G=- Ñ F _e - k ´ Ñ h (1)

where F _e and h are referred to as equivalent geopotential and geo-streamfunction,

respectively. In this paper, (1) is used to compute the HPGF in the MM5 instead of the original HPGF scheme.

In p-coordinates, the HPGF -Ñ F (x, y, p, t) has only the irrotational part, and its rotational part, (corresponding to - k ´ Ñ h ), can only be computed through the lower boundary condition at the earth's surface and expressed implicitly. The rotational part is much smaller than the irrotational part in (1), thus the small difference between two large terms over steep slopes is eliminated automatically.

There are three important advantages of using (1). The first is that the HPGF in s -coordinates is computed accurately. Based on the tests of the computed HPGF over Greenland and across Pacific Northwest of US, the major differences between the original scheme of MM5 and (1) are located over the steep slopes. Colle et al. (1999) verified the MM5 precipitation forecasts across Pacific West and found that the MM5 tends to generate too much precipitation along the steep windward slopes and not enough precipitation in the lee. Recently the MM5 was also used to simulate precipitation over Greenland. The modeled precipitation with the original HPGF scheme is excessive along the steep coastal margins especially on the southeast coast of Greenland. Tests of the simulated precipitation by using (1) in the MM5 without other changes show that the precipitation errors over the steep slopes of Greenland and Pacific Northwest of US are greatly improved.

The second advantage is that F _e can be used in synoptic analysis and model outputs directly on constant s surfaces and in the same way as F (x, y, p, t) is used in p-coordinates. The modeled results are not necessary to transfer to p-coordinates and can be compared with the observed analyses directly in s -coordinates. The artificial anomalous systems over the Tibetan Plateau and Greenland on the sea-level pressure maps and the lower tropospheric isobaric surfaces are not present, and weather systems over these high mountain regions are shown more clearly on the s surfaces. The third advantage of (1) is that the rotational part can be computed explicitly and accurately. The rotational part is small, but is related to exchange of momentum or vorticity between the mountain and atmosphere and is important in diagnosing orographic effects.