35.1.8.5 A note about spherical coordinates and extra metric terms

In the formulation of Redi diffusion and GM90 skew-diffusion, there is no need to worry about spherical versus Cartesian coordinates. The Cartesian form for the expressions transform trivially to spherical. RM98, however, prescribe a Laplacian acting on the slope vector. On the sphere, the unit vectors $\hat{\lambda}, \hat{\phi}$ are spatially dependent and so the Laplacian will pick up extra terms^35.1. These metric terms are related, though not identical, to the metric terms arising in the dissipation of momentum (a vector) as described in Section 9.8. For the purpose of completeness, it is worth presenting these metric terms, and then discussing why it may make sense to ignore them.

The two-dimensional slope vector can be written in the form

$${\bf S} = S_{\lambda} \, \hat{\lambda} + S_{\phi} \, \hat{\phi},$$ (35.48)

where the components to the slope are given by

$$S_{\lambda} - \frac{1}{a \, \cos\phi} \left( \frac{\rho_{\lambda}}{\rho_{z}} \right)$$ = (35.49)

$$S_{\phi} - \frac{1}{a} \left( \frac{\rho_{\phi}}{\rho_{z}} \right).$$ = (35.50)

The following expression for the horizontal Laplacian acting on a spherical vector can be obtained from Appendix 2 in Batchelor (1967)

 

$$\nabla_{h}^{2} {\bf S} \hat{\lambda} \left[ \left( \nabla_{h}^{2} - \frac{1}{a^{2} \, \c... ...^{2} \, \cos^{2}\phi}\right) \frac{\partial S_{\phi}}{\partial \lambda} \right]$$ =

$$\hat{\phi} \left[ \left( \nabla_{h}^{2} - \frac{1}{a^{2} \, \cos^... ... \, \cos^{2}\phi}\right) \frac{\partial S_{\lambda}}{\partial \lambda} \right],$$ + (35.51)

where

$$\nabla_{h}^{2} \alpha = \left(\frac{1}{a^2 \, \cos\phi} \right) \... ...}{\cos\phi} \, \alpha_{\lambda\lambda} + (\cos\phi \, \alpha_\phi)_\phi \right)$$ (35.52)

is the horizontal Laplacian acting on a scalar which lives on the sphere. The terms appearing in equation (34.51) in addition to $\nabla_{h}^{2} \, S_{\lambda}$ and $\nabla_{h}^{2} \, S_{\phi}$ constitute the ``metric terms.'' To see what the metric terms do, it is useful to write the tracer flux with the GM90 scheme included as well

$${\bf F}_{h} T_{z} \left( \kappa + \frac{B}{a^{2} \, \cos^{2}\phi} - B\nabla_{... ..., \cos^{2}\phi} \right) \hat{z} \partial_{\lambda} \, \wedge \, \right) {\bf S}$$ = (35.53)

$$-\nabla_{h} T \cdot \left( \kappa + \frac{B}{a^{2} \, \cos^{2}\ph... ... \cos^{2}\phi} \right) \hat{z} \partial_{\lambda} \, \wedge \, \right) {\bf S}.$$ F^z = (35.54)

In general, the metric terms are smaller than the Laplacian in those cases when the power is concentrated at the grid scale. This is the situation for which the RM98 biharmonic operator is designed. One therefore finds little motivation to include the metric terms. Even so, it is useful to look a bit more closely at how the metric terms contribute to the properties of the operator.

The first metric term, which is proportional to the slope, acts in a manner just like the $\kappa$ term from GM90. As such, this metric term provides a sign definite sink of potential energy. To gauge the strength of this sink, consider a very high latitude point $\phi = 89^{\circ}$ and a relatively large diffusivity B = 10²⁰ cm⁴/sec. In this case,

$$\frac{B}{a^{2} \, \cos^{2}\phi} = 8 \times 10^{5} cm^{2}/sec.$$ (35.55)

For the more reasonable $\phi = 45^{\circ}$ and B=10¹⁹ cm⁴/sec,

$$\frac{B}{a^{2} \, \cos^{2}\phi} = 50 cm^{2}/sec.$$ (35.56)

Both of these values should be compared to the usual $\kappa \approx 10^{7} cm^{2}/sec$ GM90 diffusivity. As such, the sink is quite small.

