Global Weak Solution of Planetary Geostrophic Equations with Inviscid Geostrophic Balance

Transcription

1 Global Weak Solution o Planetary Geostrophic Equations with Inviscid Geostrophic Balance Jian-Guo Liu 1, Roger Samelson 2, Cheng Wang 3 Communicated by R. Temam) Abstract. A reormulation o the planetary geostrophic equations PGEs) with inviscid balance equation is proposed and the existence o global weak solutions is established, provided that the mechanical orcing satisies an integral constraint. There is only one prognostic equation or the temperature ield and the velocity ield is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity proile can be accurately represented as a unctional o the temperature gradient. In particular, the vertical velocity depends only on the irst order derivative o the temperature. As a result, the bound or the L, t 1 ; L 2 ) L 2, t 1 ; H 1 ) norm o the temperature ield is suicient to show the existence o the weak solution. 2 Mathematics Subject Classiication: 35Q35, 86A1. Key words and phrases: inviscid planetary geostrophic balance, hydrostatic balance, thermal wind equation, global weak solution. 1. Introduction The planetary geostrophic equations PGEs) have played an important role in largescale ocean circulation since the pioneering work o A. Robinson and H. Stommel [6] and P. Welander [12]. This system arises as an asymptotic approximation to the primitive equations PEs) or planetary-scale motions in the limit o small Rossby number. The PGEs are considerably simpler than the PEs, but retain the dynamics necessary to represent the largescale, low-requency dynamics o the mid-latitude oceans. 1 Institute or Physical Science and Technology & Department o Mathematics, University o Maryland, College Park, MD The research o this author was supported by NSF Grant DMS jliu@math.umd.edu 2 College o Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR The research o this author was supported by NSF Grant OCE and Navy ONR Grant N rsamelson@coas.oregonstate.edu 3 Corresponding author: Department o Mathematics, University o Tennessee, Knoxville, TN wang@math.utk.edu

2 The PGEs with viscous geostrophic balance has been analyzed at the PDE level in recent articles. See [1, 8, 9] or relevant discussions. One distinguishing eature o the PGEs is that there is only one prognostic equation in the system or the temperature ield; the velocity ield is diagnostically determined by the planetary geostrophic balance. The addition o a diusion term in the geostrophic balance equation is or the sake o simplicity in mathematical analysis, due to the lack o regularity or the velocity ield by a straightorward manipulation. In this article, we consider the original ormulation o the PGEs, with no viscous term in the geostrophic balance equations. In such a ormulation, both horizontal and vertical velocity proiles are accurately represented as unctionals o the temperature gradient. The representation ormula or the horizontal velocity ield is based on the planetary geostrophic balance. Since every variable can be uniquely determined by the combination o its mean in vertical direction) and its vertical derivative, the horizontal velocity turns out to be the solution o a dierential equation at each ixed horizontal point, depending only on the temperature gradient. The vertical velocity can be recovered by the continuity equation. Using the special orm o the Coriolis parameter, we arrive at a two-point boundary value ordinary dierential equations O.D.E.) in the vertical direction at each ixed horizontal point or the vertical velocity, with the right hand side depending only on the irst order derivative o the temperature ield. The new ormulation is derived in Section 2 and the existence o the global in time) weak solution o the reormulated PGEs is provided in Section 3. The approach o Galerkin approximation is used. Standard energy estimate or the temperature equation gives the bound o the L, t 1 ; L 2 ) L 2, t 1 ; H 1 ) norm o the temperature variable, which in turn shows the bound o the L 2, t 1 ; L 2 ) norm o the horizontal velocity. Moreover, the L 2, t 1 ; L 2 ) norm o the vertical velocity is also uniormly bounded since it satisies the second order O.D.E., in which the orce term only involves the temperature gradient. The compactness or the time derivative o the temperature ield can be established in a similar manner. Thus the existence o the global weak solution is proven. 2

