the values of Fr -1 and Ro -1 increasing in time.

Kinetic energy and energy spectra

3.1. Quasi-horizontal motion Immediately after the passage of the rake, different initial behaviours are observed. For low towing speeds, the initial wake is laminar with quasi-horizontal motions; while for higher towing speeds, three-dimensional turbulence is observed directly behind the rake. This turbulence soon collapses into quasi-horizontal motion. In this regime, individual coherent vortices emerge and their subsequent evolution can be accurately tracked. Such observations can be quantified by measuring the energetic contribution of the vertical velocity, see figure 3. The vertical velocity is evaluated through the continuity equation, that gives, in the Fourier space, for k z =0,

û3 (k)=- k h • ûh (k) k z . (3.1)
For k z = 0, we choose û3 (k x ,k z , 0) = 0, assuming the absence of any mean vertical motion. The comparison of the vertical kinetic energy E v to the horizontal kinetic energy clearly shows that the contribution of E v to the total kinetic energy is negligible for the duration of the experiment (less than 5 %) and decreases with time (less than 2 % at the end of the experiment). The energy of the horizontal motion can be therefore considered as the total kinetic energy of the flow within a precision better than 5 %. At the final stage of decay in figure 3, the residual vertical kinetic energy is such that E h /E v ∼ 70.

Inhibition of the energy decay by rotation

The decay of the turbulent kinetic energy E(t)= u 2

x + u 2 y /2, averaged over the measurement volume, is examined in figure 4 for several experiments. The energy and time are normalized by U 2 and M/U respectively. All these experiments correspond to the same initial Reynolds number, equal to 9000, while the Coriolis parameter f and the stratification change. The most obvious effect of rotation is the inhibition of the energy decay: the exponent of the decay power law changes from ∼-1.3i n the non-rotating case to ∼-0.5f o rRo M =0.56, and for Ro M 6 0.2 there is almost no decay of kinetic energy. This change in energy decay is remarkably not influenced by stratification. For each rotation rate, two different stratifications are compared in figure 4, and no significant dependence on the Froude number is observed. The decay of kinetic energy depends only on the Rossby number. We show later in this paper that the absence of energy decay observed for Ro M 6 0.2 is associated with a k -3 h energy spectrum, the emergence of intense concentrated coherent vortices and a symmetric probability distribution of vertical vorticity. Therefor, the non-dissipative dynamics observed for Ro M 6 0.2, together with the other observations, suggest that, when the rotation rate increases, the turbulence approaches the quasi-geostrophic dynamics, in which a direct energy cascade is absent (like in twodimensional turbulence).

Notice that fluctuations of the kinetic energy are observed at late times. These occur when large vortices enter or leave the measurement volume. Such fluctuations become more important at late times as the energy is controlled by only a few vortices.

Horizontal energy spectra

The two-dimensional horizontal energy spectrum is calculated as a function of the horizontal wavenumber k h =(k 2

x + k 2 y ) 1/2 . A two-dimensional fast Fourier transform of the horizontal velocity field is performed in each horizontal plane of the measurement volume. An azimuthal average at radius k h is taken to obtain the two-dimensional energy spectrum in each plane E(k h ,z). Finally all these horizontal spectra are averaged over the vertical coordinate z to yield the two-dimensional horizontal energy spectrum E(k h )= E(k h ,z) z .

