Analysis of the semi-geostrophic shallow water equations

Physica D 237 (2008) 1461–1465 www.elsevier.com/locate/physd

Analysis of the semi-geostrophic shallow water equations M.J.P. Cullen ∗ Met Office, Fitzroy Road, Exeter, EX1 3PB, UK Available online 20 March 2008

Abstract The semi-geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the Rossby radius of deformation. The global existence of weak solutions of the Lagrangian form of the equations is known. However, uniqueness is not known except for short times for smooth initial data. Similarly, a rigorous proof of existence of solutions has not been achieved for variable rotation rate, which is important for atmospheric applications. The obstacle is the lack of regularity of solutions to the Monge–Ampere equation, which underlies the solution procedure. In this note, it is shown formally that the modified form of the Monge–Ampere equation that appears in the solution procedure for the semi-geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case. This suggests that more progress should be possible with the rigorous theory, and in particular that it should be possible to prove uniqueness. c 2008 Published by Elsevier B.V. All rights reserved. Crown Copyright PACS: 02.30.Jr; 92.05.Bc; 92.60.Aa; 92.60.Bb Keywords: Semi-geostrophic; Lagrangian maps; Energy minimisation

1. Introduction The semi-geostrophic shallow water model is an accurate approximation to the shallow water equations in the rotationdominated regime as described in [6]. If the rotation rate is a constant, Cullen and Gangbo proved, [3], that weak solutions of these equations exist if they are rewritten in ’dual’ variables corresponding to the geostrophic coordinates of Hoskins [7]. Cullen and Feldman, [5], extended this result to a weak Lagrangian form of the equations in physical coordinates. There is as yet no proof of uniqueness in either formulation. Loeper, [8], has proved uniqueness of the incompressible semigeostrophic equations by assuming greater regularity of the initial conditions. However, this regularity can only be proved to hold for short times, so uniqueness also only holds for short times. The rotation-dominated regime only holds on large horizontal scales in the Earth’s atmosphere. On these scales, the small depth of the atmosphere means that the shallow ∗ Tel.: +44 (0)1392885147; fax: +44 (0)1392885681.

E-mail address: [email protected].

atmosphere and hydrostatic approximations are accurate. The semi-geostrophic approximation should be made after the shallow atmosphere approximation as discussed in [6], chapter 4. The resulting equations are similar to those studied above, but the rotation rate is now a function of horizontal position. This means that a straightforward application of the geostrophic coordinate transformation is not possible. Cullen et al., [4], developed a formal procedure for showing the weak existence of solutions to the shallow water form of these equations. The key step was the use of a constructive procedure for finding local energy minimisers. In the constant rotation case this procedure is exactly equivalent to that developed by [3]. In the present note, it is shown that the large-scale behaviour of the shallow water semi-geostrophic equations is better controlled than that of the incompressible equations. It is then shown how this constructive procedure can exploit this. The first formal result is a proof that there is a ’solution’ to the shallow water semi-geostrophic equations in the form of a function h(t, x) where h is the fluid depth. This may be an analogue of the solution in dual variables obtained by [3]. Further work is still required to extend this to a solution of the Lagrangian equations in physical space, as carried out by [5]

c 2008 Published by Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter Crown Copyright doi:10.1016/j.physd.2008.03.014

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for constant rotation. The second application is a formal proof of uniqueness for the constant rotation case. 2. Model formulation We first consider the semi-geostrophic shallow water equations in cartesian coordinates (x, y) on a convex domain Σ ⊂ R2 . These are   ∂h ∂h , (u g , vg ) = f −1 −g , g ∂y ∂x Du g ∂h +g − f v = 0, Dt ∂x (1) Dvg ∂h +g + f u = 0, Dt ∂y ∂h + ∇ · (hu) = 0. ∂t Here, h is the fluid depth, u = (u, v) are velocity components, D/Dt ≡ ∂/∂t + u · ∇, g is the acceleration due to gravity and f is the Coriolis parameter representing the rotation rate. These equations are an accurate approximation √ to the full shallow water equations on scales larger than gh/ f and Lagrangian timescales greater than f −1 . The nature of the solution procedure can be illustrated by rewriting the second and third equations in the form given in [9]: ∂ Qu + g (∇h) = f 2 ug ∂t  ∂vg ∂vg  f2 + f f   ∂x ∂y Q= ∂u g  . ∂u g 2 f − f −f ∂x ∂y

Loeper [8] analysed an incompressible version of the semigeostrophic equations given by   ∂p ∂p −1 (u g , vg ) = f − , , ∂y ∂x Du g ∂p + − f v = 0, (5) Dt ∂x Dvg ∂p + + f u = 0, Dt ∂y ∇ · u = 0. The equivalent equation to (3) is now    ∂p ∇ · Q−1 ∇ = ∇ · (Q−1 f 2 ug ). ∂t