3 2.1) 2. Reormulation o the Inviscid Planetary Geostrophic Equations The non-dimensional PGEs can be written as T t + v )T + w T z = ) z 2 T, Rt 1 Rt 2 k v + p = F, p z = T, v + z w =, where T represents the temperature, v = u, v) the horizontal velocity, w the vertical velocity, and p the pressure. The term k v corresponds to the Coriolis orce with depending only on the latitude y. As a typical example used in geophysical literatures, its β plane approximation is given by = +βy. The parameters 1/Rt 1, 1/Rt 2 stand or the horizontal and vertical heat conductivity coeicients. The operators,,, stand or the gradient, perpendicular gradient, divergence and Laplacian in horizontal plane, respectively. For simplicity, we set κ 1 = 1/Rt 1, κ 2 = 1/Rt 2. The orcing term F = F x, F y ) appearing in the geostrophic balance equation 2.1) 2 comes rom the wind stress at the ocean surace, which is a boundary layer approximation. It may or may not depend on the vertical variable z. For simplicity, we assume in this article F = F x, y) = F x x, y), F y x, y) ). The discussion o a general case can be carried out in the same ashion and does not add any mathematical diiculty. See the relevant reerences on both the physical and mathematical descriptions o the PGEs in [2, 3, 4, 5, 6, 8, 9, 1, 12], etc. The computational domain is taken as M = M [, ], where M is the surace o the ocean. The boundary condition at the top and bottom suraces are given by 2.2) w = w = and κ 2 T z = T, at z =, and κ 2 T z =, at z =, where the term T represents the heat lux at the surace o the ocean. Usually T can be taken as either a ixed heat lux unction or o the orm T = αt θ ), where θ is a reerence temperature. Both boundary conditions can be dealt with in an eicient way. In this article or simplicity we choose T as a given lux. On the lateral boundary section M [, ], the ixed boundary condition is prescribed or the temperature ield 2.3) T = T lb, on M [, ], 3

4 where T lb is a given distribution. As will be shown later, the purpose o the choice o a Dirichlet boundary condition is to acilitate the analysis o the system at the PDE level, although the no-lux boundary condition or the temperature ield is physically more relevant. The boundary condition 2.3) can also be viewed as an approximation such that the disturbance o oceanic circulation motion is ar away rom the lateral boundary. For simplicity o the presentation, we set the homogeneous proile T lb = in the theoretical and numerical analysis. There is no real change or the non-homogeneous case. The normal component o the vertically averaged horizontal velocity turns out to have a vanishing lux 2.4) v n =, on M, which is compatible with the continuity equation 2.1) 4. We recall that the average in the vertical direction) o any 3-D ield g is given by gx, y) = 1 H gx, y, z) dz. See [4, 8] or a detailed explanation or the choice o this nonlocal boundary condition in the case where no viscosity is present in the geostrophic balance equation. We now derive an equivalent ormulation o the system o PGEs 2.1)-2.4). The key point in this reormulation is that both the horizontal and vertical velocity variables can be determined by the irst order derivative o the temperature ield. This makes valid the analysis o the well-posedness or the system. 2.5) The horizontal velocity ield is the solution o the ollowing system z u = T y, zv = T x, ux, y) = u e, where u e is explicitly given below. vx, y) = v e = yf x + x F y y Equation 2.5) is derived rom the geostrophic equation and hydrostatic equation. Taking the vertical derivative o the geostrophic balance equation k v+ p = F gives the thermal wind equation, 2.6) v z = p z + z F = T, i.e. u z = T y, v z = T x, where the hydrostatic balance p/ z = T and the independence on z o F and = y) were used. In other words, the proile v z can be expressed by the temperature gradient. 4

5 Meanwhile, integrating the geostrophic balance equation k v + p = F in the vertical direction and dividing by H, we ind 2.7) k v + p = F. Applying the curl operator to 2.7) results in 2.8) y )v + v y + u x ) = F = y F x + x F y. Moreover, the continuity equation v + z w = and the boundary condition or the vertical velocity w, ) = w, ) = show that the averaged horizontal velocity ield is divergence-ree, i.e., 2.9) v =. The combination o 2.8) and 2.9) yields 2.1) v e = vx, y) = yf x + x F y. y By taking the tangential part o 2.7) and applying 2.4), one obtains the vertically averaged tangential pressure gradient and thus the pressure, aside rom an arbitrary constant) on the boundary rom the tangential component o the orcing F : 2.11) p τ = F τ, where τ is the unit tangential vector on the boundary. Integrating this relation around the boundary, one obtains the constraint that the line integral o F around the boundary must be zero: 2.12) F τ dl =. M This means that the orcing must not give a net torque on the luid. From 2.9), we can ind a 2-D mean stream unction ψx, y) or the vertically averaged velocity ield, such that u, v) = y ψ, x ψ). Moreover, the boundary condition 2.4) indicates that ψ is a constant on the lateral boundary. For simplicity o the discussion, we take ψ = on M. In addition, we denote γ 1 y ), γ 2 y ) by the x-coordinates o the 5

