MAXIMUM PRINCIPLES FOR THE PRIMITIVE EQUATIONS OF THE ATMOSPHERE

Transcription

1 DISCETE AND CONTINUOUS Website: htt://aisciences.org DYNAICAL SYSTES Volume 7, Number, Aril AXIU PINCIPLES FO THE PIITIVE EQUATIONS OF THE ATOSPHEE Brian D. Ewald and oger Témam, Institute for Scientific Comuting and Alied athematics Indiana University awles Hall 831 E. Third Street Bloomington, IN Université de Paris-Sud et Centre National de la echerche Scientifique Laboratoire d Analyse Numérique d Orsay Bâtiment Orsay Cedex, France Abstract. In this article, maximum rinciles are derived for a suitably modified form of the equation of temerature for the rimitive equations of the atmoshere; we consider both the limited domain case in Cartesian coordinates and the flow of the whole atmoshere in sherical coordinates. Introduction. The rimitive equations of the atmoshere and of the ocean have been studied, from the mathematical viewoint, in Lions, Temam, and Wang [6], [7]; see also [8], [9] for the couling of the atmoshere and the ocean. Concerning the equations of temerature, the classical methods used for the maximum rincile aly to the temerature equation for the ocean but they seemingly do not aly to the temerature equation for the atmoshere. This does not aear to be due to a mathematical technicality but rather to a difference of structure of the equations, the water being incomressible and the air being comressible. On hysical grounds, it was suggested to us to consider a modified temerature equation for the atmoshere, namely the equation for the otential temerature θ, whose definition is recalled below. We have been able, in this way, to derive the desired estimates. The aim of this article is to introduce the otential temerature equation and to derive L estimates for θ which rovide, afterwards, ositivity and L estimates for the classical temerature T. The article is organized as follows: in Section 1, we recall the rimitive equations of the atmoshere PEs and introduce the otential temerature and the corresonding equations. In Section, we rovide the weak formulation of the PEs in a limited domain with suitable, hysically reasonable boundary conditions, and we establish the existence of weak solutions. Then, maximum rinciles are established in Section 3 by a combination of the truncation Stamacchia method for the ositivity and classical methods for the L bound. In Section 4, we describe the similar results for the whole atmoshere and resent the changes in the roofs which are necessary in this case. 000 athematics Subject Classification. 35Q35, 86A10, 37N10, 76U05. This work was suorted in art by the grant NSF-DS and by the esearch Fund of Indiana University. 343

2 344 BIAN D. EWALD AND OGE TÉA The second author T wishes to thank Joe Tribbia for suggesting that he consider the otential temerature and for introducing him to the thermodynamics of the atmoshere. 1. The Primitive Equations of the Atmoshere. In this section, we recall the rimitive equations and the associated boundary conditions for a limited region of the atmoshere. We also introduce the otential temerature and the corresonding equation The rimitive equations. We first recall the equation of temerature for the atmoshere see [10], beginning on.71: dt c dt T ω = dq + D. 1.1 dt In this equation, T is the actual atmosheric temerature;, the ressure, is used as the vertical coordinate; and ω = d is the vertical velocity in the ressure coordinate system. Both c and are constants the secific heat of the atmoshere at dt constant ressure and the gas law constant, resectively, and Q is the heat alied to the system. In 1.1, D reresents a dissiation term. As in [6], we choose D = L T T see below, and we arrive at the following modification of equation 1.1: where dt c dt T d dt L TT = Q E, 1. d dt = t + v + ω, [ g L T = µ T + ν T Here, T is the average temerature over the isobar with ressure ; see footnote following equation 1.1 of [8]. We will need to make the assumtion that there exist ositive constants T and T such that 0 < T T T. Both and are alied only in the horizontal direction. We consider to have values from somewhere high in the atmoshere to 1 near the earth s surface; i.e., = is an isobar in the high atmoshere, = 1 is an isobar slightly above the earth and the oceans, and we study the dynamics of the atmoshere in the region 0 < 1. We then introduce the otential temerature defined as It is imortant here to observe that θ = T 0 dt c dt T d dt = c ] Substituting θ for T, we transform 1. into [ dθ c dt µ 0 g T θ ν T dθ dt. 1.4 θ ] = Q E. 1.5 In this article, we will rimarily be concerned with the temerature equation in this form.

