Turbulence closure modelsand
sediment transport routinesin
ROMS
John C. Warner, U.S. Geological Survey
Christopher R. Sherwood U.S. Geological SurveyHernan G. Arango IMCS, Rutgers Richard Signell U.S. Geological SurveyMeinte Blaas IGPP / CESR / IoE, UCLABradford Butman U.S. Geological Survey
Outline• Turbulence Closure Models (focus on GLS)
– Available models in ROMS – Background of two-equation models– GLS method
• Numerical implementation• Applications
– Open channel flow(2)– Surface mixed layer deepening– Estuary (idealized + realistic)
• Sediment Transport Routines– Overview of routines– Applications
• Summary
Turbulence Closure Models in ROMS
Reynolds Averaged Navier Stokes Equations
Continuity
Momentum
Transport
Equation of State 000 TTβ)s(sα1ρρ
unknowns u, v, w, , temp, sal, ssc,
j
iij
ji
iljijl
j
ij
i
x
Uuu
xg
x
PU
x
UU
t
U
00
12
0
i
i
x
U
jj
jjj x
uxx
Ut
jsj
jjj x
Ssu
xx
SU
t
S
su j ju ijuu
Available turbulence closures in ROMS
• Background value
• Analytical expression• BVF - Brunt_Vaisala frequency
• LMD - Large / McWilliams / Doney
• Two equation models:– MY25 - Mellor/Yamada 2.5
– GLS - Umlauf/Burchard Generic Length Scale
Available turbulence closures in ROMS
• Background value– ocean.in
! Vertical mixing coefficients for active tracers: [1:NAT,Ngrids]
AKT_BAK == 5.0d-6 5.0d-6 ! m2/s
! Vertical mixing coefficient for momentum: [Ngrids].
AKV_BAK == 5.0d-6 ! m2/s
– mod_mixing.FDO itrc=1,NAT
MIXING(ng) % Akt(Imin:Imax,Jmin:Jmax,1:N(ng)-1,itrc) = & & Akt_bak(itrc,ng)
END DO
MIXING(ng) % Akv(Imin:Imax,Jmin:Jmax,1:N(ng)-1) = Akv_bak(ng)
Available turbulence closures in ROMS
• Analytical– cppdefs.h
#define ana_vmix
– analytical.F, ana_vmix# elif defined SED_TEST1
DO k=1,N(ng)-1 ! vonkar*ustar*z*(1-z/D) DO j=JstrR,JendR DO i=IstrR,IendR Akv(i,j,k)=0.025_r8*(h(i,j)+z_w(i,j,k))* & & (1.0_r8-(h(i,j)+z_w(i,j,k)) / (h(i,j)+zeta(i,j,knew))) Akt(i,j,k,itemp)=Akv(i,j,k)*0.49_r8/0.39_r8 Akt(i,j,k,isalt)=Akt(i,j,k,itemp) END DO END DO END DO
Available turbulence closures in ROMS
• BVF
• !-----------------------------------------------------------------------• ! Set tracer diffusivity as function of the Brunt-vaisala frequency.• ! Set vertical viscosity to its background value.• !-----------------------------------------------------------------------
– cppdefs.h– #define BVF_MIXING /* Activate Brunt-Vaisala frequency mixing */
Available turbulence closures in ROMS
• LMD– cppdefs.h
/* Options for the Large/McWilliams/Doney interior mixing */# define LMD_MIXING
#undef LMD_SKPP /* surface boundary layer KPP mixing */#undef LMD_BKPP /* bottom boundary layer KPP mixing */#undef LMD_NONLOCAL /* nonlocal transport */
#undef LMD_RIMIX /* diffusivity due to shear instability */#undef LMD_CONVEC /* convective mixing due to shear instability */#undef LMD_DDMIX /* double-diffusive mixing */
Available turbulence closures in ROMS
• MY25– cppdefs.h
#define MY25_MIXING
• GLS– cppdefs.h
#define GLS_MIXING
• For either MY25 or GLS – cppdefs.h
#define KANTHA_CLAYSON
or #define CANUTO_A
or #define CANUTO_B
#define N2S2_HORAVG
Two Equation Models
Transport equation for Reynolds Stresses
Pij
ij
Bij
ij
ij
ij
ijij
ijij
ijij
ijjiij
Skc
PDc
BBc
PPc
kuuk
c
5
4
3
2
1
3
2
3
2
3
2
3
2
lB
q
1
3
5120320
20800
500
77600
52982
5
4
3
2
1
..
