An Unconditionally Stable Scheme for the Shallow Water Equations*

810 MONTHLY WEATHER REVIEW VOLUME 128

MOSHE ISRAELI Computer Science Department, Technion, Haifa, Israel

NAOMI H. NAIK AND MARK A. CANE Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York

(Manuscript received 24 September 1998, in ®nal form 1 March 1999)

ABSTRACT A ®nite-difference scheme for solving the linear shallow water equations in a bounded domain is described. Its time step is not restricted by a Courant±Friedrichs±Levy (CFL) condition. The scheme, known as Israeli± Naik±Cane (INC), is the offspring of semi-Lagrangian (SL) schemes and the Cane±Patton (CP) algorithm. In common with the latter it treats the shallow water equations implicitly in y and with attention to wave propagation in x. Unlike CP, it uses an SL-like approach to the zonal variations, which allows the scheme to apply to the full primitive equations. The great advantage, even in problems where quasigeostrophic dynamics are appropriate in the interior, is that the INC scheme accommodates complete boundary conditions.

1. Introduction is easy to code and boundary conditions for the discre- The two-dimensional linearized shallow water equa- tized equations are fairly natural to impose. At the other tions represent the evolution of small perturbations in end of the spectrum, the CP algorithm is speci®cally the ¯ow ®eld of a shallow basin on a rotating sphere. designed with the characteristics of the physics of the Our interest in these model equations arises from our equatorial ocean dynamics in mind. By separating the interest in solving for the motions in a linear beta-plane free modes into the eastward propagating Kelvin mode deep ocean. If the strati®cation is assumed to be purely and the remaining westward propagating modes, the a function of depth, then the solution can be found as shallow water system can be solved semiimplicitly with a sum over N vertically standing normal modes, each very large time steps. The major drawback to this ap- of which satis®es a shallow water system. Moreover, proach is that it is prohibitively awkward to treat re- this is the wave operator even in nonlinear problems alistic boundaries. and is used in both semi-Lagrangian and semiimplicit Here we present a new scheme that has neither of the schemes. For example, it could readily be used in the disadvantages of these schemes: it is not restricted by standard semiimplicit setup as the solution procedure a CFL condition nor are there restrictions on the basin for the wave operator. geometry. The unconditional stability of the scheme Two successful, but very different, approaches to means that only accuracy, not computational stability, solving the shallow water system are standard leapfrog determines an appropriate time step. Since the time step with centered space differences and the Cane±Patton in present ocean circulation codes is restricted by the (CP; Cane and Patton 1984) scheme. The leapfrog CFL associated with inertia-gravity waves of no im- scheme is the standard explicit time-stepping scheme portance for the large-scale ocean circulation, this is used routinely for generic advection problems. As for potentially of great value. all explicit schemes, a Courant±Friedrichs±Levy (CFL) The new scheme, Israeli±Naik±Cane (INC), combines condition restricts the allowed time step, but the scheme concepts from semi-Lagrangian (SL) methods (e.g., Staniforth and CoÃteÂ 1991) and the CP scheme. It is entirely natural to envision solving a tracer equation, * Lamont-Doherty Earth Observatory Contribution Number 6014. dC/dt ϭ 0, at a point x and time t by following a characteristic back to a ``take-off point'' x* at the previous time t Ϫ⌬t. Here SL extends this strategy to nonpassive Corresponding author address: Dr. Naomi Naik, Lamont-Doherty quantities such as momentum and pressure. In doing so Earth Observatory, Columbia University, 106 D Oceanography, Pal- isades, NY 10964. for a shallow water system, SL is applied only to the E-mail: [email protected] advective operator, which is split off from the wave

᭧ 2000 American Meteorological Society MARCH 2000 ISRAELI ET AL. 811 operator. The latter is usually solved implicitly. The SL along the lines of constant x Ϫ t and x ϩ t, respectively, method thus does not follow the true characteristics of that is, the system. In the INC scheme we ®rst split off the zonal direction part of the wave operator and treat it in (u ϩ h)(x, t) ϭ (u ϩ h)(x Ϫ⌬t, t Ϫ⌬t) a semi-Lagrangian manner, and then, as in CP, solve the ⌬t ␷ ϩ F ϩ Q)(x Ϫ s, t Ϫ s) dsץmeridional part implicitly. INC somewhat resembles the ϩ ( f␷ Ϫ ͵ y method of bicharacteristics (e.g., Roe 1994), the im- 0 portant difference being this implicit step. In INC, as (u Ϫ h)(x, t) ϭ (u Ϫ h)(x ϩ⌬t, t Ϫ⌬t) in CP, the different treatment of the x and y directions is motivated by the anisotropy of planetary waves on a ⌬t .␷ ϩ F Ϫ Q)(x ϩ s, t Ϫ s) dsץϩ ( f␷ Ϫ rotating sphere (on a beta plane); that is, they propagate ͵ y in the x direction but are standing in the y direction. 0 These two equations can be combined to easily ®nd 2. Method u(x, t) and h(x, t) given u and h at the previous time .(y␷ for time in the interval (t Ϫ⌬t, tץ and ␷ and The nondimensional linearized shallow water system To solve the problem, we must relax the assumption y␷) for a whole time interval. Weץ in two space dimensions can be written as of knowing ␷ (and approximate the integrals along characteristics, assum- (h ϭ F (1ץu Ϫ f␷ ϩ ץ tx ing we know ␷ at the beginning of the time step, t ϭ tn, and at the end of the time step, t ϭ tnϩ1. The trap- (h ϭ G (2ץty␷ ϩ fu ϩץ ezoidal rule yields a second-order approximation of the

