Etudes et évaluation de processus océaniques par des hiérarchies ...

86 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
A. Wirth / Ocean Modelling 9 (2005) 71–87 83
tel-00545911, version 1 - 13 Dec 2010
ocean floor by a no-slip boundary condition. The boundary condition in the horizontal direction
is periodic. In the middle of the domain a vertical (downward) force, with a circular Gaussian
profile, is added ðGðx 1 ; x 2 Þ ¼ expð 32 10 6 m 2 ðx 2 1 þ x2 2ÞÞÞ. Such flow leads to a well studied
instance of a non-linear boundary layer at the lower boundary, the so called Hiemenz flow (see
e.g. Schlichting, 1968, pp. 96–99). Seven experiments are performed with varying spatial resolution.
The number of grid points in the domain (including boundary points) are 8·8, 16·16,
32·32, 64·64, 128·128, 256·256, 512·512. The buffer zone above and below the fluid area is
250 m, each. The friction parameters are m ¼ j ¼ 1:0 m 2 /s and the time-step is Dt ¼ 60 s in all
calculations.
The integration started from a vanishing velocity field and was performed for t max ¼ 2:16 10 5 s.
For the 256·256 resolution run the integration was continued until t long ¼ 3:00 10 5 s. Almost no
differences are visible between the velocity fields at t max and t long (see Figs. 8–10).
A contour plot of the horizontal velocity component for the lowest and the highest resolution
run is given in Fig. 7.
At the lower boundary a Hiemenz flow is generated, and the boundary-layer thickness is about
200 m. We like to mention that in order to capture the non-linear dynamics in the (non-linear)
boundary layer about 10 grid points have to be within the boundary layer.
The horizontal velocity along x ¼ 250 m and the vertical velocity along x ¼ 500 m, as a function
of depth are shown in Fig. 8.
The horizontal velocity component clearly exhibits Gibbs oscillations at the lower (no-slip)
boundary, the amplitude of the oscillation however decreases linearly with resolution. These
oscillations are exposed in Fig. 9.
A linear decrease of the amplitude in the Gibbs oscillations reveals a discontinuity in the first
derivative at the boundary. This comes at no surprise as the horizontal velocity component is
extended skew symmetrically across the lower boundary. This forces the second derivative to
vanish at the boundary which is inconsistent to the real dynamics (see Schlichting, 1968, pp. 96–
99). No such oscillation occur in the vertical velocity component at the lower boundary (Fig. 10),
as a vanishing second derivative is consistent with the real dynamics (see Schlichting, 1968, pp.
96–99). Gibbs oscillations are absent for both velocity components at the upper (free-slip)
boundary (Fig. 8).
Fig. 7. Contour plot of the horizontal velocity, in the lowest resolution run (8·8) (left) and the highest resolution run
(512·512) (right). Contour intervals are drawn every 0.005 m/s, starting from )0.0375 m/s, dashed lines show negative
values.