1

SCHISM numerical formulation

Joseph Zhang 2 Horizontal grid: hybrid 3 4 Side 2 3 Side 2 Side 1 Side 3 Side 1

1 Side 4 2 1 Side 3 2 3 2 1

1 Counter-clockwise convention 3 2 3 ys 2 2 x i 2 1 s N 1 i i 1 2 Ni

N i 1 Ni 1 i • Internal ‘ball’: starting elem is arbitrary • Boundary ball: starting elem is unique Vertical grid (1): SZ 3 Terrain-following zone

s zone S zone SZ zone z s=0 Nz

hc h hs s S-levels kz

s=-1 kz-1

Z-levels 2

1

 z(1  s )  h s  ( h - h ) C ( s ) ( - 1  s  0)  cc tanh ( s- 1/ 2) tanh(  / 2)  sinh(sf ) ff C(s ) (1 - b )   b (0   b  1; 0   f  20)  sinhff 2tanh( / 2) h min( h , hs ) S-coordinates 4

-3 f=10 , any b f=10, b=1

hc

f=10, b=0 f=10, b=0.7 Vertical grid: SZ 5

z=-hs

The bottom cells are shaved – no staircases! Vertical grid (2): LSC2 via VQS

Dukhovskoy et al. (2009)

Procedure for VQS

• Different # of levels are used at different depths • At a given depth, the # of levels is interpolated from the 2 adjacent master grids • Thin layers near the bottom are masked • Staircases appear near the bottom (like Z), but otherwise terrain following What to do with the staircases?

A

u u P 1 B 2 Q R 1

2

LSC2= Localized Sigma Coordinates with Shaved Cells

(ivcor =1 in vgrid.in)

*No masking of thin layers (c/o implicit scheme)

Vertical grid: LSC2 8 z (m) z

Degenerate prism Vertical grid (3) 9 • 3D computational unit: uneven prisms • Equations are not transformed into S- or s-coordinates, but solved in the original Z-space  • Pressure gradient d z •Z-method with cubic spline (extrapolation on surface/bottom)

nk nk k k (whole level) Nz  N -1 ……. z uj,k+1 vj,k+1  uj,k+1 vj,k+1 k Ci,k

……. k-1 Ci,k k-1 k-1 k +1 wi,k-1 wi,k-1 b kb Polymorphism

Zhang et al. (2016) Bathymetry (m) Polymorphism

flood z (m) z

x (m)

S (PSU)

*Make sure there is no jump in Cd between 2D/3D zones Semi-implicit scheme 12

nn1 -  Continuity -uunn1dzdz(1)0 t --hh 풖풏+ퟏ − 풖∗ = 풇 − 𝑔휃훻휂푛+1 − 𝑔(1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Momentum Δ푡 풛

n1  nnnu 1  τw , at ;z b.c. (3D)  z  un1 nnn  -u 1, at ,zh  nn C ||u  z b Db

Du uun1 - Advection: ≈ * ; u (x , y , z , t  t ,  t ) Dt t *

 Implicit treatment of divergence, pressure gradient and SAV form drag terms by-passes most severe stability constraints: 0.5 ≤≤1

 Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, radiation stress… Eulerian-Lagrangian Method (ELM) 13

 ELM: takes advantage of both Lagrangian and Eulerian methods

 Grid is fixed in time, and time step is not limited by Courant number condition

 Advections are evaluated by following a particle that starts at certain point at time t and ends right at a pre-given point at time t+t.