(6)

The result of [2] shows that solutions of (5) exist in dual variables which are characterised by positive definite Q. Again Q may not be strictly positive definite, and thus (6) may not be uniformly elliptic. There is now no undifferentiated term as there is in (3) to control the large-scale response. Loeper, [8], achieved uniform ellipticity by making additional assumptions on the initial data. This extra regularity can only be sustained for a short time. By doing this, he was able to prove uniqueness of the solutions. In the present paper we demonstrate that the extra control available in (3) should be enough to prove uniqueness in the shallow water case. 3. Solution procedure for the shallow water semi-geostrophic equations

(2)

Eq. (2) can be rewritten as a single equation for ∂h ∂t by using the fourth equation of (1), giving    ∂h ∂h − ∇ · ghQ−1 ∇ = −∇ · (hQ−1 f 2 ug ). (3) ∂t ∂t If f is a constant, Q can be rewritten using the first equation of (1) as   ∂ 2h ∂ 2h 2 g  f + g ∂x2 ∂ x∂ y  . (4) Q= 2  ∂ h ∂ 2h  2 g f +g 2 ∂ x∂ y ∂y Eq. (3) is elliptic if Q is positive definite, which will be assured if 12 f 2 x2 + gh is convex. The theorem of [3] shows the existence of a solution to Eq. (1) with f constant after reformulation in dual variables. This solution satisfies the convexity requirement. However, it does not have sufficient regularity to prove that Q is strictly positive definite, so that (3) may not be uniformly elliptic. The large-scale response to perturbations on the right-hand side is still controlled because of the first term on the left-hand side.

We now reformulate the formal solution procedure for Eq. (1) with smoothly varying f set out in [4], using Lagrangian maps as introduced by [5]. We will then show that the extra large-scale control described above allows a formal proof that we can find a converged solution h(t, x) by time stepping. First recast the definition of the stationary energy principle ˜ ([4], Theorem 2). Given a state h(x), u(x), ˜ v(x), ˜ and energy E˜ defined by Z  1 ˜ 2 (7) E˜ = h(u˜ + v˜ 2 ) + g h˜ 2 dΣ , 2 consider the effect on E˜ of a virtual Lagrangian displacement expressed by a Lagrangian map FΞ (x) satisfying FΞ (x) = x + Ξ = x + (ξ, η), ˜ FΞ #h(x) = h Ξ (x), u Ξ (FΞ (x)) − u(x) ˜ = f η,

(8)

vΞ (FΞ (x)) − v(x) ˜ = − f ξ. The # symbol indicates the push-forward of a measure, as in [5]. Calculate a change to the energy by substituting h Ξ , u Ξ and vΞ into (7). The second equation of (8) corresponds to (10) of [4], and the third and fourth to (11) of [4]. Derivatives of f do not enter if the displacement Ξ is infinitesimal. Then Theorem 2 of [4] states that E˜ is stationary with respect to these variations if ˜ (v, ˜ −u) ˜ = g f −1 ∇ h.

(9)

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For physical reasons, we seek minimisers rather than stationary points. It is shown in [4], Eq. (24), that this corresponds to the positive definiteness of the matrix Q defined in (2) and can be written as a condition on h˜ called ‘involutive’. In particular ∇ h˜ has bounded variation. ˜ u, Now suppose we are given h, ˜ v˜ not satisfying (9), the energy minimisation problem can then be stated as finding a Lagrangian map FΞ satisfying (8) where h Ξ , u Ξ , vΞ is an energy minimiser, so that in particular (vΞ , −u Ξ ) = g f

−1

∇h Ξ .

(10)

