\[fig:validation\] and table \[tab:validation\] we compare these theoretical results with thresholds obtained by interpolating growth rates from numerical simulations, as shown in figure \[fig:valid\_slopes\_rates\]. The parameters are as previously stated for capillary waves in section \[sec:phys\_par\], except that we increase the viscosity to $\nu_d=10^{-4} m^2s^{-1}$. For $\ell=6$, the error in the threshold is about 1% for a resolution of $128^3$ and only about 0.1% for $256^3$.

[ccccc]{}

  -----------------------
   Spherical mode $\ell$
              
  -----------------------

  : Comparison between theoretical and computed thresholds $a_c$ for various $\ell$ values and numerical resolutions.[]{data-label="tab:validation"}

&

  ------------
   Resolution
        
  ------------

  : Comparison between theoretical and computed thresholds $a_c$ for various $\ell$ values and numerical resolutions.[]{data-label="tab:validation"}

&

  -------------
   Theoretical
   $m\,s^{-2}$
  -------------

  : Comparison between theoretical and computed thresholds $a_c$ for various $\ell$ values and numerical resolutions.[]{data-label="tab:validation"}

&

  -------------
    Numerical
   $m\,s^{-2}$
  -------------

  : Comparison between theoretical and computed thresholds $a_c$ for various $\ell$ values and numerical resolutions.[]{data-label="tab:validation"}

& $\dfrac{|\Delta a_c|}{a_c}$(%)\
6 & $128^3$ & 0.2604  & 0.2640  & 1.38\
5 & $128^3$ & 0.1857  & 0.1886  & 1.52\
4 & $128^3$ & 0.09415 & 0.0919  & 2.39\
6 & $256^3$ & 0.2604  & 0.2607  & 0.08\
5 & $256^3$ & 0.1857  & 0.1864  & 0.34\
4 & $256^3$ & 0.09415 & 