Michel De Lara

Preferences Yielding the "Precautionary Effect"

Keywords: value of information, uncertainty, learning, precautionary effect, support function. JEL Classification: D83

 has provided the most operative tool to exhibit the precautionary effect. Epstein's Theorem holds true when the difference of two convex functions is either convex or concave, which is not a straightforward property, and which is difficult to connect to the primitives of the economic model. Our main contribution consists in giving a geometric characterization of when the difference of two convex functions is convex, then in relating this to the primitive utility model. With this tool, we are able to study and unite a large body of the literature on the precautionary effect.

Introduction

Consider an agent taking two successive decisions to maximize his expected utility under uncertainty. As illustrated in Figure 1, after his first decision, a signal is revealed that provides information about the state of nature. The observation of the signal allows the decision-maker to revise his prior and the second decision is taken accordingly. Assuming that the first decision is a scalar representing consumption, the precautionary effect holds when initial consumption is less in the prospect of future information than without (no signal). The example above is a stereotype of sequential decisions problems with learning were focus is put on comparison of the optimal initial decisions with different information structures. For instance, should we aim at more reductions of current greenhouse gases emissions today whether or not we assume some future improvement of our knowledge about the climate? Economic analysis has identified effects that go in opposite directions and make the conclusion elusive. This article proposes a characterization of utility functions such that the precautionary effect holds for all signals.

Seminal literature in environmental economics ( [START_REF] Arrow | Environmental preservation, uncertainty, and irreversibity[END_REF], (Henry, 1974a,b)) focused on the irreversible environmental consequences carried by the initial decision and showed that the possiblity of learning should lead to less irreversible current decisions ("irreversibility effect"). Henry; Arrow and Fisher consider additive separable preferences, and so do [START_REF] Freixas | The irreversibility effect[END_REF], [START_REF] Fisher | Quasi option value: Some misconceptions dispelled[END_REF], [START_REF] Hanemann | Information and the concept of option value[END_REF]. Epstein in [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF] studies a more general nonseparable expected utility model, and derives a condition that identifies the direction of the precautionary effect. His contribution remains the most operative tool. Yet, the conditions under which this result holds are difficult to connect to the primitive utility model. Further contributions have insisted on the existence of an opposite economic irreversibility since environmental precaution imply sunk costs that may constrain future consumption ( [START_REF] Kolstad | Fundamental irreversibilities in stock externalities[END_REF], [START_REF] Pindyck | Irreversibilities and the timing of environmental policy[END_REF], [START_REF] Fisher | Global warming, endogenous risk, and irreversibility[END_REF]). Risk neutral preferences are studied in [START_REF] Ulph | Global warming, irreversibility and learning[END_REF] for a global warming model. Assuming time separability of preferences, the papers [START_REF] Gollier | Scientific progress and irreversibility: an economic interpretation of the "precautionary principle[END_REF] and [START_REF] Eeckhoudt | Optimal consumption and the timing of the resolution of uncertainty[END_REF] examine risk averse preferences. Gollier, Jullien, and Treich identified conditions on the second-period utility function for the possibility of learning to have a precautionary effect with and alternatively without the irreversibility constraint. By this latter, we mean that the domain of the second decision variable depends on the first decision.

The driving idea for linking the effect of learning and the value of information is the observation that, once an initial decision is made, the value of information can be defined as a function of that decision. This is the approach of Jones and Ostroy who define the value of information in this way in their paper [START_REF] Robert | Flexibility and uncertainty[END_REF] where they formalize the notion of flexibility in a sequential decision context, and relate its value to the amount of information an agent expects to receive. Whenever the second-period value of information -namely the value of information measured after an initial commitment is made -is a monotone function of the initial decision, optimal initial decisions can be ranked. All this is recalled in Sect. 2, where we also extend the approach in [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF] and [START_REF] Robert | Flexibility and uncertainty[END_REF] to non necessarily finite sets.

In [START_REF] Robert | Flexibility and uncertainty[END_REF], the monotonicity property of the second-period value of information is related to convexity in the prior of a difference of maximal payoffs. This is convexity is far from being granted since the difference of two convex functions is generally not convex. Our main result consist in giving a general condition under which a difference of maximal payoffs exhibits convexity in the prior. This is the object of Sect. 3 where, first, we provide a geometric characterization and, second, carry it to the utility model. With this tool, we are able to study in Sect. 4 a large body of the literature on the precautionary effect.

The precautionary effect: statement and recalls

We first give a formal statement of the precautionary effect, then recall and extend some results in the literature, upon which we shall elaborate our main contribution in the next section.

We shall assume that all sets are Borel spaces, endowed with the Borel σ-algebra generated by the open subsets, that all mappings have appropriate measurability and integrability properties needed to perform mathematical expectations operations, and that all probabilities are regular to ensure the existence of conditional distributions [START_REF] Bertsekas | Stochastic Optimal Control: The Discrete-Time Case[END_REF], [START_REF] Kallenberg | Foundations of Modern Probability[END_REF].

Problem statement

Consider an agent taking two successive decisions as in Figure 1. The initial decision a is a scalar belonging to an interval I of R; the following and final decision b belongs to a set B(a) which may depend on the initial decision1 a (B(a) is a subset of a fixed set B). Uncertainty is represented by states of nature ω ∈ Ω with prior P on the Borel σ-field F, and by a random variable X : (Ω, F) → (X, X). Partial information on X is provided by means of a signal

(random variable) Y : (Ω, F) → (Y, Y). A utility function U(a, b, x) is given, defined on I × B × Ω. The expected utility maximizer solves 2 max a∈I E max b∈B(a) E[U a, b, X | Y ] .
(1)

Thus, the second decision b is taken knowing Y .

The evaluation of expected utility right after the first decision a has been taken is conditional on the signal Y and defined as follows:

V Y (a) := E max b∈B(a) E[U a, b, X | Y ] .
(2)

With this notation, the program of the Y -informed agent is max a V Y (a). Let us assume, for the sake of simplicity, that an optimal solution exists and is unique, denoted by āY .

A signal Y is said to be more informative than a signal

Y ′ if the σ-field σ(Y ) := Y -1 (Y) contains σ(Y ′ ). It is equivalent to say that Y ′ is a measurable function of Y , namely Y ′ = f (Y ) where f : (Y, Y) → (Y ′ , Y ′ ).
The precautionary effect is said to hold whenever the optimal initial decision is lower with more information that is, if when the signal Y is more informative than the signal Y ′ , then āY ≤ āY ′ .

Precautionary effect and second-period value of the information monotonicity

We shall recall a sufficient condition under which the presence of learning affects the first optimal decision in a predictable way (see [START_REF] Robert | Flexibility and uncertainty[END_REF], (De Lara andDoyen, 2008, p.229), (De Lara and[START_REF] Lara | Precautionary effect and variations of the value of information[END_REF]).

Let us compare the programs of the Y -informed and Y ′ -informed agent by writing

max a V Y (a) = max a {V Y ′ (a) + V Y (a) -V Y ′ (a) } . (3)
It appears that the decision maker who expects more information optimizes the same objective as the less informed decision maker plus what we shall coin the second-period value of information

∆V Y Y ′ (a) := V Y (a) -V Y ′ (a) (4)
which depends on his initial decision. The more-informed agent initial optimal decision achieves a trade-off: it can be suboptimal from the point of view of the less-informed decision maker but compensates for this by an increase of the second-period value of information.

Proposition 1 Assume that the programs max a V Y (a) and max a V Y ′ (a) have unique3 optimal solutions āY and āY ′ . Whenever the second-period value of the information is a decreasing function of the initial decision, namely

∆V Y Y ′ : a → V Y (a) -V Y ′ (a) is decreasing, (5) then āY ≤ āY ′ . (6)
In the case where Y ′ is constant (no information since σ(Y ′ ) is the trivial σ-field {∅, Ω}), the effect of learning is precautionary in the sense that the optimal initial decision is lower with information than without.

Second-period value of information and Epstein functional

We extend the approach of Epstein in [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF] and [START_REF] Robert | Flexibility and uncertainty[END_REF] to non necessarily finite sets. We denote by P(X) the Borel space of probability measures on X, with its Borel σ-field; the same holds for P(Y).

Following Epstein, let us define what we shall coin the Epstein functional by4 :

J(a, ρ) := sup b∈B(a) E ρ U a, b, • = sup b∈B(a) X U a, b, x dρ(x) , ∀ρ ∈ P(X) . (7)
Denote by ν and ν ′ the unconditional distributions of the signals Y and Y ′ on their image set Y. The conditional distribution of X knowing Y is a mapping P Y X : Y → P(X). The following result may be found in [START_REF] Robert | Flexibility and uncertainty[END_REF]. We give its proof in the general case of non necessarily finite sets.

Proposition 2 Assume that 1. for any pair of initial decisions a

1 ≥ a 0 , ρ ∈ P(X) → J(a 1 , ρ) -J(a 0 , ρ) is convex (resp. concave), 2. Y is more informative than Y ′ .
Then, the value

∆V Y Y ′ (a) = V Y (a) -V Y ′ (a) of substituting Y for Y ′ , is increasing (resp.
decreasing) with initial decision a ∈ I. Hence, Proposition 1 applies.

Proof. We have:

V Y (a) = E P max b∈B(a) E P [U a, b, X | Y ] by definition (2) = E ν max b∈B(a) E P Y X [U a, b,
• ] by using the conditional distribution = E ν J(a, P Y X ) by ( 7). Let a 1 ≥ a 0 , and suppose that ϕ(ρ) = J(a 1 , ρ) -J(a 0 , ρ) is convex in ρ ∈ P(X). We have [START_REF] Dellacherie | Probabilités et potentiel[END_REF], [START_REF] Artstein | Sensors and information in optimization under stochastic uncertainty[END_REF], [START_REF] Artstein | Gains and costs of information in stochastic programming[END_REF]

∆V Y Y ′ (a 1 ) -∆V Y Y ′ (a 0 ) = E ν J(a 1 , P Y X ) -J(a 0 , P Y X ) -E ν ′ J(a 1 , P Y ′ X ) -J(a 0 , P Y ′ X ) = E ν ϕ(P Y X ) -E ν ′ ϕ(P Y ′ X ) ≥ 0 since Y is more informative than Y ′ . Indeed, it is known (see
) that if Y is more informative than Y ′ , then E ν [ϕ(P Y X )] ≥ E ν ′ [ϕ(P Y ′ X )
] for all convex function ϕ on P(X). 5 The converse holds for concave. 2

3 Conditions on the primitive utility for the precautionary effect

Epstein's condition for the precautionary effect in [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF] relies upon convexity (or concavity) of ρ → ∂J ∂a (ā Y , ρ). In the same vein, Jones and Ostroy rely upon convexity of the mapping ρ ∈ P(X) → J(a 1 , ρ)-J(a 0 , ρ) in [START_REF] Robert | Flexibility and uncertainty[END_REF]. With the expression (7) of J(a, ρ), it is not easy to see how this relates to the primitive of the model, namely the utility U.