.
.
.
..
c
c
c
c
c
CAKC
pug
B
x
UuuP
i
jij
30
1
2 3
Pressure-strain correlation
dissipation Triple correlation
Reynolds Stress transport
kji
kij
jik
Tkji uux
uux
uux
uuu
l
i
l
j
ij
ji
ijji
mijlmmjilml
l
jli
l
ijl
jil
jiljill
ji
x
u
x
u
x
pu
x
pu
ugug
uuuu
x
Uuu
x
Uuu
uux
uuuuuUx
uut
2
1
1
2
0
0
j
i
i
jij
i
llj
j
lliij
x
U
x
US
x
Uuu
x
UuuD
2
1
ijij 3
2
1
2
3
Transport equation for Reynolds Stresses:scaled + boundary layer approximation
Pij
ij
Bij
ij
ij
ij
j
i
i
jijji
ijji
mijlmmjilml
l
jli
l
ijl
jil
lji
lij
jil
TCjill
ji
x
U
x
Ukckuu
kc
ugug
uuuu
x
Uuu
x
Uuu
uux
uux
uux
uux
uuUx
uut
3
2
2
1
3
2
1
2
51
0
Scaling by q3/
BL: - neglect rotation - neglect gradients parallel to boundary
Algebraisation of second moment clsoures:eddy viscosity and diffusivity
z
US
kcuw M
2302
z
US
kcw H
2302
z
USlquw M
zSlqw H
2
2
1qk
l
kc
2330
/
))-C(B A(GA -
/BA - AS
hH
3212
112
1631
61
h
hh/-
M GAA-
G))S-C(AAAA(BS
21
2211131
1
91
1918
k
lNGh 2
22
Table 2. Kantha and Clayson (1994) stability function parameters
So now need 2 equations: one for q (or k)one for l (or )
or
MV SlqK
HH SlqK
“k” “e” notation “q” “l” notation
HH Sk
K
2
2
MV Sk
K
2
2
eddy viscosityeddy viscosity
eddy diffusivityeddy diffusivity
Parameter A1 A2 B1 B2 C2 C3
Value 0.92 0.74 16.6 10.1 0.7 0.2
Two equation turbulence closuresMY25 (Mellor, Yamada 1982)
k - (Rodi, 1980)
k - (Wilcox, 1988)
εBPq
zSlq
z
q
xU
q
t qi
i
222
222
εBPz
kK
zx
kU
t
k
k
M
ii
k
cBcPckz
K
zxU
tM
ii
2
231
εBPz
kK
zx
kU
t
k
k
M
ii
k
cBcPckz
K
zxU
tM
ii
2
231
wallqi
i FB
qBPEllq
zSlq
zlq
xUlq
t 1
3
1222
2
21sb
sbwall dd
ddlEF
2
2 ,
11
sbwall ddMIN
lEF
2
4
2
21sb
wall d
lE
d
lEF
Why does MY25 need a wall proximity function?
assume st st, no horiz grad, no B
2
21zL
lEF
where
in bottom constant stress layer : l = z, P = , q2 is const
FEB
qSq q 1
1
3230
1
21
B
FESq
Negative diffusion without a wall function
2
21zL
lEF
2
21
3
1222 1
zq
ii L
lE
B
qBPEllq
zSlq
zlq
xUlq
t
FB
qPEllq
zSlq
z q1
3
120
szbszbs
bsz dLddMINL
dd
ddL
;,;
E1 = 1.8 B1 =16.6 E2 = 1.33 Sq = 0.2
“Generic Length Scale” turbulence closure
nmp 0μ kcψ l
1-3/230μ εkcl
1/nm/n3/2p/n30μ ψkcε
Umlauf and Burchard (2003) J. Mar. Res.
