New vertical coordinates stretching function, Vstretching = 4

[arango]

A new vertical stretching function, Vstretching = 4, was added and it is denoted as Shchepetkin (2010). This is a major improvement to his 2005 function, Vstretching = 2. The new function, C(s), has a double stretching transformation:

   C(s) = [1 - COSH(theta_s * s)] / [COSH(theta_s) - 1]

with bottom refinement:

   C(s) = [EXP(theta_b * C(s)) - 1] / [1 - EXP(-theta_b)]

Notice that the bottom function is the second stretching of an already stretched transform. The resulting function is continuous with respect the control parameters theta_s and theta_b with a meaningful range of:

   0 <  theta_s <= 10.0
   0 <= theta_b <=  4.0

Due to its functionality and properties Vtransform = 2 and Vstretching = 4 are now the default values for ROMS.

Notice that with Vtransform = 2 the true sigma-coordinate system is recovered as hc goes to infinity. This is useful when configuring applications with flat bathymetry and uniform level thickness. Practically, you can achieve this by setting in ocean.in:

   THETA_S = 0.0d0
   THETA_B = 0.0d0
   TCLINE  = 1.0d+17       (a large number)

Please check ​WikiROMS for updated documentation and plots. For example, in the coarse resolution, North Atlantic application (DAMEE_4) we get:

---

WARNING:

The new vertical stretching transformation Vtransform = 2 and Vstretching = 4 may have different time-step limitations due to the thickness of the discritized vertical levels. Users need to pay attention to extreme r-factor (rx1) values near the bottom

The following testcases were changed to have the default transformation Vtransform = 2 and Vstretching = 4:

                       OLD (1,1)                      NEW (2,4) 
               theta_s  theta_b  Tcline      theta_s   theta_b  Tcline
              ---------------------------------------------------------

BASIN            3.3      0.0     1400         3.3       0.0     1400
BENCHMARK        5.0      0.4      200         0.0       0.0      400                      
BL_TEST          3.2      0.2        5         5.0       1.5       20
CANYON2D        1d-4      0.0       90         0.0       0.0    1d+16
DOUBLE_GYRE      1.0      1.0       50         0.0       0.0    1d+16
FLT_TEST2D       3.0      0.4      100         0.0       0.0    1d+16
FLT_TEST3D       3.0      0.4      100         0.0       0.0    1d+16
GRAV_ADJ        1d-4      0.0       50         0.0       0.0    1d+16
KELVIN           3.0      0.0       50         0.0       0.0    1d+16
LMD_TEST        1d-4      0.0       50         0.0       0.0    1d+16
RIVERPLUME1      3.0      0.4       50         3.0       0.0       30
RIVERPLUME2      3.0      0.4       50         3.0       0.0       30
SEAMOUNT         5.0      0.4       50         6.5       2.0      100
SOLITON         1d-4      0.0       50         0.0       0.0    1d+16
UPWELLING        3.0      0.0       50         3.0       0.0       25
WINDBASIN       1d-4      0.0       50         1.0       0.0       50

---

Many thanks to Sasha Shchepetkin for designing this nice and continuous transformation.