The second metric term, proportional to the zonal derivative of the slope, adds a term to the vertical density flux of the form

$$- \left(\frac{2 \, B \, \sin\phi}{a^{2} \, \cos^{2}\phi} \right) \nabla_{h} \rho \cdot (\hat{z} \wedge \partial_{\lambda} {\bf S}) \left(\frac{2 \, B \, \sin\phi}{a^{2} \, \cos^{2}\phi} \right) \rho_{z} \, {\bf S} \cdot (\hat{z} \wedge \partial_{\lambda} {\bf S})$$ =

$$\left(\frac{2 \, B \, \sin\phi}{a^{2} \, \cos^{2}\phi} \right) \rho_{z} \, \hat{z} \cdot (\partial_{\lambda} {\bf S} \wedge {\bf S}).$$ = (35.57)

This term has no definite sign, and so its effects on potential energy cannot be established in general. As with the constant slope metric term, this term is largest at the high latitudes. To consider its strength, let the slopes have a scale $S_{\lambda} \approx S_{\phi} \approx S$, where $\vert S\vert \le 1/100$. Also, let the contributions to the Laplacian due to zonal variations be about the same as the meridional variations: $\partial_{\lambda \lambda}S \approx \cos\phi \, \partial_{\phi}(\cos\phi \, \partial_{\phi}S)$. As such, the second metric term is large whenever

$$\left(\frac{\partial^{2} S}{\partial \lambda^{2}} \right) \times \left(2 \, \sin\phi \, \frac{\partial S}{\partial \lambda} \right)^{-1}$$ (35.58)

is small. Let $\partial_{\lambda} S \approx S/\Delta \lambda$, and $\partial^{2}_{\lambda} S \approx S/(\Delta \lambda)^{2}$, where $\Delta \lambda$ is the zonal grid spacing in radians. The question then becomes whether $2 \, \sin\phi \, \Delta \lambda$ is larger than one. If it is, then the second metric term is non-negligible. For a $3^{\circ} = 0.0524$ radian zonal resolution and $\phi \approx 90^{\circ}$, $2 \, \sin\phi \, \Delta \lambda \approx 1/10.$ For mid-latitudes and $\Delta \lambda = 1/4^{\circ}$, $2 \, \sin\phi \, \Delta \lambda \approx 1/162.$ Both of these results suggests that the second metric term is no more than 10% as large as the Laplacian term, for the cases when the scaling is relavent. Of course, when there is zero curvature in the slope field, then Laplacian vanishes when the metric term may not. But again, such a slope field is perhaps not the kind for which the RM98 scheme is designed to attack.

In summary, the added metric terms do the following:

• The first metric term is proportional to the slope, and it acts in a manner just like GM90. The latitudinally dependent diffusivity setting the scale of this term increases with increasing latitude, with largest values no larger than 10⁴ - 10⁵ cm²/sec next to the pole. This term is trivial to incorporate into the existing numerical framework from Griff ies et al. (1998).
• The second term is proportional to the zonal derivative of the slope. This term adds a sign indefinite contribution to the potential energy. It is at most 1/10 the size of the Laplacian term for fields in which the slope curvature is nonzero, with 1/100 being the a more realistic size. This term cannot be discretized using the Griff ies et al. (1998) numerical framework.

In conclusion, MOM currently ignores the metric terms, and this was also the approach taken by Roberts and Marshall (1998) (M. Roberts, personal communication, 1998). As seen from the above arguments, ignoring these terms is consistent with a desire to act on noise at the grid scale, and to leave the larger scales untouched. Recall that neglecting the analogous (not identical) metric terms in the second order momentum friction operator leads to unphysical consequences, as discussed in Section 9.3.9 and Wajsowicz (1993). Nevertheless, the angular momentum arguments which guide the form of second order momentum friction are absent for the tracer mixing operators. In the absence of other arguments, there appears little to motivate retaining the metric terms for RM98.