Michel Bergmann

Laurent Cordier

Jean-Pierre Brancher

Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition

In this paper we investigate the optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200). The objective is the minimization of the mean total drag where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equations, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced-order models during the optimization process. Benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. The key enablers to an accurate and robust POD ROM for the pressure and velocity fields are the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of different non-equilibrium modes. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% for reduced numerical costs (a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly).

Introduction

Managing the Use of Approximation Models in Optimization

During the last decade, the optimal control theory [START_REF] Gunzburger | Flow control[END_REF] has emerged as a new approach to solve active flow control and aerodynamic shape design problems. Indeed, these problems can be reduced [START_REF] Gunzburger | Introduction into mathematical aspects of flow control and optimization[END_REF] to the minimization or maximization of an objective functional J (drag or lift coefficients, concentration of pollutant, emitted noise, mixing . . . ) according to n control or design parameters c = (c 1 , c 2 , • • • , c n ) (unsteady blowing/suction velocities, heat flows, . . . ) under some constraints (Navier-Stokes equations, geometric constraints . . . ). Roughly, these optimization problems can be solved by two different class of numerical methods, on the one hand, the methods of descent type which at least require an approximation of the gradient of the objective functional, and, on the other hand, the stochastic methods whose principle consists in studying the evolution of a population of potential solutions during successive generations (genetic algorithms, simplex method . . . ). Whatever the specific class of numerical methods considered, the computational costs (CPU and memory) related to the resolution of optimization problems are so important that they become unsuited to the applications of flow control. This situation is even worse in an optimization setting where the governing equations need to be solved repeatedly, or in closed-loop control problems, for which the controller needs to determine his action in real time. Consequently, an alternative approach is necessary.

In this communication, we propose to solve the aforementioned problem of optimization by an optimal control approach in which the Navier-Stokes equations -called high-fidelity model in the multidisciplinary optimization literature -are replaced by a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) -low-fidelity model -of the dynamics for the controlled flow. The POD was originally introduced in Turbulence [START_REF] Cordier | Lecture series 2002-04 on post-processing of experimental and numerical data[END_REF] as an unbiased method of extraction of the Coherent Structures widely known to exist in a turbulent flow. Essentially, this technique leads to the evaluation of POD functions that define a flow basis, optimal in an energetic sense. Thereafter, these POD modes can be used through a Galerkin projection on the Navier-Stokes equations to derive a POD ROM of the controlled flow [START_REF] Cordier | Two typical applications of POD: coherent structures eduction and reduced order modelling[END_REF]. The POD basis is determined a posteriori using experimental or numerical data previously obtained for the configuration under study. In first approximation, the POD can be viewed as a method of information compression. Essentially, the POD algorithm try to remove "redundant" information (if any) from the data base. As a consequence, the ability of POD modes to approximate any state of a complex system is totally dependent of the information originally contained in the snapshot set used to generate the POD functions. Then, despite the energetic optimality of the POD modes, it seems difficult to build once for all, at the beginning of the optimization process, a POD ROM able to approximate correctly the different controlled states encountered by the flow along the optimal path (see the discussion in [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF]). Some kind of reactualization of the POD basis during the optimization process seems essential, the main difficulty consisting in determining the moment when a new resolution of the Navier-Stokes equations is necessary to evaluate a new POD basis. Thereafter, we will use a specific adaptive method called Trust-Region POD (TR-POD) to update the reduced-order models during the optimization process. This approach, originally introduced in [START_REF] Fahl | Trust-Region methods for flow control based on Reduced Order Modeling[END_REF], couples a trust-region method of optimization and reduced-order models based on POD (see Sect. 2). The principal advantages of this approach are, on the one hand, that the radius of the trust-region corresponding to the POD ROM does not have to be fixed by the user, but is evaluated automatically during the optimization process, and that on the other hand, there are results of convergence proving that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations.

A Prototype of Massively Separated Flows: the Cylinder Wake Flow

In this study, we are interested to control the unsteady wake flow downstream from a circular cylinder (Fig. 1). The objective is the mean drag minimization 000 000 111 111 of the wake flow by rotary oscillation of the cylinder. The flow is considered as incompressible and the fluid is supposed to be viscous and Newtonian. Wake flows dynamics are characterized [START_REF] Williamson | Vortex dynamics in the cylinder wake[END_REF] by the Reynolds number Re and by the natural Strouhal number St n at which vortices are shed in the wake of the cylinder (Fig. 8(a)). Traditionally, the Reynolds number is defined as Re = U ∞ D/ν where D is the cylinder diameter (R is the corresponding radius), U ∞ the uniform velocity of the incoming flow and ν the kinematic viscosity of the fluid. As for the natural Srouhal number, the common definition is St n = f D/U ∞ where f is the frequency characteristic of the periodic behavior of the flow. The cylinder wake is considered in the laminar regime (Re = 200). The rotary control is characterized by the non dimensional velocity γ(t) defined as the ratio of the tangential velocity V T to the upstream velocity U ∞ i.e. γ(t) = V T (t)/U ∞ . For γ = 0, the flow is naturally said uncontrolled. Contrary to the case considered in [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF] where no particular assumption was done on the variation of the control function, γ(t) is hereafter sought as an harmonic function of the form:

γ(t) = A sin (2πSt f t)
where the amplitude A and the forcing Strouhal number St f correspond to two degrees of freedom for the control. The optimal control theory is then used to determine the control vector c = (A, St f ) T which minimizes the mean time drag coefficient of the wake. For a circular cylinder, this quantity estimated over a finite horizon T equal to a few vortex shedding periods writes:

C D T = 1 T T 0 2π 0 2 p n x R dθ dt - 1 T T 0 2π 0 2 Re ∂u ∂x n x + ∂u ∂y n y R dθ dt , (1)
where n x and n y are the projections of the external normal vector n onto the cartesian basis vectors e x and e y respectively, and θ is an angle defining the curvilinear coordinate of a point on Γ c (see Fig. 1). Furthermore, let us recall (see [START_REF] Protas | Drag force in the open-loop control of the cylinder wake in the laminar regime[END_REF] for example), that in the supercritical regime of the wake flow, every mean quantity consists of two terms, the basic flow i.e., the unstable, steady, symmetric flow, and the mean flow correction which is due to the vortex shedding. Consequently, the mean time drag coefficient C D T writes:

C D T = C basic D + C 0 D , (2)
where C basic D and C 0 D represent the drag of the basic flow and the mean flow correction respectively (see Fig. 2). Of course, at a given Reynolds number, the contribution of the basic flow to drag cannot be modified. Then controlling the wake flow by rotary oscillations can only reduce the contribution of the mean flow correction to drag. If we consider as in [START_REF] Protas | Drag force in the open-loop control of the cylinder wake in the laminar regime[END_REF] that the drag of the mean flow correction field can be only positive, the minimal value of drag that can be obtained under periodic forcing conditions is that corresponding to the basic flow. In conclusion, this flow would be thus a natural 'desired' field in a flow tracking procedure of optimization. In order to validate a posteriori the control law obtained with the TRPOD algorithm, an open-loop control study is first performed numerically. The contours of the mean temporal drag coefficient, estimated over approximately 6 periods of vortex shedding after the transients have died out, are visualized in Fig. 3 in the space spanned by the forcing parameters A and St f .

