An initial value test case for the shallow water equations J. Galewsky
1
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY
R. K. Scott Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY
L. M. Polvani Department of Applied Physics and Applied Mathematics and Department of Earth and Environmental Sciences, Columbia University, New York, NY
1
Full address: Department of Applied Physics and Applied Mathematics, Columbia University, 500 West
120th St. New York, NY 10027. E-mail:
[email protected]
Numerical shallow water models (SWM) continue to be extremely important in the study of the dynamics of two-dimensional atmospheric flows as well as in the development of new numerical techniques that may be applied to more sophisticated models. In this paper, we present numerically converged solutions to a nontrivial initial value problem for the shallow water equations (SWE). We suggest that this problem can serve as a useful test case for new numerical shallow water models. The test presented here is intended to complement the standard tests for the shallowwater equations proposed by Williamson et al, (1992) (hereafter W92). Unfortunately, most of the flows in the W92 cases are unrealistically simple compared to the flows that need to be computed in typical modeling studies. Specifically, none of the W92 cases generate tight vorticity gradients or subgrid scales, features essential to many models of atmospheric dynamics. One of the most widely used W92 cases (test case 5) involves zonal flow over an isolated mountain. This test is popular because it generates a reasonably complex flow with non-trivial temporal and spatial features. Unfortunately, the mountain topography contains a cusp that is difficult for some numerical schemes to handle. Also, this test requires that additional code be added to the main SWM to handle the flow over topography. Our new test specifically addresses these shortcomings in the W92 test suite by generating complex dynamics within the context of a simple initial value problem. The test does not require that any additional code be added to the SWM beyond that required to set up the initial conditions. In vector form, the inviscid shallow water equations can be written as:
dV = −f k × V − ∇Φ dt (1) dΦ = −Φ∇ · V dt where dtd () is the material derivative, V ≡ iu + jv is the horizontal velocity vector, i and 2
j are unit vectors in the eastward and westward direction, k is the unit vector in the vertical direction, f ≡ 2Ω sin φ is the Coriolis parameter, Ω is the angular velocity of the Earth, φ is the latitude, and Φ = gh is the geopotential, where h is the fluid depth and g is the acceleration due to gravity. For the results presented here, we have set h = 10, 000 meters. The initial condition consists of a basic zonal flow, representing a typical midlatitude tropospheric jet, to which a small perturbation is added to induce the development of barotropic instability. Both the basic flow and the initial perturbation are analytically specified, allowing the complete initial condition to be reproducible in testing future shallow water models. The basic flow is a zonal jet u that is a simple function of latitude (φ):
u(φ) = u0 cos
2
π (φ − φ0 ) 2 γ
(2)
where u0 is the maximum zonal velocity, φ is the latitude in radians, φ0 is the meridional offset of the jet in radians, and γ is a nondimensional parameter that controls the width of the jet. The constants are chosen as follows: u0 = 80 ms−1 , φ0 = π/4, and γ = π/18. Once the zonal flow is specified, the geopotential height h is obtained by numerically integrating the balance equation:
gh(φ) = gh0 −
φ
tan(φ) u(φ ) dφ au(φ ) f + a
(3)
where a is the radius of the earth. In order to initiate the barotropic instability, the zonal flow is perturbed by adding a localized bump to the balanced height field in the form: ˆ sech2 (α(λ − λ0 ))sech2 (β(φ − φ0 )) Φ (λ, φ) = Φ
3
(4)
ˆ = 120 meters. The where λ is longitude (in radians), φ0 = λ0 = π/4, α = 3, β = 15, Φ initial zonal flow and perturbed height field are illustrated in Fig. 1. Because the height field is not completely balanced, gravity waves radiate away from the perturbed zone during the first several hours of the integration. After a few days, the growing barotropic instability leads to the development of tight gradients in the vorticity field. Thus, the test we propose here contains complex dynamics on two different time scales: fast gravity waves that develop on timescales of minutes and hours and slower vorticity dynamics that evolve on timescales of days. We present solutions computed with GFDL’s FMS Shallow Water Model (FMS-SWM) using the “Havana” release. FMS-SWM solves the inviscid shallow water equations using a spectral-transform method with an Robert-Asselin-filtered semi-implicit leapfrog scheme for time integration. The results described below represent a numerically converged solution of the shallow water equations, in the sense that the plots do not change as the spatial and temporal discretization are increased. Furthermore, the results are not dependent on any numerical technique. Any numerical model that claims to solve the SWE should be able to precisely reproduce these results. See Galewsky et.al, (2003) for a detailed discussion of convergence for the solutions presented here. The short timescale gravity wave dynamics are illustrated by the divergence and height fields from the first several hours of the integration. These fields are presented in Figs. 2 and 3 and show how the gravity waves rapidly radiate from the region of the unbalanced perturbation. Because we are solving the inviscid SWE, we note that these gravity waves are not strongly dissipated and that the gravity waves are active throughout the model run. The longer timescale balanced dynamics of the barotropic instability are illustrated in Fig. 4, which shows the time evolution of the vorticity (ζ) field over 90 hours. By the middle 4
of the third day of integration (84 hours), tight vorticity gradients have developed. Beyond 90 hours, however, our inviscid solutions become unstable. The addition of some form of explicit diffusion allows the integration to proceed longer, but changes the equations solved and the resulting solutions. We have computed a reference solution for the initial condition described above, with the addition of a simple diffusive operator to the SWE. The results for the vorticity field are in Fig. 5. In summary, we have computed and presented numerically converged solutions to an initial value problem for the shallow water equations in spherical coordinates. The initial condition consists of a barotropically unstable midlatidute jet and yields solutions that contain complex dynamics on two timescales: short timescale gravity waves and longer timescale vorticity dynamics. These solutions are intended to serve as a non-trivial test for the core dynamics of shallow water models. We believe that this test case is a useful compliment to the test cases of W92, in that our case provides an easily implemented initial condition, complex dynamics, tight vorticity gradients, and well-resolved reference solutions.