These spectra are represented in figure 5 for cases with Ro M 6 0.2. The spectral shape is remarkably constant in time, so the normalized spectra at successive times collapse. Furthermore a k -3 h spectral law is observed. This behaviour, in association with the very small energy dissipation and the cyclones/anticyclones symmetry is consistent with the enstrophy cascade predicted by the quasi-geostrophic theory of Charney. This has also been observed in the numerical simulations of decaying turbulence by Metais et al. (1996). Steeper spectra are generally observed in numerical computations of two-dimensional turbulence [START_REF] Mcwilliams | Statistical properties of decaying geostrophic turbulence[END_REF]) and of geostrophic turbulence (McWilliams et al. 1999) in association with the emergence of coherent vortices. Although the emergence of coherent vortices is also observed in our experiments (see § 5) we did not observe horizontal spectra steeper than k -3 h . For larger Rossby numbers, for which significant ageostrophic effects are expected, the turbulence spectra do not reach a well-established shape. As illustrated in figure 6, the energy remains confined to the large scales, and the slope at high wavenumbers decreases with time, remaining less steep than -3. The vortices interact by mutual advection and undergo merging phenomena; this leads to an increase of their horizontal size. These interactions include multipole formation (dipoles) by pairing of oppositely signed vortices and mergers of like-signed vortices. Properties of the interactions observed in numerical simulations of geostrophic turbulence (see [START_REF] Mcwilliams | The emergence of isolated vortices in turbulent flow[END_REF][START_REF] Mcwilliams | Statistical properties of decaying geostrophic turbulence[END_REF] or rotating turbulence experiments (see [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF]) are discernible in these experiments. In particular, merging of vortices generally occurs when one of the two vortices is weaker. Vortices with weaker amplitudes are more likely to be destroyed by strain, or absorbed by merging. When two close together vortices have approximately the same strength, the interaction eventually leads to the merging of the two and a symmetrization of the resulting vortex. More generally, the interactions produce shearing deformation. Strong distortion occurs, including the production of elongated horizontal filaments of vertical vorticity which leads to a reduction in the vertical enstrophy.

E(k h )/U 2 k h M Ro M = 0.14, Fr M = 0.17
As time goes on, the number of intense vortices decreases, they become sparser and the encounters occur less frequently but their horizontal size keeps growing. The final stage of evolution is well represented by figure 7(d). The flow consists of a few robust vortices, surrounded by weak vertical vorticity characterized by elongated filaments at small scales. The turbulent field becomes progressively more inhomogeneous and vortices with increasing size are separated by large unstructured areas. For very late times, the size of the structures is limited by the width of the channel, and the horizontal growth is stopped.

For moderate rotation rates (Ro M > 0.2), cyclonic vortices are found to be more abundant and stronger than anticyclonic vortices. Moreover, cyclonic vortices exhibit greater axisymmetry and are more robust in the presence of ambient strain. While a horizontal symmetrization process is observed after merging for the cyclonic vortices, the anticyclonic vortices remain elongated in the horizontal plane. This tendency toward cyclonic predominance will be discussed in § 5. By contrast, cyclones and anticyclones have the same properties in the quasi-geostrophic model, as a mathematical symmetry. This symmetry is for instance observed in the numerical simulations (see [START_REF] Mcwilliams | Statistical properties of decaying geostrophic turbulence[END_REF].

In the absence of rotation, cyclones and anticyclones are obviously found in equal numbers and strengths. They are more diffusive and less sparse than when rotation is present. Expulsion of vorticity filaments also seems to be reduced.

Antagonistic effect of rotation and stratification and three-dimensional

structure of the flow The competition between stratification and rotation is illustrated in figures 8 and 9: horizontal layering is obtained when stratification dominates, while columnar structures are obtained when rotation dominates. These figures show successive threedimensional views of the vertical vorticity fields represented by iso-surfaces, for two different rotation rates. In both cases, the initial wake of the rake is made of coherent columnar vortices, extending through the whole fluid depth of the layer.

In the absence of rotation, an instability mechanism for stratified columnar vortices, called the zig-zag instability, was proposed by Billant et al. (2000) and Billant & Chomaz (2000) to explain the layering. The vertical scale selected by this mechanism is proportional to U/N. However, Praud et al. (2005) showed that the vertical scale in the turbulent regime is not influenced by this initial instability mechanism and exhibits no dependence on the stratification. Moreover, as we will show in § 4.3, under the joint effects of rotation and stratification, the aspect ratio of these 'pancake' vortices is prescribed by the ratio f/N.

For moderate rotation rates, as in the case presented in figure 8 (Ro M =0.28, f/N =0.15), the stabilizing effect of rotation is not sufficiently strong to resist the straining deformation induced by the other vortices and vertical fragmentation occurs. The shed vortices rapidly become unstable and lose their vertical coherence under the action of the mutual advection induced by the neighbouring vorticity distribution, producing multiple layers of eddies. The flow then consists of patches of vertical vorticity localized in space, which look similar to the so-called 'pancake' vortices observed by [START_REF] Fincham | Energy dissipation and vortex structure in freely decaying stratified grid turbulence[END_REF] or Praud et al. (2005) in purely stratified turbulence experiments.

