The Hamiltonian particle-mesh method for the spherical shallow water equations

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1 ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 5: (004) Publishe online in Wiley InterScience ( DOI: /asl.70 The Hamiltonian particle-mesh metho for the spherical shallow water equations J. Fran 1 * an S. Reich 1 CWI, Amsteram, The Netherlans Department of Mathematics, Imperial College Lonon, Lonon, UK *Corresponence to: J. Fran, CWI, P.O. Box 94079, 1090 GB Amsteram, The Netherlans. jason@cwi.nl Receive: 1 December 003 Revise: 7 February 003 Accepte: 15 March 003 Abstract The Hamiltonian particle-mesh (HPM) metho is generalize to the spherical shallow-water equations, utilizing constraine particle ynamics on the sphere an Merilees pseuospectral metho (complexity O(J log J ) in the latituinal grisize) to approximate the inverse moifie Helmholtz regularization operator. The time step for the explicit, symplectic integrator epens only on a uniform physical smoothing length. Copyright 004 Royal Meteorological Society Keywors: shallow-water equations; spherical geometry; particle-mesh metho; Hamiltonian equations of motion; constraints; symplectic integration 1. Introuction In this paper, we exten the Hamiltonian particle-mesh (HPM) metho of Fran an Reich (003) an Fran et al. (00) to the shallow-water equations in spherical geometry (Williamson et al. 199). By woring with a fully Lagrangian escription, an embeing the sphere in 3, one can avoi pole-relate stepsize limitations an retain exact conservation of mass, energy an circulation. Aitionally, the metho can be mae symplectic, which has even stronger implications, an in particular implies conservation of potential vorticity. We tae Côte s (1988) three-imensional constraine formulation t x = v, t v = v g xh λx, 0 = x x R as a starting point to erive an approximation to the shallow-water equations in the form of a constraine system of orinary ifferential equations (ODEs) in the particle positions x an their velocities v, = 1,...,K.Hereg = m s is the gravitational constant, = s 1 is the rotation rate of the earth, R = m is the raius of the earth, h is the geopotential layer epth, = (0, 0, 1) T, an λ is a Lagrange multiplier to enforce the position constraint. A ey aspect of the HPM metho is the regularization or smoothing of the particle-base iscrete mass istribution over a computational gri, which yiels the layer epth. The regularization is a ifferential operator: specifically, an inverse moifie Helmholtz operator. In the present paper we approximate the Helmholtz operator using a irectional splitting an utilize a spectral technique ue to Merilees (1973) to approximate ifferential operators an their inverses over the sphere. Merilees metho uses FFTs in both longituinal an latituinal irections an requires O(J log J ) operations per smoothing step as oppose to O(J 3 ) operations for the spectral transform metho (see Spotz et al. 1998). Here J enotes the number of gri points in the latituinal irection. We note that for very fine iscretizations, or in a parallel computing environment, the FFT-base smoother may be replace by a gripoint-base approximation without significantly influencing our results. Another ey aspect of the HPM metho lies in the variational or Hamiltonian nature of the spatial truncation. This property, combine with a symplectic timestepping algorithm (Hairer et al. 00), guarantees excellent conservation of total energy an circulation (Fran an Reich, 003). These esirable properties also apply to the propose HPM in spherical geometry an we emonstrate this for a numerical test problem from Williamson et al. (199). Finally, the time steps achievable for our semi-explicit symplectic integration metho are entirely etermine by the uniform physical smoothing length an not by the longitue latitue grisize near the poles.. Description of the spatial truncation The HPM metho utilizes a set of K particles with coorinates x 3 an velocities v 3 as well as a longitue latitue gri with equal gri spacing λ = θ = π/j. The latitue gri points are offset a half-gri length from the poles. Hence we obtain gri points (λ m,θ n ), where λ m = m λ, θ n = π + Copyright 004 Royal Meteorological Society