We shall proceed in two steps to characterize utility functions such that ρ ∈ P(X) → J(a 1 , ρ) -J(a 0 , ρ) is convex. The difficulty comes from the fact that this latter function is the difference of two convex functions, hence has no reason to be convex. We shall, first, provide a geometric characterization and, second, carry it functionally to the utility model by means of so-called support functions.

A geometric characterization

The following characterization of when ρ ∈ P(X) → J(a 1 , ρ) -J(a 0 , ρ) is convex relies upon the notion of sum of subsets of a vector space. Recall that, for any subsets Λ 1 and Λ 2 of a vector space, Λ 1 + Λ 2 = {x 1 + x 2 , x 1 ∈ Λ 1 and x 2 ∈ Λ 2 } is their so called direct sum, or Minkowsky sum.

Let us define, for any initial decision a ∈ I,

Λ -(a) := {f : X → R | there exists b ∈ B(a) such that f (x) ≤ U(a, b, x) , ∀x ∈ X} (8)
the set of maximal possible random rewards when the initial decision is a. Our first main result is the following.

Proposition 3 Let a 1 > a 0 . If there exists a subset K of functions defined on X such that

Λ -(a 1 ) = Λ -(a 0 ) + K , (9)
then ρ ∈ P(X) → J(a 1 , ρ) -J(a 0 , ρ) is convex. Hence, the first hypothesis of Proposition 2 is satisfied.

Proof. The proof comes from the observation that the Epstein functional J(a, ρ) is a so-called support function6 as a function of its argument ρ. Recall that, to any set Λ of bounded measurable functions f : X → R is attached the support function σ Λ , defined on the Banach space of finite signed measures on X, by:

σ Λ (ρ) := sup λ∈Λ X λ(x)dρ(x) . (10)
Indeed, (7) may be written as

J(a, ρ) = sup b∈B(a) E ρ U a, b, • = sup λ∈Λ(a) E ρ λ = σ Λ(a) (ρ) , (11)
where

Λ(a) := {f : X → R | there exists b ∈ B(a) such that f (x) = U (a, b, x) , ∀x ∈ X} . (12)
Now, since ρ belongs to cone of positive measures (ρ ∈ P(X)), we also have that J(a, ρ) = σ Λ(a) (ρ) = σ Λ -(a) (ρ), because the polar cone of nonnegative functions on X is the set of measures (Aubin, 1982, p.31,p.107). Support functions have the nice property to transform a Minkowsky sum into a sum of functions [START_REF] Aubin | Mathematical Methods of Game and Economic Theory[END_REF]:

σ Λ 1 +Λ 2 = σ Λ 1 + σ Λ 1 . (13)
Thus, whenever Λ -(a 1 ) = Λ -(a 0 ) + K, we obtain that

J(a 1 , ρ) -J(a 0 , ρ) = σ Λ -(a 1 ) (ρ) -σ Λ -(a 0 ) (ρ) = σ Λ -(a 0 )+K (ρ) -σ Λ -(a 0 ) (ρ) = σ Λ -(a 0 ) (ρ) + σ K (ρ) -σ Λ -(a 0 ) (ρ) = σ K (ρ) .
Hence, ρ → J(a 1 , ρ) -J(a 0 , ρ) = σ K (ρ) is convex since it is a support function. 7 2

Characterization of utility functions ensuring the precautionary effect

We shall now show how the above geometric characterization ( 9) translates into a condition on the utility function U.

Consider two initial decisions a 1 > a 0 . To any mapping φ : B(a 0 ) → B(a 1 ) between second decision sets, associate the following set of minimizers

B φ (a 1 , a 0 , x) := arg min b∈B(a 0 ) U(a 1 , φ(b), x) -U(a 0 , b, x) , ∀x ∈ X . ( 14
)
and

B φ (a 1 , a 0 ) := x∈X B φ (a 1 , a 0 , x) . ( 15
)
When this latter set is not empty, there exists at least one minimizer b ∈ B(a 0 ) of U(a 1 , φ(b), x)-U(a 0 , b, x) independent of the alea x. Our second main result is the following.

Proposition 4 Assume that 1. the set Φ(a 1 , a 0 ) := {φ : B(a 0 ) → B(a 1 ) | B φ (a 1 , a 0 ) = ∅} is not empty, 7
The condition Λ -(a 1 ) = Λ -(a 0 ) + K is almost necessary for J(a 1 , ρ) -J(a 0 , ρ) to be convex in ρ ∈ P(X). Indeed, (Hiriart-Urrurty and Lemaréchal, 1993, p. 92-93)

states that the closed convex hull co(σ Λ2 -σ Λ1 ) is equal to σ Λ2 ⋆ -Λ1
, where the star-difference

Λ 2 ⋆ -Λ 1 := {x | x + Λ 1 ⊂ Λ 2 }.
Recall that the closed convex hull cof of a function f minorized by an affine function is the largest closed convex function minorizing f .

to any b

1 ∈ B(a 1 ) can be associated at least one φ ∈ Φ(a 1 , a 0 ) and one b 0 ∈ B φ (a 1 , a 0 ) such that b 1 = φ(b 0 ).
Then there exists a subset K of functions defined on X such that Λ -(a 1 ) = Λ -(a 0 ) + K.

Hence, the assumption of Proposition 3 is satisfied.

Proof. Let us define the set of functions K = ∪ φ∈Φ(a 1 ,a 0 ) K φ , where

K φ = {x ∈ X → U (a 1 , φ(b), x) -U (a 0 , b, x) for b ∈ B φ (a 1 , a 0 )} . ( 16
)
1. We first show the inclusion Λ -(a 1 ) ⊃ Λ -(a 0 ) + K. Indeed, for any φ ∈ Φ(a 1 , a 0 ) and b ∈ B(a 0 ), we have by definition of K φ and B φ (a 1 , a 0 ):

k(x) ≤ U (a 1 , φ(b), x) -U (a 0 , b, x) , ∀x ∈ X , ∀k ∈ K φ . Hence, k(x)+U (a 0 , b, x) ≤ U (a 1 , φ(b), x). Thus, Λ(a 0 )+K ⊂ Λ -(a 1
), and therefore Λ -(a 0 )+ K ⊂ Λ -(a 1 ).

We now show the reverse inclusion Λ

-(a 1 ) ⊂ Λ -(a 0 ) + K. Let b 1 ∈ B(a 1 ). By assumption, there exist φ ∈ Φ(a 1 , a 0 ) and b 0 ∈ B φ (a 1 , a 0 ) such that b 1 = φ(b 0 ). We have that U (a 1 , b 1 , x) = U (a 1 , φ(b 0 ), x) = U (a 0 , b 0 , x) + U (a 1 , φ(b 0 ), x) -U (a 0 , b 0 , x) k(x) (17)
where the function k belongs to K φ since b 0 ∈ B φ (a 1 , a 0 ).

2

The following Corollary provides practical conditions on the utility function U which ensure that the assumptions of Propositions 4 and 3 are satisfied, hence that the first hypothesis of Proposition 2 is satisfied.

What is more, the first-order condition (19) has proximities with the second-order one in [START_REF] Salanié | Option value and flexibility: A general theorem with applications[END_REF]. This opens the way for comparison between our approach and the invariance approach of Salanié and Treich.

Corollary 5 Assume that the second decision variable b belongs to B = R n and that the minimizers in (14) are characterized by the first-order optimality condition

φ ′ (b) ∂U ∂b (a 1 , φ(b), x) - ∂U ∂b (a 0 , b, x) = 0 , ∀x ∈ X . ( 18
)
Suppose that, to any vector b 1 ∈ B(a 1 ) can be associated at least one vector b 0 ∈ B(a 0 ) and one square matrix M ∈ R n×n such that

M ∂U ∂b (a 1 , b 1 , x) - ∂U ∂b (a 0 , b 0 , x) = 0 , ∀x ∈ X . ( 19
)
If, in addition, we have b

1 + M(b -b 0 ) ∈ B(a 1 ) for all b in a neighbourhood of b 0 in B(a 0 ), 8
then the assumptions of Proposition 4 are satisfied.

Proof. By assumption, to any b 1 ∈ B(a 1 ) can be associated at least one mapping defined by φ(b) = b 1 + M (bb 0 ) in a neighbourhood of b 0 (and smoothly prolongated outside). The first-order optimality condition (18) attached to ( 14) admits the solution b = b 0 by (19). 2

Analysis of examples in the literature

We shall examine a large body of the literature, and see that Corollary 5 applies in all cases and explains the precautionary effect. At first, we shall assume that the second decision set B(a) = B does not depend on the initial decision a, to concentrate on the precautionary effect and to try to disentangle it from the irreversibility effect. Second, we shall relax this assumption and attempt to point out the impact that the second decision set B(a) indeed depends upon the initial decision a.

In the sequel, we consider two initial decisions a 1 > a 0 .

Additive separable preferences

The case of additive separable preferences may be found in [START_REF] Arrow | Environmental preservation, uncertainty, and irreversibity[END_REF], (Henry, 1974a), [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF], [START_REF] Freixas | The irreversibility effect[END_REF], [START_REF] Fisher | Quasi option value: Some misconceptions dispelled[END_REF], [START_REF] Hanemann | Information and the concept of option value[END_REF] and is formalized by

U(a, b, x) = u(a, x) + v(b, x) .
It can be seen that Λ(a) defined in ( 12) may be written as Λ(a 1 ) = {u(a 1 , x)+v(b, x) , b ∈ B} = u(a 1 , x)-u(a 0 , x)+{u(a 0 , x)+v(b, x) , b ∈ B} = K+Λ(a 0 )

where K is the singleton {u(a 1 , x)-u(a 0 , x)}. Hence J(a 1 , ρ)-J(a 0 , ρ) = σ u(a 1 ,x)-u(a 0 ,x) (ρ) =

X u(a 1 , x)u(a 0 , x) dρ(x) is linear in the prior ρ, hence both concave and convex. Thus, the precautionary effect holds true in a strong sense since the initial optimal decision does not depend on the amount of information the agent expects to receive.

