Sophie Thery

Well-posedness of a non local ocean-atmosphere coupling model: study of a 1D Ekman boundary layer problem with nonlocal KPP-type turbulent viscosity profile

This paper addresses the mathematical analysis of the ocean-atmosphere coupling problem, including Coriolis force, nonlocal turbulent closure and realistic nonlinear interface conditions. We consider a simplified 1D vertical model corresponding to a coupled Ekman boundary layer problem with a nonlocal turbulent viscosity. The main difficulty lies in the nonlocal properties induced by the interface conditions. Well-posedness is first studied in stationary and nonstationary states considering generalized parameterized turbulent viscosity, and we prove that the uniqueness of the solution is not guaranteed. We then give a well-posedness criterion for the stationary state of the coupled problem for viscosity profiles that are representative to those used in the ocean and atmosphere models.

Introduction

Ocean-atmosphere (OA) interactions play a critical role for several applications, like forecasting the trajectories of tropical cyclones, seasonal weather forecasting, or climate studies. Therefore numerical modeling systems for such applications generally couple an oceanic model with an atmospheric model, with complex interface conditions (reffered to as a "bulk closure") that model these interactions. However ocean and atmosphere models have originally been constructed separately, by two distinct communities. Thus the question of the mathematical coherence of such a coupled system naturally arises, since there is no garantee that all possible associations of an atmospheric model, an oceanic model and interface "bulk" conditions will lead to a well-posed problem.

The translation of such an OA coupled model into a single global mathematical model is challenging and gives rise to specific difficulties. A first global OA coupled model has been presented and studied by [START_REF] Lions | Mathematical theory for the coupled atmosphereocean models (cao-iii)[END_REF] as a coupling of the so-called primitive equations with nonlinear interface conditions. Many studies on the well-posedness of the primitive equations (without coupling) can be found in the context of ocean or atmosphere modeling (see for example [START_REF] Lions | Models of the coupled atmosphere and ocean (cao i). i[END_REF], [Lions et al., 1993a] or [START_REF] Cao | Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics[END_REF]). OA coupled models mainly differ by the strategies they use to take into account the turbulent scales (turbulent closure) and the interactions between the two domains (bulk closure). The turbulent closure scheme considered hereafter relies on the Boussinesq hypothesis which requires the definition of a turbulent viscosity profile. The resulting model is known as the coupled Ekman layer problem [START_REF] Ekman | On the influence of the Earth's rotation in ocean-currents[END_REF]. The present study addresses the well-posedness of such an Ekman layer coupled model that takes into account the specificities brought by the bulk interface conditions. Mathematical results can be obtained for this model, which is representative of the physics and numerics of realistic models. The combination of the turbulent closure schemes with specific interface conditions from the bulk formulation leads to a nonlocal coupled problem with nonlinear interface conditions, that is, the entire viscosity profile depends directly on the solution at the interface. This nonlocality is the main difficulty in tackling the mathematical analysis. A first study of this model was proposed by [START_REF] Pelletier | Twosided turbulent surface-layer parameterizations for computing air-sea fluxes[END_REF]. In the present work, we present a first synthetic step in the analysis on this nonlocal problem in the context of OA coupling, and investigate the well-posedness of this coupled problem by searching for constraints on the turbulent viscosity profiles. As a global approach to establish well-posedness, we will adapt a method from existing work in the fluid-fluid interaction community. The type of non-local character we are dealing with here has very rarely been considered in the literature. Here we will refer to a stationary model proposed by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] which considers a two-fluid interaction, with a turbulent viscosity profile different from the one considered here. The authors of this work prove the existence of solutions and show that the uniqueness depends strongly on the viscosity profile and the regularity of the solution itself. More precisely, a fixed-point method is used to study the uniqueness of solutions and it appears that it depends on the ratio between the H 1 -norm on the solution, the derivative of the viscosity profile and its minimum. In the present paper, we study a close model and use the same fixed-point method to discuss the uniqueness of the solution. As indicated previously, our model is simplified into a 1D model thanks to several hypotheses but retains the main ingredients (the turbulent closure schemes and the bulk interface conditions) in order to be representative to the formulation of realistic models. Therefore, even if our model is simplified, the criterion that we establish contains the same ingredients and follows the same behavior as in [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF]. However, adapting this strategy in the OA context, we will show that this criterion is too restrictive. Therefore, to further investigate this issue, we will secondly examine a specific stationary model on which we will highlight that the non-uniqueness of the solutions stems from the combination between the specific interface conditions and the viscosity profiles considered.

The remainder of this paper is organized as follows. We first construct the OA model in Section 2. Starting from the primitive equations, we briefly describe the usual assumptions considered to obtain the model in each domain and the interface conditions ( §2.1). Then we describe the viscosity profiles considered, which are representative of OA coupled models ( §2.2). We also provide representative values and viscosity profiles for numerical illustration. The global problem under investigation is summarized in §2.3. The study of the well-posedness is discussed in section 3. Our global strategy is to solve the nonlocal problem using a fixed-point method. We first recall the well-posedness properties on the local problem but with nonlinear interface condition ( §3.1). Thus, in line with the work of [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] and [Chacón Rebollo et al., 2014], we prove the existence of solutions for the stationary problem and give a criteria to have the existence of a non stationary solution in the neighborhood of this stationary solution ( §3.2). We finally give a well-posedness criterion on the viscosity profiles that ensure the uniqueness of the solution in the stationary and nonstationary state ( §3.3). In this section, we consider generic parameterized viscosity profiles and apply the result to the special case of OA coupling. Whereas the resolution of the problem using fixed-point method gives a sufficient condition for the well-posedness, a necessary and sufficient condition for the well-posedness of the stationary problem is given in section 4. We give this well-posedness criterion depending on the parameterization of viscosity profiles in §4.1, and its application to a simplified problem with no Coriolis effect is discussed. Finally we apply the well-posedness criterion on viscosity profiles that are representative of those used in the OA model and conclude on the non-uniqueness of the solution ( §4.2).

2 Construction of the ocean-atmosphere coupled model

A simplified 1D coupled model

To establish a coupled model, we start from the Navier Stokes equations on which we will make simplifying assumptions, while keeping the most important ingredients to obtain a relevant coupled model. The steps involved in building our model are the following.

In each domain:

1. Starting from the Navier-Stokes equations with density stratification in both oceanic and atmospheric domains, we make the following classical hypotheses to obtain the so-called primitive equations :

• Hydrostatic hypothesis and Boussinesq approximation (the variations of the fluid density are weak)

• We consider the earth rotation, represented by the Coriolis force This leads to

B t u h `f k ˆuh ´νm ∆u h " ∇ h p ρ 0 ´∇ ¨pU b U h q B z p " ´gρ 1 (hydrostatic balance) ν m ∇ ¨uh " 0 (incompressibility) BtΦ " F Φ ´∇h ¨pu h Φq ρ " ρ eos pΦ, zq (1)
where U " pu h , wq represents the speed (wind or current), p is the pressure, ρ the density provided by an equation of state ρ eos , µ m the molecular viscosity and Φ is a general symbol for tracers (salinity, temperature...). The well-posedness of these primitive equations are widely studied, some first studies related to ocean and atmosphere model can be found in [START_REF] Lions | Models of the coupled atmosphere and ocean (cao i). i[END_REF]]- [Lions et al., 1993a], or [START_REF] Cao | Local and global well-posedness of strong solutions to the 3d primitive equations with vertical eddy diffusivity[END_REF] and we refer to [START_REF] Petcu | Some mathematical problems in geophysical fluid dynamics[END_REF] for a recent review of existing results.

2. In order to take into account the fine-scale dynamics that is not resolved by the numerical grid, equations (1) must be supplemented by "sub-grid" parameterization schemes. The classical approach to introduce these parameterizations consists in using the so-called Reynolds decomposition of each variable ϕ into a "resolved" averaged component ϕ and an "unresolved" component ϕ 1 , with ϕ 1 " 0. Using this decomposition in (1) gives for the first line

B t u h `f e z ˆuh ´νm ∆u h ´∇h ¨u1 h u 1 h ´Bz w 1 u 1 h " ∇ h p ρ 0 (2)
where terms of the form ϕ 1 φ 1 represent the effect of unresolved scales on the resolved scales. To close the system, we use a turbulent closure considering the Boussinesq hypothesis which gives w 1 φ 1 in terms of the known resolved-scale variables as:

u 1 φ 1 " ´pν t,x B x φ, ν t,y B y φ, ν t,z B z φq
T where ν t are turbulent viscosities (a.k.a. eddy-viscosity) depending on space and time and potentially other parameters. These turbulent viscosities are parameterized using different closure schemes that we will discuss in paragraph 2.2. Note ν t is strongly non-isotropic between horizontal direction and vertical direction. From fluid mechanics notations, we introduce the constraint tensor:

σ " ´pI 3 `ρ0 ¨pν t x `νm qB x u pν t x `νm qB x v 0 pν t y `νm qB y u pν t y `νm qB y v 0 pν t z `νm qB z u pν t z `νm qB z v 0 ' ( 3
)
with I 3 the identity matrix.

For more details on approximations and closure assumptions taken so far, we can refer to [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]. These assumptions are common in oceanic and atmospheric models used for climate simulations (see for example [START_REF] Madec | Nemo ocean engine reference manual[END_REF]).

At the interface The interface conditions between the ocean and the atmosphere are complicated, due to the complexity of the natural phenomena they describe. Very close to the interface, dedicated parameterizations are applied and superposed to the numerical models. These interface parameterizations rely on the [START_REF] Monin | Basic laws of turbulent mixing in the surface layer of the atmosphere[END_REF] (MO) theory that assumes constant vertical fluxes and a wall-law in this near-interface zone. The numerical counterparts to MO theory are the so-called bulk formulations, see [START_REF] Pelletier | Twosided turbulent surface-layer parameterizations for computing air-sea fluxes[END_REF] for more details. In order to formulate interface conditions consistently, we separate the near-interface zone whose flow is governed by MO theory from the rest of the domain where the primitive equations are considered. The altitude that limit the near-interface to the boundary layer is defined by δ o in the ocean and δ a with δ o ă 0 ă δ a . We assume pp α , u α q are parameterized in rδ o , δ a s for α P to, au. The ocean and atmosphere domain are defined as Ω o " rz 8 o , δ o r and Ω a "sδ a , z 8 a s. In the following, we consider the interface between the ocean and the atmosphere as a buffer zone rδ o , δ a s where MO theory applies. The interface condition at the ocean surface are given by the continuity of the constraint σ and the continuity of u. Applying the MO theory in rδ o , δ a s (which takes the form of a non-linear friction law see [START_REF] Pelletier | Twosided turbulent surface-layer parameterizations for computing air-sea fluxes[END_REF] for a detailled description), leads to the following conditions :

ρ o pν m o `νt,o pδ o , tqq B z u h,o pδ o , tq " ρ a pν m a `νt,a pδ a , tqq B z u h,a pδ a , tq (4a)
pν m a `νt,a pδ a , tqq B z u h,a pδ a , tq " C D pu ˚q }u h,a pδ a , tq ´uh,o pδ o , tq} pu h,a pδ a , tq ´uh,o pδ o , tqq (4b)
u ˚" a C D pu ˚q }u h,o pδ a , tq ´uh,o pδ o , tq} (4c)
with ν m a (resp. ν m o ) the molecular viscosity in the atmosphere (resp. in the ocean) at and u ˚the friction velocity. The coefficient C D is given by the MO theory. This type of non-linear interface condition based on friction laws is widely studied in fluid-structure interactions theory (see for example [START_REF] Bresch | Global weak solutions to a generic two-fluid model[END_REF]).