Optimization by Trust-Region Methods and POD Reduced-Order Models

In this section, only the principle of the Trust-Region Proper Orthogonal Decomposition approach for flow control is exposed. For all the details of the algorithms and in particular the proofs of convergence, the reader is referred to [START_REF] Fahl | Trust-Region methods for flow control based on Reduced Order Modeling[END_REF][START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF].

We consider that the flow control problem discussed in Sect. 1.1 can be formulated as an unconstrained optimization problem

min c∈R n J (φ N S (c), c) (3)
where J : R m × R n → R represents the objective functional and where φ N S and c respectively represent the state variables obtained by numerical resolution of the state equations and the control variables. The subscript N S means that the state equations which connect the control variables c to the state variables are the Navier-Stokes equations. Since an accurate computation of the state variables φ for given c is computationally expensive when the Navier-Stokes equations are used as the state equations, the evaluation of J during the solution of the optimization process (3) is computationally expensive. A reduction of numerical cost can be achieved by employing a POD ROM as the state equation. In such a way an approximate solution φ P OD of the state variables φ is obtained and the optimization problem (3) is then replaced by a succession of subproblems of the form

min c∈R n J (φ P OD (c), c). (4)
Usually, a POD ROM is constructed for a specific flow configuration, e.g., for an uncontrolled flow or for a flow altered by a specified control. Therefore, the range of validity of a given POD ROM is generally restricted to a region located in the vicinity of the design parameters in the control parameter space, the so-called trust-region. It is then necessary to update the POD ROM during the iterative process, the crucial point being to determine when such a reactualization must take place. Let ∆ k > 0 be the trust-region radius and c k be the control parameters obtained at an iterate k of the optimization process. To evaluate the function J (φ N S (c k ), c k ), it is necessary to determine the variables φ N S (c k ). These variables are obtained by resolution of the highfidelity model, the Navier-Stokes equations. Then, we compute snapshots that correspond to the flow dynamics forced by c k . These snapshots form the input ensemble necessary [START_REF] Cordier | Lecture series 2002-04 on post-processing of experimental and numerical data[END_REF] to generate a POD basis {Φ k i } i=1,...,NP OD (here, N P OD corresponds to the number of POD modes). This POD basis can then be used via a Galerkin projection of the Navier-Stokes equations onto the POD eigenvectors to derive a POD ROM for c k [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF]. After integration in time of this POD ROM, the state variables φ P OD (c k ) are estimated, and thus the function J (φ P OD (c k ), c k ) is evaluated. Since this POD ROM can be employed for an optimization cycle, we define

m k (c k + s k ) = J (φ P OD (c k + s k ), c k + s k ), (5)
as a model function for

f (c k + s k ) = J (φ N S (c k + s k ), c k + s k ), (6)
on the trust-region

s k ≤ ∆ k around c k .
One is then brought to solve approximately1 the corresponding trustregion subproblem defined as

min s∈R n m k (c k + s), s.t. s ≤ ∆ k . (7)
In order to estimate the quality of the presumed next control parameters c k+1 = c k + s k where s k is an approximate solution of (7), we compare the actual reduction in the true objective, ared k = f (c k + s k ) -f (c k ), to the predicted reduction obtained with the model function

pred k = m k (c k + s k ) -m k (c k ).
Essentially, it is this comparison that gives a measure for the current models prediction capability. If the trial step s k yields to a satisfactory decrease in the original objective functional in comparison to the one obtained by the model function, the iteration is successful, the trial step s k is accepted and the model m k is updated i.e. a new POD ROM is derived that incorporates the flow dynamics as altered by the new control2 c k+1 . Furthermore, if the achieved decrease in f indicates a good behavior of the model m k , the trust-region radius ∆ k can be increased. Now, if there is a limited predicted decrease compared to the actual decrease, we have the possibility to decrease slightly the value of the trust-region radius. For unsuccessful iterations, the trial step s k is not accepted, the trust-region radius ∆ k is decreased and the trust-region subproblem ( 7) is solved again within a smaller trust-region. With the contraction of the trust-region it is more likely to have a good approximation to the true objective functional with the POD ROM. The corresponding TRPOD algorithm is schematically described in Fig. 4.

Drag Minimization of the Cylinder Wake Flow by the TRPOD Algorithm

The objective of this section is to implement the TRPOD approach presented in Sect. 2 for minimizing the mean drag coefficient of the cylinder wake flow.

A Robust POD Surrogate for Drag Function

In order to simplify the future notations, one introduces the drag operator C D defined as:

C D : R 3 → R U → 2 2π 0 pn x - 1 Re ∂u ∂x n x - 1 Re ∂u ∂y n y R dθ, (8)
where U = (u, v, p) T denotes the vector corresponding to the velocity and pressure fields.

k = k + 1 k = k + 1 c0 ck ck+1 ck+1

Figure 4

Schematic representation of the TRPOD algorithm.

care is taken to the development of the POD ROM. First, a POD basis Φ i representative of the velocity fields u and v, as of the pressure field p was determined [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF]. Then, to improve the robustness of the POD ROM, the POD basis functions which represent the dynamics of the reference operating condition c, were increased by adding N neq non-equilibrium modes corresponding to new operating conditions, following the procedure described in [START_REF] Noack | A hierarchy of low-dimensional models for the transient and post-transient cylinder wake[END_REF]. In addition, a calibration procedure of the POD-Galerkin system similar to the methods recently introduced in [START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF] for the pressure model was used to take into account some modelling errors (Galerkin truncation, pressure model,. . . ).

Here, this calibration procedure corresponds to the introduction of a timedependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem (see [START_REF] Bergmann | Optimisation aérodynamique par réduction de modèle POD et contrôle optimal[END_REF] for all the details). Finally, the control function method introduced in [START_REF] Graham | Optimal Control of Vortex Shedding Using Low Order Models. Part 1. Open-Loop Model Development[END_REF] is used to determine POD basis functions with homogeneous boundary conditions. The velocity and pressure fields can then be expanded into the POD basis functions Φ i as:

U (x, t) = N gal i=0 a i (t) Φ i (x)
Galerkin modes

+ N gal +Nneq i=N gal +1 a i (t) Φ i (x) non-equilibrium modes + γ(c, t) U c (x) control function , (9)
where N gal is the number of Galerkin modes and where U c is called the control function. Mathematically, U c is determined as a solution of the Navier-Stokes equations satisfying specific boundary conditions (the procedure is fully described in [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF]).

The Galerkin projection of the Navier-Stokes equations on the space spanned by the first N gal + N neq + 1 POD modes yields [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF] to

d a i (t) d t = N gal +Nneq j=0 B ij a j (t) + N gal +Nneq j,k=0
C ijk a j (t)a k (t)

+ D i d γ d t +   E i + N gal +Nneq j=0 F ij a j (t)   γ(c, t) + G i γ 2 (c, t), (10a)
with the following initial conditions:

a i (0) = (u(x, 0), Φ i (x)). ( 10b
)
The coefficients B ij , C ijk , D i , E i , F ij and G i depend explicitly on Φ i and U c . Their expression could be found in [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF].