The above analysis is confirmed by the first-order optimality condition (19) which is

M ∂v ∂b (b 1 , x) = ∂v ∂b (b 0 , x) , ∀x ∈ X .
Since M = Id n×n and b 0 = b 1 are solutions, the precautionary effect holds true. However, when B(a) indeed depends upon a, we no longer have that Λ(a 1 ) = K + Λ(a 0 ). We also observe that the additional conditions of Corollary 5 related to the irreversibility constraints, namely B(a 1 ) ⊂ B(a 0 ) and b 1 ∈ B(a 1 ) for all b in a neighborhood of b 1 ∈ B(a 0 ), are not generaly satisfied and this may prevent the precautionary effect to hold true.

Risk neutral preferences

Examples in [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF][START_REF] Ulph | Global warming, irreversibility and learning[END_REF] present the general structure

U(a, b, x) = u(a, b) + v(a, b)x = u(a, b) + p i=1 v i (a, b)x i .
The first-order optimality condition ( 19) is

       M ∂u ∂b (a 1 , b 1 ) = ∂u ∂b (a 0 , b 0 ) M ∂v i ∂b (a 1 , b 1 ) = ∂v i ∂b (a 0 , b 0 ) , i = 1, . . . , p .
This is a system of n+np equations with n+n 2 unknown (M, b 0 ). Thus, when the dimension p of the noise is less than the dimension n of the second decision variable, the precautionary effect generally holds true.

Risk averse preferences

We shall examine three models where preferences exhibit risk aversion.

A consumption-savings problem [START_REF] Epstein | Decision making and temporal resolution of uncertainty[END_REF] A two-periods consumption-savings problem is modelled by

U(a, b, x) = u 1 (w -a) + βu 2 (ra -b) + β 2 u 3 (bx) ,
with savings a, b, and irreversibility constraint B(a) = [0, ra].

The first-order optimality condition ( 19) is

Mβxu ′ 3 (b 1 x) -βxu ′ 3 (b 0 x) = Mu ′ 2 (ra 1 -b 1 ) -u ′ 2 (ra 0 -b 0 ) , ∀x ∈ X .
If there exists a solution (M, b 0 ), this implies that there must exist constants α, γ and δ such that u ′ 3 satisfies an equation of the form

xu ′ 3 (αx) = γxu ′ 3 (x) + δ , ∀x ∈ X . A candidate is u ′ 3 (x) = x -γ , yielding a solution M = γ and b 0 = αb 1 , with the compatibility condition γu ′ 2 (ra 1 -b 1 ) -u ′ 2 (ra 0 -αb 1 ) + δ = 0.
Hence, the precautionary effect holds true.

Global warming and emissions [START_REF] Gollier | Scientific progress and irreversibility: an economic interpretation of the "precautionary principle[END_REF] With pollution emissions a, b, a two-periods model with benefits and costs of emitting pollutions is modelled by

U(a, b, x) = u(a) + v b -x(a + b) .
The first-order optimality condition ( 19) is

Mv ′ b 1 -x(a 1 + b 1 ) = v ′ b 0 -x(a 0 + b 0 ) , ∀x ∈ X .
If there exists a solution (M, b 0 ), this implies that there must exist constants α, β and M such that v ′ satisfies an equation of the form v ′ (αx + β) = Mv ′ (x) , ∀x ∈ X .

In this case, b 0 = b 1 a 0 /a 1 . The utility v(x) = γ 1-γ η + x γ 1-γ in [START_REF] Gollier | Scientific progress and irreversibility: an economic interpretation of the "precautionary principle[END_REF] precisely satisfies v ′ (αx+γη(α-1)) = α -γ v ′ (x), which explains why the precautionary effect holds true.

Eating a cake with unknown size [START_REF] Eeckhoudt | Optimal consumption and the timing of the resolution of uncertainty[END_REF] The following model from [START_REF] Eeckhoudt | Optimal consumption and the timing of the resolution of uncertainty[END_REF] is qualified of the problem of "eating a cake with unknown size" in [START_REF] Salanié | Option value and flexibility: A general theorem with applications[END_REF]:

U(a, b, x) = u(a) + v(b) + w(x -a -b) .
The first-order optimality condition ( 19) is

Mv ′ (b 1 ) -v ′ (b 0 ) = Mw ′ (x -(a 1 + b 1 )) -w ′ (x -(a 0 + b 0 )) .
If there exists a solution (M, b 0 ), this implies that there must exist constants β, κ and M such that w ′ satisfies an equation of the form w ′ (x + β) = Mv ′ (x) + κ. Then, we find that β + a 1 + b 1 = a 0 + b 0 and that Mv ′ (b 1 )v ′ (b 0 ) + κ = 0. Thus, β, κ and M must satisfy some compatibility constraint, so that the precautionary effect holds true.

Conclusion

We have provided general conditions for the precautionary effect to hold true, including a condition that bears directly on the primitive utility of the economic model. We have examined a large body of the literature, and seen how operative is this condition to explain the precautionary effect. Preferences yielding the precautionary effect for all signals appear to belong to a restricted class. This is related to the strong conditions, that we provide, needed to have a difference of maximal payoffs exhibit convexity in the prior. The connexion with the invariance approach of [START_REF] Salanié | Option value and flexibility: A general theorem with applications[END_REF] deserves to be studied and clarified.

Figure 1 :

 1 Figure 1: Decision with learning; agent takes decision a; a signal is revealed; agent takes decision b accordingly.

This may materialize "irreversibility" of the initial decision.

We shall always assume that, for the problems we consider, the sup is attained and we shall use the notation max.

Unicity is for the sake of simplicity. If the programs max a V Y (a) and max a V Y ′ (a) do not have unique optimal solutions, denote by arg max a V Y (a) and arg max a V Y ′ (a) the sets of maximizers. If ∆V Y (a) is decreasing, the upper bounds of these sets can be ranked: sup arg max a V Y (a) ≤ sup arg max a V Y ′ (a). If ∆V Y (a) is strictly decreasing, we obtain that sup arg max a V Y (a) ≤ inf arg max a V Y ′ (a) that is, āY ≤ āY ′ for any āY ∈ arg max a V Y (a) and any āY ′ ∈ arg max a V Y ′ (a).

We denote by U (a, b, •) the mapping X → R given by x → U (a, b, x).

Notice that E ν [] denotes a mathematical expectation taken on the probability space (Y, Y, ν).

It is well known that an expected utility is convex in the prior. It seems less noticed that the expected utility, seen as a function of the prior, is the support function of the set of payoffs indexed by actions.

This condition is meaningless if b 1 belongs to the interior of B(a 0 ). Hence this condition has to be verified only when an irreversibility constraint bites.

Acknowledgements. We want to thank Christian Gollier, François Salanié and Nicolas Treich for their invitation to the seminar at LERNA (Toulouse School of Economics) on 19th January 2009, and for the subsequent discussions.


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Jean-Pierre Pécau

CATERINA SFORZA, THE LIONESS OF ROMAGNA, IMMORTALIZED AS THE LADY WITH AN ERMINE BY LEONARDO DA VINCI?

If there is a female warrior figure who marked the Italian Renaissance and thus inscribed her name in letters of blood and gold in the legend of her multiple conspiracies, it is that of Caterina Sforza, the lioness of Romagna 3 . Caterina Sforza, the amazon who fought in armor, sword in hand, against the armies of the all-powerful and fearsome Borgia family

Brotherhood, developped and edited par Ubisoft Caterina Sforza was raised by her paternal grandmother Bianca Maria Visconti in the lively and refined court of the great poets and writers of the Italian Renaissance. The latter, as well as her father, transmitted to her their passion for war and hunting and taught her the art of governing 6 .

Women ferreting rabbit hunts in the Queen Mary Psalter7

The Lady with an Ermine by Leonardo da Vinci This "ermine" -which is actually an albino ferret -is an early example of a pet ferret, whose conspicuous coat was particularly prized for rabbit hunting8

Married at 10 to Girolamo Riario, 20 years her senior, nephew of Pope Sixtus IV 9 , pregnant with their first child at 15, she will therefore bear the title of Countess of Imola and Forlì. On the death of her husband, assassinated in 1488 by supporters of the new pope, Innocent VIII 10 , she was led to exercise the regency in the name of her son Ottaviano, still a minor. As a perfect strategist trained in the executive, she begins by lowering taxes and sealing alliances with neighboring states through the unions contracted for her sons. Moreover, driven by her natural inclination for the army, as an authentic condottiere, she takes charge of the military training of her troops.

Caterina Sforza played on screen by Gina McKee 11

Caterina Sforza then secretly marries a young man named Giacomo Feo, with whom she will have a son, Bernardino Carlo, a year later. The passion she feels for the young man makes the invincible Caterina Sforza bend, so much so that she ends up dismissing her son Ottaviano from the government to entrust the reins of the State to her new husband and place the parents of his at the head of the fortresses which defend the city. But Ottaviano's supporters did not give up and Giacomo Feo was assassinated by conspirators. The young widow then massacres the families and supporters of the assassins in retaliation. In the fortress of Forlì, Caterina Sforza takes over the reins of state administration, caring with as much attention as ever for her children. With her faithful apothecary, Ludovico Albertini, she developed her knowledge of medicine and pharmaceutical-naturalist and perfected beauty recipes and long-life remedies by mixing herbs and essences using processes that at that time were called alchemicals. In 1497, she obtained permission from her uncle, Duke Ludovico Maria Sforza, alias "il Moro", to marry Giovanni De'Medici, a member of the powerful Florentine Medici dynasty, then ambassador of Florence. Just a year after giving birth to a 10 Giovanni Battista Cybo-Tomasello 11 The Borgias, television series created by Neil Jordan son, the one who would later be known as Giovanni Dalle Bande Nere12 , her third husband dies of pneumonia.