Simplifying assumptions To reduce the complexity of the problem, we make the following assumptions:

• In the buffer zone sδ o , δ a r, equations are parameterized. These parameterizations are taken into account in ? C D , that depends on u ˚itself. However the role of C D is minor in our context and can be considered as constant. According to [START_REF] Large | Open ocean momentum flux measurements in moderate to strong winds[END_REF], we set C D " 1.2 ˆ10 ´3.

• We make an assumption of horizontal homogeneity, justified by the fact that in this study we are focusing on exchanges that are predominantly in the vertical direction. Therefore the terms in B x ' and B y ' are neglected, with the exception of the horizontal pressure gradient.

• It is assumed that the geostrophic winds/currents, noted u g , are known and are defined by the equilibrium

´f u g α " 1 ρ 0,α B y p α f v g α " 1 ρ 0,α B x p α
with α P to, au (α " o in the ocean, α " a in the atmosphere). This assumption allows us to decouple the different variables and to consider a condition at the outer edge of the media.

u α pz 8 α q " u g pz 8 α q (5)

Finally our model in each domain can be written as:

B t u h `f e z ˆuh ´νm ∆u h ´Bz pν t pz, u h , ...qB z u h q " f e z ˆug on pδ α , z 8 α q (6)

In [START_REF] Klein | Multiple scales asymptotics for atmospheric flows[END_REF] a rigorous derivation of the Ekman layer equation ( 6) is obtained from multiple scales asymptotic technique. Their derivation indicates that such model is relevant to describe the evolution of atmospheric horizontal velocities at large scale. From now on, the overbar notation ¨and ¨h are neglected.

Definition 2.1 Notation : Equation ( 6) is considered both in the oceanic and the atmospheric domains, denoted Ω α with α P to, au and the notation pa, bq is defined by: pa, bq :"

" sa, bs if a ă b rb, ar if a ą b
Also we define a notation for the "jump" of a variable in the vicinity of the interface:

u δa δo :" u a pδ a , tq ´uo pδ o , tq

Viscosity profiles and reference values for ocean-atmosphere coupling

In this study, we consider viscosity profiles based on the parameterizations of [START_REF] Troen | A simple model of the atmospheric boundary layer: Sensitivity to surface evaporation[END_REF]] and [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF] commonly used in ocean-atmosphere models and adapted to the Ekman layer problem by [START_REF] Mcwilliams | Ekman layer rectification[END_REF]].

In the following we will refer to the corresponding viscosity as K-Profile Parameterization (KPP) viscosity. There exists different turbulent closure schemes with different degrees of complexity [Chacón Rebollo et al., 2014]. We focus here on a closure scheme based on a so-called zeroth-order closure i.e ν is directly diagnosed from u ˚and z and does not involve additional evolution equations as is the case for parameterizations based on the turbulent kinetic energy (TKE) via the Prandtl-Kolmogorov relation. In the stationary case a TKE-based viscosity profile would depend locally on the wind shear. A mathematical analysis of a model close to (1)+( 4) is made in the stationary case by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] and [START_REF] Rebollo | An iterative procedure to solve a coupled two-fluids turbulence model[END_REF], with a TKE viscosity profile. They prove the existence of a solution and highlight issues that occur to prove the uniqueness of such solution. In this study we have a similar objective but with a KPP viscosity profile and show that the same global uniqueness issues are encountered.

Definition 2.2 The KPP viscosity profile : The KPP viscosity profile is built to be consistent with the MO theory near the interface and to connect continuously with the constant molecular viscosity outside the boundary layer [O'Brien, 1970]. ν P C1 pδ α , h α q only depends on u ˚and z, and is such that

ν α pu ˚, zq " νpu α, zq " $ & % ν m α on ph α , z 8 α q D α pu ˚, zq H ˆ1 ´z h α ˙`ν m α on pδ α , h α q (7)
with D α ě 0 for all z P pδ α , z 8 α q, H is the Heaviside function, h α depending on u ˚and ν m α the molecular viscosity. It must also satisfy

• consistency with the MO theory : D α pδq « κu αδ " ν m α with κ the Von Karman constant and B z D α pδq « κu α. with u o " λu å " λu ˚1.

• order of magnitude assumptions : |z 8 α | " |δ α | and D α pzq " ν m α for all z P pδ α , h α p1 ´ϵqq for an ϵ ! 1. Also Arya, 1981]) and r c o " ´0.7λ [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF].

-ν m o " λν m a with λ " a ρ o {ρ a « 0.03 -h α " c α u ˚with c α " r c α {|f | and r c α taken from r c a " 0.2 ([
-|f | « 5 ˆ10 ´5 s ´1 -u ˚P r10 ´3, 1r ms ´1
The range of values for u ˚is the one considered in [START_REF] Pelletier | Twosided turbulent surface-layer parameterizations for computing air-sea fluxes[END_REF]] and corresponds to "classic" values for this parameter in OA models.

Definition 2.3 Reference values for the ocean-atmosphere context : We consider a KPP viscosity profile specific to the application of OA coupling as given by [O'Brien, 1970]:

D α pu α, zq " κu α|z| ˆ1 ´z h α ˙2 (8)
To apply our results to the specific OA framework, we choose a number of fixed parameters, which we will call reference values for ocean-atmosphere coupling :

• z 8 a " 3000m and z 8 o " ´500m • δ a " 10m and δ o " 1m

• ν m a " 15.6 ˆ10 ´6m 2 s ´1 and ν m o " 5 ˆ10 ´7m 2 s ´1

• u ˚is taked such that δ α ă h α ă z 8 α for α P to, au i.e u ˚Ps3.24 ˆ10 ´3, 9.75 ˆ10 ´1r ms ´1.

Model problem

The coupled OA model studied is this paper is: or ν a B z u a pδ a q " pu ˚q2 e τ with e τ the unit vector in the direction of pu a pδ a , tq ´uo pδ o , tqq. This last condition corresponds more closely to the conditions given in realistic models, but in the context of our mathematical analysis it would provide an equivalent problem.

B t u α `Bu α ´Bz pν α pz, u ˚ptqqu α q " Bu g α on pδ α , z 8 α qˆs0, T r (9a) u α pz 8 α ,
3 Well-posedness depending on viscosity profile

Our OA coupled problem has some specific features which we separate to better identify the difficulties they induce 1. coupling model: we consider two (ocean and atmosphere) models on two distinct domain.

nonlinearity: the interface condition

ν α B z u " C D › › › u δa δo › › › u δa δo is nonlinear.
3. nonlocality: considering viscosity profiles depending on u ˚means that the global solution depends on the local value of the solution at the interface u α pδ α q.

Specificities 2 and 3 can be found in a single domain problem e.g. the atmospheric problem alone (with a given u o pδ o q condition). Since most of the difficulties encountered in coupled problems are found in the single domain case, considering the single domain problem does not make it any easier to tackle the well-posedness issues. The coupled case will therefore always be considered in this section. We will first prove the well-posedness of the coupled local version of our model (points 1+2) in §3.1. A close version of (9) has been studied by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] in the stationary state and without Coriolis effect. The authors prove the existence of solutions for TKE viscosity profiles, but the uniqueness of the solution is obtained under some restrictive conditions. In §3.2, we adapt the method from [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] to prove the existence of the full stationary problem (1+2+3). We then study the existence of non stationary solutions in a neighborhood of the stationary state using the method proposed by [Chacón Rebollo et al., 2014]. The well-posedness (existence and uniqueness) of the full problem (1+2+3) is studied in §3.3 for the stationary and non stationary cases. We propose to study the problem using a fixed-point formulation and we give a criterion on the viscosity profiles that ensures the uniqueness of the solution.

Well-posedness of the local problem

Assuming that u ˚is given and fixed, a local version of our model is obtained:

B t u α `Bu α ´Bz pν α pz, u ˚qB z uq " Bu g α on pδ α , z 8 α qˆs0, T r (10a) u α pz 8 α , tq " u g α | z 8 on s0, T r (10b) u α pz, t " 0q " u g α | t"0 on rz 8 o , z 8 a s (10c) ν o B z u o pδ o , tq " λ 2 ν a B z u a pδ a , tq on s0, T r (10d) ν a B z u a pδ a , tq " C D › › › uptq δa δo › › › uptq δa δo on s0, T r (10e)
We will now detail a proof of its well-posedness using similar techniques as [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] in the stationary case and [START_REF] Lions | Mathematical theory for the coupled atmosphereocean models (cao-iii)[END_REF] in the non-stationary case.

Definition 3.1 Notation : To simplify the writing in this section, we introduce the notation

ν Y x " › › ? νB x y a › › 2 L 2 pΩaq `λ´2 › › ? νB x y o › › 2 L 2 pΩoq
An example of such a notation is given by ν U z in (13).

Property 3.1 Well-posedness of the stationary version of system (10) : Let us suppose that ν α P C 1 pΩ α q, 0 ă ν m α ď ν α for all z P Ω α , and u g P H 2 pΩq. Then we have a unique solution u P H 2 pΩq of the stationary version of (10) with Ω " Ω o Y Ω a and u α " u| Ωα .

Proof. We first consider u r P H 2 pΩq satisfying the linear interface conditions u r α pz 8 α q " u g α | z 8 , B z u r o pδ o q " λ 2 B z u r a pδ a q and u r a pδ a q " u r o pδ o q (this choice is justified to simplify the nonlinear part in the condition (10e)). Then using the change of variable u α " r u α `ur α , the study of the well-posedness is conducted on the problem:

Br u α ´Bz pν α pz, u ˚qB z r u r α q " B z pν α pz, u ˚qB z u r α q `B pu g ´ur α q on pδ α , z 8 α q (11a) r u α pz 8 α q " 0 (11b) ν o B z r u o pδ o , tq " λ 2 ν a B z r u a pδ a q (11c) ν a B z r u a pδ a q `νa B z u r a pδ a q " C D › › › r u δa δo › › › r u δa δo (11d) Let us denote V α :" ␣ v P H 1 pΩ α q, vpz 8 α q " 0 ( and V :" V a X V o .
We first consider the equation (11a) on the domain Ω a . Multiplying by a test function v T a P V and integrating on Ω a leads to the formulation:

ż z 8 a δa v T a Br u a ´ż z 8 a δa v T a B z ν a B z r u a " ż z 8 a δa v T a B z pν α B z u r α q `ż z 8 a δa v T a Bg a ż z 8 a δa v T a Br u a `ż z 8 a δa B z v T a ν a B z r u a `va pδ a q T C D › › › r u δa δo › › › r u δa δo " ż z 8 a δa B z v T a ν α B z u r α `ż z 8 a δa v T a Bg a
with g a " u g a ´ur a P H 2 pΩ a q. The same stands for the ocean domain. Then multiplying the formulation from the oceanic part by λ ´2 and adding the formulation from the atmospheric part, we get the weak formulation of problem (10): Find a unique solution r u P V such that

f `@v, r u K D a `λ´2 @ B z v, r u K D o ˘`xB z v, ν a B z r uy a `λ´2 xv, ν o B z r uy o `CD › › › r u δa δo › › › v δa δo ¨ r u δa δo " @ ? ν a B z v, ? ν a B z u r D a `λ´2 @ ? ν o B z v, ? ν o B z u r D o ´f `@v, g K D a `λ´2 @ v, g K D o ˘@v P V (12)
with pu, vq K " p´v, uq and where x¨, ¨yα is the scalar product in Ω α . Choosing v " r u gives:

ν r U z `CD › › › r u δa δo › › › 3 " x ? ν a B z r u, ? ν a B z u r y a `λ´2 x ? ν o B z r u, ? ν o B z u r y o ´f `@r u, g K D a `λ´2 @ r u, g K D o ˘(13) with ν r U z " } ? νB x r u a } 2 L 2 pΩaq `λ´2 } ? νB x r u o } 2 L 2 pΩoq .
We bound the right-hand side using

ˇˇx ? ν a B z u, ? ν a B z u r y a `λ´2 x ? ν o B z u, ? ν o B z u r y o ˇˇď β 2 ν r U z `1 2β ν U r z
for every β ą 0 and where the notation ν U r z is defined by definition 3.1. Using Poincaré's inequality on

u α such that }u α } 2 2 ď r C 2 α }B z u α } 2 2 , we have ˇˇf `@r u, g K D a `λ´2 @ r u, g K D o ˘ˇď |f | 2 ˆβ r U `1 β G ˙ď |f | 2 ˆβ r C ν r U z `1 β G ẇith r C " maxp r C 2 a , r C 2 o q{ minpν m o , ν m a q.
The well-posedness of the weak formulation ( 12) in V is given in appendix 6.1, as a variant of the proof given by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] on a similar problem. To obtain more regularity, since we have

› › › r u δa δo › › › ď C ν r
U z and using classical method (see for exemple [Evans, 2022]), we obtain the weak solution u in H 2 pΩq .

