Dynamics of Quasi-Geostrophic Fluid Motions with Rapidly Oscillating Coriolis Force

Transcription

1 Dynamics of Quasi-eostrophic luid otions with Rapidly Oscillating Coriolis orce Hongjun ao and Jinqiao Duan 1. Department of athematics Nanjing Normal University Nanjing , China 2. Department of Applied athematics Illinois Institute of Technology Chicago, I 60616, USA duan@iit.edu Abstract An averaging principle for quasi-geostrophic fluid motions with rapidly oscillating Coriolis force is proved. This result includes comparison estimate and convergence result between quasi-geostrophic fluid motions and its averaged fluid motions. This averaging principle provides an autonomous system as an approximation for the nonautonomous quasi-geostrophic flows with rapidly oscillating Coriolis force. Key words: Quasi-geostrophic fluid flows, rapidly oscillating forcing, averaging principle athematics Subject Classification 2000): 3420, 35Q35, 86A05, 76U05 Abbreviated title: Dynamics of Quasi-eostrophic luid otions 1

2 1 Introduction eophysical flows involve multiple scales in both space and time. The quasi-geostrophic Q) equation models large scale geophysical flows. It is derived as an approximation of the rotating Navier-Stokes equations by an asymptotic expansion in a small Rossby number. The Q equation is written in terms of the stream function as in [1]:!!"%' ) +*-, ) where 354 is the meridional gradient of the Coriolis parameter, 6354 is the viscosity,,'374 is the Ekman dissipation constant, 0! is the wind forcing, and 8:9;!<16 91=<?>*@9>A<? is the Jacobian operator. oreover,! is the fluctuating component of the Coriolis force or the parameter. This component is usually fast in time. There are a few sources for this fast component [29, 30, 31]: fluctuations in the gravitational force between the Sun and the Earth; fluctuations in the geomagnetic field of the Earth; and fluctuations in the Earth s environment atmosphere, oceans and land). These fluctuations further affect the Earth s rotation and thus affect the Coriolis force. The Equation 1.1) can be rewritten in terms of the relative vorticity 1 '!C% on an arbitrary bounded planar domain *-, DE 1 as 1.2) with sufficiently regular such as, piecewise smooth) boundary H. This equation is supplemented with homogeneous Dirichlet boundary conditions for both and, namely, the no-penetration and free-slip boundary conditions proposed by Pedlosky [2], p. 34 see also [27]): I 41 on 1.3) together with an appropriate initial condition, J4K0 12 in N 1.4) The global well-posedness i.e., existence and uniqueness of smooth solution) of the dissipative model 1.2)-1.4) can be obtained similarly as in, for example, [3], [4], [5] or [6]. ost works on this fluid model are for the case of OP4 constant parameter). rannan et al [8] considered the effect of quasi-geostrophic dynamics under random forcing. Duan et al [9] and [10] obtained the existence of time periodic, time almost periodic 2

3 < 3 quasi-geostrophic response, under time periodic and time almost periodic wind forcing, respectively. We assume that fluctuating Coriolis force term! of the Q flow model 1.1) is rapidly oscillating, i.e., it has the form!8, with parameter We also assume that has a time average in a sense to be specified later. With such Coriolis force, it is desirable to understand the fluid dynamics in some averaged sense, and compare the averaged flows with the original un-averaged flows. Starting from the fundamental work of ogolyubov [11] the averaging theory for ordinary differential equations has been developed and generalized by many authors; see [12] [14] and the references therein. ogolyubov s main theorems have been further generalized in [15] to the case of differential equations with bounded operator-valued coefficients. Some problems of averaging of differential equations with unbounded operator-valued coefficients have been considered in [16] [19] in the framework of abstract parabolic equations. ore recent works on averaging are by Ilyin [20, 21]. See [28] for a survey on averaging principles of partial differential equations. The main result of this paper is an averaging principle for quasi-geostrophic motions with rapidly oscillating Coriolis force. This includes comparison estimate, stability estimate, and convergence result as ) between quasi-geostrophic fluid motions and its averaged fluid motions. 2 Averaging Principle for Quasi-eostrophic lows In this section, we consider the averaging principle for the Q flows under fluctuating Coriolis force!!e' ', with parameter. irst, we provide some preliminaries for later use. Standard abbreviations, A are used for the common Sobolev spaces, with and denoting the usual scalar product and norm, respectively, in following properties and estimates see [4]) of the Jacobian operator :H!<1!9!K * A9H <"!K!<1 <"!K 41 8: <%! ' ) *). We need the : 2.1) 2.2) 3