The destabilization of the columnar vortices that occurs for low values of f/N,i s consistent with the results of Dritschel & de la Torre Juarez (1996) and Dritschel, de la Torre Juarez & Ambaum (1999). Numerically investigating the stability of columnar vortices in quasi geostrophic flows, they observed that two-dimensional vortices of diameter L smaller than the Rossby deformation radius, L R = HN/f (H is the vertical height of the column), are unstable to three-dimensional disturbances and (1999) for L<L R /2. In the present experiments, the initial height is the depth of the fluid (H = 90 cm). We observe a rapid fragmentation of the columnar vortices produced in the wake of the rake for f/N 6 0.6. This transition corresponds to L R =1.45. The vortex size, of the order of the grid mesh 45 cm, is close to L R /3, in rough agreement with the stability results.

Fragmentation is generally followed by new attachments of like-sign vortices whose cores are separated vertically. Thus, as the processes of attachment and fragmentation continuously act, the vertical extent is highly variable in both space and time. The dominance of the cyclonic vortices is clearly observed in this process: they tend to be vertically coherent, whereas the anticyclones remain fragmented with a smaller vertical scale.

By contrast, for high rotation rates (Ro M =0.07, f/N =1.2), the wake remains quasi-two-dimensional (figure 9), with vortices strongly correlated along the vertical. These columnar vortices are very robust, as they do not lose their vertical coherence during strong interactions and close encounters. This development of vortical columns is more pronounced as f/N is increased. A progressive transition from the 'pancake' eddies of purely stratified turbulence to such regimes with vertical columns is observed as the rotation rate is increased.

Length-scale evolution

Horizontal length scales

The increase of the horizontal size of the vortices observed in figures 8 and 9 can be characterized in terms of the horizontal integral scale L h . This scale is obtained as the integral of the longitudinal autocorrelation function of the horizontal velocity, taken up to its first zero crossing. The growth of L h for various experiments is represented in figure 10. A good collapse of all the curves is observed over a large range of Rossby and Froude numbers: the growth of L h is remarkably independent of rotation and

stratification. It approximately follows a power law

L h /M ≃ 0.5t * 0.5 , (4.1)
where t * is the non-dimensional horizontal advective time: t * = tU/M. The value of the exponent is close to what is obtained in decaying three-dimensional homogeneous turbulence, as well as in purely two-dimensional turbulence (see [START_REF] Bracco | Revisiting freely, decaying two-dimensional turbulence at millenium resolution[END_REF]. Such similar behaviour of L h for very different dynamical regimes is remarkable.

For late times, the growth becomes slower and the plot becomes scattered. Indeed, the number of vortices present in the measurement area decreases with time and the correlation function fluctuates strongly depending on the relative position of these vortices: when L h ∼ 100 cm, there are less than four vortices in the field of view. Moreover the growth of L h starts to be limited by the width of the channel.

Vertical length scales

Since the horizontal velocity component dominates the flow, we estimate the vertical integral scale from the transverse autocorrelation function of the horizontal velocity. However, for a number of experiments, this function remains positive for all accessible vertical separations z, implying that the vertical integral scale is greater than the vertical extent of the measurement volume. The vertical scale is thus examined directly from the correlation function, as shown in figure 11. The competing effects of stratification and rotation on the vertical correlation is clearly visible, confirming the qualitative discussion of § 4.2: the vertical correlation clearly increases with the ratio f/N. Note however, that even when rotation is dominant over stratification (case f/N =1.2), the correlation function still significantly decreases with z,t h a ti s , the flow is not purely two-dimensional and the columns are tilted as seen in figure 9.

The vertical scale depends only on the ratio f/N. This is suggested by comparing the two curves with f/N =0.3. This is also observed in the purely stratified case (f/N = 0) where stratification does not influence the integral vertical scale (Praud et al. 2005). In the previous subsection, it was also shown that the horizontal scale does not depend on stratification, so the aspect ratio between vertical and horizontal integral scales depends only on f/N. This result, consistent with the quasi-geostrophic model, seems to apply in all the regimes of stratified rotating turbulence.