Property 3.2 Well-posedness of system (10) : We suppose a given ν α P C 1 pr0, T s, Ω α q such that 0 ă ν m α ă ν α pz, u ˚ptqq for all pz, tq P Ω α ˆr0, T s. Then, for u g P L 2 `0, T ; H 2 pΩq ˘X H 1 `0, T ; L 2 pΩq ˘, we have a unique solution u P L 2 `0, T ; H 2 pΩq ˘X H 1 `0, T ; L 2 pΩq ˘X L 8 p0, T, H 1 pΩqq of system (10) and with

› › › u δa δo › › › P Cpr0, T sq. Note also that the condition u g P L 2 p0, T ; H 3 pΩqq X H 2 p0, T ; L 2 pΩqq would imply B t › › › u δa δo › › › P Cpr0, T sq.
Proof. Let us first consider u r P L 2 `0, T ; H 2 pΩq ˘X H 1 `0, T ; L 2 pΩq ˘satisfying the linear interface conditions (10b)-(10d) and u r a pδ a q " u r o pδ o q, Then using the change of variable u α " r u α `ur α the study of the wellposedness is made on the problem:

B t r u α ptq `r Br u α ´Bz pν α pzq, u ˚q " B z pν α pz, u ˚qB z u r α pz, tqq `gα pz, tq on pδ α , z 8 α qˆs0, T r (14a) r u α pz 8 α , tq " 0 on s0, T r (14b) r u α pz, t " 0q " u 0 on Ω o Y Ω a (14c) ν o B z r u o pδ o , tq " λ 2 ν a B z r u a pδ a , tq sur s0, T r (14d) ν a B z r u a pδ a , tq `νa B z u r a pδ a , tq " C D › › › r u δa δo › › › r u δa δo on s0, T r (14e)
with g α " Bpu g α ´ur α q ´Bt u r α . Using the same method than for the stationary case, the weak formulation of the problem reads: Find

r u P L 2 p0, T ; Vq such that xv, B t r uy a `λ´2 xv, B t r uy o `f `@v, r u K D a `λ´2 @ v, r u K D o ˘`xv, νB z r uy a `λ´2 xv, νB z r uy o `CD › › › r u δa δo › › › v δa δo ¨ r u δa δo " xv, νB z r uy a `λ´2 xv, νB z r uy o `xv, gy a `λ´2 xv, gy o @v P V (15)
Let us suppose r u P V is a solution of the weak formulation. Then, using the same bound on the right-hand terms than in the stationary case:

1 2 B t r U `ˆ1 ´β 2 p1 `|f |Cq ˙ν r U z `CD › › › r u δa δo › › › 3 ď 1 2β p ν U r z `|f |Gq 1 2 r U `ˆ1 ´β 2 p1 `|f |Cq ˙ż t 0 ν r U z `CD ż t 0 › › › r u δa δo › › › 3 ď U 0 `1 2β ż t 0 p ν U r z `|f |Gq
for all β ą 0. Using Galerkin method, we can prove that there exists a unique solution of the weak formulation in u P Cpr0, T s; L 2 pΩqq X L 2 p0, T, H 1 pΩqq and upt " 0q " u 0 . The proof given in appendix 6.1 is a variant of the proof by [START_REF] Lions | Mathematical theory for the coupled atmosphereocean models (cao-iii)[END_REF] on an close problem. Moreover, since we are in one dimension, we also have

› › › u δa δo
› › › P Cpr0, T sq. For higher regularity, since the nonlinear term appears only in the boundary term, we can use a classical approach (see for example [Evans, 2022]). Then with the assumption ν α ptq ą ν m α ą 0 and u g P L 2 p0, T ; L 2 pΩqq and u g | t"0 P H 1 pΩq, we have u P L 2 p0, T ; H 2 pΩqq X L 8 p0, T ; H 1 pΩqq X H 1 p0, T ; L 2 pΩqq. Moreover, taking higher regularity such that u g | t"0 P H 2 pΩq and u r , u g P H 1 p0, T, H 1 pΩqq we obtain in particular B t u P L 8 p0, T ; L 2 pΩqq X L 2 p0, T ; H 1 pΩqq and B 2 t u P L 2 p0, T ; H ´1pΩqq that gives B t u δa δo P Cr0, T s. In §3.2 and §3.3, we will illustrate results that depend on a L 8 bound of the solution. This is why we introduce the following property: Property 3.3 A precise upper bound depending on u g : Suppose u g is constant in time 2 , and u g P L 2 `0, T ; H 2 pΩq ˘X H 1 `0, T ; L 2 pΩq ˘a solution of (10) for ν α P Cp0, T ; C 1 pΩ α qq and B t ν α P L 8 p0, T ; L 8 pΩ α qq with α P to, au we have bound on ν U z P L 8 pr0, T sq depending only on the norm of u g and ν α :

• In the stationary case, a solution u e satisfies ν U e z ď 8 27 C D › › › u g δa δo › › › 3 `νU g,e z :" M e (16)
• in the non stationary case, a solution u satisfies

ν U z ptq ď a C D ´› › › u g δa δo › › › ¯3 `Me T N N :" max αPto,au › › › › B t ν α ν α › › › › L 8 p0,T ;L 8 pΩaqq (17)
Note that the upper bound depends on the final time here, because u g is constant in time, so that u g P L 2 p0, T ; H 1 pΩqq for finite time. Other upper bounds independent of the final time can be found in the general case.

Proof. see appendix

Existence of solutions in the neighborhood of the stationary state

The proof of existence of the stationary state follows the work by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF]. The existence of the nonstationary state in the neighborhood of such existing stationary solutions has been studied by [Chacón Rebollo et al., 2014] in the single domain case, without nonlinear interface conditions and considering TKE viscosity profiles. This method has been applied to the local coupled problem with nonlinear interface conditions but with constant viscosity (points 1+2) by [START_REF] Clement | Semi-discrete analysis of a simplified air-sea coupling problem with nonlinear coupling conditions[END_REF] on a discrete in space problem. In property 3.5 we extend these results to the nonlocal coupled problem (i.e. considering specificities 1+2+3).

Property 3.4 Existence of solution for the stationary state : Under the regularity hypotheses of property 3.1, there exists at least one couple pu e , u e q P H 2 pΩq ˆR`s olution of the stationary state of (9).

Proof. We first prove the existence a solution of the weak formulation given in property 3.1 with the additional constraint u ˚" ?

C D › › › u δa δo › › ›. Let us note Ψ m : V m ˆR Ñ V m ˆR given by Ψ m pu m , u ˚q " pΦ m,u ˚pu m q, u ˚? C D › › › u m δa δo › › ›q and such that Ψ m pu m , u ˚q ¨pv m , v ˚q " Φ m,u ˚pu m q ¨pv m q `v˚p u ˚´? C D › › › u m δa δo › ›
›qq with Φ m,u ˚is Φ m given in appendix 6.1 with ν α " ν α pu ˚q. Then using same bounding than in properties 3.1 and using

› › › u m δa δo › › › 2 ď C ν pU m q z , we have: Ψ m pu m , u ˚q ¨pu m , u ˚q ě b ν pU m q z ´b ν pU m q z ´a ν U r z ´|f | ? C ? G ¯`C D › › › u m δa δo › › › 3 `pu ˚q2 ´u˚a C D › › › u m δa δo › › › ě b ν pU m q z ´b ν pU m q z ´a ν U r z ´|f | ? C ? G ¯´u ˚aC D C b ν pU m q z ě b ν pU m q z ´minpν m a , λ ´2ν m o q ´}B z u a,m } 2 `}B z u o,m } 2 ¯´a ν U r z ´|f | ? C ? G ´u˚a C D C
2This assumption is made to be coherent with the ocean-atmosphere context, other bound can be found in the more general case.

Let us define a value u max P R and take u ˚" u max and any u such that

´}B z u a,m } 2 `}B z u o,m } 2 ¯" }νpu max q} L 8 a U r z `|f | ? C ? G `um ax ? C D C
, then Ψ m pu, u max q ¨pu, u max q ě 0. Thus there exists a solution Ψ m pu m , u mq " 0 in V m ˆr0, u max r and such that u m follows the condition minpν m a , λ

´2ν m o q ´}B z u a,m } 2 `}B z u o,m } 2 ¯ď }νpu max q} L 8 a U r z `|f | ? C ? G`u max ? C D C.
Since ν P L 8 pΩq, same arguments than in the property 3.1 hold for the convergence of u m to u as well as for its regularity, and we have a solution of the weak formulation.

Property 3.5 Existence of (9) in the neighborhood of the stationary state : If there exists a solution pu e , u e q to the stationary state of (9), we can ensure the existence of a solution of ( 9) in an neighborhood of pu e , u e q if the viscosity νpz, u e q satisfies pu e q 2 N 2 δ

`?C D u e M 1 ă 2 (18)
with N δ " ν 1 α pu e , δ α q{ν α pu e , δ α q, ν 1 pzq " B u ˚ν pδ, u e q and M 1 is an upper bound of

› › › ν 1 a ? νa B z u e a › › › 2 2 `λ´2 › › › ν 1 o ? νo B z u e o › › › 2 2 for
all u e solution of the stationary version of the local problem ( 10). An upper bound of

M 1 is max ´}N a } 2 L 8 pΩq , }N o } 2 L 8 pΩq ¯Me
with M e given by ( 16).

