7.6.2 An approximate streamfunction

In general, to evaluate the vertically integrated transport passing between two points, a direct evaluation of the integral (see Section 6.2)

$$T_{ab} = \int^{b}_{a} dl \; \hat{n} \cdot {\bf U},$$ (7.66)

can be given. Although accurate and complete, this integral does not readily provide a map of transport, and so it looses much of the appeal associated with the barotropic streamfunction used with a rigid lid. Instead, for many practical situations, maps of the function

$$\Psi(\lambda, \phi) = -a \, \int^{\phi}_{\phi_{o}} \, d\phi' \, U(\lambda,\phi')$$ (7.67)

may prove useful, where the lower limit $\phi_{o}$ is taken at the southern boundary of the domain (either a wall or $-\pi/2$). The meridional derivative of $\Psi$ yields the exact zonal transport

$$-\frac{1}{a} \Psi_{,\phi} = U.$$ (7.68)

The longitudinal derivative, however, is given by

$$\frac{1}{a \, \cos\phi} \, \Psi_{,\lambda} = V - \frac{a}{\c... ...hi_{o}} \, d\phi' \, \cos\phi' \, \nabla_{h} \cdot {\bf U},$$ (7.69)

where $\cos\phi_{o} \, V(\lambda,\phi_{o}) = 0$ was used. The free surface height equation (Section 7.2.3)

$$\nabla_{h} \cdot {\bf U} = -\eta_{t} + q_{w}$$ (7.70)

indicates that $\nabla_{h} \cdot {\bf U}$ vanishes only when there is zero time tendency of the free surface height and zero fresh water flux through the surface. Hence, the longitudinal derivative leads to the meridional transport plus an error term, where the error term vanishes in the case of a steady state ( $\eta_{t} = 0$) and a zero fresh water flux. It is useful to see how large the error term might be. For this purpose, zonally integrate $\Psi_{,\lambda}$ to find

$$\Psi(\lambda,\phi) - \Psi(\lambda_{o},\phi) a \, \cos\phi \, \int^{\lambda}_{\lambda_{o}} d\lambda' \, V(\lambda',\phi)$$ =

$$a^{2} \, \int^{\lambda}_{\lambda_{o}} d\lambda' \, \int^{\phi}_{\phi_{o}} d\phi' \, \cos\phi' \, (-\eta_{t} + q_{w}).$$ - (7.71)

For example, with $-\eta_{t} + q_{w} = 1m/year$ applied over a $1000km \times 1000km$ area, the error term contributes much less than a Sv to the streamfunction. Cases where the differences are larger certainly can be constructed. But for many diagnostic purposes, the differences are negligible. By construction, $\Psi$ reduces to the barotropic streamfunction in the case of a rigid lid model where $\nabla_{h} \cdot {\bf U} = 0$. However, this is not a unique choice and alternatives do exist. For example,

$$\Psi^{*}(\lambda,\phi) = \Psi(\lambda_{o},\phi) + a \, \cos\phi \, \int^{\lambda}_{\lambda_{o}} \, d\lambda' \, V(\lambda',\phi),$$ (7.72)

gives

$$U(\lambda,\phi) -\frac{1}{a} \, \Psi^{*}_{, \phi} + a \, \cos\phi \, \int^{\lambda}_{\lambda_{o}} \, d\lambda' \nabla_{h} \cdot {\bf U}$$ = (7.73)

$$\frac{1}{a \, \cos\phi} \, \Psi^{*}_{, \lambda}.$$ V = (7.74)

$\Psi^{*}$ has the advantage that zonal derivatives give the exact meridional transport. In the end, it might be useful to plot $\Psi$ and $\Psi^{*}$ and compare. These two streamfunctions are indeed plotted in MOM's snapshots when the free surface is enabled. In snapshots, $\Psi$ is called psiU, and $\Psi^{*}$ is called psiV. The reference points are $\phi_{o}$ is the southern-most latitude and $\lambda_{o}$ is the western-most longitude. Additionally, as each streamfunction is defined only up to an arbitrary constant, it is useful to specify this constant in a manner to correspond to that resulting from the rigid lid approximation. The option  explicit_psi_normalize normalizes each streamfunction by the value at $\lambda = 300^{\circ}$ and $\phi=-20^{\circ}$, which corresponds to a point over South America. This convention corresponds to taking the Americas as the zeroth island in the rigid lid method.

A final example distributes the $(-\eta_{t} + q_{w})$ piece evenly:

$$\Psi^{**} = -\frac{a}{2} \, \int^{\phi}_{\phi_{o}} \, d\phi'... ...int^{\lambda}_{\lambda_{o}} \, d\lambda' \, V(\lambda',\phi)$$ (7.75)

yields

$$-\frac{1}{a} \, \Psi^{**}_{, \phi} + \frac{a \, \cos\phi}{2} \, \int^{\lambda}_{\lambda_{o}} d\lambda' \, (-\eta_{t} + q_{w})$$ U = (7.76)

$$\frac{1}{a \, \cos\phi} \, \Psi^{**}_{, \lambda} + \frac{a}{2 \, \cos\phi} \, \int^{\phi}_{\phi_{o}} d\phi' \, \cos\phi' \, (-\eta_{t} + q_{w}).$$ V = (7.77)

This streamfunction is not computed in MOM.