␷ ϭ Q, (3) integrals, and gives us the following formulas for u ϩץu ϩץtxyh ϩץ h and u Ϫ h at the new time step: where a partial derivative with respect to x, for example, nϩ1 nnϩ1 (␷ ϩ F ϩ Qץx. The variables u, ␷,h,and forcings F, (u ϩ h) ϭ (u ϩ h)Lyϩ ␶[( f␷ Ϫץ is denoted as G, and Q are all functions of x, y, and t. The remaining n (yL␷ ϩ F ϩ Q) ] (4ץsymbol is f, the nondimensional coriolis parameter, ϩ ( f␷ Ϫ which is a function of y (latitude), only. The dimensional ␷ ϩ F Ϫ Q) ϩ1ץu Ϫ h)nϩ1 ϭ (u Ϫ h)nnϩ ␶[( f␷ ϩ) equations and variables can be recovered by multiplying Ry h by H, (u, ␷)byU, t by T, and (x, y)byL, where U n (yR␷ ϩ F Ϫ Q) ], (5ץϩ ( f␷ ϩ ϭ c, H ϭ c 2/g, T ϭ (c␤)Ϫ1/2, and L ϭ (c/␤)1/2. The dimensional forcing terms are (U/T)F, (U/T)G, and where quantities without subscripts are evaluated at an (H/T)Q. arbitrary grid point, (x, y), with a subscript ``R,'' they To solve an initial value problem satisfying this sys- are evaluted at (x ϩ⌬t, y), and with a subscript ``L,'' tem analytically, we specify the solution (u, ␷,h)ata at (x Ϫ⌬t, y). Superscripts n and n ϩ 1 indicate quan- n nϩ1 beginning time, t 0, and ®nd an expression for (u, ␷,h) tities at t and t , respectively. For convenience, ␶ ϭ for all t Ͼ t 0. Numerically, we usually employ a time- ⌬t/2. stepping scheme, where we specify an algorithm for These explicit formulas for unϩ1 and hnϩ1 depend on advancing the solution from any time t Ϫ⌬t to a time ␷ nϩ1. To compute ␷ nϩ1, we discretize the y-momentum t. Thus, assuming a uniform grid in time, this algorithm equation (2) in time using a centered, second-order is just a recurrence relation for the solution at t ϭ tnϩ1, scheme: n i given the solution at t ϭ t , where t ϭ t 0 ϩ i⌬t. nϩ1 nnnϩ1 n ϩ1 n h u ϩ uץyyh ϩץ ␷ Ϫ ␷ The motivation for the INC scheme comes from the ϩϩf observation that, if ␷ were known, then Eqs. (1) and (3) ⌬t 22 form a hyperbolic system, which is easily diagonalized, Gnϩ1 ϩ Gn with characteristics given by lines of constant x Ϫ t and ϭ . (6) x ϩ t with characteristic variables, u ϩ h and u Ϫ h. 2 This is easy to see by simply adding and subtracting (1) Suppressing the superscript for quantities at time tnϩ1 and (3), yielding and rearranging terms with known quantities on the

:y␷ ϩ F ϩ Q right, unknown on the left ץtx)(u ϩ h) ϭ f␷ Ϫץϩץ)

n n n n .( yh ϩ fu Ϫ G Ϫ Gץ)yh ϩ fu) ϭ ␷ Ϫ ␶ץ)y␷ ϩ F Ϫ Q. ␷ ϩ ␶ ץtx)(u Ϫ h) ϭ f␷ ϩץϪץ) (7) For each ®xed latitude line, that is, y ϭ constant, these two decoupled equations propagate the quantities u ϩ To decouple Eqs. (4), (5), and (7), we solve Eqs. (4) h and u Ϫ h in opposite directions along characteristics and (5) for unϩ1 and hnϩ1, and substitute into Eq. (7). in (x, t) space with unit speed. Solving these equations Solving for u and h (again suppressing the superscript is straightforward: the right-hand sides are integrated for quantities at time tnϩ1), gives 812 MONTHLY WEATHER REVIEW VOLUME 128