In In May 1509, she died of pneumonia, aged only 46. She is buried anonymously in the convent of Santa Maria delle Murate, as she had stipulated in her will. But his nephew Cosimo I, Grand Duke of Tuscany, ordered a white marble tombstone bearing his name to be placed on his grave. Caterina Sforza, the lioness of Romagna, was indeed one of the most romantic and adventurous female characters of the Italian Renaissance. Described by her contemporaries as a sophisticated, beautiful and witty woman... They also attributed to her a temperament that was independent and determined to say the least: A velvet-gloved hand that has never been afraid to cross swords with men, as powerful and fearsome as the Borgias! 15 In addition to the fact that she rubbed shoulders with the greatest geniuses of art and culture of her time, she also unashamedly broke a number of conventions and dusted off many principles, to the point of flirting with alchemy.

14 Giuliano della Rovere At the turn of the 15th century, Caterina Sforza thus wrote more than four hundred recipes for beauty and good health -tinged with alchemical formulas -in a manuscript entitled Experimenti. Extant in a 16th century transcription produced by Lucantonio Cuppano (1507-1557) 16 , the manuscript includes instructions for making lip colors, lotions, and hair dyes (with a clear preference for a blonde or red shade), prescriptions for treating ailments ranging from fever and headache to epilepsy and infertility and recipes to improve libido and "restore" virginity... Further still, he gives instructions for producing the philosopher's stone and the quintessence: the elixir supposed to protect and cure all diseases and what is more to ensure eternal youth...

As an example, this "talc water" recipe supposed to achieve several of these objectives at once: "Talc is the star of the earth and has shining scales; it is found on the island of Cyprus and is similar in color to citrine; in a mass it looks green, and dissolved in air it looks crystalline; and it has the following virtues, not to mention others not noted in this book, which will be the alchemist's desire to discover: First, to make women beautiful, and to remove all spots or marks from the face, so that a a woman of sixty will appear to be twenty years old. Also...mixed with white wine, its powder will cure one who is poisoned and whoever drinks the powder in white wine will be protected on that day from poison and from any disease or 16 Medici, in Caterina Sforza, ed. Pier Desiderio Pasolini, v. 3, 617-618 (Rome : Loescher, 1893). Cuppano's 16th-century transcription is held in private archives.

Experimenti de la Ex[ellentissi]ma S]igno]ra Caterina da Furlj Matre de lo inllux[trissi]mo S[ignor] Giovanni de

pestilence. Also...this water turns silver into gold, and makes fake jewelry perfect and fine."

Multi-faceted recipes like this, informed by alchemical practice, abounded in modern scientific culture. They circulated in manuscripts and printed matter, among men and women, by word of mouth and through epistolary, commercial and courteous networks. Privileging direct experience, observation and application to study or theoretical explanation, the recipes contained in Caterina Sforza's Experimenti reflect the empirical and heterogeneous nature of modern science. Like the printed books of secrets that would explode in popularity in the early 15th century (described in William Eamon's Science and the Secrets of Nature), Caterina Sforza's manuscript blurs the lines between alchemy and medicine, practical and obscure, home and court. Her quest for the secrets of the Great Work is not only recorded in the manuscript she bequeaths to her son (the condottiere Giovanni Dalle Bande Nere, father of Cosimo I de'Medici), she also mentions it in letters to her apothecary, his agents, family members and other alchemy enthusiasts throughout Italy.

Caterina Sforza's activity places her at the origins of a Medici interest in alchemy and experimentation that would extend into the 17th century, while positioning her within the broader panorama of modern women's scientific activity. She was not alone in this activity: women's participation in scientific culture was wide and diverse, and the language of recipes proved useful in a variety of contexts, including the Renaissance debate over women. In 1600, Moderata Fonte, in his dialogue, Il Merito delle donnen 17 , will pit the efficacy and frankness of certain medicinal and cosmetic recipes -capable of curing disease or transforming appearance -against the impossible task of finding a recipe for teaching men to respect women as their equals 18 .

17 The value of women

  1495, the King of France, Louis XII, allied with Pope Alexander VI 13 -who succeeded three years earlier to Innocent VIII -decides to declare war on the Italian States. He first conquered the Duchy of Milan while the Pope claimed the domains of Romagna for Cesare Borgia, "the Valentino" his son. What the indomitable Caterina Sforza is obviously not willing to accept. Caterina Sforza immediately set to work to expand her troops, improve the armament and store large quantities of provisions and ammunition in anticipation of a possible siege led by the troops commanded by César Borgia, Duke of Valentinois and son of the Pope. It also strengthens the defenses of its fortresses, in particular that of Ravaldino, where it resides.

  15 https://www.nationalgeographic.fr/histoire/catherine-sforza-laguerriere-qui-defia-les-borgia Superposition and transparency of the portrait of Caterina Sforza and the Lady with an ermine

  Despite this, Caesar wasted no time in sending him to Rome after subjecting him to all sorts of humiliations. Pope Alexander VI assigned him to residence at the Belvedere Palace, a beautiful villa near Rome. But the attentions of the pope fail to quell the rebellious spirit of Caterina Sforza. She tries to flee, but, discovered, she is accused of preparing an attack against the pope by means of poisoned letters, and the rebellious Duchess is locked up in Castel Sant'Angelo, this fortress which she had defended with so much ardor a few years earlier. She only remained in prison for a short time because she was released on June 30, 1501, Louis XII having interceded on her behalf. Caterina Sforza then retired to Florence and took refuge in the villa which belonged to her third husband, Jean de Médicis. After the death of Pope Alexander VI, she tries to recover her fiefs from the new pontiff Julius II 14 . However, Imola and Forlì opposed his return and passed into the hands of a Vatican nobleman named Antonio Maria Ordelaffi. Her combative fervor subsided, Caterina Sforza spent the last years of her life alongside her sons and devoted herself to the study of alchemy.

The Lady with an Ermine is… Caterina Sforza ?

Forlì and Imola having fallen, Borgia began the siege of the fortress of Ravaldino on December 19, 1499. Niccolò Machiavelli reported that the proud and daring Caterina resisted the siege alone, admired for it throughout Italy, but no one is came to help him. Supported by more than a thousand soldiers, Caterina Sforza leads the resistance herself. She repeatedly rebuffs her enemy's peace proposals, though it may cost the lives of her children, as the legend goes. On January 12, 1500, after a succession of terrible battles, Caesar Borgia's troops invaded Ravaldino and Caterina Sforza was taken prisoner. Although she placed herself under the protection of King Louis XII of France, an ally of the pope, it seems that Cesar Borgia refused to part with his prisoner. Partly out of pride, but also because Caterina Sforza would have become his mistress shortly after Ravaldino's surrender.

  The esoteric science revealed by this painting is of such depth that one wonders where and how a man, at the end of the Middle Ages, was Caterina Sforza, the amazon who fought in armor, sword in hand, against the armies of the all-powerful and fearsome Borgia family! Superposition and transparency of the Lady with an Ermine and the Great Work of Leonardo 19 able to acquire it. What universities or academies taught such anthropogony? It was only in Italy, in the neo-Platonic Florence of Marsilio Ficino, Pico de la Mirandola and Christoforo Landino, that a similar syncretism flourished, in which the Christian doctrine of Salvation, the Greek philosophy of Nature, the profound symbolism of the Hellenistic Mysteries mingled. and Gnostics, practice of magic and theurgic therapy (Wilhelm Fraenger, The Millennial Kingdom of Jérôme Bosch, Denoël editions -1947)

	18 https://fleurtyherald.wordpress.com/2016/06/21/caterina-sforza-
	the-alchemist/

19 https://hal.science/hal-03966781v1/file/Le_Grand_Oeuvre_de_Leonardo.pdf

https://www.caterinasforza.it/en/history/history-of-caterina/

https://commons.wikimedia.org/wiki/File:Women_hunting_rabbits_ with_a_ferret.jpg

https://fr.wikipedia.org/wiki/Furet

Francesco della Rovere

John of the Black Bands


De Marez Charly

Carton Xavier

Charly De Marez
email: charly.demarez@univ-brest.fr

Interaction of an Upwelling Front with External Vortices: Impact on Cross-shore Particle Exchange

Keywords: coastal upwelling front, vortices, filaments, baroclinic instability, frontal waves, particle motion and tracking

Coastal upwellings, due to offshore Ekman transport, are more energetic at the western boundaries of the oceans, where they are intensified by incoming Rossby waves, than at the eastern boundaries. Western boundary upwellings are often accompanied by a local vortex field. The instability of a developed upwelling front and its interaction with an external vortex field is studied here with a three-dimensional numerical model of the hydrostatic rotating Navier-Stokes equations (the primitive equations). The baroclinic instability of the front leads to the growth of meanders with 100-200 km wavelength, in the absence of external vortex. On the f-plane, these waves can break into a row of vortices when the instability is intense. The beta-effect is stabilizing and strongly decreases the amplitude of meanders. Simulations are then performed with a front initially accompanied by one or several external vortices. The evolutions in this case are compared with those of the unstable jet alone. On the f-plane, when an external vortex is close to the front, this latter sheds a long filament which wraps up around the vortex. This occurs over a period similar to that of the instability of the isolated front. Cyclones are more efficient in tearing such filaments offshore than anticyclones. On the beta-plane, the filaments are short and turbulence is confined to the vicinity of the front. At long times, waves propagate along the front, thus extending turbulence alongshore. The initial presence of a vortex alley leads to a stronger destabilization of the front and to a larger cross-shore flux than for a single vortex, with many filaments and small vortices pushed far offshore. In the ocean, this cross-shore exchange has important consequences on the local biological activity.

Introduction

The Arabian Sea, in the North-Western Indian Ocean, is the seat of oceanographic processes which interact dynamically (see Fig. 1). These physical processes in turn, influence the regional biological activity of the ocean. This sea evolves under the influence of monsoon winds (north-easterlies in winter and south-westerlies in summer; [START_REF] Schott | Winter monsoon circulation of the northern Arabian Sea and Somali Current[END_REF] and of a strong evaporation rate [START_REF] Privett | Monthly charts of evaporation from the N. Indian Ocean (including the Red Sea and the Persian Gulf)[END_REF]. The strong winds induce a cyclonic regional circulation in winter and an anticyclonic regional one in summer. Moreover, these winds generate Rossby waves near the western coast of India [START_REF] Brandt | Annual Rossby waves in the Arabian Sea from TOPEX/POSEIDON altimeter and in situ data[END_REF]. These waves propagate westward and they can strengthen large, long-lived eddies, like the Great Whirl near the coast of Somalia (Vic et al., 2015) and the Ras al Hadd dipolar eddy South of Oman [START_REF] Ayouche | Structure and dynamics of the Ras al Hadd oceanic dipole in the arabian sea[END_REF] (eddies are oceanic vortices). The Arabian Sea is populated with many eddies [START_REF] Bruce | Evidence for eddy formation in the eastern Arabian Sea during the northeast monsoon[END_REF][START_REF] Saafani | Westward movement of eddies into the Gulf of Aden from the Arabian Sea[END_REF]Trott et al., 2019). Such eddies are formed in particular by the instability of coastal currents and of the regional gyre currents.

