# Terrain-Following Coordinate Transformation

Terrain-Following Coordinate Transformation

From the point of view of the computational model, it is highly convenient to introduce a stretched vertical coordinate system which essentially "flattens out" the variable bottom at . Such "" coordinate systems have long been used, with slight appropriate modification, in both meteorology and oceanography [e.g., Phillips (1957) and Freeman et al. (1972)]. To proceed, we make the coordinate transformation:

 $StartLayout 1st Row 1st Column ModifyingAbove x With caret 2nd Column equals x 2nd Row 1st Column ModifyingAbove y With caret 2nd Column equals y 3rd Row 1st Column sigma 2nd Column equals sigma left-parenthesis x comma y comma z right-parenthesis 4th Row 1st Column z 2nd Column equals z left-parenthesis x comma y comma sigma right-parenthesis 5th Row 1st Column ModifyingAbove t With caret 2nd Column equals t EndLayout$ (1)

See Vertical S-coordinate for the form of used here. Also, see Shchepetkin and McWilliams, 2005 for a discussion about the nature of this form of and how it differs from that used in SCRUM.

In the stretched system, the vertical coordinate spans the range ; we are therefore left with level upper () and lower () bounding surfaces. The chain rules for this transformation are:

 $StartLayout 1st Row 1st Column left-parenthesis StartFraction normal partial-differential Over normal partial-differential x EndFraction right-parenthesis Subscript z 2nd Column equals left-parenthesis StartFraction normal partial-differential Over normal partial-differential x EndFraction right-parenthesis Subscript sigma Baseline minus left-parenthesis StartFraction 1 Over upper H Subscript z Baseline EndFraction right-parenthesis left-parenthesis StartFraction normal partial-differential z Over normal partial-differential x EndFraction right-parenthesis Subscript sigma Baseline StartFraction normal partial-differential Over normal partial-differential sigma EndFraction 2nd Row 1st Column Blank 3rd Row 1st Column left-parenthesis StartFraction normal partial-differential Over normal partial-differential y EndFraction right-parenthesis Subscript z 2nd Column equals left-parenthesis StartFraction normal partial-differential Over normal partial-differential y EndFraction right-parenthesis Subscript sigma Baseline minus left-parenthesis StartFraction 1 Over upper H Subscript z Baseline EndFraction right-parenthesis left-parenthesis StartFraction normal partial-differential z Over normal partial-differential y EndFraction right-parenthesis Subscript sigma Baseline StartFraction normal partial-differential Over normal partial-differential sigma EndFraction 4th Row 1st Column Blank 5th Row 1st Column StartFraction normal partial-differential Over normal partial-differential z EndFraction 2nd Column equals left-parenthesis StartFraction normal partial-differential sigma Over normal partial-differential z EndFraction right-parenthesis StartFraction normal partial-differential Over normal partial-differential sigma EndFraction equals StartFraction 1 Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential Over normal partial-differential sigma EndFraction EndLayout$ (2)

where

 $upper H Subscript z Baseline identical-to StartFraction normal partial-differential z Over normal partial-differential sigma EndFraction$ (3)

As a trade-off for this geometric simplification, the dynamic equations become somewhat more complicated. The resulting dynamic equations, after dropping the carats, are:

 $StartFraction normal partial-differential u Over normal partial-differential t EndFraction minus f v plus ModifyingAbove v With right-arrow dot normal nabla u equals minus StartFraction normal partial-differential phi Over normal partial-differential x EndFraction minus left-parenthesis StartFraction g rho Over rho Subscript o Baseline EndFraction right-parenthesis StartFraction normal partial-differential z Over normal partial-differential x EndFraction minus g StartFraction normal partial-differential zeta Over normal partial-differential x EndFraction plus StartFraction 1 Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential Over normal partial-differential sigma EndFraction left-bracket StartFraction left-parenthesis upper K Subscript m Baseline plus nu right-parenthesis Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential u Over normal partial-differential sigma EndFraction right-bracket plus script upper F Subscript u Baseline plus script upper D Subscript u$ (4)

 $StartFraction normal partial-differential v Over normal partial-differential t EndFraction plus f u plus ModifyingAbove v With right-arrow dot normal nabla v equals minus StartFraction normal partial-differential phi Over normal partial-differential y EndFraction minus left-parenthesis StartFraction g rho Over rho Subscript o Baseline EndFraction right-parenthesis StartFraction normal partial-differential z Over normal partial-differential y EndFraction minus g StartFraction normal partial-differential zeta Over normal partial-differential y EndFraction plus StartFraction 1 Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential Over normal partial-differential sigma EndFraction left-bracket StartFraction left-parenthesis upper K Subscript m Baseline plus nu right-parenthesis Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential v Over normal partial-differential sigma EndFraction right-bracket plus script upper F Subscript v Baseline plus script upper D Subscript v$ (5)