3 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 345 Using θ, we arrive at the following formulation for the PEs comare to 1.33 in [6]: v v + v v + ω + fk v + Φ t t [ g ] v µ V v ν V = 0 Φ + θ = 0 v + ω = 0 θ c t + c v θ + c ω θ µ T θ ν T 0 [ g ] θ = Q E Boundary conditions. We will consider equations 1.6 in a aralleleied over the earth s surface, =, 1, where is a horizontal isobar in. We artition the boundary of as: = Γ i Γ u Γ l, where Γ i = { = 1 } corresonds to the interface with the ocean or the earth s surface, Γ u = { = } the uer surface is the boundary in the high atmoshere, and Γ l =, 1 the lateral surface is the vertical boundary. We will use the following hysically reasonable boundary conditions: ω = 0 on Γ u Γ i, v n = 0 on Γ u, ω = g ω on Γ l, v n = α Vv v on Γ i, 1.7 v = g v on Γ l, θ = g θ on Γ l, θ n = α Tθ θ θ on Γ u Γ i. c Here, g v, g θ, g ω, v, and θ are given functions. Note also that the boundary equation on Γ u Γ i, when written in terms of T rather than θ, is the classical obin tye equation T n = α TT T, with θ and T related by The Existence of Solutions. In this section, we rovide the weak formulation of equations 1.6 sulemented by the boundary conditions 1.7 and show how to rove the existence of weak solutions to these equations..1. athematical setting and weak formulation. We define the function saces { 1 } V v = v = v 1, v H 1 γ 0 v Γl = 0, v d = 0, V θ = { θ H 1 γ0 θ Γl = 0 }, V = V v V θ.

4 346 BIAN D. EWALD AND OGE TÉA Here, γ 0 : H 1 H 1 is the trace oerator. Notationally, we will write u = v, θ, û = ˆv, ˆθ, ũ = ṽ, θ, etc., for elements u, û, ũ of V. We define the inner roducts on V v, V θ, V, as follows: v, ˆv = θ, ˆθ = v ˆv d + θ ˆθ d + u, û = v, ˆv + θ, ˆθ. v ˆv d, θ ˆθ d, These roducts are equivalent to the standard H 1 -inner roducts on these saces, and with the norms v = v, v 1, θ = θ, θ 1, u = u, u 1, the saces V v and V θ are closed subsaces of H 1 and H 1. We use single arentheses and norms for the L -inner roducts and norms, e.g., 1 v, ˆv = v ˆv d, θ = θ d. To obtain the weak formulation of 1.6 and 1.7, we make the following observations: In view of the first equation of 1.7 and the third equation of 1.6, we have ω = 1 for any v V v, whence the condition 1 ω d = 1 v d = ω 1 ω = 0 v d,.1 aearing in the definition of V v. Also, writing Φ s = Φ =1 for the value of Φ at the earth s surface, we have Thus, Φ = Φ s and, for ˆv V v 1 1 = Φ s + Φ d θ d, from the second equation of Φ = Φ s + θ d, Φ s ˆv d = Φ s n h ˆv d Γ l Φ s ˆv d Γ l ˆω = Φ s d Φ s = Φ s n ˆω d Γ u Γ i ˆω d. = 0.

5 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 347 Thus, we have Φ ˆv d = 1 Now, we note that, for v, ˆv V v, [ g µ V v ˆv d ν V T = µ V v ˆv d + ν V θ d ˆv d.. ] v ˆv d g v ˆv d g µ V vn h ˆv dγ l ν V Γ l = µ V v ˆv d + ν V ν V Similarly, for θ, ˆθ V θ, µ T θ ˆθ d ν T Γ i g g αv v v ˆv dγ i. 0 = µ T θ ˆθ d µ T n h θ Γ ˆθ dγ l l g 0 + ν T θ g ν T n g = µ T θ ˆθ d + ν T ν T g c T v n ˆv dγ u Γ i v ˆv d [ g ] θ ˆθ d θ 0 ˆθ d ˆθ dγ u Γ i θ ˆθ d g θˆθ d ν T αt θ θˆθ dγ u Γ i. In view of equations.1-.4, we make the following definition: Definition.1. For u = v, θ, û = ˆv, ˆθ, ũ = ṽ, θ V, we set g v a v v, ˆv = µ V v ˆv d + ν V ˆv d, a θ θ, ˆθ g θ = µ T θ ˆθ d + ν ˆθ T d g ν T θˆθ d, c T au, û = a v v, ˆv + a θ θ, ˆθ. b v v, ˆv, ṽ = v ˆv + ω ˆv ṽ d, b θ v, ˆθ, θ = c v ˆθ + ω ˆθ θ d, bu, û, ũ = b v v, ˆv, ṽ + b θ v, ˆθ, θ,