When the vertical integral scale can be measured (for f/N 6 0.3), it often fluctuates, as can be seen in figure 12(a). The attachment of like-sign vortices at different altitudes results in a sudden increase of the integral vertical scale. By contrast, vortex fragmentation by the strain field induced by neighbouring vortices results in a sudden loss of vertical coherence and a decrease of the vertical correlation length. Nevertheless, in all the experiments a global increase of the integral vertical scale with time is observed. This tendency is partially due to viscous diffusion but mainly to the fact that attachment events are more common than fragmentation events, as already noticed in numerical simulations (see McWilliams 1989). In the horizontal directions, the merger events and the destruction of the weaker vortices always lead to an increase of L h , while the vertical integral scale can either grow or shrink. However, as can be seen in figure 12(b), for f = 0, the ratio L v /L h oscillates around a constant value, which is proportional to f/N. The results show that for the joint conditions of rotation and stratification, these eddies take the form of lens-like vortices with aspect ratio L v /L h ∼ 2f/N. Such a proportionality characterizes the geostrophic regime.

L v /L h Time × U/M Time × U/M f/N = 0.3 (hb) f/N = 0.15 (fb)
The proportionality of the aspect ratio to f/N and its time independence agree well with the results of McWilliams et al. (1999). From numerical simulation of decaying geostrophic turbulence, he found that h/r ∼0.83f/N and is globally conserved with time (he defined h and r respectively as the half-height and the radius of a vortex element). These data lead to an average height-to-width aspect ratio of vortices of 1.7f/N which is comparable to the 2f/N found in the present study.

In the purely stratified regime, the flow degenerates into an assembly of quasi-twodimensional flows in layers. An important role is then played by viscosity (see Praud et al. 2005) which limits the vertical shear and the vertical scale grows according to a diffusive process, L v ≃ 2(νt) 0.5 . A definite change in the nature of the dynamics occurs under the influence of rotation: the vertical scale is no longer controlled by viscosity; this is discussed in the next section.

Vanishing effect of the vertical gradients

A direct consequence of the increase of the vertical scale with f/N is the reduction of the vertical gradients and therefore the reduction of their contribution to the total enstrophy and to the viscous energy dissipation. This effect is illustrated well in figure 13, which shows the evolution of the contributions of ω 2 z and ω 2 h for different values of f/N, at a given initial Reynolds number (Re M = 9000). The scatter observed in the estimation of ω 2 h for the case f/N =1.2 is due to measurement noise (as the vertical gradient is weak in this case and more difficult to distinguish from noise). Nevertheless, figure 13 clearly shows that the relative contribution of the vertical gradients decreases with increasing f/N.F o rf/N = 0 the enstrophy is entirely dominated by the horizontal vorticity (then equivalent to the vertical shear) which confirms the layered structure of the flow and is consistent with numerical simulations (see [START_REF] Herring | Numerical simulation in forced stably stratified turbulence[END_REF][START_REF] Smith | Generation of slow large scales in forced rotating stratified turbulence[END_REF] and experiments on stratified turbulence (see [START_REF] Fincham | Energy dissipation and vortex structure in freely decaying stratified grid turbulence[END_REF]Praud et al. 2005). The organization of the flow into thin 'pancake' eddies localized in space and bounded by horizontal vortex sheets leads to strong vertical shear. With increasing f/N, the contribution of the horizontal vorticity progressively decreases and becomes of the same order as the vertical vorticity when f/N =1.2.

The ratio α =( ω 2 z / ω 2 h ) 1/2 can also be considered as the aspect ratio of the eddies based on the vertical to the horizontal Taylor microscales, defined as

λ h = 0.5 u 2 x + u 2 y w 2 z 1/2 , λ v = u 2 x + u 2 y ω 2 x + ω 2 y 1/2 . (4.2)
As for the aspect ratio based on the integral scale, α is approximately conserved with time in the presence of rotation: the decay of ω 2 h and ω 2 z follows approximately the same law and their relative contribution to the total enstrophy remains the same for the duration of the experiment.

The time-averaged value of α is plotted versus f/N in figure 14. This ratio is a monotonically increasing function of f/N, in agreement with other scale estimates. Note that α is not quite a linear function of f/N; this indicates some departure from the quasi-geostrophic model.

Remarkably, for f =0 (N fixed), α does not depend significantly on the Reynolds number. This is consistent with the fact that energy dissipation (mainly the product of the shear squared and viscosity) tends to zero in the limit of small viscosity. This contrasts with the purely stratified case (f = 0, also represented in figure 14) for which α definitely decreases with increasing Re:hereα ∼ Re -0.5 as described by Praud et al. (2005) so that the energy dissipation (∼Re α 2 ) is independent of viscosity. The transition between these two regimes cannot be seen in figure 14. Further experiments at low rotation rates would be needed to analyse this transition.