εBPz
kK
zx
kU
t
k
k
M
ii
FcBcPckz
K
zxU
tM
ii
231
Warner, Sherwood, Arango, and Signell (2005) Performance of four turbulence closure models implemented using a generic length scale method, Ocean Modelling 8, p. 81-113.
c2: free decay of homogenous turbulencec1: homogenous sheared grid turbulence
c3: buoyancy parameter for unstable
k: diffusion of k: diffusion of fit to law of wall
p, m, n : define
Determination of c3 buoyancy coefficient
31222/330
21 20 ccNlSkckccc H
Start with transport equation for Assume: P + B = Substitute expressions for KM, B, and can derive:
length scale limitation l < sqrt (0.56 k) / Nyields:
213 08.408.5 ccc for Kantha/Clayson stability functions
Numerical implementationgrid + limitations
Length scale limitation:2
2 560
N
k.l
k (q2) limitation: k = MAX(k, 1e-8)
30
12330
NLεkcl -/
μ
Numerical implementation
time step advective transport terms
time step with P, B
time step (F)
apply BCs, time step diff term, update values
calc length scale
calc buoyancy parameter Gh = ( L N / Q) ^2
limit Gh
calc stability functions Sm, Sh = functs (Gh)
calc eddy visc and eddy diffKm = Q L Sm, Kh = Q L Sh, Kq = Q
L Sq
MY25 GLS
εBPz
kK
zx
kU
t
k
k
M
ii
FcBcPckz
K
zxU
tM
ii
231
εBPq
zSlq
z
q
xU
q
t qi
i
222
222
2
21
3
1222 1
zq
ii L
lE
B
qBPEllq
zSlq
zlq
xUlq
t
22 NKB;MKP HM 22 NKB;MKP HM
lB
q
q
lql
1
3
2
2
; ; wall fnct (l).
q2, q2l at new time step
N
qlMINl
q
lql ited
53.0,; lim2
2
2
22lim
q
NlG ited
h
)(, hHM GfunctSS
MitedV SlqK lim
HitedH SlqK lim
2
2q
xU
ii
lqx
Ui
i2
ii x
U
ii x
kU
k, at new time step
/nm/n/p/n
μ ψkcε 12330
2
22lim
q
NlG ited
h
)(, hHM GfunctSS
MitedV SlkK lim2
HitedH SlkK lim2
nn/m p
μn NkcMINψ 202/56.0,
22 560
N
k.l
m/n1/np/n 0μ kψc
itedllim
32
1
Test Cases
1) Open channel flow (2 simulations)
to compare closures with velocity log layer
2) Mixed layer deepening
to calibrate c3 buoyancy parameter
3) Estuary
Test Case # 1: Steady Open Channel Flow:Experimental Description
L = 10000, W=1000, H=10 mZob = 0.005ubar = 1m/sS0 = 4x10-5 m/mtcr = 0.05 N/m2
E = 5x10-5 kg/m2/sPorosity = 0.90
grid spacings: dx = 100m, dy = 10m, dz = 0.25m)
dt = 30s5000 s simulated (st. st. reached)
Domain parameters
Model parameters2 simulations
Q
Q
1) and Q
2) and
zx
Test Case # 1: Steady Open Channel Flow:Analytical Results
uwzx
P
x
UU
t
U
0
1
momentum eq.
