5. Examples

In the input file filename{numo3d.in} the test cases are controlled by the input parameter ‘icase’. h-NUMO contains the following ready-to-use test cases: begin{enumerate} item test_case = ‘bump’ is the perturbation in the bottom layer for the two-layer. item test_case = ‘double-gyre’ is a double-gyre test case. item test_case = ‘lakeAtrest’ is lake-at-rest test case. end{enumerate}

We present some examples of idealised modeling studies. To run these simulations change into the Examples/ directory.

5.1. test_case = ‘lakeAtrest’: Well-balanced test

This test case is used to validate the well-balanced property of the DG scheme for the multilayer equations. We have initialized the layers fluid domain with densities $\rho_k = 1027.01037 + 0.2110\times (k-1) \frac{kg}{m^3}, \ k = 1,\ldots,N_l$ and the layer interface positions $z_k = -40/N_l$ m, where $N_l$ is the number of layers. The horizontal extend of the domain was $(x,y)\in [0,\ 2000]\times[0,\ 2000]$ m with wall boundary conditions on both ends, and the bottom topography is given by

\begin{equation} Z_b(x,y) = 3\left( 1 + \cos\left(\frac{\pi r}{250} \right)\right), \end{equation}

where $r = \sqrt{(x-1000)^2 + (y-1000)^2}$.

Cross-section. — Figure 5.1 Cross-section of the free surface at time t = 3 hours

Infinity norm. — Figure 5.2 Infinity norm of the free surface solution over 5 days

5.2. test_case = ‘bump’: Baroclinic wave propagation

This test case consists of a two-layer system. To test whether our model captures the wave propagation speeds correctly, we consider a small perturbation in the interface between layers of the lake-at-rest case with a flat bottom topography. The layer initial interface positions are $z_0 = 0$ m, $z_1 = -20 + 0.5\left( 1 + \cos\left(\frac{\pi r}{250} \right)\right)$ m and $z_2 = -40$ m. This test can be run with no-slip or free-slip boundary conditions, and more on these boundary conditions are in Sec.ref{sec:boundary_conditions}.

5.3. test_case = ‘double-gyre’ Double-gyre test

This test case consists of an idealized double-gyre benchmark test [3] to validate h-NUMO ability to simulate the mesoscale and submesoscale processes and compare the results with HYCOM. The domain is a closed rectangular ocean basin with a flat bottom. The forcing is spatially varying wind stress with intense western boundary currents, which, together with Coriolis force, results in a counter-clockwise circulation in the northern part and a clockwise circulation in the southern part of the domain.

The horizontal extend is $D = 2000\ km$ in both the zonal and the meridional direction. The depth of the basin is $10\ km$ consisting of two layers, with the upper and lower layers initially having $:math:1.5km and $8.5\ km$ depths respectively. The densities in the layers are $\rho_1 = 1027.01037\ \ kg/m^3$ , $\rho_2 = 1027.22136\ \ kg/m^3$. The Coriolis force is prescribed using a beta-plane approximation centered at $45^\circ$ N, with a parameter $f = f_0 + \beta(y-D/2)$, where $f_0 = 9.3 \times 10^{-4} \ s^{-1}$ and $\beta = 2 \times 10^{-11} \ m/s$. We consider two different values of the horizontal viscosity $\nu = 50 \ m^2/s$ and $\nu = 500 \ m^2/s$, the dimensionless bottom drag coefficient in the linear bottom stress is $c_d = 10^{-7} \ s^{-1}$, and we assume no shear stress between layers. The system is forced by a purely zonal wind stress $\tau = (\tau_x,\ \tau_y)$, where $\tau_x = -\tau_0\cos(2\pi y/D)$, $\tau_y = 0\ N/m^2$, and $\tau_0 = 0.1 \ N/m^2$. We considered two different boundary conditions: free-slip and no-slip. Each model year consists of 360 days, divided into 12 months, with 30 days per month.