A second main feature of the Arabian Sea are two western boundary upwellings (near Somalia and South of Oman), as the wind blows parallel to the coast and northwestward in summer [START_REF] Sastry | Upwelling and upward mixing in the Arabian Sea[END_REF][START_REF] Bruce | Some details of upwelling off the somali and arabian coasts[END_REF][START_REF] Elliott | Some features of the upwelling off Oman[END_REF][START_REF] Currie | Circulation and upwelling off the coast of south-east arabia[END_REF][START_REF] Shi | The Oman upwelling zone during 1993, 1994 and 1995[END_REF][START_REF] Fischer | Mesoscale eddies, coastal upwelling, and the upperocean heat budget in the Arabian Sea[END_REF][START_REF] Piontkovski | Coastal upwellings and mesoscale eddies of the western Arabian Sea: some biological implications[END_REF]Vic et al., 2017). In upwellings, the Ekman drift due to the wind pushes the surface water offshore leading to an outcropping of deeper and colder water from depths of 150 m. This cold water (which can be 5 • C cooler than the Arabian Sea surface water) is rich with nutrients, favoring the local biological activity. And indeed, this upwelling region is an intense fishing area. When the upwelling is developed (in August/September), the sea surface height near the coast can lie 30 cm below the offshore sea surface and a northeastward current, the Oman Coastal Current, flows offshore. Western boundary upwellings are more energetic than their eastern boundary counterparts because they absorb energy from incoming Rossby waves. The temperature and salinity fronts which bound the upwelling region offshore are not steady. They often meander and they may form cold filaments. Capes play an important role in the offshore growth and protrusion of cold filaments from upwellings [START_REF] Currie | Circulation and upwelling off the coast of south-east arabia[END_REF][START_REF] Shi | The Oman upwelling zone during 1993, 1994 and 1995[END_REF][START_REF] Meunier | Influence of bottom topography on an upwelling current: Generation of long trapped filaments[END_REF]. These filaments can extend offshore over more than 250 km and bring nutrients into the oligotrophic surface ocean [START_REF] Manghnani | Advection of upwelled waters in the form of plumes off oman during the southwest monsoon[END_REF]. South of Oman, the upwelling lasts for the whole summer and interacts with the local eddy field (see Fig. 5 of [START_REF] Shi | The Oman upwelling zone during 1993, 1994 and 1995[END_REF].

The upwelling front and current can form meanders and eddies, either by instability or via the interaction with the eddies (see Fig. 5 of [START_REF] Sastry | Upwelling and upward mixing in the Arabian Sea[END_REF]. The evolution of an upwelling front and coastal current in the presence of external vortices, and its comparison with the evolution of an unstable front in the absence of such vortices, is the subject of the present paper, with application to the Arabian Sea (again, see Fig. 1).

The structure and stability of upwelling systems, with fronts and currents, have been the subject of many previous studies (Yoshida, 1955;[START_REF] Hidaka | Physical oceanography of upwelling[END_REF][START_REF] Pedlosky | Longshore currents and the onset of upwelling over bottom slope[END_REF][START_REF] Mccreary | A linear stratified ocean model of the coastal undercurrent[END_REF][START_REF] Csanady | On the structure of transient upwelling events[END_REF], to name a few). The first studies aimed at understanding the spin-up of the upwelling as the wind starts to blow and before it becomes a well established front associated with a geostrophic current. In particular, the role of the frictional boundary layers in the formation of the upwelling was investigated [START_REF] Pedlosky | Longshore currents and the onset of upwelling over bottom slope[END_REF]. The importance of the topographic constraint on the deep boundary layer was analyzed (Pedlosky, 1978a,b).

Analytical solutions of the density and velocity structure were provided for a time varying wind stress forcing the upwelling. In particular, the formation of a sub-surface countercurrent was studied with regard to the vertical mixing of heat and momentum [START_REF] Mccreary | A linear stratified ocean model of the coastal undercurrent[END_REF]. For an upwelling south of a zonal coast, an eastward baroclinic current builds up as the offshore front steepens. When the wind stress has a longshore variation, Kelvin waves propagate along the coast, away from the upwelling region. The alongshore flow deviates from geostrophy. Ageostrophic velocities perpendicular to the coast then form a two-cell vertical circulation [START_REF] Suginohara | Upwelling front and two-cell circulation[END_REF].

In our study, we investigate the instability of the geostrophic current associated with the offshore upwelling front, or how it interacts with an external eddy field; we assess the consequences of these interactions on cross-shore transport. The interaction between a zonal current and a single external eddy has been the subject of several studies. [START_REF] Pratt | Dynamics of potential vorticity fronts and eddy detachment[END_REF] studied the growth and detachment of an eddy from a potential vorticity front in a one-layer model. [START_REF] Stern | On the interaction of a vortex with a shear flow[END_REF] considered the interaction between a single eddy and a zonal potential vorticity front in a one-layer model (with finite or infinite radius of deformation). They showed that, in the linear stage of the interaction, the surface integral of vorticity in the meander of the front is equal and opposite to the area integral of the eddy vorticity. This leads to a dipolar effect which can advect the eddy along the front. Vandermeirsch et al. (2003a,b) determined the conditions under which an eddy can cross a potential vorticity front meridionally, in one and two-layer models. The existence of a stagnation point (a hyperbolic point in the flow field) was found as a necessary condition for this crossing. This condition is identical for the detachment of an eddy from an unstable front [START_REF] Capet | Nonlinear regimes of baroclinic boundary currents[END_REF]. These latter authors found that the circulation of the detached eddy was proportional to that enclosed within a wavelength of the meandering front, extending offshore to the stagnation point. This paper is organized as follows. Section 2 describes the numerical model and the analysis tools. In particular, the novelty of this paper lies in the use of a three-dimensional model (see supplementary information for the presentation of the 3D evolution of some simulations discussed in this study). Section 3 details the numerical results of the upwelling current instability, in the absence of vortices. The results are analyzed with respect to the characteristic wavelengths of the meanders growing on the current and of the inshore and offshore transports. Section 4 considers the nonlinear evolution of the front in the presence of a single vortex offshore. Cyclone-anticyclone asymmetry is considered since the flow is not quasi-geostrophic. Section 5 considers the situation where multiple vortices exist South of the front, as observed in the Arabian Sea. Section 6 characterizes the cross-shore exchanges with particle tracking. Conclusions follow.

Methods

The numerical simulations

In this section, we present the idealized model runs performed for this study. The aim of these runs is to study the instability of an upwelling front and to simulate the interaction of an upwelling front with one or several vortices.

Numerical setup and domain

The simulations are carried out in a 3D hydrostatic primitive equation framework. They are performed using the Coastal and Regional Ocean COmmunity (CROCO) model (Shchep-etkin & McWilliams, 2005). This model solves the hydrostatic primitive equations for the velocity, temperature, and salinity, using a full equation of state for seawater [START_REF] Shchepetkin | Accurate Boussinesq oceanic modeling with a practical, "stiffened" equation of state[END_REF]. The simulations performed integrate the primitive equations for 1 year. The numerical settings are similar to previous simulations performed in an idealized context (see e.g. [START_REF] Ménesguen | Destabilization of an oceanic meddy-like vortex: energy transfers and significance of numerical settings[END_REF][START_REF] De Marez | Study of the stability of a large realistic cyclonic eddy[END_REF]: horizontal advection terms for tracers and momentum are discretized with fifth-order upwind advection schemes (UP5); the explicit horizontal viscosity and diffusivity are set to zero, since the UP5 scheme damps dispersive errors; the vertical advection is discretized with a fourth-order centered parabolic spline reconstruction (Splines scheme). Further discussion about these parameterizations can be found in [START_REF] Klein | Upper ocean turbulence from high-resolution 3D simulations[END_REF] or [START_REF] Ménesguen | Destabilization of an oceanic meddy-like vortex: energy transfers and significance of numerical settings[END_REF]. Vertical mixing of tracers and momentum is done using a K-profile parametrization (KPP, [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF] there is no bottom friction. Simulations have 32 terrain-following vertical levels, which are stretched such that the vertical resolution is ∆z ∼ 8 m at the surface, and ∆z ∼ 120 m at the bottom. The horizontal resolution is 5 km. The domain is square, with a length L domain = 2000 km or L domain = 3000 km in vortex alley simulations (see details below). The bottom is flat and 2000 m deep. We set the Coriolis parameter to f = f 0 = 10 -4 s -1 or f = f 0 + βy, with β = 2 × 10 -11 m -1 s -1 and y the meridional coordinate. The northern and southern boundaries are closed, with a 10 km wide sponge layer to avoid the generation of spurious boundary dynamics. A zonal periodic condition is chosen at the eastern and western boundaries such that the domain is a zonal channel.

We initialize an analytical background stratification N (z), which fits the average ambient stratification in the five major oceanic basins, similarly as in de Marez et al. (2020a,c):

N (z) = N 0 + N 1 e z/z h , (1)
with z < 0 the vertical coordinate, N 1 = 9 × 10 -3 s -1 , z h = 150 m, and N 0 = 7 × 10 -3 s -1 .

Integrating this stratification from the surface (where ρ(z = 0) = 1030 kg m -3 ), gives the ambient density background ρ(z); the temperature background T (z) is obtained by inverting the TEOS-10 equation of state for seawater [START_REF] Mcdougall | Getting started with teos-10 and the gibbs seawater (gsw) oceanographic toolbox[END_REF] and assuming a constant salinity background S(z) = 35 psu. The model is initialized with these temperature and salinity background profiles.

Components of simulations

Upwelling front. We chose to study the evolution of the upwelling front once it is established by the wind forcing. This established front is the initial condition of our simulations, which are then run in free decay. These simulations determine the further, unforced, evolution of this front. Not adding surface forcing in the simulations allows us to analyze specifically the front instability or the front-vortex interaction as a classical initial-value problem. Our simulations have an application to the ocean when the time scales of the evolution are shorter than those of the forcing. For other cases, simulations with forcing, or with a relaxation towards a prescribed state, will be performed in a follow-up of this study. It must be noted that free-decay simulations have often been used in studies of oceanic flow stability, since they produce instability waves which depend only on the current and not on the forcing.