linear stress
0Z
zLn
uU
*
H
zzuKM 1*
sm
H
z
z
HLn
uu /.* 0620
1 0
0
H
z
c
uk 1
20
2
*= 0.013 m2/s2 at z = 0
shear velocity
velocity
eddy viscosity
turbulent kinetic energy
z
UK
H
z
xgH
H
zu M
112
*
Simulation 1Depth and Flow
Q
Test Case # 1: Steady Open Channel Flow
z
UK
H
z
xgH
H
zu
M
1
12
*
Test Case #2 : Surface mixed layer deepening
L = 5000, W=1000, H=50 mZos = 0.005u*surf = 0.01 m/sN0 = 0.01 /s
grid spacings: dx = 250m, dy = 100m, dz = 0.25m)
dt = 30s30 days simulated
Domain parameters
Model parameters
zxz
Dm
Means to confirm c3 buoyancy parameter
Test Case #2 : Surface mixed layer deepening
2/12/1*05.1 tNuD osm
Mixed layer depth, Dm
(Price, 1979)
Critical Richardson No. controls evolution of mixed layer deepening
Test case #3 : Idealized estuary
L = 100000, W=1000, H=5-10 mZob = 0.005River ubar = 0.08m/sTidal ubar = 0.40 m/sS0 = 5x10-5 m/mtcr = 0.05 N/m2 ws = 0.5mm/sE = 1x10-4 kg/m2/sPorosity = 0.90
grid spacings: dx = 500m, dy = 100m, dz = 0.25-0.5 m
dt = 30s20 days (~st. st. reached)
Domain parameters
Model parameters
Test case #3 : Idealized estuary
L = 100000, W=1000, H=5-10 mZob = 0.005River ubar = 0.08m/sTidal ubar = 0.40 m/sS0 = 5x10-5 m/mtcr = 0.05 N/m2 ws = 0.5mm/sE = 1x10-4 kg/m2/sPorosity = 0.90
grid spacings: dx = 500m, dy = 100m, dz = 0.25-0.5 m
dt = 30s20 days (~st. st. reached)
Domain parameters
Model parameters
Realistic estuary - Hudson RiverJohn C. Warner W. Rockwell Geyer
James A. Lerczak
200 along channel cells
20 lateral cells
20 vertical levels
Model parameters
Initial salinity
distributionlevel free surface
zero velocity
salt distribution
dt = 30s
z0 = 0.005
Simulate: - tides, salt, suspended-sediment
- for 50 days (days 100 – 150 , 2003)
Initial parameters
Operational parameters
• Northern end: – Measured Q– Salinity = 0
• Southern end:– Tidal boundary using observed
free surface only
– Salinity gradient condition
Boundary conditions
salbndry = salj=1+ dSdx
t11
t00
t1
t1
d
tb
t1t
01t
0 ηhηh0.5
VvC
Δy
ηηgΔtvv
Free surface model results
Model-data comparison at site N3 (22 km)
Comparison of vertical structure of salt and velocity (k-)
Neap tide
Spring tide
Model
Model
Observed
Observed
Comparison of 3 closures
Sediment transport routines in ROMS
BBL and Sediment• Bottom boundary layer subroutines - enhance bottom
stress to include the average affect of surface waves on the mean currents
(mb_bbl.F and sg_bbl.F)
• Sediment transport subroutine – transport multiple classes of suspended sediment and track evolution of multi-layered bed framework (sediment.F)
• User can specify :
1) just BBL
2) just Sediment
3) or both BBL + Sediment
Wave - Current BBL Physics• Increased turbulence• Enhanced drag• Enhanced mean stress• Increased maximum stress• Moveable bed roughness• Input:
– Current speed and direction at reference height
– Wave orbital velocity, period, and direction
– Bottom sediment characteristics
• Output:– Apparent drag coefficient– Wave-maximum shear stress– Bedform geometry
current/(current+wave) m
ean/
(cu
rren
t+ w
ave) Non-linear enhancementwave-mean bottom stress
Grant and Madsen (1986) Ann. Rev. Fluid Mech. 18:265-305
W-C Bottom Boundary Layer Routines
SG_BBL