2 90 J. Fran an S. Reich (n 1/) θ, m = 1,...,J, n = 1,...,J, an the gri imension is J J. All particle positions satisfy the holonomic constraint x x = R (1) where R > 0 is the raius of the sphere. Differentiating the constraint (1) with respect to time immeiately implies the velocity constraint x v = 0 () We convert between Cartesian an spherical coorinates using the formulas x = R cos λ cos θ, y = R sin λ cos θ, z = R sin θ an ( λ = tan 1 y ) (, θ = sin 1 z ) x R Hence we associate with each particle position x = (x, y, z ) T a spherical coorinate (λ,θ ). The implementation of the HPM metho is greatly simplifie by maing use of the perioicity of the spherical coorinate system in the following sense. The perioicity is trivial in the longituinal irection. For the latitue, a great circle meriian is forme by connecting the latitue ata separate by an angular istance π in longitue (or J gri points). See, for example, the paper by Spotz et al. (1998). Let ψ mn (x) enote the tensor prouct cubic B-spline centre at a gri point (λ m,θ n ),ie ( ) λ ψ mn λm (x) ψ cs λ where ψ cs (r) is the cubic spline ( ) θ θn ψ cs θ 3 r + 1 r 3, r 1, ψ cs (r) 1 6 ( r )3, 1 < r, 0, r > (3) an (λ, θ) are the spherical coorinates of a point x on the sphere. In evaluating Equation (3) it is unerstoo that the istances λ λ m an θ θ n are taen as the minimum over all perioic images of the arguments. With this convention the basis functions form a partition of unity, ie ψ mn (x) = 1 (4) hence satisfying a minimum requirement for approximation from the gri to the rest of the sphere. The graient of ψ mn (x) in 3 can be compute using the chain rule an the stanar formula x = 1 R ˆθ θ + 1 R cos θ ˆλ λ with unit vectors [ ] cos λ sin θ ˆθ = sin λ sin θ cos θ, ˆλ = [ ] sin λ cos λ 0 Let us assume for a moment that we have compute a layer epth approximation Ĥ mn (t) over the longitue latitue gri. Maing use of the partition of unity (4), a continuous layer epth approximation is obtaine: ĥ(x, t) = mn Ĥ mn ψ mn (x) (5) Computing the graient of this approximation at particle positions x, the Newtonian equations of motion for each particle on the sphere are given by the constraine formulation t x = v, (6) t v = v g x ψ mn (x ) Ĥ mn (t) λ x, (7) 0 = x x R (8) To close the equations of motion, we efine the geopotential layer epth Ĥ mn (t) as follows. We assign to each particle a fixe mass m which represents its local contribution to the layer epth approximation an essentially equiistribute the particles (in the physical metric) over the sphere at the initial time t = 0. First, we compute A mn = ψ mn (x ). (9) We fin that A mn is not approximately constant but rather A mn cos(θ n ) const ie, A mn is proportional to the area of the associate longitue latitue gri cell on the sphere. Secon, we efine the particle masses an obtain m = H mn ψ mn (x ) (10) H mn 1 m ψ mn (x ) A mn which provies us with the esire layer epth approximation. The area coefficients (9) an the particle masses (10) are only compute once at the beginning of the simulation. During the simulation the layer

3 Hamiltonian particle-mesh metho for shallow water equations 91 epth is approximate over the longitue latitue gri using the formula H mn (t) = 1 m ψ mn (x (t)) (11) A mn A crucial step in the evelopment of an HPM metho is the implementation of an appropriate smoothing operator S over the longitue latitue gri. We will erive such a smoothing operator in the subsequent section. For now we simply assume the existence of a symmetric linear operator S an efine smoothe gri functions via S : {A mn } {Ã mn } an S : {M mn } { M mn }, respectively, where M mn (t) = m ψ mn (x (t)) We now replace the efinition (11) by H mn (t) = M mn (t) Ã mn (1) an finally introuce Ĥ mn (t) via S : { H mn } {Ĥ mn }. This approximation is use in Equation (7) an closes the equations of motion..1. Conservation properties The HPM metho conserves mass, energy, symplectic structure, circulation, potential vorticity an geostrophic balance. Trivially, since the mass associate with each particle is fixe for the entire integration, the HPM metho has local an total mass conservation. Furthermore, Equation (4) implies /t M = 0, an the same will hol for M mn (t) for appropriate S. This implies the conservation of H mn mn (t)ã mn by Equation (1). Since the HPM particles are accelerate in the exact graient fiel of Equation (5), one can efine an auxiliary continuum flui whose particle an velocity fiels initially interpolate the x (0) an u (0), evolve this flui uner the continuous approximate layer epth (5), an the HPM metho will remain embee in the auxiliary flow. In Fran an Reich (003) we show that the auxiliary flui satisfies a circulation theorem an, via Stoes theorem, conserves vorticity. The auxiliary flui also conserves potential vorticity, an this permits us to approximate PV by simply assigning the PV of the auxiliary flui to the particles, ie q (0) = q(x, 0), an avecting this with the particles. Generalize potential enstrophy conservation follows when a function f (q) is approximate in a manner analogous to Equations (5) an (11). See Fran an Reich (003) for full etails. The equations of motion (6) (8) efine a constraine Hamiltonian system that conserves the total energy (Hamiltonian) H = m v v + g H mnãmn. Note that H mnãmn = M mnã 1.Thesymplectic structure of phase space is given by ω = m v x + m x ( x ) (13) The symplectic structure may be also be embee the auxiliary flui, allowing one to pull bac to label space by writing the particle flow as a function of the initial conitions. One consequence of this is another statement of potential vorticity conservation. See Briges et al. (004) for a iscussion of symplecticity an PV. See also Cotter an Reich (004) for a iscussion of the preservation properties of HPM for aiabatic invariants an the implications of this for geostrophic an (in vertical or 3D moels) hyrostatic balance relations. 3. The smoothing operator To complete the escription of the HPM metho, we nee to fin an inexpensive smoothing operator that averages out fluctuations over the sphere on some length scale. Following Merilees pseuospectral coe (Merilees, 1973), we compute erivatives by employing one-imensional fast Fourier transforms (FFTs) along the longituinal an the latituinal irections as summarize, for example, by Fornberg (1995) an Spotz et al. (1998). This allows us essentially to follow the HPM smoothing approach of Fran an Reich (003) an Fran et al. (00), which achieves smoothing by inverting a moifie Helmholtz operator ( ) = 1 x using two-imensional FFT over a regular gri in planar geometry. More specifically, one can easily solve moifie Helmholtz equations separately in the longituinal an latituinal irections an apply an operator-splitting iea to efine a two-imensional smoothing operator. The ey observation is that such a smoothing operator is cheaper to implement than the inversion of a moifie Helmholtz operator using spherical harmonics. We woul lie to mention that our smoothing operator is ifferent from the filtering achieve in the pseuospectral metho of Spotz et al. (1998), which is neee to control aliasing in a pseuospectral metho. We use the following specific technique to achieve approximately uniform smoothing over the sphere with a physical smoothing length. In the lateral irection we use the moifie Helmholtz operator lat ( ) = 1 R θ

