AVERAGING METHOD APPLIED TO THE THREE-DIMENSIONAL PRIMITIVE EQUATIONS

. In this article we study the small Rossby number asymptotics for the three-dimensional primitive equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-called renormalisation group method.


1.
Introduction. The geophysical fluids are influenced by rotational and stratification effects. The study of the limit of the equations describing these flows, as the rotation and the stratification are very important, is a problem of major interest from the theoretical and computational points of view.
In this article we study the small Rossby number asymptotics for the threedimensional primitive equations of the ocean and the atmosphere. When a small parameter, related to the Rossby number, goes to zero, the solution undergoes fast oscillations which we would like to eliminate by an averaging method. In order to average the exact solution, we use the so-called renormalisation group method, which was introduced by Schochet in [22,23]. The form of the method that we use here is due to Ziane [32]. The method was introduced in a physical context by Chen, Goldenfeld and Oono [6] and used in a mathematical context, for rotation fluids and geophysical flows by Chemin [7], Embid-Majda [9], Genier [12] and many others. Many more articles on the subject of the renormalisation group method are available in the physics and mathematical literatures; we mention here the works of Gallagher [10,11], of Babin, Mahalov, Nicolaenko [3]- [5], of Moise, Temam, Ziane [16]. In the context of ODEs Temam and Wirosoetisno applied the method to higher orders [24].
This article is organized as follows: in Section 2 we introduce the three-dimensional primitive equations and recall some important results on the global-in-time existence of weak and very strong solutions. In Section 3 we recall the idea of the renormalisation group method, given in an abstract context. In Section 4 we apply the renormalised group method to the primitive equations, we construct an approximate solution and we study the error between the exact solution and the approximate solution. The Appendix contains a technical result regarding the way we can bound some small denominators, result necessary for the error estimates.
2. The three-dimensional primitive equations. In this section we introduce the three dimensional primitive equations written in a non-dimensional form and we recall the available results on the global in time existence and regularity of the solutions. The equations are considered on the domain The primitive equations read: where (u, v, w) is the three-dimensional velocity, p is the pressure, ρ is the density and ε is the Rossby number. Here ν v and ν ρ are the non-dimensional eddy viscosities, N is the Burgers number and (S u , S v , S ρ ) is a forcing term. The variable ρ is the perturbation of the density from a stably-stratified profile, the full density of the fluid being given by ρ full = ρ 0 +ρ + ρ, (2.2) whereρ is the density stratification profile which is assumed to be linear since the Brunt-Väisälä frequency is assumed to be constant. The total pressure is given by p full = p 0 +p + p, (2.4) where p 0 ,p, p are respectively in hydrostatic equilibrium with ρ 0 ,ρ and ρ. For simplicity, we assume periodic boundary conditions for the perturbation variables. The variables of the system are of two types: u, v, ρ are the diagnostic variables for which we prescribe an initial condition while p and w are the prognostic variables that can, at each instant of time, be determined in terms of the prognostic variables.

AVERAGING THE THREE-DIMENSIONAL PRIMITIVE EQUATIONS 5683
The vertical velocity w is determined in terms of U = (u, v, ρ) from the incompressibility condition (2.1) 4 and the periodicity and antisymmetry (below) conditions in x 3 , The pressure is determined from the hydrostatic relation (2.1) 3 , up to the surface pressure p s , Moreover, we assume the following symmetry properties on the variables: these assumptions are typical in numerical simulations of stratified turbulence (see e.g. [1]). In order for these symmetries to persist, we need S u , S v to be even in x 3 and S ρ to be odd in x 3 . We introduce the following function spaces: In (2.5) and elsewhere, we denote byḢ m per (M), with m ≥ 0 integer, the functions of H m per (M) with zero average on M.
2.1. Variational formulation of the problem. The variational formulation is the following: Given t * > 0 arbitrary, and In (2.6) we introduced the following forms: We also define the linear operators 12) and the bilinear operator Problem (2.6) can be thus written as an abstract evolution equation: (2.14) The existence of weak solutions for (2.6) was proved in [14], while the existence and uniqueness, globally in time, of strong solutions was proved in [8] and [13]. The high order regularity of the solution of (2.6) was proved in [18]. All these results are collected in the following theorem: Theorem 2.1. Given U 0 ∈ H and S ∈ L ∞ (R + , H), there exists at least one solution U of (2.6) with initial condition (2.7) such that If U 0 ∈ V and S ∈ L ∞ (R + , H), there exists a unique solution of (2.6)-(2.7) such that 3. The renormalisation group method. The averaging method we will use in what follows is known as the renormalisation group method. This allows us to study the asymptotic solutions of an equation which can be written in the following general form: where ε is a small parameter, L is a diagonalizable, antisymmetric linear operator and F is a non-linear operator. The fact that L is antisymmetric explains why the solutions display large oscillations when ε is small. This problem has two natural where we denoted V (s) = U (εs). We start by writing a naive perturbation expansion for V , We substitute (3.3) into (3.2) and we finally derive and so on. From (3.4) we find V 0 (s) = e −Ls U 0 . Using the variation of constants formula to (3.5), we obtain where the term F r which is independent of time is called resonant and the remaining, time-dependent term F n is called non-resonant. We define and we can write V 1 (s) = e −Ls sF r (U 0 ) + εF np (s, U 0 ) . (3.8) We thus find our leading-order approximate solution, In (3.9) we remove the term εs by searching for a functionŪ having U 0 +εsF r (U 0 ) as first order Taylor expansion. This justifies to introduce the first order renormalised group equation 10) and to consider the first-order approximate solutioñ U 1 (s) = e −Ls Ū (s) + εF np (s,Ū (s)) . (3.11) The main issue now is to solve equation (3.10) and to compare the approximate solution (3.11) to the exact solution of (3.2) and to prove that the error is of order ε in an interval of time s of order O( 1 /ε). For more details on this method, see e.g. [15], [17], [24], [19].