n u ϭ ␶ f␷ ϩ c (8) 1 ␦␷Ϫϩf 2 ␷ n yy 2 y␷ ϩ d , (9) ΂΃␶ץh ϭϪ␶ where 11 ϭ [ funnϩ fc ϩ ␦ (h nnϩ d ) Ϫ (G ϩ G n)] Ϫ ␷ n. y 2 1 ␶␶ n nn c ϭ {(u ϩ h)LRϩ (u Ϫ h) 2 3. Compute ␦y␷. nϩ1 nϩ1 n 4. Find u ϭ u and h ϭ h : (yL␷ ϩ F ϩ Qץϩ ␶[2F ϩ ( f␷ Ϫ n n u ϭ ␶ f␷ ϩ c h ϭϪ␶␦y␷ ϩ d . n {[(yR␷ ϩ F Ϫ Qץϩ ( f␷ ϩ Note that, for the discretization in the x direction, the 1 n nn ``take-off'' or ``departure'' points, x ϩ⌬t and x Ϫ d ϭ {(u ϩ h)LRϪ (u Ϫ h) i,j i,j 2 ⌬t, do not necessarily fall on grid points. Piecewise ␷ ϩ F ϩ Q)n cubic interpolation in x is used to obtain the expressionsץϩ ␶[2Q ϩ ( f␷ Ϫ yLthat need to be evaluated at these points, ensuring that n yR␷ ϩ F Ϫ Q) ]}. this step does not introduce an instability (StaniforthץϪ ( f␷ ϩ and CoÃteÂ 1991). Note that the scheme is computationally Substituting these formulas into the y momentum equa- undemanding. All steps are explicit except for the so- tion (7), and rearranging terms, lution of the one-dimensional equation in step 1. For a second-order approximation, that may be accomplished 1 .(2␷ Ϫϩf 2 ␷ with a tridiagonal solve (pentadiagonal for fourth orderץ y ΂΃␶ 2 1 nn nn n 3. Boundary conditions [( y(h ϩ d ) Ϫ (G ϩ Gץϭ [ fu ϩ fc ϩ ␶ To be relevant to the ocean, we need to solve the 1 shallow water system on a bounded domain. The interior Ϫ ␷ n. (10) algorithm is easily modi®ed to incorporate no-¯ow ␶ 2 boundary conditions. The boundary condition ␷ ϭ 0is This formula for ␷ is simply a second-order ordinary used at northern and southern boundary points in Eq. n differential equation (ODE) in y. Once we have found (10). In addition, ␦y(d ) must be known at the point next ␷, we obtain u and h using Eqs. (8) and (9). To complete to the boundary, so ␦y␷ is needed at the boundary. This the description of the scheme, we need discretizations is obtained by a second-order one-sided approximation, 2 y␷. On an A-grid, where u, ␷, and h are consistent with conservation. At the eastern and westernץ y␷ andץ of de®ned at the same grid points, second-order approxi- boundaries, the simplest algorithm is derived by spec- 2 y␷ are de®ned as ifying u and using only the outgoing characteristic toץ y␷ andץ mations to obtain h. At the eastern boundary, for example, there is def 1 ␷ ഠ ␦␷ϭ (␷ Ϫ ␷ ) and only the characteristic from the left, which determines ץ yy2⌬y i,jϩ1 i,jϪ1 the boundary condition on u ϩ h. Assuming no normal ¯ow at the boundary, u ϭ 0, which then gives a formula def 1 .2␷ ഠ ␦␷ϭ (␷ Ϫ 2␷ ϩ ␷ ). for h directlyץ yyyi,j⌬y2 ϩ1 i,j i,jϪ1 Time-stepping algorithmÐ To summarize, the solution algorithm is the following. Modi®cations at boundary points Time-stepping algorithmÐInterior points Western boundary points 0. Assume un, ␷ n, hn are known. 1W. Find dn: n n 1. Find c and d : n n d ϭϪ(u Ϫ h)R

1 n cn ϭ (I ϩ I ) ϩ ␶F, Ϫ ␶[F Ϫ Q ϩ ( f␷ ϩ ␦y␷ ϩ F Ϫ Q)R]. 2 12 2W. Find ␷ ϭ ␷ nϩ1 by solving an ODE in y for each nn :␷ ϩ F ϩ Q) value of xץI1 ϭ (u ϩ h)LyLϩ ␶( f␷ Ϫ 1 1 n d ϭ (I Ϫ I ) ϩ ␶Q, ␦␷yyϩ ␦ y( f␷) Ϫ ␷ 2 12 ␶ 2 ␷ ϩ F Ϫ Q). 11ץI ϭ (u Ϫ h)nnϩ ␶( f␷ ϩ 2 RyR ϭ [␦ (dnnϩ h ) Ϫ (G ϩ G n)] Ϫ ␷ n. ␶␶y 2 2. Find ␷ ϭ ␷ nϩ1 by solving an ODE in y for each value of x: 4W. Find u ϭ unϩ1 and h ϭ hnϩ1: MARCH 2000 ISRAELI ET AL. 813

