An Unconditionally Stable Scheme for the Shallow Water Equations*
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810 MONTHLY WEATHER REVIEW VOLUME 128 An Unconditionally Stable Scheme for the Shallow Water Equations* 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 CoÃte 1991) and the CP scheme. It is entirely natural to envision solving a tracer equation, * Lamont-Doherty Earth Observatory Contribution Number 6014. dC/dt 5 0, at a point x and time t by following a char- acteristic back to a ``take-off point'' x* at the previous time t 2Dt. 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 q 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 2 t and x 1 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 1 h)(x, t) 5 (u 1 h)(x 2Dt, t 2Dt) a semi-Lagrangian manner, and then, as in CP, solve the Dt meridional part implicitly. INC somewhat resembles the 1 ( fy 2]y 1 F 1 Q)(x 2 s, t 2 s) ds E y method of bicharacteristics (e.g., Roe 1994), the im- 0 portant difference being this implicit step. In INC, as (u 2 h)(x, t) 5 (u 2 h)(x 1Dt, t 2Dt) in CP, the different treatment of the x and y directions is motivated by the anisotropy of planetary waves on a Dt 1 ( fy 2]y 1 F 2 Q)(x 1 s, t 2 s) ds. rotating sphere (on a beta plane); that is, they propagate E 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 and y and ]yy for time in the interval (t 2Dt, t). The nondimensional linearized shallow water system To solve the problem, we must relax the assumption in two space dimensions can be written as of knowing y (and ]yy) for a whole time interval. We approximate the integrals along characteristics, assum- ] u 2 fy 1]h 5 F (1) tx ing we know y at the beginning of the time step, t 5 tn, and at the end of the time step, t 5 tn11. The trap- ]tyy 1 fu 1]h 5 G (2) ezoidal rule yields a second-order approximation of the ]txyh 1]u 1]y 5 Q, (3) integrals, and gives us the following formulas for u 1 h and u 2 h at the new time step: where a partial derivative with respect to x, for example, n11 nn11 is denoted as ]x. The variables u, y,h,and forcings F, (u 1 h) 5 (u 1 h)Ly1 t[( fy 2]y 1 F 1 Q) G, and Q are all functions of x, y, and t. The remaining n symbol is f, the nondimensional coriolis parameter, 1 ( fy 2]yLy 1 F 1 Q) ] (4) which is a function of y (latitude), only. The dimensional (u 2 h)n11 5 (u 2 h)nn1 t[( fy 1]y 1 F 2 Q) 11 equations and variables can be recovered by multiplying Ry h by H, (u, y)byU, t by T, and (x, y)byL, where U n 1 ( fy 1]yRy 1 F 2 Q) ], (5) 5 c, H 5 c 2/g, T 5 (cb)21/2, and L 5 (c/b)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 1Dt, y), and with a subscript ``L,'' tem analytically, we specify the solution (u, y,h)ata at (x 2Dt, y). Superscripts n and n 1 1 indicate quan- n n11 beginning time, t 0, and ®nd an expression for (u, y,h) tities at t and t , respectively. For convenience, t 5 for all t . t 0. Numerically, we usually employ a time- Dt/2. stepping scheme, where we specify an algorithm for These explicit formulas for un11 and hn11 depend on advancing the solution from any time t 2Dt to a time y n11. To compute y n11, 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 5 tn11, scheme: n i given the solution at t 5 t , where t 5 t 0 1 iDt. n11 nnn11 n 11 n y 2 y ]yyh 1]h u 1 u The motivation for the INC scheme comes from the 11f observation that, if y were known, then Eqs. (1) and (3) Dt 22 form a hyperbolic system, which is easily diagonalized, Gn11 1 Gn with characteristics given by lines of constant x 2 t and 5 . (6) x 1 t with characteristic variables, u 1 h and u 2 h. 2 This is easy to see by simply adding and subtracting (1) Suppressing the superscript for quantities at time tn11 and (3), yielding and rearranging terms with known quantities on the (]1]tx)(u 1 h) 5 fy 2] yy 1 F 1 Q right, unknown on the left: n n n n (]2]tx)(u 2 h) 5 fy 1] yy 1 F 2 Q. y 1 t(]yh 1 fu) 5 y 2 t(]yh 1 fu 2 G 2 G ). (7) For each ®xed latitude line, that is, y 5 constant, these two decoupled equations propagate the quantities u 1 To decouple Eqs. (4), (5), and (7), we solve Eqs. (4) h and u 2 h in opposite directions along characteristics and (5) for un11 and hn11, and substitute into Eq. (7). in (x, t) space with unit speed.