4.
Averaging the three-dimensional primitive equations. As announced before, in this section we are interested in applying the renormalization group method described in Section 3 to the three-dimensional primitive equations. The first step is to deduce the renormalised group system (3.10) that corresponds to the primitive equations and to study the well-posedness of this system. Thus, we first introduce the fast time s = t/ε in system (2.1). Since all the functions we are working with are (space) periodic, they admit Fourier series expansions. Thus, we write any wavevector k is henceforth understood to live in Z M . Thus the primitive equations written in the fast time variable s = t/ε and in Fourier modes, read For k 3 = 0 we obtain the k-component of the diagnostic variables p and w in terms of the prognostic variables, Putting (4.3) into (4.2), we obtain The k-components of the operators A, L and B are thus For k 3 = 0, we know that k 1 u k + k 2 v k = 0, which corresponds to We also know that ρ k = 0. We introduce the following notations: and for all k ∈ Z M we writek = (k 1 , k 2 ) and denotẽ We note that the unknowns u k , v k are not independent due to the constraint k 1 u k + k 2 v k = 0. We also remark here that v ⊥ k +k (k ∧ v k )/|k| 2 = 0. In this notation, the operators read In order to deduce the renormalized group system, we need to compute, as in (3.7), e Lτ F(e −Lτ U 0 ) mode by mode to find the resonant F r and the non-resonant part F n of F. We recall that in our case We need the eigenvalues and the eigenvectors of L to compute the terms in (4.14).

MADALINA PETCU, ROGER TEMAM AND DJOKO WIROSOETISNO
If |k| = 0, we have iω ± k = ±sgn(k 3 )i and the corresponding eigenvectors are For k 3 = 0, we introduce the following vectors in order to have unified notations for all cases: Equation (2.14), written in terms of the new variable V , becomes We now compute, mode by mode, the resonant and the non-resonant parts ofF and write the renormalised group system in terms of the new variable.
We compute eL τF (e −Lτ V ) for k 3 = 0. Since eL τS and eL τÃ (e −Lτ V 0 ) are fully resonant, the only term we need to compute is eL τB (e −Lτ V, e −Lτ V ). The k-mode of equation (4.20) reduces to an equation for v 0 k : is fully resonant. The only term that could have a nonresonant part is Thus, we replace U by Q −1 eL τ V in (4.22) and we notice that the only possible resonant terms come from the interactions ω + j + ω + l = 0 and ω − j + ω − l = 0 since ω + j + ω − l = 0 would imply that j 3 and l 3 have the same sign, which contradicts the fact that j 3 + l 3 = 0.
The coefficient corresponding to ljk and similar notations are used for all the coefficients. With these notations, the resonant part of (4.22) is Proof. Relation (4.25) follows from direct computations.
Proposition 4.1 implies that the renormalised group equation will not contain terms in v + and v − . Writing (4.21) with U replaced by we find the following renormalised group equation for the case k 3 = 0, Collecting the nonlinear terms in (4.26), we find the following relation for the case k 3 = 0, We also need to do the same kind of computations for the case k 3 = 0. In order to simplify the writings, we denote We need to compute the resonant parts of the terms eL τÃ (e −Lτ V ) and eL τB (e −Lτ V, e −Lτ V ).