 The process of finding the starting point of the path (foot of characteristic line) is called

backtracking, which is done by integrating dx/dt=u3 backward in 3D space and time.  To better capture the particle movement, the backward integration is often carried out in small sub-time steps

 Simple backward Euler method

 2nd-order R-K method

 Interpolation-ELM does not conserve; integration ELM does

 Interpolation: numerical diffusion vs. dispersion ut 1 x   Characteristic line t+t t advection

Flow dispersion diffusion Operating range of SCHISM 14

…. for a fixed grid

( 풖 + 𝑔ℎ)∆푡 퐶퐹퐿 = ∆푥

• ELM prefers larger t (truncation error ~1/t) • As a result, SCHISM requires CFL>0.4 • Convergence: CFL=const, t0 • For barotropic field applications: [100,450] sec • For baroclinic field applications: [100,200] sec (e.g., start with 150 sec) ELM with high-order Kriging 15

• Best linear unbiased estimator for a random function f(x) • “Exact” interpolator • Works well on unstructured grid (efficient) • Needs filter (ELAD) to reduce dispersion N f( x ,)(||) yxyK-123  iixx i1 Cubic spline Drift function Fluctuation K is called generalized covariance function Khhhhh(), -ln, , 23 Minimizing the variance of the fluctuation we get

N  i  0 i1 N 2-tier Kriging cloud  iix  0 i1 N  iiy  0 i1 x f(,) xi y i f i

• The numerical dispersion generated by kriging is reduced by a diffusion ELAD filter (Zhang et al. 2016) Semi-implicit scheme 16

nn1 -  Continuity -uunn1dzdz(1)0 t --hh 풖풏+ퟏ − 풖∗ = 풇 − 𝑔휃훻휂푛+1 − 𝑔(1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Momentum Δ푡 풛

n1  nnnu 1  τw , at ;z b.c.  z  un1 nnn  -u 1, at ,zh  nn C ||u  z b Db

Du uun1 - ≈ * ; u (x , y , z , t  t ,  t ) Dt t *

 Implicit treatment of divergence and pressure gradient terms by-passes most severe CFL condition : 0.5 ≤≤1

 Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, radiation stress… Galerkin finite-element formulation 17

A Galerkin weighted residual statement for the continuity equation:

휂푛+1 − 휂푛 න 휙 푑Ω + 휃 − න 훻휙 ∙ 푼푛+1푑Ω + න 휙 푈푛+1푑Γ + 1 − 휃 − න 훻휙 ∙ 푼푛푑Ω + න 휙 푈푛푑Γ = 0, 푖 Δ푡 푖 푖 푛 푖 푖 푛 Ω Ω Γ Ω Γ (𝑖 = 1, ⋯ , 푁푝)

휙푖: shape/weighting function; U: depth-integrated velocity; :implicitness factor

Need to eliminate Un+1 to get an equation for  alone. We’ll do this with the aid from momentum equation:

풖풏+ퟏ − 풖∗ = 풇 − 𝑔휃훻휂푛+1 − 𝑔(1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Δ푡 풛 Shape function (global) 2D case 18

The unknown velocity can be easily found as:

퐻2 푼푛+1 = 푮ෙ − 𝑔휃Δ푡 훻휂푛+1 퐻෩

퐻 푮ෙ = 푼∗ + ∆푡 푭 + 흉 − 𝑔(1 − 휃)퐻훻휂푛 (explicit terms) 퐻෩ 푤

퐻෩ = 퐻 + (휒 + 훼|풖|퐻)Δ푡 3D case 19 Vertical integration of the momentum equation

푼풏+ퟏ − 푼∗ 푛+1 푛 푛+1 휶 ub = 푭 − 𝑔휃훻휂 − 𝑔 1 − 휃 훻휂 + 흉푤 − 휒풖푏 − 훼 풖 푼 Δ푡 db 푧푣 풛풗 Assuming: න |풖|풖푑푧 ≅ 풖 න 풖푑푧 ≡ 풖 푼휶 z=-h −ℎ −풉