If f is constant, the theorem of [3] proves the existence of a unique minimising map. For variable f , as discussed in [4], this statement is only well defined if Ξ can be proved to be infinitesimal. We improve the time-stepping argument in [4], Section 4.3, to show formally that a solution h(t) to (1) exists for variable f . The time-stepping procedure gives a sequence of depth fields h(nδt, x) and associated Lagrangian maps F(nδt, x). At each time step, an energy minimisation problem as discussed above is solved. To start with, suppose we are given h(0, x) such that −gh is involutive and the energy E(0) is finite. Calculate u g , vg from h. The involutivity condition on h(0, x) implies that u g (0, x) − u g (0, x) and vg (0, x) are bounded, where u g (0, x) represents a global mean. Given finite energy, we can set |u g (0, x), vg (0, x)| ≤ U , for some U . Now establish the time-stepping procedure. Given h(t, x), u g (t, x), vg (t, x) and a time step δt, set ˜ h(x) = h(t, x), (11) (v(x), ˜ −u(x)) ˜ = vg (t, x) cos( f δt) + u g (t, x) sin( f δt),
− u g (t, x) cos( f δt) + vg (t, x) sin( f δt) . Calculate the energy E˜ using (7). Minimise the energy by finding a Lagrangian map F(δt, x) satisfying (8). This is found by the constructive procedure set out in [4]. Following [4], Eq. (90), define the residual χ as ˜ χ(x) = (v(x), ˜ −u(x)) ˜ − g f −1 ∇ h(x), = (v(x)(cos( ˜ f δt) − 1) + u(x) ˜ sin( f δt), u(x)(1 ˜ − cos( f δt)) + v(x) ˜ sin( f δt)) ,

(12)

which is O(δt). Now make an infinitesimal displacement parallel to χ , so set Fα (x) = x + α f −1 χ .

(13)

Set Fα #h = h α . Assuming this infinitesimal displacement is sufficiently smooth, the second equation of (8) can be rewritten in the Eulerian form as ∂h = −∇ · (h(Fα (x) − x)) = −∇ · (hα f −1 χ ),

(14)

where ∂h = h α (x) − h(x). Then, as in Eq. (93) of [4], we have: Z δE = − f (Fα − x) · χ hdΣ , Z = − αχ · χ hdΣ . (15)

Thus the energy is strictly decreased by the displacement while χ 6= 0. Since the energy is bounded below by zero, iterating this displacement must reach a state with χ = 0. Write the displacement that achieves this as FΞ . We now need to show that FΞ (x) − x → 0 as δt → 0, so that the assumption of infinitesimal displacements can be justified. Using (8), similar calculations to Eq. (19)–(22) of [4] show that the effect of the displacement Fα on χ is δχ = −Q · (Fα (x) − x) −

g ∇(∂h). f

(16)

Taking the dot product with χ , using (13), and integrating over Σ give Z Z χ · δχ hdΣ = − α f −1 (χ · Q · χ ) hdΣ Z − g f −1 χ · ∇(∂h)hdΣ . (17) Integrating the second term on the right-hand side by parts and using (14) gives Z Z χ · δχ hdΣ = − α f −1 (χ · Q · χ ) hdΣ Z − gα −1 (∂h)2 dΣ . (18) Since Q is assumed positive definite, we have Z Z g(∂h)2 dΣ ≤ −α χ · δχ hdΣ .

(19)

The overall minimisation of the energy is achieved by a sequence of n = O(α −1 ) displacements Fα . Summing over these to give the converged displacement FΞ which gives χ = 0 we have Z Z n α −1 Σi=1 g(∂h i )2 dΣ ≤ χ 2 hdΣ . (20) Then standard inequalities give !2 Z Z n 1 X −1 ∂h i dΣ ≤ χ 2 hdΣ , α g n i=1 so that Z Z 1 2 g(∂h) dΣ ≤ χ 2 hdΣ . αn

(21)

(22)

where now ∂h = h Ξ (x) − h(x) and FΞ #h = h Ξ . Since αn = O(1), this gives ∂h = O(δt). In the solution procedure set out in the second half of p. 146 of [4], we divide a O(1) time interval into O(1/δt) steps of length δt. Thus the total change to h over the interval will be bounded, and the sequence {h(nδt, ·)} will converge to some h(t, ·) because of the involutivity condition. The involutivity condition also means that {∇h(nδt, ·)} converges almost everywhere to ∇h(t, ·). This argument represents an advance beyond that given in [4]. Since the energy E is a function of h, this means that the change to the energy over a time step is O(δt). The solution h(δt) is an energy minimiser, so χ = 0. Applying

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the displacement −FΞ to this solution (thus reversing the calculation done to find h(δt)) shows that the energy is changed 1 by O(FΞ (x)−x)2 . Thus the L 2 norm of (FΞ (x)−x) is O(δt) 2 . This is enough to show that the displacement during each time step is infinitesimal, showing that the argument used to derive it is self-consistent. Since a semi-geostrophic solution can be entirely characterised by a function h(t, x), these arguments suggest the existence of some form of solution to the equations. In the constant f case, this procedure is equivalent to that used by the authors of [3]. In their case, the solution h(t, x) could be characterised as a solution of the equations in dual variables. We now point out that these arguments are not enough to show the existence of a Lagrangian map F over the time interval (0, 1) in the sense of [5]. This is because the estimate 1 O(δt) 2 is not enough to give convergence of the composition of displacements FΞ (δt, ·)◦FΞ (2δt, ·)··. In addition, some spatial ˜ regularity is required to prove that the condition FΞ #h(x) = h Ξ (x) holds. In the constant f case, [3,5] obtained the necessary results by using the dual variable formulation. 4. Uniqueness of solutions for constant rotation rate Start by considering the formulation of the shallow water Eq. (1) and the incompressible Eq. (5) in dual variables. The dual formulation used by [2,3] is written in Cartesian coordinates X = (X, Y ), using the notation of [6]: ∂σ + ∇ · (σ U) = 0, ∂t U = (∇ R(X) − X)⊥ ,