MADALINA PETCU, ROGER TEMAM AND DJOKO WIROSOETISNO
For the linear term we compute which implies that the resonant part is Remark 4.1. From formula (4.29) we easily notice that the operatorÃ r is still coercive.
We also need to compute (eL τB (e −Lτ V, e −Lτ V )) k for k 3 = 0. We write and we compute the resonant part for each term B i k with i = 1, 2. We find where c is a notation for the cyclic sum j+l=k and n indicates the cyclic sums for which j 3 = 0, l 3 = 0, k 3 = 0. After similar computations forB 2 k , the resonant part of the nonlinear term is a three-dimensional vector having the following components: (4.35) These coefficients also have the following properties: Interchanging l and j in (4.32)-(4.34) and using properties (4.36)-(4.38), the renormalised group system finally reads 4.1. Study of the renormalised system. In what follows we consider the case where three-wave resonances are not possible, i.e. ω ± j + ω ± l = ω ± k for j + l = k. In the Appendix we see that this scenario cannot happen when the Burgers number N lies outside a certain quasi-resonant set. In this case the renormalised equation is In system (4.40) we notice that the last equation decouples completely from the first two equations so we can start by studying the well-posedness of this equation. We also notice that the first two equations on (v + , v − ) are bilinear and can be written with A ± , B ± (v 0 ), S + , S − given mode-by-mode by equations (4.40) 1 -(4.40) 2 .
As announced earlier, we start by studying equation (4.40) 3 for v 0 k . Introducing the quasi-geostrophic potential vorticity q k = |k| N v 0 k as new unknown, equation (4.40) 3 becomes This equation is known as the three-dimensional quasi-geostrophic equation and it was studied by Babin, Mahalov and Nicolaenko in [2]. We thus have: where by K m we denote a constant depending on ν 0 but independent of the initial condition and t m (q(0)) is a time depending on the H m -norm of the initial data q(0).
Knowing these results on the regularity of v 0 , we can obtain the existence and uniqueness of a solution (v + , v − ) in L ∞ (R + , (Ḣ m (M)) 2 ), for all m > 0, provided the initial condition (v + (0), v − (0)) and the forcing term (S + , S − ) are inḢ m (M) 2 . Gathering the informations on v 0 with the results on (v + , v − ), we are actually able to prove the following result: Moreover, if r > 0 is a fixed arbitrary constant, then

46)
where b m is a constant independent of the initial data.
Proof. Taking into account the results on v 0 , it remains to prove the regularity results on (v + , v − ). By direct computations we can prove that which together with the coercivity of the operator A ± implies the results (4.45)-(4.46) for m = 0.
The results (4.45)-(4.46) for an arbitrary m ∈ N, m ≥ 1 follow from a recursive argument. Thus, we suppose that we have where by b m−1 we denote a constant independent of the initial condition (v + , v − ).
The same kind of estimate is available for the case ω s1 l = ω s2 k . We can thus bound the most difficult terms from ( . Introducing (4.50) and (4.51) into (4.49), we find We need to compareŨ 1 to the exact solution V, which satisfies (3.2), meaning that we need to evaluate the error W (s) =Ũ 1 (s) − V (s). The error satisfies − εB(e −sL F np (s,Ū ), e −sL F np (s,Ū )) − e −sL ∇Ū F np (s,Ū ) · F r (Ū ). (4.55) 4.2.1. L 2 error estimates. In order to obtain the L 2 error estimates we take the scalar product in (L 2 (M)) 3 of (4.54) with W and using the anti-symmetry property of L and the coercivity property of A, we find 1 2 where V is the dual space of V.
In order to bound the trilinear terms in the rhs of (4.56), we use the following result (the proof can be found in [20]): into R, and the following inequalities hold: Thanks to Young's inequality, (4.56) and (4.60) imply  From (4.58) we find that (4.63) Using (4.63) and the fact that e −sL conserves all Sobolev norms, inequality (4.62) implies (4.64) We need to bound |F np (s,Ū )| H 2 and |∇Ū F np (s,Ū ) · F r (Ū )| V . We start estimating F np (s,Ū ). We recall that F n (s,Ū ) = A nŪ +B n (Ū ,Ū )+S n , with A n , B n respectively given by the time-dependent terms in (4.28) and (4.30).
In order to estimate we need to bound the integrals in time of the terms e iτ ω ± k , e iτ (ω ± j +ω ± l ) and e iτ (ω ± j +ω ± l +ω ± k ) . The integral j + ω ± l | is estimated as |I 2 (j, l)| ≤ 2 when ω ± j and ω ± l have the same sign and as follows when ω ± j and ω ± l have opposite signs. The integral , is estimated using Theorem 4.6 proved in the Appendix. Using the estimates for I 1 (k), I 2 (j, l) and I 3 (j, l, k), we are now able to bound F np (s,Ū ) in H 2 . Since A np (s,Ū ) contains only terms of type I 1 (k), we find In bounding B np (s,Ū ), the most difficult terms to estimate are those of the type where f = k∈Z M |Ū k ||k| 8 e ik·x , g = k∈Z M |Ū k | |k| 5 e ik·x and µ and γ are as in Theorem 4.6.
Since H m (M) is a multiplicative algebra for m ≥ 2, we find and if m = 0, 1, then We also remark that |S np | m ≤ |S| m , ∀m.
This allows us to conclude that Thus, for the first term in (4.70) we have For the second term in (4.70) we apply an argument similar to the one used in (4.66) Relation (4.72) leads us to We only need to find bounds for |F r (Ū )| H m , ∀m ≥ 1, in order to conclude. Since while for the nonlinear operator B r we find, using similar arguments as in (4.51) with h 1 = k |k||Ū k |e ik·x and h 2 = k |Ū k |e ik·x .

