38.6 A note about areas on the sphere

The area of a grid box on the sphere is given by

$$A = a^{2} \, \int^{\lambda_{2}}_{\lambda_{1}} \; d\lambda \, \int^{\phi_{1}}_{\phi_{2}} \; \cos\phi \, d\phi.$$ (38.2)

In a grid-point model with uniform resolution, this area is often approximated with

$$A_{approx} = a^{2} \, \Delta \lambda \, \Delta\phi \, \cos\overline{\phi},$$ (38.3)

where

$$\overline{\phi} = \frac{\phi_{1} + \phi_{2} }{2}$$ (38.4)

represents the midpoint latitude. The exact area of the finite sized grid box is given by

$$a^{2} \, \Delta \lambda \, \int^{\phi_{1}}_{\phi_{2}} \; d (\sin\phi)$$ A_exact =

$$a^{2} \, \Delta \lambda \, \Delta (\sin\phi).$$ = (38.5)

It is useful to determine the error made in the approximate expression through the use of some trigonometry:

$$\Delta (\sin\phi) \sin\phi_{2} - \sin\phi_{1}$$ =

$$2 \, \cos \overline{\phi} \, \sin (\Delta \phi/2)$$ =

$$\approx \Delta \phi \, \cos \overline{\phi} \, ( 1 - (\Delta \phi)^{2}/24 )$$ (38.6)

Hence,

$$a^{2} \, \Delta \lambda \, \Delta (\sin\phi)$$ A_exact =

$$\approx A_{approx} \, (1 - (\Delta \phi)^{2}/24).$$ (38.7)

As a result, the area of the grid box is overestimated by the amount $(\Delta \phi)^{2}/24$ when using the approximate expression. For a model with $\Delta \phi = 4^{\circ} = 0.0698 rad$, the leading error is

$$(\Delta \phi)^{2}/24 = 0.0002 = 0.02 \%$$ (38.8)

It is important to note that the above error estimate is a lower bound for the case of a grid where the cosine factor does not represent the cosine of the latitude at the center of the grid box (as in a Gaussian grid). Atmospheric models at GFDL use the exact areas, not the approximate values. Hence, care must be taken to use the same area weights between an atmospheric model and MOM. Unless the weights are identical, flux conservation is not possible. Currently, the approximate area method is used for computing diagnostics in MOM. It is clear that for the purpose of ocean-only diagnostics, the differences are quite minor and can be safely ignored. When time permits, these approximate area calculations will be replaced by the correct ones.