6 intersection points between M and y = y. The mean stream unction ψ and the mean velocity u can be determined by the kinematic relationship and ormula 2.1): 2.13) ψ e x, y) = x γ 1 y) F y dx, u e x, y) = y ψ e x, y), with γ 1 y) being a point on M. Evaluating ψ at another boundary point γ 2 y), y) with the same y-value, we obtain an additional constraint on the orcing: 2.14) γ2 y) γ 1 y) F y dx =, since ψ is identically on the lateral boundary. Constraint 2.14) amounts to saying that the average orcing across the domain at a ixed y must not give a torque on the luid. The combination o 2.6), 2.1) and 2.13) leads to the system 2.5). By the representation ormula valid or any 3-D variable g: 2.15) gx, y, z) = g z x, y, z 1 ) dz 1 + gx, y) 1 H g z x, y, z 1 ) dz 1 dz, the solution o 2.5) can be expressed explicitly using an integration ormula: 2.16a) ux, y, z) = T y x, y, z 1) dz 1 + u e x, y) + 1 H T y x, y, z 1) dz 1 dz, 2.16b) vx, y, z) = T x x, y, z 1) dz 1 + v e x, y) 1 H T x x, y, z 1) dz 1 dz, with u e, v e given by 2.1) and 2.13). The vertical velocity can be calculated by integrating the horizontal divergence o the horizontal velocity ield, due to the incompressibility condition 2.17) wx, y, z) = vx, y, s) ds. The substitution o 2.16) into 2.17) gives 2.18) wx, y, z) = 2 y )T x x, y, z 2 1 ) dz 1 dz 2 1 H z + H ) y )T x 2 x, y, z 1 ) dz 1 dz. 6

7 Thereore, w can be expressed as a unctional o the temperature gradient, like the horizontal velocity v. The vertical velocity can also be represented as the solution o a dierential equation. By taking the vertical derivative o the continuity equation 2.19) v z + zw 2 =, combined with 2.6), we arrive at 2.2) 2 zw = x u z ) y v z ) = x T y ) T ) x y )T x y =. 2 It can be observed that the second order derivatives or the temperature ield cancel each other due to the special orm o the Coriolis parameter = y). Thereore, the vertical velocity w can be reormulated as the solution o the ollowing system o second order O.D.E.s 2.21) 2 zw = y)t x 2, w =, at z =,, in which the right hand side includes only the irst order derivative o the temperature. This key point is crucial to the analysis presented below. We then have the ollowing ormulation, where the velocities are expressed as unctionals o the temperature gradient. Temperature Transport Equation 2.22a) T t + v )T + w T z = ) κ 1 + κ 2 z 2 T, T z = T T, at z =, z =, at z =, T =, on M [, ] ; Recovery o the horizontal velocity 2.22b) z u = T y, zv = T x, ux, y) = u e, vx, y) = v e ; 7