4 < < H!< A9 for all the area of the domain, and the Young s inequality [22] for nonnegative num- with bers with positive.. We also recall the Poincaré inequality [22] for < <!K ) 2.3) 2.4) We further rewrite the Q flow model 1.2). rom ) we get. Thus 1.2) can be rewritten as 1! 5 *-, D0 2.6) et * ' *-, *+ Then by a result in [16], we know that is a sectorial operator, and hence generates an analytic semigroup in. We will give a sufficient condition to ensure the smallest eigenvalue of to be positive. Consider the eigenvalue equation. We have the following energy estimate 5, *!K!K ), *+ ) I, * ) *), * ) ) : *,* ) 2.7) where the Poincaré inequality 2.3) is used and where arbitrary positive constants satisfy. Therefore, when, 3 2.8) 4

5 we could choose and such that the coefficients in 2.7) are positive, then we have : *,* ) 2.9) where 3 4 is a constant depending on, and the condition 2.8), the smallest eigenvalue model 1 8 is a dissipative dynamical system [24]. of!!. So, when, and satisfy is positive. In this case, the Q flow 50 1 Now, we define the fractional power of as follows [16] for 4 :? where The corresponding domain is a anach space with the norm defined by Now we recall some definitions and useful results for later use. Proposition 2.1 [16] Suppose is sectorial and its spectrum 3 3D4 parts: Re. Then the following estimates are valid for where are positive constants. = 3@4 has positive real 2.10) 2.11) Proposition 2.2 [16] iven two sectorial operators and in,with, 3@4 3 4 Re and Re. et the operator * be bounded in for some Then for every, we have, and the corresponding norms in and are equivalent. We now turn to the averaging principle for the Q flow model. 5

6 3 et be a large dimensionless parameter. Setting A we obtain the equation in the so-called standard form 0 * We assume that has a time average in and suppose that where is positive constant, H % *I 4! as 0 1. ore precisely, let AJ ) 2.13) Remark: We note that in the situation here we can not use the method of [20, 21] directly. Now, we consider the averaged equation 8 * ) Next, we give the existence of absorbing set see the following lemma) for 2.12) with P initial value E in the space. The existence of absorbing set for 2.14) with initial value 0 is similar. The definition of an absorbing set is in, for example, [23, 24, 25]. These sets are certain balls with radius large enough. This means that for every bounded set 2 for 3 In addition, the semigroup is uniformly bounded in these spaces, that is, given any ball, in particular, the ball, there exists a ball such that 2 y increasing we may assume that * A for 3@4 for 3@ where is a positive constant. emma 2.3 et E 2.8) be satisfied and "!, 41, and let the dissipativity condition, where is the smallest eigenvalue of and is the 6

7 Sobolev embedding constant in 2 : estimate for the solution of 2.12) with initial value 8 : where is some positive function of 0. for PROO. ultiplying 2.12) by, integrating over. Then we have the following A and integrating by parts, we get 8 y Cauchy inequality, Soblove inequality and Proposition 2.2, we obtain So, that is K * * * where is the smallest eigenvalue of we know condition of emma 2.3, there exists * we get There exists 3 4 such that for every is a constant depends only on where is a constant depends on and 2 * *, we have * * 2.15) under 2.8)). Under the. Using ronwall inequality,. We go back to 2.4), we have 41 and m.. 7

8 ultiplying 2.12) by, integrating over and integrating by parts, we obtain y Cauchy inequality, Soblove inequality and Proposition 2.2, we have where depends on and. y uniform ronwall inequality, we have for Take, the proof of emma 2.3 is completed. Now, we consider the averaging principle in the space. iven a point E in, we compare the trajectories solutions), of system 2.12) and 2.14) 41 starting from this initial point. Consider their difference on the interval, being arbitrary but fixed. We suppose for the moment that. Then the E* difference satisfies the equation 8 *D8 * We first have some estimates on the nonlinear terms. *I 2.16) emma 2.4 The nonlinear operator is a bounded ipschitz map in the following sense: where and PROO. Since * 8 E* 8 E* are some positive constants. 0*D8 >8' >* >= * 8 * * *? >* * * > * 2 >= 2 K 2.17) 2.18) 2.19)