n u ϭ 0, h ϭϪ␶( f␷ ϩ ␦y␷) ϩ d . 1 ␦␷Ϫϩf 2 ␷ Eastern boundary points yy ΂΃␶ 2 1E. Find dn: 1 n n nn nn n d ϭ (u ϩ h) ϭ [ fu ϩ fc ϩ ␦y(h ϩ d ) Ϫ (G ϩ G )] L ␶ ϩ ␶[F ϩ Q ϩ ( f␷ Ϫ ␦ ␷ ϩ F ϩ Q)n]. y L 1 n nϩ1 Ϫ ␷ . 2E. Find ␷ ϭ ␷ by solving an ODE in y for each ␶ 2 value of x: 4Wc. Find u ϭ unϩ1 and h ϭ hnϩ1: 1 n ␦␷yyϪ ␦ y( f␷) Ϫ ␷ u ϭ 0, h ϭϪ␶␦ ␷ ϩ d . ␶ 2 y Eastern boundary points 11 nn n n ϭ [␦ (d ϩ h ) Ϫ (G ϩ G )] Ϫ ␷ . n n ␶␶y 2 1Ec. Find c and d : n 4E. Find u ϭ unϩ1 and h ϭ hnϩ1: c ϭ 0, n I ϭ (u ϩ h)nnϩ ␶( f␷ Ϫ ␦␷ϩ F ϩ Q) u ϭ 0, h ϭ ␶( f␷ Ϫ ␦y␷) ϩ d . 0 y For grid points near boundaries, and for ⌬t Ͼ⌬x, an 1 dn ϭ (I ϩ I ) ϩ ␶Q, ``R'' or ``L'' point can fall outside of the boundary. In 2 02 this case, the characteristic is traced back in time to nn when it crossed the boundary. The values of u, ␷, and I2 ϭ (u ϩ h)LyLϩ ␶( f␷ Ϫ ␦␷ϩ F ϩ Q). h are not known at this intermediate time, Ã,t so we do 2Ec. Find ␷ ϭ ␷ nϩ1 by solving an ODE in y for each a constant extrapolation from the values at the point on value of x: the boundary at time tn. We also tried a linear extrapolation from the values at times tn and tnϪ1, and even 1 ␦␷Ϫϩf 2 ␷ added a ``correction'' step, using the linearly extrapo- yy ␶ 2 lated version as a ``predictor,'' but after many numerical ΂΃ experiments, concluded that the constant in time ex- 1 ϭ [ funnϩ fc ϩ ␦ (h nnϩ d ) Ϫ (G ϩ G n)] trapolation was suf®cient. See appendix A for details ␶ y and formulas. 1 Ϫ ␷ n. ␶ 2 4. Conservation 4Ec. Find u ϭ unϩ1 and h ϭ hnϩ1: Assuming conservative boundary condition on ␷ con- n sistent with the closed boundary condition at the north- u ϭ 0, h ϭ ␶␦y␷ ϩ d . ern and southern boundaries of the domain, the new scheme conserves mass in a semiperiodic domain. For 5. Accuracy and stability a domain with east±west boundaries, the boundary condition given above, which follows a single characteristic In this section, we compare the analytic dispersion into the domain, is nonconservative. Conservative relation for the shallow water equations on an equatorial boundary conditions can also be used. For ⌬t ϭ⌬x, ␤-plane (with boundary condition: u, ␷,hbounded as these can be written as the following. y → Ϯϱ), with the numerical dispersion relation of the scheme. Suppose u ϭ uÄei(kxϪ␻t), ␷ ϭ ␷Ä ei(kxϪ␻t) , and h ϭ Time-stepping algorithmÐ h˜ei(kxϪ␻t). Conservative boundary conditions The analytic dispersion relation is most easily ob- Western boundary points tained by recalling that a single equation for ␷ may be derived from the nondimensional shallow water equa- n n 1Wc. Find c and d : tions: n c ϭ 0, 22 2 .xy␷ ϩ ␷)t Ϫ f ␷ t Ϫ ␷ ttt ϩ ␷ x ϭ 0ץץ) nn I0 ϭ (u Ϫ h) ϩ ␶( f␷ ϩ ␦␷y ϩ F Ϫ Q) Using ␷ ϭ ␷Ä ei[kxϪ␻t], this equation becomes Ϫ1 n k d ϭ (I02ϩ I ) ϩ ␶Q, 222 2 .y␷Ä ϩ ␻ Ϫ k ϪϪf ␷Ä ϭ 0ץ 2 ΂΃␻ nn I2 ϭ (u Ϫ h)RyRϩ ␶( f␷ ϩ ␦␷ϩ F Ϫ Q). Note that, due to our scaling, since f ϭ Tfˆ ϭ Tf 0 ϩ nϩ1 2Wc. Find ␷ ϭ ␷ by solving an ODE in y for each (T␤L)y, then df/dy ϭ 1. On an equatorial ␤-plane, f 0 value of x: ϭ 0, and the nondimensional dispersion relation is 814 MONTHLY WEATHER REVIEW VOLUME 128

