First excited mode at the Ice-Ocean Boundary: Impact of Initial Conditions and Lewis Numbers

Date: December 08, 2024

In our previous 3D simulations with constant initial values for temperature and salinity, we observed that the first excited mode led to an unstable density profile, with heavier layers overlying lighter ones, for all Lewis numbers greater than 1. Here, we explore this instability by modifying the initial conditions for temperature and salinity, ranging from a uniform profile in the vertical direction to a stratified one. Additionally, we examine the effects of varying Lewis numbers.

We focus on the region very close to the boundary; therefore, our simulation domain is $L_x = L_y = 1 , \text{meter}$ and $L_z = 0.5 , \text{meters}$. The spatial resolution is set to 128 in each of the three directions. The spatial grid is uniform in $x$ and $y$, while refined in $z$ near the boundary. The time step is adaptive and determined by the minimum value of $\Delta z$. The simulations run for a total of 20 seconds, with output generated every 1 second.

The dynamic viscosity ($\nu$) and thermal diffusivity ($\kappa_T$) are both fixed at $1.8 \times 10^{-6}$. The salinity diffusivity ($\kappa_S$) is varied between $1.8 \times 10^{-6}$ and $1.8 \times 10^{-8}$ to achieve different values of the Lewis number. The density profile is calculated based on the linear approximation: $\rho = \rho_0 \left[ 1 - \beta_T (T - T_0) + \beta_S (S - S_0) \right]$, where $\rho_0 = 1000 \, \text{kg/m}^3$, $T_0 = 277.15 \, \text{K}$, and $S_0 = 0 \, \text{g/kg}$. The thermal expansion coefficient is $\beta_T = 3.87 \times 10^{-5} \, \text{K}^{-1}$, and the haline contraction coefficient is $\beta_S = 7.86 \times 10^{-4} \, \text{(g/kg)}^{-1}$. Finally, the melting temperature ($T_{\text{melt}}$) is determined by the equation: $ T_{\text{melt}} = \lambda_1 S + \lambda_2$ where $\lambda_1 = -5.73 \times 10^{-2} \, \text{K/(g/kg)}$ and $\lambda_2 = 8.32 \times 10^{-2} \, \text{K}$.

The behavior of the fastest-growing modes in our system follows key principles established in the literature. According to Radko (2013), the fastest-growing modes are always vertically invariant. Furthermore, the stability and instability of the system depend on specific criteria:

Fingering Systems:
- The system is double-diffusively unstable if $1 < R_0 < \tau^{-1}$.
- The system becomes linearly stable if $R_0 > \tau^{-1}$.
Overturning Double-Diffusive Convection (ODDC) Systems:
- The system is double-diffusively unstable if $1 < R_0^{-1} < \frac{Pr + 1}{Pr + \tau}$.
- The system becomes linearly stable if $R_0^{-1} > \frac{Pr + 1}{Pr + \tau}$.

These conditions reveal one of the most remarkable properties of the low $Pr$ (Prandtl number) limit: the dramatic increase in the range of inverse density ratios that are linearly unstable to ODDC. For water, where $Pr \sim 7$ and $\tau \sim 0.01$, the maximum value of $R_0^{-1}$ is approximately $1.1$. However, as $Pr \sim \tau \ll 1$, this range increases to the order of $O(Pr^{-1})$.