Let φ N S (c) = (u N S , v N S , p N S ) T represent the state variables obtained by resolution of the Navier-Stokes equations and φ P OD (c) = (u P OD , v P OD , p P OD ) T be the corresponding values estimated with the POD ROM [START_REF] Homescu | Suppression of vortex shedding for flow around a circular cylinder using optimal control[END_REF], the objective functional is

f (c) = J (φ N S (c)) = 1 T T 0 C D (φ N S (c)) dt,
and the model function, introduced and justified in [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF], is

m k (c) = J (φ P OD (c)) = 1 T T 0 N gal +Nneq i=0 a i (t)N i dt,
where

N i = C D (Φ i ).
These two functions can then be used in a procedure of optimization coupling trust-region methods and POD reduced-order models, following the method presented in Sect. 2.

Solution of the Subproblem (7)

The convergence behavior of trust-region methods for general model functions with inexact gradient information is usually based on a sufficient decrease condition of the objective function (see [START_REF] Conn | Trust-region methods[END_REF] for example). In [START_REF] Fahl | Trust-Region methods for flow control based on Reduced Order Modeling[END_REF], these classical results were extended and it was demonstrated that the exact solution of the subproblem ( 7) is not necessary to prove global convergence of the TRPOD algorithm. Here, because of the low computational costs of solving the POD reduced-order model ( 10), an optimality system based on the POD ROM is derived (see [START_REF] Gunzburger | Introduction into mathematical aspects of flow control and optimization[END_REF] for example) and solved. By definition, this reduced-order optimality system is a system of three coupled partial differential equations [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF] formed by: 1. the state equations (10) 2. the adjoint equations

d ξ i (t) dt = - N gal +Nneq j=0 (B ji + γ(c, t) F ji ) ξ j (t) - N gal +Nneq j,k=0 (C jik + C jki ) a k (t)ξ j (t) - 1 T N i , (11a)
with terminal conditions :

ξ i (T ) = 0. ( 11b
)
3. the optimality conditions

∇ c J = 1 T T 0   N gal +Nneq i=0 L i   ∇ c γ dt, (12)
with

L i = - dξ i dt D i + ξ i   E i + N gal +Nneq j=0 F ij a j + 2γ(c, t)G i   .
This system of coupled ordinary differential equations could be solved directly using a "one-shot method". However, due to large storage and CPU costs, an iterative process described in [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF] is usually preferred. In this study, the directions of descent are estimated using the Fletcher-Reeves version of the Conjugate Gradient Method [START_REF] Nocedal | Numerical Optimization[END_REF]. The linear search parameter is computed at each iteration by the backtracking Armijo method [START_REF] Nocedal | Numerical Optimization[END_REF], in which the length of the step, along each direction of descent, checks the constraint imposed by the trust-region approach. The iterative method is stopped when two following values of the functional J are sufficiently close i.e. when |∆ J (a

)| = | J (n+1) (a) -J (n) (a)| < 10 -5 .

Numerical Results and Discussion

As it was discussed for example in [START_REF] Bewley | Flow control: new challenges for a new Renaissance[END_REF], a possible drawback of solving a minimization problem with a gradient-based optimization approach is that the algorithm may converge to the global minimum or to some other local minimum of the cost function depending on the relative position of the starting point to the minima. To alleviate this difficulty and evaluate the robustness of the TRPOD algorithm, the optimization process will be initialized starting from several different control parameters c 0 = (A ; St) chosen at random in the control parameter space retained for the open-loop control procedure (see Fig. 3). Hereafter, four different initial values are employed: c 0 = (1.0 , 0.2) T , c 0 = (1.0 , 1.0) T , c 0 = (6.0 , 0.2) T and c 0 = (6.0 , 1.0) T .

According to the TRPOD algorithm (see Fig. 4), the radius of the trustregion ∆ is automatically either increased, or decreased during the resolution of the optimization process. Figure 5 represents for the different initial control parameters c 0 , the variations of the values of the forcing amplitude and Strouhal number with respect to the iteration number. When the numerical convergence of the iterative procedure is achieved, the optimal control parameters are A = 4.25 and St f = 0.738. These values of parameters, which entirely define the optimal control law γ opt (t), are obtained in less than ten resolutions of the Navier-Stokes equations, whatever the initial condition considered (a more significant number of iterations is however represented in Fig. 5 to highlight the convergence). As it was expected by the

Figure 5

Variations of the forcing amplitude (left) and Strouhal number (right) with respect to the iteration number. Results obtained with the TRPOD algorithm. global convergence properties of the TRPOD algorithm [START_REF] Bergmann | Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF][START_REF] Fahl | Trust-Region methods for flow control based on Reduced Order Modeling[END_REF], these optimal control parameters tend towards the values predicted by an open-loop control approach ( § 1.2), and this, whatever the initial values used for the control parameters (see Fig. 6 which represents the convergence in the control parameter space). This proves the performance and the robustness of the TRPOD algorithm. = 0.94), but with a value always slightly higher. The polar curves (time evolution of the drag coefficient versus the lift coefficient) are represented for the uncontrolled and optimally controlled flow in Fig. 7(b). The shape of these curves indicates characteristically that, for the two flow regimes, the drag coefficient oscillates at a frequency equal to twice that of the lift coefficient. Finally, in Figs. 8(a)-8(c) we represent the vorticity fields of the uncontrolled flow, the optimally controlled flow, and the basic flow, respectively. The significant vortex-shedding phenomenon observed in Fig. 8(a) has been substantially reduced when the control is applied and the flow has been quasisymmetrized. The resulting flow approaches the symmetric state characteristic of the corresponding basic flow as can be awaited from the results presented in [START_REF] Protas | Drag force in the open-loop control of the cylinder wake in the laminar regime[END_REF] and the discussion in [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF]. Our results are qualitatively similar to the effects observed in [START_REF] Tokumaru | Rotary oscillatory control of a cylinder wake[END_REF] and [START_REF] He | Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation[END_REF] and confirm the arguments of [START_REF] Protas | Optimal rotary control of the cylinder wake in the laminar regime[END_REF] that the mean drag reduction is associated with control driving the mean flow toward the unstable state.

Figure 8

Vorticity contour plot of the wake for the uncontrolled (a), optimally controlled (b) and basic flow (c). The dashed lines correspond to negative values.

The Trust-Region POD algorithm originally introduced in [START_REF] Fahl | Trust-Region methods for flow control based on Reduced Order Modeling[END_REF] was used to minimize the total mean drag coefficient of a circular cylinder wake flow in the laminar regime (Re = 200). The key enablers to an accurate and robust POD ROM for the pressure and velocity fields were the extension of the POD basis functions to the pressure data, the introduction of a timedependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of different non-equilibrium modes. Finally, the optimal control parameters obtained with the TRPOD algorithm are A = 4.25 and St f = 0.738. The relative mean drag reduction is equal to 30%. Compared to previous similar studies where the Navier-Stokes equations are used as state equations in the optimal control problem, the main advantage of the TRPOD algorithm is that it leads to a significant reduction of the numerical costs because the optimization process itself is completely based on reduced-order models only. Indeed, when the state equations of the optimality system are POD ROMs instead of the Navier-Stokes equations, a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly. Now, if we compare to our preceding study [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF], where a POD ROM was coupled to an optimal control approach without any strategy for updating the reduced-order model during the optimization process, the cost reduction factors, found here, are lower. However, in this study, the use of the TRPOD algorithm mathematically proves that the solutions converge at least to a local optimum for the original high-fidelity problem, and less than ten resolutions of the Navier-Stokes equations are necessary. Due to the low computational costs involved in the optimization process and the mathematical proofs of global convergence, the TRPOD algorithm is a promising method of optimization in flow control. This approach that can easily be adapted to other configurations, should finally lead to the current resolution of unsteady, three-dimensional optimization problems for turbulent flows around complex geometries.