5
References Galewsky, J., L. M. Polvani, R. K. Scott, 2003: An initial-value problem to test numerical models of the shallow water equations. Monthly Weather Review, in review. Polvani, L. M., R. K. Scott, 2002: An initial-value problem for testing the dynamical core of atmospheric general circulation models. Monthly Weather Review., in review. Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102(1), 211–224.
6
90
60
Latitude
30
0
−30
−60
−90 −10
0
10
20
30
40 50 60 Zonal Wind (m/s)
70
80
90
100
52 9000
50
Latitude
48 9300
46
44 9600
42
40 0
10
20
30
40
50 60 Longitude
70
80
90
100
Figure 1: The initial conditions: (A) Zonal profile of the initial zonal wind (B) The inital balanced height field with a superimposed, unbalanced perturbation. Contour interval of 100 meters.
7
6 Hours 90
60
60
30
30 Latitude
Latitude
4 Hours 90
0
0
−30
−30
−60
−60
−90
0
100
200 Longitude
−90
300
0
100
90
60
60
30
30
0
0
−30
−30
−60
−60
−90
0
100
200 Longitude
300
10 Hours
90
Latitude
Latitude
8 Hours
200 Longitude
−90
300
0
100
200 Longitude
300
Figure 2: Divergence field during the first 10 hours, illustrating gravity wave propagation from the initial perturbation. Results from a T170 model with a 15 second time step. Contour interval is 3 × 10−7 sec−1 . Negative contours are dashed.
8
6 Hours 90
60
60
30
30 Latitude
Latitude
4 Hours 90
0
0
−30
−30
−60
−60
−90
0
100
200 Longitude
−90
300
0
100
90
60
60
30
30
0
0
−30
−30
−60
−60
−90
0
100
200 Longitude
300
10 Hours
90
Latitude
Latitude
8 Hours
200 Longitude
−90
300
0
100
200 Longitude
300
Figure 3: Height field anomaly (defined as deviations from h0 , the balanced initial height field) during the first 10 hours, illustrating gravity wave propagation from the initial perturbation. Results from a T170 model with a 15 second time step. Contour interval is 20 meters. Negative contours are dashed.
9
Latitude
60 50 40 30
24 Hours 0
50
100
150
200
250
300
350
100
150
200
250
300
350
100
150
200
250
300
350
100
150
200
250
300
350
100
150 200 Longitude
250
300
350
Latitude
60 50 40 30
48 Hours 0
50
Latitude
60 50 40 30
72 Hours 0
50
Latitude
60 50 40 30
84 Hours 0
50
Latitude
60 50 40 30
90 Hours 0
50
Figure 4: The time evolution of the vorticity (ζ) field. The resolution is T341 with a 30 second time step. Contour interval is 2 × 10−5 sec−1 . Negative values of ζ are dashed.
10
70
Latitude
60 50 40 30 4 Days 20
0
50
100
150
200
250
300
350
100
150
200
250
300
350
100
150
200
250
300
350
100
150 200 Longitude
250
300
350
70
Latitude
60 50 40 30 5 Days 20
0
50
70
Latitude
60 50 40 30 6 Days 20
0
50
70
Latitude
60 50 40 30 7 Days 20
0
50
Figure 5: Vorticity field from a solution to the initial value problem with ordinary diffusion. Computed with FMS-SWM at T170, ν = 1.0 × 105 . Contour interval is 2 × 10−5 sec−1 . Negative values of ζ are dashed.
11