We simulate the presence of an upwelling front by initializing a zonal jet of a given velocity profile, similarly as in [START_REF] Barth | Short-wave length instabilities on coastal jets and fronts[END_REF]; this velocity profile has the form:

u up = V up 0 e z/H e -y-y up 0 L 2 , (2)
with V up 0 the initial velocity of the upwelling jet, y up 0 = L domain /2 the latitude of the upwelling jet, and H and L the vertical and zonal extent of the upwelling jet, respectively. This current is then geostrophically adjusted with the density field and the sea surface height. This leads to a temperature front separating cooler waters North from warmer waters South.

In the simulations discussed in this study, we set V up 0 = 0.2 m s -1 , H = 200 m, and L = 50 km. These values correspond to those chosen in previous studies (e.g. [START_REF] Barth | Short-wave length instabilities on coastal jets and fronts[END_REF]Vic et al., 2017) for the study of upwellings.

Sinusoidal perturbation. To trigger the upwelling front instability, we add a sinusoidal perturbation along the front, directly in the initialization of a few experiments. This disturbance idealizes the observed perturbations due to the surrounding flow, or to the atmospheric influence, on the upwelling front. It is usual to choose a single wave as a perturbation in linear, normal-mode theory of flow stability. This disturbance is here a temperature anomaly centered around y 0 , and of the form:

T pert = 0.5 e -z/500 e -y-y up

0 50 10 3 2 cos( 2π λ pert x), (3)
with x the zonal direction, and λ pert = [50, 100, 250, 500, 1000] km. Varying the wavelength allows the determination of the fastest growing disturbance. The choice of a perturbation deeper than the mean flow is motivated by the likeliness that it is triggered by neighboring currents which are deeper than the upwelling: vortices, or marginal sea outflow currents, or baroclinic Rossby waves. A density anomaly is then computed assuming that the perturbation has no signature in salinity. The velocity field associated with this density anomaly is then obtained via the thermal wind balance.

Vortex. To address the main objective of this article (the interaction of an upwelling front with a pre-existing vortex field), we initialize one or several surface-intensified vortices, South of the upwelling front (offshore), in the vortex-front interaction simulations. When several vortices are initialized, they are aligned along the coast offshore of the front. These vortices are initialized in azimuthal velocity (or in vorticity) as in de Marez et al. (2020a,c). For each vortex, we set the initial profile of surface vertical vorticity:

ω(r) = ±ω 0 e -r R α , (4)
with the sign depending on the vortex polarity, r = (x -x c ) 2 + (y -y c ) 2 the radial coordinate referenced at the center of the vortex (x c , y c ), and α (usually an integer) the steepness parameter.

The surface azimuthal velocity of the vortex can be computed using v θ (r, 0) = 1 r r dr ω r.

In general it takes the complicated form v θ (r) = (ω 0 R 2 /(αr)) γ((2/α), (r/R) α ), where γ(s, x)

is the incomplete gamma function. In the well-known case of a Gaussian vertical vorticity,

with α = 2, the azimuthal velocity is v θ (r) = ω 0 R 2 r [1 -e -r 2 /R 2 ].
In this case, this velocity grows linearly with r, for r R, similarly to Rankine vortices. Far from the center, it decays at best as 1/r (that is, slower than for a Rankine vortex). To avoid the presence of spurious velocity at the edges of the domain, we apply a Hanning window on v θ to make it smoothly tend to zero at r > 3 R. The horizontal velocity decreases at depth such that v θ (x, y, z) = v θ (x, y, 0) e -z/Hvortex , thus defining the height of the vortex H vortex . We de-note (u, v) the horizontal Cartesian components of the velocity of the vortex. The pressure anomaly field P (x, y, z) corresponding to this velocity field is computed via the gradient wind equation:

2J(u, v) + f (∂ x v -∂ y u) = 1 ρ ∆ h P , (5)
with J(u, v) = ∂ x u∂ y v -∂ y u∂ x v the Jacobian operator, and ∆ h the horizontal Laplacian operator. From P we determine the density and the temperature anomalies of the vortex.

These anomalies are computed for as many vortices as we want, at positions x c , and y c . Note that such vortices are robust during the whole simulation.

We set H vortex = 1000 m, as surface-intensified vortices are mostly about 1000 m deep in this part of the ocean [START_REF] Chaigneau | Vertical structure of mesoscale eddies in the eastern South Pacific Ocean: A composite analysis from altimetry and Argo profiling floats[END_REF][START_REF] Pegliasco | Main eddy vertical structures observed in the four major Eastern Boundary Upwelling Systems[END_REF][START_REF] Keppler | Observed Characteristics and Vertical Structure of Mesoscale Eddies in the Southwest Tropical Pacific[END_REF][START_REF] De Marez | On the 3D structure of eddies in the Arabian Sea[END_REF], and α =24, so that the vortex has a profile close to a Rankine vortex (also called a top-hat vortex, as in e.g. de Marez et al. ( 2020c)). For our study, we vary other initial parameters as R = [50, 100] km, and

V vortex 0 = [0.2, 0.4, 0.6] m s -1
for both cyclonic and anticyclonic vortices. For each set of parameters, we also vary the initial distance between the vortices and the upwelling front as d/R = [2.0, 3.0, 4.0, 5.0], thus setting y c .

In simulations with an isolated vortex, we set x c = L domain /2. For simulations with a zonal vortex alley (i.e. an alley of vortices of alternated polarities, at a given latitude, with a constant spacing between their centers) we also vary the distance between each vortex d btw /R = [2.0, 3.0, 4.0] and the number of vortices N = [2,4,6]. In these cases, the center (in the zonal direction) of the vortex alley is at the center of the domain L domain /2. Therefore

L domain = N × d btw .
A summary of the three kinds of simulations we ran, with the different parameters we varied, is shown in Fig. 2.

Diagnostics

Modal decomposition

To analyse the growth of perturbations on the upwelling front, we decompose its deviation from a straight front in zonal Fourier series. We apply this analysis to the surface temperature field at each time step

T (x, y)-< T (y) >= j=∞ j=0 [c j (y) cos(jx) + s j (y) sin(jx)] (6)
where < T (y) > is the straight temperature front corresponding to u up (y, z = 0). Then, to examine only the modes of the upwelling front, we average c j and s j in a 100 km wide meridional band. This gives the time varying amplitude of each mode c 2 j + s 2 j y (t). In the following, we present and discuss the wavelength of each mode λ = L domain /j, with L domain the size of the domain (2000 km).

The frontogenesis function

To characterize the evolution of buoyancy gradients, we calculate the frontogenesis function F associated with the buoyancy. It is defined as follows [START_REF] Hoskins | The mathematical theory of frontogenesis[END_REF]:

F (u, b) = ∂ x u(∂ x b) 2 + (∂ x v + ∂ y u)∂ x b∂ y b + ∂ y v(∂ y b) 2 .
The opposite of the function F indicates the tendency of the buoyancy gradients to steepen:

-F (u, b) = d dt |∇b| 2 .

Particle advection

We ran particle advection simulations, using the outputs of CROCO simulations, with the set of python classes Parcels (Parcels stands for "Probably A Really Computationally Efficient Lagrangian Simulator"). This software simulates the advection of an ensemble of particles, using a given 2D or 3D velocity field. This tool has been widely used in the past few years, and it is fully described in Lange & van Sebille (2017); [START_REF] Delandmeter | The Parcels v2.0 Lagrangian framework: new field interpolation schemes[END_REF] and in references therein.

The surface velocity field is used here, and the forward advection is computed with a fourth order Runge-Kutta scheme (time spacing dt = 5 minutes). Particles are initially set at t =0 days on a regular grid, at 100 ≤ x ≤ 1900 km and 500 ≤ y ≤ 1500 km, with a spacing between particles of 25 km. Thereby, a total of 2993 particles are advected, throughout the year of simulation.

Analysis of the upwelling front instability

Modal analysis of growing perturbations

Firstly we studied the instability of the straight upwelling front alone (in the absence of any initial vortex). Only a sinusoidal perturbation was added along the front, to trigger its instability (see section 2.1.2 and Fig. 2(left)). The modal amplitude of the growing perturbations (the unstable waves on the front) for various perturbation wavelengths, when one neglects the β-effect (i.e. on the f -plane), is shown in Fig. 3(a). We distinguish the time series of amplitudes for a mode with a given wavelength initially (λ pert ), from those for the mode which is finally the most unstable one (λ max ), and has this wavelength (see also figure caption). If the initial wave grows the fastest, then λ pert =λ max .

When they are forced initially, waves shorter than the most unstable one flso grow but decay rapidly; thus they do not reach large amplitudes. They dominate at short time for fast jets and may thus be related to horizontal shear flow instability. On the contrary, longer waves reach larger amplitudes with time, either within a month or within 8 months.

In particular, waves with 200-250 km wavelength are the most unstable over a 1-3 months period; they are followed in amplitude by 500 km long waves. With an internal radius of deformation close to 50-60 km in this oceanic region [START_REF] Chelton | Geographical Variability of the First Baroclinic Rossby Radius of Deformation[END_REF], a wavelength of 2πR d = 300-350 km is characteristic of baroclinic instability. Therefore, the growth of these long waves in the first months of simulation can be related to the development of baroclinic instability along the jet. When long waves grow after a longer period (e.g. 8 months), their origin is rather to be searched in wave-wave interaction, since the flow is not forced.

A similar modal analysis but now including the β-effect is shown in Fig. 4(a). The β-effect renders flows zonal so that long waves are damped (see e.g. [START_REF] Flierl | Vortex formation by unstable oceanic jets[END_REF], or the wellknown Phillips model of baroclinic instability (Vallis, 2017)). As a result, the most unstable waves in our simulations lie near ∼100 km wavelength. In fact, the dimensionless value of β is

βL 2 /U = L 2 /L 2 Rhines .
For the same geophysical value of β = 2 × 10 -11 m -1 s -1 , this damping effect will be weak for a fast jet such as the Gulf Stream (U = 1 m s -1 ), but stronger for the jet associated with a typical upwelling front (in our simulations, U = 0.2 m s -1 ). Therefore meanders on the upwelling front are expected to have a shorter wavelength than those of intense jets such as western boundary currents.