AVERAGING THE THREE-DIMENSIONAL PRIMITIVE EQUATIONS 5699
For m = 0, 1, we have while for m ≥ 2 we find Returning to T 2 , we find Similar estimates can be deduced for the term B np (s, F r (Ū ),Ū ). Returning to (4.64) and recalling the fact that the renormalized group system is globally well-posed in all Sobolev spaces, provided the initial data is regular enough, we find that we can bound R ε as |R ε | V ≤ c(N, µ, |S| 9 , |U 0 | 10 ). (4.79) Thus, (4.61) can be written as where f 1 (s) = εc Ũ 1 2 Ũ 1 2 H 2 and g 1 (s) = ε 2 c|R ε | 2 V . From (4.79) we know that g 1 is an L ∞ (R)-function. Recalling formula (4.53) forŨ 1 as well as relation (4.69), we also find that f 1 is an L ∞ (R)-function. Applying Gronwall lemma to (4.80), we conclude with the following result: , the difference between the exact solution U of (2.6) and the approximate solutionŨ 1 in (4.53) satisfies where k and k are constants depending on N , L 1 , L 2 , L 3 , µ, |U 0 | H 10 and |S| H 3 .

4.2.2.
H m -error estimates, for m ≥ 1. In order to estimate the H m -norm of the error, we take the L 2 -scalar product of equation (4.56) by (−∆) m W . We obtain (4.81) The first term from the right hand side of (4.81) is estimated as We need to bound R ε in the H m−1 -norm: Using (4.69), we find where c(N, µ, |S| m+1 , |U 0 | 9+m ) is a constant depending on h, µ, |S| m+1 and |U 0 | 9+m . Using (4.65) we know and using (4.74)-(4.77), we find The last terms in (4.83) is bounded using (4.67) and (4.68). It only remains to bound |B(U,Ũ )| H m and we do this similarly to (4.66). The most difficult terms in B(U,Ũ ) are the terms containing δ i j l 3 and they are bounded as follows: with f = k∈Z M |k||U k |e ik·x and g = k∈Z M |k||Ũ k |e ik·x . For m ≥ 2, H m is a multiplicative algebra and we continue estimating T 3 as For m = 0 we find and for m = 1 we have We can thus conclude with the following estimate: ≤ c|U | m+2 |Ũ | m+2 for m = 0, 1. (4.86) Using (4.86) and (4.69), we find The same arguments are obtained for |B(e −sLŪ , e −sL F np (s,Ū ))| m−1 . We can thus conclude that ≤ c(N, µ, |S| m+1 , |U 0 | m+9 ) for m = 1, 2. (4.87) In order to be able to estimate the last three terms in (4.81), we need to be able to bound terms of the form b(U,Ũ , (−∆) m U # ). We use the following lemma: 3 . Then the following inequality holds: Proof. Relation (4.89) is obvious. For (4.88), we estimate as follows: (4.90) For the first sum, we find The second sum is bounded as
We continue estimating the terms as follows For every T > 0 we can find an ε T > 0 such that for all ε ≤ ε T we have |W (s)| 1 ≤ c 1 /2c, which implies that estimate (4.98) holds globally on the interval [0, T ]. We proceed similarly for m = 2. We can thus conclude with the following theorem on the error estimates in H 1 or H 2 . , we have For all T > 0 there exists ε T > 0 such that for all ε ≤ ε T , the error between the exact solution U of (2.6) and the approximate solutionŨ 1 given by (4.53) satisfies |Ũ 1 (t) − U (t)| m ≤ ε 2 ke tk , ∀t ∈ [0, T ], (4.99) where k and k are constants depending on N , µ, |S| m+1 , |U 0 | m+9 , L 1 , L 2 and L 3 .