6 348 BIAN D. EWALD AND OGE TÉA where ω = 1 v d. e c v, ˆv = e Φ θ, ˆv = d v v, ˆv = ν V d θ θ, ˆθ = ν T fk v ˆv d, 1 θ d ˆv d,.7 eu, û = e c v, ˆv + e Φ θ, ˆv. g Γ i αv v v ˆv dγ i, g αt θ θˆθ dγ u Γ i,.8 du, û = d v v, ˆv + d θ θ, ˆθ. Thus, we arrive at the weak formulation of 1.6 sulemented with the boundary conditions 1.7, namely: To find ut V + g u, where g u = g v, g θ, such that v θ t, ˆv + c t, ˆθ + au, û + bu, u, û + eu, û.9 du, û = Q E, ˆθ, for all û V... Some bounds and inequalities. In order to establish the existence of solutions u to.9, we need some basic bounds and inequalities. Lemma.1. For v, ˆv, ṽ H 1, θ, ˆθ H 1, we have v i a v v, ˆv µ V v ˆv + ν V c 1 ˆv, ii a v v, v µ V v v + ν V c, iii a θ θ, ˆθ θ µ T θ ˆθ + ν T c 1 ˆθ + ν Tc 3 θ ˆθ, iv a θ θ, θ µ T θ θ + ν T c ν T c 3 θ, v b v v, ˆv, ˆv = 0, if v V v, vi b v v, ˆv, ṽ v ˆv L ṽ, vii b θ v, θ, θ = 0, if v V v, viii b θ v, θ, ˆθ v θ L ˆθ, ix d v v, ˆv ν V α V c 1 v v L ˆv, g g x d v v, v = ν V Γ i αv v v dγ i ν V Γ i αv v dγ i, g where ν V Γ i αv v v dγ i ν V α V c 1 v L v, g and ν V Γ i αv v dγ i ν V α V c v, xi d θ θ, ˆθ ν T α T c 1 θ θ L ˆθ, g g xii d θ θ, θ = ν T αt θ θ dγ u Γ i ν T αt θ dγ u Γ i, g where ν T αt θ θ dγ u Γ i ν T α T c 1 θ L θ,

7 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 349 g and ν T αt θ dγ u Γ i ν T α T c θ, xiii e c v, ˆv c 4 v ˆv, xiv e c v, v = 0, xv e Φ θ, v c 5 θ v, if v V v, g1 g0, g1 where c 1 =, c = c3 =, c 4 = f L, and c T c 5 = 1 1. Proof. Since i-iv, vi, viii-xiii are clear, we omit the roofs of these. For v, we see that bv, ˆv, ˆv = v ˆv + ω ˆv ˆv d = 1 v ˆv + ω ˆv d = 1 ˆv v ˆv v + ω ˆv d = 1 n h ˆv v dγ l 1 ˆv v d Γ l + 1 ω ˆv dγ u Γ i 1 ω ˆv d = 0, since the boundary terms are zero and v + ω = 0; the roof of vii is similar. For xiv, k v v, so that k v v = 0. For xv, we see that 1 e Φ θ, v = θ d v d 1 = θ d v d = Γ l 1 = 1 1 since the boundary term is zero. Now, 1 θ d θ d 1 n h v dγ l θ d v d θ d v d, 1 1 θ d,