Momentum equation applied to the bottom boundary layer

풖풏+ퟏ − 풖∗ 휕 휕풖 풃 풃 = 풇 − 𝑔휃훻휂푛+1 − 𝑔 1 − 휃 훻휂푛 − 훼 풖 풖풏+ퟏ + 휈 Δ푡 풃 풃 풃 휕푧 휕푧

Therefore

푛+1 1 ∗ 푛 𝑔휃∆푡 푛+1 풖푏 = 풖푏 + 풇풃∆푡 − 𝑔(1 − 휃)∆푡훻휂 − 훻휂 1 + 훼|풖푏|∆푡 1 + 훼|풖푏|∆푡

Now still need to find U 3D case: locally emergent vegetation 20

We have 푼훼 = 푼n+1

So: ෡ 푛+1 𝑔휃퐻Δ푡 푛+1 푼 = 푮ퟏ − 훻휂 1 + 훼|풖|∆푡

∗ ෡ 푛 ∗ 푼 + 푭 + 흉풘 ∆푡 − 𝑔 1 − 휃 퐻∆푡훻휂 − 휒෤∆푡(풖푏 + 풇푏∆푡) 푮1 = 1 + 훼|풖|∆푡

퐻෡ = 퐻 − 휒෤∆푡

휒 휒෤ = 1 + 훼|풖풃|∆푡 3D case: locally submerged vegetation 21 Integrating from bottom to top of canopy: 푼훼 휕풖 푧푣 = 푼∗훼 + 푭휶∆푡 − 𝑔휃퐻훼∆푡훻휂푛+1 − 𝑔(1 − 휃)퐻훼∆푡훻휂푛 − 훼∆푡 풖 푼훼 + ∆푡휈 ቤ 푧 푧 휕푧 푣 푣 −ℎ 푼∗훼 = න 풖∗푑푧, 푭휶 = න 풇푑푧 −ℎ −ℎ Inside vegetation layer, the stress obeys exponential law (Shimizu and Tsujimoto 1994)

휕풖 휈 ≡ 푢′푤′ = 푹 푒훽2(푧−푧푣), (푧 ≤ 푧 ) 휕푧 0 푣

훽2 is a constant found from empirical formula:

푁 퐻 − 퐻훼 휒|풖푏| 훽 = 푣 −0.32 − 0.85 log 퐼 퐼 = 2 퐻훼 10 퐻훼 𝑔퐻

휕풖 푧푣 Therefore 휈 ቤ = 훽휒풖푛+1 (훽 = 푒훽2 푧푣−푧푏 − 1) 휕푧 푏 𝑔휃−ℎ 퐻෡훼∆푡 훼 푛+1 푼∗훼 + 푭휶∆푡 + 훽휒෤∆푡 풖∗ + 풇 ∆푡 − 𝑔(1 − 휃)퐻෡훼∆푡훻휂푛 푼 = 푮3 − 훻휂 푏 푏 훼 훼 1 + 훼|풖|∆푡 푮3 = 퐻෡ = 퐻 + 훽휒෤∆푡 1 + 훼|풖|∆푡 Finally: 훼|풖|∆푡 푛+1 푛+1 훼 푼 = 푮ퟐ − 𝑔휃퐻നΔ푡훻휂 퐻ന = 퐻 − 휒෤Δ푡 − 푐퐻෡ 푐 = 1 + 훼|풖|∆푡 General case 22

In compact form:

푼푛+1 = 푬 − 𝑔휃퐻෱Δ푡훻휂푛+1

퐻2 , locally 2D 퐻෩ 퐻෱ = 퐻෡ , locally 3D emergent 1 + 훼|풖|∆푡 (Note the difference in 퐻ෙ between 2D/3D) 퐻ന, locally 3D submerged

푮෱, 2D 푬 = ൞ 푮ퟏ, 3D emergent 푮ퟐ, 3D submerged Finite-element formulation (cont’d) 23 Finally, substituting this eq. back to continuity eq. we get one equation for elevations alone: I1

푛+1 2 2 푛+1 න 휙푖휂 + 𝑔휃 Δ푡 퐻ෙ훻휙푖 ∙ 훻휂 푑Ω I3 Ω

푛 푛 푛+1 푛 푛+1 = න 휙푖휂 + 휃Δ푡훻휙푖 ∙ 푬 + (1 − 휃)Δ푡훻휙푖 ∙ 퐔 푑Ω − 휃Δ푡 න 휙푖푈෡푛 푑Γ푣 − (1 − 휃) Δ푡 න 휙푖푈푛 푑Γ − 휃Δ푡 න 휙푖푈푛 푑Γത푣 + 퐼7