Intermittency and asymmetry of the vorticity distribution

We have discussed the natural tendency of rotating stratified turbulence to form coherent vortices that can be identified and tracked for periods of time that are long compared with their typical turnover times. Figure 7 provides an example. When rotation is present, most observed vortices are monopoles but dipoles or tripoles can be temporarily formed. As time goes on, the number of structures decreases, and those remaining tend to dominate over the background vorticity.

An objective statistical measure of vortex dominance and therefore of the intermittent character of the flow is the flatness,

Ku = ω 4 ω 2 2
(5.1) which can be interpreted as the inverse of the area fraction occupied by the vortices. Its evolution is represented in figure 15 for f =0 and for Ro M =0. 07. In all the experiments, an initial value of Ku close to 7 is obtained. This relatively large value is due to vortex shedding behind the rake creating compact coherent vortices. In the absence of rotation, Ku decreases to reach a value close to 3, the characteristic value for a Gaussian field. When rotation is present, Ku increases from a starting value of 7, and can reach values greater than 20. The large values of the flatness characterize a highly intermittent field, in which the vorticity is concentrated and occupies a relatively small volume of the turbulent field.

The intermittent character of the flow and the cyclone/anticyclone asymmetry can be best seen from the probability distributions of the vertical vorticity for several different initial Rossby numbers (figure 16). For Ro M 6 0.2w eo b s e r v eas t r o n g intermittency with no asymmetry, similar to that obtained with a quasi-geostrophic model. Cyclones and anticyclones are found in equal strengths and proportions: the vorticity distributions have big, almost symmetric tails, corresponding to a relatively large probability of occurrence for values much larger than the root mean square vorticity. It is important to notice that this behaviour is associated with the absence of significant energy dissipation (see § 3.2). All these properties are in agreement with quasi-geostrophic dynamics, as discussed for instance by [START_REF] Mcwilliams | The vortices of geostrophic turbulence[END_REF] and McWilliams et al. (1999).

For larger Rossby numbers (Ro M > 0.2) a clear dominance of cyclonic vorticity is observed. The regions of strong cyclonic vorticity are highly concentrated, corresponding to the positive tails in the probability distribution. By contrast, the anticyclonic part of the vorticity distribution drops steeply. This result is consistent with the qualitative observations of columnar cyclonic vortices amidst deformed, non-axisymmetric and fragmented anticyclonic vortices. This asymmetry is inconsistent with the quasigeostrophic model, but is predicted by models including some higher-order corrections. The bias for cyclones is for instance predicted by a surface geostrophic model (Hakim A local Rossby number, Ro local =(ω 2 z ) 1/2 /f , can be associated with any region in these probability distributions. It can be seen in figure 16, that for the asymmetric cases corresponding to Rossby numbers of order 1, the local Rossby number exceeds unity in the cyclonic vortex cores but seems to 'saturate' at values close to unity for the anticyclonic regions. Here the absolute vorticity -f + ω z is zero and locally there is no effective rotation, a condition that is not favourable to vortex persistence.

We recover here results from linear stability of rotating flows which predict unstable strong anticyclones (Ro local > 1) and possibly stable weak anticyclones (Ro local 6 1). This destabilization of strong anticyclones for Ro ∼ O(1) is indeed consistent with the centrifugal stability arguments that yield a stability/instability crossover for anticyclones when ω z (r)=f [START_REF] Kloosterziel | An experimental study of unstable barotropic vortices in a rotating fluid[END_REF][START_REF] Pedley | On the stability of viscous flow in a rapidly rotating pipe[END_REF]. Instability criteria were also found in the more general or complex flows by [START_REF] Cambon | Stability analysis and large eddy simulation of rotating turbulence with organized eddies[END_REF] and [START_REF] Leblanc | On the three dimensional instabilities of plane flows subjected to coriolis force[END_REF][START_REF] Fincham | 3D measurement of vortex structures in stratified fluid flows[END_REF] but all these stability studies lead to the conclusion that strong anticyclonic vortices are unstable while weak anticyclones and cyclones are stable. Our results are consistent with this conclusion.