8 350 BIAN D. EWALD AND OGE TÉA and so e Φ θ, v θ v. θ d 1 θ d 1 v d 1 1 d v d.3. Existence of solutions. We rove the following theorem about the existence of solutions to.9. Theorem.. For any u 0 sufficiently regular, there exists a solution u to.9 satisfying u0 = u 0. Proof. We will have use for an à riori estimate obtained by setting û = u g u in.9, where u is a certain solution, so that u = û + g u : 1 d dt ˆv + c d dt ˆθ + a v ˆv, ˆv + a θ ˆθ, ˆθ + e Φ ˆθ, ˆv = Q E, ˆθ + d v ˆv, ˆv + d θ ˆθ, ˆθ a v g v, ˆv a θ g θ, ˆθ b v ˆv, g v, ˆv b v g v, g v, ˆv b θ ˆv, g θ, ˆθ b θ g v, g θ, ˆθ e c g v, ˆv e Φ g θ, ˆv + d v g v, ˆv + d θ g θ, ˆθ gv t, ˆv gθ t, ˆθ. Therefore, by using Lemma.1, we have where 1 d dt ˆv + c d dt ˆθ + µ V ˆv ˆv + ν V c + µ T ˆθ + ν T c ˆθ + ν V α V c ˆv + ν T α T c ˆθ ξ, ξ = ν T c 3 ˆθ + c 5 ˆθ ˆv + Q E ˆθ + ν V α V c 1 v L ˆv g v + ν T α T c 1 θ L ˆθ + µ V g v ˆv + ν V c 1 ˆv g θ + µ T g θ ˆθ + ν T c 1 ˆθ + ν T c 3 g θ ˆθ + g v L ˆv + g v L g v ˆv + c g θ L ˆv ˆθ + c g θ L g v ˆθ + c 5 g θ ˆv + c 4 g v ˆv + ν V α V c 1 v g v L ˆv + ν T α T c 1 θ g θ L ˆθ.1

9 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 351 ν T c 3 ˆθ + 3c 5 µ V ˆθ + µ V 6 ˆv + 1 Q E + 1 ˆθ + ν Vα V c 1 v L c + ν Vα V c ˆv + ν Tα T c 1 θ L 4 c + ν Tα T c ˆθ + 3µ V 4 g v + µ V 6 ˆv + ν Vc 1 g v c + ν Vc + ν Tc ˆv + µ T g θ + µ T ˆθ + ν Tc 1 g θ c ˆθ + ν Tc 3 g θ + ν Tc 3 ˆθ + g v L ˆv + 1 g v L g v + 1 ˆv + 1 ˆv + c g θ L ˆθ + 1 g v + c g θ L ˆθ + 3c 5 g θ + µ V µ V 6 ˆv + c 4 g v + 1 ˆv + ν Vα V c 1 v g v L c + ν Vα V c ˆv 4 + ν Tα T c 1 θ g θ L c + ν Tα T c ˆθ. 4 Then.11 and.1 give us d d dt ˆv + c dt ˆθ + µ V ˆv ˆv + ν V c + µ T ˆθ + ν T c ˆθ + ν V α V c ˆv + ν T α T c ˆθ K 1 + K ˆθ + K 3 ˆv,.13 where the K i s are constants deending on the data. Hence, d dt ˆv + c ˆθ K 1 + K 4 ˆv + c ˆθ, and Thus, d e K 4t ˆv + ˆθ K 1 e K4t. dt t ˆvt + c ˆθt e K4t ˆv0 + c ˆθ0 + K 1 e K4t s ds..14 Then, by integrating.14 from 0 to t 1, we obtain 0 ˆv L 0,t 1;L + c ˆθ L 0,t 1;L ˆv 0 + ˆθ 0 t1 e K4t dt + K 1 t1 t e K4t s ds dt..15

10 35 BIAN D. EWALD AND OGE TÉA Also, by integrating.13 from 0 to t 1, we see that t1 t1 ˆvt 1 + c ˆθt 1 + µ V ˆvs ds + ν V c ˆvs ds 0 0 t1 t1 + µ T ˆθs ds + ν T c ˆθs ds 0 t1 t1 + ν V α V c ˆvs ds + ν T α T c ˆθs ds 0 0 ˆv 0 + c ˆθ 0 + tk 1 + K ˆv L 0,t 1;L + K 3 ˆθ L 0,t 1;L..16 Thus, in view of.15 and.16, we can bound each of ˆv L 0,t 1;L, ˆv ˆθ L L 0,t 1;L, 0,t 1;L, ˆθ ˆv L L 0,t 1;L, 0,t 1;V, ˆθ L 0,t 1;V, ˆv L 0,t 1;L, and ˆθ L 0,t 1;L by quantities deending only on the data. Now that we have the above à riori estimates, we can roceed with the roof of the theorem. The roof uses the Galerkin method, and since this method is standard, we will just resent an outline of the roof. Let ψ j = ψj v, ψθ j, for j = 1,,... be a comlete orthonormal basis in V. Then we look for aroximate solutions u m t = m j=1 g jmtψ j which satisfy, with u m = v m, θ m : vm t, θm ψv j + c t, ψθ j + au m, ψ j + bu m, u m, ψ j + eu m, ψ j du m, ψ j = Q E, ψj θ.17, for j = 1,,..., m, u m 0, ψ j = u 0, ψ j, for j = 1,,..., m. Solving for u m is then just solving a system of differential equations for the g jm. It is easy to see that the à riori estimates above aly as well to each u m, so that the u m are all in a subset of L 0, t 1 ; L L 0, t 1 ; V bounded indeendently of m. We are then able to ass to the limit using standard comactness methods and we obtain a solution u to.9. emark.1. Since is bounded from below away from 0 > 0 and from above 1, the roerties of θ are the same as those of T in [6], as one can exect. 3. aximum Princiles. In this section, we will establish maximum rinciles for T, or, what is the same thing, for θ. First, we will use the truncation method of Stamacchia to establish the ositivity of θ. Then we will use a classical method to bound θ from above Positivity. We use Stamacchia s method to show ositivity of θ. We make, as is usual, the following additional hyotheses on the aroriate boundary conditions, forcing terms, and initial conditions: Q E 0, g θ 0, 3.1 θ 0, θ 0 0. We have the following theorem: 0