3.2. Spatial structure of the front with growing perturbations 3.2.1. In the absence of β-effect

When short waves are forced initially, they can grow even if they are shorter than the linearly most unstable wave. This growth will only be transient. These short waves saturate in amplitude without breaking and then slowly decay. Longer waves break and form vortices of about 200 km diameter (see Fig. 3(b)). We note that northward extending (anticyclonic) meanders do not form vortices while southward (cyclonic) meanders do. This North-South asymmetry of meanders on a front can be explained by ( 1) the difference in local deformation radius (smaller to the North than to the South, if the upper ocean layer is thinner to the North); for baroclinic instability, a smaller deformation radius favors the growth of shorter waves (and conversely); and (2) the curvature vorticity V /r ∂ r V , which contrary to the geostrophic vorticity is parity biased (here V is the jet velocity magnitude in the meander, and r is the radius of curvature of the meander). These arguments can explain the growth of longer waves than those initialized, South of the jet, thus leading to the breaking of meanders into vortices. Such asymmetries do not appear in quasi-geostrophic simulations of jet instability.

The most unstable wave is determined by computing its growth during the early stage of the evolution. When this wave is initialized (e.g. λ pert = 250 km), it grows up to breaking the front, generating vortices of ∼200 km diameter. Since the most unstable perturbation is initialized everywhere, vortices form both north and south of the jet. Along the central latitude, the jet comes back to a quasi zonal state once the vortices have detached. This effect has been explained in previous studies [START_REF] Baey | Ocean jet instability: a model comparison[END_REF] and it can be clearly seen on Fig. 3(c).

If a longer wave is initialized, shorter waves emerge via nonlinear wave-wave interactions, and finally vortices of ∼200 km diameter form. These nonlinear wave interactions also generate other features such as filaments and smaller vortices (see Fig. 3(d)).

In the presence of β-effect

When β-effect is added, long waves do not grow and shorter waves dominate the evolution of the flow. This leads to the generation of filaments and of small vortices (of ∼50 km diameter). The most unstable wave form vortices of ∼100 km diameter. Since the β-effect renders flows zonal, meanders grow less north and south, and the perturbations remain confined near the front axis, see Fig. 4(b,c,d). Even long waves imposed initially on the front bifurcate towards smaller meanders. Still, they do not form very small scale features and thus the flow pattern is spatially more regular.

Upwelling front evolution in the presence of a single vortex

Now that the instability of the front, without external vortex, has been studied, we investigate the influence of a single vortex on this same front. The vortex is initialized south of the front and it replaces the perturbation added to the front in the previous section (see section 2.1.2 and Fig. 2(middle)). This vortex deforms the front along its own spinning motion. Here, we do not perform a Fourier analysis of the front perturbation; indeed, the wavelength which is the most unstable on the straight front in the absence of external vortex, is now perturbed by the presence of the vortex. Performing this analysis here would in particular reflect the vortex size. For the simulations discussed here, the vortex position, polarity, radius, and velocity are varied. The evolution of the front on the f -plane and on the β-plane is now presented. The position of the front is chosen as the isotherm corresponding to the average of the northern and southern temperature after 100 (resp. 200) days of simulated time in Fig. 5 (resp. Fig. 6).

The reference cases

To further study the influence of the vortex parameters on the front stability, we define reference cases, for which the vortex can have either polarity, a radius R = 100 km (corresponding to the radius of vortices formed by the instability of the front previously described), a distance to the front d = 2R, and an azimuthal velocity V vortex 0 = ±0.2 m s -1 initially. The front positions in these cases are shown by solid color lines in Figs. 5 and6.

In this case, on the f -plane, the presence of a vortex south of the front induces a meridional deformation of the front in its vicinity, but not over the whole length of the domain, during the first three months of simulation. At later times (200 days), this filament extrudes offshore and detaches from the front. Also, the most unstable wave of the front grows, leading to small scale perturbations along the whole front.

After 3 months, we observe that if the vortex is cyclonic, the front is more deformed than if the vortex were anticyclonic. This can be explained by the following. Though their relative vorticities are initially antisymmetric, their Ertel potential vorticities are asymmetric. Indeed, even at first order beyond quasi-geostrophy the absolute vorticity multiplies the density anomaly, and we can write this first-order Ertel potential vorticity anomaly as

δQ = ω + (f 0 + ω)b/N 2 ,
where b is the buoyancy anomaly associated with the vortex, and N the Brunt-Väisälä frequency N 2 = -∂b ∂z . δQ characterizes the vortex ability to remotely influence its environment.

Since ω and b are antisymmetric at first order for cyclones and anticyclones, δQ is larger in modulus for cyclones. Again at first order, this potential vorticity anomaly is conserved in time. [START_REF] Stern | On the interaction of a vortex with a shear flow[END_REF] have shown that in two-dimensional flows, a vorticity front develops a meander with equal and opposite circulation to that of the vortex deforming it.

Applying this principle here explains why fronts extrude longer and/or larger filaments when nearing a cyclone than an anticyclone; this can be seen after 100 simulated days. After 200 days, the perturbation breaks down non-linearly and becomes spatially more convoluted and intricate. These nonlinear effects can also lead to the production of long offshore filaments by fronts under the influence of anticyclones, but this effect is weaker than after 100 days.

Meanders and filaments are much smaller in the presence of β-effect than in its absence.

Sensitivity to the vortex radius

The influence of the vortex radius R on the flow evolution is studied for a vortex lying at South of the front). The growth of a longer filament in the presence of a larger vortex is also associated with a weaker growth of the 200 km long wave on the front; the evolution of the front under the influence of the vortex supersedes the unstable evolution of the front (alone).

At long times, the long frontal wave recovers its strength of the jet instability case, for both small and large vortices. In the presence of β-effect, cyclonic vortices drift northwestward and thus come closer to the front while anticyclonic vortices drift away from the front on the β-plane, explaining the difference in the generated filament lengths. Again, the zonal straightening effect of β on streamlines prevails, so that shorter filaments are generated on the β-plane than on the f -plane.

Sensitivity to the vortex intensity

The influence of the vortex intensity V vortex 0 is studied for a vortex lying at a distance d = 2R from the front, with a radius R = 100 km initially, see Figs. 5(c,d) and 6(c,d). When

V vortex 0 increases, so does the vortex dynamical influence on the front. Thus, the filaments growing from the front increase in size.

On the f -plane, after 100 days, the filaments wrap up around the vortex. Though the perturbation is concentrated around the quasi-steady vortex, this interaction leads to much offshore export of coastal fluid. Again, cyclones disturb the front more deeply than anticyclones do. After 200 days, filaments still protrude far offshore on the f -plane, but forced waves have now propagated along the front axis so that much offshore-coastal fluid exchange occurs, all along the front axis.

On the β-plane, short waves develop on the front, with smaller amplitude than on the fplane. The most intense external vortices induce more small scale turbulent features. Again, more small-scale features are induced by cyclones than by anticyclones. After 100 days, these small-scale features remain trapped between the front and the vortex and do not drift far offshore. This can be related to the flow straightening by the β-effect, and to the westward vortex drift which, for the front and its meanders, displaces the external deformation field.

After 200 days, turbulence has developed from its state at 100 days. It extends all along the front and the inshore-offshore fluid exchange is amplified. Still, no coastal fluid is advected offshore beyond the vortex. Here, the meridional gradient of upper layer vortex stretching (across the front) is positive, which increases the westward Rossby wave speed so that frontal meanders are located more to the West than on the f -plane.

Sensitivity to the vortex distance to the front

The influence of the distance between the vortex and the front d/R is studied for a vortex with a radius R = 100 km, and an azimuthal velocity V vortex 0 = ±0.2 m s -1 initially, see Figs. 5(e,f) and 6(e,f). The same observations as in the previous subsection hold here.

On the f -plane, after 100 days, long waves grow and large meanders break into offshore filaments when the vortex is close to the front (d/R =2 or 3). After 200 days, shorter waves develop and affect the whole length of the front. Here again, cyclones induce more pronounced deformations and longer filaments than anticyclones. When d/R = 4 or 5, very little deformation of the front occurs, both on the f -and on the β-plane. As a consequence, vortices lying 400 km away from the upwelling front influence it very little, as could be anticipated from the calculation of the velocity field. On the β-plane, the stabilizing effect of the planetary vorticity gradient confines the waves and the turbulence to the vicinity of the front.

Interaction of an upwelling front with a vortex alley

Though the interaction of a single vortex with an upwelling front is an important step in our study, and though analyzing this case provides important and general conclusions, sea surface measurements in the Arabian Sea show that the southern Omani upwelling is rarely bordered by only one vortex; multiple vortices surround the front (see e.g. Fig. 1).

Therefore, we generalize the previous case to that of a vortex alley along the front (see section 2.1.2 and Fig. 2(right)). We use the physical vortex parameters of the most significant case previously analyzed: V vortex 0 = 0.2 m s -1 , R = 100 km, and d/R = 2. Here we vary the alongshore distance between the vortices d btw , the number of vortices N , and the presence or absence of β-effect.

Sensitivity to the number of vortices

We present the case with d btw /R = 4, which shows the highest efficiency for filament production, for N = 2 -6 vortices, see Figs. 7(a,b) and 8(a,b). On the f -plane, at t = 100 days, the effect for each vortex is comparable to that of a single vortex, i.e. long filaments protrude offshore from the front. The velocity field field resulting from the addition of the vortices is stronger at the center of the domain, and thus the filaments are longer there.

Filaments are advected southward between an anticyclone (to the West) and a cyclone (to the East). On the contrary, the front is pushed northwards and forms a wide meander to the North, between a cyclone (to the West) and an anticyclone (to the East). At t = 200 days, the filaments have been cut, and some of them are brought closer to the front by the velocity field of the zonal jet. We can also see (as in the reference cases, see section 4.1), the development of shorter waves, which now grow on the wide meanders. This state is comparable to that of the nonlinear evolution of the unstable front in the absence of external vortices initially. Note a difference between the present case and that with a single external vortex: here, the front is shifted North, which did not occur with only one vortex. On the β-plane, again, shorter and less prominent meanders and filaments are produced after 100 days. These waves break into a turbulent field, in the vicinity of the front, after 200 days.