• Modifed Grant-Madsen w-c model (Styles & Glenn, 2000)
• Formally related to three-layer eddy viscosity profile
• Ripple roughness (Styles & Glenn, 2000)
• Immobile sediment roughness gets default value
• No skin friction / form drag partitioning; no sediment stratification
• Contributed by Rich Styles and Scott Glenn
MB_BBL
• Empirical DATA2 wave-current solution (Soulsby, 1995)
• Ripple geometry for sand or silty beds
• Nikuradse, saltation, ripple, and/or biogenic roughness (Combination of methods)
• Faster than SG_BBL• No skin friction / form drag
partitioning; no sediment stratification
• Contributed by Meinte Blass
model “flow chart”
set_vbc.F
#if defined bbl_model #else
sg_bbl.F mb_bbl.F
hydrodynamic routines(advection-diffusion)
sediment.F(deposition and erosive fluxes,bed evolution)
#if defined sediment #else
bottom drag:ocean.in
rdrgrdrg2
Zo
waves data:SWAN.nc, ana_waves
T, Dir, Amp / Ub
surficial sed data:ana_sediment,forcing.nc, orsediment.F
bottom(i,j, isd50)idens)iwsed)itauc)
sediment data:ana_sediment or initial.ncbed (i,j,k, thick) botom(i,j, isd50)
age) idens)poro) iwsed)diff) itauc)
bed_frac(i,j,k,ised) irlen)bed_mass(i,j,k,ised) irhgt)
izdef)….. )
bustr, bustrcwmaxbvstr, bvstrcwmaxbottom(i,j, irlen)
irhgt)
bustrbvstr
Sediment transport – bed layers
Activelayerthick
Activelayerthick
Erosion ceb ττfor
Deposition
502
cw1a
Dk
ττkz
502
cw1a
Dk
ττkz
Rule: create a newlayer for depositionif top layer > 5mm
z
Cw
t
Cs,i
i
iaiis,
ice,
wii
i
dep_fluxz*frac*poro1*ρ
;1τ
τ*frac*poro1*E*dt
MIN
eros_flux
Harris, Wiberg 1997
Example application : Massachusetts Bay
multibeam backscatter intensity
Surficial mean grain size distribution- binned 2:6
Surficial sediment characteristics
http://pubs.usgs.gov/of/2003/of03-001/index.htm
Wentworth grade scale
Phi
23456
d ws tce E
mm kg/m3 mm/s N/m2 kg/ m2 s
0.25000 2650 27.00 0.190 5.00E-06
0.12500 2650 8.70 0.140 5.00E-06
0.06250 2650 2.40 0.090 5.00E-06
0.03125 2650 0.62 0.061 5.00E-06
0.01560 2650 0.15 0.038 5.00E-06
Initial conditions:5 sediment classes
8 bed layers (5 cm ea.) Equal fractions
http://pubs.usgs.gov/of/2003/of03-001/htmldocs/nomenclature.htm
Sed particle property calcs
http://woodshole.er.usgs.gov/staffpages/csherwood/sedx_equations/RunSedCalcs.html
Google: Sherwood USGS
+ other Sediment transport applets
activate sediment and bbl
activate sediment
cppdefs.h
activate bbl
activate source of wave data
specify input file and output parameters
activate output
ocean.in
identify name ofsediment.in file
sediment.in example input file
Establish sediment parameters
set grid size
mod_param.F
set :number of bed layersnumber of cohesive
sediment classesnumber of non-cohesive
sediment classes
Initialize sediment arrays
bed(i,j,k, MBEDP)
bed_frac(i,j,k, ised)
bed_mass(i,j,k, ised)
bottom(i,j, MBOTP)
ana_sediment
ithck = 1 ! layer thicknessiaged = 2 ! layer ageiporo = 3 ! layer porosityidiff = 4 ! layer bio-diffusivity
isd50 = 1 ! mean grain diameteridens = 2 ! mean grain densityiwsed = 3 ! mean settle velocityitauc = 4 ! critical erosion stressirlen = 5 ! ripple lengthirhgt = 6 ! ripple heightibwav = 7 ! wave excursion amplitudeizdef = 8 ! default bottom roughnessizapp = 9 ! apparent bottom roughnessizNik = 10 ! Nikuradse bottom roughnessizbio = 11 ! biological bottom roughnessizbfm = 12 ! bed form bottom roughnessizbld = 13 ! bed load bottom roughnessizwbl = 14 ! wave bottom roughnessiactv = 15 ! active layer thicknessishgt = 16 ! saltation height
