2D Atlantic Meridional Ocean Circulation

This describes a rectangular 2D model of the Atlantic meridional ocean circulation (AMOC).

Governing Equations

In a domain of dimensions $L \times D$, the governing equations are given by

\[\begin{split}\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= -\frac{1}{\rho_0} \nabla p + \nu \nabla^2 \mathbf{u} - \frac{\rho g}{\rho_0} e_z\\ \nabla \cdot \mathbf{u} &= 0\\ \frac{\partial T}{\partial t} + (\mathbf{u} \cdot \nabla) T &= \kappa_T \nabla^2 T\\ \frac{\partial S}{\partial t} + (\mathbf{u} \cdot \nabla) S &= \kappa_S \nabla^2 S\end{split}\]

with boundary conditions

\[\begin{split}x &= 0, L &:~& u = w = \frac{\partial T}{\partial x} = \frac{\partial S}{\partial x} = 0\\ z &= 0 &:~& u = w = \frac{\partial T}{\partial z} = \frac{\partial S}{\partial z} = 0\\ z &= D &:~& u = w = 0, T = T_S(x), \frac{\partial S}{\partial z} = \sigma Q_S(x)\end{split}\]

where $T_S(x)$ is a prescribed temperature distribution along the surface. The parameter $\sigma$ is the strength of the surface fresh-water flux and $Q_S(x)$ represents its spatial structure. See Symbols for the definition of all quantities used here.

Nondimensional formulation

In TransiFlow, we implement the following nondimensionalized equations

\[\begin{split}\frac{\partial \tilde{\mathbf{u}}}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde \nabla) \tilde{\mathbf{u}} &= -\tilde \nabla \tilde p +\frac{1}{\mathrm{Gr}^{1/2}}\tilde \nabla^2 \tilde{\mathbf{u}} + (\tilde T - \tilde S)e_z\\ \tilde\nabla \cdot \tilde{\mathbf{u}} &= 0\\ \frac{\partial \tilde T}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde T &= \frac{1}{\mathrm{Pr} \mathrm{Gr}^{1/2}} \tilde \nabla^2 \tilde T\\ \frac{\partial \tilde S}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde S &= \frac{1}{\mathrm{Pr} \mathrm{Le} \mathrm{Gr}^{1/2}} \tilde \nabla^2 \tilde S\end{split}\]

with boundary conditions

\[\begin{split}\tilde x &= 0, A_x &:~& \tilde u = \tilde w = \frac{\partial \tilde T}{\partial \tilde x} = \frac{\partial \tilde S}{\partial \tilde x} = 0\\ \tilde z &= 0 &:~& \tilde u = \tilde w = \frac{\partial \tilde T}{\partial \tilde z} = \frac{\partial \tilde S}{\partial \tilde z} = 0\\ \tilde z &= 1 &:~& \tilde u = \tilde w = 0, \tilde T = \tilde T_S(x), \frac{\partial \tilde S}{\partial \tilde z} = \sigma Q_S(x)\end{split}\]

Here $x$ and $z$ are scaled by $D$, and hence $A_x = L / D$. The other quantities are scaled using $u = \hat u \nu / D$, $t = \hat t D^2 / \nu$, $p = \hat p (\mu \nu / D^2) - \rho_0 g z$, $T = \Delta T \hat T$, $S = \Delta S / \lambda \hat S$ with $\lambda = \alpha_S \Delta S / (\alpha_T \Delta T)$ and additionally $\hat u = \tilde u \mathrm{Gr}^{1/2}$, $\hat t = \tilde t \mathrm{Gr}^{-1/2}$, $\hat p = \tilde p \mathrm{Gr}$, $\hat T = \tilde T$, $\hat S = \tilde S$. Moreover, the Prandtl number is given by $\mathrm{Pr} = \nu / \kappa_T$, the Rayleigh number by $\mathrm{Ra} = (\alpha_T g \Delta T D^3) / (\nu \kappa_T)$, the Grashof number by $\mathrm{Gr} = \mathrm{Ra} / \mathrm{Pr}$ and the Lewis number by $\mathrm{Le} = \kappa_T / \kappa_S$. All other quantities are defined in Symbols.

Parameters

These are the relevant parameters in the parameters dictionary for this problem type.

Parameter name	Default value	Notes
'Problem Type'		Set to 'AMOC'
'Rayleigh Number'	1.0	Unused if Gr is defined
'Prandtl Number'	1.0
'Grashof Number'	Ra / Pr	Overrides Ra if defined
'Lewis Number'	1.0
'Asymmetry Parameter'	0.0	Used to switch branches
'Freshwater Flux'	0.0	Magnitude of $Q_S$
'Temperature Forcing'	0.0	Magnitude of $T_S$