Proof. The proof is based on the method developed by [Chacón Rebollo et al., 2014], where the authors consider TKE-type viscosity profiles which bring nonlinearity in the main equations. We adapt this method to our nonlocal problem and briefly recall the steps:

1. Suppose a stationary solution pu e , u e q P H 2 pΩq for a source term u e,g

Define

Ψpu, u ˚q "

# B t u α `Bpu α ´ue,g α q ´Bz pνpzqB z u α q, u α pt " 0q ´ue α , u α pz 8 α q ´ue,g α pz 8 α q λ 2 ν a B z u a pδ a q ´νo B z u o pδ o q, ν a pδ a qB z u a pδ a q ´CD › › › u δa δo › › › u δa δo , u ˚´? C D › › › u δa δo › › › +
then Ψ is continuous from X :" L 2 `0, T ; H 2 pΩq ˘XH 1 `0, T ; L 2 pΩq ˘ˆL 2 p0, T q to Y :" L 2 `0, T, L 2 pΩq ˘, ˆH1 pΩqL 2 p0, T q 4 and differentiable, DΨpu e , u e q is continuous from X to Y

Prove that

DΨpu e , u e qpv, v ˚q "

$ ' ' ' ' & ' ' ' ' %
B t v α `Bpv α q ´Bz pν α pz, u e qB z v α `v˚ν1 α pz, u e qB z u e α q, v α pt " 0q, v α pz 8 α q, λ 2 ν a B z u a pδ a q ´νo B z u o pδ o q `v˚p ν 1 a pu e qB z u e a pδ a q ´λ2 ν 1 o pu e qB z v e o pδ o qq, ν a pδ a qB z u a pδ a q `v˚ν1 a pu e qB z u e a pδ a q ´?C

D u e ´´ v δa δo ¨eτ ¯eτ ` v δa δo ¯, v ˚´? C D › › › v δa δo › › › ¨eτ , / / / / . / / / / - with u τ " u e δa δo { › › › u e δa δo › ›
› and ν 1 " B u ˚ν pu ˚, z, B z u e q, is continuous and invertible from X to Y 4. Use the inverse theorem to obtain the existence of the solution around pu e , u e q.

Finally, our work is to show that the differential of Ψ is continuous and invertible from X to Y. To ensure the continuity of DΨ we have to suppose N α P L 8 pΩq. To show the invertibility of DΨ, we have to prove that the linear model

$ ' ' ' ' ' & ' ' ' ' ' % B t v α `Bv α ´Bz pνpu e qB z vq " B z pv ˚pB u ˚να pu e , zqB z u e qq `Φ on pδ α , z 8 α qˆs0, T r v α pz 8 α , tq " Φ 8 on s0, T r v α pz, t " 0q " Φ 0 on Ω o Y Ω a ν o B z v o pδ o , tq `v˚ν1
o pu e qB z u e o pδ o q " λ 2 ν a B z v a pδ a , tqq `λ2 v ˚pν 1 a pu e qB z u e a pδ a q `ΦI,1 on s0, T r ν a B z v a pδ a , tq `v˚ν1 a pu e qB z u e a pδ a q "

? C D u e ´´ v δa δo ¨eτ ¯eτ ` v δa δo ¯`Φ I,2 on s0, T r (19a) v ˚" a C D v δa δo ¨eτ `Φ˚( 19b)
is well-posed on X for all Y " pΦ, Φ 8 , Φ 0 , Φ I,1 , Φ I,2 , Φ ˚q P Y. We use a fixed point problem to treat the nonlocality. For a given stationary solution pu e , u e q of (9), we want to show that P : L 2 p0, T q Ñ L 2 p0, T q

given by P pv ˚q " ? C D v δa δo ¨eτ with v solution of (19a) is a contraction mapping. Consider u ˚, v ˚P L 2 p0, T q and u, v the corresponding solution of (19a), we pose w ˚" u ˚´v ˚and w " u ´v. Then, by linearity, pw ˚, wq is solution of (19a) with Y " 0. Thus an a priori estimate can be computed:

B t W `νW z `aC D u e › › › w δa δo › › › 2 `cospθq 2 `1˘" ´w˚A 1 ´w˚A 2 A 1 " @ B z w a , ν 1 a B z u e a D `λ´2 @ B z w o , ν 1 o B z u e o D A 2 " w T a ν 1 a B z u e pδ a q ´λ´2 w T o ν 1 o B z u e o pδ o q
with ν 1 pzq " B u ˚ν pz, u e q. We have the following bound A 2 " pu e q 2 pN a pδ a qw a pδ a q ´No pδ o qw o pδ o qq ¨eτ N α pzq :" ν 1 α pzq{ν α pzq

To simplify, we suppose N a pδ a q " N o pδ o q " N δ and

|w ˚A2 | ď |w ˚|pu e q 2 › › › w δa δo › › › N δ ď 1 2 a C D u e › › › w δa δo › › › 2 `1 2 b C ´1 D pu e q 3 |w ˚|2 N 2 δ
The term in A 1 can be bounded using same procedure and

|w ˚A1 | ď ν W z `|w ˚|2 4 ˆ}N a ? ν a B z u e a } 2 2 `1 λ 2 }N o ? ν o B z u e o } 2 2 ˙ď ν W z `|w ˚|2 4 N 2 ν U e z
Injecting these bounds in the a priori estimate

B t W `pcospθq 2 `1 2 q a C D u e › › › w δa δo › › › 2 ď |w ˚|2 ˆpu e q 3 N 2 δ 4 ? C D `M1 4
İntegrating in p0, tq leads to the final bound:

ż t 0 C D › › › w δa δo › › › 2 ď 1 2 ˆpu e q 2 N 2 δ `?C D u e M 1 ˙ż t 0 |w ˚|2 (20)
This first result highlights the difficulties associated with the nonlocal nature of the equation. Indeed the wellposedness of the linearized but nonlocal problem strongly depends on the ratio between the variations of the solution and those of the viscosity profile. We apply this result to the KPP profile defined by 2.2 to illustrate the problem it generates.

Application to ocean-atmosphere KPP profile With KPP viscosity profiles, we can approximate N δ " pu e q ´1 and we suppose, as an example, that u g,e is constant in space. Then using the upper bound given by ( 16), a criterion for the existence of a solution in the neighborhood of the stationary solution is:

1 `1 u e max αPto,au }N α pu e q} 2 L 8 pΩq ˆ2? C D 3 } u g,e a o } ˙3 ă 2 (21)
Considering KPP viscosity profiles given by definition 2.3, we compute numerically }N a pu e q} L 8 pΩq which turns out to be of the order of 10 4 (see figure 2). For a given u g,e and a corresponding coupled solution to the stationary case pu e , u e q, (21) is satisfied if } u g,e a o } ă 0.1 ˆpu e q 1{3 Ps10 ´2, 10 ´1rm.s ´1, that would be very small compared to the values used in the physical application. The profile of ν is not sufficiently smooth, especially near h α , to guarantee that P is a contraction mapping in the order of magnitude that interesting us. Note that the upperbound on M 1 can be computed in a different way from what is done here. In section 4, we give an exact computational bound M 1 but the regularity issue shown here is still relevant.

Well-posedness criterion on viscosity profiles

In this paragraph, we study the well-posedness (existence and uniqueness) of the coupled OA model in the stationary and nonstationary cases. The global strategy is to express the model as a fixed-point formulation (see [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] for a stationary problem and [START_REF] Rebollo | An iterative procedure to solve a coupled two-fluids turbulence model[END_REF] for a nonstationary discrete-in-time problem). Here we adopt an other point of view and derive a criterion on viscosity profile that ensures the well-posedness. We recall that the model can be split as:

$ ' ' ' ' ' & ' ' ' ' ' % B t u α `Bu α ´Bz pν α pz, u ˚qB z u α q " Bu g α on pδ α , z 8 α qˆs0, T r u α pz 8 α , tq " u g α | z 8 on s0, T r u α pz, t " 0q " u g α | t"0 on rz 8 o , z 8 a s ν o B z u o pδ o , tq " λ 2 ν a B z u a pδ a , tq on s0, T r ν a B z u a pδ a , tq " C D › › › uptq δa δo › › › pu a pδ a , tq ´uo pδ o , tqq on s0, T r (22a) u ˚ptq " a C D › › › uptq δa δo › › › (22b)
where ν only depends on time, through the parameter u ˚. Using a Banach fixed-point theorem on u ˚, we give a criterion on ν α that ensures the existence and uniqueness of a coupled solution pu ˚, uq.

Property 3.6 Well-posedness of the linear and nonlocal problem : We suppose ν α pz, u ˚q is C 1 for the two variables. Then for a given u g in the space given by properties 3.2, there exists u max pu g q ą 0 such that

› › › u δa δo › › › ă u max for any solution u of the local problem. Moreover if max zPΩα › › › › › νpz, u ˚q ´νpz, v ˚q a νpz, u ˚qνpz, v ˚q › › › › › ď L |u ˚´v ˚|3{2 0 ď L ď ˆbC 1{6 D M ˙´1 (23) with M " sup tPp0,T q ´}? ν a B z u a } 2 2 `λ´2 } ? ν o B z u o } 2 2
¯for all u solution of the local problem, then there exists a unique coupled solution of ( 22) such that

• in the stationary case: pu, u ˚q P H 2 pΩq with u ˚ď u max .

• in the nonstationary case: pu, u ˚q P L 2 `0, T ; H 2 pΩq ˘X H 1 `0, T ; L 2 pΩq ˘ˆC 1 r0, T s with u ˚ptq ď u max @t.

Since it is difficult to translate the previous conditions in terms of the norm of ν α , another more restrictive condition can be derived as

}B u ν α } L 8 pΩq ď ν m α ? C D CM (24)
with C " pC a `λ2 C o q with C α " ? 2|Ω α |{ν m α . An upper bound for such M can be found in property 3.3. Proof. The existence of such u max is given by the bound computed in appendix 6.2 and requires }ν α pu ˚q} P L 8 pΩq. We first prove the property in the stationary case. From property 3.1, if u ˚is known and fixed, then the problem (22a) is well-posed. So a solution of ( 22) is solution of the fixed-point problem u ˚" P pu ˚q where P :"

u ˚ÝÑ ? C D › › › u δa δo › › ›
where u is the stationary solution of (22a). We pose u ˚, v ˚ď u max and u the solution of (22a) obtain with ν " νpu ˚q and v the solution (22a) obtained with νpv ˚q ": µ. We define w :" u´v and w ˚:" u ˚´v ˚. Since u ˚, v ˚ď u max then ν, µ P L 8 pΩq. We want to find a condition to ensure that P is a contracting mapping, which means ?

C D ˇˇ› › › u δa δo › › › ´› › › v δa δo › › › ˇˇă |u ˚´v ˚| " |w ˚|.
Subtracting equations (22a) on pv, µq to those on pu, νq gives Bw ´Bz pνB z w `pνpzq ´µpzqqB z vq " 0 in pδ α , z 8 α q wpz 8 , tq " 0 wpt " 0q " 0 νB z w o pδ o q `pν o ´µo q B z v o pδ o q " λ 2 νB z w a pδ a q `λ2 pν a ´µa q B z v a pδ a q νB z w a pδ a q `pν a ´µa q B z v a pδ a q "

C D ´› › › u δa δo › › › u δa δo ´› › › v δa δo › › › v δa δo
Let η α " ν α ´µα ? ν α µ α for α P ta, ou. Then

|xB z w α , pν α ´µα q B z v α y| " |x ? ν α B z w α , η α pzq ? µ α B z v α y| ď }η} L 8 pΩαq } ? ν α B z w α } 2 } ? µB z v α } 2 ď ϵ 2 } ? ν α B z w α } 2 2 `1 2ϵ }η} 2 L 8 pΩαq } ? µB z v α } 2 2
As previously in §3.1, taking ϵ " 2, adding the inequality on the atmospheric part and the oceanic part multiplied by λ ´2 gives an a priori estimate

C D ´ u δa δo ´ v δa δo ¯¨´› › › u δa δo › › › u δa δo ´› › › v δa δo › › › v δa δo ¯ď 1 4 max ´}η a } 2 L 8 pΩαq , }η o } 2 L 8 pΩαq ¯µV z
It can be shown that, for all x, y P R 2 , we have |}x} ´}y}| 3 ď px ´yq ¨p}x} x ´}y} yq. Suppose that there exists L ą 0 such that }η α pzq} L 8 pΩαq ď L|w ˚|3{2 for all z P Ω. An upper bound in L 8 of µV z can be found using properties 3.3 and is noted M.