Figure 1

 1 Figure 1Controlled flow configuration.

Figure 2

 2 Figure 2 Variation with the Reynolds number of the mean drag coefficient. Contributions and corresponding flow patterns of the basic flow and unsteady flow.

Figure 3

 3 Figure 3 Variation of the mean drag coefficient with A and St f at Re = 200. Numerical minimum: (Amin, St f min ) = (4.25, 0.74) ; [9]: (A, St f ) = (3., 0.75) ; [10] (not shown): (A, St f ) = (3.25, 1.13) ; [5]: (A, St f ) = (2.2, 0.53).

  By definition, C D (U ) = C D (t) where C D represents the instantaneous drag coefficient. The velocity component u and pressure p present in the relation (8) can be obtained either by resolution of the Navier-Stokes equations, or by estimation using a POD ROM. In this study, a special Initialization: c0, Navier-Stokes resolution, J0. k = 0. ∆0 Construction of the POD ROM and evaluation of the model objective function mk Solve the optimality system based on the POD ROM under the constraints ∆k ck+1 and mk+1 Solve the Navier-Stokes equations and estimate a new POD basis Jk+1 Evaluation of the performance (Jk+1 -Jk)/(mk+1 -mk)

  c0 = (6.0 , 1.0) T .

Figure 6

 6 Figure 6Evolution of the control parameters during the optimization process for the different initial values c0.

Figure 7 (

 7 a) represents the time evolutions of the .

Figure 7

 7 Figure 7Time evolutions of the aerodynamic coefficients for the basic flow, uncontrolled flow (γ = 0) and optimally controlled flow (γ(t) = γopt(t)). Control was started at time t = 0.

  (a) Uncontrolled flow (γ = 0). (b) Optimally controlled flow (γ(t) = A sin(2πSt t) with A = 4.25 and St = 0.738). (c) Basic flow (γ = 0).

Following the trust-region philosophy[START_REF] Conn | Trust-region methods[END_REF], it is sufficient to compute a trial step s k that achieves only a certain amount of decrease for the model function.

Since a new snapshot ensemble is available, a new POD basis can then be determined, and finally, a new POD ROM can be derived.

Recently, numerical evidence were brought[START_REF] Bergmann | On the generation of a reverse Von Kármán street for the controlled cylinder wake in the laminar regime[END_REF] that, for the circular cylinder wake flow at Re = 200, a partial control restricted to an upstream part of the cylinder surface could lead to a mean flow correction field with negative drag.

Acknowledgments

The authors acknowledge M. Braza (Institut de Mécanique des Fluides de Toulouse) and D. Ruiz (ENSEEIHT) to kindly provide us with an original version of their Matlab Navier-Stokes solver. Stimulating and fruitful discussions with the low-dimensional modelling and control team at the Technische Universität Berlin, in particular Bernd R. Noack, are acknowledged.


Pierre G Valla
email: pierre.valla@univ-grenoble-alpes.fr

Pietro Sternai
email: pietro.sternai@unimib.it

Matthew Fox
email: m.fox@ucl.ac.uk

How climate, uplift and erosion shape the Alpine topography

Keywords: Mountain geodynamics, erosion & sediment yield, topographic evolution, climate and glaciations, geodetic uplift, modeling, geochronology

Decades of scientific research across the European Alps quantify the vast array of processes shaping the Earth's surface. Developments in thermochronometry and terrestrial cosmogenic nuclides constrain spatial patterns of rock exhumation, surface erosion and topographic changes. These can be compared to sediments eroded from the Alps and preserved in surrounding sedimentary basins or collected from modern rivers. Erosion-driven isostatic uplift explains up to around 50% of the modern geodetic rock-uplift rates, revealing the importance of internal (tectonics, deep-seated geodynamics) and external (glacial rebound and topographic changes) processes. We highlight recent methodological and conceptual developments that have contributed to our present view of the European Alps, and suggest steps needed to fill gaps in our understanding.

INTRODUCTION

Mountain topography lies at the interface between the Lithosphere and the Hydrosphere/Biosphere, and its long-term evolution results from the interplay between internal and external driving mechanisms (Fig. 1A, e.g. [START_REF] Whipple | The influence of climate on the tectonic evolution of mountain belts[END_REF].

Internal forcing involves crustal thickening from tectonic shortening and deeper processes such as lithospheric delamination or sub-lithospheric mantle flow. External forcing is mainly characterized by climate, whose variability controls erosion and the building-melting of ice caps and glaciers, as well as the biota evolution and base-level changes, which all operate to redistribute material across the Earth's surface. Key components of this system are surface processes (Fig. 1A) which are central in regulating the interactions between internal and external drivers [START_REF] Champagnac | Late-Cenozoic relief evolution under evolving climate: A review[END_REF]. Surface processes act in space and time directly on the Lithosphere (e.g. mass redistribution affecting the crustal stress field and thermal structure) and the Hydrosphere/Biosphere (e.g. erosion modulating rock weathering and carbon burial).

They shape mountain topography and relief, with indirect feedbacks on tectonics (topographic effects on the lithospheric stress and thermal state) and climate (orographic precipitation and large-scale atmospheric circulation controlled by mountain topography). A quantitative characterization of the mechanisms that control mountain topographic evolution is challenging, since they are intrinsically linked but also operate at different spatial and temporal scales (10 1 -10 6 meters or years), with thresholds and non-linear processes involved [START_REF] Champagnac | Late-Cenozoic relief evolution under evolving climate: A review[END_REF].

