1. Dynamic Height (a.k.a. Geopotential) Jessie Sagona Physical Oceanography November 20, 2007

2. What is Dynamic Height? Definition: The amount of work required to move a unit mass of water vertically from sea level to a given point. Or, the gravitational potential energy per unit mass. Definition: The amount of work required to move a unit mass of water vertically from sea level to a given point. Or, the gravitational potential energy per unit mass. = ∫α dp = ∫g dz = ∫α dp = ∫g dz Equivalent to geopotential in the atmosphere (but opposite sign). Equivalent to geopotential in the atmosphere (but opposite sign). Variable designation: Φ or D Variable designation: Φ or D Units: J/kg or m 2 /s 2 (don ’ t let the word “ height ” in the name fool you!) Units: J/kg or m 2 /s 2 (don ’ t let the word “ height ” in the name fool you!)

3. AtmosphereOceanUnits Geopotential (Φ) Dynamic Height (D) J/kg Geopotential Difference (ΔΦ) Dynamic Height Anomaly (ΔD) J/kg Geopotential Height (Z) ? Geopotential depth? m

4. Geopotential Height Z = (1/g 0 )∫g dz from sea level to a given height (or depth). Z is also equal to Φ/g 0. Z = (1/g 0 )∫g dz from sea level to a given height (or depth). Z is also equal to Φ/g 0. g 0 is defined as 9.80665 m/s 2, the value of g at 45 º N. g 0 is defined as 9.80665 m/s 2, the value of g at 45 º N. Gravity is a function of both height and latitude. Geopotential height takes this into account and allows you to use a constant 9.80665 for “ g ”. Gravity is a function of both height and latitude. Geopotential height takes this into account and allows you to use a constant 9.80665 for “ g ”.

5. Why is this better than geometric height? The PE of a parcel raised to a geometric height (recall g varies) is equal to the PE of a parcel raised to geopotential height. The PE of a parcel raised to a geometric height (recall g varies) is equal to the PE of a parcel raised to geopotential height. Lines of constant geopotential have constant PE  geopotential height is numerically proportional to PE. Lines of constant geopotential have constant PE  geopotential height is numerically proportional to PE. Note that Z is smaller than geometric height in the mid-latitudes. Note that Z is smaller than geometric height in the mid-latitudes.

7. Use, and misuse, of Φ and Z Geopotential height is the standard vertical coordinate in both the atmosphere and the ocean. Geopotential height is the standard vertical coordinate in both the atmosphere and the ocean. White et al. (2004) note that models often assume that surfaces of constant geopotential are spherically uniform across the planet  not true. White et al. (2004) note that models often assume that surfaces of constant geopotential are spherically uniform across the planet  not true. The difference between geometric and geopotential height may be small near the surface for both atm. and ocean, but it should not be neglected when precise calculations are attempted! The difference between geometric and geopotential height may be small near the surface for both atm. and ocean, but it should not be neglected when precise calculations are attempted!

8. References NASA Goddard Earth Science Data website http://daac.gsfc.nasa.gov/PIP/shtml/geopotential_height. shtml Accessed 11-16-07. NASA Goddard Earth Science Data website http://daac.gsfc.nasa.gov/PIP/shtml/geopotential_height. shtml Accessed 11-16-07. http://daac.gsfc.nasa.gov/PIP/shtml/geopotential_height. shtml http://daac.gsfc.nasa.gov/PIP/shtml/geopotential_height. shtml AMS Glossary http://amsglossary.allenpress.com/glossary/search?id=d ynamic-height1 Accessed 11-16-07. AMS Glossary http://amsglossary.allenpress.com/glossary/search?id=d ynamic-height1 Accessed 11-16-07. http://amsglossary.allenpress.com/glossary/search?id=d ynamic-height1 http://amsglossary.allenpress.com/glossary/search?id=d ynamic-height1 White, A.A. et al. White, A.A. et al. Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi- hydrostatic and non-hydrostatic. Quarterly Journal of the Royal Meteorological Society, 131, pp. 2081-2107. 2005.