9 * * 2.17) and 2.18) are obtained by direct estimates. Here the equivalence of norms and is used. Now we get back to equation 2.16). Inverting the linear operator we come to an equivalent integral equation Using 2.11) and 2.17), the the inequality Here. 8! *D8 * ) -norm of the first term in the right hand side satisfies 8 *D *? * et us estimate the second term in the right hand side of 2.20). Since!!*I 1! *- 1!! 2.21) then! *I Since "*!!E*+ 1!*I *+ 2.22) 9

10 here * * ' 2 * is used. Integrating by parts and using 2.11) we have!*- Since! 2!!!! *I 1!!*I 1!!*I!! *I 1! *!*-! K!*I!*I 2! 1! % % K! *I!*I 2.23)!! K K Similarly,!*+!*I 1 y 2.13) and emma 2.3, we have!*i 1! * 10

11 * 2.24) or any, let be so large that for. et be so small that for 3 then inequality is valid. Then if if Since does not depend on, we let 4 and then 4. We obtain! *I 4 when ) Thus, by 2.20) 2.25) we obtain the following inequality: *?? where We need the following fact. emma 2.5 [16] et Then where the function. 41 and for 41 % * is increasing and! A?% as ) Applying this lemma to the inequality 2.26) on 4, we obtain E 2.27) We thus have proved the proximity of solutions of 2.12) and 2.14) in, 4K assuming that the trajectory with initial condition 8 stays in the ball on the interval 41. et be so small that the right-hand side of 2.27) are less than, where is defined earlier in this section when we discuss absorbing sets. Suppose that the trajectory 11

12 3 leaves the ball during the interval However, on the interval and let be the first momemt where both trajectories stay in the ball * and what we have proved so far shows that the inequality is valid. * In particular, it is valid for. This together with the inequality, which holds by the hypothesis of the following theorem and the property of the semigroup, gives the contradiction E* * Therefore we have the following main result in this paper. Theorem 2.6 Averaging Principle for Quasi-eostrophic luid otions) et 41, and let the dissipativity condition 2.8) be satisfied and "!, where is the smallest eigenvalue of and is the Sobolev embedding constant in 2. Assume that the right-hand side of equation 2.12) has an average in the sense of 2.13). et 3@4 be arbitrary and fixed. 4K. 4K If, that is, the initial values coincide and belong to the 41 absorbing ball, then for, where E E* is defined in 2.27). E 4 as 41 This theorem gives comparison estimate and convergence result as ) between the Q flows and averaged Q flows, on finite but large time intevals., 3 Summary We have obtained an averaging principle, Theorem 2.6, for quasi-geostrophic fluid motions with rapidly oscillating Coriolis force characterizied by a large dimensionless parameter ). We have derived comparison estimate and proved convergence result as ) between quasi-geostrophic fluid motions and its averaged fluid motions. The averaged fluid model is an autonomous system while the original quasi-geostrophic fluid system with rapidly oscillating Coriolis force is a nonautonomous system. Thus the 12

13 averaging principle obtained in this paper provides an autonomous fluid system as an approximation for the original nonautonomous fluid system. Acknowledgements. A part of this work was done while J. Duan was visiting the Oberwolfach athematical Research Institute, ermany and the athematics Institute, University of Warwick, England, and while H. ao was visiting Illinois Institute of Technology, Chicago, and Institute of athematics and Its Applications, innesota, USA. This work was partly supported by the NS rant DS and by the grant of NNS of China No , Natural Science oundation of Jiangsu Province No. K and Scientific Research oundation for Returned Overseas Chinese Scholar of Jiangsu Education Commission. References [1] J. Pedlosky, eophysical luid Dynamics, 2nd ed., Springer-Verlag, erlin, New York, [2] J. Pedlosky, Ocean Circulation Theory. Springer Verlag, erlin, [3] V. arcilon, P. Constantin and E. S. Titi, Existence of solutions to the Stommel- Charney model of the ulf stream, SIA J. ath. Anal ), [4] V. P. Dymnikov and A. N. ilatov, athematics of Climate odeling, irkhauser, oston, A, [5] J. Wu, Inviscid limits and regularity estimates for the solutions of the 2D dissipative Quasigeostrophic equations, Indiana Univ. ath. J ), no. 4, [6] A.. ennett and P. E. Kloeden, The dissipative quasigeostrophic equation, athematika, ), [7] P. Cessi and. R. Ierley, Symmetry-breaking multiple equilibria in quasigeostrophic, wind-driven flows, J. Phys. Oceanography, ), [8] J. rannan, J. Duan and T. Wanner, Dissipative quasigeostrophic dynamics under random forcing, J. ath. Anal. Appl., ),

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