The European Alps are a classic example of a mid-latitude convergent mountain belt, extending over 1000 km (Fig. 1B) and forming an arc-shape which can be divided into three main sectors: Western, Central and Eastern Alps [START_REF] Schmid | Tectonic map and overall architecture of the Alpine orogen[END_REF]. The Alpine orogeny is the result of continent-continent collision between the European and Adriatic plates since the Late Eocene. The main topographic construction and rock exhumation, i.e. unroofing history or a rock's path towards Earth's surface, began at ca. 35 Ma or earlier mostly driven by crustal thickening [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF][START_REF] Schmid | Tectonic map and overall architecture of the Alpine orogen[END_REF]. The main drainage organization and major drainage divide between Alpine sectors (Fig. 1B) were established relatively early in the orogeny, following the main tectonic structures (Figs. 1B), and are strongly influenced by the Early-Oligocene to Early-Miocene exhumation of crystalline massifs. The overall Alpine topography reached high elevations during the early collisional stages, with Early-Oligocene elevations similar to present-day in the Western Alps (as revealed by palynology; [START_REF] Fauquette | Quantifying the Eocene to Pleistocene topographic evolution of the southwestern Alps, France and Italy[END_REF], and the high topography of the Central Alps was acquired during mid-Miocene times (from stable-isotope paleoaltimetry; [START_REF] Campani | Miocene paleotopography of the central Alps[END_REF]. It has been suggested that the topography of the Eastern Alps developed during Late Oligocene [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF], but this has not yet been confirmed quantitatively. As a mid-latitude mountain range, and given their spatial extent, the European Alps are characterized by a variety of climatic regimes, with high spatial variability in precipitation and temperature. This climatic setting leads to various geomorphic processes (fluvial, hillslope, glacial) which control erosion (i.e. surface mass removal by both mechanical and chemical processes), sediment export to forelands or intramountain deposition. During the Late Cenozoic, global climate evolved towards cooler conditions and increased variability [START_REF] Zachos | Trends, rhythms, and aberrations in global climate 65 Ma to present[END_REF]. The onset of glaciation since ca. 3 Ma for the Northern Hemisphere also impacted the European Alps, with extensive glacier coverage during glacial periods (Fig. 1B; [START_REF] Ehlers | Quaternary glaciations-extent and chronology: part I: Europe[END_REF]. Cyclic glacial/interglacial conditions and associated transient geomorphic responses have shaped the modern Alpine topography and relief. Today, tectonic horizontal shortening from plate convergence appears only active in the Eastern Alps, while the Western and Central Alps are subject to limited shortening and even extension in some areas [START_REF] Serpelloni | Kinematics, seismotectonics and seismic potential of the eastern sector of the European Alps from GPS and seismic deformation data[END_REF]. However, geodetic measurements (GPS, leveling) show that modern rock-uplift rates (i.e. vertical surface rock velocity, relative to a reference base level) are faster in the Western and Central Alps than in the Eastern Alps [START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF]. tectonics, climate and topography/relief in mountain evolution. This complex system involves interactions between the Lithosphere and the Hydrosphere and Biosphere, with surface processes regulating the interactions (details on these are presented in the original figure of [START_REF] Champagnac | Late-Cenozoic relief evolution under evolving climate: A review[END_REF]. (B) Modern topography of the European Alps (90-m resolution DEM) with Last Glacial Maximum (LGM) ice extent (white lines; [START_REF] Ehlers | Quaternary glaciations-extent and chronology: part I: Europe[END_REF], major Alpine tectonic lineaments [START_REF] Schmid | Tectonic map and overall architecture of the Alpine orogen[END_REF] and swath profile (thick and dashed blue lines).

Here, we review some key pieces of evidence constraining the topographic evolution of the European Alps. We present how methodological developments, especially regarding topographic, geochronologic and modeling methods, have quantified longterm erosion and relief development. Such a quantitative framework is needed to assess the relative contributions of internal and external forcing in the evolution of the European Alps, and to diagnose the potential drivers for modern rock-uplift patterns observed along the Alpine arc.

OLIGOCENE-MIOCENE EVOLUTION OF THE ALPS

The main Alpine collisional phase started at ca. 35 Ma, with the rapid development of mountainous topography, major drainage reorganization [START_REF] Lu | Evaluating igneous sources of the Taveyannaz formation in the Central Alps by detrital zircon U-Pb age dating and geochemistry[END_REF], and onset of sediment production on both the pro-(northern) and retro-(southern) sides of the orogen [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF][START_REF] Fox | The exhumation history of the European Alps inferred from linear inversion of thermochronological data[END_REF]. Sedimentary basins surrounding the European Alps offer a crucial archive to reconstruct the evolution of sediment yield during mountain building. The main challenges when using sediment records as proxies for long-term erosion history are (1) sediment preservation and possible remobilization after deposition or the recycling of sedimentary rocks during orogenesis,

(2) changes in the river drainage patterns (i.e. inferred link between sediment deposits and original relief sources), and (3) chemical erosion and the importance of dissolved load in the total erosion budget. Figure 2A presents a compilation of erosion products for the European Alps [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF], showing two main periods in the sediment yield history (35-15 Ma and 15-0 Ma, Fig. 2A). There is a significant increase in sediment yield between ca. 30 and 25 Ma, reflecting topographic building and relief development allowing the onset of active geomorphic processes and efficient sediment production. Between 25 and 15 Ma, sediment yields remained high and have been punctuated by short pulses which are proposed to reflect changes in tectonic forcing and movements of the drainage divide [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF]. The mobility of the drainage divide is also evidenced by antagonistic trends in sediment discharge between the Northern and Southern Alps. This 25-15 Ma period is considered the main tectonic constructional phase of the European Alps (especially for the Eastern Alps). The 15-5 Ma phase is characterized by a significant decrease in sediment yield just before 15 Ma for both the Western/Central and Eastern Alps, followed by steady sediment flux for the Western and Central Alps. For the Eastern Alps, a minor increasing trend can be observed during the mid-Miocene. Finally, the most striking observation from [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF] is the significant increasing trend beginning at ca. 5 Ma (Miocene/Pliocene transition, Fig. 2A), observed for the entire European Alps but apparently more important for the Western and Central Alps. Exhumation of Western (thick colored lines) and Central (dashed colored lines) Alps, extracted from geometric reconstruction of bedrock thermochronometric isoages [START_REF] Vernon | Increase in late Neogene denudation of the European Alps confirmed by analysis of a fission-track thermochronology database[END_REF]. Note the overall increase trend (grey envelope) in exhumation since ca. 5 Ma, similar to sediment yield trends shown in (A). (C) Spatial distribution of Alpine exhumation from linear inversion of thermochronomotric data. 12-10 Ma (left) and 2-0 Ma (right) time windows are presented for illustration of the temporal variability (insets show temporal resolution for each time window). Black lines indicate main Alpine massifs and tectonic lineaments [START_REF] Schmid | Tectonic map and overall architecture of the Alpine orogen[END_REF]. Based on [START_REF] Fox | The exhumation history of the European Alps inferred from linear inversion of thermochronological data[END_REF].

Sediment yield records indicate that Alpine orogeny has experienced major changes in topographic and erosion histories. However, given the large-scale spatial integration of sediment records, assessing the spatio-temporal patterns in erosion at massif scale has remained challenging. Thermochronometry records the time since a rock passed through an effective closure temperature, and can provide a direct quantification of rock exhumation towards Earth's surface driven by erosion and tectonic unroofing. In addition, the thermal field of the upper crust is also sensitive to rock uplift and surface topography (i.e. amplitude and wavelength). Low-temperature thermochronometry (apatite and zircon thermochronometers, i.e. with closure temperatures <250 °C) can be used to quantify rock exhumation, at a timescale provided by the respective rock cooling ages. Detrital thermochronometry, from modern river sediments or past sediment records, provides an integrated overview of Alpine long-term erosion. Despite fragmentary records for the early construction stages, detrital thermochronometry confirms erosion pulses during the Oligocene but suggests an overall steady erosion over the European Alps since ca. 15 Ma at rates of 0.1-0.4 km Ma -1 [START_REF] Bernet | Steady-state exhumation of the European Alps[END_REF]. Within this apparent steady setting, detrital thermochronometry has also revealed major changes in sediment provenance that reflect re-organization of river drainage patterns for the Eastern Alps (around 20 Ma) and Western/Central Alps (around 13-10 Ma), in agreement with sediment records [START_REF] Kuhlemann | Post-collisional sediment budget history of the Alps: tectonic versus climatic control[END_REF].