C 1{3 D ˇˇ› › › u δa δo › › › ´› › › v δa δo › › › ˇˇď L 2{3 M 1{3 |w ˚| Then P is a contraction mapping if C 1{6 D L 2
M ă 1 and we deduce (23). Another bound can be obtained taking ϵ " 1 that gives :

1 2 ν W z ď 1 2 max ´}η a } 2 L 8 pΩαq , }η o } 2 L 8 pΩαq ¯µV z
Suppose now there exists L 1 ą 0 such that }η α pzq} L 8 pΩαq ď L 1 |w ˚|. Then, using the trace theorem we would have

ˇˇ› › › u δa δo › › › ´› › › v δa δo › › › ˇˇ2 ď CpL 1 q 2 M|w ˚|2
Thus P is a contraction mapping if C D CpL 1 q 2 M ă 1. An upper bound for L 1 is obtained using }η α pzq} L 8 pΩαq ď }ν α pz, u ˚q ´να pz, v ˚q} L 8 pΩαq {ν m α and gives the conditions (24). For the nonstationary case, we consider V ˚:" tu P C 1 pr0, T squ and P :"

# V ˚ÝÑ V ů˚Ý Ñ ? C D › › › u δa δo › › ›
where u is the solution of (22a)

Using the same notations than in the stationary case, from property 3.

2 if u ˚, v ˚P V ˚then › › › u δa δo › › › , › › › v δa δo › › › P V ˚.
Using the same computation and integrating in p0, tq, then P is a contracting mapping in L 3 r0, T s if (23) is verify. The second inequality lies for P a contracting mapping in L 2 pr0, T sq.

Generally speaking, the condition given above on the regularity of ν depends on the inverse of M(which is an upperbound of ν U z ). This result is consistent with the one given by [START_REF] Bernardi | A model for two coupled turbulent fluids. part i: analysis of the system[END_REF] where the uniqueness of the solution is obtained for }u} H 1 0 pΩq }B u ν α } L 8 ď ν m α . This regularity is very restrictive and cannot be expected in the context of OA coupling. Therefore the fixed-point method as it is proposed here does not seem to be well adapted to fully answer the question of the well-posedness of the coupled problem.

Application of property 3.6 to KPP viscosity profiles As an example, suppose u g is constant in time and space. First, to ensure that }B u ν α } L 8 pΩq exists, let impose u max " minpz 8 α {c α q αPto,au « 1. Then, source terms considered to have P pV ˚q ď u max must satisfy: } u g a o } ď ? C D ´1 « 30ms ´1, which is consistent with the order of magnitude expected in the OA context. Now, using upper bound given in property 3.3 a criterion on ν α to ensure the well-posedness of the stationary case of (10) is

}B u ν α } 8 ă ? 27ν m α C D a 8pC a `λC o q › › › u g δa δo › › › ´3{2 « 2.5 ˆ10 ´6 › › › u g δa δo › › › ´3{2 (25)
For particular viscosity profiles as given in definition 2.2, we can prove that

}B u ν α } 8 " 2κ|h α |{3 ? 3 ď 2κ|z 8 α |{3 ? 3.
Then the obtained bound is approximately

› › › u g δa δo › › › ď 10 ´6m.s ´1.
The small size of the interval on

› › › u g δa δo › ›
› is due to the strong variation of ν especially close to h α . In the nonstationary case a criterion on ν α to ensure the wellposedness of ( 22) is of the same order of magnitude or smaller. In the context of OA coupling, order of magnitude expected to › › › u g δa δo › › › are approximately 10 ms ´1. Order of magnitude on › › › u g δa δo › › › to ensure well-posedness of ( 22) are too restrictive and we see the limit of this fixed point method applied on the OA framework.

4 Well-posedness of the stationary problem for KPP viscosity profiles In §3.2, the existence of a solution of (9) in the stationary case has been proved in a general parametrized viscosity profile νpz, u ˚q. However, in §3.3 the uniqueness of the solution is proved only for very smooth profiles of ν.

In this section we discuss the existence and uniqueness of a solution for viscosity profiles used in realistic OA coupled models as given by definition 2.2. To do so, we seek to explicit these stationary solutions considering constant in time and space geostrophic wind/current. Consideration of KPP viscosities leads us to consider values of u ˚P I ˚for which viscosities have a physical meaning. Note that we will rewrite the interface condition as ν a pδ a , u ˚qB z u a pδ, tq " ? C D u ˚ u δa δo . The obtained system is equivalent to (9). Followings results will also hold to the original system (9) since the criterion given here is a necessary and sufficient condition for the well-posedness.

In the following, we will use the following change of variable to reduce the system (9) to a complex-variable system :

Φ α " 1 ? 2 ˆ1 i 1 ´i˙ˆu α v α ˙" ˆΦα,1 Φ α,2
Using this change of variable in ( 9) gives

ˆif 0 0 ´if ˙Φα ´Bz pν α pz, u ˚q B z u α q " ˆif 0 0 ´if ˙Φg α on pδ α , z 8 α q (26a) Φ α pz 8 α , tq " Φ g α (26b) ν o B z Φ o pδ o , tq " λ 2 ν a Φ a pδ a , tq (26c)
ν δ a B z Φ a pδ a , tq " u ˚aC D pΦ a pδ a , tq ´Φo pδ o , tqq (26d) u ˚" a C D › › Φ δ a ´Φδ o › › (26e) with }Φ} " b Φ T Φ.
Note that the change of variable conserves the norm i.e. }u} 2 " }Φ} 2 . We have a pair of equation systems that can be translated as ϕ α,1 " ϕ α,2 and }u} " }Φ} " ? 2|ϕ|. Finally, we obtain the system

$ ' ' & ' ' % if ϕ α ´Bz pν α pu ˚, zqB z ϕ α pzqq " if ϕ g α
on pδ α , z 8 α q ϕ α pz 8 α q " ϕ g α pz 8 α q ν o pu ˚, δ o q B z ϕ o pδ o q " λ 2 ν a pu ˚, δ a q B z ϕ a pδ a q ν a pu ˚, δ a q B z ϕ a pδ a q " ? C D u ˚pϕ a pδ a q ´ϕo pδ o qq

(27a) with u ˚" a 2C D |ϕ a pδ a q ´ϕo pδ o q| (27b)
where we have to consider f ą 0 and f ă 0 to taking into account the two systems of equations.

Well-posedness criteria for the stationary problem

The previous section was considering the coupled system as a fixed-point problem, and the obtained well-posedness criterion is too restrictive to be used in the OA context. Let assume that, in the stationary case, the problem can be solved with some given viscosities that allow us to explicit solution of ( 27), and thus lead to a necessary and sufficient well-posedness criterion for existence and uniqueness of solutions.

Definition 4.1 Interval of u ˚: Values of u ˚are limited by their physical definitions and by the choice of our parameterization giving h α dependent on u ˚and with the constraint h α P pδ α , z 8 α q. Under these constraints we define I ˚as the interval of u ˚corresponding to physically acceptable values

I ˚" su min , u max r :" ȷ max ˆδa c a , δ o c o ˙, max ˆz8 a c a , z 8 o c o ˙" (28)
Depending on the reference values given by definition 2.3, this gives I ˚«s0.001, 0.7r.

Definition 4.2 Notation : For a given νpu ˚, zq, solutions of " if ϕ α ´Bz pν α pu ˚, zqB z ϕ α pzqq " 0 on pδ α , z 8 α q ϕ α pz 8 α q " 0 (29) are given by A α ψ α pz, u ˚q where A α is a constant and ψ is a generating solution of the first line of ( 29). We define S α pu ˚q :" ψpδ α , u ˚q ν α pδ α , u ˚qB z ψpδ α , u ˚q (30) Some profiles of S α have been given in [START_REF] Thery | Analysis of schwarz waveform relaxation for the coupled ekman boundary layer problem with continuously variable coefficients[END_REF] for different types of viscosity profiles.

Property 4.1 Well-posedness criterion : Let define

F : $ & % I ˚ˆr2πr Ñ C pu, θq Ñ ue iθ ? 2C D `1 ´u? C D S a puq `λ2 u ? C D S o puq ˘(31)
Then problem ( 26) is well-posed on I Ă I ˚if and only if F is injective on I ˆr2πr. So, we want to determine the largest interval

I Ă I ˚such that ˇˇˇB u ˆˇˇˇu b C ´1 D ´u2 S a puq `λ2 u 2 S o puq ˇˇˇ˙ˇˇˇą 0 @u P I Ă I ˚(32)
If ϕ g α are constant we have the equality:

ϕ g a ´ϕg o " F pu ˚, argpϕ a pδ a q ´ϕo pδ o qqq (33)
Proof. Let pose ψ p α a particular solution of the main equation of (27a) such that ψ p α pz 8 α q " ϕ g α pz 8 α q , and solutions in each domain can be written as u α " A α ψ α pzq `ψp α . Using interface conditions and definitions of S α , we have:

ψ p a pδ a q ´ψp o pδ o q " ψ p a pδ a q ´ψp o pδ o q `Aa ψ a pδ a qp1 ´λ2 S ´1 a S o q " ψ p a pδ a q ´ψp o pδ o q `u˚a C D `ϕδ a ´ϕδ o ˘Sa p1 ´λ2 S ´1 a S o q `ϕδ a ´ϕδ o ψp a pδ a q ´ψp o pδ o q " pϕ δ a ´ϕδ o qp1 ´aC D u ˚Sa `λ2 a C D u ˚So q (34) Let u ˚" ? 2C D |ϕ δ a ´ϕδ o | and θ " argpϕ δ a ´ϕδ
o q, which gives (33). Define G the application from R 2 into R 2 as G : pu ˚, θq Ñ p|F pu ˚, θq|, argpF pu ˚, θqqq. Then G is a diffeomorphism from I Ă I ˚ˆr0, 2πr to F pI, r0, 2πrq if detpJ G pu ˚, θqq ‰ 0 for all pu ˚, θq P I ˆr0, 2πr with J G the Jacobian matrix of G. Then

J G pu, θq " ˆBu |F pu, θq| B θ |F pu, θq| B u pargpF pu, θqqq B θ pargpF pu, θqqq ˙" ˆBu |F pu, θq| 0 B u pargpF pu, θqqq 1 Ṫhen detpJ G qpu, θq " B u |F pu, θq|. If ϕ g
α is constant, it is a particular solution of (27a) and we obtain (33) from (34).

