Vertical Buoyancy Balance

The verical buoyancy profiles within ocean basins are adjusted via advection & diffusion. If we assume that isopycnals (i.e. surfaces of constant buoyancy) are relatively flat, horizonatal advection has a small contribution to the buoyancy tendencies, and the dominant balance is between vertical advection and diffusion, as first described by Munk (1966).. Following Jansen and Nadeau (2019), we here relax the strict requirement for flat isopycnals by implementing a residual-advection diffusion equation, which is horizontally averaged along isopycnals, such that horizontal buoayncy advection vanishes by construction.

The residual advection-diffusion equation can be derived by starting with the Boussinesq continuity equation in isopycnal space

\[\partial_t\sigma + \nabla_h\cdot\left(\sigma\vec{u}\right)=-\partial_b\left(\sigma\mathcal{B}\right)\]

Here we define \(\mathcal{B}\) as the diabatic buoyancy tendency, including small scale diapycnal mixing processes that cannot be resolved by the model. \(\sigma\equiv\partial_b z\) is the isopycnal “thickness”—a measure of the depth range covered by a given buoyancy class. We can now transform the continuity equation into depth space, by integrating from the minimum modeled buoyancy (\(b_{min}\)) to buoyancy \(b\)

\[\partial_t z=-\nabla_h\cdot\int_{b_{min}}^b\sigma\vec{u}db^\prime - \sigma\mathcal{B}\]

We next express the diabatic forcing term via the convergence of a vertical buoyancy flux (\(F^b\)):

\[\begin{split}\begin{aligned} \sigma\mathcal{B}&=-\partial_b F^b \\ F^b&= \begin{cases} F_s^b, & b\ge b_s(x,y) \\ -\kappa\partial_z b, & b_{bot}(x,y)\lt b\lt b_s(x,y) \\ 0, & b\le b_{bot}(x,y) \end{cases} \end{aligned}\end{split}\]

where \(F^s\) is the surface buoyancy flux.

We can now integrate \(\partial_t z\) zonally and meridionally across the basin, assuming zero flux across the zonal boundaries (consistent with a basin bounded by continental landmasses):

\[\partial_t\int_{x_1}^{x_2}\!\int_{y_1}^{y_2}zdxdy = \int_{x_1}^{x_2}\!\int_{b_{min}}^b \sigma vdb^\prime \left.dx\right|_{y_2}^{y_1} + \partial_b\int_{y_1}^{y_2}\!\int_{x_1}^{x_2}F^bdxdy\]

Defining

\[\begin{split}\begin{aligned} \Psi\left(y, b\right) &\equiv -\int_{x_1}^{x_2}\!\int_{b_{min}}^b \sigma vdb^\prime dx \\ \langle X \rangle &\equiv \frac{1}{A_o}\int_{x_1}^{x_2}\!\int_{y_1}^{y_2}Xdxdy \\ A_o &\equiv \int_{x_1}^{x_2}\!\int_{y_1}^{y_2}1dxdy \\ \end{aligned}\end{split}\]

where \(\Psi\left(y, b\right)\) is the meridional overturning streamfunction in bouyancy space, and \(\langle X \rangle\) is the horizontal spatial mean of an arbitrary quantity \(X\), we get

\[\begin{aligned} \partial_t\langle z \rangle &= \frac{1}{A_o}\left[\Psi\left(y_2, b\right) - \Psi\left(y_1, b\right)\right] + \partial_b\langle F^b \rangle \end{aligned}\]

Dividing by \(\langle \sigma \rangle\), we can derive an equation for \(b(\langle z\rangle)\):

\[\begin{split}\begin{aligned} \left.-\partial_t b \right|_{\langle z\rangle} &= -\frac{1}{A_o}\left[\Psi\left(b\left(\langle z \rangle\right), y_2\right) - \Psi\left(b, y_1\right)\right]\partial_{\langle z\rangle}b-\partial_{\langle z\rangle }\langle F^b \rangle \\ &\equiv -w^\dagger\partial_{\langle z\rangle}b-\partial_{\langle z\rangle }\langle F^b\rangle \end{aligned}\end{split}\]

where we’ve defined \(w^\dagger\) as the residual upwelling due to convergence of volume flux along isopycnals below \(b\left(\langle z\rangle\right)\).

Dropping the spatial averages in our notation, defining the surface forcing \(\mathcal{B}_s\equiv -\partial_b F^s\) and effective vertical diffusivity \(\kappa_{eff}\equiv\left(\frac{A_1}{A_o}\right)^2\kappa\) where \(A_1\) is the area of the non-incropping part of the isopycnal surace, the equation can be reduced to (see Jansen and Nadeau (2019) for details):

\[\begin{aligned} \partial_t b \approx -w^\dagger\partial_z b+\partial_z(\kappa_{e\!f\!f}\partial_z b)+\mathcal{B}_s \end{aligned}\]

This final equation, solved by the column model module, is equivalent to the 1-dimensional vertical advection-diffusion equation, except for some re-interpretation of the variables and the introduction of an effective diapycnal diffusivity that accounts for the reduced area of isopycnals incropping into the bottom.