Bedrock thermochronometry provides direct quantification of erosion and topographic history. Since the 1970's, over 3000 bedrock cooling ages (including multithermochronometers) have been acquired across the Alps, providing dense datasets for extracting exhumation patterns in space and time [START_REF] Vernon | Increase in late Neogene denudation of the European Alps confirmed by analysis of a fission-track thermochronology database[END_REF][START_REF] Fox | The exhumation history of the European Alps inferred from linear inversion of thermochronological data[END_REF]. Bedrock thermochronometry suggests early onset of erosion in the Eastern Alps (Tauern window and Austroalpine units) and Southern Alps (Bergell and Adamello massifs), with Early-to mid-Miocene erosion pulses linked to tectonic shortening and crustal thickening, followed by overall moderate erosion magnitudes since the mid-Miocene. A Late-Miocene erosion increase has also been documented for the Southern Alps, while this has not been observed with bedrock thermochronometry in the Eastern Alps. In the Western and Central Alps, thermochronometric data highlight exhumation contrasts, with mid-Miocene erosion onset linked to the exhumation of the External Crystalline Massifs (Aar-Gotthard, Mont-Blanc, Belledonne-Pelvoux; [START_REF] Schmid | Tectonic map and overall architecture of the Alpine orogen[END_REF] and within more internal parts of the orogen (Lepontine Dome), followed by an apparent major increase in erosion during the Late Miocene (Fig. 2B). This ca. 5-Ma erosion signal, similar to the sediment record (Fig. 2A), has raised long-lasting discussions about the potential contributions of tectonics vs. climate in late-stage erosion dynamics of the European Alps. For the Western and Central Alps, both hypotheses have been postulated with an orogen response to (1) a climate shift at the Miocene-Pliocene transition, with enhanced climatic variability and possibly increased precipitation favoring efficient geomorphic processes and sediment production/export (e.g. [START_REF] Vernon | Increase in late Neogene denudation of the European Alps confirmed by analysis of a fission-track thermochronology database[END_REF], (2) deep-seated geodynamic processes such as lithospheric slab detachment (e.g. [START_REF] Fox | Rapid exhumation in the Western Alps driven by slab detachment and glacial erosion[END_REF]. For the Eastern Alps, limited post-Miocene rock uplift and erosion has been documented, although not recorded by thermochronometry, and related to changes in regional tectonics (i.e. inversion of Pannonian Basin; e.g. Ruszkiczay-Rüdiger et al., 2020 and references therein).

Recent numerical developments in thermal(-kinematic) models and inversion approaches (e.g. [START_REF] Fox | The exhumation history of the European Alps inferred from linear inversion of thermochronological data[END_REF]; Fig. 2C) have allowed researchers to include multithermochronometers for assessing bedrock erosion histories. For the Western and Central Alps, these methods have revealed a more complex erosion framework. There is evidence for mid-Miocene onset of high erosion rates (Fig. 2C) with tectonic uplift from crustal thickening, but also for a subsequent decrease in erosion towards the Late Miocene-Early Pliocene. Temporal erosion trends from bedrock thermochronometry (Fig. 2C) and sediment yield records (Fig. 2A) slightly differ for the Late Miocene.

The progressive exhumation and exposure of crystalline and highly-resistant rocks at this time could have caused an overall decrease in bedrock erosion (lower erodibility) while increasing the relative abundance of crystalline clasts (better preservation) in the sediment record. Finally, inversion of bedrock thermochronometry reveals a major increase in erosion since ca. 2 Ma (Fig. 2C) for the Western/Central Alps, although the resolution of both current thermochronometric data and imaging of the Earth's interior via inversion of seismic data cannot be used to distinguish between either tectonics or climate forcing, nor to recognize feedbacks triggering this erosion increase [START_REF] Fox | Rapid exhumation in the Western Alps driven by slab detachment and glacial erosion[END_REF].

ALPINE TOPOGRAPHY & PLIOCENE-QUATERNARY GLACIATION

Alpine landscapes present typical glacial landforms with glacial cirques, U-shaped wide, steep and deep valleys (Fig. 1B), but also "hidden" landscape features such as overdeepnings which form major lakes and sediment infills in the present-day topography. Although the Quaternary geomorphic imprint of glacial erosion is obvious, key questions remain regarding its timing, magnitude and spatial variability.

Is landscape transition from fluvial to glacial landforms a rapid process that occured during the early glaciations? How variable are spatial patterns and rates in glacial erosion between different glacial periods? Are fluvial features (such as inner gorges and hanging valleys) markers of post-glacial landscape re-adjustment or do they evolve through multiple glacial/interglacial cycles? Numerical outcomes suggest that glacial erosion in the Western/Central Alps has propagated from low to high elevations during the successive glacial periods, as the landscape evolved from fluvial to glacial conditions [START_REF] Sternai | Spatial and temporal variations of glacial erosion in the Rhône valley: implications for relief development and sediment production[END_REF]. However, quantifying via observations the impact of Plio-Quaternary glaciation on Alpine erosion and topography has remained difficult due to (1) the relatively short timescales involved (1-2 Ma for the Quaternary and 10-100 ka for individual glacial/interglacial cycles) compared to the current resolution of thermochronometric methods, and (2) the preservation of, and/or access to, continuous sedimentary records or geomorphic markers for individual glaciations.

For the European Alps, the onset of major glaciation follows the Northern Hemisphere glaciation (ca. 3 Ma), with a major environmental and stratigraphic change reported at ca. 0.9 Ma (e.g. [START_REF] Muttoni | Onset of major Pleistocene glaciations in the Alps[END_REF]. Previous Plio-Quaternary glacial phases would have been of limited extent, leaving only scarce sediment records in the internal parts of the Alpine massifs. The mid-Pleistocene transition (MPT, ca. 1.2 Ma) promoted global climate change with the switch from lowamplitude short (symmetric 40-ka) to high-amplitude long (asymmetric 100-ka) glacial/interglacial cycles. For the European Alps, this MPT change would have resulted in the development of extensive and long-lasting glaciers that reached the Alpine forelands [START_REF] Muttoni | Onset of major Pleistocene glaciations in the Alps[END_REF].

In the Western/Central Alps, there is quantitative evidence for the impact of glaciation on Alpine topography (Fig. 3A). In the Swiss Central Alps, [START_REF] Haeuselmann | Abrupt glacial valley incision at 0.8 Ma dated from cave deposits in Switzerland[END_REF] have used cosmogenic 26 Al/ 10 Be dating of buried cave sediments to quantify the Aare valley deepening with respect to the cave system. Dating results show two valley deepening periods over the Plio-Quaternary, with limited deepening (at ~0.1 km Ma -1 ) until ca. 0.9 Ma followed by abrupt valley deepening (at >1 km Ma -1 ). In the upper Rhône valley (Swiss Western Alps), [START_REF] Valla | Significant increase in relief of the European Alps during mid-Pleistocene glaciations[END_REF] used apatite 4 He/ 3 He thermochronometry to quantify the late-stage bedrock cooling along the valley flank.