Property 4.2 Particular case with no Coriolis force pf " 0q : Let consider the simpler case where the Coriolis force is neglected (f " 0) in the system (27a). Then (27a) can be solved3 for every ν and the well-posedness criterion ( 32) is an analytic expression on u ˚. Under the hypothesis of KPP viscosity profiles (definition 2.2) we found that the problem ( 27) is well-posed on the interval Proof. If f " 0 the resolution of (27a) gives ν α pz, u ˚qB z u α " x α with x α a constant. The interface condition can be written as λ ´2x o " x a . Then r ϕ α pzq " ´Xα ş pz,z 8 α q x α ν ´1 α pz 1 qdz 1 where X a " 1 and X o " ´1, which leads to (36). Under the hypotheses of definition 2.2, and especially ν m α ! D α pu ˚, zq for all z P pδ α , hαp1´ϵqq, and using ν m o " λν m a , we can approximate:

I " ȷ u min ,
S α « ´Xa |z 8 α ´hα | ν m α ´Sa `λ2 S o « z 8 a ´λz 8 o ´ph a ´λh o q ν m a B u `´S a `λ2 S o ˘« ´ca ´λc o ν m a (37)
Note that we find that S " ´Sa `λ2 S o ą 0 and condition (32) can be rewritten as ˇˇ?CD ´1 `Bu `pu ˚q2 Spu This result illustrates the fact that there is non-uniqueness of solutions, and this non-uniqueness is inherent in the global KPP viscosity profiles (with a global shape such that ν α " ν m α on pδ α , h α q). For viscosity profiles given by definition 2.2, figure 1 presents the computed S α , the profile of | ϕ g a o | " |F pu ˚q| for u ˚P I ˚, and their respective approximations given by (36).

Remark. According to 3.5, there exists a nonstationary solution of (9) in a neighborhood of the stationary solution u e P I if

1 `?2C D u e ¨› › › › › ν 1 a a ν 3 a pu e q 2 › › › › › 2 2 `λ´2 › › › › › ν 1 o a ν 3 o pu e q 2 › › › › › 2 2 'ă 2 (38)
For KPP viscosity given in definition 2.2, we can numerically compute

› › › › ν 1 α ? ν 3 α › › › › 2 « 1.4ˆ10 6 , then criterion is satisfied
for u e ă 10 ´5ms ´1. Again, even with an exact value of M 1 , the orders of magnitude for which the criterion is satisfied are not relevant in the OA context.

Application to KPP viscosity profiles

The resolution of (27a) cannot be explicitly computed for a general viscosity profiles if f ‰ 0. Here we propose to consider KPP viscosity profiles as given by definition 2.3. The parameterization of ν in the context of OA coupling is generally given by a third-order polynomial in z to suit the hypotheses of the KPP viscosity profile. In order to simplify the resolution of (27a), we propose to approximate the viscosity by a second-order polynomial. This approximation can be justified by the concave profile of the viscosity profiles in the turbulent zone pδ α , h α q. We consider this simplification to compute the profile of ϕ α , the solution of the linear problem (26) only. However, a third-order viscosity profile being necessary to ensure C 1 regularity of ν in h α , we will assume here that considering only C 0 regularity on ν to compute B z ϕpδq{ϕpδq does not have a too large impact (but this issue has to be considered). To remain consistent with the OA context, we build a viscosity profile that verify the KPP viscosity profile hypotheses (see definition 2.2) except the C 1 continuity in h α .

Definition 4.3 Definition and hypotheses on ν approx : To guarantee that the assumptions of definition 2.2 are satisfied, and to ensure a relevant approximation, we will make the approximation that ν approx α ph α q " ν α ph α q " ν approx α pz " 0q " ν α pz " 0q " ν m α . One degree of freedom remains to approximate ν, which must guarantee the concave shape of ν approx , and thus we add the condition ν approx pδ α q « νpδ α q. We denote this last degree of freedom A α such that:

ν approx α " $ & % ν α " ν m α on ph α, z 8 α q D approx α pzqH ˆ1 ´z h α ˙`ν m α P P 2 pRq on p0, h αq (39a) with D approx α pzq " A α zpz ´hα q ą 0, A α « ´κu ˚|δ α ´hα | ´1 ă 0 (39b)
Two examples of such approximations are proposed here:

1. to ensure ν approx pδ α q " νpδ α q for all u ˚, we take A α " ´να pδq pδ α pδ α ´hα qq ´1.

2. to ensure }ν approx } L 8 pΩq pu ˚q " }ν} L 8 pΩq pu ˚q for all u ˚, we take A α " ´4 }ν} L 8 pΩq {h 2 α Property 4.3 Resolution of the equations on each subdomain : We suppose f ‰ 0, then for KPP viscosity profiles given by definition 2.2, using ν approx as given in (4.3), the resolution of the differential equation gives:

ν α pδ α qS α " X α h α ? 1 `4µ α ξ α p1 `ξα q pG ά `βα P ά qP δ α `pG ὰ ´βα P ὰ qP ´δ α pG ά `βα P ά qG δ α ´pG ὰ ´βα P ὰ qG ´δ α (40)
with X a " 1, X o " ´1, and

η α " X α ˜d1 `4ν m α h 2 α |A α | ¸´1 ξ " ´1 2 ´1 `a1 `if {A α βα " ς α tanhpς α pz 8 α ´hα qq h α η α ξ α pξ α `1q ζ " a if {ν m α P α " 2 F 1 pξ `1, ´ξ, 1, p1 ´˘ηq{2q G α " 2 F 1 pξ `2, 1 ´ξ, 2, p1 ´˘ηq{2q
where 2 F 1 is the hypergeometric function (see [Olver, 2010]). Under the hypotheses given in definitions 2.2 and 4.3, an asymptotic approximation of ( 40) is given by

ν α pδ α qS α « δ α ˆln ˆδα h α ˙´p1 `ξα q ´1˙(

41)

Proof. Using approximation (39), we can now solve the system (29). For each u ˚, we separate each domain into two parts:

• the free zones, where interface turbulence has no impact, pz P ph α, z 8 α qq :

p φ n α pzq " A n α,l ´eςαz ´eςαp2z 8 α ´zq ¯Bz p φ n α " ς α A n α,l ´eςαz `eςαp2z 8 α ´zq with ς α " c if ν m α
• the turbulent zone, close to the interface pz P p0, h αqq :

φ n α pzq " A n α,t P α pr α pzqq `Bn α,t P α p´r α pzqq B z φ n α pzq " X α ξ α pξ α `1q h α ? 1 `4µ α `An α,t G α pr α pzqq ´Bn α,t G α p´r α pzqq with X o " ´1, X a " 1, r α pzq " X α p1 ´2z{h α q{ ? 1 `4µ α , µ α " ν m α {p|a 2,α |ph ˚q2
α q, et P α prpzqq " P 0 ξα prpzqq. For a justification of this result see [START_REF] Thery | Analysis of schwarz waveform relaxation for the coupled ekman boundary layer problem with continuously variable coefficients[END_REF]. The Legendre polynomials P and G can also be written in terms of hypergeometric function P α prpzqq " 2 F 1 pξ α `1, ´ξα , 1, p1 ´rpzqq{2q, and G α pηpzqq " 2 F 1 pξ α 2, 1 ´ξα , 2, p1 ´rpzqq{2q.

Results in property 4.3 are then obtained considering a solution in C 1 pΩ α q and using some asymptotic expansions mostly based on the fact that µ ! 1. See appendix 6.3 for the full computation to obtain ( 40) and (41). for all u ˚P I and α P to, au.

• If there exists a solution w ˚of ÿ αPto,au

pδ α ´hα q ´1 ˆ2 ´u˚B u ν a pδ a , u ˚q ν a pδ a , u ˚q `u˚B u A a pu ˚q A a pu ˚q ˙" 0 (44)
in I ˚, then there exists at least two solutions of ( 27) on any interval containing w ˚. Moreover w ˚is of the order of 2u min :

for the first approximation that ensures ν approx pδq " νpδq and for any KPP viscosity profile ν.

for the second approximation that ensures }ν approx } L 8 pΩq pu ˚q " 2κ|h α |{3 ? 3 (which corresponding to › › ν Obrien › › L 8 pΩq pu ˚q with ν Obrien given by ( 8)) Proof. The proof is given in appendix 6.3. It is based on the study of the sign of B u |F | given by (31) to follow the criterion (43). It shows that assuming (43) for u ˚P I allows to ensure that B u |F | ą 0 whatever the sign of f . Also we show that the sign of B u |F | changes when (44) occurs.

Finally, for KPP viscosity profiles, we have the non-uniqueness of solution for physically relevant values of u ˚.

Even if an approximation is made on viscosity profiles for the computations, it seems that the well-posedness issues are inherent in the global KPP viscosity profiles and the OA orders of magnitude. For the reference OA values from definition 2.3, we plot in figure 3 the S α as given by ( 40) and the corresponding profile of | ϕ g a o | " |F pu ˚q| for u ˚P I ˚, and their respective approximations given by (41). Note that, according to the computation in appendix 6.3, the inflection point w ˚goes to 0 if |δ α | goes to 0, and then the equivalent problem considering an interface with zero thickness would be well-posed. But the non zero thickness of the interface zone is necessary to be representative of the interface conditions used in the realistic OA coupled models.

Conclusion

In this paper we construct a global OA coupled model considering realistic boundary conditions and taking into account the numerical strategy used by the actual implemented models. Our coupled model can be described as a nonlocal Ekman boundary layer problem with parameterized turbulent viscosity profiles and nonlinear interface conditions. The nonlocal property lies in the dependency of the turbulent viscosities to the jump of the solution around the interface. The existence of solutions has been proved on a close problem in the stationary case, and it was shown that the uniqueness of the solution is possible only for viscosity profiles with very low variations. We adapt this method, based on a fixed-point problem, to our model and discuss on the application in the OA context. We give a criterion on the viscosity profiles that ensures the well-posedness of the problem in the stationary and nonstationary cases. This criterion implies the same conclusion that the uniqueness of the solution can be ensured for viscosities with slow variations, which it is not relevant in the OA framework. For the stationary problem, when it is possible to solve the main equation, we give a sufficient and necessary well-posedness criterion that ensures existence and uniqueness of a solution. We first apply this criterion to the problem where the Coriolis effect is neglected. This gives us a well-posedness criterion for every parameterized turbulent viscosities. Applying the study on KPP viscosity profiles, which are representative to those used in the OA models, shows that the solution u ˚is unique on an interval restricted by physical validity assumptions. We then apply this well-posedness criterion considering the Coriolis effect and KPP viscosity profiles. We show that there is no uniqueness of the solution for an interval of physically relevant solutions. This non uniqueness is valid not only for a specific viscosity profile but for a general viscosity profile which follows the hypothesis imposed in OA context.

This paper is a synthetic work on the well-posedness properties on a simplified but somewhat realistic OA coupled problem. It confirms that, even in a simplified model, the regularity issues involved by the nonlocal behavior of the problem remain valid, both for the stationary and the non stationary cases. Giving a sufficient condition on the viscosity profiles to ensure the well-posedness, it highlights that the naive resolution based on a fixed-point problem is not adapted to the OA framework. Indeed the regularity of the specific viscosities considered in this framework does not satisfy the given necessary condition. Also, on the stationary problem where some more precise computation can be made, results show that the consideration of the Coriolis effect is indispensable and can change the nature of the solution. The non uniqueness issues are also relevant for viscosities derived from oceanic and atmospheric models. This issue does not concern only the coupled model but could also appear if we consider one domain with this interface condition (e.g. the atmosphere forced by the ocean). The non uniqueness of solution is related to the combination between the viscosity profiles, the interface condition and the thickness of the interface zone, as used in the realistic OA coupled models.