11 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 353 Theorem 3.1. Under the assumtions 3.1, any solution u = v, θ to.9 satisfies θt 0, for all t [0, t 1 ]. Consequently, T t 0, for all t [0, t 1 ]. Proof. We decomose θ = θ + θ into its ositive and negative arts. We then multily the equation for θ in.9 by θ, and integrate over. The following terms aear. The first term is c Then, we have the term c v θ + ω θ = c = c t θ θ d = c d θ d. dt θ d θ v θ v, ω n d c + ω θ d v + ω θ d, and this term vanishes, since, on Γ u Γ i, v, ω n = ±ω = 0; on Γ l, θ = θ g 0 so that θ = 0; and in all of, v + ω = 0. The horizontal dissiation term for θ reads µ T θ θ d = µ T n h θ θ dγ l µ T Γ l = µ T θ d, θ θ d since, as before, θ = 0 on Γ l. The vertical dissiation term for θ reads ν T 0 = ν T ν T = I J. [ g 0 g θ θ ] g θ θ d θ 1 0 d = θ d

12 354 BIAN D. EWALD AND OGE TÉA Now, I = ν T = ν T = ν T = ν T 0, 0 g T 1 θ d = 0 g T θ 0 g T θ d =1 =0 g θ αt θ θ g + θ =1 αt θ θ d =0 g g αt θ θ + θ αt θ θ + θ d since θ 0. Also, J = ν T = ν T = ν T Therefore, we have ν T = 1 + [ g θ + θ 1] c [ 0 θ ] θ d c g [ θ + θ ] [ ] θ c θ d c [ g ] θ θ d. c 0 ν T The final term is simly [ g [ g θ Collecting all these terms, we find c c Q E θ d 0. d dt θ + µ T θ d + ν T ν T θ ] θ d ] θ d. g = θ d 3. g θ c T d. d Thus, dt θ c θ, where c = ν Tg. Therefore, θ c 3 T t e ct θ 0, and since θ 0 = 0, we obtain θ 0.

13 AXIU PINCIPLES FO THE PIITIVE EQUATIONS Uer bounds. In this section we establish a uniform uer bound for θ using a classical maximum rincile method. For this, we will need to make more exlicit the θ-equation in 1.6 and calculate the derivatives in the ν T -term. We find: [ 0 g ] ν T θ = α θ + β θ + γθ, 3.3 where g α = ν T T 0, β = α T T +, c γ = α + c T c T. c 3.4 As in 1.1 of [8], we assume that T = c = constant. 3.5 c Using this, we may rewrite β as β = α 1 + c 0, 3.6 and γ as γ = α 1 + c c. 3.7 c g Thus, we have γ γ γ, where we have set γ = ν T and γ = ν T g1 0 c 1 + c. We may therefore rewrite the θ-equation of 1.6 as c c dθ dt µ T θ α θ β θ γθ = Q E. 3.8 If we set θ = e λt θ, where λ > 0 will be secified later, then 3.8 yields c d θ dt µ T θ α θ β θ + λ γ θ = Q E e λt. 3.9 Consider a solution to 3.9 from time t = 0 to time t = t 1, with global maximum θ max at some oint x [0, t 1 ]. We consider the following cases: x / [0, t 1 ], that is, x is in the interior of [0, t 1 ]. In this case, since θ has a maximum at x, we see that d θ dt = 0, θ 0, θ θ 0, and = 0 at x. Thus, at x, 3.9 becomes λ γ θ max Q E L e λt, Therefore, rovided we choose λ γ, we may write θ θ max Q E L e λt γ. 3.10