Ω Γ푣 Γ Γഥ푣

I5 I6 I4 sources (i=1,…,Np)

Notes: • When node i is on boundaries where essential b.c. is imposed,  is known and so no equations are

needed, and so I2 and I6 need not be evaluated there • When node i is on boundaries where natural b.c. is imposed, the velocity is known and so the last term on LHS is known  only the first term is truly unknown! • b.c. is free lunch in FE model (cf. staggering in FV) • The inherent numerical dispersion in FE nicely balances out the numerical diffusion inherent in the implicit time stepping scheme! Shape function 24

 Used to approximate the unknown function

 Although usually used within each individual elements, shape function is global not local! N u ui i,  i ( x j ) d ij i1

 Must be sufficiently smooth to allow integration by part in weak formulation

 Mapping between local and global coordinates

 Assembly of global matrix

NMN L udL  ud   i i ji i j m 0, jN 1,..., imi111  m x

1 j-1 j j+1 N x

1 j-1 j j+1 N Conformal Non-conformal (Elevation, velocity) (optional for velocity: indvel=0 -> less dissipation and needs further stabilization (MB* schemes) Matrices: I1 25 Ni Igt Hd ()ˆˆ nn1221 퐻ෙˆ 1''   iij j1  j i34(j) Ni 3 2 n1 1 dil', gt     AHil 퐻ෙˆ ' j  j, lj 124 A jl11 j i i’ is local index of node i in j. il' reaches its max when i’=l, and so does I1. 3 i’ i'0 so diagonal is dominant if the averaged friction-reduced i '1 depth ≥0 (can be relaxed) Mass matrix is positive definite and symmetric! JCG or PETSc solvers work fine > (i’,2) (i’,1) i ' 26 Matrices: I3 and I5

I Uˆˆnn11 d   U d  3 i n v i n ij n j v ij j L Nz -1  i ij ˆˆnn11 ij  zij, k 1()() u n ij , k 1  u n ij , k j4 k kbs

Flather b.c. (overbar denotes mean) j

nn11 uugˆnn- h - / ()  ()U n ij ghij nn11 IL3 -ijiijj - 2() ()  26 j 

L Nz -1 Similarly Iˆ Uˆ n d  ij  z()() uˆˆ n  u n 5 i ' n ij   ij , k 1 n ij , k 1 n ij , k j j4 k k ij bs 27 Matrices: I7

Related to source/sink

Bi: ball of node i Um: elements that have source/sink 3 Sj: volume discharge rate [m /s] 28 Matrices: I4

Has the most complex form

i34(j) Ni A 3 IttdtAtd -nnnn (1 )(1)  (1UGUG ) ˆˆˆ - ˆ  j  ' 3,','  iiij li lj iji j E   jl12 1

Most complex part of E is the baroclinicity g  f  - d at elements/sides c z 0 Baroclinicity: z-method 29 o Interpolate density along horizontal planes B o Advantage: alleviate pressure gradient errors (Fortunato and Baptista 1996) 3 1 0 o Disadvantage: near surface or bottom Element j o Density is defined at prism centers (half levels) A o Re-construction method: compute directional derivatives along two direction at node i and 2 then convert them back to x,y o Density gradient calculated at prism centers first (for continuity eq.); the values at face centers are averaged from those at prism centers (for momentum eq.) o constant extrapolation (shallow) is used near bottom (under resolution) o Cubic spline is used in all interpolations of  o Mean density profile  () z can be optionally removed o 3 or 4 eqs for the density gradient vs. 2 unknowns – averaging needed for 3 pairs; e.g. ''' ' (x ) i, ky ()x i 1 k - ( xy 0, ) - () 1 y 0  - 1 0 ''' ' (x ) i, ky ()x i 2  k - ( xy 0, ) - () 2 y 0  - 2 0 o Degenerate cases: when the two vectors are co-linear (discard) o Vertical integration using trapzoidal rule Momentum equation: 2D case 30

The depth-averaged velocity can be directly solved once unknown elevations are found:

퐻෱ 풖풏+ퟏ = 풖∗ + (풇 + 흉 /퐻)∆푡 − 𝑔휃훻휂푛+1 − 𝑔(1 − 휃)훻휂푛 퐻 풘 Momentum equation: 3D case 31

Galerkin FE in the vertical

휂 휕 휕풖푛+1 휂 න 휑 푏풖푛+1 − Δ푡 휈 푑푧 = න 휑 품푑푧 , (푙 = 푘 + 1, ⋯ , 푁 ) 푙 휕푧 휕푧 푙 푏 푧 푧푏 푧푏

푛 푏 = 1 + 훼|풖 |Δ푡ℋ(푧푣 − 푧)

휑푙: vertical hat function; g: explicit terms

b--zzu u uun1 nn  11 n  1 b AA(N- ltk )ll- -111 (2)( lttun lll 1 1) uu nn  l 11 d uu nn -  (2) 11     zl l lbl l1 ll - k1/21 - k 1/2,1  1 bb 66zzll1 zz d -tNτgnn11+ lk gg AA ( l  ) -ll1 (2 ) ( 1 ) (2  gn1) l,1 Nz w zl l66 bl  l -1

(l kbz 1, N )

…….. N • The matrix is tri-diagonal (Neumann b.c. can be imposed easily) z • Solution at bottom from the b.c. there (u=0 for ≠0); otherwise free slip k +2 • Vegetation terms are also treated implicitly b kb+1 db z=-h kb Velocity at nodes 32

• Needed for ELM (interpolation), and can serve as a way to further stabilize the mom solver by adding some inherent numerical diffusion • Method 1: inverse-distance averaging around ball (indvel=1 or MA schemes) – i larger dissipation; no further stabilization needed • Method 2: use linear shape function (conformal) (indvel=0 or MB schemes) uuuu- 1 IIIIII 3

• Discontinuous across elements; averaging to get the final value at node II I • Generally needs velocity b.c. (for incoming info) • indvel=0 generally gives less dissipation but needs further stabilization in 1 III 2 the form of viscosity or Shapiro filter for stabilization (important for eddying regime)

Eternal struggle between diffusion and dispersion Diffusion Dispersion Shaprio filter 33

A simple filter to reduce sub-grid scale (unphysical) oscillations while leaving the physical signals largely intact (=0.5 optimal)

 4 1 4 ˆ uuuu000-  i 4, (00.5)  4 i1 0 • No filtering for boundary sides – so need b.c. there especially for incoming flow 2 3 • Equivalent to Laplacian viscosity Horizontal viscosity 34 Important for eddying regime for adding proper dissipation  u P (u)  0 d 1 4 1 6 0 A  A  n Assuming uniformity I II  I II Q 0x S 2 x0 5 2 3 I II  (like Shapiro filter) (u)  0 (u  u  u  u - 4u ) R 3 4 0 1 2 3 4 0 3AI

Bi-harmonic viscosity (ideal in eddying regime) 1a 4b

- 4u  - (2u  2u  2u  2u - 42u )  1b 1 4 4a 0 3 1 2 3 3 0 0x  2 7(u  u  u  u ) - u - u - u - u - u - u - u - u - 20u  2a 3 3b t 1 2 3 4 1a 1b 2a 2b 3a 3b 4a 4b 0 2 2b 3a

(Zhang et al. 2016) Vertical velocity 35

 Vertical velocity using Finite Volume

ˆˆnxnynznxnynz111111 SunvnwnSkkkkki11111,()() kkk 11, kkkki unvnw- k k n 3 ˆ ˆˆnn11  PsqqkkNjs( i , m ) i ,( mjs ,()/ ),( i m 20, , kjs ), i 1 m(,...,1)- kbz  m1

qˆ j,, kun j k j

i34(j) 1 3 wPsnnnn11111111--- qqS xn yn u x nv nˆ nS yn u z()/ˆˆ n 2()() v , n w n ˆˆ i, kjs 1( i m , ) iz m ,( ,ˆ ),( js  , i m ), kjs  11 i m kk 1 11 k kk 1, kk k k k ki k k nSkk11m1