Appendix

Proof of the weak formulation

Well-posedness of the stationary weak formulation (12) We define pV m q mě0 the increasing sequence of finite-dimensional Hilbert subspaces such that V " Y mě0 V m and the continuous mapping Φ m : V m Ñ V m such that

xΦ m pu m q, v m y " f `@v m , u K D a `λ´2 @ B z v m , u K m D o ˘`xB z v m , νB z u m y a `λ´2 xv m , νB z u m y o `CD › › › u m δa δo › › › v m δa δo ¨ u m δa δo ´`x ? ν a B z v m , ? ν a B z u r y a `λ´2 x ? ν o B z v, ? ν o B z u r y o ´f `@v m , g K D a `λ´2 @ v m , g K D o ˘The
existence of solution u m of Φ m pu m q " 0 is proved using a monotonicity method, see [Evans, 2022] chapter 9 for more details. We need to prove that there exists r ą 0 such that Φ m pu m , u m q ě 0 for }u m } " r. We first minimize Φ m pu m , u m q using majoration:

ˇˇx ? ν a B z u, ? ν a B z u r y a `λ´2 x ? ν o B z u, ? ν o B z u r y o ˇˇď } ? ν a B z u a } 2 } ? ν a B z u r a } 2 `λ2 } ? ν o B z u o } 2 } ? ν o B z u r o } 2 ď ? 2 b ν r U z a ν U r z and ˇˇf `@r u, g K D a `λ´2 @ r u, g K D o ˘ˇď ? 2|f | }r u a } 2 }g a } 2 `|f |λ 2 }r u o } 2 }g o } 2 ď |f | a r U ? G ď |f | r C b ν r U z ? G with r C " 2 maxp r C a , r C o q{ a minpν m a , ν m o q. Then Φ m pu m , u m q " ν pU m q z `CD › › › u m δa δo › › › 3 ´`xν a B z v m , B z u r y a `λ´2 xν o B z v, B z u r y o ´f `@v m , g K D a `λ´2 @ v m , g K D o ˘ě ν pU m q z ´?2 b ν pU m q z a ν U r z ´|f | r C b ν pU m q z " b ν pU m q z ´b ν pU m q z ´?2 a ν U r z ´|f | r C ? G ě b ν pU m q z ´minpν m a , λ ´2ν m o q ´}B z u m } 2 `}B z u a,m } 2 2 ¯´? 2 a ν U r z ´|f | a r C ? G Taking }B z u m } 2 " ´?2 a ν U r z `|f | r C ? G ¯{ minp ? ν m a ,
? ν m o q (and the corresponding norm on }u m } 2 ) and applying the Brouwer's fixed point, there exists a solution in u m P V m . Then using the a priori estimate (13) u m is bounded in V and then there exist a subsequence pu m l q lě0 and u P V, such that • pu m l q converge weakly to u in V

• pu m l q converge to u in C 0 pΩq (Morrey's inequality) Previous convergence properties gives u α,m pδ α q converge to upδ

α q. So › › › u m l δa δo › › › u m l δa δo converge to the boundary term in › › › u δa δo › › › u
δa δo and u is a solution of (12). The uniqueness is proved by showing that if we have two solutions u, u 1 P V of (12), they satisfy:

ν pU ´U 1 q z `´› › › u δa δo › › › u δa δo ´› › › u 1 δa δo › › › u 1 δa δo ¯´ u δa δo ´ u 1 δa δo ¯" 0 (45)
And since p}x} x ´}y} yq px ´yq ě 0 for all x, y P R 2 , we have uniqueness in V.

Well-posedness of weak formulation (15) As previously, we use a Galerkin method, we suppose u m " p ř c k ptqe k , ř d k ptqe k q T with e k and orthogonal basis of V m . Then the weak formulation becomes a nonlinear ODE on c k and d k . Since all terms are continuous, by Cauchy-Lipschitz theorem, there exists a unique set of solutions with c k , d k P Cr0, T r. The a priori estimate gives bounds such that u m P L 8 p0, T ; L 2 pΩqq X L 2 p0, T ; H 1 pΩqq and B t u m P L 2 p0, T ; H ´1pΩq. Using classical convergence results, there exists u P L 2 p0, T ; Vq X Cpr0, T s; L 2 pΩqq with B t u P L 2 p0, T ; H ´1pΩqq and a X P L 3{2 pr0, T sq such that

• u m converges weakly to u in L 2 p0, T, H 1 pΩqq • B t u m converges weakly to B t u in L 2 p0, T ; H ´1pΩqq • › › › u m δa δo › › › u m δa δo converges weakly to X in L 3{2 pr0, T sq • u m converges to u in L 2 p0, T ; L 2 pΩqq.
To prove that u is the solution of the weak formulation, we have to prove that X is the term

C D › › › u δa δo › › › u δa δo . Use }u m pδ, tq ´upδ, tq} 2 ď c 1 }u m ´u} L 2 pΩq ptq }B z u m ´Bz u} L 2 pΩq ptq, then u m pδq converges to upδq in L 2 pr0, T sq. It implies that › › › u m δa δo › › › u m δa δo converges to › › › u δa δo › › › u δa δo in L 3{2 pr0, T sq.
Since the nonlinearity affects the boundary terms only, and νptq is continuous and bounded, we can use the classical method to prove that u P Cpr0, T s; L 2 pΩqq is a solution of the weak formulation with upt " 0q " u 0 . The proof of the uniqueness of the solution is based on the same argument than in the stationary case. For higher regularities we use the classical approach (see for example [Evans, 2022]). Using the weak formulation on B t u m and integration gives the a priori estimate

pU m q t `f `@B t u m , u K m D a `λ´2 @ B t u m , u K m D o ˘`xB t u m , νB z u m y a `λ´2 xB t u m , νB z u m y o ` B t u m δa δo ¨CD › › › u m δa δo › › › u m δa δo " xB t u m , νB z u r y a `λ´2 xB t u m , νB z u r y o `xB t u m , gy a `λ´2 xB t u m , gy o
Passing terms in f on the right-hand side and using νB

t B z u " 1 2 B t `νpB z uq 2 ˘´1 2 B t pνqpB z uq 2 and B t u|u|u " 1 3 B t p|u| 3 q we have pU m q t `1 2 B t ν pU m q z `CD B t › › › u m δa δo › › › 3 ď C 1 `ϵ1 pU m q t `ϵ´2 1 U m ˘`C 2 `ϵ2 pU m q t `ϵ´1 2 ν U r z C3 `ϵ3 pU m q t `ϵ´1 3 G ˘`maxpN o , N a q pU m q z
with some C 1 , C 2 , ϵ 1 , ϵ 2 ą 0 and N α " }B t ν} L 8 p0,T ;L 8 pΩq . Using ν α ptq ą ν m α ą 0, B t ν P L 8 p0, T ; L 8 pΩqq and u g P L 2 p0, T, L 2 pΩqq and u g | t"0 P H 1 pΩq (then ν pU m q z pt " 0q ă 8 and › › › u 0 δa δo › › › ă 8) we obtain the u P L 2 p0, T ; H 2 pΩqqXL 8 p0, T ; H 1 pΩqqXH 1 p0, T ; L 2 pΩqq. To obtain more regularity on B t u, we proceed in a similar way by first deriving the equation w.r.t. t, then potentially problematic terms are the term in ν and the boundary term.

Using B t B z B t pνB z uq " νpB t B z uq 2 `1 2 B t νB t pB z u 2 q and B t uB t p|u|uq " B t |u| 1 2 B t p|u| 2 q `|u|pB t uq 2 " pB t |u|q 2 |u| `|u|pB t uq 2 . Then an a priori estimate is given by:

1 2 B t pU m q t `ν pU m q tz `´B t › › › u m δa δo › › › ¯2 › › › u m δa δo › › › `› › › u m δa δo › › › › › ›Bt › › › u m δa δo › › › › › › 2 ď C 2 `ϵ2 pU m q t `ϵ´1 2 ν U r tz C3 `ϵ3 pU m q t `ϵ´1 3 G t ˘`maxpN o , N a q 1 2 B t pU m q z
Suppose also B t u r P L 2 p0, T ; Vq, B 2 t u r P L 2 p0, T ; L 2 pΩqq, then terms on the right-hand side exist. Integrating on p0, tq, since other terms than B t pU m q t on the left-hand side are positive, gives an upper bound on B t u m ptq for all 0 ď t ď T .

6.2 Bound in L 8 p0, T ; H 1 pΩqq Bound on trace Suppose u P H 1 pΩq with upz 8 q " 0 Then for all β ą 0 :

› › › u δa δo › › › 2 ď `› › u δ a › › `› › u δ o › › ˘2 ď p1 `βq › › u δ a › › 2 `p1 `β´1 q › › }u δ o › › 2
Since upz 8 q " 0 the Poincaré inequality holds and we can explicit:

}u α pδq} 2 " |2 xB z u, uy| ď 2 }B z u} }u} ď ? 2|Ω α | }B z u} 2 ď ? 2|Ω α | ν m α }ν α B z u} 2 " C α }ν α B z u} 2 Choosing β " λ 2 C o {C a gives › › › u δa δo › › › 2 ď pC a `λ2 C o q ν U z .
Bound on L 8 p0, T ; H 1 pΩqq Let u " r u `ug with r u satisfying the boundary conditions r u α pz 8 α , tq " 0 for all 0 ď t ď T and r u α pz, t " 0q " 0 for all z P Ω α . Moreover we have B t r u α ptq `r Br u α ´Bz pν α pzqB z r u α q ´Bz pν α pzqB z u g α q " 0 on pδ α , z 8 α qˆs0, T r (46)

Taking this equation on Ω a and multiplying by r u a , integrating on Ω a and using the boundary conditions give xr u, B t r uy `xB z r u a , νB z u a y `r u T a ν a B z upδ a q " 0. Then replacing r u a " u a ´ug a in the last terms and using B t u g " 0 gives:

1 2 B t }r u a } 2 2
`}? ν a B z u} 2 2 `ua pδ a q ¨νa B z upδq " u g a pδ a q ¨νa B z upδq `xB z u g a , ν a B z uy Doing the same on Ω o , multiplying by λ ´2 and adding the two lines gives

B t r U `Uν `CD › › › u δa δo › › › 3 ď C D › › › u δa δo › › › u g δa δo ¨ u δa δo `› › ? νB z u g a › › › › ? νB z u a › › `λ´2 › › ? νB z u g o › › › › ? νB z u o › ›
The left-hand side is composed of positive terms thus the right-hand side can be bounded using triangular inequality. Then for all ϵ ą 0 we have the Young inequality

´› › › u δa δo › › › ¯2 › › › u g δa δo › › › ď 2 3 ϵ ´› › › u δa δo › › › ¯3 `1 3ϵ 2 › › › u g δa δo › › › 3
Using this inequality, in the stationary case we have:

1 2 U ν ď 4 27 C D › › › u g δa δo › › › 3 `1 2 ν U g z and 1 3 C D › › › u δa δo › › › 3 ď C D › › › u δa δo › › › u g δa δo ď 1 3 C D › › › u g δa δo › › › 3 `1 4 ν U g z
For the nonstationary case, after integrating on p0, tq we obtain Using the properties from [Olver, 2010] and Ψp1 `ξq " Ψpϵq " Ψp1 `ϵq ´ϵ´1 and Ψp´ξq " Ψp1 ´ϵq we can simplify σ as:

σ " 2Ψp1q `ϵ´1 ´Ψp1 `ϵq ´Ψp1 ´ϵq " ´2γ `ϵ´1 `γ ´8 ÿ k"2

p´1q k ζpkqpϵq k´1 `γ ´8 ÿ k"2 p´1q k ζpkqp´ϵq k´1 " ϵ ´1 ´8 ÿ k"2 p´1q k `1 `p´1q k´1 ˘ζpkqϵ k´1 " ϵ ´1 `2 8 ÿ k"1 ζp2k `1qϵ 2k
which gives the asymptotics (41).