8 Recovery o the vertical velocity 2.22c) 2 zw = y)t x 2, w =, at z =,. Remark 2.1. It is observed that the alternate ormulation 2.22a-c) is equivalent to the original ormulation 2.1)-2.4) o the PGEs, rom which they were derived. Indeed, to recover 2.1)-2.4) rom 2.22a-c), we need to show that φ = v + z w. A simple calculation utilizing 2.22b) and 2.22c) leads to 2.23) z φ = z v+ z w ) = x z u)+ y z v)+ 2 zw = x T y ) T ) x y )T x + y + =, ) φ = v + z w ) = v + z w =, since the average o v is divergence-ree in the horizontal plane. The combination o 2.23) and 2.24) results in the incompressibility condition. In addition, a direct calculation 2.25) k v F ) = y )v F =, indicates the existence o a mean pressure ield p such that k v + p = F. Accordingly, we deine the total pressure ield as 2.26) px, y, z) = T x, y, s) ds + px, y) 1 H T x, y, s) ds dz. Clearly, the hydrostatic balance is satisied by taking the vertical derivative o 2.26). The geostrophic balance equation can also be veriied by using the integration ormulas or v in 2.16), which comes rom the recovery equation 2.22b), combined with the horizontal gradient o 2.26): 2.27) p = T ds + p 1 H T ds dz. All the boundary conditions presented in 2.1)-2.4) are included in the system 2.22). Hence, T, u, p) is also a solution o 2.1)-2.4). This completes the proo o the ormal equivalence o smooth solutions between the two ormulations. 8

9 3. Existence o a Global Weak Solution Beore starting the discussion on the weak solutions, we introduce the ollowing unctional setting: 3.1) H = L 2 M), V = the closure o C lat,m) in H 1 M), C m lat,m) = {T C m M) T = on M [, ]}. Note that the introduction o Clat,M) m is motivated by the boundary condition or T on the lateral boundary sections. Let, ) be the inner product in L 2 M) = L 2 M [, ]), and the corresponding L 2 norm. For any positive inal time t 1 >, the unctions T, u, v, w) T L, t 1 ; H ) L 2, t 1 ; V ), u, v, w L 2, t 1 ; L 2 M) ), are called a weak solution o the original PGEs ormulated in 2.22) i 3.2a) t T φ) + ut φ x + vt φ y + wt φ z + κ 1 T ) φ) + κ 2 z T ) z φ) ) dx dz M +κ 2 M α κ 2 T x, ) θ )φx, ) dx =, φ C 1 lat,m) H 3 M), where 3.2b) ux, y, z) = vx, y, z) = T y x, y, z 1) dz 1 + u e x, y) + 1 H T x x, y, z 1) dz 1 + v e x, y) 1 H T y x, y, z 1) dz 1 dz, T x x, y, z 1) dz 1 dz, 3.2c) wx, y, z) = 2 y )T x x, y, z 2 1 ) dz 1 dz 2 1 H z + H ) y )T x 2 x, y, z 1 ) dz 1 dz. Theorem 3.1 Suppose F H 2 M ) is given and the constraint 2.14) is satisied so that u e, v e can be consistently determined. Let T = T, ) L 2 M). Then there exists at least one global weak solution or the PGEs 2.22), such that or any t 1 > 3.3) T L, t 1 ; H ) L 2, t 1 ; V ), t T L 4 3, t1 ; H 2 M) ), u, v, w L 2, t 1 ; L 2 M) ), z u, z v, 2 zw L 2, t 1 ; L 2 M) ), 9

10 3.4) t T, t) κ1 T, s) 2 + κ 2 z T, s) 2) ds T 2 + C t, or < t < t 1, with C = α θ ) 2 dx, M 3.5) v dz =, H in the sense o distribution, v dz n =, on M, w =, at z =,. Proo. The proo can be accomplished by the Galerkin procedure. A standard energy estimate is used to obtain the uniorm bound or the L, t 1 ; L 2 M)) and L 2, t 1 ; H 1 M)) norms o the temperature ield. Let {Φ j } j 1 H 2 M) be the eigenvectors o the diusion operator corresponding to the eigenvalues λ j, j = 1, 2,..., such that ) κ1 + κ 2 z 2 Φj = λ j Φ j, λ j, 3.6) Φ j z = αφ j κ 2, at z =, Φ j z =, at z =, Φ j =, on M [, ]. The diusion operator A = κ 1 + κ 2 z 2 with the given boundary condition is a sel-adjoint linear operator and admits a compact inverse. Then {Φ n } n 1 deines a complete orthogonal basis in L 2 M). To seek a weak solution o the reormulated PGEs deined in 3.2), we ind an approximate solution {T m } such that m 3.7) T m x, z; t) = βj m t)φ j x, z), 3.8) d ) ) ) ) ) Tm, Φ j + vm T m, Φ j + wm T m, z Φ j + κ1 Tm, Φ j + κ2 z T m, z Φ j dt α +κ 2 T m x, ) θ )Φ j x, ) dx =, j = 1, 2,..., m, κ 2 M 3.9) u m x, y, z) = v m x, y, z) = y T m x, y, z 1 ) dz 1 + u e x, y) + 1 H x T m x, y, z 1 ) dz 1 + v e x, y) 1 H 1 y T m x, y, z 1 ) dz 1 dz, x T m x, y, z 1 ) dz 1 dz,

