Advanced Ocean Modelling: Using Open-Source Software

5.6 Exercise 23: Coastal Upwelling in 3D 1455.6.4 Additional Exercise for the ReaderRepeat this exercise with an overall onshore (northward) wind stress. Does the eastfacing coast of the bay support coastal upwelling?5.6.5 Time-Splitting MethodsThe reader will have noticed by now that three-dimensional model simulation arefairly slow and take a long time to complete. The reason of this time-step compliancewith the CFL criterion for fast propagating surface gravity waves (Eq. 5.15).Enabling the rigid-lid condition (see Sect. 3.7) seems a solution, but this could leadto a bias in the dynamics predicted. Another method is to use different numericaltime steps in different parts of the model code, a method called time splitting.Tothisend, small time steps only need to be applied within the S.O.R. iteration involvingthe surface pressure field, whereas all other parts such as advective and diffusivechanges can employ a much larger time step. Note that the latter still needs to satisfythe stability criterion Eq. (5.14).The first step of the time-stepping procedure is to predict the density field andfirst-guess changes of the velocity field, Δu, Δv and Δw, using an “internal” timestep Δt i being a multiple of the “external” time step Δt e ; that is,Δt i = mΔt ewhere the multiplier m is a positive integer. In a second step and prior to the S.O.R.iteration, the vector (u ∗ , v ∗ , w ∗ ) is initialised with the current velocity field (u n ,v n , w n ). After this, the S.O.R. iteration is repeated m times whereby the first-guessvelocity is updated using equal fractions of the first-guess velocity change:u ∗ ← u ∗ + Δu/mv ∗ ← v ∗ + Δv/mw ∗ ← w ∗ + Δw/mAn important rule when using a time-splitting method is that the internal timestep should be not more than about 20 times the external time step, provided that allstability criteria are met. Otherwise, the external and internal modes inherent withthe dynamics could drift apart from each other leading to a bias in the predictions.The author has tested the time-splitting method for this exercise achieving a threefoldreduction of the total simulation time. A modified code for Exercise 23 usingthe time-splitting method is contained in the folder “Miscellaneous/Time Splittingfor Exercise 23” on the book’s ftp site. The following exercises, however, do notmake use of this method.