k Solving for uÄ and h˜ in the ®rst two equations, sub- 22 ␻ Ϫ k Ϫϭ2n ϩ 1, n ϭ 0, 1, . . . (11) 2 y ϭ 1 and␮␷Ä for ( fץ/f ץ ␻ stituting into the third, using Ϫ ␦ 2)␷Ä gives the numerical dispersion relation for the → → y where the boundary conditions (␷Ä 0asy Ϯϱ) scheme: Ã imply that eigensolutions⌿n of the␷Ä equation satisfy 2 2 ÃÃ 1 ␩ Ϫ 1 (yn)⌿⌿(2n ϩ 1) n . (␩22Ϫ 2␩ cos␪ ϩ 1) ϩ ␮(␩ Ϫ 1ץy Ϫϭ) ␶␩2 ϩ 1 For the special case of the Kelvin wave, ␷ ϭ 0, the ΂΃ dispersion relation is ϩ 2i␩ sin␪ ϭ 0, (15) ␻ ϭ k, (12) or, dividing though by 2i␩: which, for convenience, can be thought of as a special 1 ␥ (cos␥ Ϫ cos␪) tan ϩ ␮ sin␥ ϩ sin␪ ϭ 0. (16) case of Eq. (11) with n ϭϪ1. ␶ 2 2 As a generalization to the bounded domain, the eigenfunctions of the␷Ä equation satisfy As in the continuous case, the special case of the Kelvin wave, ␷ ϭ 0, has a separate dispersion relation, since 2 2 y)␷Ä ϭ ␮␷Ä , solving for uÄ and h˜ in the ®rst two equations impliesץ f Ϫ ) see, for example, Cane and Sarachik (1979). The dis- that uÄ ϭ h˜ ϭ 0 unless persion relation above becomes, after multiplication by ␥ ϭ ␪. (17) ␻:

3 2 For convenience, as in the analytic case, we can treat ␻ Ϫ k ␻ Ϫ k ϭ ␮␻. (13) this solution as special case of Eq. (16) with ␮ ϭϪ1. Neutral solutions of the assumed form exist only if the However, we must remember that ␮ ϭϪ1isnot an 2 2 solutions of the cubic are all real; otherwise, there will eigenvalue of the operator ( f Ϫ ␦y )␷Ä , and this case be a conjugate pair of solutions implying growing and must be analyzed separately. decaying waves. There will be three real roots if the We would like to ®nd conditions under which the discriminant scheme is stable for all values of ⌬t by deriving a cubic equation for g tan( /2), and ®nding when all three k22(k ϩ ␮) 3 ϭ ␥ D(␮) ϭϪ (14) roots are real, in which case |␩| ϭ 1. First note that the 427 discrete Kelvin wave solution, ␮ ϭϪ1, is stable. So is not positive. We note that ␮ Ͼ 0 is required to avoid we consider only values of ␮ corresponding to eigen- 2 2 instability as k → 0. For ␮ Ն 1.5 a maximum is obtained values of the discrete operator ( f Ϫ ␦y)␷Ä . at the left end point, k ϭ 0, of the interval of interest Using also s ϭ tan(␪/2), ␶Ã ϭ ␶ 2, and ␮Ã ϭ ␮␶ 2 gives and it is negative, implying stability for all k. For 0 Յ rise to ␮ Յ 1.5, an interior maximum is obtained, of magnitude g3 ϪϪ␶Ã sg22[s 2 ϩ ␮Ã (1 ϩ s )]g Ϫϭ␶Ã s 0. (18) (1 Ϫ ␮)/4. Thus we may conclude that a necessary and suf®cient condition for stability at all wavenumbers is We compute the discriminant of the cubic,D(␮Ã ) . Set- ␮ Ն 1. ting, without loss of generality, ␮Ã ϭ ␶Ã (1 ϩ 2nÃ), we get Another approach that goes over to the discrete case, ␮Ã ϭ ␶Ã for nÃ ϭ 0. Then like the continuum case the gives a proof of suf®ciency. Observe that, when ␮ ϭ discriminant factors: 1, the discriminant factors as D(␶Ã ) ϭϪ[(2 ϩ ␶Ã )s 2 Ϫ ␶Ã 2]2[4␶Ã ϩ (1 ϩ ␶Ã )2s 2]/108; 2 2 2 D(1) ϭϪ(1 Ϫ 2k ) (4 ϩ k )/108. this quantity vanishes for s 2 ϭ ␶Ã 2 /(2 ϩ ␶Ã ) and is negative Thus D(1) vanishes for k 2 ϭ ½ and is negative else- elsewhere. where. Furthermore For nÃ ± 0 the discriminant is a complicated expression dif®cult to evaluate but we observe that the dif- D(1) Ϫ D(␮) Ն 0 ference D(␶Ã ) Ϫ D(␮Ã ) is a cubic in nÃ and the coef®cients for ␮ Ն 1 so that the last inequalities are suf®cient for of all powers of nÃ are positive, thusD(␮Ã ) is negative stability. for positive nÃ, in agreement with the continuous case. Some of the above steps can be applied to obtain the Thus a suf®cient condition for stability for all ␶ is again numerical dispersion relation. De®ne ␩ ϭ eϪi␥, and ␣ ␮ Ն 1. ϭ eϪi␪, where ␥ ϭ ␻⌬t and ␪ ϭ k⌬x. Then, if ⌬t is a In conclusion we recall that ␮ are the eigenvalues of multiple of ⌬x, no interpolation is needed in the x-di- the associated eigenproblem for the eigenfunctions in rection, and y, so that the above conditions should be satis®ed by the smallest eigenvalue. Therefore when a numerical ␷Ä(ץ␩ Ϫ ␣)(uÄ ϩ h˜) ϭ ␶(␩ ϩ ␣)( f Ϫ) y scheme is used also in y, or when interpolation is needed Ϫ1 Ϫ1 y)␷Ä in x, the discrete analog of␮Ã should be considered. Inץ␩ Ϫ ␣ )(uÄ Ϫ h˜) ϭ ␶(␩ ϩ ␣ )( f ϩ) either the discrete or continuous case, stability can be yh˜ ϩ fuÄ) ϭ 0. guaranteed when ␮ Ն 1. For the continuous case, withץ)(␩ Ϫ 1)␷Ä ϩ ␶(␩ ϩ 1) MARCH 2000 ISRAELI ET AL. 815