11 3.1) w m x, y, z) = 2 y ) x T m x, y, z 2 1 ) dz 1 dz 2 1 H z + H ) y ) x T m 2 x, y, z 1 ) dz 1 dz, 3.11) T m t= = P m T, where P m is the orthogonal projection operator in L 2 M): P m : L 2 M) Span {Φ 1,..., Φ m }. The scheme 3.8) and 3.11) proposes an initial value problem or a system o m O.D.E.s, with the velocities determined by 3.9) and 3.1). Thereore, it is straightorward to conclude the local in time) existence o the approximate solution. To get the global in time) solution, the energy estimates are necessary. We observe that the approximated velocity ield u m = v m, w m ) satisies 3.12) v m + z w m =, v m n M [,]= v e n M [,]=, w m z=, H =, which comes rom the construction 3.9), 3.1) and the homogeneous Dirichlet boundary condition or T m on the lateral boundary. As a result, we ind by integration by parts that 3.13) vm T m, T m ) + wm T m, z T m ) =. Multiplying 3.8) by β m j t) and adding up the resulting equations leads to 3.14) 1 d 2 dt T m 2 + κ 1 T m 2 + κ 2 z T m 2 α = κ 2 Tm, ) θ ) T m, ) dx M κ 2 = α Tm, ) θ ) T m, ) dx M α T 2 m, 2 ) dx + α θ ) 2 dx M 2 M α θ ) 2 dx. 2 M Applying the Gronwall inequality to 3.14) results in 3.15) T m, t) t κ 1 T m 2 + κ 2 z T m 2 ) ds T 2 + C t, with C = α M θ ) 2 dx, 11

12 which in turn indicates that 3.16) T m a bounded set o L, t 1 ; L 2 M) ) L 2, t 1 ; H 1 M) ). Moreover, by the recovery ormulation 3.9), 3.1), we have 3.17) u m, v m, w m a bounded set o L 2, t 1 ; L 2 M) ), z u m, z v m, 2 zw m a bounded set o L 2, t 1 ; L 2 M) ). Furthermore, we need an estimate or t T m so that we can obtain compactness and a strong convergence result. Consider T H 2 M) given by T = βj Φ j. Equation 3.8) shows that 3.18) t T m, T ) = t T m, P m T ) = ) ) ) ) v m T m, P m T + wm T m, z P m T + κ1 Tm, P m T + κ2 z T m, z P m T α +κ 2 T m x, ) θ )P m T, ) dx. κ 2 M Regarding the nonlinear term, we have 3.19) M v m T m P m T dx v m L 2 T m L 3 P m T L 6 C v m T m 1/2 T m 1/2 H 1 P m T H 2. It is observed that 3.2) P m T z = α P m T κ 2, at z =, P m T =, on M [, ], P m T z =, at z =, since P m T = m βj Φ j and each Φ j satisies the boundary condition in 3.6). Consequently, an application o the elliptic regularity or P m T gives 3.21) P m T H 2 C AP m T ). In more detail, we have 3.22) AP m T m ) ) = A β m m j Φ j = β j AΦ j = λ j βj Φ j, by 3.6), A T = A ) β j Φ j = β j AΦ j = λ j βj Φ j, by 3.6), 12

