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## Results

### 1. Gaussian hill with high deformation

This test case consists of the linear advection of a Gaussian hill function $\omega_{0}(x,y)$
$\omega_{0}(x,y) = \frac{1}{2\pi\sigma^{2}}e^{-\frac{(x-x_{0})^{2}}{2\sigma^{2}}}e^{-\frac{(y-y_{0})^{2}}{2\sigma^{2}}},$
where $\sigma = 0.2$, $x_{0}=0.01 – \pi$, $y_{0} = 2.0$. The advecting vector field $\vec{u}$ is given by
$\vec{u} = \nabla\times\psi,$
with
$\psi(x,y) = -\cos(x-\pi) + \frac{1}{2}y^{2} + 1.$
The computational domain is $\Omega = [-2\pi,2\pi)\times[-3,3)$, with periodic boundaries.

The video below shows the evolution from $t=0$ to $t=4$ and then the advection velocity field is reversed from $t=4$ to $t=8$. With this test we can show the reversibility of the solver, since we can retrieve the initial Gaussian hill. For this simulation 40000 triangular elements of polynomial degree $p=4$ have been used. The time step is $\Delta t = 10^{-2}$.

Conservation of the total enstrophy $\mathcal{E}(t):=\int_{\Omega}\omega(x,y,t)^{2}$ and total vorticity $\mathcal{W}(t):=\int_{Omega}\omega(x,y,t)$ are shown in the figures below. Enstrophy (left) and total vorticity (right) error for the passive advection of Gaussian hill with 40000 triangular finite elements, basis functions of polynomial degree $p = 4$, and time steps $\Delta t = 0.01$.