The predominance of cyclonic vorticity for moderate rotation rates (Ro ∼ O(1)) was previously observed in purely rotating turbulence, either in numerical simulation [START_REF] Bartello | Coherent structures in rotating three-dimensional turbulence[END_REF][START_REF] Smith | Transfer of energy to two-dimensional large scale in forced, rotating three-dimensional turbulence[END_REF][START_REF] Godeferd | Direct numerical simulatins of turbulence with confinement and rotation[END_REF] or in experiments [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF]. A similar emergence of coherent vortices, with dominance of cyclones at moderate Rossby numbers, has been also obtained by [START_REF] Longhetto | Coherent vortices in rotating flows: a laboratory view[END_REF] with the same device as for the non-stratified case. Moreover, like [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF] they observed that when the Rossby number gets smaller, the symmetry is restored. Note that strong anticyclones, with ω z <f , can be produced just behind the rake as K árm án vortex streets, but they do not persist in the established turbulence.

In the stratified case with no rotation, there is no reason for the flow to exhibit any cyclonic/anticyclonic asymmetry; this is confirmed by the measurements. Furthermore, the probability distribution function then exhibits a quasi-Gaussian behaviour, which marks the absence of intense vortices. This demonstrates one of the profound differences between the stratified and stratified rotating cases.

Conclusion

The influence of rotation and stratification on decaying grid turbulence has been studied with our experimental system. We have covered the parameter space from the homogeneous to the strongly stratified case (Fr ∼ 10 -2 ) and obtained strong rotation effects (Ro ∼ 10 -2 ), while the Reynolds number was kept sufficiently large (about 10 4 ) to approach the inertial regime. This is confirmed by the invariance of the results with respect to changes in the Reynolds number. The scanning correlation imaging velocimetry technique provides time-resolved velocity fields in a volume, with direct access to scales at which energy is dissipated by viscosity.

An important effect of rotation is the inhibition of the energy decay as illustrated in figure 4. As rotation increases the dynamics undergoes a transition from a highly dissipative regime (f =0, N = 0) to a non-dissipative regime at low Rossby number. We find that for Ro M 6 0.2 Ro 6 0.02 based on the velocity r.m.s. and the integral scale), the energy decay becomes negligible. However, the dissipation is influenced by Ekman layers. This is the dominant source of dissipation in the unstratified rotating case, as found for instance by [START_REF] Longhetto | Coherent vortices in rotating flows: a laboratory view[END_REF] during experiments with the same device as we used. We find that this effect also controls dissipation for the final decay in the stratified cases with the strongest rotation, when the vertical integral scale corresponds to the depth of the tank. However, the combined effects of stratification and rotation cause the dissipation to be significantly lower than in the non-stratified case, due to the confinement of the Ekman layer pumping.

This non-dissipative regime is also associated with a k -3 h energy spectrum and a symmetric probability distribution of vertical vorticity which involves tails corresponding to the emergence of intense vortices. These features are consistent with the quasi-geostrophic model of [START_REF] Charney | Geostrophic turbulence[END_REF] which predicts, in the limit of small Rossby and Froude numbers, the absence of a direct energy cascade and a direct cascade of enstrophy, as in two dimensional turbulence. Some of the properties observed in this regime can alternatively be described in terms of wave theories. For instance, [START_REF] Cambon | Advanced in wave turbulence: rapidely rotating flows[END_REF] obtained the emergence of k -3 h energy spectrum in purely rotating turbulence by the nonlinear interactions of dispersive waves. However, the emergence of intense vortices and the high values of the kurtosis (figures 7 and 15) are typical of quasi-geostrophic dynamics and are not consistent with quasi-Gaussian wave theories. These observations tend to confirm that, in this regime, the effect of inertial-gravity waves remains negligible and that the spectrum is associated with the enstrophy cascade.

These experiments provide the possibility of investigating the breakdown of this non-dissipative quasi-geostrophic regime, as the Rossby number is increased. The first ageostrophic effect is the systematic suppression of the intense anticyclonic vortices with local Rossby number w 2 z 1/2 /f larger than unity, as shown in figure 16. The absolute vorticity in such vortex cores vanishes. We find that anticyclonic vortices with higher local Rossby numbers, such as those produced directly in the wake of the rake, are quickly destroyed.