(23)

det D 2 R = σ, σt=0 = σ 0 . The existence of weak solutions means, using the formulation of [5], that, for any φ ∈ C1c ([0, T )) × R2 :   Z ∂φ(t, X) + U(t, X) · ∇φ(t, X) σ (t, X)dtdX ∂t (0,T )×R2 Z + σ 0 (X)φ(0, X)dX = 0. (24) R2

We assume that σ 0 has compact support Σ 0 , as in [3]. [3] showed that the support Σ t of σ (t, ·) is also compact We suppose that σ1 (t, X) and σ2 (t, X) are two solutions of (2). We wish to prove that σ1 = σ2 in the weak sense, that for any φ ∈ C1c ([0, T )) × R2 , Z Z σ1 (t, X)φ(0, X)dX = σ2 (t, X)φ(0, X)dX. (25) R2

R2

Using (23) and writing det D 2 R1 = σ1 , det D 2 R2 = σ2 , we have ∂ (σ1 − σ2 ) + ∇ · (σ1 U1 ) − ∇ · (σ2 U2 ) = 0, ∂t U1 − U2 = (∇ R1 (X) − ∇ R2 (X))⊥ .

∇ · (σ1 (U1 − U2 )) + ∇ · ((σ1 − σ2 )U2 ).

(27)

It is the first of these terms that has prevented the derivation of a proof of uniqueness. The second term can be controlled by the arguments used to prove uniqueness of solutions to the transport equation for a BV velocity field by [1]. To control the first term, it is necessary to show that kU1 − U2 k ≤ Ckσ1 − σ2 k.

(28)

Uniqueness is proved in [8] for the case where 0 < σl < σ 0 (X) < σu < ∞ and in addition σ 0 has a bounded modulus of continuity. We consider the case where only the first assumption is made. Under this condition, R is C 1,α for 0, α < 1. This is not sufficient for Loeper’s argument to work, and the best estimate is that 1

kU1 − U2 k ≤ Ckσ1 − σ2 k 2 .

(29)

We now demonstrate that a better estimate can be made in the shallow water case. The definition (8) of the displacement FΞ used to construct the solution in the previous section can be written using the dual variables (X, Y ) as FΞ (x) = x + Ξ = x + (ξ, η), ˜ FΞ #h(x) = h Ξ (x), X Ξ (FΞ (x)) − X˜ (x) = 0, YΞ (FΞ (x)) − Y˜ (x) = 0.

(30)

In the construction of a solution by time stepping, Eq. (12) represents the transport of σ in R2 for a time step δt, and the construction of FΞ represents the resulting change to the optimal map from R2 back to Σ . The main estimate of the previous section shows that kh(δt, ·) − h(0, ·)k L 2 is O(δt). In Eq. (27), the difference between σ1 and σ2 is generated by transport by the bounded velocity field U1 − U2 . Thus given σ1 and U1 , we obtain σ2 by transporting it with a bounded velocity field for a small time δt. Now calculate U2 . We have to show that kU1 − U2 k L 2 = O(δt). The argument in the previous section proves that kh 1 − h 2 k L 2 = O(δt). Solutions of (1) satisfy (X, Y ) = (x, y) + f −2 g∇h with 21 (x 2 + y 2 ) + f −2 gh convex. The Legendre duality for solutions of (1) (see [6], p. 80) shows that 1 2 (x + y 2 ) + f −2 gh + R = x X + yY. 2

(31)

If a change to h is O(δt), the convexity condition means that the change to ∇h and hence (X, Y ) is also O(δt) in the L 2 norm. Thus the change to R and to (x, y) as a function of (X, Y ) is also O(δt) in L 2 . Thus we have kU1 − U2 k L 2 = O(δt) as desired, and can use Loeper’s argument to prove uniqueness. Acknowledgements

(26)

The second and third terms on the left-hand side of the first equation of (26) can be written as

Some of this work was done during a visit to the University of Wisconsin sponsored by NSF grant DMS-0354729. Discussions with Mikhail Feldman during the visit clarified many of the issues discussed here.

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