Accepted Manuscript published at Elements Vol. 17, pp. 41-46 (DOI: 10.2138/gselements.17.1.41)

Using geothermal constraints and thermal-kinematic modeling, their results highlight a quiescent erosion phase during Plio-Quaternary followed by subsequent valley incision (i.e. topographic change by spatially-focused erosion) at 1 km Ma -1 since ca.

1 Ma (Fig. 3A). These outcomes not only point towards a major erosional shift since around 1 Ma for the Western/Central Alps, but also reveal a topographic change with significant relief increase that is interpreted as glacial valley deepening. Such a topographic response to glaciation has not been observed or with limited magnitude for the Eastern and Southern Alps, despite similar glacial landforms with deep and wide U-shaped valleys [START_REF] Sternai | Preglacial topography of the European Alps[END_REF]. Pre-glacial topographic reconstructions have been attempted using different methods, such as the geophysical relief approach [START_REF] Champagnac | Late-Cenozoic relief evolution under evolving climate: A review[END_REF] or by computing a steady-state fluvial topography [START_REF] Sternai | Preglacial topography of the European Alps[END_REF] with subsequent modifications by glacial processes. Although these models rely on a number of untestable (but plausible) assumptions (e.g. constant drainage network throughout the Quaternary), they provide useful first-order estimates for evaluating glacial topographic changes in the European Alps (Fig. 3B) and the associated isostatic response to non-steady erosional unloading (Fig. 5).

Moreover, these results raise new contradictory observations and questions: topographic changes appear similar to slightly more pronounced for the Eastern Alps (Fig. 3B) than in the Western and Central Alps, whereas bedrock thermochronometry suggests significantly different trends for long-term erosion (Figs. 2C and3B). Such observations cannot be explained by horizontal shortening, which has been limited in the Western/Central Alps and is ongoing in the Eastern Alps. One alternative mechanism could be the occurrence of deep geodynamic forcing (e.g. sub-lithospheric mantle flow) sustaining relatively high steady erosion in the Western and Central Alps compared to the Eastern Alps [START_REF] Fox | Rapid exhumation in the Western Alps driven by slab detachment and glacial erosion[END_REF][START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF]. The observed differences (Fig. 3B) between Plio-Quaternary erosion estimates and Quaternary topographic changes would call for further research to quantify the respective contributions from "steady" (i.e. driven by rock uplift) erosion and "non-steady" topographic evolution of the European Alps. [START_REF] Haeuselmann | Abrupt glacial valley incision at 0.8 Ma dated from cave deposits in Switzerland[END_REF] and Rhône (bedrock low-temperature thermochronometry, converted into valley floor paleo-elevations using thermal-kinematic modeling, green lines; [START_REF] Valla | Significant increase in relief of the European Alps during mid-Pleistocene glaciations[END_REF] valleys. Red dashed line indicates onset of major Alpine glaciation from stratigraphic evidence (Po River Basin; [START_REF] Muttoni | Onset of major Pleistocene glaciations in the Alps[END_REF]. (B) Swath (see Fig. 1B for location) profiles of modern topography (Fig. 1B, grey envelope), 2-0 Ma erosion rate (Fig. 2C, red line and envelope) and topographic change over the last 1 Ma [START_REF] Sternai | Preglacial topography of the European Alps[END_REF] blue line and envelope).

MODERN ROCK UPLIFT AND EROSION: CAUSES AND IMPLICATIONS

The modern European Alps are characterized by limited shortening in the Western and Central Alps, and by ongoing active shortening in the Eastern Alps. In contrast, geodetic (leveling, GPS/GNSS) rock-uplift rates, averaged over the last 10 to 100 years, are highest in the Western and Central Alps (up to around 2 mm a -1 , Fig. 5; [START_REF] Nocquet | Present-day uplift of the western Alps[END_REF][START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF]. What is driving the observed spatial patterns of rock uplift in the European Alps? Erosion rates have been invoked to explain these rock-uplift patterns, but their spatial and temporal quantification is required to recognize whether the European Alps are actually experiencing or not surface uplift.

Modern erosion for the European Alps has been estimated using sediment yield (river and reservoir gauges; [START_REF] Hinderer | River loads and modern denudation of the Alps-a review[END_REF] and cosmogenic 10 Be (riverine sediments, [START_REF] Delunel | Late-Pleistocene catchment-wide denudation patterns across the European Alps[END_REF] for the main drainage basins across the Alps. Modern sediment yield data cover the last decades and combine physical and chemical erosion. The spatial distribution of modern Alpine erosion shows a ~3 fold difference in erosion between the Western/Central Alps and the Eastern Alps, which is interpreted as reflecting enhanced chemical erosion of carbonate sedimentary rocks that are abundant in the external mountainous parts of the Western/Central Alps [START_REF] Hinderer | River loads and modern denudation of the Alps-a review[END_REF]. Cosmogenic 10 Be-derived erosion yields millennial integration timescales and presents similar patterns, with higher erosion in the Western/Central Alps (~2-3 fold difference, Fig. 4) compared to the Eastern Alps. Moreover, both erosion datasets show no evidence for a modern climatic control (i.e. present-day precipitation patterns) on the spatial erosion distribution, but they rather reveal a significant slope/relief control on erosion which reflects intense glacial pre-conditioning of the Alpine topography as well as ongoing glacier retreat [START_REF] Hinderer | River loads and modern denudation of the Alps-a review[END_REF][START_REF] Delunel | Late-Pleistocene catchment-wide denudation patterns across the European Alps[END_REF]. Millennial to modern erosion patterns indeed follow an expected geomorphic response since the last glacial maximum (ca. 20 ka ago), characterized by high post-glacial erosion rates and transient hillslope and fluvial topographic readjustment. It remains debated how long landscapes take to switch from glacial to fluvial conditions, and this response may take multiple interglacial periods (e.g. [START_REF] Montgomery | Preservation of inner gorges through repeated Alpine glaciations[END_REF][START_REF] Leith | Signatures of Late Pleistocene fluvial incision in an alpine landscape[END_REF].