Even for the weakest rotation (largest Rossby number) used in our experiments the vertical shear (or horizontal vorticity) is controlled by the geostrophic balance: it is found to be in a ratio f/N with the horizontal shear and is independent of viscosity (see figure 14). By contrast, in isotropic homogeneous turbulence, and in non-rotating stratified turbulence, the vertical shear is related to the Taylor microscale and is controlled by viscous dissipative effects.

In this transitional regime between the non-dissipative regime (Ro 6 0.02) and the non-rotating stratified turbulence, we observe that energy dissipation depends only on rotation, with no dependence on stratification, as shown in figure 4. By contrast the vertical shear is still controlled by the geostrophic balance and therefore depends on stratification (it scales with f/N alone). Therefore the dissipation cannot be approximated from the vertical shear (νω 2 h ) as in the non-rotating stratified case and other components have to be taken into account. In this transitional regime, unlike in the non-dissipative regime (Ro → 0, Fr → 0), we cannot exclude significant wave effects and wave theories could be relevant to interpret some of the results observed.

In this regime, the decrease of the energy dissipation with increasing rotation rate and its independence from the stratification may indicate that the mechanisms of energy transfer are close to the ones observed in purely rotating turbulence [START_REF] Waleffe | Inertial transfert in the helical decomposition[END_REF][START_REF] Cambon | Energy transfer in rotation turbulence[END_REF]). Energy dissipation is necessarily controlled by nonlinear energy transfers between modes. Nevertheless, some aspects of the dynamics may be accounted for well by linear theories, as proposed by [START_REF] Hanazaki | Linear process in stably and unstably stratified rotating turbulence[END_REF] for rotating stratified turbulence. Using the rapid distortion theory (RDT) he has shown that the integrated quantities depend only on f/N, a result in agreement with our findings on the vertical scale. The high anisotropy ratio between horizontal and vertical kinetic energy seems also to be consistent with the RDT for initially horizontal turbulence. Indeed, equations (5.09)-(5.11) of [START_REF] Hanazaki | Linear process in stably and unstably stratified rotating turbulence[END_REF] give, for the long-time asymptotics and for f/N =0.3, a ratio E h /E v ∼ 75 close to our estimate E h /E v ∼ 70, for the final stage of the decay in figure 3. This large ratio is the consequence of the particular initial conditions with purely horizontal velocity (zero 'wave energy'). This corresponds to our experimental forcing by a rake of vertical plates. We expect that this initial condition would be forgotten through a phase of three-dimensional turbulence, but this agreement with the RDT theory may indicate that the system keeps track of

  Fi gure 1. Side, top and front views of the experimental set-up.

Figure 3 .

 3 Figure 3. Comparison of the horizontal kinetic energy, E h , and the estimated vertical kinetic energy, E v . Re M = 18 000, Fr M =0.34, Ro M =1.13.

  Ro M = 0.14, Fr M = 0.09 (ec) Ro M = 0.14, Fr M = 0.17 (gc) Ro M = 0.56, Fr M = 0.09 (fc) Ro M = 0.56, Fr M = 0.17 (hc)f = 0, Fr M = 0.09 (bc) f = 0, Fr M = 0.17 (ic) Time × U/MFi gure 4. Decay of mean kinetic energy for several parameters. The initial Reynolds number is the same for the different experiments, Re M = 9000.

Figure 6 .

 6 Figure 6. Two-dimensional energy spectra at different stages of the experiment in the ageostrophic case. Re M = 9000, Ro M =0.56, Fr M =0.09.

Fi gure 7 .

 7 Vertical vorticity field in a horizontal plane at four successive times (t 1 = 377 s, t 2 = 619 s, t 3 = 915 s and t 4 = 3000 s). The grid was towed from right to left. Re M = 4500, Fr M =0.04, Ro M =0.07. The positive values of vertical vorticity are in light grey, negative in dark grey. The size of the represented domain is 250 × 250 cm 2 . 4. General description of the flow 4.1. Vortex structure and evolution Once the regime of horizontal motion is established, long-lived patches of vertical vorticity ω z , similar to those observed in two-dimensional turbulence, are quickly formed. In figure 7, fields of vertical vorticity are shown at four stages of development during a single experiment.