Research of an inflexion point on S α We decompose ξ α and σ α as

ξ α « 1 ´iY ´1 α ´Y ´2 α , Y α " 4A α f σ α « ´1 `iY α
According to the well-posedness criterion (32), we are searching for a solution of ˇˇˇB u ˆu2 ´aC D ´1 ´Na X a ´No X o ¯2 `u2 pN a Y a `No Y o q 2 ˙ˇˇˇ" 0 (49) with X α " lnpδ α {h α q `1, N a " u ˚|δ a |{ν δ a and N o " λ 2 u ˚|δ o |{ν δ o where ν δ α " ν α pδ α q. The derivative on u ˚gives u ˚N 1 α " N α R α with R α " 1 ´u˚`νδ α ˘1 {ν δ α , X 1 α " ´1{u ˚.nd Y 1 α " 4A 1 α {f " Y α A 1 α {Aα. Equation ( 49) can be rewritten as B u `u2 X 2 `pu ˚q2 Y 2 ˘" 0 with X " b C ´1 D ´Na X a ´No X o and Y " N a Y a `No Y o . Then we look for a solution of X pX `uX 1 q `Y pY `uY 1 q " 0. We have In the order of magnitude of ocean-atmosphere framework, we have X α P r1 `lnpδ{z 8 q, 1s «s ´5, 1r.

X `u˚X1 " b C ´1 D ´Na X a ´No X o ´Na R a X a `Na ´No R o X o `
• If δ α {h α ě e ´1 « 3 :

obviously X " ˆbC ´1 D ´Na X a ´No X o ˙ą 0 -X `uX 1 ě 0 if 1 ´Xα p1 `Dα q ą 0, because X α ă 0, an equivalent condition is upν δ α q 1 {ν δ α ď 2 ´X´1 α . Then X `uX 1 ě 0 if u ˚pν δ α q 1 {ν δ α ď 2 . • If δ α {h α ă e ´1 « 3 :

-X " ˆbC ´1 D ´Na X a ´No X o ˙ą 0 because N α X α P r0, κ ´1s is small enough compared to b C ´1 D .

since X α Ps0, 1s, we have X `uX 1 ě 0 if X ´1 α ě 1 ´u˚p ν δ α q 1 {ν δ α i.e 0 ď u ˚pν δ α q 1 {ν δ α .

Finally the order of magnitude of X P Especially, when u ˚is not too large (for example, h a ă z 8 a {2 gives |N a Y a | ą 10 3 ), the order of magnitude of Y is much larger than the order of magnitude of X. So if there exists w ˚such that Ypw ˚q `w˚Y pw ˚q " 0 and that the sign of Y `uY changes in w ˚, then there exists a solution of X pX `u˚X1 q `Y pY `u˚Y1 q " 0 close to w ˚. Also, because A α ă 0 by definition, if 2 ´u˚`ν then Y pY `uY 1 q ě 0, which gives the sufficient condition.

Figure 1 :

 1 Figure 1: Left panel: computed Sα (plain lines) and their approximations given by (36) (dashed lines). Curves are superimposed. Right panel: corresponding profile of ˇˇ ϕ g a o ˇˇ" |F pu ˚q|

˚q˘ˇą 0 .

 0 Then the problem is well-posed if B u `pu ˚q2 Spu ˚q˘ą 0 or ˇˇB u `pu ˚q2 Spu ˚q˘ˇă ? C D ´1. Since ν m a ! u ˚?C D , we can consider only B u `pu ˚q2 Spu ˚q˘ą 0. The expression from (37) gives us the the root

Figure 2 :

 2 Figure 2: Left panel: νa given by (8) and ν approx a given by (39) for different values of u ˚. The continuous line corresponds νa, the dashed line to ν approx for the first approximation proposed in definition 4.3, and the dotted line for the second approximation proposed in definition 4.3. Right panel: the corresponding values of }Na} L 8 pΩaq (black), }Buν} L 8 pΩaq (dark grey) and }ν} L 2 pΩaq (grey).

  Property 4.4 Well-posedness properties for KPP viscosity profiles : The well-posedness criterion (32) applied to the problem (27), and using result (41), gives the following well-posedness properties:• A sufficient condition to ensure well-posedness of (27) on I Ă I ˚is 1 ď u ˚Bu ν α pδ α , u ˚q ν α pδ α , u

Figure 3 :

 3 Figure 3: For KPP viscosity profiles with reference values (39). Left panel: Sα. Given by the first approximation (39).1 in continuous line, the second approximation, (39).2 in dashed line. Black lines correspond to the atmospheric part and grey lines to the oceanic part. Right panel: the corresponding profiles of ˇˇ} ϕ g } a o ˇˇ" |F pu ˚q| with the same line style. Dot-dashed line and doted line are superimposed.

  p1 ´Xa p1 `Ra qq `No p1 ´Xo p1 `Ro qq andY `u˚Y1 " N a Y a `No Y o `Na R a Y a `Na Y a u ˚A1 a A a `No R o Y o `No Y o u

  close to the order of magnitude of b C ´1 D . And since N α « κ ´1 and |1 ´u˚p ν δ α q 1 {ν δ α | is small (by definition, to be consistent with KPP assumption), then the order of magnitude of X `uX 1 is close to b C ´1 D . First remark that the sign of f does not impact the sign of Y pY `uYq. The order of magnitude of |N a Y a | " 4|δ a |u ˚A{|ν δ a f | « 4u ˚|h a ´δa | ´1|f | ´1 P r24, 8r.

  B z u o pδ o , tq " λ 2 ν a B z u a pδ a , tq on s0, T rs (9d) ν a B z u a pδ a , tq " C D }u a pδ a , tq ´uo pδ o , tq} pu a pδ a , tq ´uo pδ o , tqq

			tq " u g α pz, tq	on s0, T r (9b)
			u α pz, t " 0q " u g α pz 8 α , t " 0q	on rz 8 o , z 8 a s (9c)
			ν o on s0, T r (9e)
			u ˚ptq "	a C D }u a pδ a , tq ´uo pδ o , tq}	on s0, T r	(9f)
	with B "	ˆ0 f ´f 0	˙. Nonlocal aspects come from the interface condition, that depends on u ˚which itself depends
	on the jump of the solution around the interface. The parameter u ˚thus makes it possible to group together all the nonlocal aspects of the problem. An equivalent formulation would be to replace (9e) by ν a B z u a pδ a q " ? δa C D u ˚ u δo

  " 2Ψp1q ´Ψp1 `ξq ´Ψp´ξq and Ψ the digamma function. These asymptotics are given in the ocean part. Finally using Γp1 `ξq " Γpξqξ: Bplnpµq ´σq is dominating. Indeed if ζ ą 1 (that is the case if f ą 1e ´5) then tanhpζpz 8 ´h˚q q « 1 for reasonable value of u ˚and B « ζh ˚! µ ´1. If f ! 1e ´5 we would have B « h ˚{pz 8 ´h˚q ! µ ´1 for h ˚ă z 8 .

	with σ pG ´`βP ´q «	1 µΓpξ `2qΓp1 ´ξq	`ςα tanhpς α pz 8 α ´hα qq	h α η α ξ α pξ α `1q	σ ´lnpµq Γpξ `1qΓp´ξq
					"		1 Γpξ `2qΓp1 ´ξq ˆ1 µ	`ζhplnpµq η tanhpζpz 8 ´hq ´σ	˙"	1 σpξ `2qΓp1 ´ξq ˆ1 µ	`Bplnpµq ´σq ṗG
	´`βP ´qG δ «	1 Γpξ `2q 2 Γp1 ´ξq 2		h δ ˆ1 µ	`Bplnpµq ´σq ṗG
	´`βP	´q	P δ ξpξ `1q	«	1 Γpξ `2qΓp1 ´ξq ˆ1 µ	`Bplnpµq ´σq	˙σ ´lnpδ{hq ξΓp´ξqpξ `1qΓpξ `1q
					"		lnpδ{hq Γpξ `2q 2 Γp1 ´ξq 2 ˆ1 µ ´σ	`Bplnpµq ´σq	ẇith
	B "		tanhpζ ζh pz 8 ´h˚q q	. Moreover
					pG `´βP `q «1	´ςα tanhpς α pz 8 α	´hα qq	α h η α ξ α pξ α `1q	" 1 ´B{pξpξ `1qq
			pG `´β α P ὰ qG ´δ a «p1 ´B{pξp1 `ξqqq ˆ1 `p1 ´ξqp2 `ξq	2h δ	ṗG
			`´βP	`q P ξpξ `1q ´δ	«p1 ´B{pξp1 `ξqq ˆ1{pξp1 `ξqq	´δ h ˚Ṫhe
	term `So 1 µ
							φpzq B z φpzq	« h a	1 `4µ	δ h	plnpδ{hq ´σq « δ plnpδ{hq ´σq
							r U	`1 2	ż t 0	ν U z ď	ż t 0	C D	4 27	´› › › u g δa δo	› › › ¯3	`1 2	ż t 0	ν U g z

This choice of u ˚can be justified by the construction of u ˚in the interface buffer zone, and is detailed in[Pelletier et al.,

2021]

Remark that we consider f " 0 in the equation, but hα still depending on f and we use a realistic value of f for hα definition

Acknowledgements

The author thanks to Florian Lemarié (Inria, Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, France) and Eric Blayo (Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, France) for fruitful discussions on the physical OA coupled model and for their critical feedback on the manuscript.

We want a bound for }νB z u} 2 in L 8 p0, T q. Taking the same notation r u and multiplying by B t r u on Ω a :

As previously, we can do the same on the oceanic part, then multiply by λ

This ensures a bound on ν U z ptq and

Computations for the stationary problem with KPP viscosity profiles

Resolution of the equation on each subdomain Considering the equation in the proof of property 4.3, to have C 1 pΩ α q regularity, especially in h α, we have to add the constraints:

A α,l " A α,t P α pr α ph α qq `Bα,t P α p´r α ph α qq e ςαhα ´eςαp2z 8

Finally: φ α pzq " A α,t rP α pr α pzqq `Πα P α p´r α pzqqs B z p φ α pzq " X α ξ α pξ α `1q h α?

1 `4µ α rG α pr α pzqq ´Πα G α p´r α pzqqs noting η " rphq " ´rp0q, P `" P pηq and P ´" P p´ηq:

Asymptotics on hypergeometric function Considering KPP viscosity profiles as in definition 2.2 and approximation (39), we get the order of magnitude:

for reasonable value of f ă 7e ´5 or u ˚not too close from u min . Then we can pose ξ α « ´1 `ϵα with ϵ α P C and |ϵ α | ă 1.

The first inequality allows to write η α « ´Xα p1 ´2µ α q and r α pδ α q " ´ηα p1 ´2δ α h α q. We focus on the computation on the oceanic part; by symmetry the atmospheric part follows the same principle. In the rest of the proof, indices o are omitted. We can use the following asymptotic taken from [Olver, 2010] and [Barton, 1965] :