 $StartFraction normal partial-differential upper C Over normal partial-differential t EndFraction plus ModifyingAbove v With right-arrow dot normal nabla upper C equals StartFraction 1 Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential Over normal partial-differential sigma EndFraction left-bracket StartFraction left-parenthesis upper K Subscript upper C Baseline plus nu right-parenthesis Over upper H Subscript z Baseline EndFraction StartFraction normal partial-differential upper C Over normal partial-differential sigma EndFraction right-bracket plus script upper F Subscript upper C Baseline plus script upper D Subscript upper C$ (6)

 $rho equals rho left-parenthesis upper T comma upper S comma upper P right-parenthesis$ (7)

 $StartFraction normal partial-differential phi Over normal partial-differential sigma EndFraction equals left-parenthesis StartFraction minus g upper H Subscript z Baseline rho Over rho Subscript o Baseline EndFraction right-parenthesis$ (8)

 $StartFraction normal partial-differential upper H Subscript z Baseline Over normal partial-differential t EndFraction plus StartFraction normal partial-differential left-parenthesis upper H Subscript z Baseline u right-parenthesis Over normal partial-differential x EndFraction plus StartFraction normal partial-differential left-parenthesis upper H Subscript z Baseline v right-parenthesis Over normal partial-differential y EndFraction plus StartFraction normal partial-differential left-parenthesis upper H Subscript z Baseline normal upper Omega right-parenthesis Over normal partial-differential sigma EndFraction equals 0$ (9)

where

 $ModifyingAbove v With right-arrow equals left-parenthesis u comma v comma normal upper Omega right-parenthesis$ (10)
 $ModifyingAbove v With right-arrow dot normal nabla equals u StartFraction normal partial-differential Over normal partial-differential x EndFraction plus v StartFraction normal partial-differential Over normal partial-differential y EndFraction plus normal upper Omega StartFraction normal partial-differential Over normal partial-differential sigma EndFraction$ (11)

The vertical velocity in coordinates is

 $normal upper Omega left-parenthesis x comma y comma sigma comma t right-parenthesis equals StartFraction 1 Over upper H Subscript z Baseline EndFraction left-bracket w minus StartFraction z plus h Over zeta plus h EndFraction StartFraction normal partial-differential zeta Over normal partial-differential t EndFraction minus u StartFraction normal partial-differential z Over normal partial-differential x EndFraction minus v StartFraction normal partial-differential z Over normal partial-differential y EndFraction right-bracket$ (12)

and

 $w equals StartFraction normal partial-differential z Over normal partial-differential t EndFraction plus u StartFraction normal partial-differential z Over normal partial-differential x EndFraction plus v StartFraction normal partial-differential z Over normal partial-differential y EndFraction plus normal upper Omega upper H Subscript z$ (13)

## Vertical Boundary Conditions

In the stretched coordinate system, the vertical boundary conditions become:

top ():

 $StartLayout 1st Row 1st Column left-parenthesis StartFraction upper K Subscript m Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential u Over normal partial-differential sigma EndFraction 2nd Column equals tau Subscript s Superscript x Baseline left-parenthesis x comma y comma t right-parenthesis 2nd Row 1st Column left-parenthesis StartFraction upper K Subscript m Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential v Over normal partial-differential sigma EndFraction 2nd Column equals tau Subscript s Superscript y Baseline left-parenthesis x comma y comma t right-parenthesis 3rd Row 1st Column left-parenthesis StartFraction upper K Subscript upper C Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential upper C Over normal partial-differential sigma EndFraction 2nd Column equals StartFraction upper Q Subscript upper C Baseline Over rho Subscript o Baseline c Subscript upper P Baseline EndFraction 4th Row 1st Column normal upper Omega 2nd Column equals 0 EndLayout$ (14)

and bottom ():

 $StartLayout 1st Row 1st Column left-parenthesis StartFraction upper K Subscript m Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential u Over normal partial-differential sigma EndFraction 2nd Column equals tau Subscript b Superscript x Baseline left-parenthesis x comma y comma t right-parenthesis 2nd Row 1st Column left-parenthesis StartFraction upper K Subscript m Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential v Over normal partial-differential sigma EndFraction 2nd Column equals tau Subscript b Superscript y Baseline left-parenthesis x comma y comma t right-parenthesis 3rd Row 1st Column left-parenthesis StartFraction upper K Subscript upper C Baseline Over upper H Subscript z Baseline EndFraction right-parenthesis StartFraction normal partial-differential upper C Over normal partial-differential sigma EndFraction 2nd Column equals 0 4th Row 1st Column normal upper Omega 2nd Column equals 0 EndLayout$ (15)

Note the simplification of the boundary conditions on vertical velocity that arises from the coordinate transformation.