Modeling of sediment transport in Mass Bay
Justification:• Relocation of Boston sewage outfall
(Sept 2000) • Habitats – fisheriesPurpose• simulate tidal currents and transport of
sediment due to combined tides and storm forcing
• determine transport pathways of sediment in Mass Bay (relative contribution of storms, tides, etc)
• test of numerical transport algorithms, bed model
Methods• Conduct 70 day simulation of currents
and sediment transport, driven by tides and 6 repeating storm events
Butman, Valentinehttp://woodshole.er.usgs.gov/project-pages/coastal_mass/html/intro.html
20 vertical sigma layers68x68 horizontal orthogonal curvilinear
Grid
Storm forcing
Use pattern of October 1996 event
To reperesent a “typical” storm
Repetition of October 1996 eventto simulate 6 storms
SWAN output at peak of storm
SWAN inputs:
27 deg
Wind 15 m/s
Swell 6m, 11s
Boston buoy
Mass Bay– Tides + Storm
initial evendistribution of
5 sediment classes:2, 3, 4, 5, and 6 phi
Comparison of Observed – Model surficial sediment distribution
Modelled Observed
Issues with negative sediment
TS_U3HADVECTION MPDATA-Positive definite
Simulation of river sediment dispersal on shelf
EAST_WALLWEST_WALLNS_PERIODICsvstr = -0.05 N/m2
west Qsource = 500 m3/swest Tsource = 1kg/m3
Summary
Recent advancements to the model:
• Multiple two-equation turbulence closure schemes (GLS, Umlauf and
Burchard 2003)
• Sediment transport algorithms
- suspended sediment transport
- bed framework
- transport multiple grain sizes
• Interaction between sediment and wave/current modules
- Styles/Glenn (sg_bbl.F) – existing
- Soulsby (mb_bbl.F) – (Blaas, UCLA)
• MPDATA positive definite horizontal advection scheme
• Tidal elevation only boundary condition
Conclusions• ROMS has many options for turbulence closure:
Analytical, BVF, KPP, MY25, GLS
• GLS method provides a canonical form to both recover existing models and to develop new models.
• Performance of GLS reveals:
– Model correctly simulates the bulk response of Hudson River estuary (L, strat, and ds/dx) to tidal spring/neap and fresh water inflow variations
– Model simulates the overall stratification well, but the vertical structure is more diffuse (mixed) in the model.
– 3 turbulence closures of k-, k-, and k-kl produce consistent results for salt transport
• Performance of sediment routines qualitatively reproduce observed surficial sediment distribution
• MPDATA advection ensures positivity of tracer values, but is less accurate.
Future directions• Turbulence closures :
– continue evaluations /comparisons– compare to LES simulations
• Sediment transport :– suspended-sediment stratification effects in wave bl.– mixed grain bed mechanics (cohesive v. non-cohesive)– gravity-driven transport in bbl– aggregation / dissaggregation– wetting / drying– bioturbation in sediment layers– bedload transport (with wave effects)– radiation stresses– one layer BBL module
arrivederci !