6.2 Streamfunction and volume transport

The barotropic streamfunction is specified only to within a constant. As such, only differences are physically relevant. In particular, consider the vertically integrated advective transport between two points

$$T_{ab} = \int^{b}_{a} dl \; \hat{n} \cdot \int_{-H}^{0} dz \; {\bf u}_{h},$$ (6.7)

where dl is the line element along any path connecting the points a and b, and $\hat{n}$ is a unit vector pointing perpendicular to the path in a rightward direction when facing the direction of integration. As written, T_ab has units of volume/time, and so it represents a volume transport. The definition of the barotropic streamfunction and Stokes' Theorem renders

$$\int^{b}_{a} dl \; \hat{n} \cdot {\bf U}$$ T_ab =

$$\int^{b}_{a} dl \; \hat{n} \cdot \hat{z} \wedge \nabla_{h} \psi$$ =

$$-\int^{b}_{a} dl \; \nabla_{h} \psi \cdot (\hat{z} \wedge \hat{n})$$ =

$$-\int^{b}_{a} dl \; \nabla_{h} \psi \cdot \hat{t}$$ =

$$\psi_{a} - \psi_{b},$$ = (6.8)

where

$$\hat{t} = \hat{z} \wedge \hat{n}$$ (6.9)

is a unit vector tangent to the integration path, pointing in the direction of integration from point a to point b. Therefore, the difference between the barotropic streamfunction at two points represents the vertically integrated volume transport between the two points. It is for this reason that the barotropic streamfunction is sometimes called the volume transport streamfunction. Note that Bryan (1969) defined the barotropic streamfunction with an extra factor of the Boussinesq density $\rho_{o}$, such than $\psi_{bryan} = \rho_{o} \; \psi_{mom}$. Hence, the barotropic streamfunction of Bryan has the dimensions of mass/time rather than volume per time, and so it represents a mass transport streamfunction. Since MOM assumes a Boussinesq fluid, the difference is trivial.