14 356 BIAN D. EWALD AND OGE TÉA x is at time t = 0. In this case, we simly bound θ by the initial condition, θ θ 0 L = θ t=0 L x is at time t = t 1, but in not on. In this case, we have d θ dt 0, θ 0, θ θ 0, and becomes λ γ θ max Q E L e λt, and thus, using again λ γ, we see that θ θ max Q E L e λt x is in Γ u Γ i, for some time t 0, t 1 ]. = 0 at x, and 3.9 γ. 3.1 Then, at x, θ 0, and we have the boundary condition n θ n = α Tθ θ θ, c which in terms of θ at the oint x is Thus, θ n = α T θ α T + c θ max. θ θ max α T θ L α T c x is in Γ l, for some time t 0, t 1 ]. Then, we have simly θ θ max g θ L, 3.14 where g θ = e λt g θ. So, in any case, we can bound θ max by a certain function of the data, namely, { } Q E L θ max K = max γ, θ 0 L, αt θ L, α T + g θ L, 3.15 or, as well, θ max K = max { Q E L γ 1c, θ 0 L, αt θ L, g θ L α T + 1c }, 3.16 since the L -norms of θ 0, θ, and g θ are bounded by those of θ 0, θ, and g θ. Therefore, we have the following theorem, where we have used λ = γ : Theorem 3.. If u = v, θ is a solution to.9 on the interval [0, t 1 ] and all of the data Q E, θ 0, θ, and g θ are uniformly bounded, then we can bound θ max, the maximum value of θ, by θ max Ke γ t That is, we may bound the maximum value of T by 1 T max Ke γ t 1.

15 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 357 emark 3.1. There is some flexibility in the modelling of the dissiation term D in 1.1, and the choice D = L T T is not the only one which is used see a discussion about this in [1]. Hence, assume that we relace the fourth equation of 1.6 by c θ t + c v θ + c ω θ µ T θ αθ, θ + γθ = Q E, 3.18 with α, γ 0; more hyotheses on α and γ would be needed to guarantee existence, but the emhasis here is not on existence of solutions. If we also assume that 3.1 holds and furthermore that Q E = 0, then we can rove a stronger form of 3.17 with no growth in time, namely, for all t 0, 0 θ K = max { θ 0 L, θ L, g θ L } This estimate can be obtained by using the Stamacchia version of the maximum rincile using truncations: we multily 3.18 by θ K +, integrate over, and integrate by arts using the Stokes formula. We arrive at a differential inequality of the form c d dt θ K + γ θ K +, with the same γ as before; this imlies, at all times t, θ K + t θ 0 K + e γ t, and the right-hand side of this last inequality vanishes since θ 0 K a.e. einterreting 3.18 in terms of T, we find a somewhat comlicated exression for D: T c t +c v T + c ω T ω T µ T T + where βθ, = c αθ,. [ [ αθ, βθ, 0 T ] ] 0 T + γt = QE = 0, 4. The Primitive Equations on the Shere. In this section, we will consider the rimitive equations on the whole shere of the earth. Once we overcome the difficulties related to the use of the geometrical tools, the results and the roofs are very similar. Therefore, we will only emhasize the main differences with the receding results athematical setting. We consider first the mathematical and geometrical setting necessary for the rimitive equations on the shere. We define S = S a to be the shere with radius a in 3. We will use the sherical coordinates λ, ϕ, r in 3, where λ is the latitude, ϕ is the longitude, and r is the radius. Thus, S a = {r = a}. We set = S, 1, and T q,r = T q S T r, 1 = T q S is its tangent sace at the oint q, r. We have = Γ i Γ u, where Γ i = S { 1 } and Γ u = S { }. The iemannian metric for defined for q S and r, 1 reads g q, r; v, ω, ˆv, ˆω = g S q; v, ˆv + ω ˆω,