␷ ϭ 0 at the endpoints, all eigenvalues are greater than or equal to 1. The smallest eigenvalue for a discrete 2 2 -y ) can be guarץ approximation to the operator ( f Ϫ anteed to be greater than one with a little care in the discretization of the coriolis term, f 2. For example, on an equatorial ␤ plane with a uniform grid with grid size 2 y givenץof ⌬y, and the second-order approximation to by 1 ,( 2␷ ഠ ␦␷ϭ (␷ Ϫ 2␷ ϩ ␷ץ yyyi,j⌬y2 ϩ1 i,j i,jϪ1 the y grid should be arranged so that the equator (a zero of f) lies half-way between grid points with the analytic 2 2 2 operator f ϭ y being approximated as (yj) . Using the Courant±Fischer minimax thereom (see Golub and

VanLoan 1989), the smallest eigenvalue, ␭min of a symmetric n ϫ n matrix, A, can be written as xTAx ␭ (A) ϭ min . min T 0±x∈Rn xx Therefore, for any two symmetric matrices, A and B, xT(A ϩ B)x ␭ (A ϩ B) ϭ min min T x±0 xx xTTAxzBz Ն min ϩ min TT x±0 xxz±0 zz

ϭ ␭min(A) ϩ ␭ min(B).

The discrete operator Ϫ␦yy can be written as a symmetric tridiagonal matrix A and f 2 can be written as a diagonal matrix B. The smallest eigenvalue of A is given by

4 ⌬y ␭ (A) ϭ sin2 min ⌬y2 ΂΃2 and the smallest eigenvalue of B is given by

⌬y 2 ␭ (B) ϭ . FIG. 1. Dispersion diagrams for ⌬t ϭ 0.2. min ΂΃2 The lower bound on ␮ for the discrete operator then follows, since identical and the curves for the Yanai, or mixed mode, n ϭ 0, are almost identical. Note that, in these plots, 1 ⌬y 22⌬y ␻⌬t ϭ ␲ and k⌬x ϭ ␲ correspond to the highest fre- ␭min(A ϩ B) Ն 1 ϪϩϾ1. quency represented on the grids, that is, the two grid- 32 2 []΂΃ ΂΃ point wave. A comparison of the analytic dispersion diagrams to For larger time steps, there is a greater discrepancy the discrete dispersion diagrams for the special case of between the discrete and analytic curves, see Figs. 2 an equatorial ␤ plane ( f ϭ y) gives a good indication and 3, but they are arguably close enough for most of the spatial and temporal accuracy constraints appro- climate-scale ocean dynamics. (For a fast baroclinic priate for the dynamics of interest. In Fig. 1 we plot the mode in the ocean with a grid spacing of ⌬x ϭ ½,or curves for a few values of the wavenumber n for the 1.6Њ, a wavenumber of ␲/4 corresponds to a wavelength special case, ⌬t ϭ⌬x ϭ 0.2. For this relatively ®ne of about 1200 km and ⌬t ϭ ½ corresponds to about 18 resolution, there is very good agreement between the h.) For ⌬t ϭ 2, the curves are close only for the Rossby analytic and discrete curves for the Rossby waves of waves with very small k and even for these the maxima moderate wavenumber in x (see lower panel) and even of the curves (distinguishing regions of eastward and for the gravity waves (upper portion of upper panel). westward propagation of energy) are way off, see Fig. For the equatorial Kelvin wave, n ϭϪ1, the curves are 4. In the following section we explore the effect of this 816 MONTHLY WEATHER REVIEW VOLUME 128

FIG. 2. Dispersion diagrams for ⌬t ϭ 0.5. FIG. 3. Dispersion diagrams for ⌬t ϭ 1.0. discretization error for the time and length scales rel- In the meridional direction, we use a relatively high ative to the fastest baroclinic (internal) modes of the resolution of ¼Њ (scaled, ⌬y ϭ 0.05) so that errors due ocean. to the discretization in y will be negligible compared with those in x and t. The zonal and time resolutions 6. Experiments are varied. A scaled ⌬x ϭ 1 corresponds to a 5Њ zonal resolution and a scaled ⌬t ϭ 1 corresponds to a time In this section we illustrate the stability and accuracy step of 37.3 h. of the INC scheme for time and length scales appropriate In this unforced initial value problem we expect the to ocean dynamics with a few simple experiments. initial perturbation to propagate westward. Because we are initializing with a Gaussian, we expect to see dispersion in the wake of the initial bump associated with a. Midlatitude Gaussian bumpÐPeriodic domain the higher frequency waves. A midlatitude Gaussian perturbation of h is allowed Figures 5 and 6 show plots of the propagation of the to propagate in a semiperiodic domain, from 20ЊNto anomaly at 2-year intervals, for various values of ⌬t 60ЊNona90Њ wide beta plane centered at 40ЊN. We and ⌬x. The actual solution is shown in Fig. 5a (obtained assume an internal wave speed of 3.17 m sϪ1, corre- by the leapfrog method, checking for convergence by sponding to a horizontal scale L ഠ 425 km. successively reducing ⌬x, ⌬y, and ⌬t). MARCH 2000 ISRAELI ET AL. 817

