# Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

## TL;DR

This paper analyzes how thermohaline circulation in a spherical shell transitions to multiple equilibrium states, identifying conditions for different transition types and deriving exact formulas for transition numbers.

## Contribution

It provides the first exact formulas for transition numbers in spherical thermohaline circulation and characterizes the nature of bifurcations depending on system parameters.

## Key findings

- Transition type depends on the sign of the transition number.
- Exact formulas for transition numbers are derived for specific spherical harmonic degrees.
- Numerical results show different transition behaviors based on the Lewis number.

## Abstract

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $l_c$, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $l_c=1$ and $l_c=2$ cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $Le<1$ but only continuous transition for $Le>1$. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2$l_c$ dimensional sphere and consists entirely of degenerate steady state solutions.