16 358 BIAN D. EWALD AND OGE TÉA where g S is the iemannian metric for the shere S. That is, we set g ij = a 0 sin λ 0 = g λ, ϕ, r;,. x i x j The unit vectors in T are e λ = 1 a λ, e ϕ = 1 a cos λ ϕ, e r = 1 a r. We now define some function saces on. Let C be the sace of all smooth functions from into, C T be the sace of all smooth vector fields on, C S be the sace of smooth functions on S, C T S be the sace of smooth vector fields on S, and C T T S be the sace of smooth functions v : T S with each vλ, ϕ, r Tλ,ϕ S. For v T S, we write v = v λ e λ +v ϕ e ϕ. Consider v, ˆv C T S and θ C S. Then we define the following oerations on v, ˆv, and θ: vˆv = 1 v λ ˆvλ a λ + vϕ ˆv λ cos λ ϕ + vϕˆv ϕ tan λ e λ + 1 v λ ˆvϕ a λ + vϕ ˆv ϕ cos λ ϕ vϕˆv λ tan λ e ϕ, v θ = vλ θ a λ + vϕ θ a cos λ ϕ, grad θ = 1 θ a λ e λ + 1 θ a cos λ ϕ e ϕ, div v = 1 v λ a cos λ λ + vϕ, ϕ 1 θ = a cos λ θ + 1 θ cos λ λ λ cos λ ϕ v = 1 a v λ + sin λ v ϕ cos λ ϕ vλ cos e λ λ + 1 a v ϕ sin λ v λ cos λ ϕ vϕ cos λ, e ϕ. Here, vˆv and v θ are the covariant derivatives of ˆv and θ in the direction of v, and is the Lalace-Beltrami oerator on the shere for the scalar function θ; for the vector function v, v denotes the Lalacian of v as defined by A. Lichnerowicz [5], namely, for the shere of radius a, v i = k k v i 1 a vi, where k and k are the covariant and contravariant derivatives, resectively, in the kth direction. 1 Note that we have the following Stokes formula on the shere: v ˆv d = eλ v eλ ˆv + eϕ v eϕ ˆv + 1a v ˆv d. 1 We recall that, for vector functions, and contrarily to the Cartesian case, there are several nonequivalent definitions of the Lalacian of a vector function tangent to a manifold, and they differ in the lower order terms.

17 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 359 We also consider the following inner roducts: v, ˆv = θ, ˆθ = eλ v eλ ˆv + eϕ v eϕ ˆv + v r ˆv r + v ˆv d grad θ grad ˆθ + θ r ˆθ r + θ ˆθ d. Here we define V v = { v H 1 T T S } div v d = 0, 1 V θ = H 1, V = V v V θ. On the shere, instead of 1.6, the rimitive equations read v t + vv + ω v + fk v + Φ [ g µ V v ν V Φ + θ = 0 div v + ω = 0 θ c t + c v θ + c ω θ µ T θ ν T 0 v [ g ] = 0 ] θ = Q E. 4.1 Here we have furthermore relaced the vertical variable r by the ressure variable as before. The aroriate boundary conditions become: ω = 0 on Γ u Γ i, v n = 0 on Γ u, v n = α Vv v on Γ i, θ n = α Tθ θ θ on Γ u Γ i. c 4.

18 360 BIAN D. EWALD AND OGE TÉA We redefine a v, a θ, b v, and b θ from.5-.8 as follows: a v v, ˆv = µ V eλ v eλ ˆv + eϕ v eϕ ˆv + v ˆv d g v +ν V ˆv d, a θ θ, ˆθ = µ T grad θ grad ˆθ g θ d + ν ˆθ T d g 4.3 ν T θˆθ d, c T b v v, ˆv, ṽ = vˆv + w ˆv ṽ d, b θ v, ˆθ, θ = c v ˆθ ˆθ + w θ d, and with these new definitions, the weak formulation of 4.1 sulemented with boundary conditions 4. is: To find ut V, such that v θ t, ˆv + c t, ˆθ + au, û + bu, u, û + eu, û 4.4 du, û = Q E, ˆθ, for all û V. We have the following lemma corresonding to Lemma.1: Lemma 4.1. For v, ˆv, ṽ H 1, θ, θ H 1, we have i a v v, ˆv c v 1 v, ˆv, ii a v v, v c v v, iii a θ θ, ˆθ c T 1 θ, ˆθ, iv a θ θ, θ c T θ ν T c 3 θ, v b v v, ˆv, ˆv = 0, if v V v, vi b v v, ˆv, ṽ v θ L ṽ, vii b θ v, θ, θ = 0, if v V v, viii b θ v, θ, ˆθ v θ L ˆθ, ix d v v, ˆv ν V α V c 1 v v L ˆv, g g x d v v, v = ν V Γ i αv v v dγ i ν V Γ i αv v dγ i, g where ν V Γ i αv v v dγ i ν V α V c 1 v L v, g and ν V Γ i αv v dγ i ν V α V c v, xi d θ θ, ˆθ ν T α T c 1 θ θ L ˆθ, g g xii d θ θ, θ = ν T αt θ θ dγ u Γ i ν T αt θ dγ u Γ i, g where ν T αt θ θ dγ u Γ i ν T α T c 1 θ L θ, g and ν T αt θ dγ u Γ i ν T α T c θ, xiii e c v, ˆv c 4 v ˆv, xiv e c v, v = 0,

