Turbulence closure models and sediment transport routines in ROMS John C. Warner, U.S. Geological...

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 !