4 9 J. Fran an S. Reich In the longituinal irection a uniform physical smoothing length is obtaine with lon ( ) = 1 R cos θ λ an, using a secon-orer operator splitting, the complete smoothing operator can schematically be written as = 1 lon ( /) 1 lat ( ) 1 lon ( /) Upon implementing these operators using FFTs, we obtain a iscrete approximation S over the longitue latitue gri which was use in the previous section to efine the layer epth Ĥ mn. Figure 1. Stereographic projection of 500 mb geopotential height fiel on ay 5, Test case 7. Contours by 50 m from 9050 (blue) to (re) A movie is available from the supplementary materials page 4. Time iscretization an numerical experiments Since the equations of motion (6) (8) are Hamiltonian, it is esirable to integrate them with a symplectic (a) J=18 (b) J=56 contour from 9050 to 1050 by 50 (c) J=384 contour from 9050 to 1050 by 50 () contour from 9050 to 1050 by 50 contour from 9050 to 1050 by 50 Figure. Comparison of Day 5 solution for (a) J = 18, (b) J = 56, (c) J = 384, () T13 reference solution

5 Hamiltonian particle-mesh metho for shallow water equations 93 metho, as this implies long-time approximate conservation of energy, symplectic structure (an hence PV) an aiabatic invariants such as geostrophic balance. Therefore, the following moification of the symplectic RATTLE/SHAKE algorithm (Hairer et al. 00) suggests itself: [ v n+1/ = (I + t ) 1 v n g t ] x ψ mn (x n )Ĥ mn (t n ) λ n xn, x n+1 v n+1 = x n + tvn+1/ 0 = x n+1 x n+1 R, = (I t )v n+1/ x g t ψ mn (x n+1 )Ĥ mn (t n+1 ), v n+1 = v n+1 R x (x n+1 v n+1 ) The first three equations, solve simultaneously, lea to a scalar quaratic equation in the Lagrange multiplier λ n for each. The roots correspon to projecting the particle to the near an far sies of the sphere, so the smallest root is taen. The last two equations upate the velocity fiel an enforce Equation (). Hence the above time-stepping metho is explicit. One can show that the metho also conserves the symplectic two form (13) an hence is symplectic (Hairer et al. 00). To valiate the HPM metho, we integrate Test Case 7 (Analyze 500 mb Height an Win Fiel Initial Conitions) from Williamson et al. (199) withthe initial ata of 1 December 1978 (T13 truncation), over an interval of 5 ays. All calculations were one in Matlab, using mex extensions in C for particle-mesh operators. We have observe no significant, long-live clustering of particles. The iscretization parameters (number of latituinal gripoints J, total number of particles K, smoothing length, an time step size of t) for the various (a) J=18 (b) J=56 contour from -00 to 00 by 5 (c) J=384 contour from -100 to 100 by 1.5 () contour from -50 to 50 by 6.5 contour from 9050 to 1050 by 50 Figure 3. Stereographic projection of error in geopotential on ay 5 for (a) J = 18, (b) J = 56 an (c) J = 384. The reference solution is reprouce in ()