(,...,1)k- kNbz ෍ |푄 | = ෍ |푄 | n In compact form: 푗 푗 k+1 푗∈푆+ 푗∈푆− k+1 This is the foundation of mass conservation and monotonicity in transport solvers Pˆ Bottom b.c. b un1 n x v n  1 n y  w n  1 n z , ( k  k ) k k k k i, k kt b Sˆ k Surface b.c. not enforced (closure error) Inundation algorithms 36  Algorithm 1 (inunfl=0)  Update wet and dry elements, sides, and nodes at the end of each time step based on the newly computed elevations. Newly wetted vel., S,T etc calculated using average  Algorithm 2 (inunfl=1; accurate inundation for wetting and drying with sufficient grid resolution)  Wet  add elements & extrapolate velocity  Dry  remove elements  Iterate until the new interface is found at step n+1  Extrapolate elevation at final interface (smoothing effects)

dry A n B wet

Extrapolation of surface Transport: upwind (1) 37 S j k TT   ˆ  3 ()()uTQT    h  t  z  z S T Ti,k Tzˆ, z k-1 T  0, - zh (sources: bdy_frc, flx_sf and flx_bt) T n i* Finite volume discretization in a prism (i,k): mass conservation i34(j)+2 mm1 5 TTTTmmmm1111-- TTii- ^ m i, ki- 1,,,ki ki k 1 Vuij S j TV ji i QA kii ki k t - *,,, 1 '- m≠n, t’ ≠ t; tzz' jV i, ki- 1/k 2, 1/ 2 i34(j) Ti=Ti,k; 3 TTmm- tSk'( ), kN (1,ji ,) u is outward normal  h jjbz d j j1 ij velocity Focus on the advection first t ' mm1 QuS TTQiij j - T n  * jjj Vi jV QQQ0|  || | From continuity equation jjj j j Sj- S S+: outflow faces; S -: inflow faces  Ti , j S Upwinding Tj*   - Tj , j S Transport: upwind (2) S 38 j k Drop “m” for brevity (and keep only advection terms) S Ti,k m1 t'''    t   t TTQTQTQTQTi i -n | j | i -  | j | j  1 -  | j | i   | j | j  VVV -  - ij S j  S   i j  S  i j  S k-1 S (1- )TTij

Max. principle is guaranteed if

′ ∆푡 푉푖 ≥ ′σ푗∈푆− |푄푗| ≥ 0 ֜ ∆푡 − 1 푉푖 σ푗∈푆− |푄푗| Courant number condition (3D case) Global time step for all prisms Implicit treatment of vertical advective fluxes to bypass vertical Courant number restriction in shallow depths

푉 ∆푡′ ≤ 푖 σ − |푄 | (Modified Courant number condition) 푗∈푆 퐻 푗

Mass conservation • Budget – vertical and horizontal sums • movement of free surface: small conservation error • Includes source/sink Solar radiation 39

total solar radiation 1 H ˆ --D// dD12 d QH- zH , ( )(0) Re(1R)e 0Czp 

0 H (0) Budget Qdzˆ  - 0C p The source term in the upwind scheme (F.D.)

HHAmmmm-- HH () ˆ m Vi i,, ki kii 1,, ki 1 k -- VQii 0,0CzCpi kp  H m  0 ik, b

*Assume the heat is completely absorbed by the bed (black body) to prevent overheating in shallow depths 2 Transport: TVD 40

Solve the transport eq in 3 sub-steps: t t m1 n m m m m ˆ m Horizontal advection Ci  Ci  | Q j | (C j - Ci ) - Q j j , (m  1,..., M ) V - V (explicit TVD method or WENO) i jSH i jSH ~ t ~ ~ t M 1 Vertical advection (implicit TVD2) Ci  Ci  |Q j | (C j - Ci ) - Q j ( j  j ), ( j  kb,.., Nz ) V - V i jSV i jSV n1 n1 n1 ~ A t  C   C   t n C  C  i  -   F dV, (k  k ,.., N ) i i       h b z Remaining (implicit diffusion) Vi  z i,k  z i,k-1 Vi   Vi