initial westward propagation of the Northern Hemi- sphere perturbation continues until it hits the western boundary. This creates a coastal Kelvin wave propagating equatorward. An equatorial Kelvin wave carries the perturbation eastward along the equator. Then Ross- by waves spread the perturbation slowly back into the domain from the eastern boundary in the characteristic parabolic shape, propagating faster at lower latitudes, slower at higher latitudes. We illustrate the correct mod- eling of this behavior by the INC scheme using the Atlantic Basin, arti®cially closed at 30ЊS and 60ЊN. In Fig. 7, the perturbation is shown at monthly intervals, from left to right, continued on the line below.

c. Forced motion, equatorial Paci®c Finally, we test the new INC scheme in our domain of primary interest, the equatorial Paci®c. We ®rst compare the INC scheme to both the CP and leapfrog (LF) schemes for a simpli®ed equatorial Paci®c basin. For a typical equatorial Paci®c density structure, the ®rst baroclinic mode has an equivalent depth of 84 cm, Kelvin wave speed of 287 cm sϪ1, horizontal length scale of 3.19Њ, and timescale of 1.43 days. We force this mode with the Florida State University analysis (Legler and O'Brien) of monthly wind stress anomolies from 1964 to 1991. Due to constraints of the CP methodology, we approximate the basin geometry, representing Australia and North America by rectangular regions in the south- west and northeast corners, respectively. All three schemes were run on a 2Њ by ½Њ grid. The ®ner meridional resolution is required to resolve the equatorial dynamics. The CP scheme is run with a time step of 5 days and the LF a time step of 2 h. For the INC scheme, we have seen that the zonal resolution dictates a natural time step, for which the scaled value of ⌬t equals the scaled value of ⌬x. Since ⌬x ϭ 2Њ/3.19Њ ഠ 0.627, this implies a time step of approximately 22 h. The anom- FIG. 4. Dispersion diagrams for ⌬t ϭ 2.0. olous pressures at the end of 1991, computed by the CP, LF, and INC schemes are shown in Figs. 8a±c, respectively. The solutions are in close agreement near the Note that Figs. 5c and 6a are almost identical, show- equator and in the interior, and differ near boundaries, ing again that it is time accuracy that dictates the overall due to varying boundary treatment. In Figs. 9a,b, we accuracy at these resolutions. Figure 6c shows the dis- also show the INC solution using time steps of a half persive nature of the scheme for a larger time step, ⌬t and a full CP time step of 5 days. Clearly the INC ϭ 1, as expected from the dispersion diagram in Fig. scheme solution is not very accurate at the CP time step, 3. All of the above experiments were with ⌬t an integer but the 2.5 day time step solution is better. The time multiple of ⌬x, so that take-off points always correspond step of the LF scheme is limited by the CFL restriction to grid points. When ⌬t is not an integer multiple of imposed by the high resolution in the meridional direc- ⌬x, diffusion is introduced due to the interpolation of tion. In addition, near the western boundary, the LF values from grid points to take-off points. This diffusion scheme also requires the addition of a small amount of is largest for the case where ⌬x ϭ 2⌬t, where take-off horizontal diffusion. In this case we use a diffusion points are half-way between grid points (see Fig. 6b). (Laplacian) coef®cient of 2 ϫ 102 m2 sϪ1 in the meridional direction, and 2 ϫ 103 m 2 sϪ1 in the zonal direc- b. Midlatitude Gaussian bumpÐBounded domain tion. Although the CP scheme has the advantage of the When we introduce eastern and western boundaries, larger time step, it is limited to simple geometries. The we expect a more complicated sequence of events. The INC scheme is not, and we illustrate the results of re- 818 MONTHLY WEATHER REVIEW VOLUME 128

FIG. 5. Propagation of a midlatitude depth perturbation in a periodic domain. MARCH 2000 ISRAELI ET AL. 819

FIG. 6. Propagation of a midlatitude depth perturbation in a periodic domain, ⌬t ϭ 1. 820 MONTHLY WEATHER REVIEW VOLUME 128

FIG. 7. Propagation of a depth perturbation in the North Atlantic. MARCH 2000 ISRAELI ET AL. 821

FIG. 8. Comparison of CP, leapfrog, and INC schemes in the equatorial Paci®c. peating the above experiment in a basin with more re- ef®ciency of explicit schemes are limited by the CFL alistic geometry. The ®nal anomolous pressure ®eld is conditions and require time steps on the order of an shown in Fig. 9c. hour for the fastest baroclinic mode. The CFL is imposed by gravity waves, and this time step is an order of magnitude smaller than typically needed for the ac- 7. Conclusions curate computation of Rossby waves. The only con- The existing schemes for solving the shallow water straint on the time step of the INC scheme arises from equations to which INC should be compared fall into accuracy considerations, which may allow an order of two major categories, explicit and implicit in time. The magnitude improvement in ef®ciency. The other implicit 822 MONTHLY WEATHER REVIEW VOLUME 128

