Modelling advection with MOHID Contents What is advection

Modelling advection with MOHID

Contents • What is advection? • The importance of advection in coastal areas • Methodology implemented in MOHID to simulate • • advection Code trip; Test cases – – 1 D and 2 D cyclic channel; Symetric estuary Freshwater cylinder Density front

What is advection ? k i j

The importance of advection in coastal waters • Estuary mouths – horizontal advection of momentum; • Coastal Upwelling – vertical and horizontal advection of heat; • Baroclinic instabilities – horizontal advection of mass, of heat and of momentum;

Aveiro Mouth

Upwelling in schematic coast Schematic slope

Upwelling scenario – more important because is the scenario more common in summer

Density front Ajustamento geostrófico segundo um modelo analítico baseado nas equações de gravidade reduzida a) Instante inicial (velocidade perpendicular à frente); b) instante em que é atingido o equilíbrio velocidade paralela à frente.

Baroclinic instability w (f+w)/H= const. w • Balanço de forças (explicação de engenheiro) • Vorticidade potencial (explicação de oceanógrafo) H H

Modelling a density front with Mohid

Modelling a density front with Mohid

Some details

Methodology implemented in Mohid to compute advection Central Differences 2 nd order upwind (1 D Quick) 3 th order upwind (1 D Quickest)

Problems related with higher order schemes Ok Vi Problems Vi-1>>Vi Or Vi-1<<Vi =0

Problems related with higher order schemes Lack of positivity – production of ripples near sharp gradients

Total Variation Diminishing TVD schemes implemented in Mohid if r>0 • Min. Mod : =min(1, r) • Van Leer : =2 r/(1+r) • MUSCL : min(2, 2 r, (1+r)/2) • Super. Bee : max(min(1, 2 r), min(r, 2)) A value for function of the spatial variability of P is compute in a way that the problems describe in the earlier slide • PPM : max(min(Aux, 2/(1 -Cr), 2 r/Cr)) a = 0. 5 + (1 - 2. *abs(cr))/6 b = 0. 5 - (1 - 2. *abs(cr))/6 Aux = a + b * r Else =0 endif r<0=> =0 in all schemes r>0

1 D channel test - TVD

Code Trip • The nuclear subroutine is Compute. Advection. Face: – The advective flux in each face can depend of the property value in the follow compute points (i-2, i-1, I, i+1) – This subroutine compute for each face the coefficients a, b, c and d

Test cases • Already tested – 1 D and 2 D cyclic channel; – Symetric estuary • Need more testing – Fresh water cylinder – Density front

Depth = 20 Water m, 20 layers Fresh Cylinder Exercise 30 km x 30 km, dx = 1 km • • Latitude 52º N Cylinder of 10 m deep and 3 km of radius. Outside the cylinder salinity is 34. 85 psu Within the cylinder the salinity and the density variability is given by: Run period 144 h Bottom rugosity =0 Null turbulent diffusion or minimal values to avoid instabilities A relaxation boundary condition for the water level and salinity. Relax in the boundary the water level to 0 and salinity to 34. 85 psu. A four-point-wide relaxation zone:

Conclusions • The TVD with a Superbee limitation looks to be so far the best method. However, in extreme cases it generates wiggles;

Future Work • Introduce the cross-derivates in the Quick and Quickest methods.

Bibliography • Pietrzak, J. (1997). The use of TVD limiters forward-in-time upstream-biased • • • advection schemes in ocean modeling. Monthly Weather Review. Volume 126, 812830, 1997. Burchard, H. , and K. Bolding, GETM - a general estuarine transport model. Scientific documentation, Tech. Rep. EUR 20253 EN, European Commission, 2002. James I. D. (1996). "Advection schemes for shelf sea models. " Journal of Marine Systems 8: 237 -254. James, I. D. (2000). "A high-performance explicit vertical advection scheme for ocean models: how PPM can beat the CFL condition. " Applied Mathematical Modelling, 24(1): 1 -9. Tartinville, B. , E. Deleersnijder, P. Lazure, R. Proctor, K. G. Ruddick and R. E. Uittenbogaard (1998). "A coastal ocean model intercomparison study for a threedimensional idealised test case. " Applied Mathematical Modelling, 22(3): 165 -182. Shchepetkin, A. , & J. C. Mc. Williams, 1998: Quasi-monotone advection schemes based on explicit locally adaptive dissipation. Monthly Weather Review, 126, 1541 -1580.