• 1st step: explicit TVD or WENO3 • 2nd step: Taylor expansion in space and time

n1/ 2 n1 n1 n1 t C n1 C j  C jup   j  j  C jup  r [C] jup - [ ] jup 2 t Duraisamy and Baeder (2007) Space limiter Time limiter IMPORTANT: built-in monotonicity constraint is very important for higher-order transport; use of additional diffusion would degrade order of accuracy 2 TVD 41     t 1  p  ~ ~ | Q j | 1   -  j (Ci - C j ) V - 2   r  ~ i jSV   pSV p  M 1 Ci   Ci t  q 1  | Q j | (  - j ) 2V  - s i jSV qSV q ~ ~ | Qq | (Cq - Ci ) - qSV  Upwind ratio rp  ~ ~ , p  SV | Q p | (Ci - C p ) ~ M 1 (Ci - Ci ) | Q p |  pS - Downwind ratio V • Iterative solver converges very fast sq  ~ M 1 , q  SV nd | Qq | (Cq - Cq ) • 2 -order accurate in space and time • Monotone    1  p  • Alleviate the sub-cycling/small t for 1  -  j  0 transport that plagued explicit TVD 2   r  TVD condition:  pSV p  • Can be used at any depths!

t  q 1 | Q j | (  - j )  d  0 (iterative solver) 2V  - s i jSV qSV q Turbulence closure (itur=3) 42 퐷푘 휕 휕푘 = 휈휓 + 휈푀2 + 휅푁2 + 푐 훼 풖 3ℋ 푧 − 푧 − 휖 퐷푡 휕푧 푘 휕푧 푓푘 푣

퐷휓 휕 휕휓 휓 = 휈 + 푐 휈푀2 + 푐 휅푁2 + 푐 훼 풖 3ℋ(푧 − 푧) − 푐 휖퐹 퐷푡 휕푧 휓 휕푧 푘 휓1 휓3 푓휓 푣 휓2 푤푎푙푙

Essential b.c. Natural b.c. k 1 u  0, - zhor kzh - , or k z ()cz02    0 -nzh , lzh0 -, or zl 0 pmn  ()()ckzh - , or 0   - nz, zl0

FE formulation

• k,  and diffusivities are defined at nodes and whole levels • Use natural b.c. first and essential b.c. is used to overwrite the solution at boundary • Neglect advection • The production and buoyancy terms are treated explicitly or implicitly depending on the sum to enhance stability Turbulence closure (itur=3) 43

nn11 -zkll11 kn l 11 n n n  A(l- Nkzl )22( - k l ) - k l - kt lk111/ l 2   6 zl1 nn11 -zklln k11 l n n n  -1 A(l- kkbl ) - k l 22( - k l) ---111/ kt lk 2 l   6 zl

z 22n l1 MNP  l1/ 2 n 2 0 3 1/ 2- 11  1zl1 nn A()l- Nz t -(c )(2) k  lk k ll1 z n l1/ 2 l1 MN22P (2)kknn11 6 n  l1/2 ll1 6kl1/ 2

z 22n l MNP  l-1/ 2 n 2 0 3 1/ 2- 11 zl n n1 A(l- kbl )( - tc )(2 l k lk k    -1 ) z n l-1/ 2 l M2P Nk 21 k(2) 1 nn 6 n  l-1/ 2 ll-1 6kl-1/ 2

(,,)l kbz N Turbulence closure 44

nn11 -zl11n11 n n n  l l A(l- Nz ) 2 l   l1 - 2  l -  l  1 - (  ) l  1/ 2  t  6 zl1 nn11 -zln11 n n n  l l-1 A(l- kb ) 2 l   l---1 - 2  l -  l 1  (  ) l 1/ 2  t  6 zl n n zl1 22 c13 MQ c N   2 l1/ 2 k l1/ 2 - n zl1 2 2nn 1 1 A()l- Nz t c MQ c  N (2   )  n  1 3l1/ 2 ll 1 6kl1/ 2