  Fi gure 8. Evolution of the three-dimensional structure of the vertical vorticity component represented by iso-surfaces at t = 280 s, t = 600 s, t = 1100 s and t = 2000 s for f/N =0.15. The vorticity value is 35 % of the maximum of |ω z |. The positive values of ω z are in dark grey while the negative values are in light grey. The rake is towed from right to left. Re M = 4500, Ro M =0.28, Fr M =0.04. The size of the represented volume is 250 × 250 × 45 cm 3 and the vertical axis is stretched three times to enhance the visualization. break down. Dritschel & de la Torre Juarez (1996) found instability for L<L R /3and Dritschel et al.

Figure 10 .

 10 Figure 10. Evolution of the horizontal integral scale, L h for various experiments.

Figure 11 .

 11 Figure 11. The transverse velocity correlation function f (z), for various experiments at t × U/M ∼ 11, Re M = 4500.

Figure 12 .

 12 Figure 12. Evolution of (a) the vertical integral scale L v and (b) the aspect ratio of the structure. Re M = 4500.

Figure 13 .

 13 Figure 13. Vertical vorticity, ω 2 z , and horizontal vorticity ω 2 h decay for increasing values of f/N. Re M = 9000. (a) f/N =0, (b) f/N =0.3, (c) f/N =0.6, (d) f/N =1.2.

Figure 14 .

 14 Figure 14. Ratio α =( ω 2 z / ω 2 h ) 1/2 versus f/N, averaged over the time interval [200...3000 s]. Experiments with different Re M are used for each value of f/N.

Figure 15 .

 15 Figure 15. Ku(t)= ω 4 / ω 2 2 for Ro M =0.07 (gb) and f =0 (bc).

Figure 16 .

 16 Figure 16. Normalized probability distribution of vertical vorticity for several values of f/N at t × U/M = 50. Gaussian distributions with equivalent variances are plotted in dotted lines. In the four cases, the Froude Fr M is equal to 0.09, and only the rotation is changed: (a) Ro M =0.07 (gb), (b) Ro M =0.14 (ec), (c) Ro M =0.28 (hb), (d) Ro M =0.56 (fc). The value of the normalized Coriolis parameter f/ w 2 z 1/2 , corresponding to Ro = 1, is represented by a vertical dashed line. For the two higher rotation rates (a, b), this value is outside the plotting range and is not represented: (a) f/ w 2 z 1/2 =1 3a n df o r( b) f/ w 2 z 1/2 =8.57. et al. 2002) (Muraki, Snyder & Rotunno 1999). In numerical simulations of this model, for moderate Rossby number Ro ∼ O(0.1), Hakim et al. (2002) indeed observed a distinct dynamical asymmetry favouring strong compact cyclones and weak broad anticyclones.A local Rossby number, Ro local =(ω 2 z ) 1/2 /f , can be associated with any region in these probability distributions. It can be seen in figure16, that for the asymmetric cases corresponding to Rossby numbers of order 1, the local Rossby number exceeds unity in the cyclonic vortex cores but seems to 'saturate' at values close to unity for the anticyclonic regions. Here the absolute vorticity -f + ω z is zero and locally there is no effective rotation, a condition that is not favourable to vortex persistence.

  Table 1 summarizes the different experiments performed and figure2represents them in a diagram of non-dimensional parameters. The horizontal axis represents the purely stratified cases without rotation, while the vertical axis represents homogeneous fluid cases. For the experiments, stratification dominates over rotation (f/N <1) for all cases except ga, gb, gc, gd, for which f/N =1.2.More relevant non-dimensional parameters can be defined from the turbulent velocity and horizontal integral length scale, Re

† Present address: I.M.F.T., Allíe du Prof. Camille Soula 31400 Toulouse, France. ‡ Present address: Dept of Aerospace and Mechanical Engineering, University of Southern California, RRB 101, 854 W 36th Place, Los Angeles, CA 90089, USA.

the initial conditions. This point should be investigated by comparison with forcing by a grid instead of a rake of vertical plates and by direct measurement of energy transfers.

Further work would be required to study the transition between the energyconserving dynamics and the dissipative dynamics of non-rotating stratified turbulence, with a more detailed analysis of non-geostrophic effects like front formation and wave-vortex interactions.

The authors would like to thank Claude Cambon and the referees for their constructive comments on the manuscript. The author also thank the support of Henri Didelle and René Carcel for the design and realization of the set-up. The work presented here has been supported by the European Commission project HYDRIV under contract HPRI-1999-50019 of the FP5 Improving Human Potential programme)