Both modern geodetic rock-uplift rates and erosion-rate patterns (derived from sediment yield and cosmogenic 10 Be) present a similar increasing trend from the Western to Central Alps followed by a decrease towards the Eastern Alps, suggesting the existence of a functional relationship, with the proposed hypothesis of erosiondriven rock uplift for the European Alps [START_REF] Champagnac | Erosion-driven uplift of the modern Central Alps[END_REF]. However, while patterns do correlate, modern erosion rates are generally lower than modern rockuplift rates (Fig. 4), implying that the isostatic response to erosional mass removal Accepted Manuscript published at Elements Vol. 17, pp. 41-46 (DOI: 10.2138/gselements.17.1.41) cannot explain all the observed rock-uplift rates. This discrepancy may result from the different spatial and temporal scales covered by erosion and rock-uplift datasets, which can be problematic when extrapolating yearly to decadal sediment yield, climatic data and geodetic rock-uplift estimates to thousand-year timescales which are representative of cosmogenic 10 Be-derived erosion rates. An alternative explanation for the observed discrepancy between modern rock-uplift and erosion patterns, if any relationship between them should exist, could be that modern rock uplift integrates different external or internal contributions along the European Alps [START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF]. In Figure 5, we evaluate the spatial patterns of both the modern geodetic uplift and the respective contributions of external (i.e. deglaciation rebound and erosioninduced elastic adjustment) and internal (dynamic uplift from mantle flow) forcing mechanisms. These estimates are based on various assumptions, such as the sublithospheric mantle viscosity and lateral/depth variations, the timing and spatial variability in deglaciation or the importance of topographic change vs. steady background erosion for erosional unloading across the European Alps (see extended discussion in [START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF]. For the Eastern Alps, the combination of erosional Basin are likely to involve, respectively, dynamic uplift and subsidence in the Eastern Alps (Fig. 5). Modern limited rock uplift in the Eastern Alps thus appears to us as the result of a combination of opposing forcings. For the Western and Central Alps, the isostatic response to deglaciation and erosional unloading contributes up to around 50% of the observed geodetic rock uplift (Fig. 5). Given the limited tectonic shortening occurring in these regions, deeper mechanisms involving lithospheric and sub-lithospheric mantle flow (and related dynamic uplift) must be at play. Convective processes from lithospheric slab detachment below the Western Alps are particularly debated [START_REF] Lippitsch | Upper mantle structure beneath the Alpine orogen from high-resolution teleseismic tomography[END_REF][START_REF] Zhao | Continuity of the Alpine slab unraveled by high-resolution P wave tomography[END_REF], since the occurrence, timing and spatial extent of such event(s) are still poorly constrained. For the Central Alps, the sub-lithospheric mantle flow contribution to rock uplift appears significant (Fig. 5), and can explain the high observed rock-uplift rates (up to 2 mm a -1 ) when combined with the isostatic adjustments to external forcing. catchment outlets for cosmogenic 10 Be-derived erosion estimates over millennial timescales [START_REF] Delunel | Late-Pleistocene catchment-wide denudation patterns across the European Alps[END_REF]. Colored lines are 2-0 Ma erosion estimates from linear inversion of thermochronometric data [START_REF] Fox | The exhumation history of the European Alps inferred from linear inversion of thermochronological data[END_REF]. Black and white lines are LGM ice extent and major Alpine tectonic lineaments, respectively. external (erosional adjustment from topographic changes and deglaciation rebound, blue plain and dashed lines respectively with grey envelopes; [START_REF] Sternai | Preglacial topography of the European Alps[END_REF][START_REF] Spada | Glacio-isostatic adjustment in the Po plain and in the northern Adriatic region[END_REF] and internal (dynamic uplift, red line and light-grey envelope; [START_REF] Zhao | Continuity of the Alpine slab unraveled by high-resolution P wave tomography[END_REF] forcing mechanisms. The different contributions are sourced from [START_REF] Sternai | Present-day uplift of the European Alps: Evaluating mechanisms and models of their relative contributions[END_REF].

SUMMARY & OUTLOOK

Our review of Late-Cenozoic evolution of the European Alps is based on different methodologies, ranging from sediment yield analyses (modern and past records), geochronology (mainly low-temperature thermochronometry and terrestrial cosmogenic nuclides) and geodesy or geophysics combined with numerical modeling. This method diversity allows us to assess the different spatial and temporal scales involved with Alpine erosion and topographic evolution. The existing data show a complex spatio-temporal evolution of the European Alps, with onset of topographic construction in the Early Oligocene, and significant tectonic controls on erosion and topographic building via crustal thickening and drainage pattern changes until the mid-Miocene. Plio-Quaternary erosion and topographic evolution appear to be controlled by climatically-driven geomorphic processes, with major glaciation impact on topography since ca. 1 Ma in the Western and Central Alps, but apparently not in the Eastern Alps. In addition, there is a spatial contrast in both modern erosion and geodetic rock uplift between the Western/Central and Eastern Alps. This strongly suggests that the late-stage evolution of the European Alps is reflecting the interplay between external (climate) and internal (solid Earth) mechanisms. Future studies need to provide higher resolution in thermochronometric data for late-stage erosion of slowly-eroding regions, where current data only provide average erosion histories over long periods. This will be possible with the recent development of very lowtemperature thermochronometers. In addition, further geomorphic markers and sediment archives need to be investigated and dated to improve the existing chronology for the progressive (or not) Alpine topographic evolution and for glacier fluctuations (timing and extent) during previous glacial/interglacial cycles of the Plio-Quaternary. Such improvements would provide a quantitative framework for the recent erosion history of the Alps, which is required to estimate the isostatic response to erosional unloading, considering both steady background erosion and topographic changes, in addition to deglaciation. Finally, sub-lithospheric mantle flow and potential slab detachment are likely to contribute to the modern geodetic rock uplift.

Higher-resolution tomographic models would provide important information for further constraining these contributions across the European Alps.

Figure 1 .

 1 Figure 1. Alpine Topography & Relief. (A) Sketch of the interplays and feedbacks between

Figure 2 .

 2 Figure 2. Erosion & Sediment Fluxes. (A) Late-Cenozoic sediment budgets for the Eastern and Western/Central (including Southern) Alps, after Kuhlemann et al. (2002). (B)

Figure 3 .

 3 Figure 3. Plio-Quaternary Erosion & Relief Development. (A) Paleo-elevation (proxy for valley incision) of the Aare (dating of cave sediments, grey ellipses and black dashed lines;

  response and deglaciation rebound (external forcing) matches the geodeticallymeasured uplift, suggesting isostatic adjustment could be the only mechanism for uplift in this region. However, this scenario is unlikely since (1) the Eastern Alps are still experiencing shortening and associated tectonic uplift, as also suggested by local examples of inversion tectonics since ca. 3 Ma (e.g. Ruszkiczay-Rüdiger et al., 2020); in addition, (2) mantle upwelling below and sediment loading within the Pannonian

Figure 4 .

 4 Figure 4. Modern Rock Uplift & Erosion. Spatial distribution (30-km resolution) of modern geodetic rock uplift (Sternai et al., 2019) over decadal timescales. Colored circles are

Figure 5 .

 5 Figure 5. Modern Geodetic Uplift & Potential Uplift Contributions. Swath (see Fig. 1B for location) profiles of modern geodetic rock uplift (Fig. 4, black line and dark-grey envelope),

ACKNOWLEDGMENTS P.G.V. acknowledges funding from the French ANR-PIA program (ANR-18-MPGA-0006) and the Swiss SNSF (PP00P2_170559). P.S. was supported by the Italian Ministry of Education MIUR (Rita Levi Montalcini, DM 694-26/2017, and Dipartimenti di Eccellenza 2018-2022 grants). M.F. is supported by NERC (NE/N015479/1). The authors warmly thank B. Salcher, K. Whipple and an anonymous reviewer for their constructive and insightful reviews, as well as the editors A. McCarthy and O. Müntener for their invitation and suggestions.