n 0 3 1/ 2- 1zl1 nn  1  1 (c ) c21 k l Fwall (2 l l ) l1/ 2 6 n n zl 22 c13 MQ c N   2 l-1/ 2 k l-1/ 2 - n zl 22nn11 A()l- kb t c M c N (2) n  13Q l-1/ 2 ll-1 6kl-1/ 2

n 0 3 1/ 2- 1zl nn  1  1 (c ) c21 k l Fwall (2 l l- ) l-1/ 2 6

(,,)l kbz N GOTM turbulence library (itur=4) 45

 General Ocean Turbulence Model

 One-dimensional water column model for marine and limnological applications

 Coupled to a choice of traditional as well as state-of-the-art parameterizations for vertical turbulent mixing (including KPP)

 Some perplexing problems on some platforms – robustness?

 Bugs page

 Division by 0? Newer versions may have resolved this issue

 Works better than itur=3 otherwise

www.gotm.net Radiation stress 46

Sxx etc are functions of wave energy, which is integrated over all freq. and direction WBBL (Grant-Madsen) 47

Wave-current friction factor

Ratio of shear stresses

angle between bottom current and dominant wave direction

tb: current induced bottom stress Uw: orbital vel. amplitude w: representative angular freq.

The nonlinear eq. system is solved with a simple iterative scheme starting from =0 (pure wave), c=1. Strong convergence is observed for practical applications Hydraulics 48

Q Faces 1 2 (b) 1 1’

2 2’

3 3’

4 4’

5 5’ 1-1’ etc are node pairs

The 2 ‘upstream’ and ‘downstream’ reference nodes on each side of the structure are used to calculate the thru flow Q Hydraulics 49

The differential-integral eq. for continuity eq. is modified as

And for the momentum eq. • Add b as a horizontal boundary in backtracking • At b and also for all internal sides in b, the side vel. is imposed

from Q – like an open bnd (add to I3 and I5) SCHISM flow chart 50

Initialization end or hot start yes forcings no

backtracking Completed? output Turbulence closure Update levels & eqstate

Transport Preparation (Mom+cont.) Momentum w eq. Most expensive part solver Numerical stability 51

 Semi-implicitness circumvents CFL (most severe) (Casulli and Cattani 1994)

 ELM bypasses Courant number condition for advection (actually CFL>0.4) 2  Implicit TVD transport scheme along the vertical bypasses Courant number condition in the vertical

 Explicit terms ght'   Baroclinicity  internal Courant number restriction (usually not a problem)  1 t x  0.5  Horizontal viscosity/diffusivity  diffusion number condition (mild) x2 2 3  Upwind, TVD transport or WEON : horizontal Courant number condition  sub-cycling

 Coriolis – “inertial modes” in eddying regime (Le Roux 2009; Danilov 2013); needs to be stabilized by viscosity/filter

nnn111 20uuu1- 2 - 3  Lateral b.c. and nudging 52 bctides.in param.in

Variable Type 1 (*.th) Type 2: Type 3 Type 4 Type 5 Type -1 Type -4, -5 Nudging/spon (*[23]D.th) (uv3D.th); ge layer near nudging boundary

 elev.th: constant Tidal elev2D.th.n elev2D.th.n Must =0 Time history; amp/phases c: time- and c: uniform space- combination along bnd varying of 3&4 along bnd S&T, tracers [MOD]_?.th Relax to Relax to i.c. [MOD]_3D. : relax to constant for for inflow th.nc: relax time history inflow to time- and (uniform space- along bnd) varying for inflow values along bnd during inflow u,v flux.th: via Via Tides uv3D.th.nc: uv3D.th.nc: Flather (‘0’ Relax to discharge discharge (uniform time- and combination for ) uv3D.th.nc (<0 for (<0 for amp/phase space- of 3&4 (but (2 separate inflow!) inflow!) along bnd) varying tidal relaxations along bnd amp/phases for in & (in lon/lat for vary along outflow) ics=2) bnd) 4H’s of SCHISM

o Hybrid FE-FV formulation o Hybrid horizontal grid o Hybrid vertical grid o Hybrid code (openMP-MPI)