39.9.2 Zonally integrated circulation and its streamfunction

Now consider the zonally integrated water transport

 

$${\mathcal V}(\phi,s,t) a \, \cos\phi \, \int d\lambda \; z_{s} \, v$$ = (39.128)

$${\mathcal W}(\phi,s,t) a \, \cos\phi \, \int d\lambda \; (z_{t} + z_{s} \, \dot{s}) = a \, \cos\phi \, \int d\lambda \; (w - \vec{u}_{h} \cdot \nabla_{s}z).$$ = (39.129)

In these expressions, the zonal integration is taken with both the latitude $\phi$ and generalized vertical coordinate s held fixed. The component ${\mathcal V}$ represents the zonally integrated water transport moving in the meridional direction. The weighting with the specific thickness can best be understood in a finite difference context, for which a weighting $z_{s} \, \delta s$ implies that ${\mathcal V}$ is the meridional transport (in units of length²/time) between the surfaces s and $s+\delta s$. In the continuum, the units of ${\mathcal V}$ are $length^{2}/time \times length/s$. The component ${\mathcal W}$ represents the zonally integrated time tendency for the changes in height of a constant ssurface, plus the specific thickness weighted velocity of water moving across a constant s surface. The units for ${\mathcal W}$ are length²/time.

In the following, the zonal integration will be taken either within a particular basin enclosed by meridional boundaries on which the zonal velocity is assumed to vanish, or for a zonally symmetric domain such as the World Ocean for which the zonal velocity is periodic. As such, the two components of the zonally integrated transport are related by

$$\partial_{s} \, {\mathcal W} a \, \cos\phi \, \int d\lambda \; \partial_{s} \, (z_{t} + z_{s} \, \dot{s})$$ =

$$-a \, \cos\phi \, \int d\lambda \; \nabla_{s} \cdot (\vec{u}_{h} \, z_{s} )$$ =

$$-a \, \cos\phi \, \int d\lambda \; \frac{1}{a \, \cos\phi} (z_{s} \, v \, \cos\phi)_{\phi}$$ =

$$-a^{-1} \, \partial_{\phi} \, {\mathcal V},$$ = (39.130)

where the thickness equation (39.129) has been used. The assumptions regarding the zonal velocity allow for the elimination of the $\int d\lambda \, (z_{s} \, u)_{\lambda}$ term. Hence, the two dimensional zonally integrated circulation is divergent-free.

The assumption of zero zonal velocity on the meridional boundaries prompts some further comments. In closed basins, a no-normal flow boundary condition does not imply a zero zonal velocity next to the meridional water-land boundaries. In MOM, there is an unambiguous distinction made between bottom and side boundaries, which is allowed by the use of step-topography. Bottom boundaries generally employ the no-normal flow condition relevant for an inviscid fluid, with an option available to add a bottom boundary layer. Side boundaries, however, always use the no-slip condition which means that all components of the velocity vanish next to side boundaries. Therefore, the elimination of the $\int d\lambda \, (z_{s} \, u)_{\lambda}$ term is always valid in MOM.

Since the two components of the zonally integrated water transport satisfy the zero divergence condition

 

$$\partial_{s} \, {\mathcal W} + a^{-1} \, \partial_{\phi} \, {\mathcal V} = 0,$$ (39.131)

the zonally integrated transport can be determined by an overturning streamfunction

  

$$-\partial_{s} \psi {\mathcal V}$$ = (39.132)

$$a^{-1} \, \partial_{\phi} \psi {\mathcal W}.$$ = (39.133)

It is important to remember that the latitudinal derivative $\partial_{\phi}$ is taken at fixed generalized vertical coordinate s, and for the generalized vertical derivative $\partial_{s}$, the latitude is held fixed. In words, the vertical convergence of the streamfunction at constant latitude gives the zonally integrated transport across that latitude. Likewise, the latitudinal divergence on constant s surfaces yields the zonally integrated transport across the s surface plus the zonally integrated time tendency for the height of the surface. It is useful to state these results another way, in terms of finite differences. The difference between the streamfunction at two points at the same latitude, yet on different s surfaces, represents the total meridional transport of fluid (in units of volume/time) crossing this latitude in between the two s surfaces. Likewise, the difference between the value of the streamfunction at two points at the same s surface, but at different latitudes, equals the total transport of fluid (in units of volume/time) crossing this particular s surface, plus a contribution due to the time tendency for the height of the s surface. An alternative interpretation of the latitudinal divergence on constant s surfaces follows from the equivalence $z_{t} + z_{s} \, \dot{s} = w - \vec{u}_{h} \cdot \nabla_{s}z$.

Integration of the equation (39.135) at a fixed latitude yields

 

$$\psi(\phi, s,t) = \psi(\phi,s_{o},t) - \int_{s_{o}}^{s} ds' \; {\mathcal V}(\phi,s',t),$$ (39.134)

where s_o is a reference value for the generalized vertical coordinate. Likewise, a latitudinal integral of equation (39.136) along a fixed surface s yields

 

$$\psi(\phi, s,t) = \psi(\phi_{o},s,t) + a \, \int_{\phi_{o}}^{\phi} d\phi' \; {\mathcal W}(\phi',s,t),$$ (39.135)

where $\phi_{o}$ is a reference latitude. It is now possible to develop two equivalent expressions for the overturning streamfunction. The first is found through substituting the expression (39.137) into (39.138)

 

$$\psi(\phi, s,t) = \psi(\phi_{o},s_{o},t) - \int_{s_{o}}^{s} ds' \... ...\phi_{o},s',t) + a \, \int_{\phi_{o}}^{\phi} d\phi' \; {\mathcal W}(\phi',s,t).$$ (39.136)

The second expression is determined through substituting equation (39.138) into (39.137)

 

$$\psi(\phi, s,t) = \psi(\phi_{o},s_{o},t) - \int_{s_{o}}^{s} ds' \... ...\phi,s',t) + a \, \int_{\phi_{o}}^{\phi} d\phi' \; {\mathcal W}(\phi',s_{o},t).$$ (39.137)

Note that the streamfunction has dimensions volume/time, and so it represents the volume transport of water in the $(\phi,s)$ plane.