ADCIRC (Advanced Circulation) Model

Advanced Circulation Model (ADCIRC) is based on the finite element codes that solve the shallow water equation on unstructured triangular grids.

Two and three dimensional finite element model used for hydrodynamic circulation problems.

Solution strategy: Finite element method (FEM) for spatial, and finite difference method (FDM) for temporal.

Versions of ADCIRC

3D-VS (currently used)

The three-dimensional model, ADCIRC 3D-VS (Velocity Solution) applies a mode-splitting technique to solve the vertical profile of horizontal velocity u and v (Luettich et al., 2002)

2D-DIThe ADCIRC 2DDI is a two-dimensional depth-integrated model that solves sea surface elevation and depth-averaged velocity U and V. (Luettich et al. 1992)

ADTRANSThe ADCIRC 2D Tranport is a two-dimensional depth-integrated model that solves sea surface elevation , depth-averaged velocity U and V, and depth-averaged concentration (S, T or C) (Scheffner, 1999)

σ=aa−b z−η

H

surface-following “” coordinate system

= a = 1 at the free surface and = b = -1 at the bottom, H = h +

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1819

20 21

(h+)/H

-

la

ye

rExample: 21 -layers

z=(x,y,t)

z=-h

=a

=b

Chain rule relates derivatives along the z-level to the stretched -level

∂∂ x z

= ∂∂ xσ

−[ σ−b a−b

∂ η∂ x z

σ−a a−b

∂ h∂ x z ] ∂∂ z

∂∂ y z

= ∂∂ y σ

−[ σ−b a−b

∂η∂ y z

σ−a a−b

∂ h∂ y z ] ∂∂ z

∂∂ z

=a−b

H∂∂σ

H ≡hη

∫∂ z=H

a−b∫ ∂ σ

Where:

Governing Equations

Continuity Equation

Horizontal Momentum Equations∂u

∂ tu

∂ u∂ x

v∂ u∂ y

ω a−bH ∂u

∂ σ− fv =−g

∂ η∂ x

m xσ−bxσ

a−bH ∂

∂ σ τ zx

ρ0

∂ v∂ t

u∂ v∂ x

v∂ v∂ y

ω a−bH ∂ v

∂ σ fu =−g

∂ η∂ x

m y σ−b yσ

a−bH ∂

∂ σ τ zy

ρ0

Vertical Momentum Equation

a−bH ∂ p

∂ z−ρg=0

∂H∂ t

∂ UH ∂ x

∂ VH ∂ y

=0

U , V = depth averaged vertical velocities

u = u(x,y,,t) ; v = v(x,y,,t) ; = (x,y,,t) ;

Governing EquationsSolving Vertical Velocity

ωk1−ωk=−1

a−b ∫k

k1

∂ η∂ t

∂ u H ∂ x

∂ v H ∂ y ∂ σ

ωadj=ωσ−ωσ η [ σ−b HLa−b

a−b 2 HLa−b

]

k = node number over vertical element

The solution k will satisfy the bottom boundary condition only. In order to satisfy the free surface, the adjoint correction is applied based on Luettich and Muccino (2001), Muccino et al. (1997) and Pandoe and Edge (2003):

() is the misfit of surface boundary condition at the free surface ,

L is the weight of the relative contribution of the boundary conditions versus the interior solution

Salinity, Temperature and Sediment Transport∂C∂ t

u∂C∂ x

v∂C∂ y

ω−ωs ∂C∂ σ

=

∂∂ x D h

∂C∂ x ∂∂ y D h

∂C∂ y a−b

H 2∂∂ σ [Dv

∂C∂σ ]SS

SS = the source/sink terms; C = salinity [psu], temperature [C] or sediment concentration [g/l]Dv = vertical mixing coefficient computed using the second moment turbulence closure scheme of Mellor and Yamada (1982)

SS=a−b ρo C p

Q pen

H

a−bH D v

∂T∂ σ = Q ns

ρo C pat z

SS = E − D at z -h

Q pen= 0 . 45 Q short exp − γ h

Governing Equations

Temperature:

Sediment Concentration:

E and D represent erosion and deposition flux of suspended sediment, respectively

ρ x , y , σ =ρN [T x , y , σ , S x , y , σ ]Density: International Equation of State of Sea Water, IES80

= penetrating solar radiation [W/m2]

ρ x , y , σ = ρ [C x , y , σ ] (Soulsby, 1997)

Baroclinic and Barotropic Pressure gradient in the equation of motion is generated by two factors:

water surface slope (baroclinic) and water density difference (baroclinic)

0 v

z

=

0

z

v0

+

z

v

vertical shear =  Baroclinic + Barotropic

Governing Equations

∂ p z

∂ x= ∂

∂ xg∫

z

η

ρ dz

= g∫z

η∂ ρ∂ x

dz gρ η∂ η∂ x

bx =g H

a−b ∫σ

a

[∂ ρ N

∂ x− σ−b

H∂ η∂ x

σ−a

H∂h∂ x ∂ ρN

∂ σ]dσ

dy

h

H

a

yH

b

yba

Hgb N

aN

y

∫

Pressure grad. = Baroclinic (bx) + Barotropic

Final form Baroclinic terms in -coordinate system:

N is a normalized in situ density (Robertson et al, 2001)