### Global Warming - GCM Model-E - Issues

While working with Model-E, I noticed a few anomalies that I thought were worth recording. Think of these as nits - if I find something big, it will get its own page.

Leap Years | Tilt of the Earth

### Leap Years

Model-E ignores leap years and uses years with only 365 days.

I ran a small simulation and determined that this produces no significant error. Basically, loosing one cold day in the northern hemisphere is balanced by loosing a hot day in the southern.

### Computing the Tilt of the Earth

When running a climate model, it is important to know how much solar radiation (heat) is provided to each location on the planet. These values vary depending on the day of the year because that depends on the orientation of the planet's axis. In the 1600's, Johannes Kepler developed the equations that determine the angle of the axis with respect to the Sun. These equations are fairly complex to derive because the Earth's orbit is actually an ellipse, not a circle.

The model uses Kepler's equation correctly ... except to find the angle of the Perihelion (the closest distance between the Earth and Sun). For that, they used the wrong equation. As a result, their calculations are off by one day.

Method MA Angle of Perihelion Vernal
Equinox
Summer
Solstice
Autumnal
Equinox
Winter
Solstice
Model-E 0.030551958 1.75049825 Mar 19/20 Jun 20/21 Sep 21/22 Dec 20/21
Correct 0.046738227 2.67790317 Mar 20/21 Jun 21 Sep 22/23 Dec 21/22
Note - These dates were computed using only 365 day years.

It isn't so much that the model gives the wrong results (a one day error should be insignificant) as it is that they used the wrong equation. If they can't get something this simple correct, then what about the rest of the model?

### Technical Details

Kepler's method involves dividing a circle into equal parts. The angular location of the planet on this circle is the Mean Anomaly (MA). Using Kepler's equation, the Eccentric Anomaly (EA) is determined (using guess and try - there is no deterministic solution). Another equation computes the True Anomaly (TA) which is then used to determine how much the Earth's axis is pointing toward the Sun.
 TA True anomaly Angle seen from the Sun with respect to the Perihelion EA Eccentric anomaly Projection of the TA angle onto a circle MA Mean anomaly Angle on a circle - 2Pi * ( day/365 ) e Eccentricity Relationship between a circle and an ellipse
 ```MA = [ (day of year/days per year) * 360 degrees ] - MA of the Perihelion MA = EA - e * Sin(EA) Kepler's equation Tan(TA/2) = Sqrt[(1+e)/(1-e)] * Tan (EA/2) ```
Given the angle of the Vernal Equinox, as seen from the Sun with respect to the Perihelion (the closest distance between the Earth and Sun), it is possible to compute the Mean Anomaly of the Perihelion which is then used to adjust the Mean Anomaly of the other days of the year.

The problem is that the model used the True Anomaly (TA) of the Vernal Equinox directly in Kepler's equation without first converting the TA to the EA (using the third equation above). Logically, this is a pretty big error. Practically, it produces only a one day error ... for the Earth.

It should be noted that these subroutines could be used in other programs for other purposes. When applied to other planets, or perhaps the Moon, this error could be very significant.

Reference with details and proofs related to these equations.

Author: Robert Clemenzi