13 which along with the orthogonality o {Φ j } j 1 in L 2 M) leads to 3.23) AP m T ) 2 = m λ 2 β j j 2 Φ j 2 λ 2 β j j 2 Φ j 2 = A T 2. The combination o 3.21) and 3.22) results in 3.24) P m T H 2 C A T C T H 2. The substitution o 3.24) into 3.19) leads to 3.25) M v m T m P m T dx C v m T m 1/2 H 1 T m 1/2 T H 2 C T m 3/2 H 1 T m 1/2 T H 2. Similarly, we have the ollowing estimates 3.26) w m T m z P m T dx w m L 2 T m L 3 z P m T L 6 M C T m 3/2 H 1 T m 1/2 T H 2, 3.27) ) ) Tm, P m T + z T m, z P m T C T m H 1 T H 2, 3.28) M α κ 2 T m x, ) θ )P m T, ) dx C T m H 1 T H 2, By the combination o 3.16), 3.17), 3.25)-3.28), we arrive at 3.29) t T m a bounded set o L 4/3, t 1 ; H 2 M) ), or any t 1 > and independent o m. The estimates 3.16), 3.17), 3.29) imply the existence o T L, t 1 ; L 2 ) L 2, t 1 ; H 1 ) and u, v, w L 2, t 1 ; L 2 ) and a subsequence {T m, v m, w m } such that T m T weakly in L 2, t 1 ; H 1 ), 3.3) T m T weak-star in L, t 1 ; L 2 ), v m, w m v, w weakly in L 2, t 1 ; L 2 ), 13

14 With the use o 3.29), 3.3) and Aubin s compactness theorem, we also have 3.31) T m T strongly in L 2, t 1 ; L 2 ). Then it is standard to pass to the limit in 3.9)-3.11) and prove that the limit unction T, v, w) is indeed a weak solution as deined in 3.2). The details are omitted or brevity. The proo or the irst part o Theorem 3.1 is completed. Furthermore, we multiply 3.15) by φt), where φ D, t 1 ) ), φt), and integrate in time: 3.32) t1 t Tm, t) κ 1 T m, s) 2 + κ 2 z T m, s) 2 ) ds ) φt) dt t1 T 2 + C t ) φt) dt. Using the weak convergence 3.3) we pass the lower limit in this inequality and obtain 3.33) t1 t T, t) κ 1 T, s) 2 + κ 2 z T, s) 2 ) ds ) φt) dt t1 T 2 + C t ) φt) dt, or all φ D, t 1 ) ), φ. This amounts to saying that the energy inequality 3.4) is satisied or almost every t [, t 1 ]. For the second part, we note that, by a direct calculation using the representation ormulas or the horizontal velocity ield in 3.2b), 3.34) vx, y, z) dz = v e x, y), x, y) M. This leads to the irst identity o 3.5), due to the ree divergence o v e given by ormulas 2.1) and 2.13). Moreover, the second identity o 3.5) is valid since v e satisies the speciied boundary condition on the lateral boundary. The third identity o 3.5) is also ound by direct veriication using ormula 3.2c). This completes the proo o Theorem 3.1. Reerences [1] C.-S. Cao, E. Titi, 23): Global well-posedness and inite-dimensional global attractor or a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 2),

15 [2] A. Colin de Verdiere, 1988): Buoyancy driven planetary lows, J. Mar. Res., 46, [3] J. Pedlosky, 1984): The equations or geostrophic motion in the ocean, J. Phys. Oceanogr., 14, [4] J. Pedlosky, 1987): Geophysical Fluid Dynamics, 2nd edition; Springer Verlag, New York. [5] N. A. Phillips, 1963): Geostrophic motion, Rev. Geophys., 1, [6] A. Robinson, H. Stommel, 1959): The oceanic thermocline and associated thermohaline circulation, Tellus, 11, [7] R. Salmon, 199), The thermocline as an internal boundary layer, J. Mar. Res., 48, [8] R. Samelson, R. Temam, S. Wang, 1998): Some mathematical properties o the planetary geostrophic equations or large scale ocean circulation, Appl. Anal., 7 1-2), [9] R. Samelson, R. Temam, S. Wang, 2): Remarks on the planetary geostrophic model o grye scale ocean circulation, Dierential Integral Equations, 13, [1] R. Samelson, G. Vallis, 1997): A simple riction and diusion scheme or planetary geostrophic basin models, J. Phys. Oceanogr., 27, [11] R. Temam, 1984): Navier-Stokes Equations: Theory and Numerical Analysis, North Holland, Amsterdam. [12] P. Welander, 1959): An advective model o the ocean thermocline, Tellus, 11,