Sensitivity to the distance between vortices

The influence of the distance between vortices within the vortex alley d btw /R is studied for a vortex alley of 6 vortices (i.e. N=6, this corresponds to cases with the most intense deformation of the front), see Figs. 7(c,d) and 8(c,d). We do not detail each figure individually, but we note that for short times, the larger the inter-vortex distance is, the stronger the offshore mass transport is. Indeed, the alongshore extent of the front deformation and the filament widths increase with d btw . At longer time, the northward displacement of the front is larger for small d btw because the vorticity dipoles which are generated are more intense (the distance between the centers of the two vortices in the dipole being smaller).

On the contrary, on the β-plane, the front show more long wave (200 km length) instability when d btw is larger (the vortices triggering longer waves). Again, the β-effect reduces the amplitude of meanders and intensifies turbulence in the vicinity of the front.

6. Discussion: cross-shore exchange of particles

We discuss here the impact of vortices and of upwelling destabilization on the transport of fluid particles in the domain. In particular, we present the export and import of particles across the y-position of the upwelling, to assess the role of the different dynamical elements in the particle transport off-shore (from North to South) and on-shore (from South to North).

This point is key to the local dynamics of upwelling rich regions because this can trigger phytoplankton blooms [START_REF] Shi | The Oman upwelling zone during 1993, 1994 and 1995[END_REF][START_REF] Liao | Potential new production in two upwelling regions of the western arabian sea: Estimation and comparison[END_REF].

To do so, we ran particle advection simulations (see details in section 2.4) in 4 particular CROCO simulations, which are the most representative of the upwelling-vortex interactions behavior. For further use, we define onshore as shorewards of the front (i.e. North of the front), and offshore conversely. The particle evolution with time is presented for an upwelling front with a sinusoidal perturbation (a), in the presence of a cyclone (b), of an anticyclone (c), or of a vortex alley (d), on the f -plane (Fig. 9) or on the β-plane (Fig. 10).

For the perturbed front, the particles drift onshore and offshore nearly equally on the f -plane, while the cross-shore exchange is strongly reduced on the β-plane (as expected).

When the front faces a single cyclone, on the f -plane, offshore particles are trapped in a meander which wraps counterclockwise and shore-ward, after 100 days. After 200 days, this meander undergoes instability and produces shorter scale meanders; it breaks and gives birth to many small vortices inshore of the front. The offshore meander at 100 days produces a long-wave perturbation on the front, leading to the offshore motion of onshore particles. This meander also breaks into smaller fragments (small vortices and filaments). Such fragments are seen on Chlorophyll-a images of the Oman upwelling front (such an image can be found in lecture 17 of Lisa Beal's oceanography course: https://bealagulhas.rsmas.miami.edu/teaching/courses/lecture-seventeen/index.html). On the β-plane, the cross-shore displacement of particles, and their meridional flux, are strongly reduced, but small-scale patches of displaced particles are still present on both sides of the front. When the front faces an anticyclone on the f -plane, a large meander of coastal water is pulled offshore and carries passive particles across the front after 100 days. This meander breaks into a coherent, medium scale cyclone, next to the anticyclone. After 360 days, both the vortex interactions and the front breaking lead to the production of small-scale filaments and vortices offshore. This phenomenon is considerably weakened on the β-plane where only three small vortices containing coastal water are found offshore after 200 days. After 360 days, particles have been displaced inside the offshore region, but little cross-front exchange has taken place. In the presence of a vortex row, on the f -plane, the intrusion and extrusion of fluid via the front meanders and the subsequent filaments lead to substantial cross-front exchange, in particular via small-scale features. On the β-plane, the cross-front flux is still the largest of all simulations, visually.

The median particle export distance for each case, in the onshore and offshore direction, and the percentage of exported particles are shown in Fig. 11. On the f -plane, the front with a vortex alley or with a cyclone displace more particles than the unstable front or the front with an anticyclone, but the front with a cyclone advects particles only close to the front. On the β-plane, much fewer particles are advected across the front and the export distance is strongly reduced. Again, the front with a cyclone or with a vortex alley advect more particles. The latter case is the most efficient to horizontally stir particles. Finally, the frontogenesis function computed in the front-vortex alley case (Fig. 12) indicates that frontogenetic tendency is large around the small vortices and formed filaments.

For a comparison of these results with the upwelling and vortices off Oman (see Fig. 1), we can see that the cyclone lying offshore of the front (at 17 • N, 57 • E) pulls a long filament away from the front. On the SST map (Fig. 1(a)) cold water extends offshore at least over 300 km. South of the main cyclone, a smaller cyclone, containing cold water, is found.

Another filament, shorter than the former, is torn away from the upwelling front by the anticyclone lying at 20 • N and 60 • E. On the SST and SSH maps, contrary to our model, more vortices lie farther offshore of those close to the front. They also carry cold water away from the coast and make it recirculate.

These results suggest that the impacts of upwelling front instability or front interaction with a vortex field on biological activity are expected to be important. Firstly, nutrient rich onshore waters will be exported offshore by the meanders which then break into filaments and into small vortices. These flow structures will be the seat of intense biological activity (blooms) favored by the further vertical uplift of nutrients to the surface, related to frontogenesis. Secondly, exchange of coastal and deep water species will be achieved by these horizontal exchanges, leading to a possible modification of local ecosystems, and to possible competitive exclusion of species, in an otherwise protected environment for them.

a=

  distance d = 2R from the front, with an azimuthal velocity V vortex 0 ±0.2 m s -1 initially, see Figs. 5(a,b) and 6(a,b). Under these conditions, wider vortices lead to longer filaments offshore both after 100 and 200 days. On the f -plane, smaller vortices yield more wave amplification downstream, and in particular more coastal intrusion of open ocean fluid (i.e

Figure 1 :

 1 Figure 1: (a) Bathymetry in the Arabian Sea and adjacent gulfs from ETOPO2 (Smith & Sandwell, 1997), dashed rectangle shows the geographic position of panels (b,c). (b) Sea Surface Temperature and (c) Sea Surface Height (SSH) along the Omani coast in the Arabian Sea on 05/08/2020. Arrows show the geostrophic velocity derived from SSH. Cold filaments resulting from the steering of the upwelling by mesoscale eddies are seen near 56 • E 16 • N. Data are from Operational Mercator global ocean analysis and forecast system (downloaded on Copernicus website).

Figure 2 :

 2 Figure 2: Scheme of the three kinds of simulations we performed and the different physical parameters we varied.

Figure 5 :;

 5 Figure 5: Position of the upwelling front (initially L = 50 km, H = 200 m, and V up 0 = 0.2 m s -1 ) at t = 100 days, in simulations with an isolated vortex of parameters (a,b) d/R=2, V vortex 0

Figure 7 :

 7 Figure 7: Position of the upwelling front (initially L = 50 km, H = 200 m, and V up 0 = 0.2 m s -1 ) at t =

Figure 9 :

 9 Figure 9: Snapshots of particle advection simulations with an upwelling front of parameters L = 50 km, H = 200 m, and V up 0 = 0.2 m s -1 , and (a) a sinusoidal perturbation (λ pert = 250 km), (b, resp. c) an isolated cyclonic (resp. anticyclonic) vortex (R = 100 km, H vortex = 1000 m, V vortex 0

Figure 10 :

 10 Figure 10: Same as Fig. 9 but with β-effect added in simulations.

Figure 11 :

 11 Figure 11: Median export distance of particles after one year of advection in different simulations on the f -plane (left) or the β-plane (right). Errorbars show the standard deviation of export distance, and width of boxes indicate the percentage of exported particles (see the size bar in each panel). Note that scales are different in each panel. All simulation have initially an upwelling front with L = 50 km, H = 200 m, and V up 0 = 0.2 m s -1 ; "wave" corresponds to simulations with a sinusoidal perturbation (λ pert = 250 km), "isolated CE (resp. AE)" corresponds to simulations with an isolated cyclonic (resp. anticyclonic) vortex (R = 100 km, H vortex = 1000 m, V vortex 0

Figure 12 :

 12 Figure 12: Time evolution of -F in the simulation with an upwelling front of parameters L = 50 km, H = 200 m, and V up 0 = 0.2 m s -1 , and a vortex alley (R = 100 km, H vortex = 1000 m, V vortex 0

Figure 6: Same as Fig. 5 at t = 200 days

Acknowledgments

This work was funded by the Direction Générale de l'Armement (DGA) via a full grant for Charly de Marez's PhD. This work is also a contribution to PRC 1069 'Meso and submesoscale vortices in the Atlantic and Indian Ocean', funded by CNRS and RFBR. Simulations were performed using the HPC facilities DATARMOR of "Pôle de Calcul Intensif pour la Mer" at Ifremer, Brest, France. Model outputs are available upon request.

Conclusions

We have studied the evolution of an unstable upwelling front and of a front in the presence of vortices offshore. We have not considered a stable front in the presence of external vortices, because upwelling fronts are naturally unstable. The characteristic wavelengths and coherent structures produced by these evolutions have been determined. Their impact in terms of horizontal fluxes of particles has been studied. In particular, wavelengths of 100 or 200 km appear on the front, and small-scale filaments and vortices are finally produced.

Frontogenesis is strong around them. Mesoscale vortices and smaller scale features advect particles across the front. The timescales of 3-6 months considered here can be made shorter by considering faster frontal currents (e.g. in the linear approximation, a 0.4-1 m s -1 frontal jet will reduce the timescales by factors of 2 to 5, leading to periods of about 1 month for the formation of meanders and filaments). Numerical simulations using particles advection have shown that a vortex alley is the most efficient perturbation to the upwelling front, in terms of cross-front transport (compared with a single vortex, or with the instability of the front alone).

Still, this study remains idealized in terms of flow conditions. First and foremost, it has only considered the inertial (unforced) evolution of the front. We will extend these results in a following study by adding wind stress and/or the front relaxation to a prescribed state, to assess the energetic balance between atmospheric forcing and mean flow instability. Bottom topography and coastal irregularities should also be added as they can alter or favor the formation of filaments and of vortices from upwelling fronts [START_REF] Meunier | Influence of bottom topography on an upwelling current: Generation of long trapped filaments[END_REF]. The wind variability is also essential in the evolution of upwelling fronts, by either weakening them or amplifying them. Further studies with nested models of the Arabian Sea and of the region south of Oman will be done in the near future. They will benefit from in situ measurements performed during the Physindien 2019 experiment South of the Sultanate of Oman, for their validation.

Participation in the study

The authors declare their participation in the study, CdM for conception, numerical work and contribution to the paper writing, XC for conception, participation in the analyses and