6 94 J. Fran an S. Reich runs are liste in the table below: J K (m) t(s) A stereographic projection of the geopotential fiel in the Northern Hemisphere is shown in Figure 1 for the J = 384 simulation, an agrees quite well with the solution shown in Figure 5.13 of Jaob-Chien et al. (1995). In Figure we give a comparison of the solutions obtaine for J = 18, J = 56, an J = 384 with the T13 reference solution. 1 Theerrorinthe geopotential fiels for these same cases is compare in Figure 3. The reaer will note that there is an error in the geopotential at time t = 0 alreay. This error is ue to the fact that the geopotential is etermine by the particle masses m. The mass coefficients are assigne initially with a certain approximation error. As a result, small-scale gravity waves may be present in the initial 1 The reference solution was compute using the spectral transform metho an Gaussian quarature points in the latituinal irection (Jaob-Chien et al. 1995). A least squares extrapolation of the l errors reporte in Jaob-Chien et al. (1995), assuming just polynomial convergence of the spectral metho, preicts an error of for the T13 truncation J=18 J=56 J=384 normalize l (h) hours Figure 4. Error growth in the l -norm of the geopotential height fiel t = 178 t = 864 t = 43 normalize energy error hours Figure 5. Variation in total energy over a 30-ay simulation

7 Hamiltonian particle-mesh metho for shallow water equations 95 state, even though these have been filtere out of the reference ata. These unbalance gravity waves also account for the fine-scale fluctuations observe in Figure 1. Figure 4 shows the growth of error in the l -norm for the geopotential height over the 5-ay perio, for J = 18, J = 56 an J = 384. We observe approximately first-orer convergence. (A numerical approximation of the orer exponent base on the given ata gave p 1.3.) We attribute the low-orer convergence to the complex ynamics of this test case. Higher-orer convergence woul be expecte for other test cases from Williamson et al. (199). As pointe out in Section, mass an potential enstrophy are preserve to machine precision by the HPM metho. Figure 5 illustrates the energy conservation property of the HPM metho. For this simulation, we chose a coarse iscretization of J = 18, an integrate over a long interval of 30 ays using step sizes of t = 43 sec, t = 864 sec an t = 178 sec. The relative energy errors observe at ay 30 were , an , respectively. Note the relatively large errors right at the beginning of the simulation. These are ue to the imbalance of the numerical initial ata an the rapi subsequent ajustment process. Acnowlegments J. Fran is fune by an Innovative Research Grant from the Netherlans Organization for Scientific Research (NWO). S. Reich acnowleges partial financial support by EPSRC Grant GR/R09565/01. References Briges TJ, Hyon P, Reich S Vorticity an symplecticity in Lagrangian flui ynamics. Proceeings of the Royal Society of Lonon (submitte). Côte J A Lagrange multiplier approach for the metric terms of semi-lagrangian moels on the sphere. Quarterly Journal of the Royal Meteorological Society 114: Cotter CJ, Reich S Aiabatic invariance an applications to MD an NWP. BIT (to appear). Fornberg B A pseuospectral approach for polar an spherical geometrics, SIAM Journal on Scientific Computing 16: Fran J, Reich S Conservation properties of smoothe particle hyroynamics applie to the shallow water equations. BIT 43: Fran J, Gottwal G, Reich S. 00. A Hamiltonian particle-mesh metho for the rotating shallow-water equations. In Lecture Notes in Computational Science an Engineering, vol. 6 Springer: Heielberg, Hairer E, Lubich C, Wanner G. 00. Geometric Numerical Integration: Structure-Preserving Algorithms for Orinary Differential Equations. Springer: Heielberg. Jaob-Chien R, Hac JJ, Williamson DL Spectral transform solutions to the shallow water test set. Journal of Computational Physics 119: Merilees PE The pseuospectral approximation applie to the shallow water equations on the sphere. Atmosphere 11: Spotz WF, Taylor MA, Swarztrauber PN Fast shallow-water equations solvers in latitue longitue coorinates. Journal of Computational Physics 145: Williamson DL, Drae JB, Hac JJ, Jaob R, Swarztrauber PN A stanar test set for the numerical approximations to the shallow water equations in spherical geometry. Journal of Computational Physics 10: 11 4.