FIG. 9. Comparison of INC solutions using a long time step and a more realistic equatorial Paci®c.

schemes (semiimplicit/semi-Lagrangian, fully implicit) or no net gain in ef®ciency. The INC scheme, by de- usually require more work (either for the interpolations, coupling the zonal and meridional dependencies, retains or for the solution of very large linear systems) in ex- the per time step ef®ciency of the explicit schemes, change for an increase in the time step, often with little while eliminating the CFL restriction on the size of the MARCH 2000 ISRAELI ET AL. 823

time step. As can be seen from the analysis and some simple experiments presented here, the time step is con- strained by zonal accuracy considerations. Thus we conclude that the INC scheme is an ef®cient alternative to the standard schemes, giving an order of magnitude in speed with little loss in accuracy. More- over, INC can be used for the wave solve of more general systems such as those with nonlinearities and used in semi-Lagrangian and semiimplicit schemes. FIG. A1. Schematic of characteristic intersecting the western boundary. Acknowledgments. This work was supported by NOAA Grant UCSIO-10075411, NASA Grant NAG5- 6315, and the generous support of the Vetlesen Foun- dation. points that are within a distance ⌬t of an ``x-boundary.'' For this case, the length of the path of integration along APPENDIX the left and right characteristics, must be reduced. See Near-Boundary Formulas Fig. A1, where the left characteristic intersects the west- If ⌬t Ͼ⌬x, characteristics may intersect a boundary ern boundary. in the time interval (t, t ϩ⌬t). This occurs at all grid The general equations can be written as

nϩ1 nnϩ1 n y␷ ϩ F ϩ Q)] L ץy␷ ϩ F ϩ Q) ϩ ( f␷ Ϫ ץu ϩ h) ϭ (u ϩ h)LLϩ ␶ [( f␷ Ϫ)

nϩ1 nnϩ1 n y␷ ϩ F Ϫ Q)] R ץy␷ ϩ F Ϫ Q) ϩ ( f␷ ϩ ץu Ϫ h) ϭ (u Ϫ h)RRϩ ␶ [( f␷ Ϫ)

nϩ1 nnϩ1 nnϩ1 nnϩ1 n ,( h ϩ fu ϩ fu ) ϭ ␶(G ϩ Gץyyh ϩץ)␷ Ϫ ␷ ϩ ␶

where 1 ␶Ј 22␷ Ϫ f ϩϩ␷ץ y ΂΃␶␶ ␶ ␶L ϭ |x Ϫ xL|/2 ␶R ϭ |x Ϫ xR|/2. 1 nn nn n [( y(h ϩ d ) Ϫ (G ϩ Gץϭ [ fu ϩ fc ϩ Note that quantities evaluated at the boundary are at an ␶ intermediate time. These quantities are obtained by a 1 constant extrapolation in time. Ϫ ␷ n. ␶␶ The algorithm can now be rewritten as the following. 4M. Find u ϭ unϩ1 and h ϭ hnϩ1: Modi®ed time-stepping algorithmÐInterior points n n , y␷ ϩ dץ y␷ ϩ c h ϭ ␶Јf␷ Ϫ ␶ץu ϭ ␶ f␷ Ϫ ␶Ј 1M. Find cn and dn: where ␶ ϭ (␶ R ϩ ␶ L)/2 and ␶Јϭ(␶ L Ϫ ␶ R)/2.

1 REFERENCES cn ϭ (I ϩ I ) ϩ ␶F ϩ ␶ЈQ, 2 12 Cane, M. A., and E. S. Sarachik, 1979: Forced baroclinic ocean motion. Part III: An enclosed ocean. J. Mar. Res., 37, 355±398. nn y␷ ϩ F ϩ Q) L , and R. J. Patton, 1984: A numerical model for low-frequency ץI1 ϭ (u ϩ h)LLϩ ␶ ( f␷ Ϫ equatorial dynamics. J. Phys. Oceanogr., 14, 1853±1863. 1 Golub, G. H., and C. F. Van Loan, 1989: Matrix Computations. 2d n d ϭ (I12Ϫ I ) ϩ ␶ЈF ϩ ␶Q, ed. The Johns Hopkins University Press, 642 pp. 2 Roe, P., 1994: Linear bicharacteristic schemes without dissipation. nnICASE Tech. Rep. 94-65, pp. [Available from Institute for Com- ,y␷ ϩ F Ϫ Q). R puter Applications in Science and Engineering, Mail Stop 32C ץI2 ϭ (u Ϫ h)RRϩ ␶ ( f␷ ϩ NASA LARC, Hampton, VA 23681-0001.] nϩ1 Staniforth, A., and J. CoÃteÂ, 1991: Semi-Lagrangian integration 2M. Find ␷ ϭ ␷ by solving an ODE in y for each schemes for atmospheric modelsÐA review. Mon. Wea. Rev., value of x: 119, 2206±2223.