19 AXIU PINCIPLES FO THE PIITIVE EQUATIONS 361 xv e Φ θ, v c 5 θ v, if v V v, g1 g0, g1 where c 1 =, c = c3 =, c 4 = f L, c T c 5 = 1 1 c v = min{µ V, ν V c }, and c T = min{µ T, ν T c }., c v 1 = max{µ V, ν V c 1 }, c T 1 = max{µ T, ν T c 1, ν T c 3 }, Proof. We exlicitly rove only v here, as the roofs of the other arts remain substantially unchanged from before. b v v, ˆv, ˆv = = 1 = 1 = 1 = 0. vˆv + ω ˆv ˆv d v ˆv + ω ˆv d div ˆv v ˆv div v + ω ˆv ˆv div v + ω d + d ˆv ω 1 dγ u Γ i = From this lemma, it is easy to rove, as before: Theorem 4.. For any u 0 sufficiently regular, there exists a solution u to 4.4 satisfying u0 = u aximum rinciles. In the sherical case, the roofs of ositivity and boundedness are easier than in the revious case, since the lack of the vertical boundary only simlifies matters. Theorem 4.3. Under the assumtions Q E 0, θ 0, θ 0 0, any solution u = v, θ to 4.4 satisfies θt 0, for all t [0, t 1 ]. Consequently, T t 0, for all t [0, t 1 ], as well. The roof is the same as in the revious case, excet that any term involving the vertical boundary Γ l is now absent, so we omit it here. Theorem 4.4. If u = v, θ is a solution to 4.4 on the interval [0, t 1 ], and all of the data Q E, θ 0, θ are uniformly bounded, then we can bound θ max, the maximum value of θ, by θ max Ke γ t 1, where { } Q E L K = max γ, θ 0 L, αt θ L. α T + 1c Again, the roof is the same as in the revious case, excet that the case where x, the oint at which θ achieves its maximum, is on the vertical boundary Γ l is absent, and so we again omit the roof.

20 36 BIAN D. EWALD AND OGE TÉA EFEENCES [1] T. Aubin, Nonlinear Analysis on anifolds. onge-amere Equations, Sringer, New York, 198. [] J. G. Charney,. F. Fjørtaft, and J. von Neumann, Numerical integration of the barotroic vorticity equation, Tellus 1950, [3] J. G. Charney, Integration of the rimitive and balance equations, Proc. Intern. Sym. Numerical Weather Prediction, Tokyo 196. [4] G. Haltiner and. Williams, Numerical Weather Prediction and Dynamic eteorology, nd edn., Wiley, New York, [5] A. Lichnerowicz, Proagateurs et Commutateurs en relativité génerále, Presses Universitaires de France, Institut des Hautes Etudes Scientifiques, Publication athématique, vol [6] J. L. Lions,. Témam, and S. Wang, New formulations of the rimitive equations of the atmoshere and alications, Nonlinearity 5 199, [7], On the equations of the large-scale ocean, Nonlinearity 5 199, [8], odels of the couled atmoshere and ocean, and Numerical analysis of the couled models of atmoshere and ocean, in Comutational echanics Advances, 1 ed. J. T. Oden 1993, [9], athematical study of the couled models of atmoshere and ocean, J. ath. Pures Al , [10] Kevin E. Trenberth, ed., Climate System odeling, Cambridge University Press, 199. [11] W.. Washington and C. L. Parkinson, An Introduction to Three-Dimensional Climate odeling, Oxford University Press, Oxford, [1] Qingcun Zun Q. C. Zeng, athematical and Physical Foundations of Numerical Weather Prediction, Science Press, Beijing, 1979 in Chinese. eceived November address: bewald@indiana.edu address: temam@indiana.edu